Plastic strains

Little error is introduced using the idealized stress—strain diagram (Eig. 4a) to estimate the stresses and strains in partiady plastic cylinders since many steels used in the constmction of pressure vessels have a flat top to their stress—strain curve in the region where the plastic strain is relatively smad. However, this is not tme for large deformations, particularly if the material work hardens, when the pressure can usuady be increased above that corresponding to the codapse pressure before the cylinder bursts. 

Plastic Stra.ln Ra.tlo. The plastic strain ratio is the ratio of strains measured in the width over the thickness directions in tensile tests. This ratio characterizes the abhity of materials to resist thinning during forming operations (13). In particular, it is a measure of the abhity of a sheet material to resist the thinning and failure at the base of a deep drawn cup. The plastic strain ratio is measured at 0°, 45°, and 90° relative to the rolling direction. These three plastic strain ratios Rq, R, and R q, are combined to obtain the average strain ratio, cahed the R or the R value, and its variation in strain ratio, cahed... 

Forming Limit Analysis. The ductihty of sheet and strip can be predicted from an analysis that produces a forming limit diagram (ELD), which defines critical plastic strains at fracture over a range of forming conditions. The ELD encompasses the simpler, but limited measures of ductihty represented by the percentage elongation from tensile tests and the minimum bend radius from bend tests. 

The normality conditions (5.56) and (5.57) have essentially the same forms as those derived by Casey and Naghdi [1], [2], [3], but the interpretation is very different. In the present theory, it is clear that the inelastic strain rate e is always normal to the elastic limit surface in stress space. When applied to plasticity, e is the plastic strain rate, which may now be denoted e", and this is always normal to the elastic limit surface, which may now be called the yield surface. Naghdi et al. by contrast, took the internal state variables k to be comprised of the plastic strain e and a scalar hardening parameter k. In their theory, consequently, the plastic strain rate e , being contained in k in (5.57), is not itself normal to the yield surface. This confusion produces quite different results. 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

The choice (5.77) for the evolution equation for the plastic strain sets the evolution equations for the internal state variables (5.78) into the form (5.11) required for continuity. The consistency condition in the stress space description may be obtained by differentiating (5.73), or directly by expanding (5.29)... 

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. 

Since the yield function is independent of p, the yield surface reduces to a cylinder in principal stress space with axis normal to the 11 plane. If the work assumption is made, then the normality condition (5.80) implies that the plastic strain rate is normal to the yield surface and parallel to the II plane, and must therefore be a deviator k = k , k = 0. It follows that the plastic strain is incompressible and the volume change is entirely elastic. Assuming that the plastic strain is initially zero, the spherical part of the stress relation (5.85) becomes... 

An explicit relation for the plastic strain rate may be obtained by using (5.80) through (5.82). The partial derivatives of/are, from (5.92)... 

Naghdi, P.M. and Trapp, J.A., On the Nature of Normality of Plastic Strain Rate and Convexity of Yield Surfaces in Plasticity, J. Appl. Mech. 62, 61-66 (1975). 

In this case the shear stress (in the plastic deformation region) depends on how much plastic strain y that has accumulated and the current rate of deformation y. For example, in shock compression the shear stress t behind the shock front (where y is nominally zero) is a function of y only, as given by the implicit relationship... 

Here t has no memory of the rate of deformation in getting to the final plastic strain y. This differs from a path-dependent law, in which the strength (for example) depends in detail on the path followed in getting to the current macroscale state. 

The initial strain hardening rate (df/dy)o is given as a function of plastic strain rate for strain rates up to 10 s (which includes shock compression to 5.4 GPa) as [38]... 

For example, a 10 GPa (total strain = 0.06) shock wave in copper has a maximum total strain rate 10 s [21] the risetime would thus be (eje) 0.6 ns. For uniaxial-strain compression, y averaged over the entire shock front. The resolution of the shock wave in a large-scale, multidimensional finite-difference code would be computationally expensive, but necessary to get the correct strength f behind the shock. An estimate of the error made in not resolving the shock wave can be obtained by calculating dt/dy)o with y 10 s (the actual plastic strain rate) and y 10 s (the plastic strain rate within the computed shock wave due to a time step of 0.06 qs). From (7.41) with y = 10 s (actual shock wave) and y = 10 s (computation) ... 

