A second-order strain gradient fracture model for the brittle materials with micro-cracks by a multiscale asymptotic homogenization

Abstract

In this work, applying a second-order multiscale asymptotic homogenization, an effective fracture model is established for the brittle materials with periodic distribution of micro-cracks. The novel second-order strain gradient fracture model based on the multiscale asymptotic technique is rigorously derived without any phenomenological assumptions, and the fourth-, sixth-, and eighth-order effective elastic tensors of the fracture criterions are obtained by the first-order and second-order multiscale unit cell functions. The significant features of the novel model are: (i) the first-order, second-order strain gradient effect and microstructure size ξ included in the fracture criterion and (ii) the strain energy and the Griffith criterion for micro-crack extensions obtained by the high-order multiscale asymptotic homogenization. Finally, the effectiveness of the proposed model is compared with the direct numerical simulations (DNS), experimental data and some typical fracture problems including Mode I crack plate, rectangular plate with two symmetric V-notch and a holed plate are also evaluated. These examples show that the second-order strain gradient fracture model is valid for solving the brittle materials with periodic distribution of micro-cracks.

Appendices

Appendix A: Theoretical analysis for the multiscale asymptotic expansion

In order to compare ${\mathbf{u}}_{1}^{\xi } ({\mathbf{x}})$ and ${\mathbf{u}}_{2}^{\xi } ({\mathbf{x}})$ with the source solutions of (1), we take ${\mathbf{u}}^{\xi } ({\mathbf{x}}) - {\mathbf{u}}_{1}^{\xi } ({\mathbf{x}})$ into the problem (1) of this work, and has the following equality:

$$\begin{aligned} & L_{\xi } \left( {{\mathbf{u}}^{\xi }({\mathbf{x}}) - {\mathbf{u}}_{1}^{\xi } ({\mathbf{x}})} \right) =f_{i} ({\mathbf{x}}) \\ & \quad \times \left[ {\frac{\partial }{{\partial y_{j} }}} \left( {L_{{ijk\alpha_{2} }}({\mathbf{y}})\varphi_{m}^{{\alpha_{1} k}} ({\mathbf{y}})} \right) +L_{{i\alpha_{1} m\alpha_{2} }} ({\mathbf{y}})\right. \\ & \quad \left.+ L_{{i\alpha_{1} kj}} ({\mathbf{y}}) {\frac{{\partial \varphi_{m}^{{\alpha_{2} k}} ({\mathbf{y}})}}{{\partial y_{j} }}}\right]\frac{{\partial^{2} u_{k}^{(0)} }}{{\partial x_{{\alpha_{1} }} \partial x_{{\alpha_{2} }} }} + \xi L_{ijkl}({\mathbf{y}})\varphi_{m}^{{\alpha_{1} k}} ({\mathbf{y}})\\ & \quad \frac{{\partial^{3} u_{k}^{(0)} }}{{\partial x_{j} \partial x_{l}\partial x_{{\alpha{1} }} }} = O(1) \end{aligned}$$

(36)

where $L_{\xi } = - \frac{\partial }{{\partial x_{j} }}\left[ {L_{ijkl} (\frac{{\mathbf{x}}}{\xi })\frac{{1}}{{2}}(\frac{\partial }{{\partial x_{k} }} + \frac{\partial }{{\partial x_{l} }})} \right]$. It can be found that the residual (36) is the order $O(1)$. However, $\xi$ is a fixed parameter rather than tending to zero for practical engineering computation. The error $O(1)$ can be not accepted for scientists who want to accurately obtain the microscopic information of the periodic composites.

Further, taking ${\mathbf{u}}^{\xi } ({\mathbf{x}}) - {\mathbf{u}}_{{2}}^{\xi } ({\mathbf{x}})$ into source Eqs. (1) of this work yields to

$$ \begin{aligned} &L_{\xi } \left( {{\mathbf{u}}^{\xi } ({\mathbf{x}}) - {\mathbf{u}}_{{2}}^{\xi } ({\mathbf{x}})} \right) \hfill \\ &\quad= - \frac{\partial }{{\partial x_{j} }}(L_{ijkl}^{\xi } ({\mathbf{x}})\frac{{\partial u_{k}^{\xi } ({\mathbf{x}})}}{{\partial x_{l} }}) \hfill \\ &\qquad+ \frac{\partial }{{\partial x_{j} }}\left\{ {L_{ijkl} ({\mathbf{y}})\frac{\partial }{{\partial x_{j} }}\left[ {{\mathbf{u}}^{(0)} ({\mathbf{x}}) + \xi \left( {{\varvec{\varphi }}^{kl} ({\mathbf{y}})\varepsilon_{kl}^{(0)} ({\mathbf{x}})} \right)} \right.} \right. \hfill \\ &\qquad+ \left. {\left. {\xi^{2} \left( {{{\varvec{\Phi}}}_{p}^{kl} ({\mathbf{y}})\frac{\partial }{{\partial x_{p} }}\varepsilon_{kl}^{(0)} ({\mathbf{x}})} \right)} \right]} \right\} = O(\xi ), \hfill \\ \end{aligned} $$

