PeriFast/Dynamics: A MATLAB Code for Explicit Fast Convolution-based Peridynamic Analysis of Deformation and Fracture

Abstract

We present PeriFast/Dynamics, a compact and user-friendly MATLAB code for fast peridynamic (PD) simulations for deformation and fracture. PeriFast/Dynamics uses the fast convolution-based method (FCBM) for spatial discretization and an explicit time marching scheme to solve large-scale dynamic fracture problems. Different from existing PD solvers, PeriFast/Dynamics does not require neighbor search and storage, due to the use of the Fast-Fourier transform and its inverse to compute the integral operator. Run-times and memory allocation are independent of the number of neighbors inside the PD horizon, leading to faster computations and lower storage requirements. The governing equations and discretization method are briefly reviewed, the code structure explained, and individual modules described in detail. A demonstrative example on dynamic brittle fracture in 3D, with multiple crack branching events, is solved using three different constitutive models: a bond-based, an ordinary state-based, and a correspondence model. The small differences between results with the three different constitutive models are explained. Users are provided with a step-by-step description of the problem setup and execution of the code. PeriFast/Dynamics is a branch of the PeriFast suite of codes, and is available for download at the GitHub link provided in reference [1].

Appendix. Constitutive models included in PeriFast/Dynamics

• 1. Linearized bond-based elastic material model

This model is basically the linearized version of the micro-elastic solid (see [47]). The internal force density for this material is

$${\varvec{L}}\left({\varvec{x}},t\right)= {\int }_{{\mathcal{H}}_{x}}{\chi }_{\mathrm{B}}{\chi }_{B}^{{\prime}}\mu {\varvec{f}}\left({\varvec{x}},{{\varvec{x}}}^{\boldsymbol{{\prime}}},t\right)d{V}_{x{^{\prime}}}={\int }_{{\mathcal{H}}_{x}}{\chi }_{\mathrm{B}}{\chi }_{B}^{{\prime}} \lambda {\lambda }^{{\prime}}\mathbf{C}\left({\varvec{\xi}}\right){\varvec{\eta}}d{V}_{x{^{\prime}}}$$

(A-1)

where ${\varvec{\xi}}$ Is the bond vector, ${\varvec{\eta}}$ is the relative displacement and $\mathbf{C}\left(\xi \right)=\alpha \omega \left(\left|{\varvec{\xi}}\right|\right)\frac{{\varvec{\xi}}\otimes{\varvec{\xi}}}{{\left|{\varvec{\xi}}\right|}^{2}}$ with $\alpha = \frac{12E}{\pi {\delta }^{4}}$ and $\left(\left|{\varvec{\xi}}\right|\right)=\frac{1}{\left|{\varvec{\xi}}\right|}$.

And the strain energy density is

$$W\left({\varvec{x}},t\right)=\frac{1}{2}{\int }_{{\mathcal{H}}_{x{^{\prime}}}}{\chi }_{\mathrm{B}}{\chi }_{B}^{^{\prime}} \lambda {\lambda }^{^{\prime}}{\varvec{\eta}}.\left(\frac{1}{2}\mathbf{C}\left({\varvec{\xi}}\right){\varvec{\eta}}\right)d{V}_{x{^{\prime}}}$$

(A-2)

The convolutional form of the internal force density and strain energy density for linearized bond-based models (Eqs. (A-1) and (A-2)) used in this work are [23]

$${L}_{i}={\int }_{{\mathcal{H}}_{x}}{\chi }_{\mathrm{B}}{\chi }_{B}^{^{\prime}} \lambda {\lambda }^{^{\prime}}{\mathrm{C}}_{ij}{\eta }_{j}d{V}_{x{^{\prime}}}= {\int }_{{\mathcal{H}}_{x}}{\chi }_{\mathrm{B}}{\chi }_{B}^{^{\prime}} \lambda {\lambda }^{^{\prime}}{\mathrm{C}}_{ij}\left({u}_{j}^{^{\prime}}-{u}_{j}\right)d{V}_{{x}^{^{\prime}}}={\chi }_{\mathrm{B}}\lambda \left\{\left[{\mathrm{C}}_{\mathrm{ij}}*({\chi }_{\mathrm{B}}\lambda {u}_{j})\right]-[{\mathrm{C}}_{\mathrm{ij}}*{\chi }_{\mathrm{B}}\lambda ]{u}_{j}\right\}$$

