Plastic yielding

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 tougher materials the minimum thickness required by equation 15 can become excessive. In such cases the /-integral test is an attractive alternative. Because this test considers the stress distribution around the crack inside the plastic 2one, it can be used to obtain a vahd toughness measurement in a thinner specimen, because more extensive plastic yielding does not invaUdate the analysis. The equivalent of equation 15 for the /-integral test is... 

Plastic Forming. A plastic ceramic body deforms iaelastically without mpture under a compressive load that produces a shear stress ia excess of the shear strength of the body. Plastic forming processes (38,40—42,54—57) iavolve elastic—plastic behavior, whereby measurable elastic respoase occurs before and after plastic yielding. At pressures above the shear strength, the body deforms plastically by shear flow. 

The process zone is a measure of the yield stress or plasticity of the material in comparison to its brittleness. Yielding within the process zone may take place either plastically or by dimise microcracking, depending on the brittleness of the material. For plastic yielding, / is also referred to as the plastic zone size. 

This competition between mechanisms is conveniently summarised on Deformation Mechanism Diagrams (Figs. 19.5 and 19.6). They show the range of stress and temperature (Fig. 19.5) or of strain-rate and stress (Fig. 19.6) in which we expect to find each sort of creep (they also show where plastic yielding occurs, and where deformation is simply elastic). Diagrams like these are available for many metals and ceramics, and are a useful summary of creep behaviour, helpful in selecting a material for high-temperature applications. 

We can perform a similar analysis for plastic yielding. A panel with the section shown in Fig. 27.5 yields at a load... 

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. 

The types of wear and failure that occur in metallic gears are distinctive and have been subjected to close analysis over many years. The more common types of failure are abrasion, scuffing, pitting, corrosion, plastic yielding and... 

TKPP Cone. Density Apparent Plastic Yield Point... 

W. L. Johnson and K. Samwer, A Universal Criterion for Plastic Yielding pf Metallic Glasses with a (T/Tg)2S Temperature Dependence, Phys. Rev. Lett., 95, 195501 (2005). 

Kronecker delta-function 5y Deformation at plastic yield... 

Yield strength - The stress at which a material exhibits a specified limiting permanent set. Determined by a measurable value of plastic yielding of the material above which the material is considered to be damaged and below which the damaging effects are considered to be negligible. 

A new ASME code for calculating high pressure vessels (Sect VIII Div. 3) is based on the formulae to determine the internal pressure pcompi-pi for complete plastic yielding through the full wall with some assumptions, e.g. perfectly elastic-plastic material behavior and the GE-hypothesis [2]. 

Figure 4.3-3. Ratio of complete plastic yielding to elastic deformation as a function of the diameter ratio.

The above mentioned ASME code [3] uses the type of calculation presented above and selects a safety factor of S=2 to complete plastic yielding according to the GE hypothesis or S=1.732 with respect to complete plastic yielding according to the shear hypothesis Hence the permissible internal pressure is ... 

As a final result of the explanations about strengthening measures the admissible static internal pressure for thick-walled cylinders is compared in Fig. 4.3-7 for different design strategies according to the equations (4.3-9), (4.3-10), (4.3-12) and (4.3-13) and the explained assumptions and optimisations. In the case of the monobloc (A), the two-piece shrink fit and the autofrettaged cylinders the maximum stress at the inner diameter stays within the elastic limit (00.2). Comparatively much larger is the admissible pressure when complete plastic yielding occurs as shown for the collapse pressure (pCOmpi pi. = Pcoii D). 

Bingham number Bm V- M-V yield stress viscous stress Flow of Bingham plastics = yield number, Y... 

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