(37)

It is easy to find that the residuals of (37) are the order $O(\xi )$. In other words, the second-order multiscale approximate solutions are equivalent to the solutions of source problem (1) in $O(\xi )$-order pointwise sense. And, it is important to obtain the accurate solutions of the periodic composites for the actual engineering computations. This is also why we consider the second-order multiscale asymptotic expansion in this work.

Appendix B: Symmetric and anti-symmetric property of first- and second-order cell functions

The micro-cracks of the structure with length l are supposed to be straight, and the cell configurations are symmetric about the middle plane, as shown in Fig. 

Fig. 31

Refined meshes in unit cell with d = 0.5

31. The symmetric and anti-symmetric features of first- and second-order cell functions are shown in Figs.

Fig. 32

First-order cell functions $\varphi_{i}^{kl} ({\mathbf{y}})$ with d = 0.5

32 and

Fig. 33

Second-order cell functions $\Phi_{ip}^{kl} ({\mathbf{y}})$ with d = 0.5

33 with normalized micro-crack d = 0.5.

The proof of the symmetric and anti-symmetric features for the first-order functions $\varphi_{i}^{kl} ({\mathbf{y}})$ can be found in Rao et al. [42] and Cao [80]. Similarly, the symmetry and anti-symmetry for the second-order unit cell functions $\Phi_{ip}^{kl} ({\mathbf{y}})$ can also obtained Cao [80]. The variational forms of (13) are

$$ \begin{aligned} &\int_{{\Theta_{s} }} {\left( {L_{{ijk\alpha_{2} }} ({\mathbf{y}})\varphi_{k}^{{\alpha_{1} m}} ({\mathbf{y}}) + L_{ijkl} ({\mathbf{y}})\varepsilon_{ykl} ({{\varvec{\Phi}}}_{m}^{{\alpha_{1} \alpha_{2} }} ({\mathbf{y}}))} \right)} \varepsilon_{yij} ({\mathbf{v}}) \hfill \\&\quad + (L_{{i\alpha_{1} m\alpha_{2} }} ({\mathbf{y}}) + L_{{i\alpha_{1} kj}} ({\mathbf{y}})\frac{{\partial \varphi_{k}^{{\alpha_{2} m}} ({\mathbf{y}})}}{{\partial y_{j} }} \\&\quad- \frac{{1}}{{\left| {\Theta_{s} } \right|}}\overline{L}_{{i\alpha_{1} m\alpha_{2} }} ){\mathbf{v}}_{i} d{\mathbf{y}} = 0,\;\;\forall {\mathbf{v}} \in S(\Theta_{s} ) \hfill \\ \end{aligned} $$

(38)

where $\varepsilon_{ykl} ({\mathbf{v}}) = \frac{{1}}{{2}}(\frac{{\partial v_{k} }}{{\partial y_{l} }} + \frac{{\partial v_{l} }}{{\partial y_{k} }})$.

In addition, the function spaces are defined by

$$ S(\Theta_{s} ) = \left\{ {{\mathbf{v}}|{\mathbf{v}} \in \left[ {H^{1} (\Theta_{s} )} \right]^{2} ,{\mathbf{v}}\;{\text{is}}\;{\text{periodic}}\;{\text{in}}\;\Theta_{s} ,\;\int_{{\Theta_{s} }} {{\mathbf{v}}d{\mathbf{y}} = 0} \;} \right\} $$

where $H^{1} (\Theta_{s} )$ represents the Sobolev space [Cao [80]].

Considering ${{\varvec{\Phi}}}_{p}^{kl} ({\mathbf{y}})$, and we introduce a vector valued ${\overline{\varvec{\Phi }}}_{1}^{11} (y_{1} ,y_{2} )$ given by

$\overline{\Phi }_{11}^{11} (y_{1} ,y_{2} ) = \Phi_{11}^{11} (1 - y_{1} ,y_{2} )$, $\overline{\Phi }_{12}^{11} (y_{1} ,y_{2} ) = - \Phi_{12}^{11} (1 - y_{1} ,y_{2} )$.

