Finite Strain Shape Memory Alloys Modeling - Scuola di Dottorato in ...

3.4 Constitutive Equations - Hyperelastic MaterialStresses result from the deformation of the material, and it is necessary to expressthem in terms of some measure of this deformation such as, for instance, the strain.These relationship, known as constitutive equations, obviously depend on the type ofmaterial under consideration and may be dependant or independent on time. Forexample, the behavior of viscous materials is clearly entirely dependent on strainrate. Generally, constitutive equations must satisfy certain physical principles. Theequations must obviously be objective, that is, frame invariant. The constitutiveequations herein described are established for hyperelastic material, whereby stressesare derived from a stored elastic energy function. Although there are a number ofalternative material descriptions that could be introduced, hyperelasticity is aparticularly convenient constitutive equation because of its simplicity and, as aconsequence, it constitutes the basis for more complex material models such aselastoplasticity.3.4.1 HyperelasticityMaterials for which the constitutive behavior is only a function of the current state ofdeformation are generally known as elastic. A so-called perfectly elastic material isby definition a material which produces locally no entropy (Trusdell and Noll, 1992).In other words, the term ‘perfectly’ is used for a certain class of materials which hasthe special merits that for every admissible process the internal dissipationDintiszero (naturally, damage, viscous mechanisms and plastic deformations are excluded).Using as strain measure the deformation gradient F associated with a particle X inthe current configuration and its conjugate first Piola-Kirchhoff stress measure P thebasic material elastic relationship is given byP=P( F( X), X )(3.89)52