Eulerian Strain Tensor Research Articles

On the determination of the volume dependence of Grüneisen parameters in cubic and non-cubic solids

Abstract For a material of orthorhombic symmetry under hydrostatic stress, three generalized Gruneisen parameters relative respectively to the three principal directions of the deformation can be defined. Finite strain expressions for these parameters are derived in terms of the Lagrangian strain tensor and the frame-indifferent analogue of the Eulerian strain tensor. The expressions require for their evaluation the thermal expansion coefficients, the elastic moduli and their pressure and temperature derivatives, and the specific heat of the material, so that there are no arbitrary constants or “curve fitting.” In the case of cubic materials, it is possible to determine the unique Gruneisen parameter as a function of volume directly. For non-cubic materials, the volume dependence of the Gruneisen parameters is calculated using a quasi-harmonic finite strain model of equation of state. Numerical applications are given for some cubic and hexagonal metals and the results are discussed.

Strain-tensor components expressed in terms of lattice parameters

Expressions are developed for the components of the linear Lagrangian, linear Eulerian, finite Lagrangian (Green's), and finite Eulerian (Almansi's) strain tensors in terms of a crystal's lattice parameters before and after a deformation. The development has been undertaken with the concepts and notations of linear algebra.

Application of finite strain theory to non-cubic crystals

Abstract The application of finite strain theory to a crystal of orthorhombic or higher elastic symmetry, which is subjected to a purely extensional deformation parallel to the crystallographic axes, yields stress-strain relations and effective elastic constants for hydrostatic stresses. Alternative formulations of the theory are obtained when the free energy is written as Taylor series in E, the invariant analogue of the Eulerian strain tensor, or η, the Lagrangian strain tensor. In most cases, the formulation in terms of E provides a better approximation when the Taylor series are truncated at the second or third order. In the case of cubic crystals, the stress-strain relations reduce to the Birch equation. For non-cubic crystals, the P-V relations calculated using the non-cubic and Birch equations will differ due to the effects of elastic anisotropy. A comparison between the second-order approximations of the non-cubic stress-strain relations and the cubic Birch equation suggests that the difference in volume will be less than 1% for most materials. The difference in volume is reduced when the third-order approximations are used. When the third-order terms are retained, the stress-strain relations calculated using the Eulerian formulation agree with measured linear compression data for quartz to 150 kbar ( P K 0 = 0.4). For zinc, the calculated pressure-volume relation agrees with the shock-wave Hugoniot up to 750 kbar ( P K 0 = 1.2), although both calculated and Hugoniot P-V relations disagree with X-ray compression data. At pressures greater than 300 kbar, the calculated axial ratio of zinc approaches that for other hexagonal metals ( c a ≅ 1.63).

Invariant finite strain measures in elasticity and lattice dynamics

Abstract Some vagueness in the literature concerning the proper measures of strain which may be used in finite elastic strain theory and lattice dynamics is discussed. The requirements for straindependent quantities to be invariant under changes of frame of references are brieffy reviewed, and it is pointed out that the common practice of writing strain-dependent quantities explicity in terms of the Lagrangian strain η is sufficient, but not necessary, for them to be invariant. Invariance is assured if any one of a class of invariant strain tensors is used for this purpose. The use of the non-invariant Eulerian strain tensor e in some applications has not usually led to difficulties because of the restricted situations which have been considered. Applications to more general situations would require the use. of an invariant strain measure. An analogous invariant strain tensor can be defined which reduces to the Eulerian strain tensor in the case of isotropic strain.