Lecture 1: Bond based truss solution

For a truss everything is a scalar. The norms are given for better illustration.

\[i\]

• point

\[j\]

• neighbor

\[\xi = x_j-x_i\]

\[\eta = u_j - u_i\]

Bond force

\[f_{ij} = c\omega_{ij} s\frac{\xi+\eta}{||\xi + \eta||} = c\omega_{ij} s\]

\[s = \frac{||\xi + \eta|| - ||\xi||}{||\xi||} = \frac{\eta}{\xi}\]

\[\rho u_i = \frac{1}{2}\sum_j (f_{ji}-f_{ij})V_j+b_i\]

point based is solved like this

Matrix based

\[\begin{bmatrix} K_{11} & \cdots & K_{1j} & 0& ...&0\\ \vdots & \ddots & \\ K_{j1} & & K_{jj} &&&\\ 0 & && \\ \vdots & & &&\ddots\\ 0& & \cdots&&&K_{nn} \end{bmatrix}\begin{bmatrix}u_1\\\vdots\\u_j\\u_{j+1}\\\vdots \\u_n\end{bmatrix} = \mathbf{F}_{internal}\]

\[K_{ij} = -\frac{c}{|\xi_{ij}|} \omega_{ij}V_j \quad \text{for } i \ne j,\quad K_{ii} = -\sum_{j \ne i} K_{ij}; i=1\,...\,n \quad \text{and} \quad j=i+1\,...\,n_{neighbors,i}+1\]

\[\mathbf{F}_{internal}\]

is the force density in $\left[\frac{N}{m^3}\right]$

1D [27] $\rightarrow$ $c = \frac{2E}{A\delta^2}$ 2D For FEM

\[\mathbf{K}=\frac{EA}{L}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}\]

Assumptions

$\delta = L,\, n_{neighbors,i}=1,\, \omega_{ij}=1,\, V_i=V=const.$

\[V=AL\]

\[c=0.5\frac{2E}{AL^2}\]

\[K_{11}=-K_{12}=-0.5\frac{c}{\xi} V=-0.5\frac{c}{L} V=-\frac{E}{AL^3}AL=-\frac{E}{L^2}\]

\[\mathbf{K}=\frac{E}{L^2}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}\]

bring it in local form of the stiffness matrix the forces and not the force densities the stiffness has to be multiplied by $V$.

\[^{forces}K_{11}=-^{forces}K_{12}=-\frac{E}{L^2}AL=\frac{EA}{L}\]

\[\mathbf{K}=\frac{EA}{L}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}\]

Analysis of matrix behavior

using LinearAlgebra
E = 1
V = 1
L = 1
np = 8
nn = 2
delta = 1
omega=ones(np, np)
##
c=zeros(np)
c .= 2*E/delta^2
c[4]=1.5 .*c[4]
K=zeros(np,np)
for iID in 1:np
  for jID in -nn:nn
    if jID != 0 && iID + jID > 0 && iID + jID < np + 1
      xi = L*abs(jID)
      K[iID, iID + jID] -= 0.5 * c[iID] / xi * V * omega[iID, iID + jID]
      K[iID, iID]      +=  0.5 *c[iID] / xi * V * omega[iID, iID + jID]
    end
  end
end

display(K)
rank(K)
eigvals(K)
det(K)

## Damage

omega[1,2] = 0

K=zeros(np,np)
for iID in 1:np
  for jID in -nn:nn
    if jID != 0 && iID + jID > 0 && iID + jID < np + 1
      xi = L*abs(jID)
      K[iID, iID + jID] -= 0.5 * c[iID] / xi * V * omega[iID, iID + jID]
      K[iID, iID]      +=  0.5 *c[iID] / xi * V * omega[iID, iID + jID]
    end
  end
end

display(K)

Perilab

Mesh

header: x y block_id volume
0 0 1 1
1 0 1 1
2 0 1 1
3 0 1 1
4 0 1 1

Yaml

PeriLab:
  Discretization:
    Node Sets:
      Node Set 1: 1
      Node Set 2: 5
    Type: "Text File"
    Input Mesh File: "truss.txt"
  Models:
    Material Models:
      Test:
        Material Model: "Bond-based Elastic"
        Symmetry: "isotropic plane stress"
        Young's Modulus: 7000
        Poisson's Ratio: 0.3
  Blocks:
    block_1:
      Block ID: 1
      Material Model: "Test"
      Density: 2e-9
      Horizon: 2
  Boundary Conditions:
    BC_1:
      Variable: "Displacement"
      Node Set: "Node Set 1"
      Coordinate: "x"
      Value: "100*t"
      Type: Dirichlet
    BC_2:
      Variable: "Displacement"
      Coordinate: "x"
      Node Set: "Node Set 2"
      Value: "0.1*t"
      Type: Dirichlet
  Solver:
    Material Models: True
    Initial Time: 0.0
    Final Time: 1.0
    Number of Steps: 20
    Static:
      Show solver iteration: true
      Residual tolerance: 1e-7
      Solution tolerance: 1e-8
      Residual scaling: 7000
      m: 550
      Maximum number of iterations: 100
  Outputs:
    Output1:
      Output Filename: "truss"
      Output File Type: Exodus
      Number of Output Steps: 20
      Output Variables:
        Displacements: True
        Number of Neighbors: True
        Forces: True