Seminar 4: From bond-based to state-based I (Theory)

Definition of states

• A state is not in general a linear function of $\xi$ .
• A state is not in general a continuous function of $\xi$.
• The real Euclidean space $V$ is infinite-dimensional, while the real Euclidean space $\mathcal{L}_2$ (the set of second order tensors) has dimension 9

Definitions

scalar state

\[\underline{a}\langle \boldsymbol{\xi}\rangle\]

vector state

\[\underline{\mathbf{A}}\langle \boldsymbol{\xi}\rangle\]

Shape tensor

\[\mathbf{K} = \underline{\mathbf{X}}*\underline{\mathbf{X}} = \int_{\mathcal{H}}\underline{\omega}\langle \boldsymbol{\xi}\rangle\underline{\mathbf{X}}\langle \boldsymbol{\xi}\rangle\otimes\underline{\mathbf{X}}\langle \boldsymbol{\xi}\rangle dV\]

• is positive definite

Figure taken from [14]

Constitutive Models

\[\boldsymbol{\xi} = \mathbf{x}'-\mathbf{x}\]

Deformation vector state field

\[\underline{\mathbf{Y}}[\mathbf{x},t]\langle \boldsymbol{\xi}\rangle=\mathbf{y}(\mathbf{x}+\boldsymbol{\xi},t)-\mathbf{y}(\mathbf{x},t)\]

A material is elastic if there exists a differentiable scalar valued function $W(·) : \mathcal{V} \rightarrow \mathbb{R}$ such that

\[\underline{\mathbf{T}}= \hat{\underline{\mathbf{T}}}(\underline{\mathbf{Y}})= \nabla W(\underline{\mathbf{Y}})\]

\[W\]

is the strain energy density function.

Ordinary state-based

Ordinary and elastic means

• It is mobile
• There exists a scalar-valued function $w$

\[W(\underline{\mathbf{Y}})= w(\underline{y})\qquad\text{and}\qquad \underline{y}=|\underline{\mathbf{Y}}\]

• For this $w$

\[\underline{t}(\underline{y})= \nabla w(\underline{y})\]

PD solid elastic

Concept uses Lamé coefficients $\boldsymbol{\sigma} = 2G\boldsymbol{\varepsilon} + K \; \operatorname{tr}(\boldsymbol{\varepsilon}) I$

\[\underline{e}\langle\boldsymbol{\xi}\rangle=|\mathbf{F}\boldsymbol{\xi}| - |\boldsymbol{\xi}|=\varepsilon_{ij}\frac{\xi_i\xi_j}{|\boldsymbol{\xi}|}\]

\[\varepsilon_{ij}=\frac12(u_{i,j}+u_{j,i})\]

\[\underline{e}^d\langle\boldsymbol{\xi}\rangle=\varepsilon_{ij}^d\frac{\xi_i\xi_j}{|\boldsymbol{\xi}|}\]

\[W=\frac{\alpha}{2}\int_{\mathcal{H}}\underline{\omega}\langle\boldsymbol{\xi}\rangle(\underline{e}^d\langle\boldsymbol{\xi}\rangle)^2dV_{\boldsymbol{\xi}}\]

under assumption of a spherical non-local domain.

\[W = \frac{\alpha m}{15}\varepsilon_{ij}^d\varepsilon_{ij}^d\]

compared with the strain energy density of the classical model.

\[\Omega=G\varepsilon_{ij}^d\varepsilon_{ij}^d\]

\[\alpha=\frac{15G}{m}\]

\[\underline{x}=|\underline{\mathbf{X}}|=|\boldsymbol{\xi}|\quad\underline{y}=|\underline{\mathbf{Y}}|\]

\[\underline{e}=\underline{y}-\underline{x}=|\boldsymbol{\eta}|\]

\[\underline{y}-\underline{x}\neq|\boldsymbol{\eta}|\]

\[\underline{t}=|\underline{\mathbf{T}}|\]

Weighted volume

\[m_V = \int_{\mathcal{H}} \underline{\omega}\langle \boldsymbol{\xi}\rangle \underline{x} \underline{x} dV\]

