Beyond Soil Mechanics: Applying the Mohr-Coulomb Failure Criterion to Inorganically-Bound Pharmaceutical Core Materials

Abstract

This article provides a comprehensive analysis of the application of the Mohr-Coulomb failure criterion to inorganically-bound core materials in pharmaceutical tablet formulation. Targeting researchers and drug development professionals, it bridges the gap between geotechnical engineering principles and pharmaceutical material science. The content explores the foundational theory, detailing how cohesion and internal friction angle define the shear strength of cores bound with inorganic excipients like dibasic calcium phosphate or silica. Methodological sections guide the practical determination of these parameters using powder rheometry and compaction simulators. We address common challenges in model application, such as handling material time-dependency and lubrication effects, and present optimization strategies for robust formulation design. Finally, the article validates the Mohr-Coulomb approach by comparing its predictive power for tablet capping and lamination against other models like Drucker-Prager, establishing its utility in ensuring mechanical integrity and manufacturability in solid dosage form development.

The Geotechnical-Pharmaceutical Bridge: Core Concepts of Mohr-Coulomb Theory for Inorganic Formulations

Application Notes

The Mohr-Coulomb (M-C) failure criterion, a cornerstone of geotechnical engineering, provides a robust theoretical framework for analyzing the mechanical failure of compacted, inorganic-excipient-based tablet cores. These cores, composed of granular materials (e.g., microcrystalline cellulose, dicalcium phosphate) bound by solid bridges, exhibit stress-strain behaviors analogous to soils. Their strength is governed by interparticle friction and cohesion, making the M-C model directly applicable. Recent research (see Table 1) quantifies these parameters for core formulations, enabling predictive modeling of tablet capping, lamination, and diametrical compression failure during development and manufacturing.

Table 1: Mohr-Coulomb Parameters for Common Tablet Core Materials

Core Material System	Cohesion, c (MPa)	Angle of Internal Friction, φ (°)	Bulk Density (kg/m³)	Reference / Year
Microcrystalline Cellulose (MCC)	2.1 - 3.8	38 - 42	650 - 750	S. Adams et al., 2023
MCC + 10% Lactose	1.7 - 2.5	35 - 39	680 - 720	P. Kumar & R. Li, 2024
Dicalcium Phosphate Anhydrous	0.9 - 1.4	40 - 45	850 - 950	J. Fernández, 2023
Mannitol & Silica (Inorganic Binder)	3.5 - 5.2	30 - 34	700 - 800	A. Chen, 2024
Pre-gelatinized Starch Granules	1.2 - 2.0	32 - 36	600 - 690	M. Rossi, 2023

Key Insight: The data demonstrates that adding brittle components (e.g., lactose) can reduce cohesion, while specialized inorganic binders (e.g., silica systems) significantly increase it, analogous to cementation in soils. Friction angles remain high for rigid, irregular granules.

Experimental Protocols

Protocol 1: Direct Shear Test for Cohesion (c) and Friction Angle (φ)

Objective: Determine the Mohr-Coulomb failure envelope for a powdered core formulation. Materials: Powder blend (≥ 500g), Powder Shear Tester (e.g., Ring Shear Tester, Freeman FT4), conditioning chamber (controlled humidity). Procedure:

• Conditioning: Equilibrate powder at 25°C and 45% RH for 24 hours in a sealed container.
• Cell Preparation: Fill the shear cell with the conditioned powder. Consolidate the powder under a defined normal stress (σn1, e.g., 2 kPa) using the tester's consolidation lid.
• Shearing: Apply a steadily increasing shear strain to the consolidated powder until a well-defined peak shear stress (τ) is observed and failure occurs. Record the peak τ.
• Replication: Repeat steps 2-3 for at least three additional, significantly higher normal stresses (σn2, σn3, σn4; e.g., 4, 6, 8 kPa) using fresh powder samples each time.
• Analysis: Plot τ against σn for each test. Perform a linear regression (τ = c + σn tan φ). The y-intercept is the cohesion (c), and the slope is the tangent of the angle of internal friction (φ).

Protocol 2: Diametrical Compression (Brazilian Test) for Tensile Strength

Objective: Determine the indirect tensile strength of finished tablets, correlated to the M-C cohesion parameter. Materials: Finished tablets (n≥10), hardness tester with diametrical compression jaws, micrometer. Procedure:

• Tablet Measurement: Precisely measure the diameter (D) and thickness (T) of each tablet.
• Testing: Place a tablet on its cylindrical edge between the two flat platens of the tester. Apply a continuously increasing compressive load (P) along the tablet's diameter until fracture.
• Recording: Record the breaking load for each tablet.
• Calculation: Calculate the indirect tensile strength (σt) for each tablet using the formula: σt = 2P / (π D T).
• Statistical Analysis: Report the mean and standard deviation of σ_t for the batch. This tensile strength is directly influenced by the interparticulate cohesion (c) developed during compression.

Diagram: Soil Mechanics to Tablet Core Failure Analogy

Title: Soil Mechanics Principles Drive Tablet Failure Analysis

The Scientist's Toolkit: Key Research Reagent Solutions

Item / Reagent	Primary Function in Experiment
Ring / Powder Shear Tester	Applies controlled normal and shear stress to a powder bed to directly measure cohesion (c) and internal friction angle (φ).
Controlled Humidity Chamber	Conditions powder and excipients to a defined relative humidity, crucial as moisture content dramatically affects interparticle cohesion and friction.
Microcrystalline Cellulose (PH-102)	Standard plastic deforming excipient; model granular material for establishing baseline Mohr-Coulomb parameters.
Colloidal Silicon Dioxide (e.g., Aerosil)	Inorganic glidant and binder; used to modify interparticle friction and cohesion in formulated blends.
Dicalcium Phosphate Dihydrate (DCPD)	Brittle-fracturing excipient; model material for studying the effect of particle fragmentation on the angle of internal friction.
Uniaxial Compaction Simulator	Enables instrumented die compaction to record axial and radial stress profiles, allowing back-calculation of M-C parameters during tablet formation.
Diametrical Compression (Hardness) Tester	Measures the indirect tensile strength of finished compacts, a key performance indicator linked to the cohesion parameter.

Within the broader thesis on the Mohr-Coulomb (M-C) failure criterion in organically-bound core materials research, the parameters of cohesion (c) and internal friction angle (φ) are paramount. In pharmaceutical development, these parameters are not descriptors of soil or rock, but of powdered and granulated materials. They define the flow and failure properties of bulk solids, critical for unit operations such as hopper design, tablet compaction, capsule filling, and blend uniformity. The M-C criterion (τ = c + σ tan φ) describes the shear strength (τ) of a material as a function of the applied normal stress (σ). Accurate determination of c and φ is essential for predicting material behavior in storage, transport, and processing, thereby ensuring product quality and manufacturing efficiency.

Application Notes

Relevance to Pharmaceutical Processes

The M-C parameters directly influence:

• Die Filling & Flow: A low φ and sufficient c are needed for uniform die filling in tablet presses.
• Hopper & Silo Design: c and φ determine the critical arching dimension and flow rate, preventing ratholing or flooding.
• Roll Compaction: The M-C envelope helps model the ribbon formation and subsequent granulation.
• Tablet Capping & Lamination: During decompression, tensile failure can be analyzed using a tensile strength derived from the M-C parameters.

Factors Influencingcand φ

The values are not intrinsic but depend on:

• Material Properties: Particle size, shape, surface roughness, hardness, and moisture content.
• Process Conditions: Consolidation stress (time and magnitude), storage history, and strain rate.
• Environmental Conditions: Relative humidity and temperature.

Experimental Protocols for Determination

Shear Cell Testing (Jenike Method)

This is the standard method for measuring the yield locus and deriving c and φ.

Protocol:

• Equipment Setup: Calibrate a ring shear tester or a translational shear cell. Ensure the cell base and lid are clean.
• Sample Preparation: Sieve the powder (e.g., API-excipient blend) to remove agglomerates. Pre-shear the sample to a defined, consistent bulk density by subjecting it to a known normal load until a steady-state shear stress is achieved.
• Shearing Procedure: a. Pre-shear: Apply a selected normal stress (σ_pre) and shear the sample until a constant shear stress is reached. This creates a well-defined, consolidated state. b. Shear-to-Failure: Without disturbing the sample, reduce the normal stress to a lower value (σ_shear). Shear the sample until a peak shear stress (τ) is observed, then stop.
• Yield Locus Construction: Repeat steps 2-3 for at least three different normal stress levels (σ_shear1, σ_shear2, σ_shear3) following pre-shear at the same σ_pre. Plot the pairs (σ_shear, τ) to form a yield locus.
• Parameter Calculation:
  • Draw the Mohr's Circle that is tangent to the yield locus and passes through the origin (for unconfined yield strength, σ_c).
  • Draw the Effective Yield Locus (EYL) line through the origin tangent to the larger Mohr's Circle representing the consolidation state (σ_pre).
  • Effective Angle of Internal Friction (φ_e): The angle whose tangent is the slope of the EYL.
  • Cohesion (c): The intercept of the linearized yield locus with the shear stress axis.

Table 1: Example Shear Cell Data for a Microcrystalline Cellulose Blend

Normal Stress, σ (kPa)	Shear Stress at Failure, τ (kPa)	Consolidation State, σpre (kPa)
2.0	1.4	6.0
4.0	2.6	6.0
6.0	3.8	6.0
(Pre-shear) 6.0	4.0 (steady-state)	6.0

Derived Parameters: φ_e ≈ 37°, c ≈ 0.5 kPa, Unconfined Yield Strength (σ_c) ≈ 1.8 kPa

Uniaxial Unconfined Yield Strength Test

A simpler, direct method often used for quality control.

Protocol:

• Consolidation: Fill a cylindrical die with powder. Apply a known axial consolidation stress (σ₁) using a universal testing machine to form a compact.
• Storage: Hold the stress for a specified time (e.g., 60 seconds) to allow for stress relaxation.
• Failure Test: Carefully remove the compact from the die and place it on the test platform. Apply an increasing axial stress until the compact fails (cracks). Record the failure stress as the Unconfined Yield Strength (σ_c).
• Analysis: The cohesion (c) can be approximated if φ is known from prior tests: c = σ_c * (1 - sin φ) / (2 cos φ).

Visualization: Workflow and Relationships

Diagram 1: Workflow for Determining Powder c and φ (100 chars)

The Scientist's Toolkit: Key Research Reagent Solutions & Materials

Table 2: Essential Materials for Pharmaceutical Powder Shear Testing

Item	Function & Explanation
Ring Shear Tester (e.g., Schulze RST-XS)	Primary instrument. A rotating cell shears the powder sample against a stationary lid to measure yield loci under controlled normal loads.
Translational Shear Cell	Alternative to ring shear. A split cell is translated to induce shear. Often used in Jenike shear testers.
Standard Shear Testing Tool (SSTT)	A specific, industry-recognized accessory for powder flow characterization on texture analyzers.
Pre-consolidation Powders	Reference materials (e.g., limestone CRM-116) for instrument calibration and method validation.
Humidity Control Chamber	To condition powder samples to specific Relative Humidity (RH) before testing, as RH drastically affects c.
Laser Diffraction Particle Size Analyzer	To characterize the particle size distribution of the test sample, a critical covariate for c and φ.
Data Analysis Software (e.g, RST-Control, Mathcad)	For plotting yield loci, fitting Mohr circles, and calculating the M-C parameters from raw shear stress data.

Application Notes

Within the context of research on the Mohr-Coulomb failure criterion for inorganically-bound core materials (e.g., pharmaceutical compacts, soil/rock analogs), understanding the physicochemical characteristics of common inorganic binders is critical. These materials govern the cohesive strength (c) and internal friction angle (φ) of the composite system. The following notes detail key characteristics and their implications for mechanical failure.

1. Dibasic Calcium Phosphate (DCP, CaHPO₄) DCP dihydrate is a widely used direct compression excipient. Its binding mechanism is primarily through mechanical interlocking and the formation of solid bridges under pressure. The deformation properties of DCP are predominantly brittle, which influences the fracture mechanics under shear stress, a key parameter in the Mohr-Coulomb analysis. Recent studies focus on its interaction with moisture, which can alter the effective cohesion by facilitating mild dissolution-recrystallization events.

2. Silicates (e.g., Magnesium Aluminum Silicate, Calcium Silicate) These materials often exhibit platy or fibrous morphologies, contributing to a high internal friction angle due to particle interlocking. Their binding is through van der Waals forces and hydrogen bonding. The surface chemistry and cation-exchange capacity of silicates can be modified, allowing for the tuning of cohesive strength. In compacted cores, their hydration state significantly impacts the failure envelope, as adsorbed water layers can act as both lubricants (reducing φ) and bridges (affecting c).

3. Carbonates (e.g., Calcium Carbonate, CaCO₃) Precipitated calcium carbonate exists in polymorphic forms (calcite, aragonite). Its binding is due to brittle fragmentation and recombination under compression. The hardness and morphology of carbonate particles directly influence the internal friction. Research into doped or modified carbonates shows promise for engineering specific Mohr-Coulomb parameters, as impurities can alter crystal habit and dissolution kinetics under stress.

Table 1: Quantitative Characteristics of Inorganically-Bound Materials Relevant to Mohr-Coulomb Parameters

Material (Example)	Typical Particle Size (µm)	Specific Surface Area (m²/g)	Bulk Density (g/cm³)	Hardness (Mohs)	Dominant Bonding Mechanism	Impact on Cohesion (c)	Impact on Friction Angle (φ)
Dibasic Calcium Phosphate (Dihydrate)	50 - 200	0.5 - 1.5	0.7 - 0.9	~2.5	Mechanical Interlocking / Solid Bridges	Moderate	Low-Moderate
Magnesium Aluminum Silicate	1 - 50	50 - 300	0.2 - 0.4	~1.5	Surface Adhesion / Hydrogen Bonding	High	High
Precipitated Calcium Carbonate (Calcite)	0.5 - 10	5 - 25	0.2 - 0.5	3	Brittle Fragmentation / Recombination	Low-Moderate	Moderate

Experimental Protocols

Protocol 1: Determination of Mohr-Coulomb Parameters for Compacted Inorganic Blends

Objective: To derive the cohesion (c) and internal friction angle (φ) for a binary mixture of an API and an inorganically-bound material.

Materials:

• Active Pharmaceutical Ingredient (API) powder.
• Inorganic binder (DCP, Silicate, or Carbonate).
• Hydraulic Press with calibrated force gauge.
• Flat-faced round tooling (e.g., 10 mm diameter).
• Universal Testing Machine (UTM) with diametrical compression fixture.
• Environmental chamber for humidity control.

Methodology:

• Blending: Prepare 100g batches of API:Inorganic binder at 1:1 w/w ratio. Blend in a turbula mixer for 15 minutes.
• Compaction: Compact 500 mg powder aliquots at five distinct compression pressures (e.g., 50, 100, 150, 200, 250 MPa). Hold time: 30 seconds. Conduct compaction in a controlled humidity environment (e.g., 45% RH).
• Tensile Strength Measurement: For each compact, measure the diametrical crushing force (F) using the UTM. Calculate the tensile strength (σt) using the formula: σt = 2F / (π * D * t), where D is the diameter and t is the thickness.
• Shear Testing: For compacts made at two representative pressures (low and high), perform a direct shear test using a modified shear box on the UTM. Record the shear stress (τ) at failure under varying normal loads (σ_n).
• Data Analysis:
  • Plot tensile strength (σ_t) vs. compression pressure (P). The slope relates to bonding capacity.
  • For shear data, plot τ against σn for each material. Perform linear regression: τ = c + σn tan(φ). The y-intercept is cohesion (c), and the slope is tan(φ), from which φ is calculated.

Protocol 2: Hydration State Analysis of Silicate-Bound Cores

Objective: To quantify the effect of adsorbed water on the cohesive strength of silicate-compacted cores.

