Perfectly plastic

Note that different perfectly plastic models for three dimensional case are considered in (Mosolov, Myasnikov, 1971). 

Anzellotti G., Giaquinta M. (1982) On the existence of the fields of stresses and displacements for an elasto-perfectly plastic body in static equilibrium. J. Math. Pure Appl. 61, 219-244. 

Erkhov M.I. (1978) Theory of perfectly plastic bodies and structures. Nauka, Moscow (in Russian). 

There is clearly no perfect plasticizer for every application. Choice depends on the performance requirements of the article being manufactured and... 

Perfectly Plastic. When A = Q then / = 0. The elastie limit surfaee in stress space is stationary, and the material is said to be perfectly inelastic. 

If the material is perfectly plastic, i.e., if the yield function is independent of k and a, then = 0 and the magnitude of the plastic strain rate cannot be determined from (5.81). Only its direction is determined by the normality condition (5.80), its magnitude being determined by kinematical constraints on the local motion. 

We imagine a finite-duration shock pulse arriving at some point in the material. The strain as a function of time is shown as the upper diagram in Fig. 7.11 for elastic-perfectly-plastic response (solid line) and quasi-elastic response generally observed (dash-dot line). The maximum volume strain = 1 - PoIp is designated... 

Figure 7.11. Quasi-elastic release (dash-dot line) from shock-compressed state. Solid line represents elastic-perfectly-plastic response.

In the numerical calculations, an elastic-perfectly-plastic ductile rod stretching at a uniform strain rate of e = lO s was treated. A flow stress of 100 MPa and a density of 2700 kg/m were assumed. A one-millimeter square cross section and a fracture energy of = 0.02 J were used. These properties are consistent with the measured behavior of soft aluminim in experimental expanding ring studies of Grady and Benson (1983). Incipient fractures were introduced into the rod randomly in both position and time. Fractures grow... 

The second physical quantity of interest is, r t = 90 pm, the critical crack tip stress field dimension. Irwin s analysis of the crack tip process zone dimension for an elastic-perfectly plastic material began with the perfectly elastic crack tip stress field solution of Eq. 1 and allowed for stress redistribution to account for the fact that the near crack tip field would be limited to Oj . The net result of this analysis is that the crack tip inelastic zone was nearly twice that predicted by Eq. 3, such that... 

Fig. 2.5. The idealized elastic/perfectly plastic behavior results in a well defined, two-step wave form propagating in response to a loading within the elastic-plastic regime. Such behavior is seldom, if ever, observed.

In the perfectly elastic, perfectly plastic models, the high pressure compressibility can be approximated from static high pressure experiments or from high-order elastic constant measurements. Based on an estimate of strength, the stress-volume relation under uniaxial strain conditions appropriate for shock compression can be constructed. Inversely, and more typically, strength corrections can be applied to shock data to remove the shear strength component. The stress-volume relation is composed of the isotropic (hydrostatic) stress to which a component of shear stress appropriate to the... 

Fig. 2.8. Idealized elastic/perfectly plastic solid behavior results in a stress tensor in which there is a constant offset between the hydrostatic (isotropic) loading and shock compression. Such behavior is only an approximation which may not be appropriate in many cases.

Perhaps the most dramatic exception to the perfectly elastic, perfectly plastic materials response is encountered in several brittle, refractory materials that show behaviors indicative of an isotropic compression state above their Hugoniot elastic limits. Upon yielding, these materials exhibit a loss of shear strength. Such behavior was first observed from piezoelectric response measurements of quartz by Neilson and Benedick [62N01]. The electrical response observations were later confirmed in mechanical response measurements of Waekerle [62W01] and Fowles [61F01]. 

Metallic glasses are almost elastic-perfectly plastic, so indentations in them are limited by the critical shear stress, not by strain-hardening as in crystalline... 

Biggs (Ref. 21) discusses responses of simple dynamic systems in great detail, including the important intermediate case of elastic, perfectly-plastic systems. He also presents dimensionless response curves for various levels of elastic-plastic response, and for several different regular pulse shapes. 

