Stress multiaxial

The stored strain energy can also be determined for the general case of multiaxial stresses [1] and lattices of varying crystal structure and anisotropy. The latter could be important at interfaces where mode mixing can occur, or for fracture of rubber, where f/ is a function of the three stretch rations 1], A2 and A3, for example, via the Mooney-Rivlin equation, or suitable finite deformation strain energy functional. 

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... 

The generation of a practical propellant failure criterion has been the object of extensive study for several years. No completely general analytical criterion seems to be forthcoming, but significant advances in the experimental characterization of ultimate properties in multiaxial stress fields promise more reasonable empirical guidelines. 

Some efforts have been undertaken to study propellant dilatation in multiaxial stress fields and even in small motor configurations. Farris (25) has conducted limited investigations along this line and has made approximate correlations between uniaxial and multiaxial tests. 

Although the uniaxial test has traditionally received the most attention, such tests alone may be insufficient to characterize adequately the mechanical capability of solid propellants. This is especially true for ultimate property determinations where a change in load application from one axis to several at once may strongly affect the relative ranking of propellants according to their breaking strains. Since the conditions usually encountered in solid rocket motors lead to the development of multiaxial stress fields, tests which attempt to simulate these stress fields may be expected to represent more closely the true capability of the material. 

Uniaxial tensile criteria can lead to gross inaccuracies when applied to situations where combined stresses lead to failure in multiaxial stress fields. Often one assumes that combined stresses have no influence and that the maximum principal stress governs the failure behavior. An improved approach applied to biaxial tension conditions relies upon a pragmatic biaxial correction factor which is applied to uniaxial data,... 

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... 

ESC is mostly a surface-initiated failure of multiaxially stressed polymers in contact with surface-active substances. These surface-active substances do not cause chemical degradation of the polymer, but rather accelerate the process of macroscopic brittle-crack failure. Crazing and cracking may occur when a polymer under multiaxial stresses is in contact with a medium. A combination of external and/or internal stresses in a component may be involved. 

Most of the above-cited work neglects the effect of stress or strain as a tensor. They mostly apply uniaxial stress or strain criteria. Unfortunately, most of the applications where ESC has been reported apply biaxial or multiaxial stresses to the polymer. Therefore, a more general model of the phenomenon of ESC is expected to account for generalized polymer-surface active agent systems, but also to account for generalized stress states in the material. 

By definition, ESC is influenced by the level of the applied multiaxial stress, ft is expected that below an assigned value of stress in a specific medium ESC will not occur. Fatigue crack growth experiments at various constant levels of A K can be applied to study the influence of the crack-tip loading on the ESC behavior in a systematic way. As an example, cast CT specimens of PMMA (Mn = 4.6 x 105) were tested in air and IPA at different levels of AK (0.6, 0.7, 0.8, 0.9 MPa -v/m). The tests were performed as described earlier, with the application of the medium after a certain time of cyclic loading in air. From the plots of the crack length ratio vs. number of cycles, the following observations can be made (Fig. 23) ... 

In practice, however, the multiaxial stress applied in a falling dart test without a preferred direction, has to be expected. This is addressed by the Gardner test of ASTM, in which a falling dart strikes centrally on a flat surface (e.g. a circular disk). The results of both test methods are important polymer design considerations. 

Owing to the multiaxial character of the problems addressed in this chapter, the field equations depend not only on time but also on the position defined by their coordinates. Finally, it is necessary to stress that the solution of viscoelastic problems requires, as in the elastic case, specification of adequate boundary conditions. In this chapter, in addition to considering both integral and differential multiaxial stress-strain relationships, some viscoelastic problems of interest in technical applications are solved. 

Equations (16.32) suggest that these relationships for a viscoelastic material under multiaxial stresses can be written as... 

At this point it can be concluded that core-shell nodules can toughen PET when loading conditions induce multiaxial stress fields (i.e., notched samples or dart tests). Precise physical phenomena involved have to be identified. This is done combining several techniques ... 

Crazing mechanisms and criteria have long been studied (Kausch, 1978 Michler, 1992), but further work is still needed because of limited number of experimental results under multiaxial stress and considerable differences in crazing behaviour under different conditions (Kawagoe, 1996). 

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. 

Consider first the geometrical character of the stress. A multiaxial stress can be characterized by the three principal stresses Sj, S2, and S3, listed in descending value. Shear yielding depends primarily upon the difference between Sj and S3. The Tresca yield condition, Sj - S3 = Y (applicable to metals), has been modified for polymers (for which the shear yield stress Y increases with hydrostatic pressure) (24). 

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. 

Hyperelastic models within finite element codes should be used carefully when a component experiences multiaxial stresses. 

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. 

Shokrieh MM, Lessard LB. Fatigue under multiaxial stress systems. In Harris D, editor. Fatigue in composites. Cambridge, England Woodhead Publishing 2003. pp. 63-109. 

Philippidis TP, Vassilopoulos AP. Fatigue strength prediction under multiaxial stress. 

One simple criterion for yielding under multiaxial stresses is known as the Tresca yield criterion. This approach recognizes that the maximum shear stress is one-half the difference between the maximum and minimum principal stresses. In terms of the uniaxial yield stress [Pg.187]

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