The Variational Explanation of Poisson’s Ratio in Bond-Based Peridynamics and Extension to Nonlinear Poisson’s Ratio

Abstract

It is commonly stated that the Poisson’s ratio associated with bond-based peridynamics is $\tfrac{1}{4}$ for three-dimensional isotropic elasticity. This manuscript critically revisits this statement from a variational perspective for both two-dimensional and three-dimensional problems. To do so, a purely geometrical description of Poisson’s ratio is considered. Unlike the commonly established treatment of the problem, the Poisson’s ratio here is calculated via minimizing the internal energy density, rather than quantifying it and comparing it to its counterpart in classical linear elasticity. The advantage of the proposed approach is threefold. Firstly, elements of Cauchy linear elasticity such as “strain”, “stress” and “elastic parameters” are entirely absent throughout the derivations here. This is particularly important since peridynamics is a non-local formulation, and therefore, using local notions such as “strain” and “stress” implies locality and is misleading. Secondly, unbound by linear elasticity, the proposed approach unlocks the limitation of the analysis to small deformations. Hence, it can be immediately applied to large deformations, resulting in a nonlinear Poisson’s ratio that is no longer constant. Thirdly, the two-dimensional analysis here is purely two-dimensional, corresponding to a two-dimensional manifold in a three-dimensional space. That is, the two-dimensional formulation is neither plane stress nor plane strain that are rather degenerate three-dimensional cases. This contribution introduces the notion of nonlinear Poisson’s ratio in peridynamics for the first time and proves that the nonlinear Poisson’s ratio at the reference configuration coincides with $\tfrac{1}{4}$ for three-dimensional and $\tfrac{1}{3}$ for two-dimensional problems.

Linearization Procedure

The purpose of this Appendix is to show how the linearization of

$$\begin{aligned} \begin{aligned} \Psi = \displaystyle \frac{1}{2} \int _{\mathcal {H}_0} \displaystyle \frac{1}{2} \, C \, |\varvec{\Xi } {}^{^{|}} | \, \big [\, {|\boldsymbol {F}\cdot \hat{\varvec{\Xi }} {}^{^{|}} |} - 1 \,\big ]^2 \, \text{ d }A {}^{^{|}} \qquad \text {with} \qquad \boldsymbol{F}=\mathrm {Diag}(\lambda ,\eta ) \,, \end{aligned} \end{aligned}$$

(27)

furnishes

$$\begin{aligned} \begin{aligned} \Psi &= \displaystyle \frac{1}{2} \int _{\mathcal {H}_0} \displaystyle \frac{1}{2} \, C \, |\varvec{\Xi } {}^{^{|}} | \, \big [\, \boldsymbol {F} : [\hat{\varvec{\Xi }} {}^{^{|}} \otimes \hat{\varvec{\Xi }} {}^{^{|}} ] - 1 \,\big ]^2 \, \text{ d }A {}^{^{|}} \qquad \text {with} \qquad \\ \boldsymbol {F} &=\mathrm {Diag}(\lambda ,\eta ) \qquad \text {and} \qquad \{ \lambda , \eta \} \approx 1 \,. \end{aligned} \end{aligned}$$

(28)

In doing so, we begin with the fact that

$$\begin{aligned} \begin{aligned} |\boldsymbol {F}\cdot \hat{\varvec{\Xi }} {}^{^{|}} | = \displaystyle \frac{\xi {}^{^{|}} }{\Xi {}^{^{|}} } \,, \end{aligned} \end{aligned}$$

(29)

for which, the linearization yields

$$\begin{aligned} \begin{aligned} \text{ Lin }\, \displaystyle \frac{\xi {}^{^{|}} }{\Xi {}^{^{|}} } = \displaystyle \frac{\xi {}^{^{|}} }{\Xi {}^{^{|}} }\Bigg |_{\boldsymbol {F}=\mathbb {I}} + \displaystyle \frac{\partial }{\partial \boldsymbol {F}}\left( {\displaystyle \frac{\xi {}^{^{|}} }{\Xi {}^{^{|}} }}\right) \Bigg |_{\boldsymbol {F}=\boldsymbol {I}} : \big {[}\boldsymbol {F}-\boldsymbol {I}\big {]}\,. \end{aligned} \end{aligned}$$

(30)

At the reference configuration, the first term reduces to unity as

$$\begin{aligned} \begin{aligned} \displaystyle \frac{\xi {}^{^{|}} }{\Xi {}^{^{|}} }\Bigg |_{\boldsymbol {F}=\boldsymbol {I}} = \displaystyle \frac{\Vert \boldsymbol {F}\cdot \varvec{\Xi {}^{^{|}} }\Vert }{\Xi {}^{^{|}} }\Bigg |_{\boldsymbol {F}=\boldsymbol {I}} = \displaystyle \frac{\Vert \boldsymbol {I}\cdot \varvec{\Xi {}^{^{|}} }\Vert }{\Xi {}^{^{|}} } = \displaystyle \frac{\Xi {}^{^{|}} }{\Xi {}^{^{|}} } = 1\,. \end{aligned} \end{aligned}$$

