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... 

Since there are only 6 independent components of stress and strain there are 36 components to S, the compliance tensor and C, the stiffness tensor. These 36 components may be further reduced using thermodynamic arguments so that there are 21 independent constants for triclinic symmetry, 13 for monodinic, 7 for tetragonal, 5 for hexagonal, 3 for cubic and 2 for isotropic materials. It is consequently more convenient to use die simplified notation of Voigt where ... 

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. 

Assuming (1) axis is the longitudinal direction of the fibre, the compliance tensor 5c takes the same form than Si, whereas, taking into account the transversal isotropy, the compliance constants have the following expressions ... 

The damage of laminate is introduced by adding the damage contribution H tensor to the compliance tensor of composite (7, 8, 9). The only non zero component of H are H22 and //gg has no influence in the present analysis) ... 

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 ... 

Here C,/ (g) and Sn (g) are the single-crystal stiffness and compliance tensors in the sample reference system. They have the following expressions ... 

In Equation (91), Sy is the compliance tensor obtained by inverting the stiffness tensor Ch ... 

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. 

The compliance tensor for background rock matrix is a general expression however, in the current work, it is defined by elastic constants. For an assumed transversely anisotropic material, the tensor is defined by five elastic constants (Ej, E2, Vi, V2, and Gt -Young s modulus in the horizontal plane. Young s modulus in the vertical plane, Poisson s ratio in the horizontal plane, Poisson s ratio in the vertical plane, and shear modulus in the vertical plane of the background rock mass, respectively). The compliance tensor for fractures is defined by ... 

For a given fracture set i, determine the equivalent compliance tensor for fractures, using... 

Equation (4). Repeat this prcx ess for all fracture sets. Combine results to obtain aggregated Construct the rock matrix compliance tensor. Combine the fracture and rock-mass compliance tensors, and invert the resulting tensor to obtain (see Equation (3)). Construct the hydroelastic and thermoelastic tensors using the previously obtained tensors and Equations (12) and (13). 

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. 

According to the principle of self-consistency (Kemeny, 1986), the compliance tensor of a jointed rock mass under tension and shear stress state can be deduced as... 

Note that, if the value of E is required for a direction OA not parallel to one of the axes OY1Y2Y3 with respect to which the compliance tensor is given, the tensor components must first be expressed with respect to a set of axes of which one is parallel to OA. 

The quantities are the components of a fourth-rank tensor called the stiffness tensor and the quantities Syi i are the components of a fourth-rank tensor called the compliance tensor (note the confusing relationship between the names and the symbols). Specification of all values of either c,y / or Sy i allows all values of the other set to be calculated. 

P = probability for fatigue life survival Pirr = irreversible deformation power per unit of volume Q = heat flux Rk = Kapitza resistance S = cross-link distance or compliance tensor T = temperature U = internal energy Us = stored stress energy... 

Jijait) is the creep compliance tensor t is the actual time... 

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. 

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