Polytopal composite finite elements for modeling concrete fracture based on nonlocal damage models

Abstract

The paper presents an assumed strain formulation over polygonal meshes to accurately evaluate the strain fields in nonlocal damage models. An assume strained technique based on the Hu-Washizu variational principle is employed to generate a new strain approximation instead of direct derivation from the basis functions and the displacement fields. The underlying idea embedded in arbitrary finite polygons is named as Polytopal composite finite elements (PCFEM). The PCFEM is accordingly applied within the framework of the nonlocal model of continuum damage mechanics to enhance the description of damage behaviours in which highly localized deformations must be captured accurately. This application is helpful to reduce the mesh-sensitivity and elaborate the process-zone of damage models. Several numerical examples are designed for various cases of fracture to discuss and validate the computational capability of the present method through comparison with published numerical results and experimental data from the literature.

Appendix A: Definition of derivatives of the damage evolution function and the equivalent strain

Derivative of the damage evolution function $\omega ' = \dfrac{{\partial \omega }}{{\partial \kappa }}$

$$\begin{aligned} \begin{aligned} \omega '&= \dfrac{{\partial \omega }}{{\partial \kappa }} = \left\{ {\begin{array}{*{20}{l}} 0&{}\quad {{\mathrm{if }\,}\kappa \le {\kappa _0}} \\ {1 - \dfrac{{{\kappa _0}}}{\kappa }\left( {1 - \alpha + \alpha \exp \left( { - \beta \left( {\kappa - {\kappa _0}} \right) } \right) } \right) }&{}{{\mathrm{otherwise}}} \end{array}} \right. \\&= \left\{ {\begin{array}{*{20}{l}} 0&{}\quad {{\mathrm{if }\,}\kappa \le {\kappa _0}} \\ {\dfrac{{{\kappa _0}}}{{{\kappa ^2}}}\left( {1 - \alpha + \alpha \exp \left( { - \beta \left( {\kappa - {\kappa _0}} \right) } \right) } \right) + \dfrac{{{\kappa _0}}}{\kappa }\alpha \beta \exp \left( { - \beta \left( {\kappa - {\kappa _0}} \right) } \right) }&{}\quad {{\mathrm{otherwise}}} \end{array}} \right. \end{aligned} \end{aligned}$$

(A.1)

Derivatives of the equivalent strain with respect to the engineering strain tensor ${\varvec{\eta }} = \dfrac{{\partial {\varepsilon _{eq}}}}{{\partial {\varvec{{\tilde{\varepsilon }} }}}}$

We first determine values of strain components and principal strain involving z-direction in the case of plane strain and plane stress condition.

• For plane strain

  $$\begin{aligned}&{{\tilde{\varepsilon }} _{xz}} = {{\tilde{\varepsilon }} _{yz}} = {{\tilde{\varepsilon }} _{zz}} = 0 \end{aligned}$$

  (A.2a)
  $$\begin{aligned}&{\varepsilon _3} = 0 \end{aligned}$$

  (A.2b)

• For plane stress

  $$\begin{aligned}&{{\tilde{\varepsilon }} _{xz}} = {{\tilde{\varepsilon }} _{yz}} = 0, \,{{\tilde{\varepsilon }} _{zz}} = - \dfrac{\nu }{{1 - \nu }}\left( {{{\tilde{\varepsilon }} _{xx}} + {{\tilde{\varepsilon }} _{yy}}} \right) \end{aligned}$$

  (A.3a)
  $$\begin{aligned}&{\varepsilon _3} = {\tilde{\varepsilon }} _{zz} \end{aligned}$$

  (A.3b)

Derivatives of the equivalent strain computed by Mazars criterion in Eq. and modified von Mises criterion are expressed as

• For Mazars criterion Principal strain in plane xy

  $$\begin{aligned}&{\varepsilon _1} = a - b \end{aligned}$$

  (A.4a)
  $$\begin{aligned}&{\varepsilon _1} = a + b \end{aligned}$$

  (A.4b)

  with $a = 0.5\left( {{{\tilde{\varepsilon }} _{xx}} + {{\tilde{\varepsilon }} _{yy}}} \right)$, and $b = \sqrt{0.5{{\left( {{{\tilde{\varepsilon }} _{xx}} - {{\tilde{\varepsilon }} _{yy}}} \right) }^2} + {\tilde{\varepsilon }} _{xy}^2}$.

