Hyperelastic materials

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. 

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). 

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)... 

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]. 

For a simple, two-dimensional, incompressible, hyperelastic material, the relationship between the membrane shear force resultant Ns and the material deformation is [Evans and Skalak, 1979]... 

Calculation of polymer flexible joints. Considered polyurethane mass PM belong to the group of elastomeric materials, thus can be described using theory of the hyperelastic materials. In simple approach of this theory, properties of polymer mass are obtained from uniaxial tension and compression tests according to (Eq. 4), presented in [21], Following equation (Eq. 4), the (Eq. 5) is proposed for calculation of uniaxial deformation of a flexible polymer joint. 

In the paper, a theory for mechanical and diffiisional processes in hyperelastic materials was formulated in terms of the global stress tensor and chemical potentials. The approach described in was used as the basic principle and was generalized to the case of a multi-component mixture. An important feature of the work is that, owing to the structure of constitutive equations, the general model can be used without difficulty to describe specific systems. 

Swieszkowski et al. studied the use of PVA-C as cartilage replacement for the shoulder joint. PVA-C was used as the articular layer of the glenoid component. The mechanical effects of using this material in the glenoid component were evaluated and a model of the cryogel as a hyperelastic material was developed to allow design modifications to limit contact stress [96]. 

A general discussion of Finite element analysis can be found in another article. This article is specifically concerned with hyperelastic materials. A common hyperelastic material of importance in adhesion studies and polymer technology is rubber. A hyperelastic material is a material that undergoes large strains and displacements with little changes... 

The mechanical behaviour of hyperelastic materials is much more complicated than that of classical materials. The material properties may differ significantly under tension, compression or shear loading conditions. Therefore, for a complete set of material data, experimental tests that cover all these modes of deformation should be carried out. 

Isotropic hyperelastic materials For this model, the strain energy density function is written in terms of the principal stress invariants I, h, h). Equation 1 becomes... 

Example 3.3 (Finite strain hyperelastic theory). The hyperelastic theory with assumptions of small strain, as shown in Example 3.1 (p. 102), can be applied to the hyperelastic material undergoing finite strain in a similar manner. In addition, the temperature field is also included. 

During deformation of hyperelastic materials, large strains of 100% or more can occur. The material behaviour is strongly non-linear. Therefore, the theory of large deformations has to be used to describe the material behaviour (see section 3.1). 

The set of all material symmetry transformations at a material point X depends on the selected reference configuration and for hyperelastic materials can be defined... 

According to Eq. (31) and definitions (28)-(30), the most general form of the second Piola-Kirchhoff stress tensor for an isotropic and hyperelastic material is ... 

In the following we analyze some elementary problems in which the deformation is homogeneous, i.e. the deformation gradient F is constant in whole body. Homogeneous deformations are equilibrium solution for all the class of hyperelastic materials for this reason they are called universal solutions [116]. 

The simple shear is an isochoric deformation that is possible in every compressible, homogeneous, and isotropic hyperelastic material. The constimtive relation (38) shows that the shear stress related to the shear strain y is given by ... 

Eihlers W, Eppers G (1998) The simple tension problem at large volumetric strains computed from finite hyperelastic material laws. Acta Mech 137 12-27... 

Figure 3,10 Various data sets plotted according to the proposal of Valanis and Landel. (Adapted from Valanis, K.C. and Landel, R.F. (1967) The strain-energy function of a hyperelastic material in terms of the extension ratios. J. Appl. Phys., 38, 2997. Copyright (1967) American Institute of Physics.)...

Consider a structural member made from isotropic hyperelastic material with variable cross-section. The material lines follow the pattern of the continuum boundaries as depicted in Figure 1. In the initial configuration the position vector P of an arbitrary point on a general material line s with respect to the right-handed rectangular Cartesian coordinate system is given by ... 

Y. Shen, K. Chandrashekhara,W.F. Breig, LR. Oliver, (2005) Finite element analysis ofV-ribbed belts using neural network based hyperelastic material model, Intematimal Journal of Non-linear Mechanics 40 875 - 890... 

Material nonlinearity may be hyperelastic or elasto-plastic. The difference between the behavior of an elastic and elasto-plastic material is seen on unloading as in the former case the unloading path coincides with the loading path whereas in the latter case a different unloading path results in permanent deformation when the load has been removed. Elasto-plastic behavior is characterized by a linear region up to the yield point, after which soflening behavior is seen. Hyperelastic materials such as elastomers exhibit nonlinear elastic response for even large strains. 

The dynamic-mechanical properties of elastomers have been studied extensively by rubber physicists and technologists for about 50 years. The principal objective in much of this work has been to relate the experimental observations to the known composition and structure of the materials. At first sight it appears that elastomers exhibit extremely complex behaviour, having time-, temperature- and strain-history-dependent hyperelastic properties. This is because elastomers are compounded for practical use and are mixtures of a hyperelastic material (the polymer) with materials exhibiting only short-range elasticity (the filler). 

---