(A-3)

$$W\left({\varvec{x}},t\right)=\frac{1}{2}{\int }_{{\mathcal{H}}_{x{^{\prime}}}}{\chi }_{\mathrm{B}}{\chi }_{B}^{^{\prime}} \lambda {\lambda }^{^{\prime}}{\eta }_{i}\left(\frac{1}{2}{\mathrm{C}}_{\mathrm{ij}}{\eta }_{j}\right)d{V}_{x{^{\prime}}}=\frac{1}{4} {\chi }_{\mathrm{B}}\lambda \left(\left[{\mathrm{C}}_{ij}*{\chi }_{\mathrm{B}}\lambda {u}_{i}{u}_{j}\right]-2\left[{\mathrm{C}}_{ij}*{\chi }_{\mathrm{B}}\lambda {u}_{i}\right]{u}_{j}+\left[{\mathrm{C}}_{ij}*{\chi }_{\mathrm{B}}\lambda \right]{u}_{i}{u}_{j}\right)$$

(A-4)

• 2. Linearized state-based elastic material model

This model is the linearized version of the native state-based linear elastic solid (see [47]). The internal force density for this material is

$${\varvec{L}}\left({\varvec{x}},t\right)= {\int}_{{\mathcal{H}}_{x}}{\chi }_{\mathrm{B}}{\chi }_{B}^{^{\prime}}\mu {\varvec{f}}\left({\varvec{x}},{{\varvec{x}}}^{\boldsymbol{^{\prime}}},t\right)d{V}_{x{^{\prime}}}={\int}_{{\mathcal{H}}_{x}}{\chi }_{\mathrm{B}}{\chi }_{B}^{^{\prime}}\mu (\underset{\_}{\mathbf{T}}\left[{\varvec{x}},t\right]\langle {\varvec{\xi}}\rangle -\underset{\_}{\mathbf{T}}\left[{\varvec{x}}\boldsymbol{^{\prime}},t\right]\langle -{\varvec{\xi}}\rangle )d{V}_{x{^{\prime}}}$$

(A-5)

where

$$\underset{\_}{\mathbf{T}}\langle {\varvec{\xi}}\rangle =\left(\frac{3k-5G}{m}\right)\omega \left(\left|{\varvec{\xi}}\right|\right)\vartheta{\varvec{\xi}}+\frac{15G}{m}\omega \left(\left|{\varvec{\xi}}\right|\right)\frac{{\varvec{\xi}}\otimes{\varvec{\xi}}}{{\left|{\varvec{\xi}}\right|}^{2}}{\varvec{\eta}}$$

(A-6)

where $k$ and $G$ here are the bulk and shear moduli, respectively, and $\vartheta$ is the a nonlocal dilation [47].

The strain energy density for this linearized state-base material model is [47]

$$W\left({\varvec{x}},t\right)=\frac{1}{2}\left(\left(k-\frac{\alpha m}{9}\right){\vartheta }^{2}+\alpha {\int}_{{\mathcal{H}}_{x}}\mu \omega \left(\left|{\varvec{\xi}}\right|\right){\varvec{\eta}}.\frac{{\varvec{\xi}}\otimes{\varvec{\xi}}}{{\left|{\varvec{\xi}}\right|}^{2}}{\varvec{\eta}}d{V}_{{x}^{^{\prime}}} \right),\alpha =\frac{15G}{m}$$

(A-7)

Note: by adopting a Poisson ratio of one-quarter, the first terms on the right-hand side of Eq. (A-6) and (A-7) vanish and the linearized bond-based model presented by Eq. (A-1) and (A-2) is recovered for the points in the bulk as explained in Section 5.6.

