Chapter 3

Constitutive Models $–$ Relations between Stress and Strain

The equations listed in Chapter 2 are universal $–$ they apply to all deformable solids.  They can’t be solved, however, unless the deformation measure can be related to the internal forces.

The constitutive model for a material is a set of equations relating stress to strain (and possibly strain history, strain rate, and other field quantities).  Unlike the governing equations in the previous chapter, these equations cannot generally be calculated using fundamental physical laws (although people are trying to do this).  Instead, constitutive models are fit to experimental measurements.

Before discussing specific constitutive models, it is helpful to review the basic assumptions that we take for granted in developing stress-strain laws.  They are listed below.

A very small sample that is extracted from the solid has uniform properties;

When the solid is deformed, initially straight lines in the solid are deformed into smooth curves (with continuous slope);

This means that very short line segments (much shorter than the radius of curvature of the curves) are just stretched and rotated by the deformation.   Consequently, the deformation of a sufficiently small volume element can be characterized by the deformation gradient;

The stress at a point in the solid depends only on the change in shape of a vanishingly small volume element surrounding the point.  It must therefore be a function of the deformation gradient or a strain measure that is derived from it.

If we accept the preceding assumptions, it means that we can measure the relationship between stress and strain by doing an experiment that induces a uniform strain in a suitable sample of the material.  According to our assumptions, the stress should also be uniform, and can be calculated from the forces acting on the specimen.

These are clearly approximations.  Materials are not really uniform at small scales, whether you choose to look at the atomic scale, or the microstructural scale.  However, these features are usually much smaller than the size of the solid part or component, and the material can be regarded as statistically uniform, in the sense that if you cut two specimens with similar size out of the material they will behave in the same way.  A continuum model then describes the average stress and deformation in a region of the material that is larger than microstructural features, but small compared with the dimensions of the part.

3.1 General requirements for constitutive equations

You may be called upon to develop a stress-strain law for a new material at some point of your career.  If so, it is essential to make sure that the stress-strain law satisfies two conditions:

(i) It must obeys the laws of thermodynamics.

(ii) It must satisfy the condition of objectivity, or material frame indifference

In addition, it is a good idea to ensure that the material satisfies the Drucker stability criterion discussed in more detail below.  Of course, your proposed law must conform to experimental measurements, and if possible should be based on some understanding of the physical processes that govern the response of the solid.

Thermodynamic restrictions: The laws of thermodynamics impose two restrictions on stress-strain laws:

1.      The first law requires that the work done by stresses must either be stored as recoverable internal energy in the solid, or be dissipated as heat (or a combination of both).

2.      The second law requires that if a sample of the material is subjected to a cycle of deformation that starts and ends with an identical strain and internal energy (at constant temperature, or without heat exchange with the surroundings) the total work done must be positive or zero.

Objectivity: Strictly speaking, the term “objectivity” or “material frame independence” is the condition that the tensor-valued functions that relate stress to deformation measure must transform correctly under a change of basis and change of origin for the coordinate system.   A detailed mathematical derivation of the consequences of objectivity will not be given here (see for example Malvern “Introduction to the Mechanics of a Continuous Medium,” Prentice Hall).  However, you can check whether a constitutive law is objective using the following simple (analytical) test:

1.      Load the solid (quasi-statically) by subjecting its boundary to prescribed forces, to induce a Cauchy stress ${\sigma }_{ij}^{\left(0\right)}$ in the solid.

2.      Subject both the solid, and the loads acting on the solid, to a quasi-static rigid rotation, characterized by a rotation tensor ${R}_{ij}$.

3.      The constitutive law must predict that after the rotation, the stress components change to new values given by ${\sigma }_{ij}^{\left(1\right)}={R}_{ik}{\sigma }_{kl}^{\left(0\right)}{R}_{jl}$

To see this, note that since the loads rotate together with the solid, the components of traction acting normal and tangent to any interior material plane in the solid must remain constant.  With this in mind:

1.      Suppose that, just before the rotation is applied, ${T}_{i}^{\left(0\right)}={n}_{k}^{\left(0\right)}{\sigma }_{ki}^{\left(0\right)}$ denotes the traction acting on an interior material plane with normal ${n}_{i}^{\left(0\right)}$.

