Continuum mechanics/Thermoelasticity

Thermoelastic materialsEdit

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 measureEdit

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

 

Therefore,

 

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 energyEdit

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

 

Therefore,

 

This means that the internal energy depends only on the stretch   and not on the orientation of the body.

Other strain and stress measuresEdit

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

 

Therefore,

 

and

 

That is,

 

Stress PowerEdit

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

 

Therefore,

 

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 energyEdit

Recall that

 

Therefore,

 

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

 

and

 

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

 

and

 

Specific HeatsEdit

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

 

and

 

Also the equation for the balance of energy can be expressed in terms of the specific heats as

 

where

 

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

 

ReferencesEdit

  1. T. W. Wright. The Physics and Mathematics of Adiabatic Shear Bands. Cambridge University Press, Cambridge, UK, 2002.
  2. R. C. Batra. Elements of Continuum Mechanics. AIAA, Reston, VA., 2006.
  3. G. A. Maugin. The Thermomechanics of Nonlinear Irreversible Behaviors: An Introduction. World Scientific, Singapore, 1999.