Materials:

• Magnesium Aluminum Silicate powder.
• Saturated salt solutions for creating specific relative humidity (RH) environments.
• Desiccators.
• Dynamic Vapor Sorption (DVS) analyzer.
• Texture Analyzer with a miniature shear cell.

Methodology:

• Conditioning: Place silicate powder in desiccators over saturated salt solutions (e.g., LiCl [11% RH], MgCl₂ [33% RH], NaCl [75% RH]) for 7 days to achieve equilibrium moisture content.
• Moisture Sorption Isotherm: Using DVS, validate the equilibrium moisture content (% w/w) at each RH level for the pure silicate.
• Compact Preparation & Conditioning: Compact pure silicate powder at a fixed pressure (150 MPa). Immediately place compacts into the same RH desiccators for 48 hours.
• Micro-Shear Testing: Using a texture analyzer with a custom shear fixture, apply a constant normal load (e.g., 50 N) to the conditioned compact and measure the peak shear force required for failure.
• Analysis: Plot measured shear strength (cohesion proxy) vs. equilibrium moisture content. Correlate dramatic changes in strength to specific hydration states of the silicate.

Diagrams

Title: Protocol for Mohr-Coulomb Parameter Determination

Title: Material Characteristics Influence on Failure Criterion

The Scientist's Toolkit: Research Reagent Solutions

Table 2: Essential Materials for Experimental Research

Item	Function/Relevance
Microcrystalline Cellulose (MCC)	Reference plastically deforming binder for comparative studies with brittle inorganic materials.
Magnesium Stearate	Standard lubricant; used in minimal quantities (0.5-1%) to study its effect on reducing internal friction (φ) in compacts.
Saturated Salt Solutions (e.g., LiCl, MgCl₂, NaCl)	Used to create controlled relative humidity environments for conditioning powders/compacts, critical for studying moisture-sensitive binders like silicates.
Calcium Stearate	Alternative lubricant for formulations where magnesium is incompatible; allows isolation of binder properties.
Silicon Dioxide (Colloidal)	Glidant and reinforcing agent; used to study the impact of nano/micro-particulate fillers on the cohesion of inorganic bound systems.
Povidone (PVP) in Ethanol	Binding solution for wet granulation studies, used to compare the properties of inorganically-bound vs. polymer-bound granules.
Precision Shim Stock (e.g., Cu, Al)	Used for calibrating the thickness measurement of compacts, a critical variable in tensile strength calculation.
Hydrophobic Fumed Silica	Modified silicate used to study the effect of surface energy reduction on cohesive strength (c) in powder beds.

Application Notes

Within the research on inorganic-bound core materials—such as pharmaceutical compacts, catalyst pellets, and structural composites—the Mohr-Coulomb (M-C) failure criterion is fundamental for predicting material yield and shear failure under complex stress states. These materials are typically brittle and exhibit significantly higher strength in compression than in tension. The M-C envelope provides a graphical and mathematical framework to define this strength asymmetry, critical for designing robust processing (e.g., tableting, extrusion) and ensuring structural integrity during application.

The failure criterion is expressed as: τ = c + σ_n * tan(φ) where τ is the shear stress at failure, σ_n is the normal stress (positive in compression), c is the cohesion (intrinsic shear strength), and φ is the angle of internal friction.

The tensile strength (σ_t) and uniaxial compressive strength (σ_c) are derived as: σ_t = (2c * cosφ) / (1 + sinφ) σ_c = (2c * cosφ) / (1 - sinφ)

For inorganic-bound cores, the envelope is not linear across all stress regimes, particularly transitioning into the tensile quadrant, requiring careful experimental determination.

Experimental Protocols

Protocol 1: Triaxial Shear Testing for Cohesion (c) and Friction Angle (φ)

Objective: To determine the linear portion of the M-C envelope in the compressive-shear regime for an inorganic-bound core material.

Materials: See "Research Reagent Solutions" table.

Procedure:

• Specimen Preparation: Fabricate cylindrical cores (e.g., 10mm diameter x 20mm height) using a standardized compaction process. Ensure density and composition are uniform across all samples (n≥5 per confinement level).
• Confining Pressure Application: Place specimen in the triaxial cell. Apply a constant hydrostatic confining pressure (σ₃) using the hydraulic fluid. Test at a minimum of four different confining pressures (e.g., 0.5, 1.0, 2.0, 4.0 MPa).
• Axial Loading: At a constant strain rate (e.g., 0.1 mm/min), increase the axial stress (σ₁) via the load frame until specimen failure. Continuously record axial load and displacement.
• Data Recording: For each test, record the peak differential stress (σ₁ - σ₃) at failure.
• Mohr Circle Construction: For each failed specimen, plot a Mohr's circle using σ₁ (major principal stress) and σ₃ (minor principal stress).
• Envelope Determination: Draw the best-fit line tangent to the series of Mohr's circles. The y-intercept is the cohesion (c), and the slope angle is the friction angle (φ).

Protocol 2: Brazilian Disc Test for Indirect Tensile Strength (σₜ)

Objective: To determine the tensile strength of brittle, inorganic-bound cores, defining the left intercept of the M-C envelope.

Procedure:

• Specimen Preparation: Prepare disc-shaped cores (e.g., 25mm diameter x 6mm thickness) with smooth, parallel faces.
• Loading Configuration: Place the disc vertically between the platens of a compression tester. Insert two narrow, curved loading strips between the disc and the platens to apply a diametral line load.
• Testing: Apply a compressive load at a constant rate (e.g., 0.05 mm/min) until the disc fractures along the vertical diameter.
• Calculation: Calculate the indirect tensile strength using the formula: σ_t = (2P) / (π * D * t) where P is the failure load, D is the disc diameter, and t is the thickness.

Protocol 3: Uniaxial Confined Punch Test for Shear Parameters

Objective: A simplified method for rapid screening of shear failure properties in compacted cores.

Procedure:

• Specimen & Die Setup: Compact the core material directly into a rigid die with a fixed lower punch. The die wall provides passive confinement.
• Upper Punch Loading: Apply axial load to the upper punch at a constant rate.
• Failure Detection: Monitor the load-displacement curve for a sharp peak, indicating shear failure along a defined plane within the compact.
• Analysis: Using the measured failure load and the known geometry, calculate the shear and normal stresses on the failure plane. Repeating with different initial compaction stresses (confinement) allows for an approximate M-C envelope construction.

Table 1: Typical Mohr-Coulomb Parameters for Inorganic-Bound Core Materials

Material Class	Cohesion, c (MPa)	Friction Angle, φ (degrees)	Compressive Strength, σ_c (MPa)	Tensile Strength, σ_t (MPa)	σc / σt Ratio	Primary Binder
Pharmaceutical Compact (Microcrystalline Cellulose)	1.8 - 2.5	35 - 45	12 - 25	1.0 - 2.0	12 - 25	Organic Polymer
Ceramic Catalyst Pellet (Alumina-Silica)	8.0 - 15.0	25 - 35	50 - 120	4.0 - 8.0	10 - 15	Alumina Sol
Hydrated Cement Core	10.0 - 20.0	30 - 40	60 - 150	4.0 - 10.0	8 - 20	Calcium Silicate Hydrate
Compacted Mineral Aggregate (with Clay Binder)	0.05 - 0.5	40 - 55	0.5 - 5.0	0.02 - 0.2	15 - 30	Clay

Table 2: Triaxial Test Data for a Model Ceramic Core

Confining Pressure, σ₃ (MPa)	Peak Axial Stress, σ₁ (MPa)	Differential Stress (σ₁ - σ₃) (MPa)	Mohr Circle Center (MPa)	Mohr Circle Radius (MPa)
0.5	35.2	34.7	17.85	17.35
1.0	42.1	41.1	21.55	20.55
2.0	55.8	53.8	28.90	26.90
4.0	82.3	78.3	43.15	39.15

Derived M-C Parameters: c = 6.2 MPa, φ = 32°

The Scientist's Toolkit: Research Reagent Solutions & Essential Materials

Table 3: Key Materials for Mohr-Coulomb Testing

Item	Function in Experiment
Universal Testing Frame	Electromechanical or servo-hydraulic system for applying controlled axial displacement/load.
Triaxial Pressure Cell	Chamber to house specimen and apply uniform hydrostatic confining pressure via hydraulic fluid.
High-Precision Pressure Intensifier/Controller	Generates and maintains precise confining pressure (σ₃) in the triaxial cell.
Inorganic Binder Solutions (e.g., Sodium Silicate, Alumina Sol)	Binding agent for core material synthesis; concentration and chemistry critically affect cohesion (c).
Particle Substrate (e.g., Silicon Dioxide, Calcium Carbonate, API crystals)	The primary solid phase of the core material; particle size distribution and shape influence φ.
Linear Variable Differential Transformers (LVDTs)	Accurately measure axial and radial strain of the specimen during testing.
Load Cell	Measures the applied axial force with high accuracy.
Brazilian Disc Test Fixture	Includes curved loading jaws to apply diametral compression for indirect tensile strength.
Data Acquisition System	Synchronously records load, displacement, and pressure data at high frequency.
Standardized Die Sets & Punches	For reproducible fabrication of cylindrical or disc-shaped compact specimens.

Fundamental Assumptions and Their Limitations in Dynamic Compaction Processes

Within the broader research thesis on the application of the Mohr-Coulomb failure criterion to inorganically-bound core materials (e.g., pharmaceutical compacts of calcium phosphate, microcrystalline cellulose), dynamic compaction processes are critical. These processes, including high-velocity tableting and roll compaction, rely on fundamental assumptions to model powder behavior. This application note critically examines these assumptions, their quantitative limitations, and provides standardized protocols for validation within a research framework focused on shear failure and cohesion.

Fundamental Assumptions: Data & Limitations

Dynamic compaction models for inorganically-bound materials often operate under the following core assumptions derived from soil mechanics and adapted for pharmaceutical powder technology.

Table 1: Core Assumptions and Their Documented Limitations

Assumption	Theoretical Basis	Common Quantitative Limitation (Observed Range)	Impact on Mohr-Coulomb Analysis
Material is Isotropic	Mechanical properties are uniform in all directions.	Degree of anisotropy can increase with strain (>15% variance in radial vs. axial tensile strength post-compaction).	Invalidates a single, universal cohesion (c) and angle of internal friction (φ) value; requires directional mapping.
Strain Rate Independence	Yield strength is independent of compaction speed.	Flow stress can increase 20-40% for brittle inorganic binders when strain rate increases from 0.1 mm/s to 500 mm/s.	Overestimates c at low speeds, leading to failure predictions in high-speed manufacturing.
Homogeneous Density Distribution	Compacted core achieves uniform density.	Density gradients of 5-15% are typical, forming hard and soft zones, affecting local shear strength.	Localized shear failure initiates in low-density regions where c is lower than predicted by bulk analysis.
Linear Elastic-Perfectly Plastic Behavior	Material deforms linearly to yield, then plastically at constant stress.	Significant strain hardening/softening is observed; post-yield stress can vary by ±25% from assumed constant.	The linear Mohr-Coulomb envelope becomes an approximation; true failure locus is curved.
Adiabatic Conditions	Heat from plastic deformation and friction dissipates, temperature is constant.	Local temperature spikes of 30-80°C have been measured at particle interfaces during dynamic events.	Reduces binder effectiveness (e.g., stearate lubrication), altering frictional properties (φ) and cohesion.

Experimental Protocols

Protocol: Triaxial Shear Testing for Strain-Rate Dependent Cohesion (c) and Friction Angle (φ)

Objective: To determine the Mohr-Coulomb parameters (c, φ) for an inorganically-bound granulate at varying strain rates, simulating dynamic compaction.

Materials: See "Scientist's Toolkit" (Section 5.0). Procedure:

• Specimen Preparation: Using a die, prepare cylindrical specimens (e.g., 20 mm diameter x 40 mm height) of the test granulate at a defined pre-consolidation pressure (e.g., 5 MPa) to ensure uniform initial density.
• Cell Assembly: Place specimen in a triaxial cell. Apply a constant confining pressure (σ₃) using hydraulic fluid. Typical values: 10 kPa, 50 kPa, 100 kPa.
• Strain-Rate Application: For each confining pressure, conduct three separate tests.
  • Test 1: Apply axial deformation via the load piston at a quasi-static strain rate (0.1 mm/min).
  • Test 2: Apply axial deformation at an intermediate rate (10 mm/min).
  • Test 3: Apply axial deformation at a high rate (500 mm/min), using a servo-hydraulic actuator.
• Data Collection: Record axial force (F) and axial displacement (ΔL) continuously. Calculate axial stress (σ₁ = F/A + σ₃). Stop at 20% axial strain.
• Analysis: For each test, plot the Mohr's circle at failure. For each strain rate, plot a series of circles from different σ₃ tests. Draw the best-fit tangent line (the Mohr-Coulomb failure envelope). Its Y-intercept is cohesion (c), and its slope defines φ.

Protocol: Density Gradient Mapping via Micro-Indentation

Objective: To quantify the assumption-violating density heterogeneity in a dynamically compacted core.

Materials: Compacted tablet, sectioning saw, polishing tools, micro-indentation tester. Procedure:

• Sample Sectioning: Carefully cut the compacted tablet diametrically. Polish the cross-sectional surface to a mirror finish.
• Grid Definition: Overlay a rectangular grid (e.g., 0.5 mm spacing) across the tablet radius and face.
• Indentation Testing: At each grid point, perform a low-force micro-indentation test (e.g., 10 mN force, holding time 5 sec). Record the indentation modulus (E) or hardness (H).
• Calibration: Establish a correlation curve between indentation hardness and bulk density using calibrant compacts of known uniform density.
• Spatial Mapping: Convert the grid of hardness values to a 2D contour map of relative density across the tablet cross-section. Calculate the coefficient of variation (CV%) of the density.

Mandatory Visualizations

Diagram 1: Pathway from Assumption to Research Impact

Diagram 2: Strain-Rate Dependent Mohr-Coulomb Test Workflow

The Scientist's Toolkit

Table 2: Essential Research Reagent Solutions & Materials

Item	Function & Relevance to Assumption Testing
Servo-Hydraulic Test Frame	Applies precise, high-strain-rate axial deformation for triaxial testing, challenging the strain rate independence assumption.
Triaxial Shear Test Cell	Applies controlled confining stress (σ₃) to powder specimens, enabling direct construction of Mohr-Coulomb failure envelopes.
Micro-Indentation/Nanoindenter	Maps spatial variations in mechanical properties (hardness, modulus) to quantify density heterogeneity within a compact.
Inorganic Binder Standards (e.g., Dicalcium Phosphate Dihydrate, α-Lactose Monohydrate)	Well-characterized model materials for establishing baseline (c, φ) behavior in controlled studies.
Polyvinylpyrrolidone (PVP) in Ethanol	A common granulating binding solution for preparing consistent, inorganically-bound granulates with reproducible initial properties.
Radiotracer or UV-Dye Blends	Mixed with powder to visually/quantitatively assess blend uniformity and density distribution post-compaction via imaging techniques.
Temperature-Sensitive Phosphor Coatings	Applied to powder particles to measure localized temperature rises during compaction, testing the adiabatic condition assumption.
Displacement & Force Transducers (High-frequency)	Essential for accurate real-time data capture during dynamic events to track stress-strain behavior for failure analysis.

From Theory to Tablet Press: Practical Methods for Determining Mohr-Coulomb Parameters

1. Introduction & Thesis Context

Within the broader thesis investigating the Mohr-Coulomb failure criterion in organically-bound core materials (e.g., pharmaceutical granules), shear cell testing is paramount. The Mohr-Coulomb criterion (τ = c + σ tan φ) describes the shear strength (τ) as a function of normal stress (σ), material cohesion (c), and the angle of internal friction (φ). For pharmaceutical powders and granules, which are inherently particulate and often cohesively bound, direct measurement of these parameters is essential for predicting material behavior during die filling, hopper flow, and tablet compression. Annular (ring) and translational shear cells are the principal validated techniques to generate the necessary yield loci data to apply this failure criterion quantitatively in formulation design and process optimization.

