The strain energy function

The corresponding stress components as defined in the deformed state are zz, where 

The work done (per unit of initial undeformed volume) in an infinitesimal displacement from the deformation state where X, X2 and A3 change to Ai + dAi, X2 + (U2 and A3 + dA3 is 

For an elastic material the work done can be equated to a change in the stored elastic energy U. In the case of rubbers, it is usual to consider a reversible isothermal change of state at constant volume, so that the work done can be equated to the change in the Helmholtz free energy A, i.e. A17 = A 4. Here U is 

It would be unreasonable for a physical quantity such as energy to depend on the choice of axes. The use of principal extension ratios, with values independent of the axis set, goes some way to ensuring that this is not the case. However, the choice of subscripts 1, 2 and 3 is arbitrary, so the chosen form must be a S3onmetric function of Ai, A2 and A3. For simplicity it should also become zero when Ai = A2 = A3 = 1, i.e. for zero strain. A further requirement is that for small strains we should obtain Hooke s law for simple tension and the equivalent equation for simple shear. 

To obtain a stress-strain relationship from this equation we invoke Equation (2.1), together with the assumption that rubber is incompressible, i.e. there is no change in volume on deformation, which is true to a good approximation. For example, consider extension under a tensile force/in the x direction. This gives 

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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... 

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. 

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. 

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... 

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) ... 

Because the strain energy function for rubber is valid at large strains, and yields stress-strain relations which are nonlinear in character, the stresses depend on the square and higher powers of strain, rather than the simple proportionality expected at small strains. A striking example of this feature of large elastic deformations is afforded by the normal stresses tn,(22,133 that are necessary... 

The topology of a network rubber that is crossiinked in the swollen state differs from that crossiinked in an unswollen state. The initial boundary conditions describing the chain conformation used in deriving the strain-energy function can be modified to allow for these differences. The resulting equations have been used to elucidate possible reasons for the deviations of stress-strain data from statistical theory. 

The strain energy function (equation 7.39) follows directly from equation 7.111 assuming AH = 0 as in the ideal rubber case. The function for a swollen sample will be modified by the inclusion of the term. This term that is essentially a scaling factor relating the unswollen dimensions to the swollen dimensions for a stretched sample, is given by... 

The work done appears as strain energy, so that eqn 3.29 is often known as the strain energy function when referring to a unit volume of rubber. In this case it may be written... 

For a certain rubber, it was found by experiment that in uniaxial extension by up to 100% the strain energy function was accuratefy given by the Mooney equation 3.N.5.2, with Cj 300 kPa and C2 100 kPa. Find the tensile stress, based on the original cross-sectional area, required to extend a bar of this rubber by 100%. If an approximate prediction of this stress is obtained by applying the Gaussian approximation (eqn 3.N.5.1) to this material, d the magnitude of the error which results. 

While the term a ln(A A2A3) is not important in the mechanical response, because of the incompressibility assumption, it may be important in swelling [61]. We also note that some of the molecular models include this logarithmic term. Then, the principal stresses o- in any deformation can be related through the strain energy function and deformations as follows ... 

Mooney-Rivlin material Mooney and Rivlin developed a hyperelastic constitutive material model. The strain energy function is written in the form of series expansion of... 

Continum mechanics is also used to account for the observed stress-strain behavior exhibited by elastomers The most general form of the strain energy function (which vanishes at zero strain) is the power series... 

There is assumed to exist a function of x, W(x), known as the strain energy function (per unit undeformed volume), if a is the engineering stress, i.e. the extending force normalized by division by initial cross-sectional area, then W(X) obeys at each X... 

Thus, from torque and normal forces in torsional experiments, the VL function derivative can be obtained. Typical data for natural rubber are presented in Figures 30, 31, 32. The figures illustrate the sequence that would be used to obtain the VL ftmction. First, obtain torque and normal force data (Fig. 30) and use equations 47 and 48 to obtain the strain energy function derivatives Wi and W2 (some typical results shown in Fig. 31). Finally, data of the sort shown in Figure 31 are used to obtain the VL fimction derivative w X). Figure 32 shows such data obtained from torsional measurements on natural rubber samples cross-linked to different extents (102). 

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. 

We wish to calculate the strain-energy function for a molecular network, assuming that this is given by the change in entropy of a network of chains as a function of strain. 

A very useful and simple form for the strain-energy function has already been given as follows ... 

In addition to these theoretical considerations, which suggest that we need not be restricted to squares of extension ratios in formulating the strain energy function, it has been found by experimentalists that there is high sensitivity to experimental error when small values of the strain invariants I and f are involved. It is therefore natural to postulate that the only constraint on the form U is that imposed by the invariance of U with respect to the axis lables, which implies that U i, X-i, A3) should be a symmetric function of the extension ratios, i.e. invariant to any permutation of the indices 1,2, 3. 

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