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5.7.3 Constitutive Modeling of Yield Response
Here, we provide basic constitutive relations for the parameters which define an elastoplastic material. We also demonstrate how these various parameters affect the uniaxial stress-strain response of a material. The example in Figure 5.1↑d shows how superposition of multiple elastic-perfectly plastic bond families may create a hardening-like curve. .
Since each family behaves elastically until it yields, a family's yield threshold is generally not the value recorded on a stress-strain curve when the slope changes (Figure 5.2↑). That value may be called the apparent yield threshold , which can be related to the true yield threshold by assuming a linear elastic stress-strain relationship prior to yielding. For simplicity, we assume that values are evenly distributed between an initial yield threshold and a final yield threshold , parameters which may be identified from a stress strain curve (Figure 5.3↓a-b). Beyond , the material either behaves as if it is perfectly plastic (a scenario which may be valid around the ultimate strength, for example), or it transitions to a linear hardening regime. The constitutive model thus specifies The relationships between and embodied in eq.(5.7.3-1) are illustrated graphically in Figure 5.2↑. Through this relationship, only the values of and must be specified, along with .
The family mass fractions govern the influence of each family on the overall material response. The simplest model for involves specifying the mass fraction of the first yielding family , which controls the slope of the initial post-yield response (Figure 5.3↓a), and then evenly weighting the rest of the bond families, . However, in cases where the material transitions to a linear hardening regime, we can recover this behavior by adding one more bond family, , that never yields, thus remaining elastic. The associated mass fraction for is called the elastic mass fraction and denoted ; a non-zero value for this parameter may be specified whenever we wish to include linear hardening behavior (Figure 5.3↓b). Given initial and elastic mass fractions and , the simplest constitutive assumption for the remaining 's assumes the remaining mass is evenly divided, such that The effect of the mass fraction parameters and is explored parametrically in Figure 5.3↓c and Figure 5.3↓d, respectively. In general, most ductile materials have very close to unity, which provides hardening behavior over a finite strain range. As the stress-strain behavior approaches perfect plasticity. In contrast, when , the material response becomes perfectly plastic once the final yield threshold has been exceeded. increases, a region of linear hardening is seen on a plot of the true stress against strain. For most ductile materials, is usually or on the order of .
It is also possible to refine the constitutive relations of Eqs. (5.7.3-1)-(5.7.3-2) by introducing a bias factor , which allows a geometric progression rather than uniform spacing of the apparent yield thresholds and family mass fractions. The bias factor has the effect of modifying the shape of the hardening region between and (Figure 5.3↑b). The modified constitutive relations for and take the form The mass fractions are similarly biased, where and are specified and The full set of parameters is then given by . Setting recovers the uniform distribution presented in Eqs.(5.7.3-1)-(5.7.3-2). Figure 5.3↑a-b graphically describes the influence of each parameter on simplified stress-strain curves, showing how these parameters may be extracted from experimental data.