A two-dimensional non-ordinary state-based peridynamic model based on three-body potential for elastic brittle fracture analysis

Abstract

Two-dimensional non-ordinary state-based peridynamics (PD) model based on three-body interaction potential is developed. Meanwhile, its mathematical expression in bond shear form is also derived under small deformation conditions, which reveals the ability of bond vector considering three-body potential on resisting transverse loadings. A uniaxial tensile plate example is used to demonstrate the robust ability of the proposed numerical method on eliminating Poisson’s ratio limitation. Combined with micro-potential based failure criterion, the PD model is used to simulate elastic brittle fracture behaviors for typical concrete like material specimens, including mixed mode (mode I + mode II) type fracture and crack dynamic branching cases. The simulation results coincide well with the experimental and other numerical results published previously.

Appendix: Numerical techniques

Due to geometric complexity of objects in engineering applications, the spatial discretization and time integration schemes need to be constructed to solve Eq. (1) in general. In present study, the domain of interest is uniformly discretized into a finite number of square subdomains with side length $\Delta$, each of which involves a single collocation point at its centroid point as shown in Fig. 24.

Fig. 24

Uniform discretization of PD medium domain into square subdomains

With the discretization scheme, the integral in Eq. (1) can be approximated by a finite summation, resulting in a set of discrete equations

$$ \rho {\varvec{\ddot{u}}}_{i}^{n} = \mathop \sum \limits_{j = 1}^{{N_{i} }} {\mathbf{f}}_{ij} {\Delta }V_{j} + {\mathbf{b}}_{i} , $$

(A.1)

in which ${\ddot{{\varvec{u}}}}_{i}^{n}= \ddot{{\varvec{u}}}\left({{\varvec{X}}}_{i},{t}^{n}\right)$ with ${t}^{n}$ the ${n}^{th}$ time step of $\Delta t$, i.e., ${t}^{n}=n\Delta t$. ${N}_{i}$ is the total number of collocation points within the horizon of point ${{\varvec{X}}}_{i}$. $\Delta {V}_{j}$ denotes the volume (area in 2D case) of subdomain-j. It could be found that Eq. (A.1) has a similar form to the linear momentum equilibrium equation in PD lattice model [see Chapter 8 in Gerstle (2015)]. As opposed to the continuum version of PD model, like the proposed one, the lattice-based PD model considers the material body to be essentially composed of a finite number of discrete interacting lattice particles, and hence, some mechanical quantities, SED for instance, take discrete sum- rather than integral representation and the geometry of analogue of horizon defined therein depends on the selected lattice. In addition, the already-discretized solid model can be directly used as the basis of formulating PD model. Notice that the SED equalization strategy for determination of PD material parameters is also suitable for the lattice-based approach.

When calculating the PD force exerted on point ${{\varvec{X}}}_{i}$ shown in Eq. (A.1), the subdomains near its horizon boundary (e.g., the Subdomain-j in Fig. 24) actually belong partially to the horizon (within the red dotted circle in Fig. 24). Therefore, it is necessary to correct their volumes to improve the accuracy of the numerical method. In the present study, an algorithm, called PA-HHB (Le et al. 2014; Hu et al. 2010; Seleson 2014), with linearly decreasing function near horizon boundary is adopted to evaluate the volume fraction ${v}_{ij}$, which is continuously varied from 1 to 0.

$$ v_{{ij}} \left( {\varvec{\xi _{{ij}} }} \right) = \left\{ \begin{gathered} \begin{array}{*{20}{l}} {1,} & {{\text{if}}\;\left| {\varvec{\xi _{{ij}}} } \right| \le \delta - \frac{\Delta }{2}} \\ {\frac{1}{2} + \frac{{\delta - \left| {\varvec{\xi _{{ij}} }} \right|}}{\Delta },} & {{\text{if}}\;\delta - \frac{\Delta }{2} \le \left| {\varvec{\xi _{{ij}}} } \right| < \delta + \frac{\Delta }{2}} \\ \end{array} \hfill \\ \begin{array}{*{20}{l}} {0,} & {{\text{if}}\;\left| {\varvec{\xi _{{ij}}} } \right| \ge \delta + \frac{\Delta }{2}} \\ \end{array} \hfill \\ \end{gathered} \right.. $$

(A.2)

To overcome the surface effect (Le and Bobaru 2018) for material points close to free surfaces or material interfaces, a correction factor based on the energy strategy (Kilic 2008) will be given for the proposed PD numerical method. The correction procedures start with calculation of $\mathbf{W}\left({\varvec{X}}\right)$ vectors for each material point firstly:

$$ {\mathbf{W}}\left( {\varvec{X}} \right)^{T} = \left\{ {\begin{array}{*{20}c} {W_{1} } & {W_{2} } \\ \end{array} } \right\}, $$

(A.3)

where ${W}_{1}$,${W}_{2}$ are evaluated based on Eq. (4) for 2D case under prescribing displacement fields ${\mathbf{u}}_{1}=\left\{\begin{array}{cc}{\varepsilon }_{0}{x}_{1}& 0\end{array}\right\}$ and ${\mathbf{u}}_{2}=\left\{\begin{array}{cc}0& {\varepsilon }_{0}{x}_{2}\end{array}\right\}$, respectively. ${\varepsilon }_{0}$ assigned a value of 0.001 would be reasonable according to Ref. (Kilic 2008).

Secondly, calculating the ${\mathbf{W}}_{c}\left({\varvec{X}}\right)$ vector at point ${\varvec{X}}$:

$$ {\mathbf{W}}_{c} \left( {\varvec{X}} \right)^{{\text{T}}} = \left\{ {\begin{array}{*{20}c} {W_{1c} } & {W_{2c} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\frac{{W_{\infty } }}{{W_{1} }}} & {\frac{{W_{\infty } }}{{W_{2} }}} \\ \end{array} } \right\}, $$

(A.4)

in which ${W}_{\infty }$ is obtained by setting ${\varepsilon }_{11}={\varepsilon }_{0}, {\varepsilon }_{22}=0, {\varepsilon }_{12}=0$ (or ${\varepsilon }_{11}=0, {\varepsilon }_{22}={\varepsilon }_{0}, {\varepsilon }_{12}=0$) in the right member of the last equal sign in Eq. (14).

Then, the average coefficient vector ${\mathbf{W}}_{ijc}$ between points ${{\varvec{X}}}_{i}$ and ${{\varvec{X}}}_{j}$ can be defined as

$$ {\mathbf{W}}_{ijc}^{T} = \left\{ {\begin{array}{*{20}c} {W_{ijc1} } & {W_{ijc2} } \\ \end{array} } \right\} = \frac{{{\mathbf{W}}_{c} \left( {{\varvec{X}}_{i} } \right)^{T} + {\mathbf{W}}_{c} \left( {{\varvec{X}}_{j} } \right)^{{\text{T}}} }}{2}. $$

(A.5)

Finally, a scalar correction factor ${R}_{ij}$ defined for the PD interaction between points ${{\varvec{X}}}_{i}$ and ${\mathbf{X}}_{j}$ is

$$ R_{ij} = \left( {\left( {\frac{{\xi_{1} }}{{W_{ijc1} }}} \right)^{2} + \left( {\frac{{\xi_{2} }}{{W_{ijc2} }}} \right)^{2} } \right)^{ - 1/2} , $$

(A.6)

In the present study, the numerical inaccuracies caused by the spatial discretization can also be corrected by this coefficient.