An asymptotic finite plane deformation analysis of the elastostatic fields at a crack tip in the framework of hyperelastic, isotropic, and nearly incompressible neo-Hookean materials under mode-I loading

Abstract

In this work, stress and displacement fields were computed around a crack tip in the case of nearly incompressible and isotropic neo-Hookean material. The constitutive equation was linearized, so that the Cauchy stress tensor could be written as a sum of two components: the linear response in term of elastic Hooke’s law and the nonlinear one. Based on this decomposition, an asymptotic analysis has been developed, the fields of linear elastic fracture mechanics (LEFM-theory) are the zero-order terms of the asymptotic expansion. The validity of the proposed theory has been checked in the case of a mode-I crack problem. A numerical model was constructed using a finite element method. It has shown that the computed fields arising from this theory are qualitatively in agreement with those of the finite element simulations.

Appendices

Appendix A

In Cartesian coordinates, the determinant of the deformation gradient tensor is defined as follows:

$$\begin{aligned} {J}=\frac{1}{6} \in _{{ijk}} \in _{{pqr}} {F}_{{ip}} {F}_{{jq}} {F}_{{kr}} \end{aligned}$$

(A.1)

where $\in _{{uvw}}$ is the Levi-Civita permutation symbol.

The components of the deformation gradient tensor are defined as follows:

$$\begin{aligned} {F}_{{rs}} = \delta _{{rs}} + {H}_{{rs}} \end{aligned}$$

(A.2)

where $\delta _{{rs}}$ is the symbol of Kronecker and ${H}_{{rs}}$ denotes the components of the displacement gradient tensor.

Substituting (A.2) into (A.1) leads to:

$$\begin{aligned} J=\frac{1}{6}\in _{{ijk}} \in _{{pqr}} \left( {\delta _{{ip}} +{H}_{{ip}} } \right) \left( {\delta _{{jq}} +{H}_{{jq}} } \right) \left( {\delta _{{kr}} + {H}_{{kr}} } \right) . \end{aligned}$$

(A.3)

Equation (A.3) can be developed as follows:

$$\begin{aligned} J= & {} 1+\frac{1}{6}\left( {\in _{{ijk}} \in _{{pjk}} {H}_{{ip}} +\in _{{ijk}} \in _{{iqk}} {H}_{{jq}} +\in _{{ijk}} \in _{{ijr}} {H}_{{kr}} } \right) \nonumber \\&+\frac{1}{6}\left( {\in _{{ijk}} \in _{{pqk} } {H}_{{ip}} {H}_{{jq}} +\in _{{ijk}} \in _{{pjr}} {H}_{{ip}} {H}_{{kr}} +\in _{{ijk}} \in _{{iqr}} {H}_{{jq}} {H}_{{kr}} } \right) +\frac{1}{6}\in _{{ijk}} \in _{{pqr}} {H}_{{ip}} {H}_{{jq}} {H}_{{kr}}\nonumber \\ \end{aligned}$$

(A.4)

where

$$\begin{aligned} \frac{1}{6}\left( {\in _{{ijk}} \in _{{pjk}} {H}_{{ip}} +\in _{{ijk}} \in _{{iqk}} {H}_{{jq}} +\in _{{ijk}} \in _{{ijr}} {H}_{{kr}} } \right) ={H}_{11} +{H}_{22} +{H}_{33} =\hbox {tr } {\mathbf{H}} \end{aligned}$$

(A.5)

and

$$\begin{aligned} \frac{1}{6}\left( {\in _{{ijk}} \in _{{pqk}} {H}_{{ip}} {H}_{{jq}} +\in _{{ijk}} \in _{{pjr}} {H}_{{ip}} {H}_{{kr}} +\in _{{ijk}} \in _{{iqr}} {H}_{{jq}} {H}_{{kr}}} \right) =\frac{1}{2}\left[ {\left( {\hbox {tr } {\mathbf{H}}} \right) ^{2}-\hbox {tr}\left( {{\mathbf{H}}^{2}} \right) } \right] . \end{aligned}$$

(A.6)