The increased impulse capacity of a structure is proportional to the square root of the increase in the area under the resistance-deflection curve. The effect of mass can be easily shown with the following equation for the impulse capacity of a ductile element with large allowable deflection and a perfectly plastic resistance function (as shown in Figure 4b). 

In blast analyses, the resistance is usually specified as a nonlinear function to simulate elastic, perfectly plastic behavior of the structure. The ultimate resistance, (R ) is reached upon formation of a collapse mechanism in the member. When the resistance is nonlinear, the dynamic equilibrium equation becomes ... 

It is clear that e = 1 for the normal impact of perfect elastic spheres, while e = 0 for the normal impact of perfect plastic spheres. 

Fig. 5a,b Schematic representation of a the tip-sample contact upon high loading b the according compliance curve. In the case of perfectly plastic response the unloading curve is identical to the vertical line intersecting with the abscissa at hmax. In general, some viscoelastic recovery occurs and the residual impression depth hy is smaller than hmax. The difference hc—hy represents the extent of viscoelastic recovery. Ap and Ae denote the dissipated and the recovered work, respectively. Ap=0 for perfect elastic behaviour, whereas Ae=0 for perfect plastic behaviour. The viscoelastic-plastic properties of the material may be described by the parameter Ap(Ap+Ae) l. The contact strain increases with the attack angle 6. Adapted from [138]... 

A plastic material is defined as one that does not undergo a permanent deformation until a certain yield stress has been exceeded. A perfectly plastic body showing no elasticity would have the stress-strain behavior depicted in Figure 8-15. Under influence of a small stress, no deformation occurs when the stress is increased, the material will suddenly start to flow at applied stress a(t (the yield stress). The material will then continue to flow at the same stress until this is removed the material retains its total deformation. In reality, few bodies are perfectly plastic rather, they are plasto-elastic or plasto-viscoelastic. The mechanical model used to represent a plastic body, also called a St. Venant body, is a friction element. The... 

For k 1, isothermal conditions prevail, while /< -c 1 when the situation is adiabatic. The characteristic timescale to for the present study is defined as the time to attain the material toughness A[r for a given loading rate, i.e., to = Kf1/K. The characteristic length Lq is taken as the size of the plastic zone of a perfectly plastic material with yield stress s0 so that L0 = (fQ/so)2 [57]. For k 1, heat conduction needs to be accounted for and this condition results in the estimation of... 

Most crystalline materials which can undergo a large permanent strain without fracture deform in a complex manner that is neither viscous nor perfectly plastic. At low temperatures, such materials deform by a process... 

In this context it has to be pointed out that in the original Dugdale model the material behavior is assumed to be linearly elastic and perfectly plastic the latter assumption leads to a uniform stress distribution in the plastic zone. This may be a simplified situation for many materials to model, however, the material behavior in the crack tip region where high inhomogeneous stresses and strains are acting is a rather complex task if nonlinear, rate-dependent effects in the continuum... 

One idealized material is the elastic perfectly plastic material a typically stress strain curve is shown in Figure I7A, For this curve we can substitute Equation (32) into Equation (43) to yield ... 

For perfectly plastic materials, post>yieldtng the strain rate is a constant function of the stress and the stress is constant and never exceeds ay (Fig. ISA). The extent of pla.stic deformation p depends upon the proportionality between plastic strain rate and the stress and the how long the. stress is applied as shown in Figure 17A. The elastic perfectly plastic material is highly idealized and not many materials exhibit this type of behavior. 

FIGURE 17 (A) Stress-strain diagram of elastic perfectly plastic material and (B) stress-strain diagram of elastic material power law with strain hardening. Source. Adapted from Ref. 98. 

For elastic perfectly plastic models there is no elastic deformation in the post-yielding phase however, with the power law strain hardening there is continued elastic and plastic deformation combined. The extent of elastic and plastic deformation post-yielding can be determined by looking at. some arbitrary stress a as shown on Figure 17B. For this stress the elastic and plastic deformations are... 

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