Soil Constitutive Modelling

Soil constitutive modelling defines how soil responds to stress, strain, and environmental conditions. It provides mathematical relationships used in simulations such as finite element analysis for geotechnical problems.

What is a Constitutive Model

A constitutive model describes the relationship between stress and strain:

$$\sigma = f(\varepsilon, \dot{\varepsilon}, t, \text{history})$$

• $\sigma$ = stress tensor
• $\varepsilon$ = strain tensor
• $\dot{\varepsilon}$ = strain rate
• history = loading path dependence

Soil behavior is nonlinear, path dependent, and often irreversible, unlike simple elastic materials.

Elastic Model

The simplest assumption is linear elasticity:

$$\sigma = D \cdot \varepsilon$$

Where $D$ is the stiffness matrix derived from:

$$E = \text{Young's modulus}, \quad \nu = \text{Poisson's ratio}$$

Limitations:

• no plastic deformation
• no failure prediction
• unrealistic for most soils

Plasticity in Soils

Real soils exhibit plastic behavior. The total strain is decomposed as:

$$\varepsilon = \varepsilon^e + \varepsilon^p$$

• $\varepsilon^e$ = elastic strain
• $\varepsilon^p$ = plastic strain

Plasticity is governed by three key components:

• yield function
• flow rule
• hardening law

Yield Criterion

Defines when soil starts to yield. A common model is Mohr-Coulomb:

$$f(\sigma) = \tau - c - \sigma \tan \phi = 0$$

• $c$ = cohesion
• $\phi$ = friction angle
• $\tau$ = shear stress

This represents a failure envelope in stress space.

Flow Rule

Defines direction of plastic strain:

$$d\varepsilon^p = \lambda \frac{\partial g}{\partial \sigma}$$

• $\lambda$ = plastic multiplier
• $g$ = plastic potential function

Associated flow: $g = f$
Non-associated flow: $g \ne f$ (common in soils)

Hardening and Softening

Material properties evolve with plastic deformation:

$$f(\sigma, \kappa) = 0$$

• $\kappa$ = internal variable (e.g., plastic strain)

Types:

• hardening: strength increases
• softening: strength decreases

Advanced Models

More realistic soil behavior is captured using:

• Cam Clay models (critical state soil mechanics)
• Hardening Soil model
• Hypoplastic models

Example: Modified Cam Clay yield surface

$$f = q^2 + M^2 p (p - p_c)$$

• $p$ = mean stress
• $q$ = deviatoric stress
• $M$ = critical state slope
• $p_c$ = preconsolidation pressure

Why It Matters

The models are used because they are essential for:

• predicting settlement and deformation
• evaluating slope stability
• simulating excavation and foundation behavior