Elastic stress distributions

It is important to be able to solve elasticity problems for a variety of dilferent loading geometries. The equations of elasticity do not possess unique solutions unless residual stresses are ignored but fortunately the principle of superposition allows one to deal with these as a separate issue. This principle can also be used to build up solutions of complicated problems by superposing a set of simpler problems. In this chapter, some simple elastic stress distributions for linearly elastic isotropic bodies will be shown and discussed. In these solutions there will be simplifying features that allows the general equations of elasticity to be bypassed. 

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Although Griffith put forward the original concept of linear elastic fracture mechanics (LEFM), it was Irwin who developed the technique for engineering materials. He examined the equations that had been developed for the stresses in the vicinity of an elliptical crack in a large plate as illustrated in Fig. 2.66. The equations for the elastic stress distribution at the crack tip are as follows. 

The stress intensity factor is a means of characterising the elastic stress distribution near the crack tip but in itself has no physical reality. It has units of MN and should not be confused with the elastic stress concentration factor (K,) referred to earlier. 

It is inherently difficult to measure the strength of a material since this is strongly influenced by the microstructure of the material, i.e., the distribution of flaws which strongly influence the propagation of cracks. This concept is illustrated in Fig. 31, where the elastic stress distribution in an ideally elastic, brittle material is seen to become infinite as the crack tip is approached. The key properties which characterize the strength of a material are ... 

Initial and Final Creep Mismatch Ratio At low temperatures, or during rapid loading, the stress in the fibers and matrix can be estimated from a simple rule-of-mixtures approach this gives the elastic stress distribution between the fibers and matrix. During creep, the stress distribution is time dependent and is influenced by both the initial elastic stress distribution and the creep behavior of the constituents. Immediately after applying an instantaneous creep load (i.e., at t = 0+), the CMR, =0+ can be found by substituting ef0 = Af (E/ Ec)a-C nf and emfi = Am (Em/Ec)[Pg.176]

First, the elastic stress distributions of the un-notched specimens are obtained from a finite element analysis. For the PI un-notched specimen, the discrepancy between the finite element and the analytical result is very small (about 0.01%), thus validating the finite element calculation in terms of accuracy through the meshing and the type of element used. Therefore a similar calculation is conducted on the G1 un-notched specimen where the span to height ratio is smaller. The mismatch on the maximum stresses at the bottom and at the top of the beam between the finite element calculations and the analytical solution is 0.74% in tension and 0.79% in compression (and remains constant upon further mesh refinement). This estimation of the stress distribution is then used for the following evaluation of the stress intensity factor. 

Another important problem in elastic stress distributions, that relates to microstructures, is the disturbance of a stress field by an inclusion with different elastic properties, i.e., there is an elastic mismatch. Extreme examples occur when the inclusion is a pore or a rigid particle. Consider a plate under uniaxial tension that contains a circular hole. Fig. 4.20. As the hole surface is free from applied stresses, o rr-o re distances from the hole, the disturbance in... 

The presence of the crack modifies the elastic stress distribution in its vicinity. 

Fig. 4.1. Elastic stress distributions in various kinds of bonded joints, (a) Lap shear, (b) Butt-tensile and scarf, (c) Cleavage, (d) Peel.

The residual stress in the coating is accounted for, but the model is not entirely valid to describe the stresses when some plastic deformation occurs. These key studies were further developed to take into account the elastic stress distribution and residual stress in the coating, which resulted in an improved equation for determining the strain energy release rate (Hutchinson and Suo, 1991 Venkataraman et al., 1993) ... 

Only elastic behaviour has been considered. Plastic deformation may be present as part of an initial shakedown process (particularly for rough contacts) but our evidence is that this occurs within a few thousand cycles and that, thereafter, the contact operates entirely elastically. Melan s theorem indicates that the build up of residual stress during shakedown can be investigated using the elastic stress distribution and we see little reason, therefore, to include the additional complexity of plastic deformation in the analysis. 

The bending capacity is usually expressed in terms of the flexural strength, ah calculated from the ultimate bending moment of the tested beam. Mu, assuming a linear elastic stress distribution. For a beam with a rectangular cross section of width 6 and height h,... 

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