The convolution structures for $\vartheta$ and $m$ are derived in [23]. Let $\mathbf{C}\left({\varvec{\xi}}\right)=30G \omega \left(\left|{\varvec{\xi}}\right|\right)\frac{{\varvec{\xi}}\otimes{\varvec{\xi}}}{{\left|{\varvec{\xi}}\right|}^{2}}$ and ${\varvec{a}}\left({\varvec{\xi}}\right)=\omega \left(\left|{\varvec{\xi}}\right|\right){\varvec{\xi}}$ with $\left(\left|{\varvec{\xi}}\right|\right)=\frac{1}{\left|{\varvec{\xi}}\right|}$, the convolution structure for internal force density and strain energy density (Eqs. (A-6) and (A-7)) obtained as [23].

$${L}_{i}={\chi }_{\mathrm{B}}\lambda \left(\left(\frac{3k-5G}{m}\right)\left(-\vartheta \left[{a}_{i}* {\chi }_{\mathrm{B}}\lambda \right]-\left[{a}_{i}*{\chi }_{\mathrm{B}}\lambda \vartheta \right]\right)+\frac{1}{m} \left({[\mathrm{C}}_{ij}*{\chi }_{\mathrm{B}}\lambda {u}_{j}]-[{\mathrm{C}}_{ij}*{\chi }_{\mathrm{B}}\lambda ]{u}_{j}\right)\right)$$

(A-8)

$$\begin{aligned}W\left({\varvec{x}},t\right)&=\frac{1}{2}\left(\left(k-\frac{\alpha m}{9}\right){\vartheta }^{2}+\frac{1}{2m}{\int}_{{\mathcal{H}}_{x}}{\chi }_{\mathrm{B}}{\chi }_{B}^{^{\prime}}\lambda {\lambda }^{^{\prime}}{\eta }_{i}{\mathrm{C}}_{ij}{\eta }_{j} d{V}_{{x}^{^{\prime}}} \right)\\&=\frac{1}{2}\left(\left(k-\frac{\alpha m}{9}\right){\vartheta }^{2}+\frac{1}{2m}{\int}_{{\mathcal{H}}_{x}}{\chi }_{\mathrm{B}}{\chi }_{B}^{^{\prime}}\lambda {\lambda }^{^{\prime}}{\mathrm{C}}_{ij}({u}_{i}^{^{\prime}}{u}_{j}^{^{\prime}}-{u}_{i}^{^{\prime}}{u}_{j}-{u}_{i}{u}_{j}^{^{\prime}}+{u}_{i}{u}_{j}) d{V}_{{x}^{^{\prime}}} \right)\\&= \frac{1}{2}\left(\left(k-\frac{\alpha m}{9}\right){\vartheta }^{2}+\frac{1}{2m}{\chi }_{\mathrm{B}}\lambda \left\{\left[{\mathrm{C}}_{ij}*{\chi }_{\mathrm{B}}\lambda {u}_{i}{u}_{j}\right]-{u}_{j}\left[{\mathrm{C}}_{ij}*{\chi }_{\mathrm{B}}\lambda {u}_{i}\right]-{u}_{i}\left[{\mathrm{C}}_{ij}*{\chi }_{\mathrm{B}}\lambda {u}_{j}\right]+{u}_{i}{u}_{j}[{\mathrm{C}}_{ij}*{\chi }_{\mathrm{B}}\lambda ]\right\}\right)\end{aligned}$$

(A-9)

The details on deriving the convolutional structure for linearized bond-based and state-based models are provided in [23].

• 3. PD correspondence hyperelastic material model

This model uses the correspondence formulation introduced in [32], and uses the classical Saint–Venant-Kirchhoff hyperelastic constitutive law. The internal force density for this material is

$${\varvec{L}}({\varvec{x}},t) ={\int}_{\mathcal{H}}{\chi }_{\mathrm{B}}{\chi }_{B}^{^{\prime}}\mu \left(\underset{\_}{\mathbf{T}}\langle {\varvec{\xi}}\rangle -\underset{\_}{\mathbf{T}\mathbf{^{\prime}}}\langle -{\varvec{\xi}}\rangle \right) \mathrm{d}{V}_{x{^{\prime}}}$$

(A-10)

where

$$\underset{\_}{\mathbf{T}}\langle {\varvec{\xi}}\rangle = \omega \left(\left|{\varvec{\xi}}\right|\right){\varvec{\sigma}}\left(\overline{\mathbf{F} }\right){\mathbf{K}}^{-1}{\varvec{\xi}}$$

(A-11)

