Finite elasticity

Knowles, J.K., 1979, On the dissipation associated with equilibrium shocks in finite elasticity, J.Elast. 9 131. 

Rivlin, R. S. Finite elastic deformation. Text for the lecture given at California Institute of Technology, Pasadena, California 1953. 

So far we have assumed that the interacting surfaces are not deformable. In reality all solids have a finite elasticity. They deform upon contact. This has important consequences for the aggregation behaviour and the adhesion of particles because the contact area is larger than one would expect from infinitely hard particles. 

For a viscoelastic solid (like an organogel), any rheological description should give a constant finite elastic modulus and infinite viscosity at zero frequency or long times. The situation is somewhat comparable to that of a cross-linked network [2. The equilibrium shear modulus for small deformations is proportional... 

The rheological properties of a dense emulsion with close-packed droplets depends on whether or not the droplets are small enough to be agitated significantly by Brownian motion. If not, because of the high packing density of the droplets, the emulsions should be elastic and have a finite elastic modulus at low frequencies. For liquids with viscosities near that of water, Brownian behavior should dominate for particle radii less than or equal to 1 fim, while non-Brownian behavior occurs when a > 10 m. 

Elastic bonding must exist At p > pc, the lattice must have finite elastic macromodulus becoming zero at p — pc + 0. 

The Tensile Stress Field, The main tool used in industry to assess the structural integrity of the propellant grain is a commercially available finite-element code that solves problems of finite elastic deformations. From numerical investigation performed with the code, one of the maximum strain locations (the critical point) is identified. This point (M) is located at the surface of the combustion chamber near the middle of the symmetry axis. 

Note that viscoelasticity always entails viscoelastic dispersion in the sense that T] and r]" are themselves a fimction of frequency [72]. The scaUng therefore no longer holds. Contrary to intuition, a finite elastic component increases the bandwidth more than it increases the negative frequency shift. An ideally elastic mediiun leads to A/ = 0 and to a nonzero AT, because energy is withdrawn from the crystal in the form of elastic waves. 

Many surfactant solutions show dynamic surface tension behavior. That is, some time is required to establish the equilibrium surface tension. If the surface area of the solution is suddenly increased or decreased (locally), then the adsorbed surfactant layer at the interface would require some time to restore its equilibrium surface concentration by diffusion of surfactant from or to the bulk liquid. In the meantime, the original adsorbed surfactant layer is either expanded or contracted because surface tension gradients are now in effect, Gibbs—Marangoni forces arise and act in opposition to the initial disturbance. The dissipation of surface tension gradients to achieve equilibrium embodies the interface with a finite elasticity. This fact explains why some substances that lower surface tension do not stabilize foams (6) They do not have the required rate of approach to equilibrium after a surface expansion or contraction. In other words, they do not have the requisite surface elasticity. 

Kofod G, Sommer-Larsen P (2005) Silicone dielectric elastomer actuators finite-elasticity model of actuation. Sens Actuators A Phys 122(2) 273-283. doi 10.1016/j.sna.2005.05.001... 

Nonlinear finite elasticity quasilinear viscoelasticity Anisotropy Biphasic poroelasticity Activation sequence... 

Backman, M.E. (1964) From the relation between stress and finite elastic and plastic strains under impulsive loading. Journal of Applied Physics, 35, 2524—2533. 

Miehe, C. and Keck, J. (2000) Superimposed finite elastic-viscoelastic-plastoelastic stress response with damage in filled mbbety polymers. Experiments, modeling and algorithmic implementation. Journal of the Mechanics and Physics of Solids, 48, 323-365. 

There is an extensive body of literature describing the stress-strain response of rubberlike materials that is based upon the concepts of Finite Elasticity Theory which was originally developed by Rivlin and others [58,59]. The reader is referred to this literature for further details of the relevant developments. For the purposes of this paper, we will discuss the developments of the so-called Valanis-Landel strain energy density function, [60] because it is of the form that most commonly results from the statistical mechanical models of rubber networks and has been very successful in describing the mechanical response of cross-linked rubber. It is resultingly very useful in understanding the behavior of swollen networks. 

Here we begin with a sample of rabber having initial dimensions l, I2, I3. We deform it by an amount A/, A/2, A/3 and define the stretch (ratio) in each direction as A, = (/, -I- A/,)//, = ///,. The purpose of Finite Elasticity Theory has been to relate the deformations of the material to the stresses needed to obtain the deformation. This is done through the strain energy density function, which we will describe using the Valanis-Landel formalism as IT(A, A2, A3). Importantly, as we will see later, this is the mechanical contribution to the Helmholtz free energy. Vala-nis and Landel assumed [60] that the strain energy density function is a separable function of the stretches A, ... 

To illustrate this last point, let us note that for a bar formed from a transversely isotropic material, (i) the constitutive equations of finite elasticity imply that the difference between S and the ambient hydrostatic pressure is a function of the values of 3z /d2, br/3Z, 3z/dR, and d /3R at the point under consideration, and (ii) the condition that deformations be isochoric implies that... 

Surtin, M. and R. Temam, On the antiplane shear problem in finite elasticity, J. Elast. 11 (1981). 

Constitutive Description of Polymer Melt Behavior K-BKZ and DE Descriptions. Although there are many nonlinear constitutive models that have been proposed, the focus here is on the K-BKZ model because it is relatively simple in structure, can be related conceptually to finite elasticity descriptions of elastic behavior, and because, in the mind of the current author and others (82), the model captures the major features of nonlinear viscoelastic behavior of polymeric fluids. In addition, the reptation model as proposed by Doi and Edwards provides a molecular basis for understanding the K-BKZ model. The following sections first describe the K-BKZ model, followed by a description of the DE model. 

