Relative strain tensor

Integral-type constitutive equations may also be written in terms of strain tensor (or relative strain tensor), instead of rate-of-strain tensor. For infinitesimally small deformations, if the response of a system can be expressed by the linear superposition of a series of separate responses at different times to a series of step changes in the input, the stress tensor a as a response can be expressed in terms of the infinitesimally small strain tensor e by... 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

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 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. 

Consequently, E has components relative to the reference configuration, and is a referential strain tensor. A complementary strain tensor may be defined from the inverse deformation gradient F ... 

Consequently, e has components relative to the current configuration and is a spatial strain tensor. If (A.18) is premultiplied by F and postmultiplied by F, it is seen from (A. 17) that... 

Experience with applying the Reynolds-stress model (RSM) to complex flows has shown that the most critical term in (4.52) to model precisely is the anisotropic rate-of-strain tensor 7 .--1 (Pope 2000). Relatively simple models are thus usually employed for the other unclosed terms. For example, the dissipation term is often assumed to be isotropic ... 

Note 1 A strain tensor is a measure of the relative displacement of the mass points of a body. 

Here X is an important parameter which determine whether the time evolution of n is dominated by the strain tensor A or by the vorticity tensor fi (cf. Eq. (42)). If X is larger than unity, the former dominates and n tends to assume a steady-state angle 0 relative to the flow direction which is determined by the equation... 

Obviously, when 0 = t, no deformation with respect to the reference configurations has taken place, and Ffxk, t t) = bg, where 6, is the Kronecker delta. The actual strain of the relative deformation tensor is better expressed by the symmetrical tensor... 

Broadly speaking, our description of continuum mechanics will be divided along a few traditional lines. First, we will consider the kinematic description of deformation without searching for the attributes of the forces that lead to a particular state of deformation. Here it will be shown that the displacement fields themselves do not cast a fine enough net to sufficiently distinguish between rigid body motions, which are often of little interest, and the more important relative motions that result in internal stresses. These observations call for the introduction of other kinematic measures of deformation such as the various strain tensors. Once we have settled these kinematic preliminaries, we turn to the analysis of the forces in continua that lead to such deformation, and culminate in the Cauchy stress principle. 

The strain tensor provides a more suitable geometric measure of relative displacements, and in the present context illustrates that the length change between neighboring material points is given by... 

There are two proper explanations, one based on physical intuition and the other based on the principle of material objectivity. The latter is discussed in many books on continuum mechanics.19 Here, we content ourselves with the intuitive physical explanation. The basis of this is that contributions to the deviatoric stress cannot arise from rigid-body motions -whether solid-body translation or rotation. Only if adjacent fluid elements are in relative (nonrigid-body) motion can random molecular motions lead to a net transport of momentum. We shall see in the next paragraph that the rate-of-strain tensor relates to the rate of change of the length of a line element connecting two material points of the fluid (that is, to relative displacements of the material points), whereas the antisymmetric part of Vu, known as the vorticity tensor 12, is related to its rate of (rigid-body) rotation. Thus it follows that t must depend explicitly on E, but not on 12 ... 

Thus the rate of change of the distance between two neighboring material points depends on only the rate-of-strain tensor E, i.e., on the symmetric part of Vu. It can be shown in a similar manner that the contribution to the relative velocity vector Su that is due to the... 

Laminar mixing depends on the strain tensor, which can be visualized as an ellipsoid formed upon straining a sphere. The strain magnitude is proportional to the relative size of the ellipsoidal axes while their relative positions to the orientational effects of the imposed flow. It can be shown that the interfacial area, A, will grow with the imposition of strain according to the relation [Erwin, 1991] ... 

In general, we measure the homogeneous strain of a solid by the relative displacement of two points and P2 separated by the vector r, keeping the coordinate system invariant (Fig. 4.9). The strain displaces the point Piix ) to the point P Xi + < j) and the point P2( i + ) to P 2 Xi H- H- u-). The vector r H- u gives the relative position of the two points of the strained solid. By analogy with equation (4.34) and (4.35), the strain tensor eexpresses the displacement u per unit... 

Here B is the standard Finger strain tensor used in continuum mechamcs, and Yfo] = 8 — B IS a relative finite strain tensor, defined in DPL, Eq. D.3-4. We note in passing that it follows from Eq. (13.5) that the quantity HfkT) QQ P, dQ is equal to oi and thus satisfies Eq. (13.6). 

Aki K, Richards PG (1980) Quantitative seismology Theory and methods. Vol. I, WH Freeman and Compaity, San Francisco Dahm T (1996) Relative moment tensor inversion based on ray theory Theory and Synthetic Tests. Geophys. J Int., 124 245-257 Dai ST, Labuz JF, Carvalho F (2000) Softening response of rock observed in plane-strain compression. Trends in Rock Mechanics, Geo SP-102, ASCE, pp 152-163... 

Here is a generalized (6N dimensional) force-torque vector, U -u (6N dimensional) is the particle translational-angular velocity relative to the bulk fluid flow evaluated at the particle centre, (3x3 matrix) is the traceless symmetric rate of the strain tensor (supposed to be constant in space). The resistance matrices Rfu (6N x6N) and Rfe (6N x 3 x 3) which depend only on the instantaneous relative particle configurations (position and orientation) relate the force-torque exerted by the suspending fluid on the particles to their motion relative to the fluid and to the imposed shear flow, respectively. Note that in ER (MR) fluids torques can be neglected. 

It is possible to gain a rather direct insight into the meaning of the diagonal coordinates Vn,V22 and F33 of the Lagrange strain tensor by a suitable choice of the material line element under consideration. Let us take first N = 1, A 2 = 0, N2 = 0. Then the relative change in length of the material line element becomes... 

The diagonal elements of the strain tensor become iderrtical with the corresponding relative change in length. 

If we consider the same deformation in a coordinate system Xi> that is rotated by 45° relative to the Xi system, the coordinate transformation results in the following strain tensor ... 

In simple extension the difference between the stress in the elongation direction and that in the direction perpendicular to the elongation direction is measured. For conciseness this stress difference will be denoted by Cg. According to Eq. (2) Og is determined by the difference between the 11-and 22-components of the strain tensor, which will be denoted by Sg. represents a tensorial strain measure determined by the first and second invariants I Ct ) and Il Ct ) of the relative Finger strain tensor. In the case of uniaxial extension these invariants can always be expressed in the ratio of the stretch ratios at times t and t , so that [ (t)/... 

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