Stress and Strain in a Fiber-Reinforced Rubber Block

Compute and visualize the strain and von Mises stress in a block of fiber-reinforced rubber under load

We wish to model a block of hyperelastic material containing fiber reinforcements. The partial differential equation governing the stress versus strain relationship for this type of system can be obtained through

SolidMechanicsPDEComponent

. Boundary conditions are implemented using

SolidBoundaryLoadValue

and

SolidFixedCondition

and the displacement of the system is then found using

NDSolveValue

Set up the variables and the model parameters for a fiber-reinforced piece of rubber that is held fixed at the left and has a pressure applied on the right. A neo-Hookean hyperelastic material model is used:

In[1]:=

vars={{u[x,y],v[x,y]},{x,y}};pars="SolidMechanicsMaterialModel""NeoHookean","Compressibility""Compressible","Thickness"1,"ShearModulus"510^8,"LameParameter"10^9,"ReinforcementDirectionVector"Normalize[{1,1}],"ReinforcementStiffnessFactor"5;

Set up the geometry:

In[2]:=

Ω=RegionDifference[Rectangle[],Disk[{1/2,1/2},1/4]]

Out[2]=

Visualize the fiber reinforcement in the geometry:

In[3]:=

StreamPlot[pars["ReinforcementDirectionVector"],{x,y}∈Ω,StreamMarkers"Segment",StreamColorFunctionNone,StreamStyleBlack]

Out[3]=

Set up the equation and solve for the displacement:

In[4]:=

displacement=NDSolveValue[{SolidMechanicsPDEComponent[vars,pars]SolidBoundaryLoadValue[x1,vars,pars,"Pressure"{4*10^8,0}],SolidFixedCondition[x0,vars,pars]},{u[x,y],v[x,y]},{x,y}∈Ω]

Out[4]=

InterpolatingFunction

Domain: {{0.,1.},{0.,1.}}

Output: scalar

[x,y],InterpolatingFunction

Domain: {{0.,1.},{0.,1.}}

Output: scalar

[x,y]

Visualize the deformed body over the original shape of the body:

In[5]:=

VectorDisplacementPlot[displacement,{x,y}∈Ω,VectorSizesFull]

Out[5]=

Compute the strain:

In[6]:=

strain=SolidMechanicsStrain[vars,pars,displacement]

Out[6]=

SymmetrizedArray

Dimensions: {2,2}

Symmetry: Symmetric[{1,2}]



Compute the stress:

In[7]:=

stress=SolidMechanicsStress[vars,pars,strain,displacement]

Out[7]=

SymmetrizedArray

Dimensions: {2,2}

Symmetry: Symmetric[{1,2}]



Compute the von Mises stress:

In[8]:=

vms=VonMisesStress[vars,pars,stress]

Out[8]=

InterpolatingFunction

Domain: {{0.,1.},{0.,1.}}

Output: scalar

[x,y]

Compute the minimum and maximum von Mises stress over the body and define a color function appropriate for visualizing the full range of stress for this system:

In[9]:=

pts=MeshCoordinates[DiscretizeRegion[Ω]];vals=(vms/.Thread[{x,y}#])&/@pts;vals=Select[vals,NumericQ];{minVMS,maxVMS}=MinMax[vals]

Out[9]=

{1.48853×

,3.86443×

}

In[10]:=

colorFunction=ColorData["Pastel"][Rescale[#5,{minVMS,maxVMS}]]&;

Now compute the displacement and stress for a range of applied pressures,

In[11]:=

dP=10^8/2;frames=Table[PrintTemporary[P];displacement=NDSolveValue[{SolidMechanicsPDEComponent[vars,pars]SolidBoundaryLoadValue[x1,vars,pars,"Pressure"{P,0}],SolidFixedCondition[x0,vars,pars]},{u[x,y],v[x,y]},{x,y}∈Ω];strain=SolidMechanicsStrain[vars,pars,displacement];stress=SolidMechanicsStress[vars,pars,strain,displacement];vms=VonMisesStress[vars,pars,stress];Labeled[Show[StreamPlot[Normalize[{1,1}],{x,y}∈Ω,StreamMarkers"Segment",StreamColorFunctionNone,StreamStyleGray,FrameFalse],VectorDisplacementPlot[{displacement,vms},{x,y}∈Ω,VectorSizesFull,FrameFalse,ColorFunctionScalingFalse,ColorFunctioncolorFunction],PlotRange{{0,2.15},{-0.1,1}},AspectRatioAutomatic,ImageSize{550,300}],Style["Deformation of hyperelastic fiber-reinforced material","Output",FontFamily"Source Sans Pro",15,Gray]],{P,0,9*10^8,dP}];frames=Join[frames,Reverse[frames]];frames=Rasterize[#1,"Image",ImageResolution144]&/@frames;

Animate the previously generated frames to visualize what happens as the applied force is varied:

In[12]:=

ListAnimate[frames,ControlPlacementTop]

Out[12]=

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Stress and Strain in a Fiber-Reinforced Rubber Block

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Stress and Strain in a Fiber-Reinforced Rubber Block

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