Strain-rate invariants

Direction-free consequences or effects are, first, a rate of change of composition dXJdt and, second, strain effects the mean strain rate at a point (1/3 X rate of change of volume), and the rate of change of shape. These are represented by , and , the first and second strain-rate invariants. Familiar effects represented in Figure 19.5 are ... 

It is evident in the foregoing examples that deviations from linear viscoelastic behavior are evoked by both large strains and large strain rates. Phenomenological constitutive equations have been developed in which one or the other has a dominant role, as described for example by memory functions which depend either on strain invariants or on strain rate invariants. - " In critical comparisons of pre-... 

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. 

It is possible, however, to obtain generalized concentration-invariant curves under straining rates close to those used in the real-scale commercial process. In [163, 164, 209] the generalized curves for PE based composites were obtained by the procedure described in [340] by carrying out nonrotationa shifts in the vertical and horizontal directions the authors sought to achieve the closest coincidence between the experimental curves in the lg t] — lg x coordinates for the base polymer and the curves for filled composites. 

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. 

If the invariants are known for some arbitrary strain-rate state, then it is clear that the three equations above form a system of equations from which the principal strain rates can be uniquely determined. This analysis is explained more fully in Appendix A. Using the principal axes greatly facilitates subsequent analysis, wherein quantitative relationships are established between the strain-rate and stress tensors. 

The relative volumetric expansion is seen to be the sum of the normal strain rates, which is the divergence of the vector velocity field. The sum of the normal strain rates is also an invariant of the strain-rate tensor, Eq. 2.95. Therefore, as might be anticipated, the relative volumetric dilatation and V V are invariant to the orientation of the coordinate system. 

Recall from the discussion in Section 2.5.4 that the stress tensor, like the strain-rate tensor, has certain invariants. For any known stress tensor, these invariant relationships can be used to determine the principal stresses. 

Show that the diagonal invariant of the strain-rate tensor is preserved. 

The (stress or strain-rate) state at a point is a physical quantity that cannot depend on any particular coordinate-system representation. For example, the stress state is the same regardless of whether it is represented in cartesian or cylindrical coordinates. In other words, the state (as represented by a symmetric second-order tensor), is invariant to the particular coordinate-system representation. 

Initial moduli at room temperature were obtained with an Instron Model 4206 at a strain rate of 2/min ASTM D638 type V specimens were used. The Instron was also used in the creep experiments, in which deformation under a 1 NPa tensile load was continuously monitored for 10 sec, followed by measurement of the recovered length 48 h after load removal. Strain dependence of the elastic modulus was determined by deforming specimens to successively larger tensile strains and, at each strain level, measuring the stress after relaxation after it had become invariant for 30 min. 

This paper will describe (1) the load to strain relationships of the tissue during confined and unconflned compression against platens of different porosity (2) their strain rate dependency (3) some aspects of the tissue s load-dissipation characteristics and (4) the time invariant or asymptotic load to strain relationship of cartilage following relaxation. 

This was taken as the position where the surface of the cartilage was just making contact with the platen without being compressed. The speed setting for the motor drive was then adjusted to compress the tissue at a desired strain rate and the experiment was begun. At selected strains the motor drive was stopped and the dissipation of the load with time was followed until a time invariant value, characteristic for a given strain, was attained. The tissue was then compressed to a new strain and the procedure repeated. 

The time Invariant load-strain curves appear to be an inherent property of the tissue. The values were independent of the porosity of the platens and the strain rates that the tissue was compressed at. Up to about 30% compression the data could be fitted, by linear regression, to a straight line. As such, the curve represents the equilibrium load-strain relationship of the swollen matrix. It should be pointed out, however, that while the strains calculated for this and the other curves are probably close to the actual strains, they nevertheless are approximations, due to the assumptions of shape of the penetrating tissue and the possible slight compression of the tissue by the rod. 

During compression, the loads Increased at Increasing rates with strain until compression ceased. Thereafter a sharp and rapid dissipation of the load was observed. This decreased at decreasing rates with time until time invariant values, characteristic for a given strain, but Independent of strain rate, were attained. During unconflned compression the loads Increased to lesser extents for the same axial displacements, and relaxation was considerably slower. Against our most porous platen, the load-strain curves had little, if any, strain rate dependence. Plots of the mean loads at each strain against the porosities of the platens resulted in a family of linear curves that extrapolated to approximately the same porosity at low loads. 

Newton s law of motion for liquids describes a linear relationship between the deformation of a fluid and the corresponding stress, as indicated in Equation 22.16, where the constant of proportionality is the Newtonian viscosity of the fluid. The generalized Newtonian fluid (GNF) refers to a family of equations having the structure of Equation 22.16 but written in tensorial form, in which the term corresponding to viscosity can be written as a function of scalar invariants of the stress tensor (x) or the strain rate tensor (y). For the GNF, no elastic effects are taken into account [12, 33] ... 

Terms on the main diagonal are strain velocities, side elements represent angle rates of initially cuboid elements. The first invariant of D, sp(D), is equal to div v, the volumetric strain rate, which must be zero for incompressible flow. 

Von Mises stress is originally formulated to describe plastic response of ductile materials. It is also applicable for the analysis of plastic failure for coal undergoing high strain rate. The von Mises yield criterion suggests that the yielding of materials begins when the second deviatoric stress invariant J2 reaches a critical value. In materials science and engineering the von Mises yield criterion can be also formulated in terms of the von Mises stress or equivalent tensile stress, a scalar stress value that can be computed from the stress tensor ... 

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 first invariant represents rate of change of volume, which is zero for incompressible fluids. The third invariant IIIo is zero for plane flows. The second invariant IIo represents a mean rate of deformation including all shearing and extensional components. It is convenient to define, for all flows, a generalized strain rate as... 

Purely viscous constitutive equations, which account for some of the nonlinearity in shear but not for any of the history dependence, are commonly used in process models when the deformation is such that the history dependence is expected to be unimportant. The stress in an incompressible, purely viscous liquid is of the form given in equation 2, but the viscosity is a function of one or more invariant measures of the strength of the deformation rate tensor, [Vy - - (Vy) ]. [An invariant of a tensor is a quantity that has the same value regardless of the coordinate system that is used. The second invariant of the deformation rate tensor, often denoted IId, is a three-dimensional generalization of 2(dy/dy), where dy/dy is the strain rate in a one-dimensional shear flow, and so the viscosity is often taken to be a specific function-a power law, for example-of (illu). ]... 

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... 

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