Nonlocal modeling of bi-material and modulus graded plates using peridynamic differential operator

Abstract

This study presents the application of the peridynamic differential operator (PDDO) on modeling of bi-material plates with/without modulus graded regions. The PDDO converts the Navier’s equilibrium equations and boundary conditions from the differential form into the integral form. The mismatch of the stiffness along the interface of two distinct materials results in an increase in the strain and stress variations, leading to the onset of cracking at the free corners of the interface. The interfacial strains and stresses can be mitigated by inserting a modulus graded layer between two different materials. The material properties in the modulus graded region is achieved through the power-law distribution. The efficacy of the proposed approach is demonstrated by considering a bi-material square plate under tension. The PDDO displacement, strain, and stress predictions are compared with the reference solutions, and good correlations are achieved. The influence of a modulus graded region with/without a pre-existing crack located between dissimilar materials is investigated for different material variations. It is noted that the PDDO performs very well on the displacement, strain, and stress predictions even if the solution domain has geometrical or material discontinuities. Moreover, modulus graded regions offer some advantages over the sharp interfaces and alleviate the strain and stress concentrations along the interface of the dissimilar materials.

Appendix

According to Eq. (8), the PD differential operator for two-dimensional domains and up to second-order derivatives may be generated by addressing a scalar field's $(f\left( {{\mathbf{x^{\prime}}}} \right) = f\left( {{\mathbf{x + \xi }}} \right))$ Taylor series expansion, in the form:

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

(48)

After multiplying each term in this formula by the PD functions, $g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}})$ with $(p_{1} ,p_{2} = 0,1,2)$ and performing integration over the family, $H_{{\mathbf{x}}}$ of point ${\mathbf{x}}$ results with

$$\begin{gathered} \int\limits_{{H_{{\mathbf{x}}} }} {f({\mathbf{x}} + {{\varvec{\upxi}}}) \, g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}A} = f\left( {\mathbf{x}} \right)\int\limits_{{H_{{\mathbf{x}}} }} {g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}A} + \frac{{\partial f\left( {\mathbf{x}} \right)}}{{\partial x_{1}^{{}} }}\int\limits_{{H_{{\mathbf{x}}} }} {\xi_{1}^{{}} g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}A} + \frac{{\partial f\left( {\mathbf{x}} \right)}}{{\partial x_{2}^{{}} }}\int\limits_{{H_{{\mathbf{x}}} }} {\xi_{2}^{{}} g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}A} \hfill \\ \quad \quad \qquad \quad + \frac{{\partial^{2} f\left( {\mathbf{x}} \right)}}{{\partial x_{1}^{2} }}\int\limits_{{H_{{\mathbf{x}}} }} {\frac{1}{2!}\xi_{1}^{2} \, g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}A} + \frac{{\partial^{2} f\left( {\mathbf{x}} \right)}}{{\partial x_{2}^{2} }}\int\limits_{{H_{{\mathbf{x}}} }} {\frac{1}{2!}\xi_{2}^{2} g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}A} + \frac{{\partial^{2} f\left( {\mathbf{x}} \right)}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }}\int\limits_{{H_{{\mathbf{x}}} }} {\xi_{1}^{{}} \xi_{2}^{{}} g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}A} . \hfill \\ \end{gathered}$$

(49)

The orthogonality property of PD functions, $g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}})$ indicated by Eq. (10) can be represented as

$$\frac{1}{{n_{1} !n_{2} !}}\int\limits_{{H_{{\mathbf{x}}} }} {\xi_{1}^{{n_{1} }} \xi_{2}^{{n_{2} }} g_{2}^{{p_{1} p_{2} }} ({{\varvec{\upxi}}}){\text{d}}A} = \delta_{{n_{1} p_{1} }} \delta_{{n_{2} p_{2} }} \;{\text{with }}(n_{1} ,n_{2} ,p_{1} ,p_{2} = 0,1,2).$$

(50)

Substituting the orthogonality conditions into Eq. (49) yields the explicit representation of the PD forms of the partial derivatives as

$$\left\{ {\begin{array}{*{20}c} {f({\mathbf{x}})} \\ {\frac{{\partial f({\mathbf{x}})}}{{\partial x_{1}^{{}} }}} \\ {\frac{{\partial f({\mathbf{x}})}}{{\partial x_{2}^{{}} }}} \\ {\frac{{\partial^{2} f({\mathbf{x}})}}{{\partial x_{1}^{2} }}} \\ {\frac{{\partial^{2} f({\mathbf{x}})}}{{\partial x_{2}^{2} }}} \\ {\frac{{\partial^{2} f({\mathbf{x}})}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }}} \\ \end{array} } \right\} = \int\limits_{{H_{{\mathbf{x}}} }} {f({\mathbf{x}} + {{\varvec{\upxi}}}) \, \left\{ {\begin{array}{*{20}c} {g_{2}^{00} ({{\varvec{\upxi}}})} \\ {g_{2}^{10} ({{\varvec{\upxi}}})} \\ {g_{2}^{01} ({{\varvec{\upxi}}})} \\ {g_{2}^{20} ({{\varvec{\upxi}}})} \\ {g_{2}^{02} ({{\varvec{\upxi}}})} \\ {g_{2}^{11} ({{\varvec{\upxi}}})} \\ \end{array} } \right\}{\text{d}}A} .$$

(51)

The PD functions can be generated in the form of polynomials by regarding Eq. (11):

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

(52)

The weight functions, $w_{{p_{1} p_{2} }} \left( {\left| {{\varvec{\upxi}}} \right|} \right)$ may be determined as $w_{{p_{1} p_{2} }} \left( {\left| {{\varvec{\upxi}}} \right|} \right) = w\left( {\left| {{\varvec{\upxi}}} \right|} \right)$ for simplicity yet it is not an obligation. By invoking this option, the orthogonality property of the PD functions demands that

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

(53)

where the PD shape matrix, ${\mathbf{A}}$, the matrix of unknown coefficients, ${\mathbf{a}}$, and the matrix of known constants, ${\mathbf{b}}$ can be given as

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

(54)

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

(55)

and

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

(56)

The determination of PD functions that are orthogonal to each term in the Taylor series expansion is achieved by obtaining the unknown coefficients after performing the operation ${\mathbf{a}} = {\mathbf{A}}^{ - 1} {\mathbf{b}}$, Eq. (48).