Conditional Lagrange Derivative with Gibbs Function

In forming constitutive relations a method of Mindlin was used. By introducing the conditional Lagrange derivative and by using the laws of thermodynamics a formula is obtained for ε. In the first law Gibbs function is used. This formula should be satisfied in case of constitutive relation.


Introduction
In solid mechanics equations containing material properties are called the constitutive relations.Constitutive relations play an essential role in most problems of the mechanics of continua.On the one hand such equations are used to take into account all observations and tests on the physical behavior of materials.On the other hand they are needed to complete the set of basic equations to perform the necessary computations.There are equations which are successfully used for several cases in applications.For example Hooke's law is suitable to metallic materials in most problems and we have accepted testing methodology to measure its material constants.However there are problems, when dissipation plays important role and such model cannot be used.
While contitutive relation is necessary and there is no generally accepted way to find it, the aim of this paper is to find a method which forms a basis for tests and experimental studies.

Variational principles and Lagrange derivative
The method we use goes back to Mindlin.In [3] stress and strain and the higher order derivatives of them are used to form a constitutive relation.Mindlin uses the variation of strain, his study is based on the virtual work and on the variation of integrals of some function W with respect to the strain tensor.Variational principles of solids contain expressions , where stress, strain and the variation of strain are denoted by σ, ε and δε , respectively.This expression can also be written in a generalized form Symbol double dots denotes the scalar product of matrices as usual.The first term in the right hand side contains the potential part of the stress while the second term refers the remaining (for example dissipation).Function D is not known generally, it is used to consider all virtual work effects.
Mindlin has investigated equation [3] ( ) ∫ ∫ By using elastic potential U he determined stress σ as a function of the strain ε, the first and the higher gradients of ε .Then he derived a new constitutive relation in such a way, which can be used for high gradient dependent elastic materials.The first term on the right hand side of (1) can be written in form where Lagrange derivative [1] is denoted by L ε ( ) .For the uniaxial case where dot and index denote derivatives, for example Then strain tensor can be expressed as a Lagrange derivative L ε with respect to ε, ( ) In mechanics we may use several types of variational principles.These contain virtual work When there are additional equations describing further material properties instead of Lagrange derivatives we should use conditional Lagrange derivaties.Assume that the additional equations have forms for (3) and for (4).

Conditional Lagrange derivative with the first law of thermodynamics
Take the first law of thermodynamics in form ( ) as the "additional" condition.The notations are: ϑ -temperature, e -internal energy, ρ -mass density, h -heat flux, r -heat source intensity, h x = ∂h / ∂x.Let us study the uniaxial case.
Then from (5) ( ) where ( ) ( ) In this case we derive [2] Here and later on small strains are assumed.Further notation are: a -strain energy, S -entropy Then we will use of complementary conditional Lagrange derivative.Now similarly to the previous case take from (6) ( ) and assume that the condition is the first law of thermodynamics ( ) is obtained.Then we derive ( ) ( ) We have two equations, multiply (10) by and add to (7).The result is (5) ( ) In the widely used standard tensile test one end of the specimen of length 0  is fixed while the other is pulled with a small constant speed v 0 .Then strain rate is apporimately 0 that is, strain rate is cosidered to be small and constant.While in tensile test constant temperature is also assumed, Eq. ( 11) should be satisfied.

Summary
Based on Mindlin's paper a method is suggested to calculate the possible forms of constitutive relations.By introducing conditional Lagrange derivatives stress and strain was calculated.Stress comes from our previous result [2] and strain was determined using Gibbs function.At the end a differential equation was obtained, which is available to analyze tensile test.