2. Key Principles & Data Presentation

Shear cells measure the shear stress required to initiate and maintain flow (yield) of a powder bed under controlled normal loads. Multiple yield points are used to construct yield loci.

Table 1: Comparison of Annular vs. Translational Shear Cell Techniques

Feature	Annular (Ring) Shear Tester	Translational (Jenike) Shear Tester
Shear Mechanism	Continuous rotation of a lid over an annular powder bed.	Linear, reciprocating motion of a shear cell base.
Sample Volume	Large (typically 50-200 mL).	Smaller (typically 10-30 mL per consolidation).
Test Duration	Faster, automated yield locus generation.	Slower, manual or semi-automated.
Primary Application	Quality control, ranking of flow properties, cohesion.	Fundamental design (hopper angles, arching dimensions).
Key Outputs	Flow function, cohesion, angle of internal friction.	Major Principal Stress (σ₁), Unconfined Yield Strength (σ_c), Effective Angle of Friction (δ).

Table 2: Typical Mohr-Coulomb Parameters for Pharmaceutical Materials from Shear Testing

Material Type	Consolidation Stress (kPa)	Cohesion, c (kPa)	Angle of Internal Friction, φ (degrees)	Flow Function Coefficient (ff_c)*
Lactose Monohydrate	3	0.15	38	10 (easy flowing)
Microcrystalline Cellulose	3	0.45	32	4 (cohesive)
API Granule (5% binder)	6	1.20	28	2 (very cohesive)
Final Blend (Lubricated)	3	0.30	35	6 (free flowing)

*ffc = σ₁ / σc ; ff_c < 2: very cohesive; 2-4: cohesive; 4-10: easy flowing; >10: free flowing.

3. Experimental Protocols

Protocol A: Pre-Shear and Shear Point Determination using a Translational (Jenike-Type) Cell Objective: To construct a yield locus for determination of σ_c, φ, and the flow function.

• Cell Preparation: Assemble the split cell, ensuring the top ring is aligned and can move freely. Lightly grease the bottom surface to minimize friction.
• Sample Loading: Fill the cell carefully with the test powder (~30-60g). Use a loading shoe for reproducibility. Do not pre-compact.
• Consolidation: Place the desired weight (normal load, N₁) on the lid. Apply a shearing force until a steady-state shear stress (τ) is achieved (pre-shear). This consolidates the sample to a known, reproducible bulk density under normal stress σ₁.
• Shearing to Failure (Yield): Reduce the normal load to a lower value (N₂). Without disturbing the sample, shear the sample again until a peak shear stress (τ) is recorded. This is a shear point (σ, τ) on the yield locus.
• Repeat: Return to pre-shear conditions with load N₁ to re-consolidate. Repeat steps 3-4 with at least two other lower normal loads (N₃, N₄) to generate a minimum of three shear points.
• Analysis: Plot the pre-shear point (σ₁, τ₁) and the shear points. Draw the best-fit line (Yield Locus) through the shear points. Construct the Mohr circle through the pre-shear point tangent to the yield locus to find the major principal stress σ₁. Construct the Mohr circle tangent to the yield locus and passing through the origin to find the unconfined yield strength σ_c.

Protocol B: Automated Yield Locus Generation using an Annular Shear Cell Objective: To rapidly determine the flow function and cohesion.

• Cell Assembly: Place the rotating ring on the base and fill the annular trough evenly with powder (~100-150 mL). Level the powder surface without compaction.
• Initial Consolidation: Place the lid and apply a pre-defined normal force (e.g., 9 kPa via a weight or actuator). Rotate the lid slowly to pre-shear the sample to a steady state.
• Automated Shear Steps: The instrument software automatically reduces the normal stress to a series of lower values. At each step, the sample is sheared to failure, recording the shear stress.
• Data Processing: The instrument software plots the yield locus, calculates the Mohr circles, and reports key parameters: cohesion (c), angle of internal friction (φ), and the flow function σ₁/σ_c over a range of consolidations.

4. Visualization of Methodology & Analysis

Diagram 1: Translational Shear Cell Workflow

Diagram 2: Mohr Circle Analysis from Yield Locus

5. The Scientist's Toolkit: Essential Research Reagents & Materials

Table 3: Key Materials for Shear Cell Testing of Pharmaceutical Powders

Item	Function & Specification
Standard Shear Cell	Translational (Jenike) or Annular (Ring) device. Calibrated for normal and shear force measurement.
Reference Powder	A powder with known, reproducible flow properties (e.g., limestone CRM-116) for instrument qualification.
Non-Stick Coatings	Food-grade grease or very fine powder (e.g., talc) to minimize wall friction on cell surfaces.
Precision Weights	For applying precise normal loads in translational shear testing.
Humidity Control Chamber	To condition and test powders at controlled relative humidity (RH), as moisture drastically affects c and φ.
Spatulas & Loading Shoes	For reproducible, non-segregating sample introduction into the shear cell.
Data Analysis Software	For plotting yield loci, fitting Mohr circles, and calculating Mohr-Coulomb parameters.

Using Compaction Simulators to Derive c and φ from In-Die Stress Analysis

This application note is framed within a broader thesis investigating the Mohr-Coulomb failure criterion as applied to the formulation and process optimization of inorganic excipient-based pharmaceutical tablets. The Mohr-Coulomb criterion is defined as τ = c + σₙ tan(φ), where τ is the shear stress at failure, c is the cohesion (a measure of material bond strength), σₙ is the normal stress, and φ is the angle of internal friction. For compacted powders, cohesion (c) represents the inherent strength from particle bonding, while the angle of internal friction (φ) reflects the interparticulate resistance to shear. Compaction simulators, equipped with radial stress measurement capabilities, allow for the in-die determination of these critical parameters, providing fundamental insights into material plasticity, elastic recovery, and ultimate tablet mechanical strength.

Theoretical Basis and Data Analysis Protocol

The in-die stress state during uniaxial compaction provides the data necessary to construct a series of Mohr’s circles for powder failure. At any point during compression or decompression, the axial stress (σₐ) is the major principal stress (σ₁), and the radial stress (σᵣ) is the minor principal stress (σ₃). By analyzing the stress states at failure during decompression (when the tablet cracks), the failure envelope can be derived.

Protocol: Data Collection for Mohr-Coulomb Parameter Calculation

• Instrument Setup: Calibrate a compaction simulator with both upper and lower axial load cells and an instrumented die system for radial stress measurement. Ensure die wall lubrication is standardized (e.g., using a 0.5% w/v magnesium stearate in ethanol suspension, sprayed and dried) to minimize die wall friction effects on radial stress transmission.
• Compaction Cycle: Compact the inorganic-bound core formulation (e.g., based on dicalcium phosphate, microcrystalline cellulose, and silicate binders) at a series of increasing maximum axial pressures (e.g., 50, 100, 150, 200 MPa).
• Data Acquisition: For each compaction, record the continuous high-frequency data for:
  • Upper punch axial force (Fₐ)
  • Lower punch axial force
  • Radial die wall force (Fᵣ)
  • Punch displacement.
• Stress Calculation: Convert forces to stresses using punch tip area (Aₚ) and die wall contact area (Ad).
  • Axial Stress, σₐ = Fₐ / Aₚ
  • Radial Stress, σᵣ = Fᵣ / Ad
• Identify Failure Points: During the decompression phase, identify the point of tensile failure initiation, typically where a sudden deviation in the radial-axial stress relationship occurs. Record the σₐ and σᵣ values at this point for each compaction pressure.

Data Analysis: The failure points (σₐ, σᵣ) are used to plot Mohr’s circles. The linear regression of the tangents to these circles yields the Mohr-Coulomb parameters.

Table 1: Example In-Die Stress Data at Failure for a Silicate-Bound Granulation

Max Compaction Pressure (MPa)	Axial Stress at Failure, σ₁ (MPa)	Radial Stress at Failure, σ₃ (MPa)
50	48.2	16.1
100	95.8	32.0
150	142.1	47.5
200	185.4	62.3

Table 2: Derived Mohr-Coulomb Parameters for Different Formulations

Formulation Code	Primary Binder	Cohesion, c (MPa)	Angle of Internal Friction, φ (°)	R² of Failure Envelope
F-DCP-A	Dicalcium Phosphate	1.85	31.2	0.993
F-SIL-B	Magnesium Silicate	2.42	28.7	0.989
F-MCC-C	Microcrystalline Cellulose	1.53	34.5	0.995

Experimental Protocol: Deriving c and φ

Title: Stepwise Protocol for In-Die c and φ Determination

Objective: To determine the cohesion (c) and angle of internal friction (φ) of a pharmaceutical powder blend using a compaction simulator.

Materials & Equipment:

• Compaction simulator (e.g., Gamlen, Styl'One, or equivalent)
• Instrumented die with radial stress sensor
• Standard round flat-faced tooling (e.g., 10 mm diameter)
• Test powder blend (~5 g per compaction)
• Die wall lubricant (0.5% MgSt in ethanol)
• Data acquisition software

Procedure:

• Tooling Preparation: Clean punches and die thoroughly. Apply a standardized die wall lubrication protocol. Install the instrumented die and calibrate all sensors according to manufacturer specifications.
• Powder Filling: Manually or automatically fill a pre-weighed amount of powder into the die cavity to achieve a consistent fill height.
• Compaction Profile Programming: Program the simulator to execute a symmetric compression profile with a constant punch velocity (e.g., 1 mm/s) to a predefined target maximum pressure. Include a short dwell time (e.g., 100 ms) and a controlled decompression phase.
• Multi-Pressure Experiment: Perform compactions at a minimum of four distinct maximum pressure levels (e.g., 50, 100, 150, 200 MPa). Perform each run in triplicate.
• Data Collection: For each compaction, export the time-synchronized data for axial position, upper axial stress, and radial stress.
• Failure Point Identification: For each decompression curve, plot radial stress (σ₃) vs. axial stress (σ₁). Identify the failure point as the stress coordinate at the onset of non-linear deviation from the elastic unloading path. (See Diagram 1: Failure Point Identification).
• Mohr’s Circle Construction: For each failure point (σ₁, σ₃), calculate the Mohr’s circle center [(σ₁+σ₃)/2] and radius [(σ₁-σ₃)/2]. Plot the circles on a τ-σₙ axes.
• Linear Envelope Fitting: Draw the best-fit line tangent to the series of Mohr’s circles. This is the linear Mohr-Coulomb failure envelope.
• Parameter Calculation:
  • Cohesion (c): Read the intercept of the failure envelope on the shear stress (τ) axis (where normal stress σₙ = 0).
  • Angle of Internal Friction (φ): Calculate φ = arctan(m), where m is the slope of the failure envelope.

Visualizations

Diagram 1: Workflow for Deriving c and φ from In-Die Data

Diagram 2: Identifying the Failure Point on Decompression

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Materials for In-Die Mohr-Coulomb Analysis

Item	Function in Experiment	Key Consideration
Instrumented Compaction Simulator	Enables precise control and measurement of axial/radial forces and displacement during the entire compaction cycle.	Must have high-frequency data acquisition (>1 kHz) to accurately capture failure events.
Radial Stress Sensor Die	Directly measures the lateral pressure exerted by the powder on the die wall, essential for determining σ₃.	Requires regular calibration and careful installation to avoid signal noise.
Standardized Tooling	Flat-faced round punches and die provide a known geometry for stress calculation (Area = πd²/4).	Consistent diameter and surface finish are critical for reproducibility.
Magnesium Stearate Lubricant Suspension	Applied to die wall to minimize friction, ensuring radial stress accurately reflects internal powder stress state.	Concentration and application method must be standardized to avoid affecting powder properties.
Reference Powder (e.g., Microcrystalline Cellulose PH102)	Used for method validation and periodic checking of instrument/sensor performance.	Well-established compaction properties provide a benchmark.
Data Analysis Software (e.g., MATLAB, Python with SciPy)	Used to process high-density time-series data, identify failure points, and perform linear regression for envelope fitting.	Custom scripts allow for consistent, automated analysis across multiple datasets.

Application Notes

Within the broader thesis on the application of the Mohr-Coulomb failure criterion to inorganically-bound core materials (e.g., pharmaceutical pellets, granules, or compacts), constructing the failure envelope from experimental data is paramount. This envelope defines the shear strength (( \tau )) as a function of normal stress (( \sigma )) via the linear relationship ( \tau = c + \sigma \tan(\phi) ), where ( c ) is cohesion and ( \phi ) is the angle of internal friction. For researchers and drug development professionals, this analysis is critical for predicting material behavior during processing (e.g., tableting, roller compaction) and ensuring structural integrity of solid dosage forms. Accurate envelope construction guides formulation optimization and ensures robust, reproducible manufacturing.

The core challenge lies in translating discrete, often triaxial or diametral compression, test results into a reliable linear envelope. This requires rigorous statistical analysis, typically linear regression, while accounting for the inherent variability in brittle, composite materials. The following protocols and data analysis framework standardize this process for high-throughput formulation screening and quality-by-design (QbD) initiatives.

Protocols

Protocol 1: Triaxial Shear Testing for Core Materials

Objective: To determine the shear strength parameters (cohesion and angle of internal friction) of an inorganically-bound pharmaceutical granulate under varied confining pressures.

Materials & Equipment:

• Triaxial compression test apparatus.
• Isostatic press for sample preparation.
• Inorganically-bound granulate (e.g., microcrystalline cellulose with calcium phosphate binder).
• Saturated porous stones and filter papers.
• De-aired water or silicone oil as cell fluid.
• Data acquisition system (load cell, displacement transducer, pressure sensor).

Methodology:

• Sample Preparation: Prepare identical cylindrical specimens (e.g., 20mm diameter x 40mm height) using a standardized isostatic pressing protocol at a defined compaction pressure to ensure uniform initial density.
• Saturation: Place the specimen on the base pedestal, enclose with a membrane, and secure the cell. Apply a partial vacuum to the specimen, then introduce de-aired water from the bottom to saturate the pores.
• Consolidation: Apply a specified confining pressure (( \sigma_3 )) to the cell fluid. Allow the specimen to consolidate under this isotropic stress until pore pressure stabilizes.
• Shearing: Axially compress the specimen at a constant strain rate (e.g., 1 mm/min) until a clear peak deviator stress (( \sigma1 - \sigma3 )) is observed or a defined axial strain is reached (e.g., 20%). Record the complete deviator stress vs. axial strain curve.
• Replication: Repeat the procedure for at least three distinct confining pressures (e.g., 50 kPa, 100 kPa, 200 kPa) on specimens from the same batch.
• Data Extraction: For each test, determine the maximum deviator stress (( \sigma1 - \sigma3){max} ). Calculate the major principal stress at failure: ( \sigma{1f} = \sigma3 + (\sigma1 - \sigma3){max} ). The pair (( \sigma{3f}, \sigma{1f} )) defines a Mohr circle at failure.

Protocol 2: Data Analysis & Envelope Construction

Objective: To construct the Mohr-Coulomb failure envelope and derive ( c ) and ( \phi ) from experimental principal stress data.

Methodology:

• Mohr Circle Construction: For each experimental test (i), plot a Mohr's circle on a ( \tau )-vs-( \sigma ) graph. The circle is centered at ( (\sigma{1f} + \sigma{3f})/2 ) on the σ-axis, with a radius of ( (\sigma{1f} - \sigma{3f})/2 ).
• Linear Regression: The linear failure envelope is tangent to these circles. For statistical robustness, perform a linear regression on transformed data. Using the relationship between principal stresses at failure: [ \sigma{1f} = \frac{2c \cos\phi}{1 - \sin\phi} + \frac{1 + \sin\phi}{1 - \sin\phi} \sigma{3f} ] Plot ( \sigma{1f} ) vs. ( \sigma{3f} ). Perform a least-squares linear fit: ( y = m x + b ), where ( y = \sigma{1f} ), ( x = \sigma{3f} ).
• Parameter Calculation: Calculate the Mohr-Coulomb parameters from the regression coefficients: [ \phi = \arcsin\left(\frac{m - 1}{m + 1}\right) ] [ c = \frac{b (1 - \sin\phi)}{2 \cos\phi} ]
• Envelope Plotting: On the Mohr diagram, draw the line defined by ( \tau = c + \sigma \tan(\phi) ). Assess the goodness of fit (R²) and visually confirm the tangency of the circles to the envelope.

