General principles of constitutive modeling

We will now introduce the topic of constitutive modeling, in a simplified setting. The purpose here is to introduce certain general principles that all constitutive models must obey. We will revisit many of the notions presented here later on, in the context of hyperelasticity.

Need for constitutive modeling

The discussion of continuum mechanics so far has been general enough to accomodate any kind of continuum body. How do we bring about material specificity in our study of continuum mechanics? To understand this better, suppose that we have a cylindrical block of material that we pull at both ends with a known force. It is intuitively clear that the response of the cylindrical block will be very different depending on whether it is made of steel or rubbber. However, the kinematical description of the process and the application of the various balance principles remain the same. So how do we add material-specific information to our theory? This is precisely what constitutive modeling is all about.

Let us now look at the same problem from a purely mathematical viewpoint. To keep the discussion simple, let us focus on the case when there is no heat exchange between the body and its environment, and there are no internal sources of heat.

Remark

Though we are considering the simple case here where there is no internal or external heat transfer, the conclusions we draw can be extended with suitable modifications to the case when they are present. Indeed, we will look into thermoelastic constitutive models later on, in the context of thermoelasticity.

In this simple case, we only need to care about the balance of mass, linear momentum and angular momentum: $$\begin{split} \text{Mass: } & \frac{D\rho(\mathsf{x},t)}{Dt} + \rho(\mathsf{x},t)\,\text{div }\mathsf{v}(\mathsf{x},t) = 0,\\ \text{Linear momentum: }& \rho(\mathsf{x},t)\frac{D \mathsf{v}(\mathsf{x},t)}{Dt} = \rho(\mathsf{x},t)\mathsf{b}(\mathsf{x},t) - \text{div }\sigma(\mathsf{x},t),\\ \text{Angular momentum: }& \sigma = \sigma^T. \end{split}$$ The equations listed above represent a set of 7 distinct equations: one for the mass density, three for the velocity, and three for the stress tensor.

Remark

Note that though the stress tensor has nine components, there are only three distinct equations that provide us with non-trivial information. To see this, let us suppose we are working within a Cartesian coordinate setting. Then, the three distinct equations are: $\sigma_{12} = \sigma_{21}$, $\sigma_{13} = \sigma_{31}$ and $\sigma_{23} = \sigma_{32}$. Notice that the diagonal equations $\sigma_{ii} = \sigma_{ii}$, where $i = 1,2,3$ do not contribute any new information. There are only three off-diagonal equations since the equation $\sigma_{ij} = \sigma_{ji}$ is identical to $\sigma_{ji} = \sigma_{ij}$.

We however have 13 unknown quantities: one for the scalar mass density, three for the spatial velocity, and nine for the Cauchy stress tensor. We thus need 6 more equations for closure.

Equivalently, we can eliminate the equation $\sigma = \sigma^T$ by considering only symmetric stress tensors. In this case, we have 6 unknowns for the stress tensor, 3 unknowns for the velocity, and 1 unknown for the mass density; this sums up to 10 unknown fields. The number of equations available reduce to 4: one for the mass density, and three for the velocity. We still need 6 equations for closure, as before. Constitutive models fill in this gap by providing a material specific set of additional equations that allow us to solve the balance principles, once we have the appropriate initial and boundary conditions.

How do we specify these six additional equations? What we control are the externally applied body forces and surface tractions. We saw earlier that the continuum responds to this by developing internal forces, namely stresses. It is clear that what differentiates one material from another is the nature of this internal response. In other words, we can, as a first attempt at a constitutive model, think of a general relation of the form $$\sigma(\mathsf{x},t) = \hat{\sigma}(\rho, \mathsf{F}, \mathsf{v}, \text{grad }\mathsf{v}, \mathsf{x}, t).$$ For now, the only restriction we have is that $\hat{\sigma}$ should yield a symmetric tensor, so that we have 6 additional equations that close the equations supplied by the balance principles. An equation of this form is called a constitutive model, or a constitutive relation.

