Deformation incompressible

That is, the determinant of E is unity. For a uniaxial extensional deformation, incompress-ibility therefore implies that A2 = A3 = A,j, where Ai is the stretch ratio in the direction of elongation. 

Demonstrate that the true stress in uniaxially deformed incompressible networks is the derivative of the free energy per unit volume FjV with respect of the logarithm of deformation A ... 

Vorticity The relative motion between two points in a fluid can be decomposed into three components rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described bv a tensor (Oy = dvj/dxj — dvj/dxi. The vector of vorticity given by one-half the... 

This chapter is a brief diseussion of large deformation wave codes for multiple material problems and their applications. There are numerous other reviews that should be studied [7], [8]. There are reviews on transient dynamics codes for modeling gas flow over an airfoil, incompressible flow, electromagnetism, shock modeling in a single fluid, and other types of transient problems not addressed in this chapter. 

When the cake structure is composed of particles that are readily deformed or become rearranged under pressure, the resulting cake is characterized as being compressible. Those that are not readily deformed are referred to as sem-compressible, and those that deform only slightly are considered incompressible. Porosity (defined as the ratio of pore volume to the volume of cake) does not decrease with increasing pressure drop. The porosity of a compressible cake decreases under pressure, and its hydraulic resistance to the flow of the liquid phase increases with an increase in the pressure differential across the filter media. 

A DEA is basically a compliant capacitor where an incompressible, yet highly deformable, dielectric elastomeric material is sandwiched between two complaint electrodes. The electrodes are designed to be able to comply with the deformations of the elastomer and are generally made of a conducting material such as a colloidal carbon in a polymer binder, graphite spray, thickened electrolyte solution, etc. Dielectric elastomer films can be fabricated by conventional... 

Any rheometric technique involves the simultaneous assessment of force, and deformation and/or rate as a function of temperature. Through the appropriate rheometrical equations, such basic measurements are converted into quantities of rheological interest, for instance, shear or extensional stress and rate in isothermal condition. The rheometrical equations are established by considering the test geometry and type of flow involved, with respect to several hypotheses dealing with the nature of the fluid and the boundary conditions the fluid is generally assumed to be homogeneous and incompressible, and ideal boundaries are considered, for instance, no wall slip. 

The foregoing models considered incompressible bodies however, this is never the case in practice. The following section discusses models that specifically consider contact between deformable solids. 

Numerous 2-D models have been developed to simulate droplet deformation processes during impact on a smooth surface. Most of these models assumed axi symmetric deformation of a spherical or cylindrical droplet. The models may be conveniently divided into two groups, i.e., compressible and incompressible. 

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) ... 

The constant Tr is called the Trouton ratio10 and has a value of 3 in this experiment with an incompressible fluid in the linear viscoelastic limit. The elongational behaviour of fluids is probably the most significant of the non-shear parameters, because many complex fluids in practical applications are forced to extend and deform. Studying this parameter is an area of great interest for theoreticians and experimentalists. 

Note 3 For an isotropic, incompressible material, // = 0.5. It should be noted that, in materials referred to as incompressible, volume changes do in fact occur in deformation, but they may be neglected. 

Note 3 Deformations and flows used in conventional measurements of properties of viscoelastic liquids and solids are usually interpreted assuming incompressibility. 

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

Note 5 For small deformations of an incompressible, inelastic solid, the constitutive equation may be written... 

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... 

Consider the vacuum forming of a polymer sheet into a conical mold as shown in Figure 7.84. We want to derive an expression for the thickness distribution of the final, conical-shaped product. The sheet has an initial uniform thickness of ho and is isothermal. It is assumed that the polymer is incompressible, and it deforms as an elastic solid (rather than a viscous liquid as in previous analyses) the free bubble is uniform in thickness and has a spherical shape the free bubble remains isothermal, but the sheet solidifies upon confacf wifh fhe mold wall fhere is no slip on fhe walls, and fhe bubble fhickness is very small compared fo ifs size. The presenf analysis holds for fhermoforming processes when fhe free bubble is less than hemispherical, since beyond this point the thickness cannot be assumed as constant. 

A unique and considerably more elaborate multiaxial test employs a thick-walled hollow sphere test specimen which may be pressurized internally or externally with a nearly incompressible liquid. Figure 20 illustrates the essential features of the test device as described by Bennet and Anderson (5). The specimen is prepared by casting propellant in a mold fitted with a sand-poly (vinyl alcohol) mandrel inside the sphere which may be removed easily after curing. A constant displacement rate instrument drives the piston to pressurize the chamber and apply large deformations. The piston s total displacement volume is transferred to... 

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 ... 

This section summarizes results of the phenomenological theory of viscoelasticity as they apply to homogeneous polymer liquids. The theory of incompressible simple fluids (76, 77) is based on a very general set of ideas about the nature of mechanical response. According to this theory the flow-induced stress at any point in a substance at time t depends only on the deformations experienced by material in an arbitrarily small neighborhood of that point in all times prior to t. The relationship between stress at the current time and deformation history is the constitutive equation for the substance. 

Finally, tensile deformations provide the same information as shear deformation as long as the incompressibility assumption is not violated. In this case, the tensile stress relaxation modulus E(t) is directly related to the shear modulus E(t) = 3G(f), and all other relationships follow accordingly. 

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