Plastic yield conditions

Here i —> i is a continuous convex function describing the plastic yield condition. The equations (5.7) provide a decomposition of the strain tensor Sij u) into a sum of an elastic part aijuicru and a plastic part ij, and (5.6) are the equilibrium equations. 

The functions v,aij,Sij v) represent the velocity, components of the stress tensor and components of the rate strain tensor. The dot denotes the derivative with respect to t. The convex and continuous function describes the plasticity yield condition. It is assumed that the set... 

Here i —> i is the convex and continuous function describing a plasticity yield condition. The function w describes vertical displacements of the plate, rriij are bending moments, (5.139) is the equilibrium equation, and equations (5.140) give a decomposition of the curvatures —Wjj as a... 

Here i —> i is the convex and continuous function describing a plasticity yield condition, the dot denotes a derivative with respect to t, n = (ni,ri2) is the unit normal vector to the boundary F. The function v describes a vertical velocity of the plate, rriij are bending moments, (5.175) is the equilibrium equation, and equations (5.176) give a decomposition of the curvature velocities —Vij as a sum of elastic and plastic parts aijkiirikiy Vijy respectively. Let aijki x) = ajiki x) = akuj x), i,j,k,l = 1,2, and there exist two positive constants ci,C2 such that for all m = rriij ... 

In this section we analyse the contact problem for a curvilinear Timoshenko rod. The plastic yield condition will depend just on the moments m. We shall prove that the solution of the problem satisfies all original boundary conditions, i.e., in contrast to the preceding section, we prove existence of the solution to the original boundary value problem. 

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... 

Although resistance to deflection and plastic yielding are obviously of first importance in choosing alternative materials, other properties enter into the selection. Let us look at these briefly. Table 27.4 lists the conditions imposed by the service environment. 

It is always very useful to be able to predict at what level of external stress and in which directions the macroscopic yielding will occur under different loading geometry. Mathematically, the aim is to find functions of all stress components which reach their critical values equal to some material properties for all different test geometries. This is mathematically equivalent to derivation of some plastic instability conditions commonly termed as the yield criterion. Historically, the yield criteria derived for metals were appHed to polymers and, later, these criteria have been modified as the knowledge of the differences in deformation behavior of polymers compared to metals has been acquired [20,25,114,115]. 

Figure 14.5b represents the uniaxial compression test, which uses samples with cylindrical or rectangular cross section. The stress and strain are defined in an analogous way to that of the tensile test. This test overcomes the disadvantages mentioned in relation to a tensile test. The stress is compressive, and consequently there is no possibility of the brittle fracture observed in tensile deformation. Plastic yield can even be seen in thermostable materials, which, under other conditions, can be brittle. In addition, the determination of the yield stress is made under conditions of stable deformation since there is no geometrical reason for the formation of a neck such as occurs in tension. A problem that can arise in this test concerns the diameter/height ratio of the sample. If this ratio is too large friction between plates and sample will introduce a constraint, and if it is very small... 

Departing from the maximum shear stress theory of plastic flow R. H. Lance and D. N. Robinson [6] developed yield conditions for fiber reinforced eomposite materials. The authors of [6] assumed that the material could flow plastically if (i) the shear stress on planes parallel to fibers, and in a direction perpendicular to them, reaches a critical value or... 

Elastoplastic materials Elastoplastic materials deform elastically for small strains, but start to deform plastically (permanently) for larger ones. In the small-strain regime, this behavior may be captured by writing the total strain as the sum of elastic and plastic parts (i.e., e = e -I- gP, where e and gP are the elastic and plastic strains, respectively). The stress in the material is generally assumed to depend on the elastic strain only (not on the plastic strain or the strain rate), and hence, no unique functional relationship exists between stress and strain. This fact also implies that energy is dissipated during plastic deformation. The point at which the material starts to deform plastically (the yield locus) is usually specified via a yield condition, which for one-dimensional plasticity may be stated as (38)... 

An altemative scheme is the von Mises yield condition. In this case, one adopts an approach with a mean-field flavor in which plastic flow is presumed to commence once an averaged version of the shear stresses reaches a critical value. To proceed, we first define the deviatoric stress tensor which is given by,... 

Strains and stresses were computed for the joined specimen cooled uniformly to room temperature from an assumed stress-free elevated temperature using numerical models described in detail previously [19, 20]. The coordinate system and an example of the finite element mesh utilized are shown in Figure 3. Elastie-plastic response was permitted in both the Ni and Al203-Ni composite materials a von Mises yield condition and isotropic hardening were assumed. 

In general, the plastic flow curve of a metallic material can be represented by a yield condition in a xmiaxial notation ... 

This generalized yield condition, which is usually referred to as the von Mises yield condition, has a simple geometrical visualization in principal stress space of o i,o 2,o 3 shown in Fig. 3.2. There the line making equal angles with the three principal stress axes represents the locus of pure mean normal stress a, along which all deviatoric stresses vanish and no plastic flow can occur. Thus, plastic flow requires a critical deviation from this line in the radial direction away from it... 

Most mechanisms of local plastic strain production are best understood in simple shear, i.e., under a local shear stress such as 022,. When this is the case, the yield condition becomes... 

At the early stage of densification where P is high, plastic yielding can be the major densification mechanism. The plastic deformation between particles may be regarded as identical to that occurring in a hardness test using an indenter. Then, if the indentation stress a,- satisfies the yielding condition. 

The maximum moment and curvature of the bottom sections of piers under El Centro seismic wave are shown in Table 3. By comparison, the longitudinal direction and transverse direction of the piers don t arrive to the yield condition. This shows that there is no plastic hinge in the piers and the bridge is still in elastic state under design rare earthquake of actual site. The same conclusion can be drawn under Taft seismic wave and artificial seismic wave. 

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