77 new thermodynamic identities among crystalline elastic material properties leading to a shear modulus constitutive law in isotropic solids

Abstract

Thermodynamics of crystalline materials is analyzed using strain volumes, an incremental tensorial state variable which is the volume per unit mass multiplied by the incremental strain. It is shown that the derivatives of the strain volume variables with respect to the stresses reduces to conventional well known isothermal, elastic, crystallographic compliances and crystallographic thermal expansion coefficients. The formulism is extended to all third order derivatives which establish 77 new thermodynamic identities: 27 are thermal and 50 are among selected stress components of elastic compliances. The stress dependence of the heat capacity is typically coupled into the crystallographic thermal expansion coefficient's temperature dependence; the temperature dependence of the elastic compliances is related to the stress dependence of the thermal expansion coefficients; stress dependent relationships among elastic compliances are also found. The paper emphasizes internal self-consistency. General triclinic identity relations which are of academic interest are applied to specific higher symmetries including orthorhombic, tetragonal and cubic; identities in higher symmetry crystals are very useful. A new generalized, constitutive law is found and applied to isotropic polycrystalline materials; the concept is based on zero shear thermal expansion coefficients. The constitutive modulus relation describes thermal and pressure properties only through an isochoric modulus.

Thermodynamic Relations among Isotropic Material Properties in Conditions of Plane Shear Stress

“…Since Gibbs’ fundamental contribution in 1876 [ 1 ], the thermodynamic theory of solids under different stress conditions has remained an active field of inquiry, with a recent intensification spurred by interest in amorphous states and glass transition, high pressure physics, and the development of artificial materials [ 2 , 3 , 4 , 5 , 6 , 7 ]. In contrast, continuum mechanics and thermoelasticity have focused more on finite deformations and field theories, traditionally shifting away from homogeneous thermodynamics [ 8 , 9 ] and the related Gibbs equation [ 10 ], in spite of the fact that these concepts are often clearer, at least for infinitesimal transformations and in uniform conditions [ 9 ].…”

“…The complete relationships among them and a full discussion of their meaning have been missing, however, due also in part to the complex stress patterns of crystals and anisotropic solids [ 1 , 17 , 18 , 19 ]. Only very recently, Burns [ 7 ] has provided a list of these relationships obtained using the Jacobian algebra of thermodynamic transformations.…”

“…This is a very important, but special case, where there is equality of the isothermal and isentropic shear compliance (or their inverse, i.e., the moduli of rigidity and ), as pointed out by Brillouin [ 12 ] and mentioned also in other work [ 2 , 4 , 19 ]. With the exception of Burns [ 7 , 20 ], this condition, however, is treated as one of general validity.…”

“…Burns has also drawn attention to the general relationships among material properties in crystals [ 7 ] and in conditions of plane shear [ 20 ], finding new explicit relationships and discussing special cases. Our contribution here represents an extension to his second work [ 20 ].…”