Principal stresses and strains

The principal strain rates are eigenvalues of the strain-rate tensor (matrix). As described more fully in Section A.21, the direction cosines that describe the orientation of the principal strain rates are the eigenvectors associated with the eigenvalues. In solving practical fluids problems, there is rarely a need to determine the principal strain rates or their orientations. Rather, these notions are used theoretically with the Stokes postulates to form general relationships between the strain-rate and stress tensors. It is perhaps worth noting that in solid mechanics, the principal stresses and strains have practical utility in understanding the behavior of materials and structures. 

From the expression for the elastic energy of an isotropic solid in terms of the principal stresses and strains, show that the von Mises yield criterion is obtained when the dilatational part is removed, leaving the elastic shear strain energy. 

Figure 13 shows a torque cell consisting of a circular shaft mounted with four strain gages on two 45° helices that are diametrically opposite to each other gages 1 and 3 mounted on the right-hand helix sense a positive strain, while gages 2 and 4 mounted on the left-hand helix sense a negative strain. The two helices define the principal stress and strain directions when the circular shaft is subjected to pure torsion. 

Figure 9.17 An elastomeric cube, (a) Undefoimed and (b) defonned states, showing principal stresses and strains.

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 crack in each layer of shell element is oriented perpendicular to the orientation of principal stresses. The membrane stress and strain vector depends on the direction of the principal stress and strain in each layer... 

Principal Stresses and Strains for Plane Stress and Plane Strain Conditions... 

In order to apply the crack nucleation approach, the mechanical state of the material must be quantified at each point by a suitable parameter. Traditional parameters have included, for example, the maximum principal stress or strain, or the strain energy density. Maximum principal strain and stress reflect that cracks in rubber often initiate on a plane normal to the loading direction. Strain energy density has sometimes been applied as a parameter for crack nucleation due to its connection to fracture mechanics for the case of edge-cracked strips under simple tension loading. ... 

In a recent attempt to bring an engineering approach to multiaxial failure in solid propellants, Siron and Duerr (92) tested two composite double-base formulations under nine distinct states of stress. The tests included triaxial poker chip, biaxial strip, uniaxial extension, shear, diametral compression, uniaxial compression, and pressurized uniaxial extension at several temperatures and strain rates. The data were reduced in terms of an empirically defined constraint parameter which ranged from —1.0 (hydrostatic compression) to +1.0 (hydrostatic tension). The parameter () is defined in terms of principal stresses and indicates the tensile or compressive nature of the stress field at any point in a structure —i.e.,... 

The principal coordinates provide an extraordinarily useful conceptual framework within which to develop the fundamental relationships between stress and strain rate. For practical application, however, it is essential that a common coordinate system be used for all points in the flow. The coordinate system is usually chosen to align as closely as possible with the natural boundaries of a particular problem. Thus it is essential that the stress-strain-rate relationships can be translated from the principal-coordinate setting (which, in general, is oriented differently at all points in the flow) to the particular coordinate system or control-volume orientation of interest. Accomplishing this objective requires developing a general transformation for the rotation between the principal axes and any other set of axes. 

In the principal coordinates, of course, there are only three nonzero components of the stress and strain-rate tensors. Upon rotation, all nine (six independent) tensor components must be determined. The nine tensor components are comprised of three vector components on each of three orthogonal planes that pass through a common point. Consider that the element represented by Fig. 2.16 has been shrunk to infinitesimal dimensions and that the stress state is to be represented in some arbitrary orientation (z, r, 6), rather than one aligned with the principal-coordinate direction (Z, R, 0). We seek to find the tensor components, resolved into the (z, r, 6) coordinate directions. 

The purpose of this chapter is to remind the reader of the basis of the theory of elasticity, to outline some of its principal results and to discuss to what extent the classical theory can be applied to polymeric systems. We shall begin by reviewing the definitions of stress and strain and the compliance and stiffness matrices for linear elastic bodies at small strains. We shall then state several important exact solutions of these equations under idealised loading conditions and briefly discuss the changes introduced if realistic loading conditions are considered. We shall then move on to a discussion of viscoelasticity and its application to real materials. 

Extensive theoretical investigations devoted to calculation of residual stresses have been carried out for metals. The principal theme of this work is assumption that residual stresses and strains are the result of differences between pure elastic and elastic-plastic deformations under fixed loading.127 128 The same mechanism, i.e., the appearance of plastic deformed zones, is responsible for the residual stresses arising during crystallization of metals, which occurs on quenching from the melt or cooling after welding. 

By symmetry, the two principal directions of stress (and strain) are in the meridian direction, Tin, and the circumferential direction 7133. The third principal stress is zero. Show that if body and acceleration forces are neglected, the following equilibrium equations are obtained for thin membranes ... 

One may try to avoid the problem by the use of the upper-convected derivative, which ensures the coincidence of the principal axes of stress and strain. But doing that, it appears that any kinetics based on the stress amplitude is improper, since materials which exhibits thickening behaviour in elongation are, to the contrary, shear-thinning. Consequently no unique dependence can be expected for these two kinematics. The determination of a single set of parameters in various flows in then bound to be a compromise. 

Let us assume that the z axis corresponds to the principal axis of the rod. In this case, the only non-null component of the strain tensor is When Lame coefficients are expressed in terms of the tensile modulus and Poisson ratio [see Eq. (4.102)], the relationship between the stress and strain tensors is given by... 

Consider now a portion of the surface of a stressed body (Fig. 16-5). Principal stresses and 02 are parallel to the surface, and 03 is zero. However, 63, the strain normal to the surface, is not zero. It has a finite value, given by the Poisson contractions due to 0j and 02 ... 

Using identical methods one can write the stress-strain equation for an orthotropic linear elastic solid in terms of the principal values of stress and strain as... 

Figures 6 through 8 show results of the strain gauge experiments on the three different molding compounds A, B, and C. Each of these figures gives the principal stresses and q2 anc The maximum shear stress Tmax as measured in the center and on the corner of the die. These results were obtained after averaging measurements on at least 10 individual die. The results are given as function of the number of cycles in THSK testing. Except for the measurement after 300 cycles the stress levels at all positions after any number of cycles are smaller for material C than for material B. The stresses for material B are comparable to those for material A. The deviant behavior after 300 cycles observed with material A shows a large reduction in stress, indicative of a loss of adhesion. However, the increase observed after 500 cycles cannot be explained if indeed the integrity of the interface has been compromised.

The second-rank tensors used in this book are property tensors, such as polarisability, and field tensors, such as stress and strain. The latter are also symmetric tensors, but their principal axes are not necessarily parallel to the crystallographic axes. 

The value of E depends upon the values of the elonents in the stress and strain tensors. Under plane stress conditions, one of die principal stresses is zero and E is equal to Young s modulus, E. However, under plane strain conditions, the strain in one of the principal axes is zero and E = E/(l — v ) where v is Poisson s ratio. For most polymers 0.3 < v < 0.5 and the values of both Gic and Kic invariably are much greater when measured in plane stress. For the purposes both of toughness comparisons and component design, die plane strain values of Gic and Kic are preferred because th are the minimum values fm any given material. In order to achieve plane strain conditions, the following criteria need to be satisfied ... 

---