Different materials behave differently; this is captured by the various choices we have in formulating a constitutive model. Thus, as mentioned before, constitutive models bring in material specificity to the balance principles. Note that the balance principles themselves are universal in the sense that they are applicable for any kind of continuum object.

An important point to keep in mind is that unlike balance principles, constitutive relations cannot be derived from purely thermodynamic considerations alone. This is because of the fact that thermodynamics, by its very nature, is the study of complex systems without any consideration of their internal constitution. It is thus outside the purview of thermodynamics to deduce facts about the internal response of the system of interest. From the viewpoint of continuum mechanics, a constitutive relation can be thought of as an additional postulate. In practice, tehre are two marjor sources to obtain constitutive relations - the first is, of course, by fitting experimental data with model certain mathematical models, and the second is to deduce the constitutive behavior from a detailed microscopic consideration of the system. In this course, we will adopt the view that constitutive relations are provided to us for various kinds of materials.

Remark

The companion course AE 731: Multiscale modeling of materials will contain some discussion of how microscopic constitutive behavior emerges starting from a detailed microscopic analysis of certain classes of solids.

Restrictions on the form of constitutive relations

Desptite the fact that thermodynamics cannot help us formulate the constitutive law associated with a given material, there are very general principles that place restrictions on the form of any constitutive relation we may use. We will now see some of the more important of these constraints.

Principle of causality

The first constraint that we will impose on constitutive behavior is that the the response of a material at a given instant of time cannot depend on what happens at a future instant of time. This obvious constraint is called the principle of causality, or the principle of determinism.

Remark

Note however that it is perfectly admissible for the response of a continuum at the current instant of time can depend on its entire history. We will, however, only consider constitutive relations whose response at a given time instant depend only on the values of the variousl field variables at the same instant.

Principle of homogeneity

A constitutive relation is said to be spatially homogeneous if the material response is the same at every point of the continuum. The simplest way to understand this is by resorting to the ball spring analogy provided earlier. The principle of homogeneity can be roughly translated as the view that the value of spring constant of a spring in a particular region of the body is the same as the value of the spring constant of another spring in a different location. In other words, we assume that the same material is present throughout the continuum.

Remark

It is possible to study to study continua that are spatially heterogeneous too. In fact, there is an important sub-branch of solid mechanics called mircromechanics where the absence of material homogeneity plays an important role. We will, however, not deal with spatially inhomogeneous continua in this course.

We will also assume that the material response of continua is homogeneous in time. This means that the material behavior does not evolve with time. Using the spring mass analogy, this simply means that the spring constants do not change with time.

It is important to note that neither spatial homogeneity nor homogeneity in time imply that the continuum does not have spatial gradients in the various physical fields, or that these fields do not evolve with respect to time. The principle of homogeneity simply states that the constitutive behavior does not depend on space and time explicitly. Thus, in place of an equation like $$\sigma = \hat{\sigma}(\Xi, \mathsf{x},t),$$ the principle of homogeneity requires that $$\sigma = \hat{\sigma}(\Xi).$$ Here $\Xi$ could represent one or more of the many kinematic fields. We will see some concrete examples shortly.

Principle of local action

The principle of local action states that material response at a given point in the continuum depends only on the values of the field variables at that point. For instance, suppose that we have a constitutive relation of the form $$\sigma = \hat{\sigma}(\Xi),$$ where $\Xi$ is some field variable, say the rate of deformation tensor. In general, what this means is that for $\mathsf{x} \in B_t$, $$\sigma(\mathsf{x},t) = \hat{\sigma}(\Xi(\mathsf{x}',t;\mathsf{x})),$$ for any $\mathsf{x}' \in B_{t}$. In other words, the response at $\mathsf{x} \in B_t$ at time $t$ could depend on the response at a different point $\mathsf{x}' \in B_{t}$. The principle of local action restricts the constitutive relations to be of the form $$\sigma(\mathsf{x},t) = \hat{\sigma}(\Xi(\mathsf{x},t)).$$ Such constitutive models are said to be local.

