Weak Form of Peridynamic Equilibrium Equations

Abstract

This chapter presents the weak form of peridynamic (PD) equilibrium equations to impose displacement and traction boundary conditions without a fictitious material layer. It permits the direct imposition of traction and displacement boundary conditions. The nonlocal deformation gradient tensor is computed in a bond-associated (BA) domain of interaction using the PD differential operator. The weak form of PD equilibrium equation is derived for the neo-Hookean material model with slight compressibility. The results are free of oscillations and spurious zero-energy modes that are commonly observed in the PD correspondence models. The fidelity of the approach is established by comparison with those of finite element analysis. Numerical results concern the finite elastic deformation of a rubber sheet with a hole under stretch.

Appendix

Under plane strain assumptions of (∂/∂x₃) = 0 and y₃ = x₃, the deformation gradient tensor reduces to F ∼ F_αβ with (α, β = 1, 2). For two-dimensional analysis, the isochoric part of Eq. (4.53) can be modified as

$$ \Psi \left({\overline{I}}_1,J\right)=\frac{\mu }{2}\left(\frac{\mathrm{tr}\mathbf{C}}{J^{n/m}}-2\right)+\frac{\kappa }{8}{\left(J-\frac{1}{J}\right)}^2 $$

(4.59a)

or

$$ \Psi \left({\overline{I}}_1,J\right)=\frac{\mu }{2}\left(\frac{\mathrm{tr}\left(\mathbf{C}\right)}{{\left(\det \mathbf{C}\right)}^{n/(2m)}}-2\right)+\frac{\kappa }{8}{\left(J-\frac{1}{J}\right)}^2 $$

(4.59b)

For a scaled right Cauchy–Green tensor aC, this equation becomes

$$ \Psi \left({\overline{I}}_1,J\right)=\frac{\mu }{2}\left(\frac{\mathrm{tr}\left(a\mathbf{C}\right)}{{\left({a}^2\det \mathbf{C}\right)}^{n/(2m)}}-2\right)+\frac{\kappa }{8}{\left(J-\frac{1}{J}\right)}^2 $$

(4.60a)

or

$$ \Psi \left({\overline{I}}_1,J\right)=\frac{\mu }{2}\left(\frac{a\mathrm{tr}\left(\mathbf{C}\right)}{a^{n/m}{\left(\det \mathbf{C}\right)}^{n/(2m)}}-2\right)+\frac{\kappa }{8}{\left(J-\frac{1}{J}\right)}^2 $$

(4.60b)

Specifying n = m = 1 makes the isochoric part invariant, and the strain energy density function becomes

$$ \Psi \left({\overline{I}}_1,J\right)=\frac{\mu }{2}\left(\frac{\mathrm{tr}\mathbf{C}}{{\left(\det \mathbf{C}\right)}^{1/2}}-2\right)+\frac{\kappa }{8}{\left(J-\frac{1}{J}\right)}^2 $$

(4.61)

Finally, it can be rewritten as

$$ \Psi \left({\overline{I}}_1,J\right)=\frac{\mu }{2}\left({\overline{I}}_1-2\right)+\frac{\kappa }{8}{\left(J-\frac{1}{J}\right)}^2 $$

(4.62)

where, ${\overline{I}}_1=\frac{\mathrm{tr}\mathbf{C}}{J}$.

Thus, the first Piola–Kirchhoff stress tensor can be obtained as

$$ \mathbf{P}=\frac{\mathrm{\partial \Psi }}{\partial \mathbf{F}}=\mu \left(\mathbf{F}-\frac{1}{2}\mathrm{tr}{\mathbf{CF}}^{-T}\right){J}^{-1}+\frac{\kappa }{4}\left({J}^2-{J}^{-2}\right){\mathbf{F}}^{-T} $$

(4.63a)

or

$$ {P}_{ij}=\frac{\mathrm{\partial \Psi }}{\partial {F}_{ij}}=\mu \left({F}_{ij}-\frac{1}{2}{F}_{mn}{F}_{mn}{F}_{ji}^{-1}\right){J}^{-1}+\frac{\kappa }{4}\left({J}^2-{J}^{-2}\right){F}_{ji}^{-1} $$

(4.63b)

