Mooney-Rivilin strain-energy function). Recall that the classical neo-Hookean free energy is with shear modulus . Another...

  

Transcribed Image Text:

Mooney-Rivilin strain-energy function). Recall that the classical neo-Hookean free energy is with shear modulus μ. Another commonly used free energy, which depends not only on the first invariant I₁ but also on the second invariant YR 12 = PR=0 (1₁ − 3), 1₁ def A² + a² + a². 2 def = is provided by the Mooney-Rivlin free energy function (Mooney, 1940; Rivlin, 1948; Rivlin and Saunders, 1951) C₁ -2 -2 ^²^3 + A3^² + A}A} = A₁² + A₂² + Az², (11 − 3) + /2² (1₂ − 3) - 2 C1 C₂ = (x² + x³ + x² − 3) + / (¹5² +Az²+A5² − 3) 2 2 with C₁ > 0 and C₂ > 0 material constants. We refer to an incompressible, isotropic, hyperelastic body with free energy of this form as a Mooney-Rivlin material. It may be shown that for a Mooney-Rivlin material in simple shear the shear modulus is given by Clearly when Ho def C₁ + C₂ > 0 C₂ = 0 and C₁ = μ0, the Mooney-Rivlin free energy (4.3) reduces to the neo-Hookean free energy (4.1). With a suitable choice of the two constants C₁ and C₂, the Mooney-Rivlin free energy func- tion often gives a slightly better fit to experimental data for rubber-like materials than the neo-Hookean free energy. This could be because the Mooney-Rivlin free energy has two adjustable parameters. (a) For a certain elastomer it has been determined experimentally that its free en- ergy for moderately large stretch levels is adequately represented by the Mooney- Rivlin free energy with C₁ = 0.6MPa and Consider a simple extension defined by C2 = 0.2MPa. A₁ = A₁ A₂ = A3=A−¹/², Mooney-Rivilin strain-energy function). Recall that the classical neo-Hookean free energy is with shear modulus μ. Another commonly used free energy, which depends not only on the first invariant I₁ but also on the second invariant YR 12 = PR=0 (1₁ − 3), 1₁ def A² + a² + a². 2 def = is provided by the Mooney-Rivlin free energy function (Mooney, 1940; Rivlin, 1948; Rivlin and Saunders, 1951) C₁ -2 -2 ^²^3 + A3^² + A}A} = A₁² + A₂² + Az², (11 − 3) + /2² (1₂ − 3) - 2 C1 C₂ = (x² + x³ + x² − 3) + / (¹5² +Az²+A5² − 3) 2 2 with C₁ > 0 and C₂ > 0 material constants. We refer to an incompressible, isotropic, hyperelastic body with free energy of this form as a Mooney-Rivlin material. It may be shown that for a Mooney-Rivlin material in simple shear the shear modulus is given by Clearly when Ho def C₁ + C₂ > 0 C₂ = 0 and C₁ = μ0, the Mooney-Rivlin free energy (4.3) reduces to the neo-Hookean free energy (4.1). With a suitable choice of the two constants C₁ and C₂, the Mooney-Rivlin free energy func- tion often gives a slightly better fit to experimental data for rubber-like materials than the neo-Hookean free energy. This could be because the Mooney-Rivlin free energy has two adjustable parameters. (a) For a certain elastomer it has been determined experimentally that its free en- ergy for moderately large stretch levels is adequately represented by the Mooney- Rivlin free energy with C₁ = 0.6MPa and Consider a simple extension defined by C2 = 0.2MPa. A₁ = A₁ A₂ = A3=A−¹/²,

Answer rating: 100% (QA)

d Shear Modulus for Simple Shear import numpy as np import matplotlibpyplot as plt Function to calcu... View the full answer