Strain-energy functions

The stored strain energy can also be determined for the general case of multiaxial stresses [1] and lattices of varying crystal structure and anisotropy. The latter could be important at interfaces where mode mixing can occur, or for fracture of rubber, where f/ is a function of the three stretch rations 1], A2 and A3, for example, via the Mooney-Rivlin equation, or suitable finite deformation strain energy functional. 

Fig. 12 The meridianal orientation function p(0) for a Gaussian distribution, the distribution function p(0)sin0,the shear strain energy function tan20sin0p(0) and the function tan20...

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). 

The analysis of Eq. (16) has led to the conclusion 7) that the strain-energy function W in the mode of uniform deformation is parabolic with a minimum potential energy in the unstrained state... 

Figure 19.7 Elastic strain energy function E c/a) for an incoherent ellipsoid inclusion of aspect ratio c/a.

There are a number of ways to model the geometry of transition metal centers. One promising treatment is based on the addition of a ligand field term to the strain energy function (Eq. 2.15)1191. 

When using the generalised Hooke s law strain energy function there are a number of possible strain definitions that can be used depending on the situation. When material deformation is very small the infinitesimal strain approach is a valid approximation with the strain defined as... 

The development of molecular constitutive equations for commercial melts is still a challenging unsolved problem in polymer rheology. Nevertheless, it has been found that for many melts, especially those without long-chain branching, the rheological behavior can be described by empirical or semiempirical constitutive equations, such as the separable K-BKZ equation, Eq. (3-72), discussed in Section 3.7.4.4 (Larson 1988). To use the separable K-BKZ equation, the memory function m(t) and the strain-energy function U, or its strain derivatives dU/dli and W jdh, must be obtained empirically from rheological data. 

The first four terms of the function are commonly found in molecular mechanics strain energy functions, and they are modified Hooke s law functions. The last term has been added to insure the proper stereochemistry about asymmetric atoms. A model is refined by minimizing the highly nonlinear strain energy function with respect to the atomic coordinates. An adaptive pattern search routine is used for the strain energy minimization because it does not require analytical derivatives. The time necessary to obtain good molecular models depends on the number of atoms in the molecule, the flexibility of the structure, and the quality of the starting model. 

In order to point out the essential difference between deformations induced by body forces and surface tractions we recall Ericksen s theorem [88]. The theorem states that homogeneous deformations are the only deformations that can be achieved by the application of surface tractions alone, considering a homogeneous and isotropic material characterized by an arbitrary strain-energy function. In other words, we cannot induce diverse non-homogeneous deformations with surface tractions. In contrast to surface tractions, application of fields that act as body forces leads to non-homogeneous deformations without any additional constraints for the material. 

As with all functions, a general form of the strain energy function for an isotropic material can be formed by a Taylor expansion. The result is ... 

We can also derive an expression on the basis of the Mooney-Rivlin strain energy function for swollen elastomers. A dry elastomer sample will undergo two types of deformation one due to swelling and the other due to extension. The strain energy function per unit volume of swollen elastomer is related to that of the dry sample by... 

Repeat Problem 3 using the Mooney-Rivlin strain energy function [equation (6-80)]. 

The crack-driving force G may be estimated from energy considerations. Consider an arbitrarily shaped body containing a crack, with area A, loaded in tension by a force P applied in a direction perpendicular to the crack plane as illustrated in Fig. 2.6. For simplicity, the body is assumed to be pinned at the opposite end. Under load, the stresses in the body will be elastic, except in a small zone near the crack tip i.e., in the crack-tip plastic zone). If the zone of plastic deformation is small relative to the size of the crack and the dimensions of the body, a linear elastic analysis may be justihed as being a good approximation. The stressed body, then, may be characterized by an elastic strain energy function U that depends on the load P and the crack area A i.e., U = U(P, A)), and the elastic constants of the material. 

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. 

Here is the elastic modulus tensor. It has 3 = 81 elements, however since the stress and strain are represented by symmetric matrices with six independent elements each, the number of independent modulus tensor elements is reduced to 36. An additional reduction to 21 is achieved by considering elastic materials for which a strain energy function exists. Physically, C2323 represents a shear modulus since it couples a shear stress with a shear strain. Cim couples axial stress and strain in the 1 or x direction,... 

The isotropic part of this expression has also been used to model the myocardium of the embryonic chick heart during the ventricular looping stages, with coefficients of 0.02 kPa during diastole and 0.78 kPa at end-systole, and exponent parameters of 1.1 and 0.85, respectively [107]. Another related transversely isotropic strain-energy function was used by Guccione et al. [108] and Omens et al. [109] to model material properties in the isolated mature rat and dog hearts ... 

A power law strain-energy function expressed in terms of circumferential, longitudinal and transmural extension ratios (A,i, A.2, and k ) was used [111] to describe the biaxial properties of sheep myocardium 2 weeks after experimental myocardial infarction, in the scarred infarct region ... 

A further phenomenological theory, which uses the concept of strain-energy functions, deals with more general kinds of stress than uniaxial stress. When a rubber is strained work is done on it. The strain-energy function, U, is defined as the work done on unit volume of material. It is unfortunate that the symbol U is conventionally used for the strain-energy function and it will be important in a later section to distinguish it from the thermodynamic internal-energy function, for which the same symbol is also conventionally used, but which is not the same quantity. 

Thus the total work done on unit volume of the rubber system during deformation, which was defined in section 6.3.3 to be the strain-energy function, is — J T dS = —T AS, with AS given by equation (6.46). 

As already noted in section 6.3.3, the strain-energy function is usually given the symbol U, in spite of the fact that it is not equal to the internal energy. Using this terminology, equation (6.46) shows that... 

The strain-energy function is actually equal to the change in the Helmholtz function A for the rubber, since A 4 = — T A5 for an isothermal change for a medium like rubber in which there is no change in the internal energy at constant temperature. 

Furthermore, to yield linear stress-strain relations at small strains, W must be initially of second order in the strains ei, C2, 3. Therefore, the simplest possible form for the strain energy function is... 

Stresses can be obtained from the derivatives of the strain energy function IT ... 

Rubber becomes harder to deform at large strains, probably because the long flexible molecular strands that comprise the material cannot be stretched indefinitely. The strain energy functions considered up to now do not possess this feature and therefore fail to describe behavior at large strains. Strain-hardening can be introduced by a simple modification to the first term in Eq. (1.18), incorporating a maximum possible value for the strain measure J, denoted Jm (Gent, 1996) ... 

Equation (1.19) reduces to Eq. (1.18) when the strains are relatively small that is, when the ratio J /Jm is small. Thus Eq. (1.19) is probably the simplest possible strain energy function that accounts for the elastic behavior to good approximation over the entire range of strains (Pucci and Saccomandi, 2002). It requires three fitting parameters, two of which are related to the small-strain shear modulus G ... 

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