Mooney–Rivlin solid

In continuum mechanics, a Mooney–Rivlin solid is a hyperelastic material model where the strain energy density function W {displaystyle W,} is a linear combination of two invariants of the left Cauchy–Green deformation tensor B {displaystyle {oldsymbol {B}}} . The model was proposed by Melvin Mooney in 1940 and expressed in terms of invariants by Ronald Rivlin in 1948. In continuum mechanics, a Mooney–Rivlin solid is a hyperelastic material model where the strain energy density function W {displaystyle W,} is a linear combination of two invariants of the left Cauchy–Green deformation tensor B {displaystyle {oldsymbol {B}}} . The model was proposed by Melvin Mooney in 1940 and expressed in terms of invariants by Ronald Rivlin in 1948. The strain energy density function for an incompressible Mooney–Rivlin material is where C 1 {displaystyle C_{1}} and C 2 {displaystyle C_{2}} are empirically determined material constants, and I ¯ 1 {displaystyle {ar {I}}_{1}} and I ¯ 2 {displaystyle {ar {I}}_{2}} are the first and the second invariant of B ¯ = ( det B ) − 1 / 3 B {displaystyle {ar {oldsymbol {B}}}=(det {oldsymbol {B}})^{-1/3}{oldsymbol {B}}} (the unimodular component of B {displaystyle {oldsymbol {B}}} ): where F {displaystyle {oldsymbol {F}}} is the deformation gradient and J = det ( F ) = λ 1 λ 2 λ 3 {displaystyle J=det({oldsymbol {F}})=lambda _{1}lambda _{2}lambda _{3}} . For an incompressible material, J = 1 {displaystyle J=1} . The Mooney–Rivlin model is a special case of the generalized Rivlin model (also called polynomial hyperelastic model) which has the form with C 00 = 0 {displaystyle C_{00}=0} where C p q {displaystyle C_{pq}} are material constants related to the distortional response and D m {displaystyle D_{m}} are material constants related to the volumetric response. For a compressible Mooney–Rivlin material N = 1 , C 01 = C 2 , C 11 = 0 , C 10 = C 1 , M = 1 {displaystyle N=1,C_{01}=C_{2},C_{11}=0,C_{10}=C_{1},M=1} and we have If C 01 = 0 {displaystyle C_{01}=0} we obtain a neo-Hookean solid, a special case of a Mooney–Rivlin solid. For consistency with linear elasticity in the limit of small strains, it is necessary that where κ {displaystyle kappa } is the bulk modulus and μ {displaystyle mu } is the shear modulus.