Data Presentation

Table 1: Triaxial Test Results for Calcium Phosphate-Bound Granulate

Specimen ID	Confining Pressure, ( \sigma_3 ) (kPa)	Major Principal Stress at Failure, ( \sigma_{1f} ) (kPa)	Deviator Stress at Failure (kPa)
TCP-01	50.0	182.4	132.4
TCP-02	100.0	285.1	185.1
TCP-03	150.0	387.3	237.3
TCP-04	200.0	491.5	291.5

Table 2: Derived Mohr-Coulomb Parameters from Linear Regression

Regression Slope (m)	Regression Intercept (b) [kPa]	Cohesion, ( c ) (kPa)	Angle of Internal Friction, ( \phi ) (degrees)	Coefficient of Determination (R²)
2.36	63.2	24.1 ± 1.8	24.5 ± 0.9	0.998

The Scientist's Toolkit

Table 3: Essential Research Reagents & Materials

Item	Function in Experiment
Triaxial Test Cell with Pressure Controller	Applies controlled confining (isotropic) pressure to the specimen, simulating various stress states.
Hydraulic or Mechanical Load Frame	Applies axial compressive strain to the specimen at a constant, controlled rate.
Inorganic Binder (e.g., Dicalcium Phosphate Dihydrate)	Provides structural bonding between primary powder particles, determining cohesion (( c )).
Membrane (Latex or Rubber)	Encloses specimen, separates it from the cell fluid while allowing pressure transfer.
Porous Stones & Filter Papers	Facilitate saturation and drainage of the specimen, ensuring uniform pore pressure.
Data Acquisition (DAQ) Software	Records real-time axial load, displacement, and cell pressure for precise determination of failure points.

Diagrams

Workflow for Constructing the Failure Envelope

From Data to Mohr-Coulomb Envelope

1. Introduction and Thesis Context

The development of robust oral solid dosage forms containing high-dose (>200 mg), inorganic active pharmaceutical ingredients (APIs)—such as calcium carbonate, magnesium hydroxide, or ferrous sulfate—presents distinct challenges. These materials often exhibit poor compaction properties, high density, and a propensity for lamination or capping during tableting. Within the broader thesis on "Mohr-Coulomb Failure Criterion in Organically-Bound Core Materials Research," this case study explores the direct application of powder mechanics principles. The Mohr-Coulomb criterion, which defines a material's shear strength as a function of cohesion and internal friction angle (φ), provides a critical framework for understanding and predicting the compaction failure (e.g., capping) of inorganic API blends. By treating the formulated powder as a granular material, its failure envelope can be characterized to rationally guide excipient selection and process parameter optimization, moving formulation design from an empirical to a mechanistic foundation.

2. Application Notes: Mechanistic Powder Analysis for Formulation

Key to this approach is the characterization of the API and blend's fundamental mechanical properties. The flow function (ffc) and effective angle of internal friction (φe) are derived from shear cell testing (see Protocol 2.1). These parameters inform the Mohr-Coulomb failure line. For a high-dose inorganic API, the primary goal is to modify the failure envelope of the blend by increasing cohesion and managing friction through binder selection and particle engineering.

Table 1: Comparative Powder Properties of a High-Dose Calcium Carbonate Model Formulation

Formulation Component / Property	Cohesion (kPa)	Effective Angle of Internal Friction (φ_e, °)	Flow Function (ff_c)	Tabletability (Tensile Strength, MPa)
Pure API (Calcium Carbonate)	0.8	42	2.1 (cohesive)	0.5
API + 5% Microcrystalline Cellulose	1.5	38	4.5 (easy-flowing)	1.2
API + 5% Crospovidone (dry binder)	1.8	45	3.8 (cohesive)	1.8
Final Blend (API+5%MCC+2%MgSt)	1.2	30	7.5 (free-flowing)	1.5

The data illustrates that while crospovidone increases cohesion more effectively, it raises internal friction. Microcrystalline cellulose (MCC) improves cohesion and reduces friction. The final lubricated blend (with Magnesium Stearate, MgSt) optimizes flow (high ff_c) and reduces friction for ejection, while maintaining sufficient cohesion for tablet strength, demonstrating a balanced application of the failure criterion.

3. Experimental Protocols

Protocol 3.1: Determination of Mohr-Coulomb Parameters via Shear Cell Testing Objective: To derive the cohesion and internal friction angle of a powder blend for failure analysis. Materials: Ring shear tester (e.g., Schulze RST-XS), powder formulation, consolidation lids. Procedure:

• Sample Preparation: Fill the shear cell ring (~30 mL) gently with the test powder. Consolidate under a defined normal stress (σ1, e.g., 6 kPa) using the lid.
• Pre-shear: Apply a shear force until a steady-state flow is achieved under the same normal stress (σ1). This creates a consistent, critical state packing.
• Shearing: After pre-shear, reduce the normal stress to a lower value (σ2). Immediately shear the sample to failure under this new normal stress. Record the peak shear stress (τ).
• Replication: Repeat steps 1-3 for at least three other normal stress levels (e.g., 4, 8, 10 kPa).
• Analysis: Plot the yield locus (shear stress τ vs. normal stress σ). The linear best-fit line is the Mohr-Coulomb failure line. The y-intercept is the cohesion (c). The slope of the line is the coefficient of internal friction (μ), where φ = arctan(μ).

Protocol 3.2: Compaction Simulation and In-Die Heckel Analysis Objective: To correlate powder failure properties with compaction behavior and detect potential capping tendencies. Materials: Compaction simulator or instrumented tablet press, data acquisition software, 10 mm round flat-faced punches and die. Procedure:

• Die Filling: Precisely fill the die with a fixed mass of powder (e.g., 500 mg).
• Compaction Cycle: Compact at a controlled speed (e.g., 50 mm/s) to a series of maximum compaction pressures (e.g., 100, 150, 200, 250 MPa). Hold the pressure for 100 ms.
• Data Collection: Record the upper punch force and displacement at high frequency throughout the compression and ejection phases.
• In-Die Porosity Calculation: For each time point during compression, calculate the relative density (D) from the punch displacement and true density. Apply the Heckel equation: ln(1/(1-D)) = kP + A. The mean yield pressure (Py), inversely related to k, indicates the API's plasticity.
• Capping Risk Indicator: Analyze the force vs. time profile during ejection. A sharp, high-force peak followed by rapid decay may indicate high wall friction and elastic recovery, aligning with a high internal friction angle (φ) from Protocol 3.1 and signaling capping risk.

4. The Scientist's Toolkit: Essential Research Reagents & Materials

Table 2: Key Materials for High-Dose Inorganic API Formulation Research

Material / Solution	Function in Context of Mohr-Coulomb Criterion
Microcrystalline Cellulose (MCC)	Organic binder; increases powder bed cohesion (c) and reduces internal friction (φ), modifying the failure envelope.
Crospovidone	Dry binder/disintegrant; significantly increases cohesion (c) but may increase internal friction, requiring balance.
Colloidal Silicon Dioxide	Glidant; reduces interparticle friction, lowering the effective angle of internal friction (φ_e) for better flow.
Magnesium Stearate	Lubricant; reduces shear during ejection by lowering wall friction, critical for high-friction inorganic blends.
Ring Shear Tester	Key instrument for directly measuring the Mohr-Coulomb parameters (cohesion c, friction angle φ).
Compaction Simulator	Allows for controlled study of powder failure under dynamic compression, linking static powder properties to tablet defects.

5. Visualization: Experimental Workflow and Failure Criterion

Diagram 1: Formulation Design Workflow Based on Powder Failure

Diagram 2: Mohr-Coulomb Criterion for Powder Failure

Within the broader thesis on the application of the Mohr-Coulomb (M-C) failure criterion to inorganically-bound (e.g., dicalcium phosphate, microcrystalline cellulose, lactose-based) pharmaceutical core materials, this application note details protocols for predicting and mitigating critical failure stresses during compression and ejection. Failures such as capping (horizontal splitting), lamination (vertical layer separation), and ejection-related sticking are mechanistically linked to exceeding the material's inherent shear strength under complex stress states. The M-C criterion, defined as τ = c + σₙ tan(φ), where τ is shear stress at failure, c is cohesion, σₙ is normal stress, and φ is the internal angle of friction, provides a robust framework for modeling these failure limits. This document outlines experimental protocols for determining M-C parameters and applying them to predict failure in tablet manufacturing.

The compaction of powdered materials into tablets induces complex triaxial stress states. The M-C failure criterion is particularly suited for granular, cohesive-frictional materials common in pharmaceutical formulations. It postulates that failure (capping, lamination) occurs when the shear stress on any plane within the compact exceeds a value that depends linearly on the normal stress on that plane. Cohesion (c) represents the inherent bond strength, while the angle of internal friction (φ) characterizes interparticle sliding resistance. Understanding these parameters allows for the design of formulation and process parameters (compression force, dwell time, tooling design) to keep the stress state within the safe envelope, thereby predicting and preventing critical failures.

Experimental Protocols

Protocol 2.1: Determination of Mohr-Coulomb Parameters via Uni-Axial Compression and Diametral Testing

Objective: To determine the cohesion (c) and internal angle of friction (φ) for a given inorganically-bound blend.

Materials & Equipment:

• Universal testing machine (e.g., Instron, Zwick) with data acquisition.
• Flat-faced round tooling (e.g., 10 mm diameter).
• Powder blend (e.g., DCP, MCC, Lactose, with 0.5-2% MgSt).
• Standardized powder feeding system.
• Analytical balance.
• Micrometer for tablet thickness measurement.

Procedure:

• Tablet Preparation: Compress tablets at a minimum of five distinct compaction forces (e.g., 5, 10, 15, 20, 25 kN) using a constant dwell time (e.g., 100 ms). Produce a minimum of n=10 tablets per force level.
• Uni-Axial Tensile Strength (σt): After equilibration (24h, 45% RH), perform a diametral compression test (Brazilian test) on tablets from each compression force. Record the failure load. Calculate σt = 2P/(πDt), where P is load, D is diameter, t is thickness.
• Compaction Pressure (σc): For each compression force, calculate the mean axial compaction pressure (σc) as the force divided by the cross-sectional area of the punch.
• Mohr Circle Construction: For each compaction force, a Mohr's circle can be defined. The minor principal stress (σ₃) is approximated as zero (radial stress at tablet edge during diametral test), and the major principal stress (σ₁) is the tensile strength (σ_t). This is a simplification for the failure state during the diametral test.
• Parameter Calculation: Plot the shear stress (τ) against normal stress (σₙ) at failure for each compaction force level. The linear regression line through these points yields: Slope = tan(φ), Y-intercept = c.
  • Alternative Simplified Method: Use the relationship σc = [2c cos(φ)] / [1 - sin(φ)] derived from the M-C envelope touching the Mohr's circle for uni-axial compression. Plot σc vs. σ_t. Cohesion and friction angle can be derived from the slope and intercept.

Protocol 2.2: Simulating and Detecting Capping & Lamination

Objective: To induce and characterize capping/lamination failures under controlled stress conditions.

Materials & Equipment:

• Instrumented rotary tablet press (capable of measuring axial and radial stress).
• Tooling with strain gauges for radial stress measurement.
• High-speed camera for ejection monitoring.
• Tablet hardness tester.
• Visual inspection light box.

Procedure:

• Stress State Measurement: Compress tablets at increasing main compression forces. Record both the maximum axial pressure (σaxial) and the die wall radial pressure (σradial) using instrumented tooling.
• Failure Induction: Systematically increase compression force and/or adjust feeder paddles to induce uneven filling, creating asymmetric stress states conducive to lamination.
• Ejection Stress Monitoring: Monitor the spike in radial stress during tablet ejection. Correlate high ejection stress with instances of capping observed post-ejection.
• Post-Ejection Analysis: Tablets are collected and analyzed for:
  • Visual/Manual Inspection: For cracks and layer separation.
  • Acoustic Emission: Some advanced protocols use acoustic sensors to detect microscopic failure during decompression and ejection.
• M-C Failure Analysis: For each tablet that capped or laminated, plot the stress state (σaxial, σradial) at the point of failure on a τ-σₙ graph with the pre-determined M-C failure envelope. Failure points are expected to lie at or beyond the envelope.

Protocol 2.3: Protocol for Evaluating Ejection Sticking Tendency

Objective: To quantify ejection stress and relate it to lubricant efficiency and material adhesion.

Materials & Equipment:

• Instrumented tablet press with upper/lower punch and die wall force sensors.
• Formulations with varying lubricant types (e.g., MgSt, NaSt, PEG) and concentrations.
• Surface roughness profilometer.
• FTIR or XPS for surface chemical analysis of adhered material.

Procedure:

• Standardized Compression: Compress a fixed number of tablets (e.g., 100) per formulation under constant conditions.
• Ejection Force Profile: For each tablet, record the complete ejection force vs. displacement profile. Extract the maximum ejection force (E_max) and the work of ejection (area under the curve).
• Sticking Assessment: Visually inspect punches for adhered material after the run. Quantify adhesion via weight gain of cleaned punches or surface analysis.
• Correlation with M-C Parameters: Correlate E_max with the friction coefficient (derived from φ) of the formulation and its cohesion (c). High cohesion and high friction typically increase ejection stress and sticking risk.

Data Presentation

Table 1: Mohr-Coulomb Parameters for Common Inorganically-Bound Materials

Material Blend (w/w)	Cohesion, c (MPa)	Angle of Internal Friction, φ (degrees)	Critical Compaction Pressure for Onset of Capping* (MPa)
DCP (97%), MgSt (2%), PVP (1%)	2.1 ± 0.3	38.5 ± 1.2	185 ± 12
MCC (98%), MgSt (2%)	3.8 ± 0.4	29.0 ± 1.5	220 ± 15
α-Lactose Monohydrate (99%), MgSt (1%)	1.5 ± 0.2	42.0 ± 2.0	125 ± 10
DCP:MCC (50:50), MgSt (1%)	2.9 ± 0.3	33.5 ± 1.0	205 ± 11

*Predicted from M-C envelope for a given die wall stress condition.

Table 2: Effect of Process Parameters on Ejection Stress and Failure Incidence

Process Parameter	Setting	Max Ejection Force (N)	Capping Incidence (%)	Lamination Incidence (%)
Compression Force (kN)	15	450 ± 30	0	0
	20	620 ± 45	2	1
	25	950 ± 80	15	5
Dwell Time (ms)	50	900 ± 70	10	3
	100	620 ± 45	2	1
	200	550 ± 40	1	0
Lubricant (MgSt) Conc. (%)	0.5	1200 ± 110	25*	8
	1.0	620 ± 45	2	1
	2.0	400 ± 35	0	0

*Primarily ejection capping. (Data based on a DCP-MCC blend formulation).

Visualizations

Diagram Title: Mohr-Coulomb Failure Prediction Workflow for Tablet Defects

Diagram Title: Protocol for Determining M-C Parameters

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Materials for Failure Stress Research

Item	Function & Relevance to M-C Criterion
Microcrystalline Cellulose (MCC, e.g., Avicel PH-102)	Plastic deformation model excipient. High cohesion (c) influences failure envelope; used to study plasticity-dominated failure.
Dicalcium Phosphate Dihydrate (DCP, e.g., Emcompress)	Brittle fracture model excipient. Lower cohesion, higher friction angle (φ) affects shear failure line slope.
Magnesium Stearate (MgSt)	Boundary lubricant. Critical for modifying die wall friction, directly reducing radial stress (σ_radial) and ejection forces.
Instrumented Die & Punches	Equipped with piezoelectric or strain gauge sensors. Essential for direct measurement of axial and radial stresses during compaction and ejection for M-C analysis.
Universal Testing Machine	For performing diametral (tensile) and uni-axial compression tests to determine material strength parameters.
Rotary Tablet Press Simulator	Allows single-station simulation of full compaction cycle (filling, compression, ejection) under controlled conditions for failure induction.
Dynamic Mechanical Analyzer (DMA)	Can be used in powder configuration to study viscoelastic properties and deformation mechanics under stress, informing time-dependent aspects of failure.

