Multiaxial stress state

The simplified failure envelopes differ little from the concept of yield surfaces in the theory of plasticity. Both the failure envelopes (or surfaces) and the yield surfaces (or envelopes) represent the end of linear elastic behavior under a multiaxial stress state. The limits of linear elastic... 

Some materials might produce a unique failure surface providing measurements could be conducted under first stretch conditions in a state of equilibrium. Tschoegl (110), at this writing, is attempting to produce experimental surfaces by subjecting swollen rubbers to various multiaxial stress states. The swollen condition permits failure measurements at much reduced stress levels, and the time dependence of the material is essentially eliminated. Studies of this type will be extremely useful in establishing the foundations for extended efforts into failure of composite materials. 

The 1-D concentric cylinder models described above have been extended to fiber-reinforced ceramics by Kervadec and Chermant,28,29 Adami,30 and Wu and Holmes 31 these analyses are similar in basic concept to the previous modeling efforts for metal matrix composites, but they incorporate the time-dependent nature of both fiber and matrix creep and, in some cases, interface creep. Further extension of the 1-D model to multiaxial stress states was made by Meyer et a/.,32-34 Wang et al.,35 and Wang and Chou.36 In the work by Meyer et al., 1-D fiber-composites under off-axis loading (with the loading direction at an angle to fiber axis) were analyzed with the... 

The stress tensor for each point of the grain can be represented by one point in principal stress space. In that space, there exists a volume where the propellant is made worthless by significant damage, even possibly a crack. A major difficulty of such a representation is the fact that the failure properties of propellant depend strongly on loading conditions (temperature and strain rate). So in this paper, for each loading condition (one strain rate and one temperature), we construct a failure surface based on experimental data for several multiaxial stress states. 

A polymer is more likely to fail by brittle fracture under uniaxial tension than under uniaxial compression. Lesser and Kody [164] showed that the yielding of epoxy-amine networks subjected to multiaxial stress states can be described with the modified van Mises criterion. It was found to be possible to measure a compressive yield stress (Gcy) for all of their networks, while the networks with the smallest Mc values failed by brittle fracture and did not provide measured values for the tensile yield stress (Gty) [23,164-166]. Crawford and Lesser [165] showed that Gcy and Gty at a given temperature and strain rate were related by Equation 11.43. 

Hyperelastic finite element analysis Accommodates complex geometries. Can handle nonlinearity in material behavior and large strains. Rapid analysis possible. Standard material models available. Does not include rate-dependent behavior. Cannot predict permanent deformation. Does not handle hysteresis. Some material testing may be required. Can produce errors in multiaxial stress states. 

It is important to note that the uniaxial loading of an off-axis plate generates a local (inherent) multiaxial stress state. It is therefore worth mentioning the investigations by Kawai and coworkers for the description of the off-axis fatigue behavior of UD and woven reinforced laminates [61,71,72] and their fatigue damage mechanics model [61]. The model is based on the nondimensional effective stress concept, which is the square root of the Tsai—Hill polynomial. 

A ratchetting criteria for multiaxial stress state, which are not provided explicitly in the design guide, was developed, and is being examined the applicability of the criteria to general components by FE Analysis. 

There is no code or standard that dictates the allowable stresses for refractory materials. Refractory suppliers do not have established criteria for acceptable stress levels. In addition there is very limited experimental information on the behavior of refractory materials under multiaxial stress states. 

Work performed during this reporting period included significant effort in two areas Bias correction using bootstrap simulation and probabilistic strength analysis in multiaxial stress states, The following two sections describe recent progress in each. 

Probabilistic Strength Analysis in Multiaxial Stress States... 

Also during this reporting period studies were carried out on the effects of multiaxial stress states on the probability of failure of brittle materials employed in structural applications. An important aspect of this is the error incurred in employing the PI A (principal of independent action) assumption where the three principal stresses are assumed to act as three independent uniaxial stress states. Batdorf in Ref. 1 has shown that the pure independence assumption can be either conservative or nonconservative when compared to more refined approaches. The PIA method involves the relationship ... 

H. Lu and W. G. Knauss, The Role of Dilatation in the Nonlinearly Viscoelastic Behavior of Pmma under Multiaxial Stress States Mec/j. Time-Dependent Mails. 2,307—334... 

