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Rubber elasticity is the ability of solid rubber to be stretched up to a factor of 10 from its original length, and return to close to its original length upon release. This process can be repeated many times with no apparent degradation to the rubber.
The Gent hyperelastic material model [1] is a phenomenological model of rubber elasticity that is based on the concept of limiting chain extensibility. In this model, the strain energy density function is designed such that it has a singularity when the first invariant of the left Cauchy-Green deformation tensor reaches a limiting value .
In continuum mechanics, an Arruda–Boyce model [1] is a hyperelastic constitutive model used to describe the mechanical behavior of rubber and other polymeric substances. This model is based on the statistical mechanics of a material with a cubic representative volume element containing eight chains along the diagonal directions.
where is the volume fraction of the fibers in the composite (and is the volume fraction of the matrix).. If it is assumed that the composite material behaves as a linear-elastic material, i.e., abiding Hooke's law = for some elastic modulus of the composite and some strain of the composite , then equations 1 and 2 can be combined to give
The polynomial hyperelastic material model [1] is a phenomenological model of rubber elasticity. In this model, the strain energy density function is of the form of a polynomial in the two invariants I 1 , I 2 {\displaystyle I_{1},I_{2}} of the left Cauchy-Green deformation tensor.
The bulk modulus (K) describes volumetric elasticity, or the tendency of an object to deform in all directions when uniformly loaded in all directions; it is defined as volumetric stress over volumetric strain, and is the inverse of compressibility. The bulk modulus is an extension of Young's modulus to three dimensions.
The Yeoh model for incompressible rubber is a function only of . For compressible rubbers, a dependence on I 3 {\displaystyle I_{3}} is added on. Since a polynomial form of the strain energy density function is used but all the three invariants of the left Cauchy-Green deformation tensor are not, the Yeoh model is also called the reduced ...
In orthogonal coordinates, the elastic energy per unit volume due to strain is thus a sum of contributions: =, where is a 4th rank tensor, called the elastic tensor or stiffness tensor [3] which is a generalization of the elastic moduli of mechanical systems, and is the strain tensor (Einstein summation notation has been used to imply summation ...