Shear strain invariants

In this case, the shear stress is linear in the shear strain. While more physically reasonable, this is not likely to provide a satisfactory representation for the large deformation shear response of many materials either, since most materials may be expected to stiffen with deformation. Note that the hypoelastic equation of grade zero (5.117) is not invariant to the choice of indifferent stress rate, the predicted response for simple shear depending on the choice which is made. 

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. 

The mathematical form of the function can be derived simply from a fit of the experimental h(y) as obtained in step shear strain for example. However, the problem is further complicated if one now takes into account flows where the two invariants differ from each other as, for example, in uniaxial elongational flows where ... 

FIGURE 6.4 Stress relaxation of synthetic 1,4-polyisoprene at the indicated shear strains. The parallel nature of the curves reflects the invariance of the time-dependence to strain (Fuller, 1988). 

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... 

The symbol I represents the strain invariants analogous to the stress invariants given as J in Eqs. (1.22e) and (1.23). The coefficients in Eq. (1.98c) are the results of the engineering shear strain being ... 

Most polymer processes are dominated by the shear strain rate. Consequently, the viscosity used to characterize the fluid is based on shear deformation measurement devices. The rheological models that are used for these types of flows are usually termed Generalized Newtonian Fluids (GNF). In a GNF model, the stress in a fluid is dependent on the second invariant of the stain rate tensor, which is approximated by the shear rate in most shear dominated flows. The temperature dependence of GNF fluids is generally included in the coefficients of the viscosity model. Various models are currently being used to represent the temperature and strain rate dependence of the viscosity. 

The most important flow process in polymer liquids is shear flow. Polymer liquids differ from simple liquids, first in that the shear viscosity is invariably extremely large, and second in that Newton s empirical equation giving a linear relationship between shear stress r and shear strain rate y with constant shear viscosity ft... 

According to more complicated criteria, for example, Huber-von Mises-Hencky, the energy of shearing strain is treated as representing the behaviour of the material and is expressed as combinations of the first invariants of the stress tensor. There are many other criteria that may be considered as more or less similar to those mentioned here. For a more detailed description of the strength criteria the reader is referred to books by Timoshenko (1953) and Fung (1965), among others. 

Energy minimization is a powerful tool for establishing global deformation characteristics of a strained film-substrate system on the basis of a particular set of assumptions and approximations, as demonstrated above. Essentially, the same assumptions and approximations can be used as a basis for a local equilibrium approach to arrive at the same results. Indeed, by exploiting the features of translational invariance and symmetry, it is revealed that some of the assumptions on which the energy analysis is based are known a priori to be true. These include the vanishing of the normal stress component Uzz, the shear stress component <7re and the shear strain components e z and egz- This approach was adopted by Freund (1993) and the main features are sketched out here for future reference. 

The strain-invariant polymer failure model (SIFT) permits the superposition of separate analyses for shear and induced peel loads. There is no interaction between the two failure mechanisms. While the need for this has been minimized through sound design practice (gentle tapering of the ends of the adherends to minimize the induced peel stresses, as explained later), the new model puts this technique on a secure scientific foundation and also accommodates any applied transverse shear loads. 

At the microlevel, initial damage in bonded joints is defined by whichever of the two strain invariants reaches its critical value first. At the ends of the overlap in a badly designed bonded joint or test coupon, the dilatational Ji failure mechanism would always occur first. There are no significant peel stresses in the interior of the joint, so the distortional invariant will be exceeded first. The visual consequences of this failure may seem to be one of dilatation because the first indication is the formation of a series of hackles formed at roughly 45° from the adherends being bonded together, as shown in Fig. 44.3. (Exactly the same model applies for in-plane-shear failures in the matrix between parallel fibers in composite laminates.) These are, in reality, failures by distortion, which result in tensile fractures perpendicular to the highest principal stress. 

Where Aj, A2, and are strains expressed as the elongational ratios in the direction of the principal axes. This equation assumes that u(Aj), u(A2), and u(A ) may in general take different values this means the coordinate axes are fixed and are not free to be chosen. This is more like the usual shear and elongational measurements where the magnitude of strain is tied to a specific coordinate system. The concept of strain invariance is not used. 

Stress and Strain Rate The stress and strain-rate state of a fluid at a point are represented by tensors T and E. These tensors are composed of nine (six independent) quantities that depend on the velocity field. The strain rate describes how a fluid element deforms (i.e., dilates and shears) as a function of the local velocity field. The stress and strain-rate tensors are usually represented in some coordinate system, although the stress and strain-rate states are invariant to the coordinate-system representation. 

An invariant-plane strain consists of a simple shear on a plane, plus a normal strain perpendicular to the plane of shear (see Section 24.1 and Fig. 24.1). This is a combination of Cases 2 and 3. The expression for Ags then follows directly from Eqs. 19.26 and 19.27, with the result that Age is proportional to c/a. Age is therefore minimized for a disc-shaped inclusion lying in the plane of shear. 

The term invariant-plane strain comes from the fact that the plane of shear in an invariant plane strain is both undistorted and unrotated. Hence the plane of shear is a plane of exact matching of the coherent inclusion and the matrix. In martensitic transformations, this matching is met closely on a macroscopic but not a microscopic scale (see Section 24.3). 

Martensitic transformations involve a shape deformation that is an invariant-plane strain (simple shear plus a strain normal to the plane of shear). The elastic coherency-strain energy associated with the shape change is often minimized if the martensite forms as thin plates lying in the plane of shear. Such a morphology can be approximated by an oblate spheroid with semiaxes (r, r, c), with r c. The volume V and surface area S for an oblate spheroid are given by the relations... 

However, important differences exist. Martensite and its parent phase are different phases possessing different crystal structures and densities, whereas a twin and its parent are of the same phase and differ only in their crystal orientation. The macroscopic shape changes induced by a martensitic transformation and twinning differ as shown in Fig. 24.1. In twinning, there is no volume change and the shape change (or deformation) consists of a shear parallel to the twin plane. This deformation is classified as an invariant plane strain since the twin plane is neither distorted nor rotated and is therefore an invariant plane of the deformation. 

In 77) the authors give dependencies of the maximum Newtonian viscosity upon amplitude of periodic strain velocity q0 = f(e) for polyethylene and polystyrene. It has been also revealed that the dependency of normalized viscosity upon the velocity of stationary shear T /r 0 = f(y q0) obtained at r. = 0 coincides with a similar dependency when acoustic treatment is employed, i.e., at e 0. In other words, the effect of shear vibrations and velocity of stationary shear upon valuer] can be divided, representing the role of the first factor in form of dependency q0(s0) and that of the second in form of dependency (n/q0) upon (y r 0) invariant in relation to e. 

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. 

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,... 

We consider these invariants independent of each other, so that we termine the shear modulus and the tensor of recoverable strains... 

Fig. 2.14. Energy as a function of shear for a generic crystalline solid. The periodicity of the energy profile reflects the existence of certain lattice invariant shears which correspond to the stress-free strains considered in the method of eigenstrains (adapted from Tadmor et al. (1996)).

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

---