The fatigue evaluation procedure is outlined in Chapter 8 in which it was mentioned how the alternating stress intensity is calculated for the general multiaxial stress state in a pressure vessel component. In addition, the effects of the so-called local structural discontinuities must be evaluated using stress concentration factors determined from theoretical, numerical or numerical techniques. These are referred to as the fatigue strength reduction factors, which generally should not exceed a value of 5. 

Since the flow stress increases due to hardening, failure by cleavage fracture may occur even after plastic yielding. In this case, a (macroscopi-cally) ductile (but microscopically brittle) cleavage fracture develops, a rather seldom case that can occur only in a multiaxial stress state (90. ... 

Strictly speaking, equivalent stresses (for example, the von Mises equivalent stress) should be used to calculate stresses and strains due to the multiaxial stress state. Furthermore, the equation iLt.cr = Kt,s is only approximately valid in the elastic region because of the transversal contraction caused by the radial and circumferential stresses. For engineering purposes, a uniaxial calculation is sufficient, especiaffy so if we consider the scatter in the material parameters. The multiaxiality of the stress state at the notch root is discussed in section 4.3. 

The present section deals with the review and extension of Schapery s single integral constitutive law to two dimensions. First, a stress operator that defines uniaxial strain as a function of current and past stress is developed. Extension to multiaxial stress state is accomplished by incorporating Poisson s effects, resulting in a constitutive matrix that consists of instantaneous compliance, Poisson s ratio, and a vector of hereditary strains. The constitutive equations thus obtained are suitable for nonlinear viscoelastic finite-element analysis. 

In order to formulate a stress-strain relationship for a multiaxial stress state, each strain component is assumed to be a linear function of the stress operators. Therefore, as in linear... 

These statistical approaches can not be applied directly to the experimental results in this study, because the fracture was occurred under biaxial stress in Disk-on-Rod tests and Piston-on-Ring tests. The statistical approaches for multiaxial stress state have been studied by several investigators. In order to determine the suitable equivalent stress more precisely, further investigation is needed for the widely variable stress states. However, it is considered that uniaxial statistical approach by eqs. (2) (5) are available for the comparison of the critical stress for maincrack formation under... 

Further investigation is needed for the application to complex multiaxial stress states and the fracture mechanical analysis of crack arrest and propagation process. However, new experimental technique of thermal shock fracture (Disk-on-Rod test), by which the indispensable information for the structural application of ceramics at high temperature, was established in the present study. 

To predict component reliability for multiaxial stress states the Batdorf the-ory ° is used. Batdorf theory combines the weakest link theory with linear elastic fracture mechanics. It includes the calculation of the combined probability of the critical flaw being within a certain size range and being located and oriented so that it may cause fracture. 

For yield to occur at all, the global stress state must contain a deviatory component, so that pure hydrostatic stress states do not result in plasticity in a uniform specimen. Indeed, in many types of material, including metals, yield is usually well described by the Von Mises criterion, in which it is considered to be independent of the hydrostatic pressure, p = (jxx + principal stresses (compare Section 14.2.1). 

Empirical criteria for the formation of crazes in multiaxial stress states, analogous to the von Mises criterion for yield [Eq. (59)], are based on the observation that crazing is absent in both compression and simple shear, which is reasonable given that one would expect cavitation to be favored by large values of p. A critical strain to craze, = A + B/p, is generally found to provide a reasonable description of craze nucleation, so that in terms of the principal stresses the criterion is given by Eq. (70), where A, B, C, and D are constants. 

This theory was first proposed by Tresca in 1865 and experimentally verified by Guest in 1900. It states that in a multiaxial stress state failure occurs when the maximum shear stress exceeds the maximum shear stress at failure in a monotonic tensile traction test. In a tensile test it is... 

It has been shown in Sect. 6.6.1 how a multiaxial stress state can modify the ductility of a material. To this purpose the triaxiality factor (TF) was introduced with Eq. (6.41) as the ratio of the hydrostatic stress to the von Mises equivalent stress. It is quite logical to think that the TF should have an effect on fatigue... 

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