Strain incompressibility

Since the yield function is independent of p, the yield surface reduces to a cylinder in principal stress space with axis normal to the 11 plane. If the work assumption is made, then the normality condition (5.80) implies that the plastic strain rate is normal to the yield surface and parallel to the II plane, and must therefore be a deviator k = k , k = 0. It follows that the plastic strain is incompressible and the volume change is entirely elastic. Assuming that the plastic strain is initially zero, the spherical part of the stress relation (5.85) becomes... 

For a rectangular rubber block, plane strain conditions were imposed in the width direction and the rubber was assumed to be an incompressible elastic solid obeying the simplest nonhnear constitutive relation (neo-Hookean). Hence, the elastic properties could be described by only one elastic constant, the shear modulus jx. The shear stress t 2 is then linearly related to the amount of shear y [1,2] ... 

We conclude that high internal stresses are generated by simple shear of a long incompressible rectangular rubber block, if the end surfaces are stress-free. These internal stresses are due to restraints at the bonded plates. One consequence is that a high hydrostatic tension may be set up in the interior of the sheared block. For example, at an imposed shear strain of 3, the negative pressure in the interior is predicted to be about three times the shear modulus p. This is sufficiently high to cause internal fracture in a soft rubbery solid [5]. 

Systematic measurements of stress and strain can be made and the results plotted as a rheogram. If our material behaves in a simple manner - and it is surprising how many materials do, especially if the strains (or stresses) are not too large - we find a linear dependence of stress on strain and we say our material obeys Hooke s law, i. e. our material is Hookean. This statement implies that the material is isotropic and that the pressure in the material is uniform. This latter point will not worry us if our material is incompressible but can be important if this is not the case. 

In order to proceed with the evaluation of the time-dependent Poisson ratio v(0, both sets of relaxation behaviour are required. Now from Chapter 2 we know the Poisson ratio is the ratio of the contractile to the tensile strain and that for an incompressible fluid the Poisson ratio v = 0.5. Suppose we were able to apply a step deformation as we did for a shear stress relaxation experiment. The derivation then follows the same course as that to Equation (4.69) ... 

Reduced dimensional parameters (strain parameters and near-neighbours diagrams) By comparing the space-filling theoretical curves and the actual values of intermetallic phases it has been observed that an incompressible sphere model of the atom gives only a rough description when discussing metallic structures. 

Note 3 The Finger strain tensor for a homogeneous orthogonal deformation or flow of incompressible, viscoelastic liquid or solid is... 

Equation relating stress and deformation in an incompressible viscoelastic liquid or solid. Note 1 A possible general form of constitutive equation when there is no dependence of stress on amount of strain is... 

Note 3 For elastomers, which are assumed incompressible, the modulus is often evaluated in uniaxial tensile or compressive deformation using X - as the strain function (where X is the uniaxial deformation ratio). In the limit of zero deformation the shear modulus is evaluated as... 

The stress-strain relations for some special cases of biaxial defonnation are derived from Eqs. (13) to (15) in the following way. Strip biaxial extension of incompressible material is defined as the mode of deformation in which one of the Xj, say X2, is kept at unity, while the other, Xt, varies. This deformation is also called pure shear . We have for it ... 

For incompressible material, the stress-strain relations for biaxial extension are given by Eqs. (13) and (14), which may be solved for bW/bli and bWjbI2 to give... 

The pressure P0 represents the arbitrary additive contribution to the normal components of stress in an incompressible system, 8i is the Kronecker delta, C[ j 1(t t) is the inverse of the Cauchy-Green strain tensor for the configuration of material at t with respect to the configuration at the current time t [a description of the motion (221)], and M(t) is the junction age distribution or memory function of the fluid. 

Incoherent Clusters. As described in Section B.l, for incoherent interfaces all of the lattice registry characteristic of the reference structure (usually taken as the crystal structure of the matrix in the case of phase transformations) is absent and the interface s core structure consists of all bad material. It is generally assumed that any shear stresses applied across such an interface can then be quickly relaxed by interface sliding (see Section 16.2) and that such an interface can therefore sustain only normal stresses. Material inside an enclosed, truly incoherent inclusion therefore behaves like a fluid under hydrostatic pressure. Nabarro used isotropic elasticity to find the elastic strain energy of an incoherent inclusion as a function of its shape [8]. The transformation strain was taken to be purely, dilational, the particle was assumed incompressible, and the shape was generalized to that of an... 

Rather surprisingly, all these kinds of deformation can be described in terms of a single modulus. This is a result of the assumption that rubber is virtually incompressible (i.e. bulk modulus much greater than shear modulus). Young s modulus E = 3G (for fdled rubbers the numerical factor may be in fact as high as 4). Indeed, these relationships by no means fully describe the complete stress strain behaviour of real rubbers but may be taken as first approximations. The shear stress relationship is usually good up to strains of 0.4 and the tension relationship approximately true up to 50% extension. 

Abstract A general theoretical and finite element model (FEM) for soft tissue structures is described including arbitrary constitutive laws based upon a continuum view of the material as a mixture or porous medium saturated by an incompressible fluid and containing charged mobile species. Example problems demonstrate coupled electro-mechano-chemical transport and deformations in FEMs of layered materials subjected to mechanical, electrical and chemical loading while undergoing small or large strains. 