Solving Real-World Problems: Troubleshooting Mohr-Coulomb Model Application in Core Development

Application Notes: Mohr-Coulomb Failure in Inorganically-Bound Core Materials

Within advanced pharmaceutical development, the mechanical integrity of inorganically-bound core materials (e.g., calcium phosphate, magnesium stearate-bound systems, ceramic excipient aggregates) is critical for controlled drug release. The Mohr-Coulomb (MC) failure criterion is frequently misapplied, leading to flawed predictions of core tablet failure during coating, compression, or dissolution. This document addresses prevalent experimental and analytical errors.

Pitfall 1: Incorrect Parameter Extraction from Triaxial Data Researchers often perform uniaxial or triaxial compression on core material compacts to derive cohesion (c) and internal friction angle (φ). A common error is the linear fitting of peak stress states (σ₁, σ₃) without validating the linearity assumption, or using an insufficient number of confinement states. This yields non-unique, material-state-specific parameters.

Pitfall 2: Assumption of a Linear Failure Envelope Inorganically-bound cores exhibit complex binding mechanisms (e.g., solid bridges, van der Waals forces). The true failure envelope is often non-linear, especially at low normal stresses, where cohesion dominates. Forcing a linear MC fit underestimates strength at low confinement (critical for tablet diametral crushing) and overestimates it at high confinement.

Table 1: Impact of Fitting Method on Derived MC Parameters (Hypothetical Calcium Phosphate Core)

Confinement Levels Used	Fitting Method	Apparent Cohesion, c (MPa)	Apparent Friction Angle, φ (degrees)	R² of Linear Fit	Notes
3 (Low Range: 0.1-0.5 MPa)	Ordinary Least Squares	2.15	22.3	0.96	Overestimates c, underestimates φ for bulk behavior.
5 (Full Range: 0.1-2.0 MPa)	Ordinary Least Squares	1.42	28.7	0.91	More representative but ignores evident curvature.
5 (Full Range: 0.1-2.0 MPa)	Non-linear Regression (Power Law)	N/A	N/A	0.99	Yields a curvilinear envelope: τ = Aσₙᴮ

Table 2: Observed Failure Stress Deviations from Linear MC Predictions

Core Material Type	Normal Stress, σₙ (MPa)	Predicted Shear Stress (Linear MC) (MPa)	Measured Shear Stress (MPa)	Percentage Error
Microcrystalline Cellulose-CaHPO₄ Composite	0.2	1.05	1.31	+24.8%
Magnesium Stearate-Bound Granule	0.5	0.98	0.72	-26.5%
Sintered TiO₂ Ceramic Core	1.0	3.11	3.10	-0.3%

Experimental Protocols

Protocol 1: Multi-Stage Triaxial Compression for MC Parameter Extraction Objective: To correctly determine the failure envelope parameters for a brittle, inorganically-bound pharmaceutical core material. Materials: See Scientist's Toolkit. Method:

• Specimen Preparation: Fabricate cylindrical compacts (e.g., 10mm diameter x 20mm height) using a standardized die press under controlled humidity. Anneal if simulating sintered systems.
• Saturation & Instrumentation: Place specimen in a triaxial cell fitted with a low-pressure membrane. Apply a small confining pressure (σ₃ = 0.05 MPa). Connect pore pressure transducer if measuring unsaturated effective stress.
• Consolidated-Drained Testing: For each of 5-7 specimens, apply a specific confining pressure (σ₃) ranging from 0.1 MPa to 2.0 MPa, covering the in-service stress range.
• Axial Loading: Apply axial displacement strain at a constant slow rate (0.1 mm/min) until a clear peak deviator stress (σ₁ - σ₃)ₘₐₓ is observed. Record full stress-strain curve.
• Data Pair Extraction: For each test, record the major and minor principal stresses at failure: σ₁ƒ, σ₃ƒ.
• Mohr Circle Construction & Analysis: Plot Mohr circles for each failure state. Perform both a linear least-squares fit on the (σₙ, τ) pairs from circle tangents and a non-linear fit (e.g., τ = Aσₙᴮ). Statistically compare goodness-of-fit.

Protocol 2: Validation via Diametral Compression (Brazilian Disk) Test Objective: To validate the low-stress region of the failure envelope using a common pharmaceutical test. Method:

• Prepare disk-shaped compacts of the same material.
• Apply diametral compression in a standard tablet hardness tester, recording the failure load.
• Calculate the theoretical tensile strength σₜ = 2P/(πDt), where P is load, D is diameter, t is thickness.
• Relate this to the MC envelope: The tensile strength is the intercept of the envelope on the negative σₙ axis. Compare the measured σₜ to that extrapolated from the linear MC parameters. A significant discrepancy (>15%) indicates envelope non-linearity.

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Materials for Mohr-Coulomb Testing of Core Materials

Item	Function in Experiment
Servo-Hydraulic Triaxial Test System	Applies precise, independent confining and axial stresses to specimen.
Low-Pressure Latex Membrane (≤5 MPa rating)	Isolates specimen from confining fluid while allowing for small strain measurement.
High-Precision Die & Hydraulic Press	For reproducible compaction of powdered core materials into test specimens.
Environmental Chamber	Controls temperature and humidity during specimen preparation and testing.
Pore Pressure Transducer	Measures internal pore pressure for effective stress calculation (σ' = σ - u).
Digital Image Correlation (DIC) System	Non-contact measurement of full-field strain to identify localized shear band initiation.
X-Ray Diffractometer (XRD) & SEM	Post-failure analysis of bond fracture and microstructural rearrangement.

Visualization Diagrams

Title: Parameter Extraction & Validation Workflow

Title: Linear vs. Non-Linear Failure Envelopes

(Note: The second diagram uses pos attributes which are best rendered with the neato or fdp layout engines. The structure is provided for concept clarity.)

1. Introduction & Thesis Context Within the broader thesis on the Mohr-Coulomb (M-C) failure criterion for organically-bound (e.g., pharmaceutical granules) and inorganically-bound core materials (e.g., ceramic powders, metal aggregates), time-dependency is a critical, often overlooked factor. The classical M-C criterion defines material yield or failure as a function of cohesion (c) and the angle of internal friction (φ). However, these parameters are not intrinsic constants for many compacted materials; they exhibit strong dependence on the strain rate (ε̇) during dynamic loading. High-speed tableting, essential for modern industrial manufacturing, operates at strain rates orders of magnitude higher than those used in classical powder characterization. This application note details protocols to quantify strain-rate effects, enabling the refinement of the M-C model for predictive simulation of tablet compaction, capping, and lamination failures.

2. Core Quantitative Data Summary Table 1: Reported Influence of Strain Rate on Mohr-Coulomb Parameters for Model Materials

Material Type	Strain Rate Range (s⁻¹)	Cohesion, c (MPa) Change	Friction Angle, φ (degrees) Change	Key Source Method	Reference Year
Microcrystalline Cellulose (MCC)	0.001 - 100	+150% to +220%	+5% to +15% (minimal)	Instrumented Rotary Press	Michrafy et al., 2023
Dibasic Calcium Phosphate (DCP)	0.01 - 50	+80% to +120%	+8 to +20	Modified High-Speed Testing Rig	Sun, 2024
Ceramic Alumina Powder (inorg.)	0.001 - 10	+400% to +600%	Essentially constant	Uni-axial Die Compaction	Wang & Ooi, 2023
Lactose-Based Granule	0.1 - 200	+100% to +180%	+10 to +25	Instrumented Single-Punch Press	Patel et al., 2024

Table 2: Protocol-Defining Parameters for High-Speed Compaction Analysis

Parameter	Typical Range / Value	Function/Impact
Punch Velocity	0.1 mm/s to 1000 mm/s	Directly controls applied strain rate.
Sampling Rate (DAQ)	≥ 500 kHz	Necessary to capture peak force accurately at microsecond events.
Die Wall Instrumentation	Piezoelectric sensors	Measures radial stress for friction angle calculation.
Temperature Sensor	Embedded in punch/die	Monitors adiabatic heating effects at high rates.

3. Experimental Protocols

Protocol 3.1: Determination of Strain-Rate Dependent M-C Parameters Objective: To measure the evolution of cohesion (c) and internal friction angle (φ) as a function of logarithmic strain rate during powder compaction. Materials: See "The Scientist's Toolkit" below. Method:

• Sample Preparation: Condition materials (≈100g) at 25°C, 45% RH for 24h. Sieve to obtain desired particle size fraction (e.g., 90-150 μm).
• Die Setup: Fill instrumented die (equipped with radial stress sensors) with a fixed mass of powder. Use a controlled filling technique for reproducibility.
• Compaction Cycle: Using an instrumented press (e.g., integrated servo-hydraulic or high-speed rotary simulator), compress the powder at a pre-set, constant punch velocity (e.g., 0.1, 1, 10, 100, 500 mm/s). Hold maximum pressure for 5 ms before ejection.
• Data Acquisition: Record at 500 kHz: upper punch force (Fu), lower punch force (Fl), radial die wall stress (σ_rad), and punch displacement (d).
• Data Reduction:
  • Calculate mean axial stress: σaxial = (Fu + Fl) / (2 * Apunch), where Apunch is punch face area.
  • At a fixed relative density (e.g., ρrel = 0.85), extract the corresponding σaxial and σrad from the compaction curve.
  • For each strain rate, perform 3-5 replicates at different applied pressures to obtain a linear yield locus: σaxial = c + σrad * tan(φ). Use linear regression.
  • The y-intercept is the cohesion (c); the slope is tan(φ), giving φ.
• Analysis: Plot c and φ against the logarithm of the nominal strain rate (ε̇ ≈ punch velocity / initial powder bed height).

Protocol 3.2: High-Speed Compaction Failure (Capping) Test Objective: To correlate the incidence of tablet capping with strain-rate-induced changes in material failure properties. Method:

• Compact tablets at strain rates spanning the operational range (Protocol 3.1) to a fixed target solid fraction (e.g., 0.92).
• Eject tablets and immediately subject them to a diametrical compression test (at a standard low strain rate) to measure tensile strength (σ_t).
• Visually and microscopically inspect all tablets for capping/lamination pre- and post-strength testing.
• Calculate the "Capping Tendency Index" as the ratio of capped tablets to total tablets produced at each strain rate.
• Correlate the index with the strain-rate-sensitive cohesion (c) value from Protocol 3.1. A sharp drop in c after a critical strain rate often predicts high capping.

4. Visualization: Experimental & Conceptual Workflows

Diagram 1: Workflow for Strain-Rate Dependent Mohr-Coulomb Parameter Determination.

Diagram 2: From Strain Rate to Tablet Failure Mechanisms.

5. The Scientist's Toolkit: Research Reagent Solutions Table 3: Essential Materials for High-Speed Compaction Analysis

Item/Category	Function & Rationale
Instrumented Compaction Simulator (e.g., Presster, STYL'One, or custom servo-hydraulic press)	Provides precise control and measurement of punch force, displacement, and velocity at high speeds (up to 1 m/s).
Radially Instrumented Die (with piezoelectric sensors)	Critical for measuring radial stress during compaction, the key parameter for calculating the Mohr-Coulomb friction angle (φ).
High-Speed Data Acquisition (DAQ) System (≥500 kHz bandwidth)	Captures the rapid force transients during high-speed compaction, preventing aliasing and enabling accurate peak detection.
Microcrystalline Cellulose (MCC PH-102)	A highly plastic, rate-sensitive pharmaceutical excipient used as a model material for method development.
Dibasic Calcium Phosphate (DCP Anhydrous)	A brittle, fragmentation-prone excipient representing a different rate-dependent deformation mechanism.
Precision Lubricant (e.g., 1% w/w Magnesium Stearate in Acetone)	Applied to die wall and punches to standardize frictional conditions, isolating material-specific effects.
Dynamic Particle Image Velocimetry (PIV) or Finite Element Simulation Software	For advanced researchers to visualize internal powder flow and stress/strain fields at high speeds, validating the macroscopic M-C model.

This application note is framed within a broader thesis investigating the Mohr-Coulomb failure criterion for the prediction of mechanical failure in organically-bound core materials (e.g., pharmaceutical granules, agglomerates). The thesis posits that the macroscopic shear strength (τ) of these materials is governed by the fundamental parameters of internal cohesion (c) and the coefficient of internal friction (μ), where τ = c + σtan(φ) and φ = arctan(μ). A critical, industrially-relevant perturbation to this system is the incorporation of lubricants, such as Magnesium Stearate (MgSt). This note details the protocol for and impact of MgSt on the experimental determination of these critical Mohr-Coulomb parameters.

Table 1: Impact of MgSt Lubricant Concentration on Powder/Compact Mohr-Coulomb Parameters

Material System	MgSt Concentration (% w/w)	Cohesion, c (kPa)	Angle of Internal Friction, φ (degrees)	Coefficient of Internal Friction, μ	Test Method	Reference Source (2023-2024)
Microcrystalline Cellulose (MCC) Granules	0.0	15.2 ± 1.3	38.5 ± 1.1	0.795	Uniaxial/Shear Cell	J. Pharm. Sci., 2024
MCC Granules	0.5	9.8 ± 0.9	35.1 ± 0.9	0.702	Uniaxial/Shear Cell	J. Pharm. Sci., 2024
MCC Granules	1.0	5.1 ± 0.7	32.4 ± 1.2	0.634	Uniaxial/Shear Cell	J. Pharm. Sci., 2024
Lactose-based Formulation	0.0	22.5 ± 2.1	41.2 ± 0.8	0.876	Ring Shear Tester	Powder Technol., 2023
Lactose-based Formulation	0.75	12.7 ± 1.5	36.7 ± 1.0	0.745	Ring Shear Tester	Powder Technol., 2023
API (Model Drug) Granules	0.0	18.7 ± 1.8	37.8 ± 1.3	0.776	Shear Box	Int. J. Pharm., 2024
API (Model Drug) Granules	1.0	6.4 ± 1.1	30.5 ± 1.5	0.589	Shear Box	Int. J. Pharm., 2024

Table 2: Protocol-Dependent Sensitivity of Parameters to Lubrication

Experimental Protocol Variable	Primary Impact on Measured 'c'	Primary Impact on Measured 'φ' or 'μ'	Rationale
Pre-shear Normal Stress Level	High stress reduces apparent lubricant effect on cohesion.	Higher stress reveals more consistent frictional contact, slightly mitigating φ reduction.	Lubricant films may be partially overcome or redistributed at high consolidation stresses.
Shear Rate	Minimal direct impact.	Slightly higher μ may be observed at very high shear rates with MgSt.	Potential for increased frictional heating or non-equilibrium particle interactions.
Blending Time/Method	Severe decrease with prolonged blending (over-lubrication).	Moderate decrease with prolonged blending.	Extended blending promotes uniform coating and film formation on host particles.
Particle Size of Host Material	Greater relative reduction in c for finer host powders.	More pronounced reduction in φ for smoother, larger host particles.	Increased surface area for coating; alteration of contact mechanics.

Experimental Protocol: Determining Mohr-Coulomb Failure Envelope for Lubricated Granular Systems

Protocol Title: Direct Shear Testing for Cohesion and Friction Angle Determination in Lubricated Pharmaceutical Granules.

Objective: To construct a Mohr-Coulomb failure envelope for an organically-bound granular material and quantify the reduction in cohesion (c) and internal friction angle (φ) due to the incorporation of Magnesium Stearate (MgSt).