Remark

There are many important classes of materials where the constitutive behavior is non-local. It is possible to extend the laws of continuum mechanics for such materials, but we will not be studying them in this course.

Example

An example of the form that a constitutive model obeying the priniciples of causality, homogeneity and locality has is given, for example, by $$\sigma(\mathsf{x},t) = \hat{\sigma}(\rho(\mathsf{x},t),\mathsf{F}(\mathsf{X},t),\text{grad }\mathsf{v}(\mathsf{x},t)),$$ where $\mathsf{X} = \phi_t^{-1}(\mathsf{x}) \in B_0$. Another example could be a relation of the form $$\dot{\sigma}(\mathsf{x},t) = \mathcal{G}(\text{grad }\mathsf{v}(\mathsf{x},t)).$$ Recall that $\dot{\sigma}$ denotes the material time derivative of the Cauchy stress tensor.

Principle of material frame indifference

One of the most important constraints, arguably, on the form of constitutive behavior is the principle of material frame indifference. This complicated sounding name is, in essesnce, identical to the observer invariance of the constitutive relations (with respect to Euclidean observer transformations). Recall that our entire development of the balance principles of continuum mechanics hinged upon the central idea that the laws of thermodynamics are observer invariant. It is evident that since constitutive laws are also part of the physical description of the material, they cannot depend on the observer. The principle of material frame indifference is essentially a statement of this fact, with one important additional constraint, namely the form invariance of the constitutive laws with respect to Euclidean observer transformations. The form invariance postulate, which is implicit in the material frame indifference postulate, states that the functional form of the the constitutive relation is also invariant with respect to Euclidean observer transformations.

Rather than explaining these in very general terms, it is perhaps better to focus on specific forms of constitutive relations and explain the consequences of the principle of material frame indifference in the process. Throughout this discussion, we will assume that $\mathcal{O}$ and $\mathcal{O}^+$ are two observers who are related by the Euclidean transformation $$\mathsf{x}^+ = \mathsf{c} + \mathsf{Q}\mathsf{x},\qquad t^+ = t.$$ Note that both $\mathsf{c}$ and $\mathsf{Q}$ are functions of time, in general.

Example

To start with, let us consider a constitutive relation of the form $$\sigma(\mathsf{x},t) = \hat{\sigma}(\mathsf{v}(\mathsf{x},t)).$$ To keep the notation simple, we will often abbreviate this as $\sigma = \hat{\sigma}(\mathsf{v})$, with the implicit assumption that the constitive relation is local. Note also that the constitutive relation, as written above, is with respect to an observer $\mathcal{O}$. The same constitutive relation takes the following form with respect to the observer $\mathcal{O}^+$: $$\sigma^+ = \hat{\sigma}^+(\mathsf{v}^+).$$ The form invariance postulate means that the specific functional form characterizing $\hat{\sigma}$, namely the relation between the Cauchy stress $\sigma$ and the spatial velocity $\mathsf{v}$ is invariant with respect to Euclidean observer transformations. We thus have $$\sigma^+ = \hat{\sigma}(\mathsf{v}^+).$$ Recall that the spatial velocity transforms as follows: $$\mathsf{v}^+ = \dot{\mathsf{c}} + \mathsf{Q}\mathsf{v} + \dot{\mathsf{Q}}\mathsf{x}.$$ The Cauchy stress tensor, on the other hand, is objective and transforms as $$\sigma^+ = \mathsf{Q} \sigma \mathsf{Q}^T.$$ To see why, note that the equation for the traction vector according to $\mathcal{O}$, $\mathsf{t} = \sigma\mathsf{n}$, where $\mathsf{n}$ is an appropriately defined normal vector, can be written with respect to $\mathcal{O}^+$ as $\mathsf{t}^+ = \sigma^+\mathsf{n}^+.$ Noting that both the traction vector and normal vector are objective, the objectivity of the Cauchy stress tensor follows. Using these results, it follows that $$\mathsf{Q}\sigma\mathsf{Q}^T = \hat{\sigma}(\dot{\mathsf{c}} + \mathsf{Q}\mathsf{v} + \dot{\mathsf{Q}}\mathsf{x}).$$ As a special case, let us consider the situation where $\mathsf{Q}(t) \equiv \mathsf{I}$. Since the foregoing relation is also valid in case, we see that $$\sigma = \hat{\sigma}(\mathsf{v} + \dot{c}),$$ for every admissible $\dot{\mathsf{c}}$. This is clearly not possible. What this means is that, according to the principle of material frame indifference, it is inadmissible to have a constitutive relation of the form $\sigma = \hat{\sigma}(\mathsf{v})$!

