A strain energy density function is proposed which is based on a generalized measure of strain. The function has the form W=(2G/n)IE+BIEm where G, B, n, and m are material constants, and IE is the first invariant of the (generalized) Lagrangean strain (λan−1)/n. The function fits data on natural rubber and on a synthetic rubber in various homogeneous stress fields up to the point of break. The powers n and m are sensibly independent of temperature, while the two moduli G and B depend linearly on temperature, over the range investigated.
A strain energy density function has been developed for compressible rubber-like materials. Its usefulness is demonstrated on hand of published data on the volume dilatation of natural rubber in simple tension.
A new four-parameter elastic potential function is proposed which represents data on the elastic deformation of rubbery materials with the same parameters in various deformation fields up to break.We wish to report on a new elastic potential function for rubbery materials which permits one to predict deformations up to break in various deformation fields if the four material parameters of the potential function are known. Our elastic potential (or strain energy density) function is based on the generalized measure of strain introduced by Seth (1). In Lagrangean coordinates, this measure of strain is expressed aswhere E is the strain, X is the stretch ratio, the as denote the three principal directions, and n is a material parameter. Seth's measure of strain is based on the realization that there is no unique definition of strain; rather, the most convenient strain measure is a property of the material and of the geometry of the deformation.The elastic potential of an isotropic material is customarily formulated in terms of three invariants of the stretch ratios. These are generally taken to be the principal invariants, I,, I2, and I3, of the right Cauchy-Green deformation tensor. However, this choice is not unique. One may choose any three symmetric functions of the stretch ratios. In a general sense one can define invariants on perfectly arbitrary functions of the stretch ratios, namely I To whom reprint requests should be addressed.
The thermodynamic description of the large principal deformations of elastomers requires four thermodynamic energy functions and their associated free energies. The significance of, and relationship between, these potential functions is discussed and their interrelations are derived. The internal energy contribution to the retractive force or the extension in an elastomer is used as an example of the application of the concepts introduced.The thermodynamic (more precisely thermostatic) description of the deformation of a solid system is more complex than that of a fluid (gas or liquid) system. In the latter, volume and pressure are the only mechanical parameters to be taken into account. In the former, the set of mechanical parameters must be enlarged, in the general case, to the 6 + 6 = 12 components of the strain and stress ten-s0rs.I This paper discusses the thermodynamic potential functions which are useful in describing the thermodynamics of large principal deformations of elastomers and the general relations between their differentials and those of various sets of independent variables. Because a n elastomer may be regarded as a homogeneous isotropic solid, its deformation can be described more simply than that of a general solid. The potential functions for the thermodynamics of elastomers introduced here are defined in a way which makes them consistent with the definitions of the potential functions in the thermodynamics of fluids. The following discussion aims to provide an exhaustive treatment of the formalism of the thermodynamics of elastomers. Potential Functionsinternal energy U as where dQ is the element of heat absorbed, and d W is the element of work done, by the system on its surroundings. If the process is conducted reversiblyThe first law of thermodynamics gives the change in thewhere T i s the (absolute) temperature and S is the entropy. Flory2 has shown that the element of elastic work, dW, done by a homogeneous isotropic system in a large principal deformation is given bywhere V is the deformed volume, the t , are the principal true stresses, and the A, are the principal extension ratios defined as the ratios of the stretched lengths, L,, to the unstretched lengths, L,o.Thus, the change in the internal energy is given by dU = TdS + V C t i d In X i i and the change in the associated (Helmholtz) free energy, defined byThe last terms in eq 4 and 6 contain both the work of extension and the work of expansion. The two must be separated because changes in volume may be induced both by the application of forces or extensions and by changes in temperature. We consider that the principal true stresses are ti = ( L i f i / V ) -P
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