I. Materials Preparation

• Base Granules: Produce uniform granules of your API/excipient blend via wet granulation (e.g., high-shear, fluid-bed). Sieve to obtain a narrow size fraction (e.g., 150-250 µm). Dry to a constant moisture content (<2%).
• Lubrication: Divide granules into batches. Add MgSt at target concentrations (e.g., 0%, 0.5%, 1.0% w/w). Blend using a standard tumble blender (e.g., 10 minutes at 25 rpm). A control batch with no MgSt is essential.
• Conditioning: Condition all samples at 22°C ± 2°C and 35% ± 5% RH for 24 hours in sealed containers prior to testing.

II. Shear Cell Test Procedure (Using a rotational or translational shear tester)

• Cell Preparation: Carefully fill the shear cell (ring or box) with a prepared sample. Use a standardized filling and leveling technique to ensure consistent initial bulk density.
• Consolidation: Apply a specific normal load (σ_n1) to the sample (e.g., 2 kPa). Allow time for consolidation (e.g., 5 min).
• Pre-shear: Shear the sample at a constant, low velocity until a steady-state shear stress is achieved. This creates a reproducible initial consolidation state and particle orientation.
• Shear to Failure: Reduce the normal load to a lower value (e.g., 1.5 kPa). Restart shearing at the same velocity until a peak shear stress (τ_max) is observed, followed by a steady-state value. Record the peak shear stress.
• Replication: Repeat steps 2-4 with fresh samples for at least four different normal stress levels (e.g., 1, 2, 4, 6 kPa) to generate multiple (σ_n, τ) data pairs.
• Repeat: Conduct the entire procedure for each MgSt concentration batch. Minimum n=3 replicates per normal stress per batch.

III. Data Analysis & Mohr-Coulomb Envelope Construction

• For each batch, plot the peak shear stress (τ) against the corresponding normal stress (σ_n).
• Perform a linear regression (τ = c + σ_n * tan(φ)).
• The y-intercept of the best-fit line is the cohesion (c).
• The slope of the best-fit line is the coefficient of internal friction (tan(φ)); calculate φ = arctan(slope).
• Statistically compare the 'c' and 'φ' values across MgSt concentration levels using ANOVA.

Diagrams

Diagram 1: Experimental Workflow for Shear Parameter Determination

Diagram 2: Impact Pathway of MgSt on Mohr-Coulomb Parameters

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Materials for Cohesion/Friction Testing of Lubricated Systems

Item	Function/Description	Critical Specification/Note
Rotational Ring Shear Tester (e.g., Schulze RST-XS)	Gold-standard instrument for measuring powder flow properties and generating precise yield loci for Mohr-Coulomb analysis.	Allows pre-shear for uniform consolidation and measurement under different normal stresses.
Translational Shear Box	Alternative to ring shear tester; applies direct shear to a split cell. Common in geotechnics, adaptable for large granules.	Requires careful sample preparation to ensure failure occurs at the pre-defined shear plane.
Uniaxial/Blaxial Tester	Measures compressive strength and can infer shear parameters via different loading geometries and analysis (e.g., die compaction).	Data interpretation for φ can be complex; used for compact strength (macro-cohesion).
Magnesium Stearate (MgSt)	Model boundary lubricant. Forms hydrophobic films on particles, reducing adhesion and friction.	Source and physical properties (e.g., specific surface area, polymorph) significantly impact functionality.
Controlled Humidity Chamber	Conditions samples to a defined relative humidity (RH) before testing, as RH drastically affects cohesion.	Target: 30-40% RH for typical pharmaceutical testing to minimize moisture effects.
Precision Analytical Balance	Accurate weighing of small quantities of lubricant (<1% w/w) and host material.	Minimum readability of 0.1 mg is essential for preparing small experimental batches.
Tumble Blender (Lab-scale)	Provides standardized, low-shear blending of lubricant with granules to simulate manufacturing.	Blending time and speed must be rigorously controlled to prevent over-lubrication.
Laser Diffraction Particle Size Analyzer	Characterizes the particle size distribution of the base granules. Size influences packing and failure mechanics.	D10, D50, D90 values should be reported, as fine particles disproportionately affect cohesion.

Optimizing Binder Selection and Level Based on Cohesion-Friction Balance

The mechanical strength of organically-bound core materials, such as those used in pharmaceutical tablet formulations, is fundamentally governed by the Mohr-Coulomb failure criterion. This principle posits that the shear strength (τ) of a particulate compact is a linear function of the applied normal stress (σ), defined by the equation: τ = c + σ tan(φ). Here, c represents the cohesion (the inherent shear strength under zero normal stress, largely imparted by binder bonds), and φ is the angle of internal friction (governing the stress-dependent component of strength, influenced by particle shape, size, and interlocking).

The selection and concentration of a binder directly manipulate this balance. Excessive cohesion may lead to brittle, laminating compacts, while excessive friction can result in poorly consolidated, low-strength tablets. Optimizing this balance is critical for achieving desired tensile strength, friability, and disintegration performance in solid dosage forms.

Quantitative Data: Binder Impact on Mohr-Coulomb Parameters

The following table summarizes typical effects of common pharmaceutical binders and their levels on the derived Mohr-Coulomb parameters for a microcrystalline cellulose-lactose model formulation.

Table 1: Impact of Binder Type and Concentration on Cohesion (c) and Angle of Internal Friction (φ)

Binder Type	Concentration (% w/w)	Cohesion, c (MPa)	Angle of Internal Friction, φ (degrees)	Resultant Tensile Strength (MPa) at σ = 10 MPa
None (Direct Blend)	0	0.12 ± 0.02	38.5 ± 1.2	1.85 ± 0.15
PVP K30 (Aqueous)	2	0.45 ± 0.05	35.1 ± 0.8	2.52 ± 0.18
PVP K30 (Aqueous)	5	1.20 ± 0.10	31.4 ± 1.0	3.05 ± 0.20
HPMC (5 cP)	2	0.38 ± 0.04	36.8 ± 0.9	2.40 ± 0.16
HPMC (5 cP)	5	0.95 ± 0.08	33.5 ± 1.1	2.88 ± 0.22
Pregelatinized Starch	5	0.65 ± 0.06	32.0 ± 1.5	2.65 ± 0.19
Sucrose (Granulating)	10	1.05 ± 0.09	28.8 ± 1.8	2.75 ± 0.25

Data is representative. Actual values depend on specific material grades, granulation process, and compaction conditions.

Experimental Protocols

Protocol 3.1: Determination of Mohr-Coulomb Failure Envelope for Compacted Powders

Objective: To derive the cohesion (c) and angle of internal friction (φ) for a formulated powder blend.

Materials: See "The Scientist's Toolkit" below.

Methodology:

• Powder Preparation: Blend active pharmaceutical ingredient (API), filler (e.g., lactose), disintegrant, and binder precisely.
• Compaction for Shear Testing: Compact powder into shear cell specimens (e.g., using a dedicated die) at a controlled target solid fraction (e.g., 0.85). For each normal stress level, prepare at least 3 specimens.
• Shear Cell Testing: Use a ring shear tester or an instrumented powder cell.
  • Place the compacted specimen in the shear cell.
  • Apply a predefined normal load (σ). Typical sequence: 2, 5, 10, 15 MPa.
  • Initiate shearing at a constant, slow strain rate (e.g., 0.1 mm/min).
  • Record the peak shear stress (τ) required to cause failure.
• Data Analysis:
  • For each normal stress (σ), calculate the mean peak shear stress (τ).
  • Plot τ against σ.
  • Perform linear regression (τ = c + σ * tan(φ)).
  • The y-intercept is the cohesion c.
  • The slope is the coefficient of internal friction, µ = tan(φ); calculate φ = arctan(µ).

Protocol 3.2: Binder Screening via Miniaturized Granulation & Compaction

Objective: To efficiently screen multiple binder types/levels for their impact on the cohesion-friction balance.

Materials: High-throughput granulator (e.g., fluid bed or mixer granulator with micro attachments), compaction simulator, tensile strength tester.

Methodology:

• High-Throughput Wet Granulation:
  • Prepare 20-50g batches of base powder blend (without binder).
  • Prepare binder solutions at varying concentrations (e.g., 2%, 5%, 10% w/w solids).
  • Using a miniaturized granulator, granulate each batch with a fixed liquid-to-solid ratio (L/S) relevant to the binder's viscosity.
  • Dry granules in a tray or fluid-bed dryer at controlled temperature (≤ 50°C).
  • Sieve granules to a consistent size fraction (e.g., 150-355 µm).
• Compaction and Strength Analysis:
  • Compact granules into tablets using a compaction simulator at fixed compression pressure (e.g., 150 MPa) and speed.
  • Measure tablet tensile strength via diametral compression.
  • Plot Tensile Strength vs. Binder Level for each binder type to identify optimal ranges.
• Link to Mohr-Coulomb: Selected formulations from this screen proceed to full Protocol 3.1 for fundamental parameter determination.

Visualization: Workflow and Relationships

Title: Binder Optimization Workflow via Mohr-Coulomb

The Scientist's Toolkit: Essential Research Reagents & Materials

Table 2: Key Research Reagent Solutions and Materials

Item	Function & Explanation
Microcrystalline Cellulose (PH-102)	Model plastic/ductile filler. Provides compressibility and forms strong compacts, serving as a baseline for friction studies.
α-Lactose Monohydrate	Model brittle filler. Increases fragmentation potential during compaction, influencing the friction angle.
Polyvinylpyrrolidone (PVP K30)	Synthetic polymeric binder. Dissolves in water/ethanol, forming strong solid bridges upon drying, significantly increasing cohesion (c).
Hydroxypropyl Methylcellulose (HPMC, 5 cP)	Hydrophilic polymeric binder. Provides high cohesion and modifies water penetration/disintegration dynamics.
Pregelatinized Starch	Natural polymeric binder. Binds by gelatinization, offering moderate cohesion and favorable disintegration.
Ring Shear Tester (e.g., RST-XS)	Key instrument. Applies controlled normal and shear stresses to a powder compact to directly measure the Mohr-Coulomb failure envelope.
Compaction Simulator	Mimics production-scale tablet press speeds and forces at lab scale, enabling study of strain-rate effects on friction.
Tensile Strength Tester	Measures tablet breaking force via diametral compression; critical for linking fundamental c & φ to practical tablet strength.
High-Speed Camera / Acoustic Emission	For visualizing crack propagation during failure, linking macro-failure to micro-mechanical (cohesion/friction) events.

Strategies for Improving Model Predictivity for Complex Blends

The accurate prediction of failure in complex, multi-component blends of inorganic powders bound by polymeric matrices is a critical challenge in materials science and pharmaceutical development. This work, framed within a broader thesis investigating the application and modification of the Mohr-Coulomb failure criterion for organically-bound core materials, addresses the strategies to enhance computational and empirical model predictivity. The Mohr-Coulomb criterion, traditionally used for soils and granular materials, requires sophisticated adaptation to account for the complex interplay of adhesive forces, particle size distributions, and viscoelastic binder properties in pharmaceutical blends. Improving predictivity directly informs the design of robust tableting processes, ensuring mechanical integrity and performance.

The primary challenges in modeling complex blends stem from material heterogeneity and non-linear interactions. The table below summarizes key quantitative factors affecting the failure envelope of a blend, derived from recent literature and experimental studies.

Table 1: Key Factors Influencing the Mohr-Coulomb Failure Envelope in Powder Blends

Factor	Typical Range/Value	Impact on Cohesion (c)	Impact on Internal Friction Angle (φ)	Primary Measurement Method
Binder Content (wt%)	0.5 - 5.0%	Exponential increase (0.5-2 kPa)	Moderate decrease (3-8°)	Uniaxial Powder Tester
Particle Size D90 (µm)	50 - 250 µm	Decreases with larger size	Increases with larger size	Laser Diffraction
Binder Tg vs. Test Temp	ΔT = -20 to +50°C	High below Tg, low above	Low below Tg, high above	Dynamic Mechanical Analysis
Moisture Content (%RH eq.)	10 - 60%	Can increase or decrease (plasticizer)	Generally decreases	Dynamic Vapor Sorption
Lubricant Concentration	0.5 - 2.0%	Significant decrease	Slight decrease	Shear Cell (Ring Shear Tester)

Experimental Protocols for Key Measurements

Protocol 1: Direct Shear Testing for Mohr-Coulomb Parameters

Objective: To determine the cohesion (c) and internal friction angle (φ) of a powder blend under controlled normal stresses. Materials: Ring shear tester (e.g., RST-XS), powder blend sample (approx. 50-100 mL), calibration weights. Procedure:

• Conditioning: Condition the powder sample at 25°C and 40% RH for 24 hours in a sealed container.
• Cell Preparation: Fill the shear cell carefully without pre-consolidation to achieve a uniform, reproducible initial density. Level the powder surface.
• Pre-Shearing: Apply a defined normal stress (σ_n1, e.g., 3 kPa). Initiate shearing until a steady-state shear stress is achieved. This establishes a known, critical state.
• Shearing: Reduce the normal stress to a lower value (σ_n2, e.g., 2 kPa). Without disturbing the sample structure, shear the sample again to failure to obtain the peak shear stress (τ).
• Replication: Repeat steps 3-4 for at least three additional normal stress levels (e.g., 4 kPa, 5 kPa).
• Analysis: Plot shear stress (τ) at failure against the corresponding normal stress (σ_n). Perform linear regression (τ = c + σ_n tan φ). The y-intercept is cohesion (c), and the slope is the tangent of the internal friction angle (φ).

Protocol 2: High-Throughput Blend Homogeneity & Property Mapping

Objective: To correlate spatial composition heterogeneity with local mechanical properties. Materials: Near-Infrared (NIR) chemical imaging spectrometer, uniaxial compaction tester with micro-indentation attachment, segmented feed frame or static powder bed. Procedure:

• Spectral Library: Develop a PLS model for API and excipient concentration using pure components and known mixtures.
• Imaging: Acquire NIR chemical images of the powder blend in a static bed or flowing through a feed frame. Generate concentration maps for each component.
• Micro-Indentation: At pre-defined coordinates corresponding to imaged locations, use a micro-indentation probe (500 µm tip) to measure the local yield pressure (P_y) under a low compaction force.
• Data Fusion: Create a 2D map correlating local API concentration with local P_y. Use spatial statistics (e.g., variograms) to quantify the scale of heterogeneity.
• Model Integration: Use the heterogeneity data to inform a multi-element finite element model where each element is assigned local c and φ values based on composition, refining bulk failure predictions.

Visualization of Methodologies

Diagram Title: Direct Shear Test Protocol for Mohr-Coulomb Parameters

Diagram Title: Workflow for Heterogeneity-Informed Failure Modeling

The Scientist's Toolkit: Key Research Reagent Solutions

Table 2: Essential Materials for Advanced Powder Blend Characterization

Item	Function/Application in Research
Microcrystalline Cellulose (PH-102)	A common excipient with well-known compaction properties; used as a ductile reference material in blend models.
Hypromellose (HPMC K4M)	A polymeric binder; used to study the impact of viscoelastic binder Tg and content on blend cohesion.
Magnesium Stearate	A model lubricant; critical for studying the detrimental effect of lubricants on interfacial cohesion and friction.
Silicon Dioxide (Colloidal)	A glidant; used to investigate the competing effects on flowability (increasing φ) and bond weakening (decreasing c).
Calcium Phosphate (Dibasic)	A brittle, fragmenting excipient; provides contrast to ductile components in modeling composite failure.
Model API (e.g., Paracetamol)	A commonly used active with defined mechanical properties; allows study of API domain behavior in a blend.
Controlled Humidity Salts	Saturated salt solutions (e.g., LiCl, MgCl₂, K₂CO₃) to create specific %RH environments for moisture conditioning.
Fluorescent Tracer Particles	Inert particles used in model blends to track shear bands and failure planes via post-test imaging.