Example

As another example, let us consider a constitutive relation of the form $$\sigma = \mathcal{F}(\text{grad }\mathsf{v}),$$ according to oberver $\mathcal{O}$. The same relation, when viewed by observer $\mathcal{O}^+$ takes the form $$\sigma^+ = \mathcal{F}(\text{grad}^+\,\mathsf{v}^+).$$ Notice how we have tacitly used the form invariance postulate in writing this expression. To compute $\text{grad}^+\,\mathsf{v}^+$, let us work in a Cartesian coordinate setting. Since the velocities transform as $$v_i^+ = \dot{c}_i + \sum Q_{ij}v_j + \sum \dot{Q}_{ij}x_j$$ as a consequene of the Euclidean observer transformation, we see that $$\begin{split} \frac{\partial v_i^+}{\partial x_j^+} &= \sum \frac{\partial v_i^+}{\partial x_k}\frac{\partial x_k}{\partial x_j^+}\\ &= \frac{\partial}{\partial x_k}\left(\sum Q_{il}v_l + \sum \dot{Q}_{il}x_l\right)Q_{jk}\\ &= \sum Q_{il}\frac{\partial v_l}{\partial x_k}Q_{jk} + \sum \dot{Q}_{ik}Q_{jk}. \end{split}$$ In deriving this equation, we have made use of the fact that $$x_i^+ = c_i + \sum Q_{ij}x_j \quad\Rightarrow\quad \frac{\partial x_k}{\partial x_j^+} = Q_{jk}.$$ The final result can be expressed in invariant form as $$\text{grad}^+\mathsf{v}^+ = \mathsf{Q}(\text{grad }\mathsf{v})\mathsf{Q}^T + \Omega.$$ Here, $\Omega = \mathsf{Q}\dot{\mathsf{Q}}^T$. It is left as a straightforward exercise to check this - expand this coordinate free result stated above using Cartesian coordinates and note that $\mathsf{Q} = \sum \mathsf{e}^+_i \otimes \mathsf{e}_i$.

The transformation rule derived above shows that that the gradient of the velocity field is not an objective tensor field; this is primarily a consequence of the additional term $\Omega$, which can be chosen arbitrarily. So a constitutive relation of the form $\sigma = \mathcal{F}(\text{grad }\mathsf{v})$ is not observer invariant.

A simple means to fix this problem is by noting that the rate of deformation tensor, $$\mathsf{d} = \frac{1}{2}\left(\text{grad }\mathsf{v} + (\text{grad }\mathsf{v})^T\right)$$ is indeed objective. In other words, $$\mathsf{d}^+ = \mathsf{Q}\mathsf{d}\mathsf{Q}^T.$$ This follows from the fact that $\Omega$ is skew symmetric, which in turn follows from the orthogonality of $\mathsf{Q}(t)$ for every $t$. Thus, if we postulate a constitutive relation of the form $$\sigma = \mathcal{F}(\mathsf{d}),$$ then this is a valid form of the constitutive law as long as the function $\mathcal{F}$ satisfies the following property $$\mathsf{Q}\mathcal{F}(\mathsf{d})\mathsf{Q}^T = \mathcal{F}(\mathsf{Q}\mathsf{d}\mathsf{Q}^T),$$ for every $\mathsf{Q} \in SO(3)$.

These examples illustrate how the requirement of observer invariance places definite restrictions on the form of the constitutive relation. We will explore some elementary, but important, constitutive relations in the context of fluids in the next chapter, followed by a discussion of elastic solids.