Elastic-plastic method

Elastic Stress Analysis - Simplified Elastic-Plastic Method. This method may be used in the case where the method indicated above shows the Pr + Pb + Q stress limits are not satisfied, but indicates that the Pl + Pb + Q range and excluding thermal effects must be less than Sps- Additionally, the effective alternating equivalent stress amplitude must include the fatigue penalty factor, ICej, which is based off of the simplified elastic-plastic criteria from the pre-2007 Section VIII, Division 2. Finally, a thermal stress ratcheting assessment must be made. 

Since these assumptions are not always justifiable when applied to plastics, the classic equations cannot be used indiscriminately. Each case must be considered on its merits, with account being taken of such factors as the time under load, the mode of deformation, the service conditions, the fabrication method, the environment, and others. In particular, it should be noted that the traditional equations are derived using the relationship that stress equals modulus times strain, where the modulus is a constant. From the review in Chapter 2 it should be clear that the modulus of a plastic is generally not a constant. Several approaches have been used to allow for this condition. The drawback is that these methods can be quite complex, involving numerical techniques that are not attractive to designers. However, one method has been widely accepted, the so-called pseudo-elastic design method. 

Methods for characterizing the elastic, plastic, and brittle properties of compacts of organic materials have been developed by Hiestand and coworkers [29-33]. These indices of tableting performance measure the mechanical properties of compacted materials. 

The finite-element method applied in the elastic-plastic mode gives satisfactory predictions of the achievable improvements from autofrettage. As the residual stresses reduce efficiently the initiation and the growth of cracks, autofrettage is especially well useful for components... 

Prediction of the restitution coefficient has been a challenging research topic for decades. Unfortunately, no reliable and accurate prediction method has been found so far. However, some useful simplified models with certain limits have been developed. One of them is the elastic-plastic impact model in which the compression process is assumed to be plastic with part of the kinetic energy stored for later elastic rebounding, with the rebound process considered to be completely elastic [Johnson, 1985]. In this model, it is postulated that (1) during the plastic compression process, a — r3/2a (2) during the compression process, the averaged contact pressure pm is constant and is equal to 3 Y and (3) the elastic rebound process starts when maximum deformation is reached. Therefore, the compressional force is... 

G. Harkegard Application of the finite element method to cyclic loading of elastic-plastic structures containing effects. Int. J. Fract. 9, 322 (1973)... 

It must be noted that the fracture mechanics framework described above only applies when plastic deformation of the material is limited. Substantial plastic deformation may accompany propagation of existing defects in structures fabricated from relatively low-strength materials, e.g., carbon steels. In these cases, the linear elastic stress intensity factor, K, does not accurately apply in structural design. Alternately, elastic-plastic fracture mechanics methods may apply. ... 

It is well known that the methods of elastic-plastic fracture mechanics provide more realistic models of cracked structures with high toughness compared with the methods of the linear elastic fracture mechanics. Ductile materials are used in structural elements not only in piping systems of power plants but in chemical industry, in aircraft propulsion systems and elsewhere [1-8], Evidently, cracked elements in chemical or power plants pose a serious threat to operation of these stmctures. Therefore, it is extremely important that the crack will not spread unstably through the pipe thickness. 

Material characteristics, such as chemistry, particle density or porosity, brittleness, elasticity, plasticity, wettability, and abrasivity, etc., play important roles in the choice of an agglomeration method. A particular chemistry may be necessary to bring about... 

There has been the question why the TPV materials with ductile thermoplastic matrix display rubber elasticity. Several models have been suggested to answer this question (41 7). Inoue group first analyzed the origin of mbber elasticity in TPVs (43). They constructed a two-dimensional model with four EPDM mbber inclusions in ductile PP matrix and carried out the elastic-plastic analysis on the deformation mechanism of the two-phase system by finite-element method (FEM). The FEM analysis revealed that, even at highly deformed states at which almost the whole matrix has been yielded by the stress concentration, the ligament matrix between mbber inclusions in the stretching direction is locally preserved within an elastic limit and it acts as an in-situ formed adhesive for interconnecting mbber particles. 

When yielding causes large departures from linearity in the force-displacement curve, such that valid Kiq data cannot be obtained, it is still possible to make geometry-independent measurements of the fracture resistance of the material, using the methods of elastic-plastic fracture mechanics. These usually require additional information to determine whether non-linearity is due to crack tip plasticity alone, or to a combination of plasticity and crack growth. Several different approaches have been developed, of which we will discuss onfy two the crack tip opening displacement (CTOD) and /-integral methods. 

Elastic-plastic fiacture mechanics has been extensively used in studies on plastics, but to data mainly tty research laboratories. The plastics industry continues to rely on impact testing, and in particular the notched Izod test, as the principal method for assessing toughness. One of the most important contributions of fracture mechanics to potymer engineering has been to provide a theoretical basis for understanding these impact test data. 

The problems of exact design for a viscoelastic polymer with non-linear properties are severe. For example, in Figure 8.1 a) the stress-strain curve is linear only at the smallest strains (below 0.2%). Most plastic parts are designed to operate at strains well above 0.2%, and in this case exact stress analysis is impossible. In practice, a safe approximate procedure known as the pseudo-elastic design method is used. The salient features of the method, which is veiy straightforward to apply, are as follows ... 

VI.7. Methods 1 and 2 are based on the criteria of the first approach above, whilst Method 3 follows the basic fracture mechanics approach or the extensions to elastic plastic fracture mechanics described later. It should be noted that whilst linear elastic fracture mechanics can be used provided that small scale yielding limits prevail, if more extensive yielding occurs then elastic plastic fracture mechanics methods should be used. Other evaluation methods are possible. Any approach suggested by the package designer is subject to the approval of the competent authority. 

The rigorous design method is based on generally accepted closed-form models. The adhesive behaviour in the models is assumed to be linearly elastic. Only the formulae used in the calculation of the temporary maximum joint resistance require the complete shear stress—shear strain curve or the elastic—plastic model of the adhesive to be known. As adhesives typically have a non-linear shear behaviour, using only the linear part of the stress—strain curve brings added conservatism to the models with respect to the actual joint resistance. 

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