Soft biological structures exhibit finite strains and nonlinear anisotropic material response. The hydrated tissue can be viewed as a fluid-saturated porous medium or a continuum mixture of incompressible solid (s), mobile incompressible fluid (f), and three (or an arbitrary number) mobile charged species a, (3 = p,m, b). A mixed Electro-Mechano-Chemical-Porous-Media-Transport or EMCPMT theory (previously denoted as the LMPHETS theory) is presented with (a) primary fields (continuous at material interfaces) displacements, Ui and generalized potentials, ifi ( , r/ = /, e, to, b) and (b) secondary fields (discontinuous) pore fluid pressure, pf electrical potential, /7e and species concentration (molarity), ca = dna/dVf or apparent concentration, ca = nca and c = Jnca = dna/dVo. The porosity, n = 1 — J-1(l — no) and no = no(Xi) = dVj/dVo for a fluid-saturated solid. Fixed charge density (FCD) in the solid is defined as cF = dnF/dV , cF = ncF, and cF = cF (Xf = JncF = dnF/d o. 

We should note that the Navier-Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity. 

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,... 

Example 7.3 Effect of Viscosity Ratio on Shear Strain in Parallel-Plate Geometry Consider a two-parallel plate flow in which a minor component of viscosity /t2is sandwiched between two layers of major component of viscosities /q and m (Fig. E7.3). We assume that the liquids are incompressible, Newtonian, and immiscible. The equation of motion for steady state, using the common simplifying assumption of negligible interfacial tension, indicates a constant shear stress throughout the system. Thus, we have... 

To determine the shear modulus and the tensor of recoverable strains, we calculate the determinants of the left-hand and right-hand sides of equation (9.74). Taking into account the incompressibility of the polymer liquid, i.e. relation AyAjy = 1, we obtain... 

Fig. 1. Schematic of a family of two-dimensional steady incompressible shear flows showing the streamline patterns at the top and the corresponding velocity components at the bottom. By varying X continuously from — 1 to +1, the flow can be varied from pure rotation (without strain) to pure strain (without rotation).

The described principle of equal force (stress) and added deformations (strains) equally applies to parallel layers of any kind, provided that their structure is isotropic. However, if any of the layers in the array is incompressible and softer than the rest, then it will expand laterally upon the force application. This is a familiar experience. When a sandwich or a layered cake is compressed, the filling sometimes leaks out from the sides, as shovm schematically in Figure 10.10. For such a situation. Equations (10.7) or (10.8) will not be an appropriate model. However, because the cellular layers retain their cross-sectional area, and because the free p>art of the expanded filling does not transmit any stress (theoretically), the stress-strain relationship of the array can still be calculated by accounting for the exuded material. 

Oakenflill et al. (1989) presented a method for determining the absolute shear modulus (E) of gels from compression tests in which the force, F, the strain or relative deformation (S/L) are measured with a cylindrical plunger with radius r, on samples in cylindrical containers of radius R, as illustrated in Figure 3-47. Assuming that the gel is an incompressible elastic solid, the following relationships were derived ... 

The complex relationship between the configurational distortion produced by a perturbation field in polymers and the Brownian motion that relaxes that distortion make it difficult to establish stress-strain relationships. In fact, the stress at a point in the system depends not only on the actual deformation at that point but also on the previous history of deformation of the material. As a consequence the relaxation between the stress and strain or rate of strain cannot be expressed by material constants such as G or /, as occurs in ideal elastic materials, but rather by time-dependent material functions, G t) and J t). It has been argued that the dynamics of incompressible liquids may be characterized by a function of the evolution of the strain tensor from the beginning up to the present time. According to this criterion, the stress tensor would be given by (3,4)... 

In formal rheology, relations between these three tensors are formulated and analyzed. Only for the two extremes of viscoelastic behaviour are such relations simple. For purely elastic materials there is a relation between the stress tensor and the strain tensor it contains the elasticity modulus and the Poisson ratio, accounting for the extent to which extension in one direction is accompamied by concomitant compression in the other two. For purely viscous fluids there is a relation between the stress tensor and the strain rate tensor. As extension in one direction is concomitant with (viscous) compression in the other two, in this case only one viscosity is required. For incompressible Newton fluids eventually an expression with only one viscosity results, see (1.6.1.131. 

Each of these different types of flows is governed by a set of equations having special features. It is essential to understand these features to select an appropriate numerical method for each of these types of equations. It must be remembered that the results of the CFD simulations can only be as good as the underlying mathematical model. Navier-Stokes equations rigorously represent the behavior of an incompressible Newtonian fluid as long as the continuum assumption is valid. As the complexity increases (such as turbulence or the existence of additional phases), the number of phenomena in a flow problem and the possible number of interactions between them increases at least quadratically. Each of these interactions needs to be represented and resolved numerically, which may put strain on (or may exceed) the available computational resources. One way to deal with the resolution limits and... 

Thus the objective here is a generally applicable simulation of steady, two-dimensional, incompressible flow between rigid rolls with film splitting. The results reported are solutions of the full Navier-Stokes system including the physically required boundary conditions. The analysis is also extended to a shearthinning fluid. The solutions consist of velocity and pressure fields, free surface position and shape, and the sensitivities of these variables to parameter variations, valuable information not readily available from the conventional approach (10). The rate-of-strain, vorticity, and stress fields are also available from the solutions reported here although they are not portrayed. Moreover, the stability of the flow states represented by the solutions can also be found by additional finite element techniques (11), and the results of doing so will be reported in the future. 

During the ignition phase, as the pressure increases, the propellant is loaded by hydrostatic pressure imposed on a biaxial tensile stress field. Because the propellant is incompressible in the ignition condition, the pressure is transmitted entirely to the case, which, being thin because of the weight requirement, presents significant hoop deformations. Therefore, a tensile strain... 

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