Combining (A.4)–(A.6) allows to rewrite the deformation gradient tensor J as follows:

$$\begin{aligned} J=1+\hbox {tr }{\mathbf{H}}+\frac{1}{2}\left[ {\left( {\hbox {tr } {\mathbf{H}}} \right) ^{2}-\hbox {tr}\left( {{\mathbf{H}}^{2}} \right) } \right] +\det {\mathbf{H}}. \end{aligned}$$

(A.7)

It is worth mentioning that $\left( {\hbox {tr } {\mathbf{H}}} \right) ^{2}$ and $\left( {\hbox {tr }{{\mathbf{H}}}^{2}} \right)$ are, respectively, quadratic and third-order terms with respect to H.

      We assume that J is linear in $\mathbf{H}$ and given by

$$\begin{aligned} J\approx 1+\hbox {tr }{\mathbf{H}}+{{o}}\left( {\hbox {tr }{\mathbf{H}}} \right) . \end{aligned}$$

(A.8)

Appendix B

Substituting Eq. (13), i.e., ${\mathbf{H}}\approx {\mathbf{H}}^{( 0 )}+{\mathbf{H}}^{\left( 1 \right) }$, in Eq. (15) allows to write:

$$\begin{aligned} {{\varvec{\upsigma }}^{{{n}}\ell }}/{\mu _{0} }= & {} \left( {\left( {{\mathbf{H}}^{( 0 )}+{\mathbf{H}}^{\left( 1 \right) }} \right) +\left( {{\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}} \right) +\left( {{\mathbf{H}}^{( 0 )}+{\mathbf{H}}^{\left( 1 \right) }} \right) \left( {{\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}} \right) } \right) {\mathrm{tr}}\left( {{\mathbf{H}}^{( 0 )}+{\mathbf{H}}^{\left( 1 \right) }} \right) \nonumber \\&-\left( {{\mathbf{H}}^{( 0 )}+{\mathbf{H}}^{\left( 1 \right) }} \right) \left( {{\mathbf{H}}^{( 0 )^{\mathrm{T}}}+ {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}} \right) . \end{aligned}$$

(B.1)

Equation (B.1) can be developed as follows:

$$\begin{aligned} {{\varvec{\upsigma }}^{{n}\ell }}/{\mu _{0} }= & {} \left( {\begin{array}{l} {\mathbf{H}}^{( 0 )}+{\mathbf{H}}^{\left( 1 \right) }+{\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+ \\ {\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}} \\ \end{array}} \right) {\mathrm{tr}}\left( {{\mathbf{H}}^{( 0 )}} \right) \nonumber \\&+\left( {\begin{array}{l} {\mathbf{H}}^{( 0 )}+{\mathbf{H}}^{\left( 1 \right) }+{\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+ \\ {\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}} \\ \end{array}} \right) {\mathrm{tr}}\left( {{\mathbf{H}}^{\left( 1 \right) }} \right) \nonumber \\&-\left( {{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}} \right) , \end{aligned}$$

(B.2)

$$\begin{aligned} {{\varvec{\upsigma }}^{{n}\ell }}/{\mu _{0} }= & {} \left( {{\mathbf{H}}^{( 0 )}+{\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{( 0 ){\mathrm{T}}}} \right) {\mathrm{tr}}\left( {{\mathbf{H}}^{( 0 )}} \right) -\left( {{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{( 0 ){\mathrm{T}}}} \right) \nonumber \\&+\left( {{\mathbf{H}}^{\left( 1 \right) }+{\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}} \right) {\mathrm{tr}}\left( {{\mathbf{H}}^{( 0 )}} \right) \nonumber \\&+\left( {{\mathbf{H}}^{( 0 )}+{\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}} \right) {\mathrm{tr}}\left( {{\mathbf{H}}^{\left( 1 \right) }} \right) \nonumber \\&-\left( {{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{( 0 ){\mathrm{T}}}} \right) + \left( {{\mathbf{H}}^{\left( 1 \right) }+{\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}} \right) {\mathrm{tr}}\left( {{\mathbf{H}}^{\left( 1 \right) }} \right) -\left( {{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}} \right) .\nonumber \\ \end{aligned}$$