Its derivative necessary in the evaluation of the incremental internal force vector ΔQ_(k) can be written as

$$ {\displaystyle \begin{array}{l}\frac{\partial {P}_{ij}}{\partial {F}_{kl}}=\mu \left(\frac{\partial {F}_{ij}}{\partial {F}_{kl}}-\frac{1}{2}\left(\frac{\left({F}_{mn}{F}_{mn}\right)}{\partial {F}_{kl}}{F}_{ji}^{-1}+\left({F}_{mn}{F}_{mn}\right)\frac{\partial {F}_{ji}^{-1}}{\partial {F}_{kl}}\right)\right){J}^{-1}\\ {}\kern1.75em -\mu \left({F}_{ij}-\frac{1}{2}\left({F}_{mn}{F}_{mn}\right){F}_{ji}^{-1}\right){J}^{-1}{F}_{lk}^{-1}\\ {}\kern1.75em +\frac{\kappa }{2}\left({J}^2+{J}^{-2}\right){F}_{lk}^{-1}{F}_{ji}^{-1}+\frac{\kappa }{4}\left({J}^2+{J}^{-2}\right)\frac{\partial {F}_{ji}^{-1}}{\partial {F}_{kl}}\end{array}} $$

(4.64a)

or

$$ {\displaystyle \begin{array}{l}\frac{\partial {P}_{ij}}{\partial {F}_{kl}}=\mu \left({\delta}_{ik}{\delta}_{jl}-\frac{1}{2}\left(2{F}_{kl}{F}_{ji}^{-1}-{F}_{mn}{F}_{mn}{F}_{li}^{-1}{F}_{jk}^{-1}\right)\right){J}^{-1}\\ {}\kern1.75em -\mu \left({F}_{ij}-\frac{1}{2}{F}_{mn}{F}_{mn}{F}_{ji}^{-1}\right){J}^{-1}{F}_{lk}^{-1}\\ {}\kern1.75em +\frac{\kappa }{2}\left({J}^2+{J}^{-2}\right){F}_{lk}^{-1}{F}_{ji}^{-1}-\frac{\kappa }{4}\left({J}^2+{J}^{-2}\right){F}_{li}^{-1}{F}_{jk}^{-1}\end{array}} $$

(4.64b)

with $\frac{\partial J}{\partial {F}_{ij}}={JF}_{ji}^{-1}$.

Under plane stress assumptions of σ₁₃ = σ₂₃ = σ₃₃ = 0, the Cauchy stress tensor σ reduce to σ ∼ σ_αβ with (α, β = 1, 2). These conditions can be equivalently imposed on the second Piola–Kirchhoff stress tensor S as S₁₃ = S₂₃ = S₃₃ = 0. The second Piola–Kirchhoff stress tensor S is related to the first Piola–Kirchhoff stress tensor as

$$ \mathbf{S}={\mathbf{F}}^{-1}\mathbf{P}=\mu \left(\mathbf{I}-\frac{1}{3}\mathrm{tr}{\mathbf{CF}}^{-1}{\mathbf{F}}^{-T}\right){J}^{-2/3}+\frac{\kappa }{4}\left({J}^2-{J}^{-2}\right){\mathbf{F}}^{-1}{\mathbf{F}}^{-T} $$

(4.65)

By using the relation C⁻¹ = F⁻¹F^−T, this expression can be simplified as

$$ \mathbf{S}=\mu \left(I-\frac{1}{3}\mathrm{tr}{\mathbf{C}\mathbf{C}}^{-1}\right){J}^{-2/3}+\frac{\kappa }{4}\left({J}^2-{J}^{-2}\right){\mathbf{C}}^{-1} $$

(4.66)

Enforcing S₁₃ = 0 and S₂₃ = 0 results in

$$ {\left({\mathbf{C}}^{-1}\right)}_{13}=0 $$

(4.67a)

$$ {\left({\mathbf{C}}^{-1}\right)}_{23}=0 $$

(4.67b)

and

$$ {\left({\mathbf{C}}^{-1}\right)}_{33}=\frac{1}{C_{33}} $$

(4.67c)

Similarly, enforcing S₃₃ = 0 leads to an additional expression as

$$ {S}_{33}=\mu \left(1-\frac{1}{3}\mathrm{tr}\mathbf{C}{\left({\mathbf{C}}^{-1}\right)}_{33}\right){J}^{-2/3}+\frac{\kappa }{4}\left({J}^2-{J}^{-2}\right){\left({\mathbf{C}}^{-1}\right)}_{33}=0 $$

(4.68)

Substituting from Eq. (4.67c) into Eq. (4.68) results in the constraint condition for C₃₃ in the form

$$ \mu \left({C}_{33}-\frac{1}{3}\mathrm{tr}\mathbf{C}\right){J}^{-2/3}+\frac{\kappa }{4}\left({J}^2-{J}^{-2}\right)=0 $$

(4.69)

Invoking Eq. (4.69) in Eq. (4.66) and performing algebraic manipulations result in the simplified form of S for plane stress analysis as

$$ \mathbf{S}=\mu \left(I-{C}_{33}{\mathbf{C}}^{-1}\right){J}^{-2/3} $$

(4.70)

Finally, the first Piola–Kirchhoff stress tensor can be obtained as

$$ \mathbf{P}=\mathbf{FS}=\mu \left(\mathbf{F}-{C}_{33}{\mathbf{F}}^{-T}\right){J}^{-2/3} $$

(4.71)

For numerical implementation, trC and J are expressed as

$$ \mathrm{tr}\mathbf{C}=\mathrm{tr}\tilde{\mathbf{C}}+{C}_{33} $$

(4.72)

and

$$ J=\sqrt{C_{33}}\tilde{J} $$

(4.73)

where $\mathrm{tr}\tilde{\mathbf{C}}={C}_{11}+{C}_{22}$ and $\tilde{J}=\sqrt{C_{11}{C}_{22}-{C}_{12}^2}$.

