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Calculation of SIFs for an edge crack in an isotropic plane under hygrothermal shock

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Abstract

In this research, the eXtended Finite Element Methode (XFEM) was used to calculate stress intensity factors (SIFs) in an isotropic 2D finite domain with a stationary edge crack under thermal shock. Fully coupled displacement, temperature and moisture fields were considered in governing equations. Fourier’s and Fick’s laws were used for heat and moisture flux respectively. Also, Soret flux was used for the caused moisture flux by heat. The SIFs were obtained for hygrothermal loadings using the interaction integral method. The discretization of the governing equations was done using the Galerkin Method, and then the Newmark time integration scheme is used to solve them. In several numerical examples, the effect of hygrothermal shock on SIFs as well as temperature and moisture distribution is studied. Results show that in addition to the effect on moisture diffusion, the Sort number affects the time history of the SIFs.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran

    Armin Gholami, Mohammad Bagher Nazari & Masoud Mahdizadeh Rokhi

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  1. Armin Gholami
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  2. Mohammad Bagher Nazari
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  3. Masoud Mahdizadeh Rokhi
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Correspondence to Masoud Mahdizadeh Rokhi.

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Appendix

Appendix

$$ \left[ M \right] = \left[ {\begin{array}{*{20}c} {\left[ {M_{1} } \right]} & {\left[ 0 \right]} & {\left[ 0 \right]} \\ {\left[ 0 \right]} & {\left[ 0 \right]} & {\left[ 0 \right]} \\ {\left[ 0 \right]} & {\left[ 0 \right]} & {\left[ 0 \right]} \\ \end{array} } \right] $$
$$ \left[ C \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ {\left[ {C_{1} } \right]} & {\left[ {C_{2} } \right]} & {\left[ {C_{3} } \right]} \\ {\left[ {C_{4} } \right]} & {\left[ {C_{5} } \right]} & {\left[ {C_{6} } \right]} \\ \end{array} } \right] $$
$$ \left[ K \right] = \left[ {\begin{array}{*{20}c} {\left[ {K_{1} } \right]} & {\left[ {K_{2} } \right]} & {\left[ {K_{3} } \right]} \\ {\left[ 0 \right]} & {\left[ {K_{4} } \right]} & {\left[ 0 \right]} \\ {\left[ 0 \right]} & {\left[ {K_{5} } \right]} & {\left[ {K_{6} } \right]} \\ \end{array} } \right] $$
$$ \left\{ \Delta \right\} = \left\{ {{\text{a}}_{h}^{{\text{u}}} ,{\text{b}}_{h}^{{\text{u}}} ,{\text{c}}_{hm}^{{\text{u}}} ,{\text{a}}_{h}^{{\text{v}}} ,{\text{b}}_{h}^{{\text{v}}} ,{\text{c}}_{hm}^{{\text{v}}} ,{\text{a}}_{h}^{{\text{T}}} ,{\text{b}}_{h}^{{\text{T}}} ,{\text{c}}_{hm}^{{\text{T}}} ,{\text{a}}_{h}^{C} ,{\text{b}}_{h}^{C} ,{\text{c}}_{hm}^{C} } \right\}^{T} $$
$$ h = 1, \ldots ,ne, m = 1, \ldots ,4 $$
$$ \left\{ F \right\} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\mathop \int \limits_{V} B_{x} S_{l} dV + \mathop \int \limits_{A} Tr_{x}^{n} S_{l} dA} \\ {\mathop \int \limits_{V} B_{y} S_{l} dV + \mathop \int \limits_{A} Tr_{y}^{n} S_{l} dA} \\ {\mathop \int \limits_{A} T_{,x} n_{x} S_{l} dA + \mathop \int \limits_{A} T_{,y} n_{y} S_{l} dA} \\ \end{array} } \\ {\mathop \int \limits_{A} \left( {V_{4} T_{,x} n_{,x} } \right)S_{l} dA + \mathop \int \limits_{A} \left( {V_{4} T_{,y} n_{,y} } \right)S_{l} dA + \mathop \int \limits_{A} \left( {c_{,x} n_{,x} } \right)S_{l} dA + \mathop \int \limits_{A} \left( {c_{,y} n_{,y} } \right)S_{l} dA} \\ \end{array} } \right\} $$
$$ \left[ {M_{1} } \right] = \mathop \int \limits_{V} \rho \left[ B \right]^{T} \left[ B \right]dV $$
$$ \left[ {C_{1} } \right] = \mathop \int \limits_{V} W_{1} \left[ {St} \right]^{T} \left[ {S1} \right]dV $$
$$ \left[ {C_{2} } \right] = \mathop \int \limits_{V} W_{2} \left[ {St} \right]^{T} \left[ {St} \right]dV $$
