Stress and Displacement Analysis of an L-Shaped Elastic Body

Solve a plane strain elasticity problem and visualize the resulting displacements and stresses in an L-shaped domain

An L-shaped elastic domain is fixed along its bottom edge and pulled by a uniform tensile load applied on the upper segment of the right boundary. The deformation is modeled using plane strain elasticity, and the resulting displacement field, stress components, and von Mises stress distribution are computed and visualized.

Define the height and thickness of the L-shaped domain:

In[1]:=

{H,W}={3,1};

Build the L-shaped domain by combining two rectangular regions:

In[2]:=

Ω=RegionUnion[Rectangle[{0,0},{W,H}],Rectangle[{W,H-W},{H,H}]];

Plot the geometry and indicate the fixed bottom edge and the applied tensile load:

In[3]:=

RegionPlotΩ,ImageSizeSmall,Epilog{Table[Arrow[{{3,y},{3.5,y}}],{y,2,3,0.2}],Thick,Line[{{0,0},{1,0}}]},



Out[3]=

Introduce the unknown displacement fields

𝑢𝑥,𝑦

and

𝑣𝑥,𝑦

, together with the stress components

𝜎

𝑥𝑥

(𝑥,𝑦),

𝜎

𝑥𝑦

(𝑥,𝑦) and

𝜎

𝑦𝑦

(𝑥,𝑦). Impose static force balance by requiring the divergence of the stress tensor to vanish, and relate stresses to displacement gradients using the linear elastic constitutive law for plane strain, parameterized by Young's modulus and Poisson's ratio:

In[4]:=

planeStrainPDEs[{u_,v_},{sxx_,sxy_,syy_},{young_,nu_}]:=Module[{lam,mu},{lam,mu}=young{nu/(1+nu)/(1-2nu),1/(2(1+nu))};{D[sxx[x,y],x]+D[sxy[x,y],y]0,D[sxy[x,y],x]+D[syy[x,y],y]0,sxx[x,y]2muD[u[x,y],x]+lam(D[u[x,y],x]+D[v[x,y],y]),sxy[x,y]mu(D[u[x,y],y]+D[v[x,y],x]),syy[x,y]2muD[v[x,y],y]+lam(D[u[x,y],x]+D[v[x,y],y])}]

Specify the boundary conditions on the different edges of the region:

In[5]:=

bcs={(*fixedbottomedge*)DirichletCondition[{u[x,y]0,v[x,y]0},y0],(*traction-freeleftedge*)DirichletCondition[{sxx[x,y]0,sxy[x,y]0},x0],(*traction-freeinnerverticaledge*)DirichletCondition[{sxx[x,y]0,sxy[x,y]0},xW&&0≤y≤H-W],(*traction-freetopedge*)DirichletCondition[{sxy[x,y]0,syy[x,y]0},yH],(*appliedtensileload*)DirichletCondition[{sxx[x,y]100,sxy[x,y]0},xH]};

Solve the coupled elasticity system over the L-shaped domain:

In[6]:=

vars={u,v,sxx,sxy,syy};{usol,vsol,sxxsol,sxysol,syysol}=NDSolveValue[Join[planeStrainPDEs[{u,v},{sxx,sxy,syy},{30*10^6,0.25}],bcs],vars,{x,y}∈Ω];

NDSolveValue

:The PDE is convection dominated and the result may not be stable. Adding artificial diffusion may help.

Visualize the displacement vectors of the deformed configuration:

In[7]:=

VectorDisplacementPlot[{usol[x,y],vsol[x,y]},{x,y}∈Ω,PlotLegendsAutomatic,VectorPointsAutomatic]

Out[7]=

Color the region by different stress components:

In[8]:=

Table[VectorDisplacementPlot[{{usol[x,y],vsol[x,y]},stress[x,y]},{x,y}∈Ω],{stress,{sxxsol,sxysol,syysol}}]

Out[8]=





Combine the normal and shear stress components into a single scalar field that represents the equivalent (von Mises) stress under plane strain conditions:

In[9]:=

vonMises=Sqrt[1/2((sxxsol[x,y]-syysol[x,y])^2+(syysol[x,y]-szz)^2+(szz-sxxsol[x,y])^2)+3sxysol[x,y]^2]/.szzν/(1-ν)(sxxsol[x,y]+syysol[x,y])/.ν0.25;

Color the deformed configuration by the von Mises stress to identify regions of stress concentration in the L-shaped domain:

In[10]:=

VectorDisplacementPlot[{{usol[x,y],vsol[x,y]},vonMises},{x,y}∈Ω,PlotLegendsAutomatic]

Out[10]=

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Stress and Displacement Analysis of an L-Shaped Elastic Body

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Stress and Displacement Analysis of an L-Shaped Elastic Body

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