Peridynamics enabled digital image correlation for tracking crack paths

Abstract

This study combines peridynamic differential operator (PDDO), digital image correlation (DIC) and the Strain Compatibility Functional (SCF) method to track crack paths. DIC provides the full-field displacements at pixel accuracy by matching digital image sub-region of the specimen surface before and after deformation using correlation functions. The PDDO is applied to the deformation field to determine the strain field and the associated SCF. In presence of a crack, the strain field of a deformed body does not satisfy the strain compatibility condition due to the crack-induced discontinuity. Following the SCF method, a regression technique is applied in the region where the strain compatibility is violated to determine the crack presence, shape and its resulting path. The accuracy of this approach is first verified by considering three different DIC challenge data sets presented by the Society of Experimental Mechanics (SEM). Concerning crack path detection, the approach is first applied to the numerically generated deformation fields corresponding to pre-existing crack configurations. Subsequently, it is applied to the measured deformation fields corresponding to experimentally induced crack propagation. All these examples indicate that the present approach successfully detects the crack paths.

Appendix

According to the 2-order TSE in a 2-dimensional space, the following expression holds

$$f\left( {{\mathbf{x + \xi }}} \right) = f\left( {\mathbf{x}} \right) + \xi_{1}^{{}} \frac{{\partial f\left( {\mathbf{x}} \right)}}{{\partial x_{1}^{{}} }} + \xi_{2}^{{}} \frac{{\partial f\left( {\mathbf{x}} \right)}}{{\partial x_{2}^{{}} }} + \frac{1}{2!}\xi_{1}^{2} \frac{{\partial^{2} f\left( {\mathbf{x}} \right)}}{{\partial x_{1}^{2} }} + \frac{1}{2!}\xi_{2}^{2} \frac{{\partial^{2} f\left( {\mathbf{x}} \right)}}{{\partial x_{2}^{2} }} + \xi_{1}^{{}} \xi_{2}^{{}} \frac{{\partial^{2} f\left( {\mathbf{x}} \right)}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} + R,$$

(27)

where $R$ is the small remainder term. Multiplying each term with PD functions, $g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}})$ and integrating over the domain of interaction (family), $H_{{\mathbf{x}}}$ results in

$$\begin{aligned} \int\limits_{{H_{{\mathbf{x}}} }} {f({\mathbf{x}} + {{\varvec{\upxi}}}) \, g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}V} & = f\left( {\mathbf{x}} \right)\int\limits_{{H_{{\mathbf{x}}} }} {g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}H_{{{\mathbf{x^{\prime}}}}} } + \frac{{\partial f\left( {\mathbf{x}} \right)}}{{\partial x_{1}^{{}} }}\int\limits_{{H_{{\mathbf{x}}} }} {\xi_{1}^{{}} g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}H_{{{\mathbf{x^{\prime}}}}} } \\ & \quad + \frac{{\partial f\left( {\mathbf{x}} \right)}}{{\partial x_{2}^{{}} }}\int\limits_{{H_{{\mathbf{x}}} }} {\xi_{2}^{{}} g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}H_{{{\mathbf{x^{\prime}}}}} } + \frac{{\partial^{2} f\left( {\mathbf{x}} \right)}}{{\partial x_{1}^{2} }}\int\limits_{{H_{{\mathbf{x}}} }} {\frac{1}{2!}\xi_{1}^{2} \, g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}H_{{{\mathbf{x^{\prime}}}}} } \\ & \quad + \frac{{\partial^{2} f\left( {\mathbf{x}} \right)}}{{\partial x_{2}^{2} }}\int\limits_{{H_{{\mathbf{x}}} }} {\frac{1}{2!}\xi_{2}^{2} g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}H_{{{\mathbf{x^{\prime}}}}} } + \frac{{\partial^{2} f\left( {\mathbf{x}} \right)}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }}\int\limits_{{H_{{\mathbf{x}}} }} {\xi_{1}^{{}} \xi_{2}^{{}} g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}H_{{{\mathbf{x^{\prime}}}}} } , \\ \end{aligned}$$

(28)

in which the point ${\mathbf{x}}$ is not necessarily symmetrically located in the domain of interaction. The initial relative position,${{\varvec{\upxi}}}$, between the material points ${\mathbf{x}}$ and ${\mathbf{x^{\prime}}}$ can be expressed as ${\mathbf{\xi = x - x^{\prime}}}$. This ability permits each point to have its own unique family with an arbitrary position.

