6. Constitutive Relations...6. Constitutive Relations 6.2.2 Coleman–Noll procedure Heat flux constitutive relation Substituting Eqn. (6.17) in Eqn. (6.15) the second law inequality

6. Constitutive Relations

Why do we need constitutive relation? •Conservation of mass: 1 equation

•Balance of linear momentum: 3 equations

• Balance of angular momentum 3 equations

•Conservation of energy: 1 equation

•Clausius-Duhem inequality: 1 equation TOTALY: 9 equations

6. Constitutive Relations The independent fields entering into these equations are: ρ (1 unknown) - density σ (6 unknown) - stress u (1 unknown) - specific internal energy

T (1 unknown) - temperature x (3 unknowns) - motion q (3 unknowns) - specific heat flux s (1 unknown) - specific entropy where we have imposed the symmetry of the stress tensor due to the constraint of the balance of angular momentum. The missing equations are the constitutive relations (response functions) that describe the response of the material to the mechanical and thermal loading imposed on it. Constitutive relations are required for u, T, σ and q

6. Constitutive Relations 6.1 Constraints on constitutive relations

Constitutive relations cannot be selected arbitrarily - they must conform to certain constraints imposed on them by physical laws and they must be consistent with the structure of the material. Constitutive relations are assumed to be governed by the following fundamental principles: I. Principle of determinism – This is a fundamental philosophical statement at the heart of science that proposes that past events determine the present. The dependent variables in a body are determined by the history of the motion of the body. The motion up to the present time determines a unique stress tensor.


II Principle of local action – Consider point P in the reference configuration. Should the motion of the whole body influence the dependent variables of P? How much of the motion and how large part of the body should we include? The principle of local action states that the material response at a point depends only on the conditions within an arbitrarily small region about that point. The motion outside a neighborhood of a particle does not influence the dependent variables. We assume that a physical variable in the vicinity of particle can be characterized by a Taylor expansion. There are nonlocal continuum theories that reject this hypothesis. In such theories, the constitutive response at a point is obtained by integrating over the volume of the body. Nonlocal theories can be very useful in certain situations, such as in the presence of discontinuities. Peridynamics is such theori.


III. Second law restrictions - A constitutive relation cannot violate the second law of thermodynamics, which states that the entropy of an isolated system remains constant for a reversible process and increases for an irreversible process. For example, a constitutive model for heat flux must ensure that heat flows from hot to cold regions and not vice versa. The application of this inequality to impose constraints on the form of constitutive relation is called Coleman–Noll procedure.

IV. Principle of material frame-indifference (objectivity) All physical variables for which constitutive relations are required must be objective tensors. An objective tensor is a tensor which is physically the same in all frames of reference.

IV. Principle of material frame-indifference (objectivity) Consider two observers in the two bases (S,O) and (S*,O*) separated by a distance c. The

base vectors in S are (e1, e2, e2) and the base vectors in S* are (e1*, e2

*, e2

*). The orthogonal

tensor Q transforms (e1, e2, e2) to (e1*, e2

*, e2

*), i.e. (e1*, e2

*, e2

*)= Q(t) (e1, e2, e2)

The two observers consider the same motion described by:


( , )X t= Xx* * *( , )X t= *Xx

We have the following relations

* = +cx Qx

*t c t= +


IV. Principle of material frame-indifference (objectivity) If a quantity is frame-indifferent the following transformations must hold: a scalar must remain unchanged under change of frame a vector v is transformed into a second-order tensor is transformed to where Q(t) is an orthogonal tensor.

* *( , ) ( , )a t a t=* x x

* *( , ) (t) ( , )t t*v = Q vx x

* *( , ) (t) ( , ) (t)t t* TS = Q S Qx x


V. Material symmetry A constitutive relation must respect any symmetries that the material possesses. For example, the stress in a uniformly strained homogeneous isotropic material (i.e. a material that has the same mechanical properties in all directions at all points) is the same regardless of how the material is rotated before the strain is applied. In addition to the five general principles we have two additional constraints: Consistency - the constitutive equations must be consistent with the equations of balance including the entropy inequality. Coordinate invariance - the constitutive equations must not depend on specific coordinate system, e.g. Cartesian, cylindrical etc.

