Complex stress

This behavior is usually analy2ed by setting up what are known as complex variables to represent stress and strain. These variables, complex stress and complex strain, ie, T and y, respectively, are vectors in complex planes. They can be resolved into real (in phase) and imaginary (90° out of phase) components similar to those for complex modulus shown in Figure 18. 

Of all the theories dealing with the prediction of yielding in complex stress systems, the Distortion Energy Theory (also called the von Mises Failure Theory) agrees best with experimental results for ductile materials, for example mild steel and aluminium (Collins, 1993 Edwards and McKee, 1991 Norton, 1996 Shigley and Mischke, 1996). Its formulation is given in equation 4.57. The right-hand side of the equation is the effective stress, L, for the stress system. 

The simplest theoretical model proposed to predict the strain response to a complex stress history is the Boltzmann Superposition Principle. Basically this principle proposes that for a linear viscoelastic material, the strain response to a complex loading history is simply the algebraic sum of the strains due to each step in load. Implied in this principle is the idea that the behaviour of a plastic is a function of its entire loading history. There are two situations to consider. 

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

Of course it should always be remembered that the solutions obtained in this way are only approximate since the assumptions regarding linearity of relationships in the derivation of equation (2.64) are inapplicable as the stress levels increase. Also in most cases recovery occurs more quickly than is predicted by assuming it is a reversal of creep. Nevertheless this approach does give a useful approximation to the strains resulting from complex stress systems and as stated earlier the results are sufficiently accurate for most practical purposes. 

Theories of the oxidation of tantalum in the presence of suboxide have been developed by Stringer. By means of single-crystal studies he has been able to show that a rate anisotropy stems from the orientation of the suboxide which is precipitated in the form of thin plates. Their influence on the oxidation rate is least when they lie parallel to the metal interface, since the stresses set up by their oxidation to the pentoxide are most easily accommodated. By contrast, when the plates are at 45° to the surface, complex stresses are established which create characteristic chevron markings and cracks in the oxide. The cracks in this case follow lines of pores generated by oxidation of the plates. This behaviour is also found with niobium, but surprisingly, these pores are not formed when Ta-Nb alloys are oxidised, and the rate anisotropy disappears. However, the rate remains linear it seems that this is another case in which molecular oxygen travels by sub-microscopic routes. 

Sophisticated design engineers unfamiliar with plastics behavior will be able to apply the information contained in this and other chapters to applicable sophisticated equations that involve such analysis as multiple and complex stress concentrations. The various machine-design texts and mechanical engineering handbooks listed in the Appendix A PLASTICS TOOLBOX and REF-... 

Maximum shear stress theory which postulates that failure will occur in a complex stress system when the maximum shear stress reaches the value of the shear stress at failure in simple tension. 

The orientation dependence of the stress alignment effect is consistent with the trigonal symmetry of the B—H complex. Stress along the [110] direction lifts the orientational degeneracy of the four BC sites about the boron while stress along the [100] direction does not. (A [111] stress also leads to a dichroism of the expected magnitude.) The sites perpendicular to... 

In the energy domain, new and efficient uses in gas lines, electric cable ducts and the like, will promote surface stabilization and endurance as well as complex stress capability of various extruded or cast systems. Such reactants as acetylene terminated polymers have yielded cross-linked cured, networks of exceptional density and durability. A diimide dianhydride combined with (3) ethynylaniline yields an acetylene terminated tetraimide. On further polymerization at 250°C, the cross-linked structure derived can be used continuously at about 230°C. When this is combined with polymer carbon fibers or filaments, an exceedingly refractory and tough binder is produced. 

So if we substitute the complex stress and strains into the constitutive equation for a Maxwell fluid the resulting relationship is given by Equation (4.21) ... 

Ratio of complex stress ( ct ) to complex strain (y ) in the forced oscillation of material M =... 

Note 5 For linear viscoelastic behaviour interpreted in terms of complex stress and strain (see notes 2 and 3)... 

Note 3 The complex stress o = crgc = cr (cos real part of the complex stress is that actually applied to the material. 

De Man (1983) has reviewed this property of fats. Consistency is defined as (1) an ill-defined and subjectively assessable characteristic of a material that depends on the complex stress-flow relation or as (2) the property by which a material resists change of shape. Spreadabil-ity, a term used in relation to consistency, is the force required to spread the fat with a knife. The definition is similar to that for hardness the resistance of the surface of a body to deformation. The most widely used simple compression test in North America is the cone penetrometer method (AOCS Method Cc 16-60, 1960). More sophisticated rheological procedures are also available. Efforts have been made to calibrate instrumental tests with sensory response. With the cone penetrometer method, penetration depth is used as a measure of firmness. Hayakawa and De Man (1982) studied the hardness of fractions obtained by crystallization of milk fat. Hardness values obtained with a constant speed penetrometer reflected trends in their TG composition and solid fat content. 

In this section, pedagogical models for the time dependence of mechanical response are developed. Elastic stress and strain are rank-two tensors, and the compliance (or stiffness) are rank-four material property tensors that connect them. In this section, a simple spring and dashpot analog is used to model the mechanical response of anelastic materials. Scalar forces in the spring and dashpot model become analogs for a more complex stress tensor in materials. To enforce this analogy, we use the terms stress and strain below, but we do not treat them as tensors. 

Yielding occurs not only in the uniaxial stress state but also in more complex stress fields where ax, experimental measurements with simple geometries. 

It can be seen that ceramic multilayer structures have been produced with increments of the hardness of up to 60 GPa, increasing the hardness by up to a factor of almost 3. Initial work in this area has developed a number of ideas, such as the effect of modulus mismatch, which in some cases give good agreement with the models suggested but in many others do not. It is suggested that at least some of this discrepancy can be accounted for by differences in the microstructure and residual stress-state of the film, both of which are often poorly characterized. Furthermore there is very little direct evidence about how these structures deform and in particular about how different layers must be strained in order to accommodate the indenter when it is pressed into the sample. Further advances in this area will require the greater use of numerical techniques to analyse the complex stress and strain behaviour under the indentation, coupled with the use of recently developed techniques that allow the localized deformation behaviour to be observed in detail. 

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