Elasticity perfect

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. 

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 ... 

Particle collisions are elastic (perfectly bouncy, with no loss of energy). 

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... 

Elastic, perfectly soluble in organic solvents (except for PVF). Resistant to sunlight, oxidizers, low thermal resistance, elevated adhesion to metals, glass, and plastics. 

At the other extreme, in the Newtonian fluid the shearing stress is proportional to the rate of shear, equation (1.1). Many materials show both elastic and viscous effects under appropriate circumstances. In the absence of the time-dependent behaviour mentioned in the preceding section, the material is said to be visco-elastic. Perfectly elastic deformation and perfectly viscous... 

However, the T)-value is often fixed to be a constant in the existing analysis for the underground engineering projects in the HB rock mass (Chen Tonon 2011, Fraldi Guarracino 2010, Li et al. 2009, Park Kim 2006, Shen et al. 2010, Zhong et al. 2009, Zhou Li 2011). In this study D is treated as a variable. To formulate the elasto-plastic analysis solution for an axisymmetrical cavern, a linear function is chosen to quantitatively describe D. Compared with the elastic perfectly-plastic and elastic-brittle-plastic results, the present analysis can objectively reflect the excavation disturbance of the surrounding rocks. 

Under the situation that the disturbance factor is independent of the stress state in the plastic zone, i.e. a fixed D-value, the presented analytical model can be degenerated to an elastic perfectly-plastic or elastic-brittle-plastic model. The corresponding analytical solutions of the elastic perfectly-plastic or elastic-brittle-plastic model can be easily derived. We only give the analytical results of the elastic perfectly-plastic and elastic-brittle-plastic models for comparison with the present analytical model. 

Linearly elastic-perfectly plastic Mohr-Coulomb model was used to model the preliminary fill, the sand cushion, and the RSW backfill. 

The Geogrid was modeled as linearly elastic-perfectly plastic material with Mohr-Coulomb failure criterion. The axial stiffness (J) and the tensile strength (T) were needed in the program and taken J = 620 kN/m and T = 70 kN/m, respectively. The interaction coefficient between the geogrid and cohesive backfill was taken 0.7. 

According to ASME PCC-2 (2006), the main purpose of FRP wrap repair is to strengthen an undamaged section of the pipe to carry the additional loads caused by the damaged or weakened section. Assuming that the repair is applied at zero internal pressure and the pipe material behaves elastic-perfectly plastic (i.e. no strain hardening), the minimum composite repair thickuess. 

In order to compare elastic design spectrum (demand) and capacity curve, first the capacity curve is transformed into an energy equivalent elastic-perfectly plastic system, then the ADRS (Acceleration Displacement Response Spectrum) representation is used, as shown in Figure 2. 

Thornton, C. (1997) Coefficient of restitntion for collinear collisions of elastic-perfectly plastic spheres. Journal of Applied Mechanics, Transactions ASME 64,383—386. 

Figure 3.2 Strain-strain characteristics for an elastic-perfectly plastic material.

The primary stress intensity in the ASME Boiler and Pressure Vessel Code is intended to prevent uncontrolled plastic deformation and to provide a nominal factor of safety on the ductile burst pressure. These limits are based on the principles of limit design. The material is assumed to be elastic-perfectly plastic. For a straight bar in tension, a load producing yield stress, Sy, results in a collapse. If it is loaded in bending, collapse does not occur until the yield moment has been increased by the shape factor of the section. 

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