Strain tensor invariants

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

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. 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

From Eq. E2.5-17 we can calculate the total viscous dissipation between the parallel plates. The second invariant of the rate of strain tensor multiplied by the viscosity gives the viscous dissipation per unit volume. From Table 2.3 we find that, for the case at hand, the second invariant reduces to y2z therefore, the total viscous energy dissipation (VED) between the plates will be given by... 

The Invariants of the Rate of Strain Tensor in Simple Shear and Simple Elogational Flows Calculate the invariants of a simple shear flow and elonga-tional flow. 

Finally, a number of commonly used constitutive equations are derived from Eq. 3.3-13 by specifying G1, G2,... instead of specifying only G1 and settingG2,... equal to zero. Moreover, in these equations, M, are allowed to be functions of the invariants of the strain or rate-of-strain tensors, since there is experimental evidence supporting this dependence (35). Examples of such usable integral co-deformational constitutive equations are ... 

GNF-based constitutive equations differ in the specific form that the shear rate dependence of the viscosity, t](y), is expressed, but they all share the requirement that the non-Newtonian viscosity t](y) be a function of only the three scalar invariants of the rate of strain tensor. Since polymer melts are essentially incompressible, the first invariant, Iy = 0, and for steady shear flows since v = /(x2), and v2 V j 0 the third invariant,... 

The free energy of the deformed isotropic body depends on the strain tensor, it is a function of three invariants of the strain tensor. The volume of the deformed body... 

The derivatives of the invariants with respect to the components of the strain tensor can easily be calculated... 

Since W is a scalar value, it depends on scalar characteristics of the strain tensors, namely on the invariants li and I2 defined as ... 

It is important to use the exact strain tensor definition, Eq. (6), to achieve rotational invariance with respect to lattice rotation the conventional linear strain tensor only provides differential rotational invariance of u in Eq. (7).hierarchy of approximations may be used for the elastic tensor 7. The most rigorous approach is to transform the bulk elastic tensor c according to... 

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... 

The shear rate is often calculated as the second invariant (the first invariant is the trace) of the rate-of-strain tensor ... 

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... 

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 ... 

The simulated dilatations involved increasing steps of imposed dilatation on the simulation cell. To permit a detailed understanding of the dilatational response of the polymer at the atomic level the entire volume of the simulation cell was tessellated into Voronoi polyhedra at each atomic site, permitting determination of strain-increment tensor elements dcy for each site from local displacement gradients by a technique described by Mott et al. (1992). Such increments of imposed dilatation at a level of 3 x 10 were applied 100 times to obtain total system dilatations of 0.3 (Mott et al. 1993b). For eaeh dilatation increment the atomic site strain-tensor increments de were obtained for each site n. The two invariants, de", the atomic site dilatation increment, and the work-equivalent shear-strain increment, dy", were obtained from the individual increments as... 

Definition (7) has been invoked within the parentheses that extend the right hand side of equation (6). Divergence represents the trace (first invariant) of the transient total strain tensor of the porous granular structure. It signifies a change in volume of this saturated porous structure. This contrasts to the deviatoric component of the total structural strain tensor, which signifies a change in shape. 

The initial considerations concerned tensile and compressive strength and these were developed later into more complicated strength criteria. Their main objective was to analytically determine how the materials fracture in various loading situations and using different strain or stress tensor components. Because of the general conditions imposed on such criteria they should be expressed by tensor invariants, and satisfy conditions of symmetry, etc. 

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)/... 

Any second-order tensor has a number of invariants associated with it. One such is the trace of the tensor, equal to the sum of its diagonal terms, applicable to any strain tensor. We define the first invariant h as the trace of the Cauchy-Green strain measure tr(C) ... 

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