Small Strain Plasticity Flow Theory

Finally, we can see the relationship between the response given by the hereditary integral form and that given by conventional creep laws such as the logarithmic form 

Therefore the logarithmic creep law employs an averaged relaxation spectrum for all elapsed time (i.e., only one time-dependent mechanism is assumed) as shown in Fig. 2.16, which may cause difficulty under real, complex situations such as the long term behavior of rock. On the other hand the power law (2.292) gives 

In this book we are considering porous materials. Therefore, the stress treated here must be an effective stress a = a + pi where a is the total stress, and p is the pore fluid pressure. Note that in this section we are using the sign convention for stresses adopted in continuum mechanics, therefore the tension stress/strain is considered positive, and the pore fluid pressure is positive, since a = a+pl (details are described in Chap. 6). In this Section we denote the stress as r instead of the effective stress a for simplicity. Readers can see that all results in this section also work for the effective stress. It should be noted that in this section the deviatoric stress is denoted as s whereas in other expositions the deviatoric stress is written as a. Similarly, the deviatoric strain is denoted as e. 

Symbols and notations used in this section are given below  

Deviatoric stress tensor Norm of deviatoric stress Lode s angle for stress Second invariant of deviatoric stress Third invariant of deviatoric stress Strain increment tensor Elastic strain increment tensor Plastic strain increment tensor Volumetric plastic strain increment tensor 

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Small Strain Plasticity Flow Theory The indicial form of D p is given by... 

The result of a simple tension experiment for a metal is schematically shown in Fig. 2.17 with axes of axial stress ai and axial sdain 1 or deviatoric stress s and deviatoric strain e. In metals the volumetric plastic strain can generally be ignored (sP = 0) therefore we can treat the behavior as a uniaxial response. On the other hand, the shearing behavior of geomaterials is inevitably accompanied by volume changes that are plastic, therefore we have to modify the original flow theory developed for metallic materials (Kachanov 2005 Lubliner 1990). Note that in small strain plasticity we assume that the plastic increment de can be decomposed into incremental elastic and plastic components ... 

Boltzmann superposition principle A basis for the description of all linear viscoelastic phenomena. No such theor) is available to serve as a basis for the interpretation of nonlinear phenomena—to describe flows in which neither the strain nor the strain rate is small. As a result, no general valid formula exists for calculating values for one material function on the basis of experimental data from another. However, limited theories have been developed. See kinetic theory viscoelasticity, nonlinear, bomb See plasticator safety. 

Although Y ions present only weak obstacles to dislocation motion [58], they are present in high concentrations and could, in theory, yield a large contribution to the flow stress. However, the crystals presently available apparently contain very small precipitates of ZrN [71] which provide stronger obstacles to slip than do unassociated Y ions [58, 72, 73]. Nonetheless, these unassociated Y ions do cause plastic instabilities, such as dynamic strain aging or the Portevin-Le Chatelier effect in... 

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