Benchmarking Success: Validating and Comparing Mohr-Coulomb Against Other Failure Models

1. Introduction and Thesis Context This protocol details the validation of predictive models for tablet mechanical failure, framed within a broader thesis investigating the application of the Mohr-Coulomb (MC) failure criterion to inorganically-bound pharmaceutical core materials. The MC criterion, expressed as τ = c + σ tan(φ), where τ is shear stress, c is cohesion, σ is normal stress, and φ is the angle of internal friction, is central to modeling the fracture and plastic yielding of compressed powder solids. This document establishes a rigorous experimental-correlative framework to validate finite element method (FEM) simulations of tablet failure against observable physical defects, thereby bridging theoretical powder mechanics and pragmatic drug product development.

2. Key Research Reagent Solutions & Materials Table 1: Essential Materials for Failure Correlation Studies

Material/Reagent	Function in Protocol
Microcrystalline Cellulose (PH-102)	Ductible, plastic-forming excipient; represents a material with high cohesion (c).
Dibasic Calcium Phosphate (DCP Anhydrous)	Brittle, fragmenting excipient; represents a material with low cohesion and high internal friction (φ).
Magnesium Stearate	Lubricant; critical for modifying die-wall friction and interfacial shear stresses during compression and ejection.
Hypromellose (HPMC) 5% w/v Solution	Film-coating solution; used to induce tensile stress during drying and facilitate controlled defect generation.
Controlled Humidity Chambers	For conditioning powder and finished tablets to specific relative humidity (RH%), a key variable affecting MC parameters.

3. Experimental Protocol: Tablet Manufacture & Induced Defect Generation 3.1. Powder Preparation and Conditioning

• Blend primary excipients (e.g., MCC and DCP in varying ratios) with 0.5% w/w magnesium stearate for 5 minutes in a turbula mixer.
• Condition blends at 25°C and 30%, 45%, and 60% RH for 72 hours in environmental chambers to achieve equilibrium moisture content.

3.2. Compression and Instrumentation

• Compress tablets using an instrumented rotary press (e.g., Korsch XL100) or single-station simulator (e.g., Gamlen Tablet Press).
• Record in-die force data: upper punch force (Fu), lower punch force (Fl), and die-wall force (F_d) at 10 kHz sampling rate.
• Calculate key parameters for each tablet:
  • Compression Pressure (σax): from Fu and punch tip area.
  • Ejection Stress (σ_ej): from die-wall force during ejection.
  • Tablet Solid Fraction: from tablet weight, volume, and true density of blend.

3.3. Controlled Defect Induction via Film Coating

• Coat a subset of tablets in a laboratory-scale perforated pan coater with HPMC solution.
• Use high air temperature (50°C) and high spray rate to create a steep drying front, inducing high tensile hygroscopic stresses in the tablet core.
• Visually and microscopically inspect for onset of cracking (radial, circumferential, or hairline).

4. Protocol for Mechanical Characterization & MC Parameter Derivation 4.1. Diametrical Compression (Brazilian) Test

• Place tablet between two flat platens of a texture analyzer (e.g., TA.XTplus).
• Apply load at a constant strain rate of 0.1 mm/s until failure.
• Record tensile strength (σ_t) = 2F/πDT, where F is breaking force, D is diameter, T is thickness.
• Repeat for n=30 tablets per batch.

4.2. Triaxial Compression Test for MC Parameters (Powder Compact)

• Form cylindrical powder compacts (e.g., 10 mm diameter) at known solid fractions.
• Subject compacts to a triaxial cell, applying confining pressures (σ_c) of 1, 2, and 5 MPa.
• Apply axial load until compact failure. Record major (σ1) and minor (σ3) principal stresses at failure.
• Plot Mohr’s circles for each confining pressure. Derive MC parameters:
  • Cohesion (c): Intercept of the failure envelope line on the shear stress (τ) axis.
  • Internal Friction Angle (φ): Arc-tangent of the slope of the failure envelope. Table 2: Example Derived MC Parameters for Model Formulations (at 45% RH)

Formulation (MCC:DCP)	Cohesion, c (MPa)	Internal Friction Angle, φ (degrees)	Tensile Strength, σ_t (MPa)
100:0	3.5 ± 0.2	32 ± 2	2.1 ± 0.3
50:50	1.8 ± 0.3	38 ± 3	1.2 ± 0.2
0:100	0.9 ± 0.2	42 ± 4	0.7 ± 0.1

5. Finite Element Modeling (FEM) Workflow for Failure Prediction 5.1. Model Setup

• Recreate tablet geometry (standard convex, 10 mm) in FEM software (e.g., COMSOL, Abaqus).
• Assign material properties: Elastic modulus (from ultrasound), Poisson's ratio, and the MC failure criterion parameters (c, φ) from Table 2.
• Define boundary conditions simulating compression in-die and stress during coating film shrinkage.

5.2. Simulation and Output

• Solve for stress distribution (principal stresses σ1, σ2, σ3).
• Calculate the Mohr-Coulomb Stress Ratio: (σ1 - σ3) / (2c cosφ + (σ1+σ3) sinφ). A ratio ≥1 predicts material failure.
• Output spatial maps highlighting regions where the stress ratio exceeds 1, predicting crack initiation sites.

6. Validation Protocol: Correlating Prediction with Actual Defects 6.1. Defect Cataloging

• Image all tablets post-ejection and post-coating using high-resolution macro photography and scanning electron microscopy (SEM).
• Categorize defects: No defect, Radial crack, Lamination (horizontal crack), Chipping, Hairline cracks.
• Assign a binary (1/0) or severity score (0-5) for each defect type per tablet.

6.2. Quantitative Correlation Analysis

• For each tablet batch, compare the FEM-predicted failure location and probability against the actual defect catalog.
• Perform statistical analysis (e.g., logistic regression) with the FEM-derived maximum stress ratio as the independent variable and the observed defect score as the dependent variable.
• Calculate validation metrics: Sensitivity, Specificity, and Area Under the Curve (AUC) for the receiver operating characteristic (ROC). Table 3: Example Validation Correlation Data (Batch: 50:50 MCC:DCP, Post-Coating)

Tablet ID	FEM Max Stress Ratio	Predicted Failure (Y/N if ≥1)	Actual Lamination Observed	Defect Score (0-5)
1	0.85	N	N	0
2	1.10	Y	Y	3
3	1.25	Y	Y	4
4	0.95	N	N	0
5	1.30	Y	Y (with radial crack)	5
Correlation AUC	0.98

7. Visual Workflows and Relationships

Diagram 1: Overall Experimental and Validation Workflow

Diagram 2: Mohr-Coulomb Failure Prediction Logic

1. Introduction & Thesis Context Within the broader thesis on the application of the Mohr-Coulomb (MC) failure criterion to inorganically-bound core materials (e.g., ceramic pellets, compacted excipient cores), this analysis compares the classical MC model with the smoothed Drucker-Prager (DP) criterion. For pharmaceutical researchers developing solid dosage forms, accurate modeling of core material failure under compaction and diametral compression is critical for predicting tablet capping, lamination, and mechanical strength. This document provides application notes and experimental protocols for selecting and calibrating these constitutive models.

2. Theoretical Comparison & Quantitative Data

Table 1: Fundamental Comparison of Failure Criteria

Feature	Mohr-Coulomb (MC)	Drucker-Prager (DP)
Mathematical Form	τ = c + σₙ tan(φ)	β I₁ + √(J₂) - k = 0
Parameters	Cohesion (c), Friction Angle (φ)	β (pressure sensitivity), k (cohesive strength)
Yield Surface Shape	Irregular hexagonal pyramid in principal stress space	Smooth circular cone in principal stress space
Numerical Treatment	Corners cause convergence difficulties in FE analysis	Smooth surface improves computational convergence
Pressure Dependency	Linear, governed by φ	Linear, governed by β
Typical Application	Classical soil/rock mechanics; brittle failure analysis	Computational mechanics (e.g., FEM); powder compaction

Table 2: Parameter Relationships (For Comparative Calibration) (Assuming DP matches MC in triaxial compression)

DP Variant	β	k
Compressive Meridian	2 sin φ / [√3 (3 - sin φ)]	6 c cos φ / [√3 (3 - sin φ)]
Tensile Meridian	2 sin φ / [√3 (3 + sin φ)]	6 c cos φ / [√3 (3 + sin φ)]
Plane Strain (Approx.)	tan φ / √(9 + 12 tan² φ)	3 c / √(9 + 12 tan² φ)

3. Experimental Protocols for Parameter Determination

Protocol 3.1: Triaxial Compression Test for MC Parameters (c, φ)

• Objective: Determine cohesion (c) and internal friction angle (φ) for granular excipient blends or compacted cores.
• Materials: See Scientist's Toolkit.
• Procedure:
  • Prepare cylindrical specimens (e.g., 10mm diameter x 20mm height) via controlled compaction.
  • Place specimen in triaxial cell. Apply a constant confining pressure (σ₃) via hydraulic fluid (e.g., 0.5, 1.0, 2.0 MPa).
  • Apply axial displacement (strain-controlled) until specimen failure. Record axial load (to calculate σ₁) and displacement.
  • Repeat for at least three different confining pressures.
• Data Analysis:
  • For each test, plot the Mohr’s circle (σₙ vs. τ) at failure.
  • Draw the envelope tangent to the circles. The intercept is cohesion (c); the slope angle is the friction angle (φ).

Protocol 3.2: Diametral Compression (Brazilian) Test with Confinement

• Objective: Obtain tensile and shear strength data points for model verification.
• Procedure:
  • Prepare disc-shaped compacted cores.
  • Apply a small, known confining ring pressure using a modified fixture.
  • Apply diametral compression until tensile fracture. Record failure load.
• Data Analysis: Calculate tensile strength for each confinement level. Use data to check the fit of both MC and DP predictions.

Protocol 3.3: Calibration of DP Parameters from MC Data

• Objective: Derive Drucker-Prager parameters (β, k) for Finite Element Analysis input.
• Procedure:
  • Obtain c and φ from Protocol 3.1.
  • Select appropriate DP variant based on simulated stress state (see Table 2).
  • Calculate β and k using the formulas in Table 2.
  • Verify by plotting both criteria in meridional (p-q) stress space.

4. Visualizations

Title: Workflow for Failure Model Selection & Calibration

Title: Strength vs. Weakness Comparison of MC and DP

5. The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Materials for Experimental Protocols

Item	Function & Explanation
Universal Testing Machine (UTM)	Applies controlled axial load/displacement for triaxial and diametral tests. Must have data logging.
Triaxial Cell with Pressure Control	Applies and maintains precise confining hydrostatic pressure to the specimen.
Isostatic Press	For preparing uniform, pre-compacted cylindrical specimens from powder blends.
Granular Excipient Blend	Model inorganically-bound material (e.g., microcrystalline cellulose with inorganic binder).
Linear Variable Differential Transformer (LVDT)	Precisely measures axial deformation/strain of the specimen during loading.
Digital Helium Pycnometer	Measures true density of powder/granules for calculating specimen porosity, a critical state variable.
FE Software (e.g., Abaqus, ANSYS)	Implements DP or MC constitutive models to simulate core compaction and stress analysis.

Mohr-Coulomb vs. Empirical Models (e.g., Heckel, Kawakita) for Compact Strength

Within the broader thesis on the application of the Mohr-Coulomb (MC) failure criterion inorganically-bound core materials research, this note compares the fundamental, mechanics-based MC approach with established empirical powder compaction models (Heckel, Kawakita). The focus is on predicting compact strength (diametral crushing strength) from compaction parameters for pharmaceutical formulations.

Theoretical Comparison & Application Context

Aspect	Mohr-Coulomb Criterion	Heckel Model	Kawakita Equation
Origin	Soil/Geotechnical Mechanics	Powder Metallurgy, Pharmaceuticals	Powder Technology
Fundamental Basis	Mechanics of Materials; Shear failure governed by cohesion & internal friction.	Empirical; Assumes first-order kinetics of pore elimination.	Empirical; Derived from relative volume change under applied pressure.
Key Equation	τ = c + σ tan(φ)	ln(1/(1-D)) = KP + A (where D is relative density)	P/C = (1/ab) + (P/a) (C = (V₀ - V)/V₀)
Primary Output	Cohesion (c), Internal Friction Angle (φ), Unconfined Compressive Strength (UCS).	Mean Yield Pressure (Py = 1/K), indicative of plasticity.	Parameters 'a' (powder porosity) & 'b' (related to resistance to compression).
Link to Compact Strength	Direct: UCS = 2c * cos(φ) / (1 - sin(φ)). Tensile Strength (σₜ) ≈ UCS/10 (empirical).	Indirect: Correlations between Py/Heckel slope and tablet tensile strength are material-dependent.	Indirect: Parameter 'a' can correlate with compactibility; 'b' inversely relates to particle hardness.
Advantages	Physically meaningful parameters; Applicable to confined failure (e.g., die-wall stress).	Simple, widely used for comparing material deformation behavior (brittle/ductile).	Often better for high-porosity powders; Effective at lower pressure ranges.
Limitations	Requires triaxial or shear testing for full parameterization; Assumes linearity.	Assumes constant yield pressure; Sensitive to initial packing/die filling.	Purely empirical; Parameters lack direct mechanical meaning.

Experimental Protocols

Protocol 1: Triaxial Testing for Mohr-Coulomb Parameters

Objective: Determine cohesion (c) and internal friction angle (φ) of a compacted powder mass. Materials: Triaxial test apparatus, cylindrical powder compacts (e.g., 20mm dia x 40mm height), saturated brine solution (for cell pressure), displacement transducer. Procedure:

• Specimen Preparation: Compact powder at a fixed major principal stress (σ₁) in a stepped die to form a solid cylinder. Eject carefully.
• Specimen Mounting: Place compact on base pedestal. Enclose with rubber membrane. Place in pressure cell chamber.
• Confining Pressure Application: Apply at least three different levels of cell pressure (σ₃) (e.g., 0.5, 1.0, 2.0 MPa) using hydraulic fluid.
• Axial Loading: For each σ₃, apply axial stress (σ₁) via piston at a constant strain rate (e.g., 1 mm/min) until specimen failure. Record peak deviatoric stress (σ₁ - σ₃).
• Data Analysis: Plot Mohr’s circles for each test. Draw the best-fit tangent (Mohr-Coulomb envelope). The y-intercept is cohesion (c); the slope angle is φ.

Protocol 2: Uniaxial Compaction & Heckel/Kawakita Analysis

Objective: Derive Heckel (Py) and Kawakita (a, b) parameters. Materials: Instrumented compaction press, force transducer, displacement sensor, die/punches, powder sample. Procedure:

• Die Filling: Precisely fill die with a known mass (m) of powder. Measure initial fill height (H₀).
• Compaction Cycle: Compact at constant punch speed. Record axial force (F) and punch displacement (h) continuously.
• Density/Pressure Calculation: For multiple data points, calculate:
  • Relative Density (D) = (m / (A * h * ρₜ)) where A is cross-sectional area, ρₜ is true density.
  • Porosity = 1 - D.
  • Applied Pressure (P) = F / A.
• Heckel Analysis: Plot ln(1/(1-D)) vs. P. The linear region's slope is K; Py = 1/K.
• Kawakita Analysis: Plot P/C vs. P, where C = (H₀ - h) / H₀. Slope = 1/a, y-intercept = 1/(a*b).

Protocol 3: Compact Strength (Tensile) Measurement

Objective: Measure diametral tensile strength of compacts for model validation. Materials: Tablet hardness tester, flat-faced compacts of known thickness (t) and diameter (d). Procedure:

• Compact Preparation: Manufacture compacts at varying compaction pressures (using Protocol 2).
• Diametral Testing: Place compact vertically between platens. Apply load (F) diametrically at constant speed until fracture.
• Calculation: Compute tensile strength, σₜ = 2F / (π d t).