(31)

Next, the derivative of ${\xi {}^{^{|}} }/{\Xi {}^{^{|}} }$ with respect to $\boldsymbol {F}$ is evaluated as

$$\begin{aligned} \begin{aligned} \displaystyle \frac{\partial }{\partial \boldsymbol {F}}\left( {\displaystyle \frac{\xi {}^{^{|}} }{\Xi {}^{^{|}} }}\right)&= \displaystyle \frac{1}{\Xi {}^{^{|}} }\displaystyle \frac{\partial \xi {}^{^{|}} }{\partial \boldsymbol {F}} = \displaystyle \frac{1}{\Xi {}^{^{|}} }\displaystyle \frac{\partial \xi {}^{^{|}} }{\partial \varvec{\xi {}^{^{|}} }} \cdot \displaystyle \frac{\partial \varvec{\xi {}^{^{|}} }}{\partial \boldsymbol {F}} \qquad \text {with} \qquad \displaystyle \frac{\partial \varvec{\xi {}^{^{|}} }}{\partial \boldsymbol {F}} = \boldsymbol {i}\otimes \varvec{\Xi {}^{^{|}} } \,, \end{aligned} \end{aligned}$$

(32)

where $\boldsymbol {i}$ is the identity tensor. Thus,

$$\begin{aligned} \begin{aligned} \displaystyle \frac{\partial }{\partial \boldsymbol {F}}\left( {\displaystyle \frac{\xi {}^{^{|}} }{\Xi {}^{^{|}} }}\right)&= \displaystyle \frac{1}{\Xi {}^{^{|}} }\displaystyle \frac{\partial \xi {}^{^{|}} }{\partial \varvec{\xi {}^{^{|}} }} \cdot \big {[}\boldsymbol {i}\otimes \varvec{\Xi {}^{^{|}} }\big {]} = \displaystyle \frac{1}{\Xi {}^{^{|}} }\displaystyle \frac{\varvec{\xi {}^{^{|}} }}{\xi {}^{^{|}} } \cdot \big {[}\boldsymbol {i}\otimes \varvec{\Xi {}^{^{|}} }\big {]}\,. \end{aligned} \end{aligned}$$

(33)

This expression, evaluated at $\boldsymbol {F} = \boldsymbol {I}$ returns

$$\begin{aligned} \begin{aligned} \displaystyle \frac{\partial }{\partial \boldsymbol {F}}\left( {\displaystyle \frac{\xi {}^{^{|}} }{\Xi {}^{^{|}} }}\right) \Bigg |_{\boldsymbol {F}=\boldsymbol {I}}&= \displaystyle \frac{1}{\Xi {}^{^{|}} }\displaystyle \frac{\partial \Xi {}^{^{|}} }{\partial \varvec{\Xi {}^{^{|}} }}\cdot \big {[}\boldsymbol {I}\otimes \varvec{\Xi {}^{^{|}} }\big {]} = \displaystyle \frac{1}{{\Xi {}^{^{|}} }^2}\varvec{\Xi } {}^{^{|}} \otimes \varvec{\Xi } {}^{^{|}} = \displaystyle \frac{\varvec{\Xi } {}^{^{|}} }{{\Xi {}^{^{|}} }}\otimes \displaystyle \frac{\varvec{\Xi } {}^{^{|}} }{{\Xi {}^{^{|}} }} = \hat{\varvec{\Xi }} {}^{^{|}} \otimes \hat{\varvec{\Xi }} {}^{^{|}} \,. \end{aligned} \end{aligned}$$

(34)

Finally, plugging this and Eq. (31) in the initial expression (30) yields

$$\begin{aligned} \begin{aligned} \text{ Lin }\, \displaystyle \frac{\xi {}^{^{|}} }{\Xi {}^{^{|}} } = 1 + \big {[}\hat{\varvec{\Xi }} {}^{^{|}} \otimes \hat{\varvec{\Xi }} {}^{^{|}} \big {]} : \big {[}\boldsymbol {F} - \boldsymbol {I} \big {]} = 1 + \big {[}\hat{\varvec{\Xi }} {}^{^{|}} \otimes \hat{\varvec{\Xi }} {}^{^{|}} \big {]} : \boldsymbol {F} - \hat{\varvec{\Xi }} {}^{^{|}} \cdot \hat{\varvec{\Xi }} {}^{^{|}} \qquad \text {with} \qquad \hat{\varvec{\Xi }} {}^{^{|}} \cdot \hat{\varvec{\Xi }} {}^{^{|}} = 1 \,, \end{aligned} \end{aligned}$$

(35)

or simply

$$\begin{aligned} \begin{aligned} \text{ Lin }\, \displaystyle \frac{\xi {}^{^{|}} }{\Xi {}^{^{|}} } = \big {[}\hat{\varvec{\Xi }} {}^{^{|}} \otimes \hat{\varvec{\Xi }} {}^{^{|}} \big {]} : \boldsymbol {F} \,. \end{aligned} \end{aligned}$$

(36)