With these expressions, Eqs. (4.71) and (4.69) can be rewritten as

$$ \mathbf{P}=\mu \left(\mathbf{F}-{C}_{33}{\mathbf{F}}^{-T}\right){C}_{33}^{-1/3}{\tilde{J}}^{-2/3} $$

(4.74)

and

$$ \mu \left(\frac{2}{3}{C}_{33}^{-2/3}-\frac{1}{3}{C}_{33}^{-1/3}\mathrm{tr}\tilde{\mathbf{C}}\right){\tilde{J}}^{-2/3}+\frac{\kappa }{4}\left({C}_{33}{\tilde{J}}^2-{C}_{33}^{-1}{\tilde{J}}^{-2}\right)=0 $$

(4.75)

Based on the displacement field obtained from the previous iteration within every load step, the value of ${C}_{33}={C}_{33}^{\ast }$ is numerically determined by using the method of bisection.

Its derivative necessary in the evaluation of the incremental internal force vector ΔQ_(k) can be written as

$$ {\displaystyle \begin{array}{l}\frac{\partial {P}_{ij}}{\partial {F}_{kl}}=\mu \left(\frac{\partial {F}_{ij}}{\partial {F}_{kl}}-\left(\frac{\partial {C}_{33}}{\partial {F}_{kl}}{F}_{ij}^{-T}+{C}_{33}\frac{\partial {F}_{ji}^{-1}}{\partial {F}_{kl}}\right)\right){C}_{33}^{-1/3}{\tilde{J}}^{-2/3}\\ {}+\mu {\tilde{J}}^{-2/3}\left({F}_{ij}-{C}_{33}{F}_{ji}^{-1}\right)\left(-\frac{1}{3}{C}_{33}^{-4/3}\frac{\partial {C}_{33}}{\partial {F}_{kl}}-\frac{2}{3}{C}_{33}^{-1/3}{F}_{lk}^{-1}\right)\end{array}} $$

(4.76a)

or

$$ {\displaystyle \begin{array}{l}\frac{\partial {P}_{ij}}{\partial {F}_{kl}}=\mu \left({\delta}_{ik}{\delta}_{jl}-\left(\frac{\partial {C}_{33}}{\partial {F}_{kl}}{F}_{ji}^{-1}-{C}_{33}{F}_{li}^{-1}{F}_{jk}^{-1}\right)\right){C}_{33}^{-1/3}{\tilde{J}}^{-2/3}\\ {}+\mu {\tilde{J}}^{-2/3}\left({F}_{ij}-{C}_{33}{F}_{ji}^{-1}\right)\left(-\frac{1}{3}{C}_{33}^{-4/3}\frac{\partial {C}_{33}}{\partial {F}_{kl}}-\frac{2}{3}{C}_{33}^{-1/3}{F}_{lk}^{-1}\right)\end{array}} $$

(4.76b)

The derivative of C₃₃ can be obtained from Eq. (4.75) as

$$ \frac{\partial {C}_{33}}{\partial {F}_{kl}}=\frac{\left(\begin{array}{l}\frac{1}{3}\mu {C}_{33}^{-1/3}{\tilde{J}}^{-2/3}\frac{\mathrm{\partial tr}\tilde{\mathbf{C}}}{\partial {F}_{kl}}+\frac{2}{3}\mu \left(\frac{2}{3}{C}_{33}^{2/3}-\frac{1}{3}{C}_{33}^{-1/3}\mathrm{tr}\tilde{\mathbf{C}}\right){\tilde{J}}^{-2/3}{F}_{ji}^{-1}\\ {}-\frac{\kappa }{4}\left({C}_{33}2{\tilde{J}}^2{F}_{lk}^{-1}+2{C}_{33}^{-1}{\tilde{J}}^{-2}{F}_{ji}^{-1}\right)\end{array}\right)}{\left(\left(\frac{4}{9}\mu {C}_{33}^{-1/3}+\frac{1}{9}\mu {C}_{33}^{-4/3}\mathrm{tr}\tilde{\mathbf{C}}\right){\tilde{J}}^{-2/3}+\frac{\kappa }{4}{\tilde{J}}^2+\frac{\kappa }{4}{C}_{33}^{-2}{\tilde{J}}^{-2}\right)} $$

(4.77)