strainStress = {
    ϵx[x, y, z] ==
     1/Y (σx[x, y,
         z] - ν (σy[x, y, z] + σz[x, y, z])),
    γxy[x, y, z] == 1/G τxy[x, y, z],
    γxz[x, y, z] == 1/G τxz[x, y, z],

    γyx[x, y, z] == 1/G τyx[x, y, z],
    ϵy[x, y, z] ==
     1/Y (σy[x, y,
         z] - ν (σx[x, y, z] + σz[x, y, z])),
    γyz[x, y, z] == 1/G τyz[x, y, z],

    γzx[x, y, z] == 1/G τzx[x, y, z],
    γzy[x, y, z] == 1/G τzy[x, y, z],
    ϵz[x, y, z] ==
     1/Y (σz[x, y,
         z] - ν (σx[x, y, z] + σy[x, y, z]))
    } /. G -> Y/(2 (1 + ν));

strainDisp={Derivative[1, 0, 0][u][x, y, z], Derivative[0, 1, 0][u][x, y, z] +
  Derivative[1, 0, 0][v][x, y, z], Derivative[0, 0, 1][u][x, y, z] +
  Derivative[1, 0, 0][w][x, y, z], Derivative[0, 1, 0][u][x, y, z] +
  Derivative[1, 0, 0][v][x, y, z], Derivative[0, 1, 0][v][x, y, z],
 Derivative[0, 0, 1][v][x, y, z] + Derivative[0, 1, 0][w][x, y, z],
 Derivative[0, 0, 1][u][x, y, z] + Derivative[1, 0, 0][w][x, y, z],
 Derivative[0, 0, 1][v][x, y, z] + Derivative[0, 1, 0][w][x, y, z],
 Derivative[0, 0, 1][w][x, y, z]};

stress = {
       σx[x, y, z], τxy[x, y, z], τxz[x, y, z],
       τyx[x, y, z], σy[x, y, z], τyz[x, y, z],
       τzx[x, y, z], τzy[x, y, z], σz[x, y, z]
       };
    strain = strainStress[[All, 1]];
    sol = stress /. Solve[strainStress, stress];
    {vec, mat} =
      CoefficientArrays[Thread[stress == sol[[1]]], strain] // Normal;
    dd = Join[Grad[u[x, y, z], {x, y, z}], Grad[v[x, y, z], {x, y, z}],
       Grad[w[x, y, z], {x, y, z}]];
    eq2 = -mat.strainDisp;
    {vec2, mat2} =
      CoefficientArrays[Thread[stress == eq2], dd] // Normal // Simplify;
    mat3 = Partition[mat2 // Simplify // Factor, {3, 3}];

cc = Array[c, {3, 3}];
coffs = Thread[Flatten[cc, 1] -> Flatten[mat3, 1]];
idd = {Inactive[Grad][u[x, y, z], {x, y, z}],
  Inactive[Grad][v[x, y, z], {x, y, z}],
  Inactive[Grad][w[x, y, z], {x, y, z}]}
e1 = cc.idd // ExpandAll