Stress and Strain Tensors

Let C i be a bounded domain with a smooth boundary L, and n = (ni,n2,n3) be a unit outward normal vector to L. Introduce the stress and strain tensors of linear elasticity (see Section 1.1.1),... 

The stress and strain tensors aij u),Sij u) are defined by the Hooke and Cauchy laws... 

Another generalization uses referential (material) symmetric Piola-Kirchhoff stress and Green strain tensors in place of the stress and strain tensors used in the small deformation theory. These tensors have components relative to a fixed reference configuration, and the theory of Section 5.2 carries over intact when small deformation quantities are replaced by their referential counterparts. The referential formulation has the advantage that tensor components do not change with relative rotation between the coordinate frame and the material, and it is relatively easy to construct specific constitutive functions for specific materials, even when they are anisotropic. 

The deformation may be viewed as composed of a pure stretch followed by a rigid rotation. Stress and strain tensors may be defined whose components are referred to an intermediate stretched but unrotated spatial configuration. The referential formulation may be translated into an unrotated spatial description by using the equations relating the unrotated stress and strain tensors to their referential counterparts. Again, the unrotated spatial constitutive equations take a form similar to their referential and current spatial counterparts. The unrotated moduli and elastic limit functions depend on the stretch and exhibit so-called strain-induced hardening and anisotropy, but without the effects of rotation. 

Obviously, the number of free indices no longer denotes the order of the tensor. Also, the range on the indices no longer denotes the number of spatial dimensions, if the stress and strain tensors are symmetric (they are if no body couples act on an element), then... 

Equation (4) expresses G as a function of temperature and state of applied stress (pressure) (o. Pa), (/(a) is given by the force field for the set of lattice constants a, Vt is the unit cell volume at temperature T, and Oj and are the components of the stress and strain tensors, respectively (in Voigt notation). The equilibrium crystal structure at a specified temperature and stress is determined by minimizing G(r, a) with respect to die lattice parameters, atomic positions, and shell positions, and yields simultaneously the crystal structure and polarization of minimum free energy. 

Using relations (2.5) and (2.6) we can determine the elasticity tensor which describes the linear relation between components of the stress and strain tensors. 2 slr.ss = CEstta n is therefore an expression of Hooke s law for anisotropic crystals... 

This is a tensor of fourth order, and in the general case it should be described by a matrix of 81 members (9x9). Since the stress and strain tensors are symmetrical and each has six independent components, the tensor of fourth order derived from them has 6x6 components. 

When analysing the stress and strain tensors at various positions through the thickness of a plate, it turns out that ... 

The state of strain in a body is fully described by a second-rank tensor, a strain tensor , and the state of stress by a stress tensor, again of second rank. Therefore the relationships between the stress and strain tensors, i.e. the Young modulus or the compliance, are fourth-rank tensors. The relationship between the electric field and electric displacement, i.e. the permittivity, is a second-rank tensor. In general, a vector (formally regarded as a first-rank tensor) has three components, a second-rank tensor has nine components, a third-rank tensor has 27 components and a fourth-rank tensor has 81 components. 

Stress and strain tensors are not matter tensors like susceptibUity or conductivity, which were covered in earlier chapters. They do not represent a crystal property, but are, rather, forces imposed on the crystal, and the response to those forces, which can have any arbitrary direction or orientation. Although the magnitude and direction of strain are influenced by the crystal symmetry, they are also determined by the magnitude... 

BETWEEN THE STRESS AND STRAIN TENSORS IN IDEAL ELASTIC SYSTEMS... 

Here only noncrystalline symmetries, which are likely to play an important role in the linear viscoelastic behavior of materials, are considered. We follow Tschoegl s approach to this subject (5). Crystalline materials and their symmetries are described in many textbooks (6,7). In order to study how the symmetry of the system affects the number of independent components of Cijki, it is convenient to reduce the number of indices of both the stress and strain tensors. Following Voigt s formulation, the reduction is made by doing 11 -> 1, 22 2, 33 3, 12 -> 4, 23 -> 5, 13 6, so that... 

In this case the relationship between the stress and strain tensors in the reference frame of the coordinate axes x, X2, x is given by... 

By comparing Eqs. (4.54) and (4.61) and taking into account the invariance of the components of the stress and strain tensors in an operation of symmetry, one obtains... 

Consequently, the number of independent components of Cy i in a system with cylindrical S3munetry is six. In this case, the relation between the stress and strain tensors can be written as... 

Then the relation between the stress and strain tensors is given by the classical expression... 

If an elastic body is under the effect of a force parallel to one of the coordinate axes, for example along the X axis, the stress tensor has a single component. The stress and strain tensors can be written in the form... 

Let us analyze the response of a material when it is compressed or expands in a single direction, for example along the Xj axis, and deformations along the X2 and X3 axes are not permitted. This situation occurs when a material is compressed in a cylinder or when an acoustic wave propagates through the material. In these conditions, the stress and strain tensors can be written as... 

The results of the previous example as well other more complex models that are analyzed in the problems section at the end of the chapter show that the relationships between the different components of the stress and strain tensors for viscoelastic materials can be established in terms of the operators P and Q. For example (9), for an elongational test... 

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... 

The nondiagonal components of the stress and strain tensors are also null. For the thermoelastic case, the stress-strain relationship is... 

The decomposition in deviatoric and dilatational components of both the stress and strain tensors are... 

To avoid these mathematical details and focus on the key concepts of tablet stress analysis this discussion will examine the simplest of viscoelastic models using the method outlined by Fluggie (97). To begin the analysis, the boundary conditions which apply to tablet compaction, will be used to set up the stress and strain tensors Equations (26) and (27). Then the dilation and distortion uations (28-31) will be used lo obtain dilation and distortion tensors. After obtaining the dilational and distortional stress and strain tensors, a Kelvin viscoelastic model will be used to relate the distortional stress to distortional strain and the dilational stress to dilational strain. 

The relaxation equations are calculated in a similar manner to the unloading equations, except that during the relaxation phase the punch stress is zero. When the punch stresses are zero, stress and strain tensors become ... 

In order to solve Cauchy s equation of motion, which is valid for any substance, a further relationship between the stress and strain tensors, or between the stress... 

The stress and strain tensors are connected by the Hooke equations. In the crystallite reference system these are the following ... 

The funchonal h) = [(Tijeij z=h> where Oij and Cij are stress and strain tensors, respectively. They should be found from the soluhon of the corresponding elashcity problem that would describe elashc equihbrium of the him... 

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 ... 

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