Strain rate principal

Since the yield function is independent of p, the yield surface reduces to a cylinder in principal stress space with axis normal to the 11 plane. If the work assumption is made, then the normality condition (5.80) implies that the plastic strain rate is normal to the yield surface and parallel to the II plane, and must therefore be a deviator k = k , k = 0. It follows that the plastic strain is incompressible and the volume change is entirely elastic. Assuming that the plastic strain is initially zero, the spherical part of the stress relation (5.85) becomes... 

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.,... 

Stokes Postulates Stokes s postulates provide the theory to relate the strain-rate to the stress. As a result the forces may be related to the velocity field, leading to viscous-force terms in the Navier-Stokes equations that are functions of the velocity field. Working in the principal coordinates facilitates the development of the Stokes postulates. 

There is always a particular set of coordinates, called the principal coordinates, for which the shear components vanish the strain-rate tensor can be written as... 

If the invariants are known for some arbitrary strain-rate state, then it is clear that the three equations above form a system of equations from which the principal strain rates can be uniquely determined. This analysis is explained more fully in Appendix A. Using the principal axes greatly facilitates subsequent analysis, wherein quantitative relationships are established between the strain-rate and stress tensors. 

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. 

Developing the stress-strain-rate relationships is greatly facilitated in the principal coordinate directions. Since isotropy requires that the constitutive relationships be independent of coordinate orientation, the principal-direction relationships can be transformed to any other coordinate directions. At every point in a flow field the strain-rate and stress state... 

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 general, the principal stress-strain-rate relationships, as stated in Eqs. 2.147, 2.148, and 2.149, can be written in tensor form as... 

Perhaps surprisingly, it turns out that the complex series of operations represented by Eq. 2.151 leads to a relatively simple result that is independent of the particular principal-coordinate directions. The stress tensor in a given coordinate system is related to the strain-rate tensor in the same coordinate system as... 

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. 

Since the principal axes are the same for the stress tensor and the strain-rate tensor, the normal strain rates are related to the principal strain rates by the same transformation rules that we just completed for the stress. Thus... 

Recall from the discussion in Section 2.5.4 that the stress tensor, like the strain-rate tensor, has certain invariants. For any known stress tensor, these invariant relationships can be used to determine the principal stresses. 

The final objective of this chapter was to develop quantitative relationships between a fluid s strain-rate and stress fields. Expressions for the strain rates were developed in terms of velocities and velocity gradients. Then, using Stokes s postulates, the stress field was found to be proportional to the strain rates and a physical property of the fluid called viscosity. The fact that the stress tensor and strain-rate tensor share the same principal coordinates is an important factor in applying Stokes s postulates. The stress-strain-rate relationships are fundamental to the Navier-Stokes equations, which describe conservation of momentum in fluids. 

Determine the principal strain rates. Since there are so many zeros in the strain rate tensor, this eigenvalue problem can be solved exactly without too much diffficulty. 

Explain why one of the eigenvalues (principal strain rates) is exactly equal to eee ... 

Given a state at a point (e.g., stress or strain rate) that can be described by a symmetric tensor in some orthogonal coordinate system, it is always possible to represent that particular state in a rotated coordinate system for which the tensor has purely diagonal components. The axes for such a rotated coordinate system are called the principal axes, and the diagonal components are called the principal components. Finding the principal states and the principal directions is an eigenvalue problem. 

Mellor and Herring (M2), observing that Eqs. (2) or (5) imply that the Reynolds-stress deviations from Jg 5,y are proportional to the strain rates (and hence that the principal axes of the stress deviation and strain rate are aligned), call these closures Newtonian. Accordingly, we denote them by MVFN. The success of the Newtonian model is remarkable, especially since for even the weakest of turbulent shear flows the principal axes are not aligned (C4). 

Critical COD Strain at break Major principal tensile strain Strain rate... 

The principal strain rates are a similar set again, and the strain-rate equations are exact. 

The material s response to nonhydrostatic stress, especially its principal strain rates e, e, and and associated strain rates in nonprindpal directions, can alternatively be seen as a response to the set of equilibrium chemical potentials. Relations of the type... 

An extensive homogeneous sample contains a homogeneous stress field with principal values cr, and (cr, + The material is Newtonian and deforms in a steady manner with principal strain rates e and —e. Energy is dissipated at a rate — a )e per second per unit volume, and energy is withdrawn from the sample at this rate uniformly throughout its volume by some unspecified process, so that the sample s temperature is also steady through time. We identify a portion in the interior of the sample that at some moment is a cube with linear dimension k. We ask where is the source of supply of energy to this cube, with power — cr )e. 

Figure 18.2 (a) A state of strain resulting from the stress state in Figure 18.1 in absence of self-diffusion. The strain rate inside the inclusion is uniform, and it is uniform again at points remote from the inclusion, with different principal values, (b) An arbitrary but convenient system for designating points around the periphery. 

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