Finite Elasticity Theory Classical Theory. The finite elasticity theories available today are very powerful and well developed from a phenomenological perspective. Because the K-BKZ (70-72) has the form of a time-dependent finite elasticity (it was developed as a perfect elastic fluid ) it is useful to briefly outline the basics of finite elasticity theory here. In the initial sections of this article, the stress and strain tensors were discussed, and it was noted that the constitutive relationships that arise between the stress and the strain include material parameters called moduli. When a material is classified as hyperelastic then the moduli are related to derivatives of the free energy function (often the Helmholtz free... 

Another important aspect of finite elasticity theory is the ability to measure the strain energy fimction derivatives Wi = dW/dli and W2 = BW/9I2. Penn and Kearsley (94) showed how this is done nsing data from torsional experiments. An interesting aspect about torsion in finite deformations is that in order to maintain the cylinder at a constant length, it is necessary to apply normal forces at the ends of the cylinder. If the cylinder is left imrestrained, it will lengthen in an effect referred to as the Poynting (95) effect, first observed early in the last century in experiments with metal wires. When a cylinder of length L is twisted by an amount... 

Finite Elasticity Theory The VL Representation. While the above description of the finite deformation behavior of elastic materials is very powerful, the limitation on it is that the material parameters W and W2 need to be determined in each geometry of deformation of interest. Hence, the torsional measurements described above only give values of Wi(/i, I2) and W2(/i, I2) for the condition of shear (torsion is anonhomogeneous shear) and that condition is/i = l2 = 3- -y. More measurements need to be made to obtain the parameters in extension, compression, etc. However, in 1967, Valanis and Landel (98) proposed a strain energy function that, rather than being a function of the invariants, is a function of the stretches Xj. The function was assumed to be separable in the stretches as... 

Finally, the power of finite elasticity theory is that once the material properties [Wi and W2 or are known, the stresses in any deformation field can be calculated. There is an extensive literature on ways to represent the material functions and, in fact, commercial finite element codes use finite elasticity theory in calculations that can be important in applications that range from the stresses in automobile tires (105) to those in earthquake bearings for large buildings (106). One feature of the K-BKZ theory to be discussed next is that it retains the structure of finite elasticity theory and includes time-dependent properties of the viscoelastic materials that were discussed in the earlier sections of this article. 

The K-BKZ Theory Model. The K-BKZ model was developed in the early 1960s by two independent groups. Bernstein, Kearsley, and Zapas (70) of the National Bureau of Standards (now the National Institute of Standards and Technology) first presented the model in 1962 and published it in 1963. Kaye (71), in Cranfield, U.K., published the model in 1962, without the extensive derivations and background thermodynamics associated with the BKZ papers (82,107). Regardless of this, only the final form of the constitutive equation is of concern here. Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U Ii, I2, t). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

Also note that the hydrostatic pressure is indeterminate because the K-BKZ is an incompressible material model. As in finite elasticity theory, the material parameters need to be obtained and, in principle, the stress response to any deformation history can be obtained. Unlike linear viscoelasticity, the integration must be carried out from —00 to t, which can lead to difficulties in numerical computer codes. This aspect of the K-BKZ theory has been discussed by (62) Larson, among others. 

The utility of the K-BKZ theory arises from several aspects of the model. First, it does capture many of the features, described below, of the behavior of polymeric melts and fluids subjected to large deformations or high shear rates. That is, it captures many of the nonlinear behaviors described above for steady flows as well as behaviors in transient conditions. In addition, imlike the more general multiple integral constitutive models (108,109), the experimental data required to determine the material properties are not overly burdensome. In fact, the information required is the single-step stress relaxation response in the mode of deformation of interest (72). If one is only interested in, eg, simple shear, then experiments need only be performed in simple shear and the exact form for U I, /2, ) need not be obtained. Furthermore, because the structure of the K-BKZ model is similar to that of finite elasticity theory, if a full three-dimensional characterization of the material is needed, some of the simplilying aspects of finite elasticity theories that have been developed over the years can be applied to the behavior of the viscoelastic fluid description provided by the K-BKZ model. One such example is the use of the VL form (98) of the strain energy function discussed above (110). The next section shows some comparisons of the material response predicted by the K-BKZ theory with actual experimental data. 

This relation, which also results from the K-BKZ model, is referred to as the Lodge-Meissner relationship (124) and results for materials with a finite elastic modulus at zero time. 

In this introduction to finite elasticity it is only necessary to develop the most elementary definition of finite strains (for a more comprehensive discussion, see [3], chapter 3). 

In the following section, the basic concepts used to describe the (finite) deformation of a simple material are briefly presented. A comprehensive introduction of finite elasticity can be found, for instance, in [97, 102, 103]. 

Drozdov AD (2007) Constitutive equations in finite elasticity of rubbers. Int J Solids Stmct 44 272-297... 

It will be convenient to discuss these various aspects separately as follows (1) behaviour at large strains in Chapters 3 and 4 (finite elasticity and rubber-like behaviour, respectively) (2) time-dependent behaviour in Chapters 5-7 and 10 (viscoelastic behaviour) (3) the behaviour of oriented polymers in Chapters 8 and 9 (mechanical anisotropy) (4) non-linearity in Chapter 11 (non-linear viscoelastic behaviour) (5) the non-recoverable behaviour in Chapter 12 (plasticity and yield) and (6) fracture in Chapter 13 (breaking phenomena). However, it should be recognised that we cannot hold to an exact separation and that there are many places where these aspects overlap and can be brought together by the physical mechanisms, which underlie the phenomenological description. 

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