Quasi-static stress-strain relation

A strength value associated with a Hugoniot elastic limit can be compared to quasi-static strengths or dynamic strengths observed values at various loading strain rates by the relation of the longitudinal stress component under the shock compression uniaxial strain tensor to the one-dimensional stress tensor. As shown in Sec. 2.3, the longitudinal components of a stress measured in the uniaxial strain condition of shock compression can be expressed in terms of a combination of an isotropic (hydrostatic) component of pressure and its deviatoric or shear stress component. 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

Indeed, carbon black-filled rubber, when loaded with time-dependent external forces, suffers a state of stress which is the superposition of two different aspects a time independent, long-term, behavior (sometimes improperly called hyperelastic ) opposed to a time dependent, short-term, response. Step-strain relaxation tests suggest that short term stresses are larger than the long term or quasi-static ones [117]. Moreover, oscillatoiy (sinusoidal) tests indicate that dissipative anelastic effects are significant, which leads to the consideration of a constitutive relation which depends not only on the current value of the strain but on the entire strain history. This assumption must be in accordance with some principles which restrict the class of rehable constitutive equations. These restrictions can be classified as physical and constitutive . The former are restrictirMis to which every rational physical theory must be subjected to, e.g., frame indifference. The latter, on the other hand, depends upon the material under consideration, e.g., internal symmetries. 

This simplified equilibration process holds strictly for a cubic crystal in a general non-cubic case, the volume, static pressure and bulk modulus have to be replaced by the strain components (related to unit-cell parameters), stress components and elastic constants, respectively. The computer program PARAPOCS[17] performs the lattice-dynamical, thermodynamical and quasi-harmonic calculations in the general tensorial formalism, and has been used to obtain all results reported below for calcite and aragonite. 

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