A set of constitutive equations is required to close to system of balance laws. These are relations between appropriate kinematic quantities and stress measures that can be assigned a physical meaning.
Deformation gradient as the strain measure
In thermoelasticity we assume that the fundamental kinematic quantity is the deformation gradient () which is given by
A thermoelastic material is one in which the internal energy () is a function only of and the specific entropy (), that is
For a thermoelastic material, we can show that the entropy inequality can be written as
At this stage, we make the following constitutive assumptions:
1) Like the internal energy, we assume that and are also functions only of and , i.e.,
2) The heat flux satisfies the thermal conductivity inequality and if is independent of and , we have
i.e., the thermal conductivity is positive semidefinite.
Therefore, the entropy inequality may be written as
Since and are arbitrary, the entropy inequality will be satisfied if and only if
Given the above relations, the energy equation may expressed in terms of the specific entropy as
Effect of a rigid body rotation of the internal energy
If a thermoelastic body is subjected to a rigid body rotation , then its internal energy should not change. After a rotation, the new deformation gradient () is given by
Since the internal energy does not change, we must have
Now, from the polar decomposition theorem, where is the orthogonal rotation tensor (i.e., ) and is the symmetric right stretch tensor. Therefore,
We can choose any rotation . In particular, if we choose , we have
This means that the internal energy depends only on the stretch and not on the orientation of the body.
Other strain and stress measures
The internal energy depends on only through the stretch . A strain measure that reflects this fact and also vanishes in the reference configuration is the Green strain
Recall that the Cauchy stress is given by
We can show that the Cauchy stress can be expressed in terms of the Green strain as
Also, recall that the first Piola-Kirchhoff stress tensor is defined as
Alternatively, we may use the nominal stress tensor
From the conservation of mass, we have . Hence,
The first P-K stress and the nominal stress are unsymmetric. Also recall that we can define a symmetric stress measure with respect to the reference configuration called the second Piola-Kirchhoff stress tensor ():
In terms of the derivatives of the internal energy, we have
The stress power per unit volume is given by . In terms of the stress measures in the reference configuration, we have
Using the identity , we have
We can alternatively express the stress power in terms of and . Taking the material time derivative of we have
Using the identities and and using the symmetry of , we have
Now, . Therefore, . Hence, the stress power can be expressed as
If we split the velocity gradient into symmetric and skew parts using
where is the rate of deformation tensor and is the spin tensor, we have
Since is symmetric and is skew, we have . Therefore, . Hence, we may also express the stress power as
Helmholtz and Gibbs free energy
Also recall that
Now, the internal energy is a function only of the Green strain and the specific entropy. Let us assume, that the above relations can be uniquely inverted locally at a material point so that we have
Then the specific internal energy, the specific entropy, and the stress can also be expressed as functions of and , or and , i.e.,
We can show that
We define the Helmholtz free energy as
We define the Gibbs free energy as
The functions and are unique. Using these definitions it can be shown that
The specific heat at constant strain (or constant volume) is defined as
The specific heat at constant stress (or constant pressure) is defined as
We can show that
Also the equation for the balance of energy can be expressed in terms of the specific heats as
The quantity is called the coefficient of thermal stress and the quantity is called the coefficient of thermal expansion.
The difference between and can be expressed as
However, it is more common to express the above relation in terms of the elastic modulus tensor as
where the fourth-order tensor of elastic moduli is defined as
For isotropic materials with a constant coefficient of thermal expansion that follow the St. Venant-Kirchhoff material model, we can show that
- T. W. Wright. The Physics and Mathematics of Adiabatic Shear Bands. Cambridge University Press, Cambridge, UK, 2002.
- R. C. Batra. Elements of Continuum Mechanics. AIAA, Reston, VA., 2006.
- G. A. Maugin. The Thermomechanics of Nonlinear Irreversible Behaviors: An Introduction. World Scientific, Singapore, 1999.