Landel-Valanis function

Other forms of the Landel-Valanis functions u are of course possible. Recently, Darijani, Naghdabadi and Kargarnovin [16] explored a number of possibilities, involving polynomial, logarithmic and exponential functions. In this scheme, strain energy functions are constructed using a set of basic functions of the principal stretches. These are ... 

Fig. 58. The VL Valanis-Landel (98) function derivative as a function of deformation for polycarbonate as determined from data of Figures 55 and 57. o 0.3 s 3.2 s A 32 s V 320 s V 3000 8. After Pesce and McKenna (146).

Thus, in this case, one may determine a single function w(X) by experiment. Eq. (8) satisfies the symmetry condition imposed by isotropy (restriction B). If its use is limited to the coordinate system whose axes are taken in the directions of the principal strains, restriction A mentioned above does not matter. Valanis and Landel deduced this form of W from the kinetic theory of network, in which the entropy change As upon deformation is represented by the sum of three components, each corresponding to the deformation in one of the Xl, X2, and X3 directions and having the same functional dependence on the argument. Thus... 

From a relevant biaxial extension experiment one can determine the function w(X) constituting the Valanis-Landel expression for W, as was done, for example, by Jones and Treloar. However, the w function so obtained is of little value unless it is tested with the stress-strain relations for other modes of deformation. This kind of test was carried out by the present authors49) and is described below. 

Eq. (13) was used to compute at for given sets of Xt and X2. The broken lines in Fig. 30 show the computed values of aj for Xj = 1.2, 2 and 3 as a function of X2-Their agreement with the directly observed values (solid lines) is very poor. Second, the Valanis-Landel expression for W [Eq. (8)] with w(X) given by Eq. (44) was substituted into Eq. (26) to calculate ax. The resulting at versus X2 curves for Xt = 1.2,... 

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). 

Shen 39) has also considered the thermoelastic behaviour of another widely used phenomenological equation of state, the so-called Valanis-Landel equation. Valanis and Landel40) have postulated that the stored energy function W should be expressible as the sum of three independent functions of principle extension ratios. This hypothesis leads to the following equation of state... 

Key Words Crosslinked Rubber, Hory-Rehner Hypothesis, Gels, Networks, Polymer, Rubber Elasticity, Scaling Theory, Solution Thermodynamics, Swelling, Valanis-Landel Function. 

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, ... 

Erom a practical viewpoint, Eq. (29.4) can be used to describe the stress-strain relation of a material if vi/(A) is known. m/(A) can be obtained in the laboratory in various ways, such as pure shear experiments as described by Valanis and Landel [60], by torsional measurements as described by Kearsley and Zapas [62] and by a combination of tension and compression experiments as also described by Kearsley and Zapas [62]. Treloar and co-workers [63] have also shown that the VL function description of the mechanical response of rubber is a very good one. The reader is referred to the original literature for these methods. 

Another point to keep in mind here is that, in most models, the description of rubber elasticity given from statistical mechanical models results in a Valanis-Landel form of strain energy density function. This will be important in the following developments. We now look at some common representations of the strain energy density function used to describe the stress-strain behavior of crosslinked rubber. 

There are two common phenomenological strain energy functions that have been used to describe the stress-strain response of rubber [58,59,64]. These are referred to as the Neo-Hookean form and the Mooney-Rivlin form and both can be written as Valanis-Landel forms, although they represent truncated forms of more general strain energy density functions. The Neo-Hookean form is a special form of the Mooney-Rivlin form, so we will begin with the latter. For a Mooney-Rivlin material the strain energy density function is written as ... 

We note that the free energy function in the Hory-Erman model is a specific form of the Valanis-Landel strain energy density function. McKenna and Hinkley [61] determined the Valanis-Landel function for the junction constraint model... 

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... 

Because of this usefulness it is important to have relatively simple experiments available to obtain the VL function. Kearsley and Zapas (97) have shown that either simple extension combined with simple compression or torsion with normal force measurements can be used to obtain the VL function. Valanis and Landel (98) used pure shear measurements. The equations for the torsional measurements arise from the relationship between w X) and Wi and W2 given by... 

E. A. Kearsley and L. J. Zapas, Some Methods of Measurement of an Elastic Strain Energy Function of the Valanis-Landel Tjrpe J. Rheol. 24, 483-500 (1980). 

Valanis and Landel [13] have introduced a strain energy function of the form... 

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.)...

Valanis and Landel (1967) proposed that for many materials, the strain-energy function is separable into the sum of the same function of each of the principal extensions a/... 

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