6. Constitutive Relations 6.2 Local action and the second law of thermodynamics

Let us consider the implications of the principle of local action and the second law of thermodynamics (principles II and III) along with constraints VI and VII for the functional forms of the constitutive relations for u, T, σ and q. Consider the specific internal energy: This is referred to as the caloric equation of state. A material whose constitutive relation depends on the deformation only through the history of the local value of F is called a simple material. A simple material without memory (depending only on the instantaneous value of F) is called an elastic simple material.

6. Constitutive Relations 6.2 Local action and the second law of thermodynamics

It is necessary for some materials to include additional internal variables that describe microstructural features (additional kinematic state variables) of the continuum such as dislocation density, vacancy density, impurity concentration, phase fraction, microcrack density etc.

Another set of possible constitutive relations are those that include a dependence on higher-order gradients of the deformation:

The result is a strain gradient theory. This approach has been successfully used to study length scale dependence in plasticity.

6. Constitutive Relations 6.2.2 Coleman–Noll procedure

Consider the second law of thermodynamics - Clausius–Duhem inequality

Substituting and expanding the divergence term, we obtain:

Rearranging, we obtain

The expression in the square brackets appears in exactly the same form in the energy equation.


Taking a material time derivative

and use of:

σ is symmetric

The argument made by Coleman and Noll is last equation must be satisfied for every admissible process. By selecting special cases, insight is gained into the relation between the different continuum fields. This line of thinking is referred to as the Coleman–Noll procedure.


Temperature constitutive relation Consider a process where the deformation is constant in time and the temperature is uniform across the body, so that ∇T = 0. In this case, Eqn. (6.15) reduces to:

The rate of change of entropy s˙ can be assigned arbitrarily. Since the sign of s˙ is arbitrary, the last Eqn. can only be satisfied for every process if:


Heat flux constitutive relation Substituting Eqn. (6.17) in Eqn. (6.15) the second law inequality reduces to

Consider a process where the deformation is constant in time

This inequality is consistent with our physical intuition: heat flows from hot to cold. This means that q must change sign in accordance with ∇T. We can therefore state in general that q must have the following functional dependence:


Start with Eqn. (6.18), and consider the case where the temperature is uniform across the body (∇T = 0). In this case, the second law inequality is:

This equation must hold for any choice of . This can only be satisfied for all if

This equation implies that all irreversibility enters through the heat flux and consequently that no irreversibility is possible under uniform temperature conditions. This is not consistent with experimental observation.


To proceed, we partition σ into two parts: an elastic reversible part that is a state variable and a “viscous,” or dissipative, part that is irreversible (plastic part).

elastic part Irreversible part

A material for which σ(v ) = 0 is called a hyperelastic material.

6. Constitutive Relations 6.2.4 Constitutive relations for alternative stress variables

Continuum formulations for solids are often expressed in a Lagrangian description, where the appropriate stress variables are the first or second Piola–Kirchhoff stress tensors. The constitutive relations for these variables can be found by suitably transforming the Cauchy stress function.

using an alternative internal energy function, , that depends on the Lagrangian strain.

The constitutive relations derived above have taken the entropy and deformation gradient as the independent state variables.

6. Constitutive Relations 6.2.5 Thermodynamic potentials and connection with experiments

The mathematical description of a process can be significantly simplified by an appropriate choice of independent state variables. A process occurring at constant entropy (s˙ = 0) is called an isentropic process. A process where F is controlled is subject to displacement control.

Consider isentropic processes under displacement control. If, in addition, the process is also reversible, it is called adiabatic.

