def disp_def_cst(u, coords=(x, y, z), h_vec=(1, 1, 1)):
"""
Compute strain measures for C-CST elasticity, as defined
in [CST]_.
Parameters
----------
u : Matrix (3, 1), list
Displacement vector function to apply the micropolar operator.
coords : Tuple (3), optional
Coordinates for the new reference system. This is an optional
parameter it takes (x, y, z) as default.
h_vec : Tuple (3), optional
Scale coefficients for the new coordinate system. It takes
(1, 1, 1), as default.
Returns
-------
strain : Matrix (3, 3)
Strain tensor.
curvature : Matrix (3, 3)
Curvature tensor.
"""
strain = sym_grad(u, coords, h_vec)
curvature = S(1) / 4 * curl(curl(u, coords, h_vec), coords, h_vec)
return strain, dual_tensor(curvature)
def navier_cauchy(u, parameters, coords=(x, y, z), h_vec=(1, 1, 1)):
"""
Navier-Cauchy operator of a vector function u.
Parameters
----------
u : Matrix (3, 1), list
Vector function to apply the Navier-Cauchy operator.
parameters : tuple
Material parameters in the following order:
lamda : float
Lamé's first parameter.
mu : float, > 0
Lamé's second parameter.
coords : Tuple (3), optional
Coordinates for the new reference system. This is an optional
parameter it takes (x, y, z) as default.
h_vec : Tuple (3), optional
Scale coefficients for the new coordinate system. It takes
(1, 1, 1), as default.
Returns
-------
navier_op : Matrix (3, 1)
Components of the Navier-Cauchy operator applied to the
displacement vector.
"""
lamda, mu = parameters
u = Matrix(u)
term1 = (lamda + 2 * mu) * grad(div(u, coords, h_vec), coords, h_vec)
term2 = mu * curl(curl(u, coords, h_vec), coords, h_vec)
return simplify(term1 - term2)
def c_cst(u, parameters, coords=(x, y, z), h_vec=(1, 1, 1)):
"""
Corrected-Couple-Stress-Theory (C-CST) elasticity operator of a
vector function u, as presented in [CST]_.
Parameters
----------
u : Matrix (3, 1), list
Vector function to apply the Navier-Cauchy operator.
parameters : tuple
Material parameters in the following order:
lamda : float
Lamé's first parameter.
mu : float, > 0
Lamé's second parameter.
eta : float, > 0
Couple stress modulus in C-CST.
coords : Tuple (3), optional
Coordinates for the new reference system. This is an optional
parameter it takes (x, y, z) as default.
h_vec : Tuple (3), optional
Scale coefficients for the new coordinate system. It takes
(1, 1, 1), as default.
Returns
-------
c_cst : Matrix (3, 1)
Components of the C-CST operator applied to the
displacement vector
References
----------
.. [CST] Ali R. Hadhesfandiari, Gary F. Dargush.
Couple stress theory for solids. International Journal
for Solids and Structures, 2011, 48, 2496-2510.
"""
lamda, mu, eta = parameters
u = Matrix(u)
term1 = (lamda + 2 * mu) * grad(div(u, coords, h_vec), coords, h_vec)
term2 = mu * curl(curl(u, coords, h_vec), coords, h_vec)
term3 = eta * lap_vec(curl(curl(u, coords, h_vec), coords, h_vec), coords,
h_vec)
return simplify(term1 - term2 + term3)
def micropolar(u, phi, parameters, coords=(x, y, z), h_vec=(1, 1, 1)):
"""
Micropolar operator of a vector function u, as defined
in [NOW]_.
Parameters
----------
u : Matrix (3, 1), list
Displacement vector function to apply the micropolar operator.
phi : Matrix (3, 1), list
Microrrotation (pseudo)-vector function to apply the
micropolar operator.
parameters : tuple
Material parameters in the following order:
lamda : float
Lamé's first parameter.
mu : float, > 0
Lamé's second parameter.
alpha : float, > 0
Micropolar parameter.
beta : float
Micropolar parameter.
gamma : float, > 0
Micropolar parameter.
epsilon : float, > 0
Micropolar parameter.
coords : Tuple (3), optional
Coordinates for the new reference system. This is an optional
parameter it takes (x, y, z) as default.
h_vec : Tuple (3), optional
Scale coefficients for the new coordinate system. It takes
(1, 1, 1), as default.
Returns
-------
disp_op : Matrix (3, 1)
Displacement components.
rot_op : Matrix (3, 1)
Microrrotational components.
References
----------
.. [NOW] Witold Nowacki. Theory of micropolar elasticity.
International centre for mechanical sciences,
Courses and lectures, No. 25. Berlin: Springer, 1972.
"""
lamda, mu, alpha, beta, gamma, epsilon = parameters
u = Matrix(u)
phi = Matrix(phi)
u_op = (lamda + 2*mu) * grad(div(u, coords, h_vec), coords, h_vec) \
- (mu + alpha) * curl(curl(u, coords, h_vec), coords, h_vec) \
+ 2*alpha*curl(phi, coords, h_vec)
phi_op = (beta + 2*gamma) * grad(div(phi, coords, h_vec), coords, h_vec)\
- (gamma + epsilon) * curl(curl(phi, coords, h_vec), coords, h_vec)\
+ 2*alpha*curl(u, coords, h_vec) - 4*alpha*phi
return simplify(u_op), simplify(phi_op)
def test_curl():
rot = curl([0, -x**2, 0])
expected_rot = Matrix([[0], [0], [-2 * x]])
assert rot == expected_rot