Part IV
Plasticity - Bulk Dissipation
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Chapter 12
1-D Thought Models of Plasticity
Plasticity described the irreversible deformation behavior of material and structural systems, when stresses in the solid reach a threshold. The stress locus is referred to as yield locus, and is often associated with the material strength. Upon reaching this yield locus energy is dissipated into heat form. Yet, in contrast to fracture energy dissipation that occurs along surfaces, plastic dissipation occurs primarily in the bulk and is thus associated with bulk dissipation due to plastic deformation. Furthermore, in contrast to the brittle behavior associated with fracture, plastic deformation occurs in a ductile fashion. The elements of the Plasticity Theory are developed in this Chapter starting with 1-D Thought Models
12.1 Ideal Plasticity
12.1.1 Rheological Toy Model for Ideal Plasticity The simplest thought-model of plasticity is shown in Figure 12.1: a (fric tional) slider (cohesion k) in series with an elastic spring (spring stiffness E) A stress o is applied to the system. This stress is in balance with the elastic spring force, whose relative length is defined by eel = -- eP; that is, o=Ee-eP) (12.1)
where aP stands for movement of the slider. As long as the applied stress magnitude, (o| = osign (), is smaller than the slider-strength k, the slider
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does not move. It is readily understood that [o] can never be greater than k; that is: f(o)=|o|-k<0 (12.2)
When the stress magnitude reaches the strength (i.e. f (o) = 0), the slider moves at an un-defined rate in the direction of the stress application, which is captured by the so-called plastic "flow rule":
=dXsign()
(12.3)
where d > 0 specifies the magnitude of the plastic flow; whereas of/oo specifies the direction of plastic flow (here collinear to the direction of load application). Moreover, a stress point which is on the yield surface, f () = 0 , will stay on the surface during plastic loading, which means:
df = d0
(12.4)
This consistency condition means that during plastic loading of the 1-D sys- tem, do = 0; and hence according to Eq. (12.1), de = deP. Following such a plastic loading, consider an unloading of the system, f () < 0, which implies that no plastic deformation will occur, d = 0, and that the stress evolves elastically, according to do = E de. That is, unloading in the plastic system occurs with the same stiffness as during loading. This is also shown in Figure 12.1.
12.1.2 1-D Thermodynamics of Ideal Plasticity
We want to capture the studied 1-D toy model response within the frame- work of thermodynamics, and specifically within the context of the Clausius. Duhem inequality (2.3), which we recall in terms of volume density quantities:
dD = ode - drb 0 [VI
(12.5)
For the 1-D system, the external work (per unit volume) is &W/|V|= ode, and the free energy volume density is /|V|= = E( - P), with |V| a reference volume to pass from work/energy quantit