(B.3)

Using both Eqs. (15) and (B.3), we can write:

$$\begin{aligned} {{\varvec{\upsigma }}^{{n}\ell }}/{\mu _{0} }\approx {{{\varvec{\Sigma }} }^{( 0 )}}/{\mu _{0} }+{{{\varvec{\Sigma }} }^{\left( 1 \right) }}/{\mu _{0} }+{{{\varvec{\Sigma }} }^{\left( 2 \right) }}/{\mu _{0} } . \end{aligned}$$

(B.4)

where ${{\varvec{\Sigma }}^{( 0 )}}/{\mu _{0} }=\left( {{\mathbf{H}}^{( 0 )}+{\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathrm{H}}^{( 0 ){\mathrm{T}}}} \right) {\mathrm{tr}}\left( {{\mathbf{H}}^{( 0 )}} \right) -\left( {{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{( 0 ){\mathrm{T}}}} \right) ,$

$$\begin{aligned} \begin{array}{l} {{\varvec{\Sigma }}^{\left( 1 \right) }}/{\mu _{0} }=\left( {{\mathbf{H}}^{\left( 1 \right) }+{\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}} \right) \hbox {tr}\left( {{\mathbf{H}}^{( 0 )}} \right) \\ \qquad \qquad \qquad + \left( {{\mathbf{H}}^{( 0 )}+{\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{( 0 ){\mathrm{T}}}+{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}} \right) \hbox {tr}\left( {{\mathbf{H}}^{\left( 1 \right) }} \right) - \left( {{\mathbf{H}}^{( 0 )} {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{( 0 ){\mathrm{T}}}} \right) ,\\ \end{array} \end{aligned}$$

and ${{\varvec{\Sigma }}^{\left( 2 \right) }}/{\mu _{0} }=\left( {{\mathbf{H}}^{\left( 1 \right) }+{\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}+{\mathbf{H}}^{\left( 1 \right) } {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}} \right) {\mathrm{tr}}\left( {{\mathbf{H}}^{\left( 1 \right) }} \right) -\left( {{{\mathbf{H}}}^{\left( 1 \right) } {\mathbf{H}}^{\left( 1 \right) {\mathrm{T}}}} \right) .$

Appendix C

We calculated the normalized displacement norm of the LEFM-theory, from Eqs. (24) and (25), as follows:

$$\begin{aligned} {U^{( 0 )}}/a=\sqrt{\left[ {u_r^{( 0 )} } \right] ^{2}+\left[ {u_\theta ^{( 0 )} } \right] ^{2}}. \end{aligned}$$

(C.1)

From Eqs. (28), (32), and (33), the normalized displacement norm of the present theory can be written as

$$\begin{aligned} {U^{\left( \ell \right) }}/a=\sqrt{u_r^2 +u_\theta ^2 }. \end{aligned}$$

(C.2)

From Eqs. (34) and (35), the normalized equivalent von Mises stress of the LEFM-theory is given by

$$\begin{aligned} {\sigma _{( \mathrm{VM} )}^{( 0 )} }/{\mu _{0} }=\sqrt{\left( {\sigma _{rr}^{( 0 )} -\sigma _{\theta \theta }^{( 0 )} } \right) ^{2}+\sigma _{rr}^{( 0 )} \sigma _{\theta \theta }^{( 0 )} +3 \sigma _{r\theta }^{( 0 )} }. \end{aligned}$$

(C.3)

We have computed the normalized equivalent von Mises stress of the present theory by using results of Eq. (36) as follows:

$$\begin{aligned} {\sigma _{( \mathrm{VM} )}^\ell }/{\mu _{0} }=\sqrt{\left( {\sigma _{rr}^\ell -\sigma _{\theta \theta }^\ell } \right) ^{2}+\sigma _{rr}^\ell \sigma _{\theta \theta }^\ell +3 \sigma _{r\theta }^\ell }. \end{aligned}$$

(C.4)