def F1(w):
u = w["disp"]
p = w["press"]
F = grad(u) + identity(u)
J = det(F)
Finv = inv(F)
return p * J * transpose(Finv) + mu * F
def A12(w):
u = w["disp"]
F = zeros_like(grad(u))
for i in range(3):
F[i, i] += 1.
F += grad(u)
J = det(F)
Finv = inv(F)
return J * transpose(Finv)
def F2(w):
u = w["disp"]
p = w["press"].value
F = grad(u) + identity(u)
J = det(F)
Js = .5 * (lmbda + p + 2. * np.sqrt(lmbda * mu + .25 * (lmbda + p) ** 2)) / lmbda
dJsdp = ((.25 * lmbda + .25 * p + .5 * np.sqrt(lmbda * mu + .25 * (lmbda + p) ** 2))
/ (lmbda * np.sqrt(lmbda * mu + .25 * (lmbda + p) ** 2)))
return J - (Js + (p + mu / Js - lmbda * (Js - 1)) * dJsdp)
def F1(w):
u = w["disp"]
p = w["press"]
F = zeros_like(grad(u))
for i in range(3):
F[i, i] += 1.
F += grad(u)
J = det(F)
Finv = inv(F)
return p * J * transpose(Finv) + mu * F
def a(u, v, w):
F, iF = deformation_gradient(w)
DF = grad(u)
dF = grad(v)
dFiF = mul(dF, iF)
DFiF = mul(DF, iF)
tr_DFiF_dFiF = ddot(transpose(dFiF), DFiF)
lnJ = np.log(det(F))
return (mu * ddot(DF, dF) - (lmbda * lnJ - mu) * tr_DFiF_dFiF +
lmbda * trace(dFiF) * trace(DFiF))
def A11(w):
u = w["disp"]
p = w["press"]
eye = identity(u)
F = grad(u) + eye
J = det(F)
Finv = inv(F)
L = (p * J * np.einsum("lk...,ji...->ijkl...", Finv, Finv)
- p * J * np.einsum("jk...,li...->ijkl...", Finv, Finv)
+ mu * np.einsum("ik...,jl...->ijkl...", eye, eye))
return L
def F2(w):
u = w["disp"]
p = w["press"].value
F = zeros_like(grad(u))
for i in range(3):
F[i, i] += 1.
F += grad(u)
J = det(F)
Js = .5 * (lmbda + p + 2. * sqrt(lmbda * mu + .25 * (lmbda + p) ** 2)) / lmbda
dJsdp = ((.25 * lmbda + .25 * p + .5 * sqrt(lmbda * mu + .25 * (lmbda + p) ** 2))
/ (lmbda * sqrt(lmbda * mu + .25 * (lmbda + p) ** 2)))
return J - (Js + (p + mu / Js - lmbda * (Js - 1)) * dJsdp)
def A11(w):
u = w["disp"]
p = w["press"]
F = zeros_like(grad(u))
eye = zeros_like(grad(u))
for i in range(3):
F[i, i] += 1.
eye[i, i] += 1.
F += grad(u)
J = det(F)
Finv= inv(F)
L = (p * J * einsum("lk...,ji...->ijkl...", Finv, Finv)
- p * J * einsum("jk...,li...->ijkl...", Finv, Finv)
+ mu * einsum("ik...,jl...->ijkl...", eye, eye))
return L
def volume(w):
dw = w["disp"].grad
F = dw + identity(dw)
J = det(F)
return J
def A12(w):
u = w["disp"]
F = grad(u) + identity(u)
J = det(F)
Finv = inv(F)
return J * transpose(Finv)
def L(v, w):
F, iF = deformation_gradient(w)
dF = grad(v)
lnJ = np.log(det(F))
return mu * ddot(F, dF) + (lmbda * lnJ - mu) * ddot(transpose(iF), dF)