The PD functions are constructed such that they are orthogonal to each term in the TS expansion as

$$\frac{1}{{n_{1} !n_{2} !}}\int\limits_{{H_{{\mathbf{x}}} }} {\xi_{1}^{{n_{1} }} \xi_{2}^{{n_{2} }} g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}V} = \delta_{{n_{1} p_{1} }} \delta_{{n_{2} p_{2} }} ,$$

(29)

with ($n_{1} ,n_{2} ,p,q = 0,1,2$) and $\delta_{{n_{i} p_{i} }}$ is a Kronecker symbol. Enforcing the orthogonality conditions in the TSE leads to the non-local PD representation of the function itself and its derivatives as

$$f({\mathbf{x}}) = \int\limits_{{H_{{\mathbf{x}}} }} {f({\mathbf{x}} + {{\varvec{\upxi}}})g_{2}^{00} ({{\varvec{\upxi}}}){\text{d}}H_{{{\mathbf{x^{\prime}}}}} } ,$$

(30)

$$\left\{ {\begin{array}{*{20}c} {\frac{{\partial f({\mathbf{x}})}}{{\partial x_{1}^{{}} }}} \\ {\frac{{\partial f({\mathbf{x}})}}{{\partial x_{2}^{{}} }}} \\ \end{array} } \right\} = \int\limits_{{H_{{\mathbf{x}}} }} {f({\mathbf{x}} + {{\varvec{\upxi}}}) \, \left\{ {\begin{array}{*{20}c} {g_{2}^{10} ({{\varvec{\upxi}}})} \\ {g_{2}^{01} ({{\varvec{\upxi}}})} \\ \end{array} } \right\}{\text{d}}H_{{{\mathbf{x^{\prime}}}}} } ,$$

(31)

$$\left\{ {\begin{array}{*{20}c} {\frac{{\partial^{2} f({\mathbf{x}})}}{{\partial x_{1}^{2} }}} \\ {\frac{{\partial^{2} f({\mathbf{x}})}}{{\partial x_{2}^{2} }}} \\ {\frac{{\partial^{2} f({\mathbf{x}})}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }}} \\ \end{array} } \right\} = \int\limits_{{H_{{\mathbf{x}}} }} {f({\mathbf{x}} + {{\varvec{\upxi}}}) \, \left\{ {\begin{array}{*{20}c} {g_{2}^{20} ({{\varvec{\upxi}}})} \\ {g_{2}^{02} ({{\varvec{\upxi}}})} \\ {g_{2}^{11} ({{\varvec{\upxi}}})} \\ \end{array} } \right\}{\text{d}}H_{{{\mathbf{x^{\prime}}}}} } .$$

(32)

The PD functions can be constructed as a linear combination of polynomial basis functions

$$\begin{aligned} g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}) & = a_{00}^{{p_{1} p_{2} }} \,w_{00} (\left| {{\varvec{\upxi}}} \right|) + a_{10}^{{p_{1} p_{2} }} \,w_{10} (\left| {{\varvec{\upxi}}} \right|)\xi_{1}^{{}} + a_{01}^{{p_{1} p_{2} }} \,w_{01} (\left| {{\varvec{\upxi}}} \right|)\xi_{2}^{{}} \\ &\quad + a_{20}^{{p_{1} p_{2} }} \,w_{20} (\left| {{\varvec{\upxi}}} \right|)\xi_{1}^{2} + a_{02}^{{p_{1} p_{2} }} \,w_{02} (\left| {{\varvec{\upxi}}} \right|)\xi_{2}^{2} + a_{11}^{{p_{1} p_{2} }} \,w_{11} (\left| {{\varvec{\upxi}}} \right|)\xi_{1}^{{}} \xi_{2}^{{}} , \\ \end{aligned}$$