Plastic strain localization and mixing due to void collapse in porous materials works in the same way, with perhaps an even greater degree of actual mixing due to jetting, and other extreme conditions that can occur at internal free surfaces in shock-loaded solids. 

A void nucleation and growth fracture model embedded in a general viscoelastic-plastic material model is representative of approaches to ductile dynamic fracture (Davison et al., 1977 Kipp and Stevens, 1976). Other approaches include employing the plastic strain as a damage variable (Johnson and Cook, 1985) so that both spall and large strain-to-failure can be treated. 

Plastic) strain after fracture, or tensile ductility. The broken pieces are put together and measured, and Cf calculated from (/ - IqI/Iq, where / is the length of the assembled pieces. 

Strain after fracture and percentage reduction in area are used as measures of ductility, i.e. the ability of a material to undergo large plastic strain under stress before it fractures. 

Dislocation motion produces plastic strain. Figure 9.4 shows how the atoms rearrange as the dislocation moves through the crystal, and that, when one dislocation moves entirely through a crystal, the lower part is displaced under the upper by the distance b (called the Burgers vector). The same process is drawn, without the atoms, and using the symbol 1 for the position of the dislocation line, in Fig. 9.5. The way in... 

In making the edge dislocation of Fig. 9.3 we could, after making the cut, have displaced the lower part of the crystal under the upper part in a direction parallel to the bottom of the cut, instead of normal to it. Figure 9.7 shows the result it, too, is a dislocation, called a screw dislocation (because it converts the planes of atoms into a helical surface, or screw). Like an edge dislocation, it produces plastic strain when it... 

For low-cycle fatigue of un-cracked components where (imax or iCT inl am above o-y, Basquin s Law no longer holds, as Fig. 15.2 shows. But a linear plot is obtained if the plastic strain range defined in Fig. 15.3, is plotted, on logarithmic scales, against the cycles to failure, Nf (Fig. 15.4). TTiis result is known as the Coffin-Manson Law ... 

Act = tensile stress range AeP = plastic strain range AK = stress intensity range N = cycles Nf = cycles to failure Cj, C2, a, b, A, m = constants = tensile mean stress ujg = tensile strength a = crack length. 

When crazing limits the ductility in tension, large plastic strains may still be possible in compression shear banding (Fig. 23.12). Within each band a finite shear has taken place. As the number of bands increases, the total overall strain accumulates. 

The thorough and persistent work on precursor decay (the dependence of Hugoniot elastic limit on propagation distance) of Duvall s Washington State University group was successful in demonstrating that precursor attenuation was due to both stress relaxation and hydrodynamic attenuation. Typical data on crystalline LiF is shown in Fig. 2.7. Observed plastic strain... 

It is instructive to describe elastic-plastic responses in terms of idealized behaviors. Generally, elastic-deformation models describe the solid as either linearly or nonlinearly elastic. The plastic deformation material models describe rate-independent behaviors in terms of either ideal plasticity, strainhardening plasticity, strain-softening plasticity, or as stress-history dependent, e.g. the Bauschinger effect [64J01, 91S01]. Rate-dependent descriptions are more physically realistic and are the basis for viscoplastic models. The degree of flexibility afforded elastic-plastic model development has typically led to descriptions of materials response that contain more adjustable parameters than can be independently verified. 

Nominal composition in weight % and characteristics 0-2% proof stress (min) (MPa) Tensile strength (MPa) Elongation (min) m Young s modulus (typical) (GPa) Fatigue limit % of T.S.) Bend radius on 2 mm Density (g/cm ) Stress for O 1% total plastic strain in 100 h (MPa) Production range... 

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