$$ \left[ {C_{3} } \right] = \mathop \int \limits_{V} W_{3} \left[ {St} \right]^{T} \left[ {St} \right]dV $$
$$ \left[ {C_{4} } \right] = \mathop \int \limits_{V} V_{1} \left[ {St} \right]^{T} \left[ {S1} \right]dV $$
$$ \left[ {C_{5} } \right] = \mathop \int \limits_{V} V_{2} \left[ {St} \right]^{T} \left[ {St} \right]dV $$
$$ \left[ {C_{6} } \right] = \mathop \int \limits_{V} V_{3} \left[ {St} \right]^{T} \left[ {St} \right]dV $$
$$ \left[ {K_{1} } \right] = \mathop \int \limits_{V} \left[ {S2} \right]^{T} \left[ D \right]\left[ {S2} \right]dV $$
$$ \left[ {K_{2} } \right] = - \mathop \int \limits_{V} \gamma_{1} \left[ {S1} \right]^{T} \left[ {St} \right]dV $$
$$ \left[ {K_{3} } \right] = - \mathop \int \limits_{V} \gamma_{2} \left[ {S1} \right]^{T} \left[ {St} \right]dV $$
$$ \left[ {K_{4} } \right] = \mathop \int \limits_{V} \left[ {S3} \right]^{T} \left[ {S3} \right]dV $$
$$ \left[ {K_{5} } \right] = \mathop \int \limits_{V} V_{4} \left[ {S3} \right]^{T} \left[ {S3} \right]dV $$
$$ \left[ {K_{6} } \right] = \mathop \int \limits_{V} \left[ {S3} \right]^{T} \left[ {S3} \right]dV $$
$$ \left[ {St} \right] = \left[ {\begin{array}{*{20}c} {N_{1} } & \cdots & {N_{4} } & {\Phi_{1} } & \cdots & {\Phi_{4} } & {\Psi_{11} } & \cdots & {\Psi_{44} } \\ \end{array} } \right] $$
$$ \left[ B \right] = \left[ {\begin{array}{*{20}c} {N_{1} } & \cdots & {N_{4} } & 0 & \cdots & 0 & {\Phi_{1} } & \cdots & \cdots & 0 & {\Psi_{11} } & \cdots & {\Psi_{44} } & 0 & \cdots & 0 \\ 0 & \cdots & 0 & {N_{1} } & \cdots & {N_{4} } & 0 & \cdots & \cdots & {\Phi_{4} } & 0 & \cdots & 0 & {\Psi_{11} } & \cdots & {\Psi_{44} } \\ \end{array} } \right] $$
$$\left[ {S1} \right] = \left[ {\begin{array}{*{20}c} {N_{1,x} } & \cdots & {N_{4,x} } & {N_{1,y} } & {\begin{array}{*{20}c} \cdots & {N_{4,y} } & {\Phi_{1,x} } & \cdots & {\Phi_{4,x} } & {\Phi_{1,y} } & \cdots & {\Phi_{4,y} } \\ \end{array} } \\ \end{array} } \right.\begin{array}{*{20}c} { \Psi_{11,x} } & \cdots & {\Psi_{44,x} } & {\Psi_{11,y} } & \cdots & {\Psi_{44,y} } \\ \end{array} ]$$
$$\left[ {S2} \right] = \left[ {\begin{array}{*{20}c} {N_{1,x} } & {N_{2,x} } & {N_{3,x} } & {N_{4,x} } & 0 & 0 & 0 & 0 & {\Phi_{1,x} } & {\Phi_{2,x} } \\ 0 & 0 & 0 & 0 & {N_{1,y} } & {N_{2,y} } & {N_{3,y} } & {N_{4,y} } & 0 & 0 \\ {N_{1,y} } & {N_{2,y} } & {N_{3,y} } & {N_{4,y} } & {N_{1,x} } & {N_{2,x} } & {N_{3,x} } & {N_{4,x} } & {\Phi_{1,y} } & {\Phi_{2,y} } \\ \end{array} } \right.\left. {\begin{array}{*{20}c} {\Phi_{3,x} } & {\Phi_{4,x} } & 0 & \ldots & 0 & {\Psi_{11,x} } & \ldots & {\Psi_{44,x} } & 0 & \ldots & 0 \\ 0 & 0 & {\Phi_{1,y} } & \ldots & {\Phi_{4,y} } & 0 & \ldots & 0 & {\Psi_{11,y} } & \ldots & {\Psi_{44,y} } \\ {\Phi_{3,y} } & {\Phi_{4,y} } & {\Phi_{1,x} } & \ldots & {\Phi_{4,x} } & {\Psi_{11,y} } & \ldots & {\Psi_{44,y} } & {\Psi_{11,x} } & \ldots & {\Psi_{44,x} } \\ \end{array} } \right]$$
$$\left[ {S3} \right] = \left[ {\begin{array}{*{20}c} {N_{{1,x}} } & {N_{{2,x}} } & {N_{{3,x}} } & {N_{{4,x}} } & {\Phi _{{1,x}} } & {\Phi _{{2,x}} } & {\Phi _{{3,x}} } & {\Phi _{{4,x}} } & {\Psi _{{11,x}} } & \cdots & {\Psi _{{44,x}} } \\ {N_{{1,y}} } & {N_{{2,y}} } & {N_{{3,y}} } & {N_{{4,y}} } & {\Phi _{{1,y}} } & {\Phi _{{2,y}} } & {\Phi _{{3,y}} } & {\Phi _{{4,y}} } & {\Psi _{{11,y}} } & \cdots & {\Psi _{{44,y}} } \\ \end{array} } \right]$$
$$\left[ D \right] = \left[ {\begin{array}{*{20}c} {\hat{\lambda } + 2\hat{\mu }} & {\hat{\lambda }} & 0 \\ {\hat{\lambda }} & {\hat{\lambda } + 2\hat{\mu }} & 0 \\ 0 & 0 & {\hat{\mu }} \\ \end{array} } \right]\; for\; plane \;strain$$

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Gholami, A., Nazari, M.B. & Rokhi, M.M. Calculation of SIFs for an edge crack in an isotropic plane under hygrothermal shock. J Braz. Soc. Mech. Sci. Eng. 45, 418 (2023). https://doi.org/10.1007/s40430-023-04299-3

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