A process at constant temperature is isothermal

It is important to note that for continuum systems, adiabatic conditions are not ensured by thermally isolating the system from its environment, which only ensures that

This does not translate to the local requirement, unless it is assumed to hold for every subbody of the body. This implies that there is no transfer of heat between different parts of the body.


The suitable energy variable is not the specific internal energy.

Helmholtz free energy The Helmholtz free energy is the appropriate energy variable for processes where T and Γ are the independent variables.

A potential closely related to the specific Helmholtz free energy is the strain energy density function W


Enthalpy, the specific enthalpy h:

is the appropriate energy variable for processes where s and γ are the independent variables. The continuum deformation measures at constant entropy are


Gibbs free energy, The specific Gibbs free energy (or specific Gibbs function) g:

It is the appropriate energy variable for processes, where T and γ are the independent variables. The continuum deformation measures at constant temperature are:

6. Constitutive Relations 6.4 Material symmetry Most materials possess certain symmetries, which are reflected by their constitutive relations. Consider, for example, the deformation of a material with a two-dimensional square lattice structure.

6. Constitutive Relations 6.4.2 Isotropic solids A simple isotropic material - an arbitrary rigid-body rotation can be applied to the reference configuration without affecting the constitutive response of the material. Crystalline materials are not isotropic at the level of a single crystal, however, at the continuum level many materials appear isotropic since the response at a point represents an average over a large number of randomly oriented single crystals (or grains). We focus here on simple elastic materials:

This relation is true for all H , so it is also true for

We see that the stress can only depend on F through the left stretch tensor. This makes sense, since V is insensitive to rotations of the reference configuration. This implies the existence of a function that depends only on the left Cauchy–Green tensor, . Thus, we can write:

6. Constitutive Relations 6.4.2 Isotropic solids

Hyperelastic solids: A simple hyperelastic material is one which possesses a strain energy density W(F) from which the stress may be obtained. The most general form of the strain energy density function for a simple hyperelastic isotropic solid is:

In order to obtain the stress from the strain energy density function, we will require expressions for certain derivatives of the principal invariants. First, the derivative of B with respect to F is: Second, the derivatives of the principal invariants of B with respect to B and F are:


Using these expressions, we can write the stress in terms of the strain energy density, the first Piola–Kirchhoff stress is:

The second Piola–Kirchhoff stress is:

The Cauchy stress is:


Some common examples of nonlinear constitutive laws for isotropic incompressible simple materials.

Neo-Hookean materials: One of the simplest possible incompressible constitutive relations, the neo-Hookean material model, has been extensively used in theoretical studies where the focus is more on developing an understanding of general continuum mechanics principles rather than obtaining results for a particular material. Motivated by experiments that show the constitutive behavior of rubber to be nearly independent of , the neo- Hookean strain energy density is defined as:

The first Piola–Kirchhoff stress tensor:

The Cauchy stress:

6. Constitutive Relations 6.4.2 Isotropic solids Moony–Rivlin materials: This incompressible material model includes a dependence on and has a strain energy density given by


The Cauchy stress:

Ogden materials: Ogden describes a general class of incompressible material models for which the strain energy density is given by a power-series:

It is easy to see that the Moony–Rivlin and neo-Hookean models are special cases of the Ogden model.

6. Constitutive Relations 6.4.2 Beyond isotropy

An example of an anisotropic, (geometrically) nonlinear material model is the Saint Venant–Kirchhoff model.

Saint Venant–Kirchhoff materials: These materials have strain energy density functions that are simply quadratic in the Lagrangian strain E:

C is a constant fourth-order tensor


The Cauchy stress:

The second Piola–Kirchhoff stress tensor:

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6. Constitutive Relations...6. Constitutive Relations 6.2.2 Coleman–Noll procedure Heat flux constitutive relation Substituting Eqn. (6.17) in Eqn. (6.15) the second law inequality