Dual-horizon peridynamics

Peridynamics is a theoretical framework within the field of continuum mechanics that is specifically designed to address deformations characterized by discontinuities, with a particular emphasis on fractures. Uniformity of the horizon is a typical prerequisite for all material points. The concept of dual-horizon is proposed in dual-horizon peridynamics to overcome the limitation of a fixed horizon, and the governing equations are reformulated accordingly.

Horizon and dual-horizon

The fundamental principles underlying dual-horizon peridynamics pertain to the notions of horizon and dual-horizon.

File:Dual-horizon peridynamics.svg

Figure 1. Fixed horizon peridynamics to dual-horizon peridynamics with variable horizons

Horizon of $𝐱$ is a finite-size neighborhood of material point $𝐱$. $$H_{𝐱}=\{{𝐱}^{\prime }|||{𝐱}^{\prime }-𝐱||\le \delta \}$$ where $\delta$ is the radius of the neighborhood.

Dual-horizon is defined as a union of points whose horizons include $𝐱$, denoted by $$H{\text{'}}_{𝐱}=\{{𝐱}^{\prime }|𝐱\in H_{{𝐱}^{\prime }}\}$$ The definition of dual-horizon implies ${𝐱}^{\prime }\in H{\text{'}}_{𝐱}\leftrightarrow 𝐱\in H_{{𝐱}^{\prime }}$.

The governing equations of conventional peridynamics $$\underset{H_{𝐱}}{\int }({𝐟}_{𝐱{𝐱}^{\prime }}-{𝐟}_{{𝐱}^{\prime }𝐱})d{V}_{{𝐱}^{\prime }}+𝐛=\rho {\ddot{𝐮}}_{𝐱}$$ where ${𝐟}_{𝐱{𝐱}^{\prime }}$ is the bond-force density between material point $𝐱$ and ${𝐱}^{\prime }$. The body force density is represented by the symbol $𝐛$, while the density is denoted by $\rho$. The displacement field is denoted by the symbol $𝐮$.

The governing equations of dual-horizon peridynamics are $$\underset{H_{𝐱}}{\int }{𝐟}_{𝐱{𝐱}^{\prime }}d{V}_{{𝐱}^{\prime }}-\underset{H_{{𝐱}^{\prime }}}{\int }{𝐟}_{{𝐱}^{\prime }𝐱}d{V}_{{𝐱}^{\prime }}+𝐛=\rho {\ddot{𝐮}}_{𝐱}$$ The dual-horizon peridynamics is a straightforward adaptation of the traditional peridynamics. It facilitates the use of peridynamics for non-uniform horizons in all material points. The present configuration bears resemblance to the non-uniform discretization technique employed in finite element methods, thereby leading to a substantial enhancement in computational efficacy.

Peridynamics versus dual-horizon peridynamics

The dual-horizon peridynamics encompasses the conventional constant-horizon peridynamics as a specific instance. The table below outlines a comparison between peridynamics and dual-horizon peridynamics.

Peridynamics vs. dual-horizon peridynamics

Model	Peridynamics	Dual-horizon peridynamics
Governing equations	$\underset{H_{𝐱}}{\int }({𝐟}_{𝐱{𝐱}^{\prime }}-{𝐟}_{{𝐱}^{\prime }𝐱})d{V}_{{𝐱}^{\prime }}+𝐛=\rho {\ddot{𝐮}}_{𝐱}$	$\underset{H_{𝐱}}{\int }{𝐟}_{𝐱{𝐱}^{\prime }}d{V}_{{𝐱}^{\prime }}-\underset{H_{{𝐱}^{\prime }}}{\int }{𝐟}_{{𝐱}^{\prime }𝐱}d{V}_{{𝐱}^{\prime }}+𝐛=\rho {\ddot{𝐮}}_{𝐱}$
BB-PD	$\begin{array}{c}{𝐟}_{{𝐱𝐱}^{\prime }}-{𝐟}_{{𝐱}^{\prime }𝐱}=c{s}_{{𝐱𝐱}^{\prime }}𝐧,\mathrm{\forall }{𝐱}^{\prime }\in H_{𝐱}\end{array}$	$\begin{array}{c}{𝐟}_{𝐱{𝐱}^{\prime }}=C\left({\delta }_{𝐱}\right)s_{{𝐱𝐱}^{\prime }}𝐧,\mathrm{\forall }{𝐱}^{\prime }\in H_{𝐱}\\ {𝐟}_{{𝐱}^{\prime }𝐱}=C\left({\delta }_{{𝐱}^{\prime }}\right)s_{{𝐱𝐱}^{\prime }}(-𝐧),\mathrm{\forall }{𝐱}^{\prime }\in H_{{𝐱}^{\prime }}\end{array}$
OSB-PD	$\begin{array}{c}{𝐟}_{{𝐱𝐱}^{\prime }}=\underset{\_}{t}⟨𝝃⟩𝐧\\ {𝐟}_{{𝐱}^{\prime }𝐱}=\underset{\_}{t^{\prime }}⟨-𝝃⟩(-𝐧),\mathrm{\forall }{𝐱}^{\prime }\in H_{𝐱}\end{array}$	$\begin{array}{c}{𝐟}_{{𝐱𝐱}^{\prime }}=\underset{\_}{t}⟨𝝃⟩𝐧,\mathrm{\forall }{𝐱}^{\prime }\in H_{𝐱}\\ {𝐟}_{{𝐱}^{\prime }𝐱}={\underset{\_}{t}}^{\prime }⟨-𝝃⟩(-𝐧),\mathrm{\forall }{𝐱}^{\prime }\in H_{{𝐱}^{\prime }}\end{array}$
NOSB-PD	$\begin{array}{c}{𝐟}_{𝐱{𝐱}^{\prime }}=\underset{\_}{𝐓}[{𝐱}^{\prime }-𝐱],\\ {𝐟}_{{𝐱}^{\prime }𝐱}={\underset{\_}{𝐓}}^{\prime }[𝐱-{𝐱}^{\prime }],\mathrm{\forall }{𝐱}^{\prime }\in H_{𝐱}\end{array}$	$\begin{array}{c}{𝐟}_{𝐱{𝐱}^{\prime }}=\underset{\_}{𝐓}[{𝐱}^{\prime }-𝐱],\mathrm{\forall }{𝐱}^{\prime }\in H_{𝐱}\\ {𝐟}_{{𝐱}^{\prime }𝐱}={\underset{\_}{𝐓}}^{\prime }[𝐱-{𝐱}^{\prime }],\mathrm{\forall }{𝐱}^{\prime }\in H_{{𝐱}^{\prime }}\end{array}$

