Objectivity of State-Based Peridynamic Models for Elasticity

Abstract

We verify the objectivity (invariance to rigid body rotations) ordinary state-based peridynamic models published in the literature that differ in their formulas. We find and explain the sources for the differences between these published formulas. We demonstrate that a primary cause leading to these differences is the way in which the peridynamic volume dilatation is defined in the different formulations. We show that the equations of motion derived from one approach apply to deformations with small or large rotations and is objective. The other approach is valid only for deformations with zero/infinitesimal rotations, thus it is not objective. We also show that both state-based models recover the correct bond-based formulations with the appropriate Poisson ratio values when the term containing the volume dilatation vanishes.

Appendices

Appendix A: Peridynamic States and Their Operations

The peridynamic states and their operations are defined in [2]. Below is a summary of the basic concepts.

Definition 1

Let ℋ be a neighborhood of radius $\delta$ centered at the origin in $\mathbb{R}^{3}$. Let $\mathcal{L}_{m}$ denote the set of all tensors of order $m$. Then a state of order $m$ is a function $\underline{\mathbf{A}} \langle \boldsymbol{\cdot} \rangle \mathcal{:H\rightarrow L}_{m}$.

A peridynamic state maps a peridynamic bond vector ($\boldsymbol{\xi}$) to a tensor of order $m$. Examples of peridynamic states include: the reference position scalar state $\underline{x}$ (mapping a bond vector to its undeformed bond length), extension scalar state $\underline{e}$ (mapping a bond vector to the bond’s elongation), force vector state $\underline{\mathbf{T}}$ (mapping a bond vector to the bond’s force density vector).

Definition 2

Let $\underline{\mathbf{A}} \in \mathcal{A}_{m}$ and $\underline{\mathbf{B}} \in \mathcal{A}_{m}$. The sum and the difference of the states $\underline{\mathbf{A}}$ and $\underline{\mathbf{B}}$ are, respectively:

$$\begin{aligned} ( \underline{\mathbf{A}} + \underline{\mathbf{B}} ) \langle \boldsymbol{\xi} \rangle =& \underline{\mathbf{A}} \langle \boldsymbol{\xi} \rangle + \underline{\mathbf{B}} \langle \boldsymbol{\xi} \rangle, \end{aligned}$$

(45)

$$\begin{aligned} ( \underline{\mathbf{A}} - \underline{\mathbf{B}} ) \langle \boldsymbol{\xi} \rangle =& \underline{\mathbf{A}} \langle \boldsymbol{\xi} \rangle - \underline{\mathbf{B}} \langle \boldsymbol{\xi} \rangle. \end{aligned}$$

(46)

Definition 3

The point product of two states $\underline{\mathbf{A}} \in \mathcal{A}_{m+p}$ and $\underline{\mathbf{B}} \in \mathcal{A}_{p}$ is a state in $\mathcal{A}_{m}$ and defined by:

$$\begin{aligned} ( \underline{\mathbf{AB}} )_{i_{1} \dots i_{m}} \langle \boldsymbol{\xi} \rangle = \underline{A}_{i_{1} \dots i_{m} j_{1} \dots i_{p}} \langle \boldsymbol{\xi} \rangle \underline{B}_{j_{1} \dots j_{p}} \langle \boldsymbol{\xi} \rangle. \end{aligned}$$

(47)

Similarly,

$$\begin{aligned} ( \underline{\mathbf{BA}} )_{i_{1} \dots i_{m}} \langle \boldsymbol{\xi} \rangle = \underline{B}_{j_{1} \dots j_{p}} \langle \boldsymbol{\xi} \rangle \underline{A}_{j_{1} \dots j_{p} i_{1} \dots i_{m}} \langle \boldsymbol{\xi} \rangle. \end{aligned}$$

(48)

Note that in this definition, the Einstein summation notation is used, i.e., the summation over all the $j_{1} \dots j_{p}$ components. The point product of two states is not necessarily commutative. If at least one of the two states is a scalar state, then their point product is commutative.

Definition 4

The dot product of two states $\underline{\mathbf{A}}$ and $\underline{\mathbf{B}}$ is:

$$\begin{aligned} \underline{\mathbf{A}} \bullet \underline{\mathbf{B}} = \int_{\mathcal{H}} ( \underline{\mathbf{AB}} ) \langle \boldsymbol{\xi} \rangle dV_{ \langle \boldsymbol{\xi} \rangle}. \end{aligned}$$

(49)

Note that the dot product of two states is not a state, unlike the point product in Definition 3.

Appendix B: Testing the Dilatation Terms of Approaches A and B in 3D

Testing conditions:

To compare the value of classical volume dilation with the peridynamic volume dilation formulas given by Eqs. (7) and (14), we subject an elastic material to a homogeneous deformation given by $( \boldsymbol{\xi} + \boldsymbol{\eta}) = \mathbf{F}. \boldsymbol{\xi}$, where $\mathbf{F}$ is the classical deformation gradient tensor. We test three cases: (i) large strains and no rotations; (ii) small strains and large rotations, and (iii) small strains and no rotations. Here are some specific values for $\mathbf{F}$:

$$\begin{aligned} \mbox{(i)} \quad \mathbf{F} =& \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 3 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 1 \end{array}\displaystyle \right ] \ \ \mbox{implies volume dilatation } \mathrm{det}(\mathbf{F})-1 = 5. \\ \mbox{(ii)} \quad \mathbf{F} =& \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0 & -1.01 & 0\\ 1.01 & 0 & 0\\ 0 & 0 & 1.01 \end{array}\displaystyle \right ]\ \ \mbox{gives volume dilatation } \mathrm{det}(\mathbf{F})-1 = 3.0301\times 10^{-2}.\\ \mbox{(iii)}\quad \mathbf{F} =& \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 1.01 & 0 & 0\\ 0 & 0.99 & 0\\ 0 & 0 & 1.02 \end{array}\displaystyle \right ] \ \ \mbox{produces volume dilatation} \mathrm{det}(\mathbf{F})-1 = 1.9898\times 10^{-2}. \end{aligned}$$

For these corresponding deformations, we compute the peridynamic dilatations given by Eqs. (7) and (14) by numerical integration. We use numerical integration in Matlab (‘integral3’ function), and the integration region is a sphere of radius 1 $(\delta=1)$. We find the corresponding peridynamic dilatations $\theta_{\mathrm{A}}$ and $\theta_{\mathrm{B}}$. For case (i) we get $\theta_{\mathrm{A}} \approx3.1557$ and $\theta_{\mathrm{B}} \approx 3.3037$, both different from the value 5 obtained above. This verifies that approached A and B are not applicable for the case of large strains. For case (ii) we get $\theta_{\mathrm{A}} \approx 1.0000\times 10^{-2}$ and $\theta_{\mathrm{B}} \approx 3.0000\times 10^{-2}$. The volume dilatation formula from approach B gives a value which is only 1% different from the classical volume dilation, while approach A cannot be used for this case. For case (iii) we obtain $\theta_{\mathrm{A}} \approx 2.0139\times 10^{-2}$ and $\theta_{\mathrm{B}} \approx 2.0139\times 10^{-2}$. In this case, both formulas give results that are about 1% different from the classical value.