I have the following equations

$$ frac { partial {N_x}} { partial {x}} + frac { partial {n_ {xy}} { partial {y}} = 0 $$

$$ frac { partial {N_y}} { partial {y}} + frac { partial {N_ {xy}}} { partial {x}} = 0 $$

Where

begin {equation}

begin {bmatrix} N_ {x} \ N_ {y} \ N_ {xy} end {bmatrix} = begin {bmatrix} A_ {11} (x) & A_ {12} (x) & 0 A_ {12} (x) & A_ {22} (x) & 0 \ 0 & A_ {66} end {bmatrix} begin {bmatrix} frac { partial {u}} { partial {x }} \ frac { partial {v}} { partial {y}} \ frac { partial {u}} { partial {y}} + frac { partial {v}} { partial {x}} end {bmatrix} end {equation}

$ A_ {ij} $ are only varied on $ x $ Direction. The boundary conditions are shown below. I'm trying to apply the complete panel BCs.

Can someone help me to model this problem. I have looked at the FEM package, but still can not solve it. Thank you in advance. I'm looking for a solution for u (x, y) and v (x, y).

Follow my code:

```
E1 = 147 10^9; E2 = 10.3 10^9; G12 = 7 10^9; nu12 = 0.27; nu21 = (E2*nu12)/E1; t = 0.127 10^-3; a = 1; b = 1; u0 = -1;
Son = {{1/E1, -nu12/E1, 0}, {-nu21/E2, 1/E2, 0}, {0, 0, 1/G12}};
Qon = Inverse(Son);
angles = {{0, 45}, {0, -45}, {0, -45}, {0, 45}};
num = Dimensions(angles)((1));
h = num*t;
pos = Table(0, num + 1);
pos((1)) = -h/2;
For(i = 2, i <= num + 1, i++,
pos((i)) = pos((i - 1)) + t;)
mA = ConstantArray(0, {3, 3});
mB = ConstantArray(0, {3, 3});
mD = ConstantArray(0, {3, 3});
For(i = 1, i <= num, i++,
T0 = angles((i, 1)) (pi/180);
T1 = angles((i, 2)) (pi/180);
theta(x_) = (2/a) (T1 - T0) Abs(x) + T0;
m = Cos(theta(x));
n = Sin(theta(x));
Q11 = Qon((1, 1));
Q12 = Qon((1, 2));
Q22 = Qon((2, 2));
Q66 = Qon((3, 3));
Qxx = m^4*Q11 + n^4*Q22 + 2*m^2*n^2*Q12 + 4*m^2*n^2*Q66;
Qyy = n^4*Q11 + m^4*Q22 + 2*m^2*n^2*Q12 + 4*m^2*n^2*Q66;
Qxy = m^2*n^2*Q11 + m^2*n^2*Q22 + (m^4 + n^4)*Q12 + -4*m^2*n^2*Q66;
Qss = m^2*n^2*Q11 + m^2*n^2*Q22 - 2*m^2*n^2*Q12 + (m^2 - n^2)^2*Q66;
Qxs = m^3*n*Q11 - m*n^3*Q22 + (m*n^3 - m^3*n)*Q12 + 2*(m*n^3 - m^3*n)*Q66;
Qys = m*n^3*Q11 - m^3*n*Q22 + (m^3*n - m*n^3)*Q12 + 2*(m^3*n - m*n^3)*Q66;
Qoff = {{Qxx, Qxy, Qxs}, {Qxy, Qyy, Qys}, {Qxs, Qys, Qss}};
mA = mA + Qoff*(pos((i + 1)) - pos((i)));
mB = mB + Qoff*(pos((i + 1))^2 - pos((i))^2);
mD = mD + Qoff*(pos((i + 1))^3 - pos((i))^3);
);
mB = mB/2;
mD = mD/3;
Needs("NDSolve`FEM`")
omega = Rectangle({-a/2, -b/2}, {a, b});
mesh = ToElementMesh(omega);
A11(x_) = mA((1, 1)); A12(x_) = mA((1, 2)); A16(x_) = mA((1, 3));
A22(x_) = mA((2, 2)); A26(x_) = mA((2, 3)); A66(x_) = mA((3, 3));
Nx(x_, y_) = A11(x) D(u(x, y), {x, 1}) + A12(x) D(v(x, y), {y, 1}) +
A16(x) (D(u(x, y), {y, 1}) + D(v(x, y), {x, 1}));
Ny(x_, y_) = A12(x) D(u(x, y), {x, 1}) + A22(x) D(v(x, y), {y, 1}) +
A26(x) (D(u(x, y), {y, 1}) + D(v(x, y), {x, 1}));
Nxy(x_, y_) = A16(x) D(u(x, y), {x, 1}) + A26(x) D(v(x, y), {y, 1}) +
A66(x) (D(u(x, y), {y, 1}) + D(v(x, y), {x, 1}));
PDEs = {D(Nx(x, y), {x, 1}) + D(Nxy(x, y), {x, 1}), D(Ny(x, y), {y, 1}) +
Nxy(x, y), {y, 1})};
gammaD = {DirichletCondition(u(x, y) == u0, x == a/2),
DirichletCondition(u(x, y) == -u0, x == -a/2)};
NDSolve({PDEs == {0, 0}, gammaD}, {u,
v}, {x, y} (Element) mesh);
```

I get the following error:

```
NDSolve::femcnmd: The PDE coefficient {{0.000508 (-8.22842*10^11 Cos(Times(<<2>>))^3 Sin(1.5708 Abs(<<1>>)) (Abs^(Prime))(x)-4.04788*10^10 Cos(1.5708 Abs(<<1>>)) Sin(Times(<<2>>))^3 (Abs^(Prime))(x))+0.000254 (<<1>>)+<<1>>+1/2 (-0.001016 (-8.22842*10^11 Power(<<2>>) Sin(<<1>>) (<<1>>^(<<1>>))(<<1>>)-4.04788*10^10 Cos(<<1>>) Power(<<2>>) (<<1>>^(<<1>>))(<<1>>))-0.000508 (2.32093*10^11 Power(<<2>>) (<<1>>^(<<1>>))(<<1>>)-7.45065*10^11 Power(<<2>>) Power(<<2>>) (<<1>>^(<<1>>))(<<1>>)+1.62623*10^10 Power(<<2>>) (<<1>>^(<<1>>))(<<1>>)+1.67953*10^10 Plus(<<3>>))-0.000508 (-2.32093*10^11 Power(<<2>>) (<<1>>^(<<1>>))(<<1>>)+7.45065*10^11 Power(<<2>>) Power(<<2>>) (<<1>>^(<<1>>))(<<1>>)-1.62623*10^10 Power(<<2>>) (<<1>>^(<<1>>))(<<1>>)+1.67953*10^10 Plus(<<3>>))),<<1>>}} does not evaluate to a numeric matrix of dimensions {1,2} at the coordinate {-0.4625,-0.4625}; it evaluated to {{0.,5.79879*10^6 (Abs^(Prime))(-0.4625)}} instead.
```

Can someone help me? Thank you all in advance.