where $𝐧=\frac{𝜼+𝝃}{\Vert 𝜼+𝝃\Vert },𝜼={𝐮}^{\prime }-𝐮,𝝃={𝐱}^{\prime }-𝐱$.

The three peridynamics models under consideration are bond-based peridynamics (BB-PD), state-based peridynamics (OSB-PD), and nonordinary state-based peridynamics (NOSB-PD), each with their respective abbreviations.

Comment on numerical implementation

Despite the introduction of the dual-horizon, the numerical implementation did not incur any additional expenses. The dual-horizon bond forces can be automatically satisfied during the computation of forces from the horizons of other material points. This concept is demonstrated through a straightforward illustration.

Force accumulation in dual-horizon peridynamics

For example: $H_{{𝐱}_0}=\{{𝐱}_1,{𝐱}_2,{𝐱}_4,{𝐱}_6\},H{\text{'}}_{{𝐱}_0}=\{{𝐱}_1,{𝐱}_2,{𝐱}_3,{𝐱}_4\}$ in Figure 1.

The bond force density on a bond between two material points ${𝐱}_i$ and ${𝐱}_j$ is denoted by ${𝐟}_{ij}$. The symbol $\Delta V_i$ represents the volume of a specific material point denoted by ${𝐱}_i$.

Internal forces from $H_{{𝐱}_0}$

• Add ${𝐟}_{01}\Delta V_0\Delta V_1$ to ${𝐱}_0$, add reaction force $-{𝐟}_{01}\Delta V_0\Delta V_1$ to ${𝐱}_1$
• Add ${𝐟}_{02}\Delta V_0\Delta V_2$ to ${𝐱}_0$, add reaction force $-{𝐟}_{02}\Delta V_0\Delta V_2$ to ${𝐱}_2$
• Add ${𝐟}_{04}\Delta V_0\Delta V_4$ to ${𝐱}_0$, add reaction force $-{𝐟}_{04}\Delta V_0\Delta V_4$ to ${𝐱}_4$
• Add ${𝐟}_{06}\Delta V_0\Delta V_6$ to ${𝐱}_0$, add reaction force $-{𝐟}_{06}\Delta V_0\Delta V_6$ to ${𝐱}_6$

Internal forces from $H{\text{'}}_{{𝐱}_0}$ are calculated when calculating bond forces in $H_{{𝐱}_1},H_{{𝐱}_2},H_{{𝐱}_3},H_{{𝐱}_4}$.

• In $H_{{𝐱}_1}$, add ${𝐟}_{10}\Delta V_1\Delta V_0$ to ${𝐱}_1$, add reaction force $-{𝐟}_{10}\Delta V_1\Delta V_0$ to ${𝐱}_0$
• In $H_{{𝐱}_2}$, add ${𝐟}_{20}\Delta V_2\Delta V_0$ to ${𝐱}_2$, add reaction force $-{𝐟}_{20}\Delta V_2\Delta V_0$ to ${𝐱}_0$
• In $H_{{𝐱}_3}$, add ${𝐟}_{30}\Delta V_3\Delta V_0$ to ${𝐱}_3$, add reaction force $-{𝐟}_{30}\Delta V_3\Delta V_0$ to ${𝐱}_0$
• In $H_{{𝐱}_4}$, add ${𝐟}_{40}\Delta V_4\Delta V_0$ to ${𝐱}_4$, add reaction force $-{𝐟}_{40}\Delta V_4\Delta V_0$ to ${𝐱}_0$

See also

• Fracture mechanics
• Continuum mechanics
• Peridynamics

External links

• Peridigm, an open-source computational peridynamics code
• website on peridynamics

This article "Dual-horizon peridynamics" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Dual-horizon peridynamics. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

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• Oseen–Frank limit
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• KOMPAS-3D

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