Hyperelasticity

A natural extension of linear elasticity is h rperelasticity.l l H rperelasticity is a collective term for a family of models that all have a strain energy density that only depends on the applied deformation state. This class of material models is characterized by a nonlinear elastic response, and does not capture yielding, viscoplasticity, or time-dependence. The strain energy density is the energy that is stored in the material as it is deformed, and is typically represented either in terms of invariants 

The distortional stretches can be obtained from the applied principal stretches by 

The two main types of hyperelastic models are the polynomial model and the Ogden model. In the polynomial model, the strain energy density is given by 

Commercial finite element software packages typically contain a number of other hyperelastic representations that can also be used. 

Hyperelastic models within finite element codes should be used carefully when a component experiences multiaxial stresses. 

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It may be noted that an elastic material for which potentials of this sort exist is called a hyperelastic material. Hyperelasticity ensures the existence and uniqueness of solutions to intial/boundary value problems for an elastic material undergoing small deformations, and also implies that all acoustic wave speeds in the material are real and positive. 

Genetically determined abnormalities in GAG or collagen synthesis lead to joints becoming easily dislocated, hyperextendable and hyperelastic ( double-jointed ),... 

The dynamic mechanical properties of elastomers have been extensively studied since the mid-1940s by rubber physicists [1], Elastomers appear to exhibit extremely complex behavior, having time-temperature- and strain-history-dependent hyperelastic properties [1]. As in polymer cures, DMA can estimate the point of critical entanglement or the gel point. 

Valanis,K.C., Landel,R.F. The strain-energy function of a hyperelastic material in terms of extension ratios. J. Appl. Phys. 38,2997-3002 (1967). 

We consider a macroscopic model of a two-dimensional, hyperelastic network, incorporating the microstructural details in a systematic manner. The model equations at equilibrium reduce to stress continuity,... 

For other models of flow of electrolytes through porous media the reader is referred to [2], [5], [6]. To take into account FCD (fixed charge density) one has to impose additional condition on the interface T (w) and the electroneutrality condition. A challenging problem is to use homogenisation methods for the case of finitely deformable skeleton, even hyperelastic. The permeability would then necessarily depend on strains. Such a dependence (nonlinear) is important even for small strain, cf. [7]. It is also important to include ion channels [8]. 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

Kliippel M, Schramm J (1999) An advanced micromechanical model of hyperelasticity and stress softening of reinforced rubbers. In Dorfmann A, Muhr A (eds) Constitutive models for rubber. A.A. Balkema, Rotterdam... 

A depressed nasal pyramid and prominent paranasal folds change facial morphology the skin is hyperelastic and can sometimes hang on the face. It is especially important to be aware of this syndrome and try to detect borderline... 

Elastic materials For elastic materials, the stress and strain are related by a unique functional relationship, which may be linear (linearly elastic material) or nonlinear (hyperelastic materials). The function between stress and strain may generally be assumed to be linear for small strains, and in the one-dimensional case, one obtains Hooke s law (37,38)... 

The investigation of the temperature dependence of dielectric characteristics of these block-copolysulfonarilates showed that the permittivity in the interval 20-250°C is stable. When higher than this temperature its increase is seen, which is explained by the transition to a hyperelastic state. 

Hyperelastic finite element analysis Accommodates complex geometries. Can handle nonlinearity in material behavior and large strains. Rapid analysis possible. Standard material models available. Does not include rate-dependent behavior. Cannot predict permanent deformation. Does not handle hysteresis. Some material testing may be required. Can produce errors in multiaxial stress states. 

Linear viscoelasticity is an extension of linear elasticity and hyperelasticity that enables predictions of time-dependence and viscoelastic flow. Linear viscoelasticity has been extensively studied for many different materials ] and can be very useful when applied under the appropriate conditions. Another added benefit of linear viscoelasticity is that it is available in all commercial finite element packages and therefore easy to use. 

The TEM observation after staining with RuOa provides us with a high definition view of the microscopic mechanisms and reveals the specific role of the POE phase. In the BD13 blend, illustrated in Fig. 19.30a, three phenomena are visible (i) interfacial debonding at the poles of PA6 particles, (ii) high elongation of hyperelastic POE particles, and (iii) cavitation in dispersed POE droplets. On the whole, since the size and number of POE particles are smaller than that of PA6 particles, the former mechanism is more frequently observed and seemed to have a leading influence on the overall volume strain. 

When the blends are subjected to tensile testing, a certain fraction of the overall strain is accommodated by conservative deformation of the material. In the PP matrix, deformation results from the combination of amorphous phase hyperelasticity and crystal plasticity, as discussed earlier (50). The PA6 phase is also capable of deforming plastically, but its flow stress in the plastic stage is much higher than that of PP. Consequently, in the PP/PA6 blends the isolated PA6 particles exhibit less... 

Sound knowledge of the joint behavior is required for a successful design of bonded joints. To characterize the bonded joint, the loading in the joint and the mechanical properties of the substrates and of the adhesives must be properly defined. The behavior of the bonded joint is investigated by finite element (FE) analysis methods. While for the design of large structures a cost-efficient modeling method is necessary, the nonlinear finite element methods with a hyperelastic material model are required for the detailed joint analysis. Our experience of joint analysis is presented below, and compared with test results for mass transportation applications. 

In a local detailed analysis, the flexible adhesive is modeled with three-dimensional solid elements to enable the refined capture of any local stress or strain gradients. The adhesive material is described as a rabber-like, nearly incompressible, hyperelastic material characterized by a strain energy function. Using U as the strain energy potential per unit of the reference volume, the form of the Ogden strain energy potential is shown in Eq. (1) jii and u are material parameters which are determined from adhesive material test data. 

At 70 °C, both systems are above Tg and behave hke rubbery" materials. No permanent strain exists after fracture. At temperatures above Tg, viscous flow is hindered due to crosslinking. However, chain segments remain flexible and the adhesive can withstand high strains. Although these strains are not as high as in a mbber, they can no longer be considered small . Hence, a hyperelastic material model must be used. The second system (Fig. 33.2, right) already shows hyperelastic properties at 23 °C due to the low strain rate. 

For flexible (mbbery) adhesives which show Tg far below room temperature, hyperelastic material models are generally used. In the hyperelastic regime, standard solutions are available which use various types of potential functions [7]. The flexible adhesive systems investigated were best fitted by a potential function which is formulated in terms of the principal stretches Aj originally suggested by Ogden [Eq. (1)] [8]. 

Table 33.2 Hyperelastic constants of flexible adhesive systems.

The preceding equations provided a reasonable foundation for predicting DE behavior. Indeed the assumption that DEs behave electronically as variable parallel plate capacitors still holds however, the assumptions of small strains and linear elasticity limit the accuracy of this simple model. More advanced non-linear models have since been developed employing hyperelasticity models such as the Ogden model [144—147], Yeoh model [147, 148], Mooney-Rivlin model [145-146, 149, 150] and others (Fig. 1.11) [147, 151, 152]. Models taking into account the time-dependent viscoelastic nature of the elastomer films [148, 150, 151], the leakage current through the film [151], as well as mechanical hysteresis [153] have also been developed. 

Lochmatter P, Kovacs G, Wissler M (2007) Characterization of dielectric elastomer actuators based on a visco-hyperelastic film model. Smart Mater Struct 16 477... 

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