Advanced elasticity/Thermoelasticity

Thermoelastic materials edit

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 edit

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 edit

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 edit

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






That is,


Stress Power edit

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 edit

Recall that




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




Specific Heats edit

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


References edit

  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.