  The non-negative part of the principal strains can be written by

  $$\begin{aligned} \left\langle {{\varepsilon _I}} \right\rangle = 0.5\left( {\left| {{\varepsilon _I}} \right| + {\varepsilon _I}} \right) ,{\mathrm{}\,}I = 1,{\mathrm{}}2,{\mathrm{}}3 \end{aligned}$$

  (A.5)

  and components of the principal strain are expressed under a tensor

  $$\begin{aligned} {\mathbf{e}} = \left[ {\left\langle {{\varepsilon _1}} \right\rangle {\mathrm{}}\left\langle {{\varepsilon _2}} \right\rangle {\mathrm{}}\left\langle {{\varepsilon _3}} \right\rangle } \right] \end{aligned}$$

  (A.6)

  By applying the chain rule, we get

  $$\begin{aligned} {\varvec{\eta }} = \dfrac{{\partial {\varepsilon _{eq}}}}{{\partial {\varvec{{\tilde{\varepsilon }} }}}} = \dfrac{{\partial {\varepsilon _{eq}}}}{{\partial {\mathbf{e}}}} \dfrac{{\partial {\mathbf{e}}}}{{\partial {\varvec{{\tilde{\varepsilon }} }}}} \end{aligned}$$

  (A.7)

  in which $\dfrac{{\partial {\varepsilon _{eq}}}}{{\partial {\mathbf{e}}}} = \dfrac{1}{{{\varepsilon _{eq}}}}\left[ {\left\langle {{\varepsilon _1}} \right\rangle \ {\mathrm{}}\left\langle {{\varepsilon _2}} \right\rangle \ {\mathrm{}}\left\langle {{\varepsilon _3}} \right\rangle } \right]$, $\dfrac{{\partial {\mathbf{e}}}}{{\partial {\varvec{{\tilde{\varepsilon }} }}}} = \left[ {\dfrac{{\partial \left\langle {{\varepsilon _1}} \right\rangle }}{{\partial {\varvec{{\tilde{\varepsilon }} }}}} \ {\mathrm{}}\dfrac{{\partial \left\langle {{\varepsilon _2}} \right\rangle }}{{\partial {\varvec{{\tilde{\varepsilon }} }}}} \ {\mathrm{}}\dfrac{{\partial \left\langle {{\varepsilon _3}} \right\rangle }}{{\partial {\varvec{{\tilde{\varepsilon }} }}}}} \right]$.

• For modified von Mises criterion The equivalent strain in Eq. 19 could be written as

  $$\begin{aligned} {\varepsilon _{eq}} = a{I_1} + b\sqrt{cI_1^2 + dJ_2^{'}} \end{aligned}$$

  (A.8)

  with $a = \dfrac{{k - 1}}{{2k\left( {1 - 2\nu } \right) }}$, $b = \dfrac{1}{{2k}}$, $c = \dfrac{{{{\left( {k - 1} \right) }^2}}}{{{{\left( {1 - 2\nu } \right) }^2}}}$, $d = \dfrac{{12k}}{{{{\left( {1 + \nu } \right) }^2}}}$. The derivatives of $\varepsilon _{eq}$ could be expressed

  $$\begin{aligned} \dfrac{{\partial {\varepsilon _{eq}}}}{{\partial {\varvec{{\tilde{\varepsilon }} }}}} = {\left[ {\dfrac{{\partial {\varepsilon _{eq}}}}{{\partial {{\tilde{\varepsilon }} _{xx}}}} \ {\mathrm{}}\dfrac{{\partial {\varepsilon _{eq}}}}{{\partial {{\tilde{\varepsilon }} _{yy}}}} \ {\mathrm{}}\dfrac{{\partial {\varepsilon _{eq}}}}{{\partial {{\tilde{\varepsilon }} _{xy}}}}} \right] ^T} \end{aligned}$$

  (A.9)

  where

  $$\begin{aligned} \dfrac{{\partial {\varepsilon _{eq}}}}{{\partial {{\tilde{\varepsilon }} _{xx}}}}= & {} a\dfrac{{\partial {I_1}}}{{\partial {{\tilde{\varepsilon }} _{xx}}}} + \dfrac{{0.5b}}{{\sqrt{cI_1^2 + dJ_2^{'}} }}\nonumber \\&\quad \left( {2c{I_1}\dfrac{{\partial {I_1}}}{{\partial {{\tilde{\varepsilon }} _{xx}}}} + d\dfrac{{\partial J_2^{'}}}{{\partial {{\tilde{\varepsilon }} _{xx}}}}} \right) \end{aligned}$$

