Material functions general elastic solid

As was discussed in some detail in chap. 2, the notion of an elastic solid is a powerful idealization in which the action of the entirety of microscopic degrees of freedom are subsumed into but a few material parameters known as the elastic constants. Depending upon the material symmetry, the number of independent elastic constants can vary. For example, as is well known, a cubic crystal has three independent elastic moduli. For crystals with lower symmetry, the number of elastic constants is larger. The aim of the present section is first to examine the physical origins of the elastic moduli and how they can be obtained on the basis of microscopic reasoning, and then to consider the nonlinear generalization of the ideas of linear elasticity for the consideration of nonlinear stored energy functions. 

Rheologists study the relations between stress (force per unit area) and strain (relative deformation) of a material, generally as a function of time scale or rate of strain. For elastic solids the modulus, i.e., stress over strain, is a characteristic parameter for pure liquids it is the viscosity, i.e., the ratio of stress over strain rate. An elastic solid regains its original shape after the stress is released and the mechanical energy used to deform it is regained a pure liquid retains the shape attained and the mechanical energy is dissipated into heat. 

In solids the strain is a function of the applied stress, providing that the elastic limit is not exceeded. For small strains, there is a linear relation between strain and stress. Since in general both the strain and stress are second-rank tensors linking material properties, the elastic constants c should be elements of a fourth-rank tensor, so that ... 

We imagine a finite-duration shock pulse arriving at some point in the material. The strain as a function of time is shown as the upper diagram in Fig. 7.11 for elastic-perfectly-plastic response (solid line) and quasi-elastic response generally observed (dash-dot line). The maximum volume strain = 1 - PoIp is designated... 

It should be noted that the elasticity modulus E is not merely a property of the solid material in the bed. In general, is a complex function of the structure of packing, material properties of packing particles, particle size, and particle contact and cohesion forces between particles. 

The mechanical properties of materials involve various concepts such as hardness, stiffness, and piezoelectric constants, Young s and bulk modulus, and yield strength. The solids are deformed under the effect of external forces and the deformation is described by the physical quantity strain. The internal mechanical force system that resists the deformation and tends to return the solid to its undeformed initial state is described by the physical quantity stress. Within the elastic limit, where a complete recoverability from strain is achieved with removal of stress, stress g is proportional to strain e. The generalized Hooke s law gives each of the stress tensor components as linear functions of the strain tensor components as... 

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