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In thermodynamic terminology, a state of purely elastic material response corresponds to an equilibrium state, and a process during which there is purely elastic response corresponds to a sequence of equilibrium states and hence to a reversible process. The second law of thermodynamics assures that the heat absorbed per unit mass can be written θds, where θ is the thermodynamic (absolute) temperature and s is…

Anisotropic solids also are common in nature and technology. Examples are single crystals; polycrystals in which the grains are not completely random in their crystallographic orientation but have a “texture,” typically owing to some plastic or creep flow process that has left a preferred grain orientation; fibrous biological materials such as wood or bone; and composite materials…

Temperature change can also cause strain. In an isotropic material the thermally induced extensional strains are equal in all directions, and there are no shear strains. In the simplest cases, these thermal strains can be treated as being linear in the temperature change θ − θ0 (where θ0 is the temperature of the reference state), writing εijthermal = δijα(θ − θ0) for the strain produced by temperature change in the…

Linear elastic isotropic solid The simplest type of stress-strain relation is that of the linear elastic solid, considered in circumstances for which |∂ui/∂Xj|<< 1 and for isotropic materials, whose mechanical response is independent of the direction of stressing. If a material point sustains a stress state σ11 = σ, with all other σij = 0, it is subjected to uniaxial tensile stress. This can be…

In the theory of finite deformations, extension and rotations of line elements are unrestricted as to size. For an infinitesimal fibre that deforms from an initial point given by the vector dX to the vector dx in the time t, the deformation gradient is defined by Fij = ∂xi(X, t)/∂Xj; the 3 × 3 matrix [F], with components Fij, may be represented as a pure deformation,…

The small strains, or infinitesimal strains, εij are appropriate for situations with |∂uk/∂Xl|<< 1 for all k and l. Two infinitesimal material fibres, one initially in the 1 direction and the other in the 2 direction, are shown in Figure 6 as dashed lines in the reference configuration and as solid lines in the deformed configuration. To first-order accuracy in components of…

Strain and strain-displacement relations The shape of a solid or structure changes with time during a deformation process. To characterize deformation, or strain, a certain reference configuration is adopted and called undeformed. Often, that reference configuration is chosen as an unstressed state, but such is neither necessary nor always convenient. If time is measured from zero at a moment when the…