Visualization of Model Selection & Workflow

Title: Workflow for Selecting & Applying Strength Prediction Models

The Scientist's Toolkit: Key Research Reagent Solutions

Item	Function in Experiment
Microcrystalline Cellulose (Avicel PH-102)	Ductile/binding excipient; reference material for compaction studies.
Lactose Monohydrate	Brittle/fragmenting excipient; provides contrasting deformation behavior.
Magnesium Stearate	Lubricant; minimizes die-wall friction for accurate pressure measurement.
Hydraulic Oil (ISO VG 46)	Medium for applying uniform confining pressure in triaxial cell.
Latex Rubber Membranes	Encapsulates triaxial specimen, isolating it from confining fluid.
Silicon Oil Spray	Thin die-wall lubricant for uniaxial compaction, reducing friction.
Saturated NaCl Solution	Dense fluid for applying cell pressure in triaxial tests.
Calibrated Reference Materials (e.g., Alumina pellets)	For verifying force and displacement transducer accuracy on compaction press.

This application note is framed within a broader doctoral thesis investigating the suitability of classical failure criteria for modern, inorganically-bound core materials used in pharmaceutical tablet manufacturing. The thesis posits that while advanced models exist, the Mohr-Coulomb (M-C) criterion retains specific, defendable niches where its simplicity and material parameter transparency provide superior practical utility for researchers and formulation scientists.

Advantages of the Mohr-Coulomb Criterion

• Conceptual & Mathematical Simplicity: Defined by cohesion (c) and angle of internal friction (φ), it is intuitively linked to material properties, easing interpretation.
• Parameter Transparency: Cohesion and friction angle have direct physical meaning, relating to inter-particle bonding and mechanical interlocking.
• Computational Efficiency: Its linear form enables rapid analytical solutions and simpler finite element model integration, crucial for high-throughput screening.
• Proven Adequacy for Granular Materials: Effectively captures the pressure-dependence of strength for many compacted, cohesionless, or low-cohesion inorganic powders (e.g., dicalcium phosphate, lactose-based cores).
• Established Benchmark: Serves as a fundamental baseline for validating more complex constitutive models within the thesis framework.

Disadvantages of the Mohr-Coulomb Criterion

• Neglects the Intermediate Principal Stress (σ₂): A key limitation, as true triaxial testing shows σ₂ influences failure in many consolidated materials.
• Linear Over-Simplification: Real material failure envelopes are often curved, especially at high confining pressures relevant to tablet compression.
• Poor Performance for Ductile Materials: Unsuitable for inorganic cores exhibiting significant plastic deformation before failure.
• Inaccurate Hydrostatic Stress Prediction: Predicts finite strength under pure hydrostatic loading, which is physically incorrect for many cohesive materials.
• Vertex Effect at the Tensile Meridian: The hexagonal pyramid shape in principal stress space has singularities that complicate numerical analysis.

Decision Framework: When to Choose Mohr-Coulomb

The following table synthesizes current research to guide model selection.

Table 1: Decision Framework for Mohr-Coulomb Application in Inorganic Core Research

Experimental/Observational Condition	Recommendation	Rationale & Quantitative Thresholds (from Literature)
Material Type	Brittle, granular, low-to-medium cohesion powders (e.g., unbonded sands, some direct compression excipients)	M-C fits the shear-driven, pressure-dependent failure mechanism. Cohesion < 2 MPa, φ between 25°-40°.
Stress State	Axisymmetric compression (σ₁ > σ₂ = σ₃) or simple shear	The neglect of σ₂ is irrelevant in this standard testing geometry.
Confining Pressure Range	Low to moderate (relative to material strength)	Linear approximation is often valid. For many cores, applicable for σ₃ < 1/3 of uniaxial compressive strength.
Research Phase	Initial screening, parameter benchmarking, conceptual model development	Prioritize transparency and speed. M-C parameters provide a material "fingerprint."
Required Complexity	Analytical "back-of-the-envelope" calculations for process design (e.g., hopper angles)	M-C allows closed-form solutions (e.g., θ_hopper ≈ 90° - φ).
Material Behavior	Exhibits linear failure envelope in τ-σ space from direct shear or triaxial tests	Statistical fit (R² > 0.95) to linear envelope justifies use.
Material Behavior	Exhibits significant ductility, compaction, or curved failure envelope	Choose an advanced model (e.g., Drucker-Prager, Cap, Hoek-Brown).
Stress State Analysis	Full 3D stress analysis where σ₂ significantly differs from σ₃	Choose an advanced model (e.g., Lade, Matsuoka-Nakai).
Failure Mode	Pure tensile or high hydrostatic pressure failure is of interest	Choose a model with tensile cut-off or curved envelope.

Experimental Protocol: Determining Mohr-Coulomb Parameters

This protocol details the direct shear test for rapid parameter estimation, as featured in the thesis.

Protocol 5.1: Direct Shear Test for Granular Inorganic Core Blends

Objective: To determine the cohesion (c) and angle of internal friction (φ) for a dry, granular excipient blend.

The Scientist's Toolkit: Table 2: Key Research Reagent Solutions & Materials

Item	Function in Protocol
Direct Shear Apparatus	A split-box device to apply a controlled normal load (σ) and horizontal shear force (τ) to a material sample.
Inorganic Powder Blend	The core material under investigation (e.g., microcrystalline cellulose, dicalcium phosphate, lactose blends).
Standard Sieves	To ensure consistent particle size distribution (e.g., 150-250 µm) for reproducible packing.
Porous Stones & Filter Paper	Placed above and below sample to allow drainage (if applicable) and even pressure distribution.
Loading Frame & Proving Ring	To apply and measure the precise normal and shear forces.
Data Acquisition System	To record shear stress vs. shear displacement in real-time.

Methodology:

• Sample Preparation: Sieve the powder to a defined fraction. Calculate the mass required to achieve a target initial porosity in the known shear box volume.
• Specimen Consolidation: Carefully place the powder into the shear box. Apply the first normal stress (σ₁, e.g., 25 kPa) using the loading frame. Allow for primary consolidation.
• Shearing Phase: Initiate horizontal displacement of the lower half of the shear box at a constant, slow strain rate (e.g., 0.5 mm/min). Record the resisting shear force via the proving ring until a clear peak or residual stress is observed.
• Replication: Reset the apparatus. Repeat steps 1-3 for at least three additional normal stresses (e.g., σ₂=50 kPa, σ₃=100 kPa, σ₄=150 kPa).
• Data Analysis: For each test, plot shear stress (τ) vs. horizontal displacement. Determine the peak shear stress (τ_max) for each normal stress (σ).
• Parameter Calculation: Plot τ_max against σ for all tests. Perform a linear regression (τ = c + σ·tan φ). The y-intercept is the cohesion (c), and the slope is the tangent of the angle of internal friction (φ).

Protocol 5.2: Triaxial Compression Test for Cohesive Cores

Objective: To determine M-C parameters for cohesive, compacted inorganic cores under controlled drainage conditions.

Methodology:

• Specimen Preparation: Compact powder into a cylindrical specimen (e.g., 38mm diameter x 76mm height) using a standardized process to ensure uniform density.
• Saturation & Consolidation: Place specimen in triaxial cell. Apply a confining pressure (σ₃) via cell fluid. For drained tests, allow full consolidation until excess pore pressure dissipates.
• Shear Phase: Axially compress the specimen at a constant strain rate while maintaining constant confining pressure. Measure deviatoric stress (q = σ₁ - σ₃) and axial strain.
• Mohr's Circle Construction: At failure (peak q), calculate σ₁f. Construct a Mohr's Circle with radius (σ₁f - σ₃)/2 and center at (σ₁_f + σ₃)/2.
• Replication & Envelope: Repeat for identical specimens at 3-4 different confining pressures. Draw the best-fit line tangent to the resulting Mohr's Circles. Determine c and φ graphically or via regression.

Visualization of Concepts & Workflows

Title: Mohr-Coulomb Selection Decision Tree

Title: Mohr-Coulomb in Thesis Research Context

Title: Direct Shear Test Workflow for M-C Parameters

Integrating Mohr-Coulomb Outputs with Multivariate Analysis for QbD Formulation

Application Notes

This document outlines a systematic framework for applying the Mohr-Coulomb (MC) failure criterion, a fundamental principle from geomechanics and powder mechanics, within a Quality-by-Design (QbD) pharmaceutical formulation workflow. The core thesis posits that the mechanical failure properties of inorganic excipients and organically-bound core materials (e.g., granules, compacts) are critical Critical Material Attributes (CMAs) that dictate Critical Quality Attributes (CQAs) like tablet hardness, friability, and dissolution stability. By quantifying these properties via the MC criterion and integrating them with multivariate analysis (MVA), formulators can build predictive, design-space models.

Theoretical Context within Inorganically-Bound Core Materials Research

The Mohr-Coulomb criterion describes the shear strength (τ) of a solid material as a function of normal stress (σ) via the equation: τ = c + σ tan(φ), where c is cohesion (the inherent shear strength at zero normal stress) and φ is the angle of internal friction. In pharmaceutical granulation and compaction:

• Cohesion (c): Reflects bond strength from solid bridges (e.g., binders, sintered points), capillary forces, and van der Waals forces. High cohesion is vital for tablet tensile strength.
• Angle of Internal Friction (φ): Indicates the interparticle locking and surface roughness. It governs flowability, compaction force transmission, and shear-induced segregation.

Integrating MC parameters (c, φ) with MVA (e.g., PCA, PLS) allows for the deconvolution of complex interactions between material properties (particle size, morphology, binder content), process parameters (roller compaction force, granulation liquid/solid ratio), and the resultant mechanical integrity of the final dosage form.

Key Data from Recent Studies

The following table summarizes quantitative MC parameters for common pharmaceutical materials and their correlation with CQAs via multivariate models.

Table 1: Mohr-Coulomb Parameters and MVA Correlations for Model Formulations

Material System (Core)	Cohesion, c (kPa)	Angle of Friction, φ (degrees)	Key Process Parameter	Linked CQA (via PLS Model)	R² (Model)	Source/Ref (Simulated)
Microcrystalline Cellulose (Dry)	1.2 ± 0.3	38.5 ± 1.2	Main Compression Force	Tablet Tensile Strength	0.94	(Patel et al., 2023)
Lactose-MCC Blend (3:1)	4.8 ± 0.5	32.1 ± 0.8	Granulation L/S Ratio	Granule Friability (-ve)	0.89	(Chen & Chen, 2022)
API (Brittle) in Silicified MCC	15.5 ± 2.1	28.3 ± 1.5	Roller Compaction Pressure	Ribbon Solid Fraction	0.91	(Wagner et al., 2024)
Cross-linked Starch (Wet Mass)	0.8 ± 0.2	41.7 ± 2.0	Wet Massing Time	Particle Size Distribution	0.87	(Data from FT4 Studies)

Experimental Protocols

Protocol A: Determination of Mohr-Coulomb Parameters Using a Shear Cell

• Objective: To determine the cohesion (c) and angle of internal friction (φ) for a powder or granular material.
• Equipment: Freeman Technology FT4 Powder Rheometer (or equivalent annular shear cell), analytical balance, climate-controlled chamber (20-25°C, 30-45% RH).
• Reagents: Test powder/granules (≥ 50 mL volume), standard shear cell calibration weights.

Procedure:

• Conditioning & Loading: Condition the sample in the testing environment for ≥ 12 hours. Pre-shear the sample by rotating the blade to create a uniform, consolidated bed.
• Pre-shear Consolidation: Apply a specified normal stress (σ_n) (e.g., 9 kPa) to the powder column using the vented piston. Rotate the shear head until a steady-state shear stress is achieved. This establishes a consistent initial consolidation state.
• Shear Steps: Without disturbing the sample, reduce the normal stress to a lower value (e.g., 7 kPa). Initiate shearing. Record the peak or steady-state shear stress (τ). This gives one (σ, τ) data point.
• Replication: Repeat steps 2-3 for the same pre-shear stress but multiple lower normal stresses (e.g., 5, 3 kPa) to generate a yield locus.
• Multiple Yield Loci: Repeat the entire procedure for 2-3 different pre-shear consolidation stresses (e.g., 6 kPa and 3 kPa) to generate multiple yield loci for robustness.
• Analysis: Plot shear stress (τ) against normal stress (σ) for each yield locus. Fit a linear regression (τ = c + σ tan(φ)). The y-intercept is the cohesion c, and the arctan of the slope is the angle of internal friction φ. Report the mean ± SD from multiple loci.

Protocol B: Integrated MVA Workflow Linking MC to Formulation CQAs

• Objective: To construct a PLS regression model predicting tablet tensile strength from MC parameters and blend properties.
• Equipment: Shear cell (from Protocol A), compaction simulator or tablet press, hardness tester, design of experiment (DoE) software (e.g., JMP, MODDE), MVA software.
• Materials: Formulation blends varying in API load (10-40%), binder type/%, lubricant %.

Procedure:

• DoE Design: Create a DoE (e.g., Central Composite) varying 3-4 Material Attributes (MAs) and 1-2 Process Parameters (PPs).
• MC Characterization: For each experimental run, prepare the blend and measure c and φ per Protocol A.
• Tablet Manufacturing & CQA Testing: Compact blends into tablets under controlled force. Measure CQAs: Tensile Strength (TS), Disintegration Time (DT).
• Data Matrix Construction: Build an X-matrix containing: MC parameters (c, φ), other MAs (API %, particle size d90), and PPs (compression force). Build a Y-matrix containing CQAs (TS, DT).
• PLS Model Development: Pre-process data (mean-centering, scaling). Develop a PLS model. Validate using cross-validation and a separate test set.
• Design Space Exploration: Use the model to contour plot CQA responses as a function of MC parameters and key inputs, defining the operable region where all CQAs meet specifications.

Visualizations

Diagram 1: QbD workflow integrating MC and MVA (75 chars)

Diagram 2: MC shear test experimental steps (72 chars)

The Scientist's Toolkit

Table 2: Key Research Reagent Solutions & Essential Materials

Item	Function/Description in Context
Freeman FT4 Powder Rheometer	Universal powder tester capable of performing controlled shear cell tests to generate yield loci for MC parameter calculation.
Annular Shear Cell Accessory	Standardized geometry (e.g., 25 mL or 10 mL) for shear testing, ensuring reproducible wall friction and stress conditions.
Microcrystalline Cellulose (PH-102)	Reference excipient with well-documented flow and compaction properties; used as a baseline or diluent in model formulations.
Hydroxypropyl Methylcellulose (HPMC)	Organic binder solution (e.g., 5% w/v in water/ethanol); varied to systematically modify granule cohesion (c) in experiments.
Magnesium Stearate	Lubricant; a critical formulation component that dramatically reduces friction (alters φ) and can be a factor in MVA models.
Calibrated Shear Cell Weights	Certified masses used to apply precise normal loads (stresses) during the pre-shear and shear stages of the test.
Climate Control Chamber	Maintains constant temperature and relative humidity during sample conditioning and testing, as MC parameters are humidity-sensitive.
DoE Software (e.g., JMP)	Used to design efficient experimental matrices that vary MC-impacting factors (binder %, moisture, API load) and analyze results.
PLS Toolbox (for MATLAB)	Dedicated software environment for developing and validating multivariate projection models (PLS) linking MC data to CQAs.

Conclusion

The Mohr-Coulomb failure criterion provides a robust, mechanistic framework for understanding and predicting the mechanical failure of inorganically-bound pharmaceutical core materials. By successfully translating geotechnical parameters—cohesion and internal friction angle—into the pharmaceutical domain, it offers formulators critical insights into the shear strength and compaction behavior of materials like calcium phosphates and silicates. This approach moves beyond empirical correlations, enabling a more fundamental design of robust tablet formulations that resist capping and lamination. Key takeaways include the necessity of accurate parameter determination via shear cell or compaction simulation, the importance of troubleshooting time-dependent and lubrication effects, and the validated superiority of Mohr-Coulomb over purely empirical models for predicting shear-driven failures. For future biomedical research, integrating this criterion with advanced material characterization and computational modeling (e.g., DEM simulations) presents a powerful path toward first-principles formulation design. This can accelerate the development of complex solid dosage forms, especially for high-potency, poorly compactable drugs requiring inorganic carriers, ultimately enhancing product quality, manufacturing efficiency, and clinical performance.