Stress Analysis in Two and Three Dimensions

The various approaches to the solution of viscoelastic boundary value problems discussed in the last chapter for bars and beams carry over to the solution of problems in two and three dimensions. In particular, if the solution to a similar problem for an elastic material already exists, the correspondence principle may be invoked and with the use of Laplace or Fourier transforms a solution can be found. Such solutions can be used with confidence but one must be cognizant of the general equations of elasticity and the methods of solutions for elasticity problems in two and three dimensions as well as any assumptions that might often be applied. To provide all of the necessary information and background for multidimensional elasticity theory is beyond the scope of this text but the procedures needed will be outlined in the following sections. 

This chapter will focus on developing the equations, assumptions and procedures one must use to solve two and three dimensional viscoelastic boundary value problems. The problem of an elastic thick walled cylinder will be used as a vehicle to demonstrate how to obtain the solution of a more difficult reinforced viscoelastic thick walled cylinder. In the process, we first demonstrate how the elasticity solution is developed and then apply the correspondence principle to transform the solution to the viscoelastic domain. Several extensions to this problem will be discussed and additional practice is provided in the homework problems at the end of the chapter. 

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The Boltzman superposition principle (or integral) is applicable to stress analysis problems in two and three-dimensions where the stress or strain input varies with time, but first the approach will be introduced in this section only for one-dimensional or a uniaxial representation of the stress-strain (constitutive) relation. The superposition integral is also sometimes referred to as Duhamel s integral (see W.T. Thompson, Laplace Transforms, Prentice Hall, 1960). 

In the study of viscoelasticity as in the study of elasticity, it is mandatory to have a thorough understanding of methods to determine principal stresses and strains. Principal stresses are defined as the normal stresses on the planes oriented such that the shear stresses are zero - the maximum and minimum normal stresses at a point are principal stresses. The determination of stresses and strains in two dimensions is well covered in elementary solid mechanics both analytically and semi-graphically using Mohr s circle. However, practical stress analysis problems frequently involve three dimensions. The basic equations for transformation of stresses in three-dimensions, including the determination of principal stresses, will be given and the interested reader can find the complete development in many solid mechanics texts. 

The calculations of the CMC distances by means of eqs. (7.3) and (7.4), and subsequent multidimensional scaling analysis, yields a 2D projection (see fig. 9.3) that differs from the EMC analysis in two respects. The reaction pathway predicted by EMC correctly shows the close correlation of the enzymatic catalyst Xg and species X. It also shows correctly the chain of linear reactions from Xi to Xt. The CMC method also shows that Xg is correlated with, but more weakly. Less importantly, the CMC method yields a wrap-around in the placing of species 7. However, the 3D representation of the multidimensional CMC object implemented by stressing the original CMC distances into three rather than two dimensions results in the correct sequencing of the species the species points lie on a 3D spiral, which when projected onto two dimensions results in the wrap-around effect. 

Fluid flow may be steady or unsteady, uniform or nonuniform, and it can also be laminar or turbulent, as well as one-, two-, or three-dimensional, and rotational or irrotational. One-dimensional flow of incompressible fluid in food systems occurs when the direction and magnitude of the velocity at all points are identical. In this case, flow analysis is based on the single dimension taken along the central streamline of the flow, and velocities and accelerations normal to the streamline are negligible. In such cases, average values of velocity, pressure, and elevation are considered to represent the flow as a whole. Two-dimensional flow occurs when the fluid particles of food systems move in planes or parallel planes and the streamline patterns are identical in each plane. For an ideal fluid there is no shear stress and no torque additionally, no rotational motion of fluid particles about their own mass centers exists. 

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