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Constitutive models (stress-strain)

Constitutive models are fundamental in understanding and predicting the behavior of soils under various conditions, a concept already introduced subtly during bachelor courses. For instance, the settlement analysis according to Kany, rooted in the theory of elasticity, and the failure analysis of slopes and foundations based on the theory of plasticity and the Mohr-Coulomb limit state, are early encounters with these models. These analytical methods, while effective, are limited in scope, typically applied to simpler geotechnical challenges involving embankments and slopes of simple geometries or simple load sequences.

To tackle more intricate problems, advanced numerical methods like the finite element method or the finite difference method are employed. In the majority of cases, both approaches - (simple) analytical and (advanced) numerical - aim to provide a statement about the expected displacements \(u_{i}\), strains \(\varepsilon_{ij}\) and stresses \(\sigma_{ij}\) for the given problem. For each point considered, this corresponds to 15 unknown variables:

  • Stress \(\sigma_{ij}\): Comprising 6 unknowns, as detailed in stress tensor.
  • Strain \(\varepsilon_{ij}\): Another set of 6 unknowns, explained in strain tensor.
  • Displacement \(u_{i}\): Consisting of 3 components (\(u_x\), \(u_y\), and \(u_z\)).

A total of 15 equations are necessary to solve these variables. From physical laws, 9 equations emerge:

  • Kinematic relations linking strain and displacement, yield 6 equations:

    \[ \varepsilon_{ij} = \dfrac{1}{2} \left( \dfrac{\partial u_i}{\partial x_j} + \dfrac{\partial u_j}{\partial x_i} \right) \]

  • The balance of linear momentum for the static case, equating the material time derivative of momentum to external forces, gives 3 equations:

    \[ \sigma_{ij,j} + b_i \rho = 0_i \]

    Here, \(\sigma_{ij,j}\) is the divergence of the stress tensor representing the internal forces, \(\rho\) denotes density and \(b_i\) the body force vector.

The final 6 equations are provided by constitutive equations which mathematically describe the specific material response by relating the stress in the material \(\sigma_{ij}\) to strain \(\varepsilon_{ij}\).

A more general definition

Constitutive models are not limited to description of stress-strain behaviour. In many applications, transport phenomena for liquids, gases or heat are of great interest. In other cases, chemical processes within the material must be taken into account. We have seen this for example for partially saturated soils, where empirical relationships have been introduced for the dependence of the degree of saturation on the suction and the relative permeability on the degree of saturation. These are also constitutive models. A more general definition is given by Jacob Fish:

"A constitutive equation demonstrates a relation between two physical quantities that is specific to a material or substance and does not follow directly from physical laws" (J. Fish, 2014, Practical Multiscaling, Wiley)

It is important to appreciate that constitutive models are a significant abstraction from reality and need to encapsulate critical aspects of the materials while omitting less crucial details. The balance between what to include and omit varies with the problem and the objective at hand. An example is Hooke's Law for linear elasticity, which simply relates stress and strain. However, the behaviour of soils is in general highly non-linear and an-elastic as was discussed for example in Chapters Triaxial test and Critical State Soil Mechanics or Section Soil behaviour under undrained cyclic shearing.

Parameters and Variables

Constitutive models, essentially equations, comprise two distinct sets: constants, known as material parameters, and state variables. The parameters in these models represent constant values, reflecting the assumption that material properties do not change during the process being modeled. Accurately determining (calibrating) these parameters is crucial for the model’s effectiveness. Calibration can be seen as an adjustment procedure that attempts to achieve a good match between the measured (observed) and calculated behaviour. This step typically uses empirical data or results from standardized laboratory tests such as triaxial tests to ensure accuracy.

In contrast, state variables represent the evolving aspects of the equations, like the current state of the soil. These variables are non-constant, changing throughout the modelled process. Additional equations, known as evolution equations, need to be formulated to describe how these state variables evolve. An example is the stress-strain relationship in soil mechanics, where stress is a state variable. For models that incorporate multiple state variables, corresponding evolution equations are necessary for each. Just as important as calibrating the parameters is determining the initial values of the state variables, which can be much more difficult than calibrating the model parameters (constants), especially for advanced constitutive models that use internal variables that are not accessible through measurements.

Outline

In what follows we will concentrate on constitutive models that relate stress and strain. We first discuss key features and challenges of constitutive models for soils and some basic features of the stiffness (matrix) linking stress to strain and wise versa:

We will then learn the basics of three (very) simple constitutive models that are frequently used in geotechnical engineering. These are namely

While introducing these models, some important features and approaches of constitutive modelling will be introduced such as the incorporation of a yield surface, a flow rule to define the direction of plastic deformation or kinematic hardening, amongst other.