2.      The traction vector and the normal to the interior plane rotate with the solid, and therefore have components ${T}_{i}^{\left(1\right)}={R}_{ij}{T}_{j}^{\left(0\right)}$ and ${n}_{i}^{\left(1\right)}={R}_{ij}{n}_{j}^{\left(0\right)}$ after rotation.

3.      By definition, ${T}_{i}^{\left(1\right)}={n}_{j}^{\left(1\right)}{\sigma }_{ji}^{\left(1\right)}$.  From (2), we see therefore that ${R}_{ik}{T}_{k}^{\left(0\right)}={R}_{kl}{n}_{l}^{\left(0\right)}{\sigma }_{ki}^{\left(1\right)}$

4.      Multiply both sides of this equation by ${R}_{ip}$ and recall that ${R}_{ip}{R}_{ik}={\delta }_{kp}$ to see that ${T}_{p}^{\left(0\right)}={n}_{l}^{\left(0\right)}{R}_{kl}{\sigma }_{ki}^{\left(1\right)}{R}_{ip}$

5.      Comparing this with (1), we conclude that ${R}_{kl}{\sigma }_{ki}^{\left(1\right)}{R}_{ip}={\sigma }_{lp}^{\left(0\right)}$.  Finally, multiplying both sides of this equation by ${R}_{jl}{R}_{np}$ we conclude that ${\sigma }_{jn}^{\left(1\right)}={R}_{jl}{\sigma }_{lp}^{\left(0\right)}{R}_{np}$, giving the required result.

Drucker Stability:  For most practical applications, the constitutive equation must satisfy a condition known as the Drucker stability criterion, which can be expressed as follows.  Consider a deformable solid, subjected to boundary tractions ${t}_{i}$, which induce some displacement field ${u}_{i}$.  Suppose that the tractions are increased to ${t}_{i}+\Delta {t}_{i}$, resulting in an additional displacement $\Delta {u}_{i}$. The material is said to be stable in the sense of Drucker if the work done by the tractions $\Delta {t}_{i}$ through the displacements $\Delta {u}_{i}$ is positive or zero for all $\Delta {t}_{i}$:

$\Delta W=\int \left\{\underset{A}{\int }\Delta {t}_{i}\frac{d\Delta {u}_{i}}{dt}\right\}dt\ge 0$

You can show that this condition is satisfied as long as the stress-strain relation obeys

$\Delta {\tau }_{ij}\Delta {\epsilon }_{ij}\ge 0$

where $\Delta {\tau }_{ij}$ is the change in Kirchhoff stress, and $\Delta {\epsilon }_{ij}=\left(\partial \Delta {u}_{i}/\partial {x}_{j}+\partial \Delta {u}_{j}/\partial {x}_{i}\right)$ is an increment in strain resulting from an infinitesimal change in displacement $\Delta {u}_{i}$.

This is not a thermodynamic law (the work done by the change in tractions is not a physically meaningful quantity), and there is nothing to say that real materials have to satisfy Drucker stability. In fact, many materials show clear signs that they are not stable in the sense of Drucker.   However, if you try to solve a boundary value problem for a material that violates the Drucker stability criterion, you are likely to run into trouble.  The problem will probably not have a unique solution, and in addition you are likely to find that smooth curves on the undeformed solid develop kinks (and may not even be continuous) after the solid is deformed.  This kind of deformation violates one of the fundamental assumptions underlying continuum constitutive equations.

A simple example of a stress-strain curve for material that is not stable in the sense of Drucker is shown in the picture.  The stability criterion is violated wherever the stress decreases with strain in tension, or increases with strain in compression. For the former we see that $\Delta {\sigma }_{11}<0$, while $\Delta {\epsilon }_{11}>0$; for the latter $\Delta {\sigma }_{11}>0$, while $\Delta {\epsilon }_{11}<0$.

In the following chapter, we outline constitutive laws that were developed to approximate the behavior of a wide range of materials, including polycrystalline metals and non-metals; elastomers; polymers; biological tissue; soils; and metal single crystals.  A few additional material models, which account for material failure, are also described in Chapter 9.