Further, a given test function ${\mathbf{v}}(y_{1} ,y_{2} )$ is defined as follows:

$$ {\overline{\mathbf{v}}}_{11}^{11} (y_{1} ,y_{2} ) = {\mathbf{v}}_{11}^{11} (1 - y_{1} ,y_{2} ),\;\;\;\;{\overline{\mathbf{v}}}_{12}^{11} (y_{1} ,y_{2} ) = - {\mathbf{v}}_{12}^{11} (1 - y_{1} ,y_{2} ) $$

Using (38), we have the following equality:

$$ \begin{gathered} \int_{{\Theta_{s} }} {\left( {L_{ijk1} ({\mathbf{y}})\varphi_{k}^{11} ({\mathbf{y}}) + L_{ijkl} ({\mathbf{y}})\varepsilon_{ykl} ({\overline{\varvec{\Phi }}}_{1}^{11} ({\mathbf{y}}))} \right)} \varepsilon_{yij} ({\overline{\mathbf{v}}}) \hfill \\ \quad+ (L_{i111} ({\mathbf{y}}) + L_{i1kj} ({\mathbf{y}})\frac{{\partial \varphi_{k}^{11} ({\mathbf{y}})}}{{\partial y_{j} }} - \frac{{1}}{{\left| {\Theta_{s} } \right|}}\overline{L}_{i111} ){\overline{\mathbf{v}}}_{i} d{\mathbf{y}} = 0, \hfill \\ \end{gathered} $$

(39)

From (38) and (39), it can be found that ${\overline{\varvec{\Phi }}}_{1}^{11} (y_{1} ,y_{2} )$ and ${{\varvec{\Phi}}}_{1}^{11} (y_{1} ,y_{2} )$ have the same variational forms. And, the second-order unit cell functions should satisfy that

$$ {{\varvec{\Phi}}}_{11}^{11} (y_{1} ,y_{2} ) = \overline{\Phi }_{11}^{11} (y_{1} ,y_{2} ) = \Phi_{11}^{11} (1 - y_{1} ,y_{2} ) $$

and.

$\Phi_{12}^{11} (y_{1} ,y_{2} ) = \overline{\Phi }_{12}^{11} (y_{1} ,y_{2} ) = - \Phi_{12}^{11} (1 - y_{1} ,y_{2} )$.

Finally, we can obtain the following relations for all the unit cell functions:

$$ \begin{gathered} \varphi_{{1}}^{{{11}}} (y_{1} ,y_{2} ) = - \varphi_{{1}}^{{{11}}} (1 - y_{1} ,y_{2} ) = \varphi_{{1}}^{{{11}}} (y_{1} ,1 - y_{2} ) \hfill \\ \varphi_{2}^{{{11}}} (y_{1} ,y_{2} ) = - \varphi_{2}^{{{11}}} (y_{1} ,1 - y_{2} ) = \varphi_{2}^{{{11}}} (1 - y_{1} ,y_{2} ) \hfill \\ \varphi_{1}^{{{12}}} (y_{1} ,y_{2} ) = - \varphi_{1}^{{{12}}} (y_{1} ,1 - y_{2} ) = \varphi_{1}^{{{12}}} (1 - y_{1} ,y_{2} ) \hfill \\ \varphi_{2}^{{{12}}} (y_{1} ,y_{2} ) = - \varphi_{2}^{{{12}}} (1 - y_{1} ,y_{2} ) = \varphi_{2}^{{{12}}} (y_{1} ,1 - y_{2} ) \hfill \\ \varphi_{1}^{{{22}}} (y_{1} ,y_{2} ) = - \varphi_{1}^{{{22}}} (1 - y_{1} ,y_{2} ) = \varphi_{1}^{{{22}}} (y_{1} ,1 - y_{2} ) \hfill \\ \varphi_{2}^{{{22}}} (y_{1} ,y_{2} ) = - \varphi_{2}^{{{22}}} (y_{1} ,1 - y_{2} ) = \varphi_{2}^{{{22}}} (1 - y_{1} ,y_{2} ) \hfill \\ \end{gathered} $$

(40)

