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I used the following equations to solve a structural problem:

{ufun,vfun, σxif, σyif, σxyif, σzif} = NDSolveValue[{

{planeStrainOperator[YM,ni]==ΓN},
     σx[x, y] == YM/((1+ni)(1 - 2ni)) ((1-ni)D[u[x, y], x] + ni D[v[x, y], y] ),
     σy[x, y] == YM/((1+ni)(1 - 2ni)) ((1-ni)*D[v[x, y], y] + ni D[u[x, y], x] ),
     σxy[x, y] == YM/((1+ni)(1 - 2ni)) (1 - 2ni)/2 (D[u[x, y], y] + D[v[x, y], x] ),
     σz[x, y]==ni(YM/((1+ni)(1 - 2ni)) ((1-ni)D[u[x, y], x] + ni D[v[x, y], y] )+
     YM/((1+ni)(1 - 2ni)) ((1-ni)*D[v[x, y], y] + ni D[u[x, y], x] )), ΓD} ,
   {u,v, σx, σy, σxy, σz}, {x, y} ∈ Ω];

Now I want to solve a termostructural problem. Can anyone help me to understand how to introduce the new terms?

For example, for the plane strain case, I would like to introduce the following relation between stresses and displacements:

planeStrainOperatorTS[Y_,ν_,CoeffDilat_,T_]:=
    {({{0,-((Y ν)/((1-2 ν) (1+ν)))},{-(Y/(2 (1+ν))),0}}.Inactive[Grad][v[x,y],{x,y}]),{x,y}
        +({{-((Y (1-ν))/((1-2 ν) (1+ν))),0},
    {0,-(Y/(2 (1+ν)))}}.Inactive[Grad][u[x,y],{x,y}]),{x,y}
        -(Y*CoeffDilat*T[x,y]/(1-2ν))*{{1},{0}},
        ({{0,-(Y/(2 (1+\[Nu])))},{-((Y ν)/((1-2 ν) (1+ν))),0}}.
        Inactive[Grad][u[x,y],{x,y}]),{x,y}
        +({{-(Y/(2 (1+ν))),0},{0,-((Y (1-ν))/((1-2 ν) (1+ν)))}}.
        Inactive[Grad][v[x,y],{x,y}]),{x,y}
        -(Y*CoeffDilat*T[x,y]/(1-2ν))*{{0},{1}}};

Thank you very much for your answer.

I am trying to mesh the region below, but an error occur: "Mesh contains no elements." Could anyone help me?

R1=5;
R2=5.5;
R3=5.7;
R4=6.0;

Theta1=0.0227;
Theta2=0.00915094;

\[CapitalOmega]=ImplicitRegion[
    !(x^2 + y^2>R2^2 && x^2 + y^2< R3^2 && 0<y<x*Tan[Theta2])&&
 (y>=0 && x^2 + y^2>=R1^2 && x^2 + y^2<=R4^2 && y<=x Tan[Theta1]),{x, y}];

Show[ RegionPlot[\[CapitalOmega]], ImageSize -> 300]

ToElementMesh[\[CapitalOmega]2, 
  "BoundaryMeshGenerator" -> {"Continuation"}]["Wireframe"]
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There is an example of a thermocouple that was presented that at the 2014 Wolfram Tech Conference. The multiphysics_wtc_2014_Oliver Rueb.nb from that page is the presentation that contains all the code.

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  • $\begingroup$ are there some new example about FE in Mathematica from 2015 to 2018? $\endgroup$ – ABCDEMMM Jul 10 '18 at 16:16
  • $\begingroup$ @ABCDEMMM, yes. $\endgroup$ – user21 Jul 11 '18 at 5:49
  • $\begingroup$ slide 9, error from Eq.91 (mma 12): NDSolveValue::femcnmd: The PDE coefficient If[y<=0.005,Subscript[TC, Al] IdentityMatrix[2],Subscript[TC, CS] IdentityMatrix[2]] does not evaluate to a numeric matrix of dimensions {2,2} at the coordinate {0.0575683,0.00377013}; it evaluated to {{{{-83,0},{0,-83}},0},{0,{{-83,0},{0,-83}}}} instead. $\endgroup$ – ABCDEMMM Jul 31 '19 at 0:25
  • $\begingroup$ @ABCDEMMM, try \[Kappa] = If[y <= hi, Subscript[TC, Al], Subscript[TC, CS]]; $\endgroup$ – user21 Jul 31 '19 at 4:34

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