$\mathbf{K}$ is the shape tensor and ${\varvec{\sigma}}$ is the first Piola–Kirchhoff (P-K) stress tensor which is in terms of $\overline{\mathbf{F} }$,the PD deformation gradient and defined based on the classical constitutive model that we use (for details on the correspondence formulation please see [32]). One issue encountered when using a PD correspondence model for problems with cracks is material instabilities in the form of zero energy modes. A number of solutions have been proposed to reduce/eliminate these zero energy modes. For a review of various strategies for stabilizing PD correspondence models please see [54]. In this work we use the method introduced in [33] in which a stabilizing term ($\underset{\_}{{\mathbf{T}}^{\mathbf{s}}}\langle {\varvec{\xi}}\rangle$) is added to the force state formulation $\underset{\_}{{\mathbf{T}}^{\mathbf{c}}}\langle {\varvec{\xi}}\rangle$ as follows:

$$\begin{array}{l}\underline{\mathbf{T}}\langle {\varvec{\xi}}\rangle =\underline{{\mathbf{T}}^{\mathbf{c}}}\langle {\varvec{\xi}}\rangle + \underline{{\mathbf{T}}^{\mathbf{s}}}\langle {\varvec{\xi}}\rangle = \omega \left(\left|{\varvec{\xi}}\right|\right) \left({\varvec{\sigma}}\left(\overline{\mathbf{F}}\right){\mathbf{K}}^{-1}{\varvec{\xi}}+\boldsymbol{ }\frac{GC}{{\omega }_{0}}\underline{\mathbf{z}}\langle {\varvec{\xi}}\rangle \right) ,\;\underline{\mathbf{z}}\langle {\varvec{\xi}}\rangle =\underline{\mathbf{Y}}\langle {\varvec{\xi}}\rangle -\overline{\mathbf{F}}{\varvec{\xi}},\\{\omega }_{0}={\int}_{\mathcal{H}}\omega (\left|{\varvec{\xi}}\right|)\mathrm{d}{V}_{{\varvec{\xi}}} ,\;C=18k/\pi {\delta }^{5}\end{array}$$

(A-12)

For the Saint–Venant Kirchhoff model used in this study we have

$$\mathbf{S}=\lambda tr\left(\mathbf{E}\right)\mathbf{I}+2G\mathbf{E}$$

(A-13)

where $\mathbf{S}$ is the second P-K stress tensor and needed to be converted to the first P-K (${\varvec{\sigma}}$)stress tensor to be used in Eq. (A-11). $\mathbf{E}$ is the Lagrangian Green strain tensor, and $\lambda$ and $G$ are the Lamé constant and shear modulus of the material. $\mathbf{I}$ is the identity tensor.

For the PD correspondence model, we use the classical formulation to compute the strain energy density as $W\left(x,t\right)=\frac{1}{2}\mathbf{S}\left({\varvec{x}},t\right):\mathbf{E}({\varvec{x}},t)$.

Let ${\varvec{a}}\left({\varvec{\xi}}\right)=\omega \left(\left|{\varvec{\xi}}\right|\right){\varvec{\xi}}$, $\omega \left(\left|{\varvec{\xi}}\right|\right)=\frac{1}{\left|{\varvec{\xi}}\right|}$ and$\beta = \frac{GC}{{\omega }_{0}}$, then the convolutional form for the internal force density in the PD correspondence model (Eq. A-10)) is

$$\begin{aligned}{L}_{i}&= {-\chi }_{\mathrm{B}}\lambda \left({\sigma }_{ij}{\mathrm{K}}_{\mathrm{jp}}^{-1}\left[{a}_{p}*{\chi }_{\mathrm{B}}\lambda \right]+\left[{\sigma }_{ij}{\mathrm{K}}_{\mathrm{jp}}^{-1}{*\chi }_{\mathrm{B}}\lambda {a}_{p}\right]\right.\\&\quad\left.-\beta \left(-2\left[{\chi }_{\mathrm{B}}\lambda *{a}_{i}\right]-2\left[{\chi }_{\mathrm{B}}\lambda *\omega \right]{u}_{i}+2\left[{\chi }_{\mathrm{B}}\lambda {u}_{i}*\omega \right]+{\overline{\mathrm{F}} }_{ij}\left[{a}_{j}{*\chi }_{\mathrm{B}}\lambda \right]+[{\chi }_{\mathrm{B}}\lambda {\overline{\mathrm{F}} }_{ij}*{a}_{j}]\right)\right)\end{aligned}$$

(A-14)

The detailed derivation of the convolutional form of the PD-correspondence model is given in [42] (see Section 10.3.4 there), and [43, 44].