and

$$ \begin{gathered} \Phi_{{{11}}}^{{{11}}} (y_{1} ,y_{2} ) = \Phi_{{{11}}}^{{{11}}} (1 - y_{1} ,y_{2} ) = \Phi_{{{11}}}^{{{11}}} (y_{1} ,1 - y_{2} ) \hfill \\ \Phi_{{{21}}}^{{{11}}} (y_{1} ,y_{2} ) = - \Phi_{{{21}}}^{{{11}}} (y_{1} ,1 - y_{2} ) = - \Phi_{{{21}}}^{{{11}}} (1 - y_{1} ,y_{2} ) \hfill \\ \Phi_{{{12}}}^{{{11}}} (y_{1} ,y_{2} ) = - \Phi_{{{12}}}^{{{11}}} (y_{1} ,1 - y_{2} ) = - \Phi_{{{12}}}^{{{11}}} (1 - y_{1} ,y_{2} ) \hfill \\ \Phi_{{{22}}}^{{{11}}} (y_{1} ,y_{2} ) = \Phi_{{{22}}}^{{{11}}} (1 - y_{1} ,y_{2} ) = \Phi_{{{22}}}^{{{11}}} (y_{1} ,1 - y_{2} ) \hfill \\ \Phi_{{{11}}}^{{{12}}} (y_{1} ,y_{2} ) = - \Phi_{{{11}}}^{{{12}}} (1 - y_{1} ,y_{2} ) = - \Phi_{{{11}}}^{{{12}}} (y_{1} ,1 - y_{2} ) \hfill \\ \Phi_{{{21}}}^{{{12}}} (y_{1} ,y_{2} ) = \Phi_{{{21}}}^{{{12}}} (y_{1} ,1 - y_{2} ) = \Phi_{{{21}}}^{{{12}}} (1 - y_{1} ,y_{2} ) \hfill \\ \Phi_{{{12}}}^{{{12}}} (y_{1} ,y_{2} ) = \Phi_{{{12}}}^{{{12}}} (y_{1} ,1 - y_{2} ) = \Phi_{{{12}}}^{{{12}}} (1 - y_{1} ,y_{2} ) \hfill \\ \Phi_{{{22}}}^{{{12}}} (y_{1} ,y_{2} ) = - \Phi_{{{22}}}^{{{12}}} (y_{1} ,1 - y_{2} ) = - \Phi_{{{22}}}^{{{12}}} (1 - y_{1} ,y_{2} ) \hfill \\ \Phi_{{{11}}}^{{{22}}} (y_{1} ,y_{2} ) = \Phi_{{{11}}}^{{{22}}} (y_{1} ,1 - y_{2} ) = \Phi_{{{11}}}^{{{22}}} (1 - y_{1} ,y_{2} ) \hfill \\ \Phi_{{{21}}}^{{{22}}} (y_{1} ,y_{2} ) = - \Phi_{{{21}}}^{{{22}}} (y_{1} ,1 - y_{2} ) = - \Phi_{{{21}}}^{{{22}}} (1 - y_{1} ,y_{2} ) \hfill \\ \Phi_{{{12}}}^{{{22}}} (y_{1} ,y_{2} ) = - \Phi_{{{12}}}^{{{22}}} (y_{1} ,1 - y_{2} ) = - \Phi_{{{12}}}^{{{22}}} (1 - y_{1} ,y_{2} ) \hfill \\ \Phi_{{{22}}}^{{{22}}} (y_{1} ,y_{2} ) = \Phi_{{{22}}}^{{{22}}} (y_{1} ,1 - y_{2} ) = \Phi_{{{22}}}^{{{22}}} (1 - y_{1} ,y_{2} ) \hfill \\ \end{gathered} $$

(41)

Appendix C: Proof for the equalities (27)

The variational equations of (9) are

$$ \int_{{\Theta_{s} }} {\left( {L_{ijkl} ({\mathbf{y}}) + L_{ijmn} ({\mathbf{y}})\varepsilon_{ymn} ({\varvec{\varphi }}^{kl} ({\mathbf{y}}))} \right)} \varepsilon_{yij} ({\mathbf{v}})d{\mathbf{y}} = 0,\;\forall {\mathbf{v}} \in S(\Theta_{s} ), $$

(42)

where $\varepsilon_{ykl} ({\mathbf{v}}) = \frac{{1}}{{2}}\left( {\frac{{\partial v_{k} }}{{\partial y_{l} }} + \frac{{\partial v_{l} }}{{\partial y_{k} }}} \right)$.

Taking ${\mathbf{v}} = {\varvec{\varphi }}^{pq} ({\mathbf{y}})$ in (42), and we have that:

$$ \int_{{\Theta_{s} }} {\left( {L_{ijkl} ({\mathbf{y}}) + L_{ijmn} ({\mathbf{y}})\varepsilon_{ymn} ({\varvec{\varphi }}^{kl} ({\mathbf{y}}))} \right)} \varepsilon_{yij} ({\varvec{\varphi }}^{pq} ({\mathbf{y}}))d{\mathbf{y}} = 0. $$

(43)

Further, choosing ${\mathbf{v}} = {{\varvec{\Phi}}}_{p}^{kl} ({\mathbf{y}})$ for (42) yields to

$$ \int_{{\Theta_{s} }} {\left( {L_{ijkl} ({\mathbf{y}}) + L_{ijmn} ({\mathbf{y}})\varepsilon_{ymn} ({\varvec{\varphi }}^{kl} ({\mathbf{y}}))} \right)} \varepsilon_{yij} ({{\varvec{\Phi}}}_{p}^{kl} ({\mathbf{y}}))d{\mathbf{y}} = 0. $$

(44)