(33)

where $a_{{q_{1} q_{2} }}^{{p_{1} p_{2} }}$ are the unknown coefficients, $w_{{q_{1} q_{2} }} (\left| {{\varvec{\upxi}}} \right|)$ are the influence functions, and $\xi_{1}^{{}}$ and $\xi_{2}^{{}}$ are the components of the vector ${{\varvec{\upxi}}}$. Assuming $w_{{p_{1} p_{2} }} (\left| {{\varvec{\upxi}}} \right|) = w(\left| {{\varvec{\upxi}}} \right|)$ and submitting the PD functions into the orthogonality equation lead to a system of algebraic equations for the determination of the coefficients as

$${\mathbf{Aa}} = {\mathbf{b}},$$

(34)

in which

$${\mathbf{A}} = \int\limits_{{H_{{\mathbf{x}}} }} {w\left( {\left| {{\varvec{\upxi}}} \right|} \right)\left[ {\begin{array}{*{20}c} 1 & {\xi_{1}^{{}} } & {\xi_{2}^{{}} } & {\xi_{1}^{2} } & {\xi_{2}^{2} } & {\xi_{1}^{{}} \xi_{2}^{{}} } \\ {\xi_{1}^{{}} } & {\xi_{1}^{2} } & {\xi_{1}^{{}} \xi_{2}^{{}} } & {\xi_{1}^{3} } & {\xi_{1}^{{}} \xi_{2}^{2} } & {\xi_{1}^{2} \xi_{2}^{{}} } \\ {\xi_{2}^{{}} } & {\xi_{1}^{{}} \xi_{2}^{{}} } & {\xi_{2}^{2} } & {\xi_{1}^{2} \xi_{2}^{{}} } & {\xi_{2}^{3} } & {\xi_{1}^{{}} \xi_{2}^{2} } \\ {\xi_{1}^{2} } & {\xi_{1}^{3} } & {\xi_{1}^{2} \xi_{2}^{{}} } & {\xi_{1}^{4} } & {\xi_{1}^{2} \xi_{2}^{2} } & {\xi_{1}^{3} \xi_{2}^{{}} } \\ {\xi_{2}^{2} } & {\xi_{1}^{{}} \xi_{2}^{2} } & {\xi_{2}^{3} } & {\xi_{1}^{2} \xi_{2}^{2} } & {\xi_{2}^{4} } & {\xi_{1}^{{}} \xi_{2}^{3} } \\ {\xi_{1}^{{}} \xi_{2}^{{}} } & {\xi_{1}^{2} \xi_{2}^{{}} } & {\xi_{1}^{{}} \xi_{2}^{2} } & {\xi_{1}^{3} \xi_{2}^{{}} } & {\xi_{1}^{{}} \xi_{2}^{3} } & {\xi_{1}^{2} \xi_{2}^{2} } \\ \end{array} } \right]dH_{{{\mathbf{x^{\prime}}}}} } ,$$

(35)

$${\mathbf{a}} = \left[ {\begin{array}{*{20}c} {a_{00}^{00} } & {a_{00}^{10} } & {a_{00}^{01} } & {a_{00}^{20} } & {a_{00}^{02} } & {a_{00}^{11} } \\ {a_{10}^{00} } & {a_{10}^{10} } & {a_{10}^{01} } & {a_{10}^{20} } & {a_{10}^{02} } & {a_{10}^{11} } \\ {a_{01}^{00} } & {a_{01}^{10} } & {a_{01}^{01} } & {a_{01}^{20} } & {a_{01}^{02} } & {a_{01}^{11} } \\ {a_{20}^{00} } & {a_{20}^{10} } & {a_{20}^{01} } & {a_{20}^{20} } & {a_{20}^{02} } & {a_{20}^{11} } \\ {a_{02}^{00} } & {a_{02}^{10} } & {a_{02}^{01} } & {a_{02}^{20} } & {a_{02}^{02} } & {a_{02}^{11} } \\ {a_{11}^{00} } & {a_{11}^{10} } & {a_{11}^{01} } & {a_{11}^{20} } & {a_{11}^{02} } & {a_{11}^{11} } \\ \end{array} } \right],$$

(36)

and

$${\mathbf{b}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right].$$

(37)

After determining the coefficients, $a_{{q_{1} q_{2} }}^{{p_{1} p_{2} }}$ via ${\mathbf{a}} = {\mathbf{A}}^{ - 1} {\mathbf{b}}$, then the PD functions $g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}})$ can be constructed. The detailed derivations and the associated computer programs can be found in [18].