where Ψ is the strain energy density function, Ī1 and Ī2 are the first and the second invariants of the deviatoric component of the left-Cauchy-Green deformation tensor, and c10 and c01 are material constants. In this work, Eq.(2) is implemented in a numerical simulation of a quasi-static micro-indentation. The basic assumption, referring to incompressibility, is that soft tissues, are composed prevalently by water which is ideally considered incompressible. Rubber is also characterized by a Poisson’s coefficients close to 0.5. Equivalent Young’s modulus for a Mooney-Rivlin solid For the case of an incompressible Mooney-Rivlin material under uniaxial elongation (simple tension), the constitutive law Eq.(2) can be written, considering the Cauchy stress in function of the stretches as, ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⋅ − ⎠ ⎞ ⎜ ⎝ ⎛ = + λ λ λ σ 1 2 2 2 01 10 c c (3) where σ is the Cauchy stress in the applied tensile direction, c10 and c01 are the material constants introduced in Eq.(2) and λ is the uni-axial stretch. According to Eq.(3) can be said that, considering small strain, the ratio σ/ε at infinitesimally small strain can be determined by the relation, ) 6 ( 01 10 c c Eeq = ⋅ + (4) The complete description of Eq.(4) is given in Appendix. In this work, the equivalent Young’s modulus Eeq, given by Eq.(4) is used to compare the results with that one provided by standard tensile tests on silicon rubber and the Boussinesq’s analytical solution [27]. MATERIAL & METHODS The material characterization procedure is schematically represented in Fig.(1). The calculation of the two material constants for a Mooney-Rivlin material model, formulated under incompressibility condition, is performed employing a non-destructive indentation test. The methodology couples the results obtained by a real experiment with that one calculated numerically by a FE simulation of indentation of a deformable sample. The coupling consists in the minimization of a cost function built considering the square residuals given by, ∑ = = N i ir S 1 2 with FEM EXP i i i f f r ( ) ( ) δ δ − = (5) 22
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