A Direct, Explicit Approach To Obtaining Finite Strain Elastic Potentials For Rubber-like Materials Via Rational Interpolation

With the fine properties of softness, wear resistance, insulativity andimpermeability, rubber-like materials are widely used in many engineeringstructures and mechanical equipment. Investigations into the mechanicalbehavior of rubber-like materials are significant for both engineering andtechnology. The highly elastic behavior of rubber-like materials can be presentedby stress-strain relation. The thermodynamic principles require that thesestress-strain relations should be derived from elastic potentials.By means of the logarithmic strain, a new direct approach is proposed toobtain elastic potentials that exactly match benchmark tests data (uniaxialextension test, biaxial extension test, plane-strain extension test and shear test)for rubber-like materials. Firstly, two one-dimensional stress-strain relations interms of rational interpolation are obtained for uniaxial case and shear case andthen two one-dimensional elastic potentials are derived via directly integratingthe stress power. Secondly, a bridging method and a matching method based onthe logarithmic strain are introduced to combine the two one-dimensionalelastic potentials in terms of a material parameter. Eventually, a unified form ofmulti-axial elastic potential is presented. Such potential in unified form ischaracterized by five parameters. Four of them are determined directly byuniaxial data and shear data, while the other is used to fit general biaxial data.The multi-axial elastic potential may match any given data for uniaxialextension test and shear test. Then we derive two universal relations between thefour tests and demonstrate that such multi-axial elastic potential match data forfour benchmark tests. Furthermore, predictions from the suggested potentials arederived for general biaxial stretch tests and compared with the extensive datagiven by R.S.Rivlin and Saunders in1951. For the first time, good agreement isachieved with these classical data.