  (A.10)
  $$\begin{aligned} \dfrac{{\partial {\varepsilon _{eq}}}}{{\partial {{\tilde{\varepsilon }} _{yy}}}}= & {} a\dfrac{{\partial {I_1}}}{{\partial {{\tilde{\varepsilon }} _{yy}}}} + \dfrac{{0.5b}}{{\sqrt{cI_1^2 + dJ_2^{'}} }}\nonumber \\&\quad \left( {2c{I_1}\dfrac{{\partial {I_1}}}{{\partial {{\tilde{\varepsilon }} _{yy}}}} + d\dfrac{{\partial J_2^{'}}}{{\partial {{\tilde{\varepsilon }} _{yy}}}}} \right) \end{aligned}$$

  (A.11)
  $$\begin{aligned} \dfrac{{\partial {\varepsilon _{eq}}}}{{\partial {{\tilde{\varepsilon }} _{xy}}}}= & {} a\dfrac{{\partial {I_1}}}{{\partial {{\tilde{\varepsilon }} _{xy}}}} + \dfrac{{0.5b}}{{\sqrt{cI_1^2 + dJ_2^{'}} }}\nonumber \\&\quad \left( {2c{I_1}\dfrac{{\partial {I_1}}}{{\partial {{\tilde{\varepsilon }} _{xy}}}} + d\dfrac{{\partial J_2^{'}}}{{\partial {{\tilde{\varepsilon }} _{xy}}}}} \right) \end{aligned}$$

  (A.12)

  with $\dfrac{{\partial {I_1}}}{{\partial {{\tilde{\varepsilon }} _{xx}}}} = 1 + \dfrac{{\partial {{\tilde{\varepsilon }} _{zz}}}}{{\partial {{\tilde{\varepsilon }} _{xx}}}}$,

  $$\begin{aligned} \dfrac{{\partial {I_1}}}{{\partial {{\tilde{\varepsilon }} _{yy}}}}= & {} 1 + \dfrac{{\partial {{\tilde{\varepsilon }} _{zz}}}}{{\partial {{\tilde{\varepsilon }} _{yy}}}}, \\ \dfrac{{\partial {I_1}}}{{\partial {{\tilde{\varepsilon }} _{xy}}}}= & {} 0,\\ \dfrac{{\partial J_2^{'}}}{{\partial {{\tilde{\varepsilon }} _{xx}}}}= & {} \dfrac{1}{3}\left( 2{{\tilde{\varepsilon }} _{xx}} + 2{{\tilde{\varepsilon }} _{zz}}\dfrac{{\partial {{\tilde{\varepsilon }} _{zz}}}}{{\partial {{\tilde{\varepsilon }} _{xx}}}} - {{\tilde{\varepsilon }} _{yy}} - {{\tilde{\varepsilon }} _{zz}}\right. \\&\left. - \dfrac{{\partial {{\tilde{\varepsilon }} _{zz}}}}{{\partial {{\tilde{\varepsilon }} _{xx}}}}\left( {{{\tilde{\varepsilon }} _{xx}} + {{\tilde{\varepsilon }} _{yy}}} \right) \right) \\ \dfrac{{\partial J_2^{'}}}{{\partial {{\tilde{\varepsilon }} _{yy}}}}= & {} \dfrac{1}{3}\left( 2{{\tilde{\varepsilon }} _{yy}} + 2{{\tilde{\varepsilon }} _{zz}}\dfrac{{\partial {{\tilde{\varepsilon }} _{zz}}}}{{\partial {{\tilde{\varepsilon }} _{yy}}}} - {{\tilde{\varepsilon }} _{xx}} - {{\tilde{\varepsilon }} _{zz}} \right. \\&\left. - \dfrac{{\partial {{\tilde{\varepsilon }} _{zz}}}}{{\partial {{\tilde{\varepsilon }} _{yy}}}}\left( {{{\tilde{\varepsilon }} _{xx}} + {{\tilde{\varepsilon }} _{yy}}} \right) \right) \\&\quad \dfrac{{\partial J_2^{'}}}{{\partial {{\tilde{\varepsilon }} _{xy}}}} = 2{{\tilde{\varepsilon }} _{xy}} \end{aligned}$$