Continuum Constitutive Modeling for Isotropic Hyperelastic Materials

The partial differential equation for isotropic hyperelastic constitutive models has been postulated and derived from the balance between stored energy and stress work done. The partial differential equation as a function of three invariants has then been solved by Lie group methods. With geometric meanings of deformations, the general solution boils down to a particular three-term solution. The particular solution has been applied for several isotropic hyperelastic materials. For incompressible materials, vulcanized rubber containing 8% sulfur and Entec Enflex S4035A thermoplastic elastomer, three coefficients have been determined from uniaxial tension data and applied to predict the pure shear and equibiaxial tension modes. For a slightly compressible rubber material, the coefficients have also been extracted from the confined volumetric test data.

A New Parameter as an Indicator of the Degree of Deformation of Coals

The deformation of coal is effected by thermal effect, pressures and tectonic stress, and the tectonic stress is the principal influence factor. However, the proposition of a useful quantitative index that responds to the degree of deformation of coals quantitatively or semi-quantitatively has been a long-debated issue. The vitrinite reflectance ellipsoid, that is, the reflectance indication surface（RIS） ellipsoid is considered to be a strain ellipsoid that reflects the sum of the strain increment caused by stress in the process of coalification. It has been used to describe the degree of deformation of the coal, but the effect of the anisotropy on the RIS ellipsoid has not yet been considered with regards to non-structural factors. In this paper, Wei＇s parameter（ε） is proposed to express the deformation degree of the strain ellipsoid based on considering the combined influence of thermal effect, pressure and tectonic stress. The equation is as follows： ε=√[（ε_1-ε_0）~2＋（ε_2-ε_0）~2＋（ε_3-ε_0）~2]/3, where ε_1=lnR_（max）, ε_2=lnR_（int）, ε_3=lnR_（min）, and ε_0=（ε_1＋ε_2＋ε_3）/3. Wei＇s parameter represents the distance from the surface to the spindle of the RIS logarithm ellipsoid; thus, the degree of deformation of the strain ellipsoid is indicated quantitatively. The formula itself, meanwhile, represents the absolute value of the degree of relative deformation and is consequently suitable for any type of deformation of the strain ellipsoid. Wei＇s parameter makes it possible to compare degrees of deformation among different deformation types of the strain ellipsoid. This equation has been tested in four types of coal： highly metamorphic but weakly deformed coal of the southern Qinshui Basin, highly metamorphic and strongly deformed coal from the Tianhushan coal mining area of Fujian, and medium metamorphic and weakly or strongly deformed coal from the Huaibei Coalfield. The results of Wei＇s parameters are consistent with the actual deformation degrees of the coal reservoirs determined by other methods, which supports the effectiveness of this method. In addition, Wei＇s parameter is an important complement to the indicators of the degrees of deformation of coals, which possess certain theoretical significance and practical values.