Average Stress Theories

The Mori-Tanaka Average Stress Theory The Mori-Tanaka equations were derived for calculating the elastic stress field in and around an ellipsoidal particle in an infinite matrix (16,17). The shape of the ellipsoid can be altered to appear more fiber-like, disk-like, or spheroidal, thus allowing for a continuous range of shapes to be considered. 

Following the theory of Watterson and White [9], we consider the volume averaged stress tensor (c) defined by... 

Chamis C.C. and Sendeckyj G. R (1968) Critique on theories predicting thermoelastic properties of fibrous composites. Journal of Composite Materials, 2, 332-358. Mori T. and Tanaka K. (1973) Average stress in matrix and average elastic energy of materials with misfitting iclusions. Acta Metallurgica, 21, 571-574. 

Voorhees analysis assumes that the creep-rupture life of a vessel under complex stressing is controlled by an equivalent stress, J, termed the shear-stress invariant. This average stress is also known as the octahedral shear stress, the effective stress, the intensity of stress, and the quadratic invariant. The theory for the biaxial-stress condition was developed by Von Mises (205), and this theory was further developed to apply to the triaxial-stress condition independently by Hencky (206, 207, 208) and by Huber (209). A derivation of the relationship between the equivalent stress, /, and the three principal stresses, /i, /2, and/s where /i > /2 > /s was given by Eichinger (210). The relationship between these stresses is ... 

The variation offs in this theory is shown in Fig. 8.9a from which the average stress in steely ean be eomputed as ... 

The modulus of the polymer is an average of the stiffnesses of its bonds. But it obviously is not an arithmetic mean even if the stiff bonds were completely rigid, the polymer would deform because the weak bonds would stretch. Instead, we calculate the modulus by summing the deformation in each type of bond using the methods of composite theory (Chapter 25). A stress d produces a strain which is the weighted sum of the strains in each sort of bond... 

There is one further point that is worth mentioning in connection with the random variable concept. We have repeatedly stressed the fact that the theory of random processes is primarily concerned with averages of time functions and not with their detailed structure. The same comment applies to random variables. The distribution function of a random variable (or perhaps some other less complete information about averages) is the quantity of interest not its functional form. The functional form of the random variable is only of interest insofar as it enables us to derive its distribution function from the known distribution function of the underlying time function X(t). It is the relationship between averages of various time functions that is of interest and not the detailed relationship between the time functions themselves. 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

Figure 1. Stress relaxation curves for three different extension ratios. Uncross-linked high-vinyl polybutadiene with a weight average molecular weight of 2 million and a reference temperature of 283 K. G is the apparent rubber elasticity modulus calculated from classical affine theory. (Solid line is data from Ref. 1).

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