Table 1 Introduction of hyperelastic constitutive model.

From: Mechanical constitutive model of stand off damping composites layered rubber core under wide strain rates

Name

Material constant

Expression of strain energy density function

Constitutive equation

M-R(2)

C10, C01

\(W = C_{10} (I_{1} - 3) + C_{01} \left( {I_{2} - 3} \right)\)

\(\sigma = 2[(1 + \varepsilon )^{2} - (1 + \varepsilon )^{ - 1} ][C_{10} + C_{01} (1 + \varepsilon )^{ - 1} ]\)

M-R(3)

C10, C01, C11

\(\begin{gathered} W = C_{10} (I_{1} - 3) + C_{01} \left( {I_{2} - 3} \right) \\ + C_{11} \left( {I_{1} - 3} \right)\left( {I_{2} - 3} \right) \\ \end{gathered}\)

\(\begin{gathered} \sigma = 2\left[ {(1 + \varepsilon )^{2} - (1 + \varepsilon )^{ - 1} } \right] \cdot \left[ {C_{10} + C_{11} [2(1 + \varepsilon ) + (1 + \varepsilon )^{ - 2} - 3} \right] \hfill \\ \;\;\;\; + \left[ {C_{01} + C_{11} \left( {2(1 + \varepsilon )^{ - 1} + (1 + \varepsilon )^{2} - 3} \right)](1 + \varepsilon )^{ - 1} } \right] \hfill \\ \end{gathered}\)

M-R(5)

C10, C01, C11, C20, C02

\(\begin{gathered} W = C_{10} (I_{1} - 3) + C_{01} \left( {I_{2} - 3} \right) \\ + C_{11} \left( {I_{1} - 3} \right)\left( {I_{2} - 3} \right) \\ + C_{20} (I_{1} - 3)^{2} + C_{02} (I_{2} - 3)^{2} \\ \end{gathered}\)

\(\begin{gathered} \sigma = 2[(1 + \varepsilon )^{2} - (1 + \varepsilon )^{ - 1} ][C_{10} + C_{11} \left( {I_{2} - 3} \right) \\ + 2C_{20} (I_{1} - 3) + (1 + \varepsilon )^{ - 1} [C_{01} + C_{11} \left( {I_{1} - 3} \right) + 2C_{02} (I_{2} - 3)]] \\ \end{gathered}\)

Yeoh

C10, C20, C30

\(\begin{gathered} W = C_{10} (I_{1} - 3) + C_{20} (I_{1} - 3)^{2} \\ + C_{30} (I_{1} - 3)^{3} \\ \end{gathered}\)

\(\begin{gathered} \sigma = 2[(1 + \varepsilon )^{2} - (1 + \varepsilon )^{ - 1} ][C_{10} + 2C_{20} [(1 + \varepsilon )^{2} + 2(1 + \varepsilon )^{ - 1} - 3] \\ + 3C_{30} [(1 + \varepsilon )^{2} + 2(1 + \varepsilon )^{ - 1} - 3]^{2} ] \\ \end{gathered}\)

Ogden(2)

μ, α

\(U = \frac{2\mu }{{\alpha^{2} }}(\lambda_{1}^{\alpha } + \lambda_{2}^{\alpha } + \lambda_{3}^{\alpha } - 3)\)

\(\sigma = \frac{2\mu }{\alpha }\left( {(1 + \varepsilon )^{(\alpha - 1)} - (1 + \varepsilon )^{{ - \frac{(\alpha - 1)}{2}}} } \right)\)

Ogden(4)

μ1, α1, μ2, α2

\(\begin{gathered} U = \frac{{2\mu_{1} }}{{\alpha_{1}^{2} }}(\lambda_{1}^{{\alpha_{1} }} + \lambda_{2}^{{\alpha_{1} }} + \lambda_{3}^{{\alpha_{1} }} - 3) \\ + \frac{{2\mu_{2} }}{{\alpha_{2}^{2} }}(\lambda_{1}^{{\alpha_{2} }} + \lambda_{2}^{{\alpha_{2} }} + \lambda_{3}^{{\alpha_{2} }} - 3) \\ \end{gathered}\)

\(\begin{gathered} \sigma = \frac{{2\mu_{1} }}{{\alpha_{1} }}\left( {(1 + \varepsilon )^{{(\alpha_{1} - 1)}} - (1 + \varepsilon )^{{ - \frac{{(\alpha_{1} - 1)}}{2}}} } \right) \\ + \frac{{2\mu_{2} }}{{\alpha_{2} }}\left( {(1 + \varepsilon )^{{(\alpha_{2} - 1)}} - (1 + \varepsilon )^{{ - \frac{{(\alpha_{2} - 1)}}{2}}} } \right) \\ \end{gathered}\)

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