Dilatation

\[\theta = \frac{3}{m_V} = \int_{\mathcal{H}}\underline{\omega}\langle \boldsymbol{\xi}\rangle \underline{x} \underline{e}\langle \boldsymbol{\xi}\rangle dV\]

\[\underline{t} = \frac{\omega\langle \boldsymbol{\xi}\rangle }{m_V}\left[3K \theta \underline{x} + 15G \underline{e}^d \right]\]

Decomposition in the devatoring and isotropic part of the strain

\[\underline{e}^d\langle \boldsymbol{\xi}\rangle = \epsilon_{ij}^d\xi_i\frac{x_j}{|\boldsymbol{\xi}|}\]

\[\underline{e}^i\langle \boldsymbol{\xi}\rangle = \epsilon_{ij}^i\xi_i\frac{x_j}{|\boldsymbol{\xi}|}\]

The force density can be determined as

\[\underline{\mathbf{T}}=\underline{t}\frac{\underline{\mathbf{Y}}}{|\underline{\mathbf{Y}}|}\]

Correspondence

\[\underline{\mathbf{Y}}\langle \boldsymbol{\xi}\rangle=\mathbf{F}\boldsymbol{\xi}\]

Then the peridynamic constitutive model corresponds to the classical constitutive model at $\mathbf{F}$ [14].

\[\mathbf{F}=\int_{\mathcal{H}}(\underline{\omega}\langle \boldsymbol{\xi}\rangle\underline{\mathbf{Y}}\langle \boldsymbol{\xi}\rangle\otimes\underline{\mathbf{X}}\langle \boldsymbol{\xi}\rangle dV )\cdot \mathbf{K}^{-1}\]

\[\mathbf{K}=\int_{\mathcal{H}}\underline{\omega}\langle \boldsymbol{\xi}\rangle\underline{\mathbf{X}}\langle \boldsymbol{\xi}\rangle\otimes\underline{\mathbf{X}}\langle \boldsymbol{\xi}\rangle dV\]

\[\boldsymbol{\sigma} = f(\mathbf{F}, t, T, ...)\]

\[\mathbf{P} = \text{det}(\mathbf{F})\boldsymbol{\sigma}\mathbf{F}^{-T}\]

\[\underline{\mathbf{T}} = \underline{\omega}\langle \boldsymbol{\xi}\rangle\mathbf{P}\mathbf{K}^{-1}\mathbf{\xi}\]

Zero-energy modes

For correspondence models, the so called zero-energy modes could occur [15]. These modes are non-physical and lead to unstable or unreasonable solutions. Several stabilization methods were published to overcome this problem [16], [17], [18], [19],[20],[21].

\[\underline{\mathbf{T}}^C=\underline{\mathbf{T}}+\underline{\mathbf{T}}^S\]

\[\underline{\mathbf{T}}^S\langle \boldsymbol{\xi}\rangle = \underline{\omega}\langle\boldsymbol{\xi}\rangle\mathbf{C}_1\underline{\mathbf{z}}\]

\[\underline{\mathbf{z}}\langle \boldsymbol{\xi}\rangle= \underline{\mathbf{Y}}\langle\boldsymbol{\xi} \rangle-\mathbf{F}\boldsymbol{\xi}\]

\[\mathbf{C}_1=\mathbf{C}\cdot\cdot\mathbf{K}^{-1}\]

x = [1,2,3,4,5]

a = [1,2,3,4,5]

integral_a = sum(a)

b=[1,2,3.1,3.9,5]

integral_b = sum(b)

display(integral_a - integral_b)

function def_grad(x,y)
    K = sum(x)
    F = sum(y)/K
    display("F*x $(F.*x')")
    display("z $(F.*x' - y)")
end

x = [1,2,3,4]
y = [0.5 1. 1.5 2]

println("Constant deformation gradient in non-local domain")
def_grad(x,y)


y = [0.5 0.9 1.6 2]
println("Non-Constant deformation gradient in non-local domain")
def_grad(x,y)

Properties of $\mathbf{K}$ and $\mathbf{F}$

using LinearAlgebra
x = [1 0;0 1;0 -1; -1 0; 1 1; -1 -1]
K = x'*x

display(K'-K)
display(rank(K))

y = [1.1 0;0 1.0;0 -1; -1 0; 1 1; -1 -1]

F = y'*x / K

display(F'-F)
display(rank(F))

strain = 0.5*(F'*F - I)
display(strain'-strain)
display(rank(strain))