Materials stress tensors

The situation is more complex for rigid media (solids and glasses) and more complex fluids that is, for most materials. These materials have finite yield strengths, support shears and may be anisotropic. As samples, they usually do not relax to hydrostatic equilibrium during an experiment, even when surrounded by a hydrostatic pressure medium. For these materials, P should be replaced by a stress tensor, <3-j, and the appropriate thermodynamic equations are more complex. 

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

V is the material velocity. a is the stress tensor. g is the acceleration of gravity. e is the internal energy per unit mass. h is the energy flux. 

The stress tensor elements a are related to the strain field Sk in the material by the equations... 

For a given deformation or flow, the resulting stress depends on the material. However, the stress tensor does take particular general forms for experimentally used deformations (see section 2). The definitions apply to elastic solids, and viscoelastic liquids and solids. 

Note 2 If the strain tensor is diagonal for all time then the stress tensor is diagonal for all time for isotropic materials. 

The stress tensor describes the forces transmitted to an element of material through its contacts with adjacent elements (78). Traction is the force per unit area acting outwardly on the material adjacent to a material plane, and transmitted through its contact with material across the plane. If the components of traction are known for any set of three planes passing through a point, the traction across any plane through the point can be calculated. The stress at a material point is determined by an assembly erf nine components of traction, three for each plane. If the orientations of the three planes are chosen to be normal to the coordinate directions of a rectangular Cartesian coordinate system, the Cartesian components of the stress are obtained ... 

Unspecified isotropic pressure term in stress tensor p for incompressible materials. 

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. 

In this section, pedagogical models for the time dependence of mechanical response are developed. Elastic stress and strain are rank-two tensors, and the compliance (or stiffness) are rank-four material property tensors that connect them. In this section, a simple spring and dashpot analog is used to model the mechanical response of anelastic materials. Scalar forces in the spring and dashpot model become analogs for a more complex stress tensor in materials. To enforce this analogy, we use the terms stress and strain below, but we do not treat them as tensors. 

The (jjj expression indicated above shows that at the crack tip (r = 0), the stress tensor components become infinite. Actually, for a material able to undergo plastic deformation, above some stress level yielding occurs and limits the stress to the corresponding value, ay. Thus, around the crack tip a zone exists in which the material is plastically deformed. Such a zone is called the plastic zone, and it is represented in Fig. 8 in the case of a crack across a plate thickness. 

For isotropic materials, a general relationship correlating the three normal components of the stress tensor and strain is expressed as... 

To close the problem, constitutive relations of powders must be introduced for the internal connections of components of the stress tensor of solids and the linkage between the stresses and velocities of solids. It is assumed that the bulk solid material behaves as a Coulomb powder so that the isotropy condition and the Mohr-Coulomb yield condition may be used. In addition, og has to be formulated with respect to the other stress components. 

Molecular theories, utilizing physically reasonable but approximate molecular models, can be used to specify the stress tensor expressions in nonlinear viscoelastic constitutive equations for polymer melts. These theories, called kinetic theories of polymers, are, of course, much more complex than, say, the kinetic theory of gases. Nevertheless, like the latter, they simplify the complicated physical realities of the substances involved, and we use approximate cartoon representations of macromolecular dynamics to describe the real response of these substances. Because of the relative simplicity of the models, a number of response parameters have to be chosen by trial and error to represent the real response. Unfortunately, such parameters are material specific, and we are unable to predict or specify from them the specific values of the corresponding parameters of other... 

The quantities r and r] in equation (8.34) depend on the invariants of the tensor rik in accordance with equation (8.32). We ought to note that the behaviour of a non-linear viscoelastic liquid in a non-steady state would be different, if a dependence of the material parameters r and r] on the tensor velocity gradients or on the stress tensor is assumed. This is a point which is sometimes ignored. In any case, if r and r) are constant, equation (8.34) belongs to the class of equations introduced and investigated by Oldroyd (1950). 

Quantities C and C2 are functions of the two invariants of the stress tensor /1 and I2 for incompressible material. 

A slightly different set of invariants that corresponds to components of the stress tensor oriented to the material direction can be constructed from the above integrity basis. This new set of invariants includes... 

The constitutive equation for a dry powder is a governing equation for the stress tensor, t, in terms of the time derivative of the displacement in the material, e (= v == dK/dt). This displacement often changes the density of the material, as can be followed by the continuity equation. The constitutive equation is different for each packing density of the dry ceramic powder. As a result this complex relation between the stress tensor and density complicates substantially the equation of motion. In addition, little is known in detail about the nature of the constitutive equation for the three-dimensional case for dry powders. The normal stress-strain relationship and the shear stress-strain relationship are often experimentally measured for dry ceramic powders because there are no known equations for their prediction. All this does not mean that the area is without fundamentals. In this chapter, we will not use the approach which solves the equation of motion but we will use the friction between particles to determine the force acting on a mass of dry powder. With this analysis, we can determine the force required to keep the powder in motion. 

Conservation of Momentum. The law of conservation of momentum can be expressed for a fluid in tensor notation and in terms of material stresses and its velocity components as ( 2 )... 

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