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5.7.1 Elastic-Perfectly Plastic Response
This section describes a reactive framework in which all loaded bonds in an elemental region break and reform simultaneously into a stressed state with a new reference configuration, resulting in elastic-perfectly plastic behavior. The theory outlined here is similar to reactive viscoelasticity (Section 5.5↑) [17, 77], although bonds now reform in a stressed rather than a stress-free state.
The elastic response of this material is achieved when bonds have not yet failed in response to loading. In this case it is assumed that the bonds belong to generation (the master constituent) whose reference configuration is represented by material points located at . When bonds break and reform at time , a new generation is formed. Consider that bonds of the generation yield based on a scalar yield measure (e.g., the von Mises stress), where is the right stretch tensor from the polar decomposition , and is the rotation tensor, assumed to also be the rotation tensor of . Thus, according to eq.(2.8.1-2), .
Let the yield threshold for the generation be given by , which represents the threshold value at which yielding begins. For generation bonds the yield criterion may thus be defined as where represents the yield surface of generation bonds whose tensorial normal is When yield thresholds are formulated in stress space, the dependence on the deformation takes the form .
Consider two consecutive generations , denoted by and , such that the bond-breaking-and-reforming reaction is . Upon breaking of the generation to form the generation the plastic consistency condition is given by , which reduces to The constitutive model for is given by where is the unit tensor along and is a non-dimensional scalar which may be determined analytically by enforcing the plastic consistency condition in eq.(5.7.1-3). When the plastic deformation is assumed to be isochoric the solution for is obtained while enforcing . For the earliest yielded generation , the preceding generation is in the elastic regime; therefore, at the start of the recursive relation in eq.(5.7.1-4).
The stress response of this solid mixture is given generically by eq.(2.8.3-6), specialized to the case where each generation is assumed to have the same constitutive model for its strain energy density, . For example, in the case of the three consecutive generations , and , the strain energy density is and the stress is where and are evaluated from ( ) and eq.(5.7.1-4), and is given in eq.(2.8.3-9). For an elastic-perfectly plastic response the bond mass fractions are constitutively assumed to satisfy where is the Heaviside unit step function. The corresponding constitutive models for may be obtained by substituting these expressions into (2.8.2-2). The onset of yielding of generations and is the time when ( , ) and ( , ), respectively. In summary during the lifetime of generation ( ) and at other times .