Elastic compliance tensor

Here g [ ] may be called the elastic compliance tensor, andl [-]maybe called the inelastic compliance tensor. Note that g is a fourth-order tensor which shares the symmetries of t. Again, (5.16) may be written as... 

If, however, one assumes uniform stress throughout the same nontextured polycrystal a similar averaging procedure can be performed over the elastic-compliance tensor using the corresponding nine elastic compliance constants Sn, S12, S33, S44, S55, Sss, S12, S23, and S31. This is known as the Reuss approximation (Reuss, 1929), after Endre Reuss (1900-1968), and it yields the theoretical minimum of the elastic modulus. 

Diffraction at high pressure also provides an opportunity to measure some combinations of elastic moduli directly, because the pressure is a stress which results in a strain that is expressed as a change in the unit cell parameters. The compressibility of any direction in the crystal is directly related to the components of the elastic compliance tensor by ... 

The strain tensor is the product of the elastic compliance tensor of the crystal by the stress tensor with components oap. For cubic crystals, where the nonzero components of the elastic compliance tensor are Sn, S12 and S44, it can be expressed1 as ... 

The elastic compliance tensor was then inverted to evaluate the elastic constants. 

Min KB, Jing L, 2003, Numerical determination of the equivalent elastic compliance tensor for fractured rock masses using the distinct element method, Int J Rock Mech Min Sci. 40(6) 795-816. 

Here Sijki are elastic compliances tensor components. Hereafter the bar denotes the spatial averaging. Without flexo- and averaged terms, the strain (4.40) is the well-known spontaneous strain. The origin of the differences like Pk Pi — Pk Pi has been discussed in details by Cao and Cross [101]. 

Here s are the elastic strains, a are the stresses, E are the electric potentials and D are the electrie displacements. [C] represents the adiabatic elastic compliance tensor at constant electric filed, [d] is the adiabatic piezoelectric tensor and [p] is the adiabatie electrie permittivity at constant stress. From these constitutive equations, it is readily known that the pie-zoelectrie element generates eleetrie signals due to mechanical motions and vice versa. 

X10. The next three rows present the viscosity rj, the surface tension, and its tenqterature dependence, in the liquid state. The next properties are the coefficient of linear thermal expansion a and the sound velocity, both in the solid and in the liquid state. A number of quantities are tabulated for the presentation of the elastic properties. For isotropic materials, we list the volume compressihility k = —(l/V)(dV/dP), and in some cases also its reciprocal value, the bulk modulus (or compression modulus) the elastic modulus (or Young s modulus) E the shear modulus G and the Poisson number (or Poisson s ratio) fj,. Hooke s law, which expresses the linear relation between the strain s and the stress a in terms of Young s modulus, reads a = Ee. For monocrystalline materials, the components of the elastic compliance tensor s and the components of the elastic stiffness tensor c are given. The elastic compliance tensor s and the elastic stiffness tensor c are both defined by the generalized forms of Hooke s law, a = ce and e = sa. At the end of the list, the tensile strength, the Vickers hardness, and the Mohs hardness are given for some elements. 

Static dielectric constant Dissipation factor Elastic stifbiess tensor Elastic compliance tensor Elastooptic 1 tensor ... 

General Static dielectric tensor Dissipation factor Elastic stiffiiess tensor Elastic compliance tensor EUastooptic tensor... 

Similarly, the fourth order elastic compliance tensor Sij i can be expressed in a contracted form S . 

Nachgiebigkeitskonstante (Reuss-Elastizitatskonstante) elastic compliance tensor Nachhaltigkeit sustained yield Nachharten post cure, postcuring nachharten post-cure Nachlauf/... 

Strains are reiated to stresses through the fourth-rank elastic compliance tensor s,... 

From the homogeneous stress hypothesis, the elastic compliance tensor of the aggregate can be expressed by equation (84). Similarly, from the hypothesis of homogeneous strain, the elastic stiffness tensor of the aggregate can be expressed by equation (85), where tjp, for example, is the direction cosine between the X axis of the aggregate and the Up axis of the crystal unit cell and are the elastic compliances and stiffnesses of the unit respectively. These expressions for averaging are, of course, two extreme approximations, which neglect the interaction between the units and do not correspond to the real situation of polycrystalline materials, as has been pointed out by several authors. ... 

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