def test_det_6(): # {{{1
"""Using a random generated matrix and
https://www.mathsisfun.com/algebra/matrix-calculator.html
"""
assert round(mat.det(_rndm), 2) == -0.45
def test_det_dia(): # {{{1
m = [[1, 0, 0], [0, 2, 0], [0, 0, 3]]
assert mat.det(m) == 6
def test_det_3(): # {{{1
"""From https://www.mathsisfun.com/algebra/matrix-determinant.html"""
m = [[6, 1, 1], [4, -2, 5], [2, 8, 7]]
assert mat.det(m) == -306
def laminate(name, layers):
"""Create a laminate.
Arguments/properties of a laminate:
name: A non-empty string containing the name of the laminate
layers: A non-empty sequence of lamina (will be converted into a tuple).
Additional properties:
thickness: Thickness of the laminate in mm.
fiber_weight: Total area weight of fibers in g/m².
ρ: Specific gravity of the laminate in g/cm³.
vf: Average fiber volume fraction.
resin_weight: Total area weight of resin in g/m².
ABD: Stiffness matrix.
abd: Compliance matrix.
Ex: Young's modulus in the x-direction.
Ey: Young's modulus in the y-direction.
Ez: Young's modulus in the z-direction.
Gxy: In-plane shear modulus.
νxy: Poisson constant.
νyx: Poisson constant.
αx: CTE in x-direction.
αy: CTE in y-direction.
wf: Fiber weight fraction.
C: 3D Stiffness matrix for the laminate in global coordinates.
"""
if not layers:
raise ValueError("no layers in the laminate")
if not isinstance(name, str):
raise ValueError("the name of a laminate must be a string")
if len(name) == 0:
raise ValueError("the length of the name of a laminate must be >0")
layers = tuple(layers)
thickness = sum(la.thickness for la in layers)
fiber_weight = sum(la.fiber_weight for la in layers)
ρ = sum(la.ρ * la.thickness for la in layers) / thickness
vf = sum(la.vf * la.thickness for la in layers) / thickness
resin_weight = sum(la.resin_weight for la in layers)
wf = fiber_weight / (fiber_weight + resin_weight)
# Set z-values for lamina.
zs = -thickness / 2
lz2, lz3 = [], []
C = lpm.zeros(6)
for la in layers:
ze = zs + la.thickness
lz2.append((ze * ze - zs * zs) / 2)
lz3.append((ze * ze * ze - zs * zs * zs) / 3)
zs = ze
C = lpm.add(C, lpm.mul(la.C, la.thickness / thickness))
C = lpm.clean(C)
S = lpm.inv(C)
Ntx, Nty, Ntxy = 0.0, 0.0, 0.0
ABD = lpm.zeros(6)
H = lpm.zeros(2)
c3 = 0
for la, z2, z3 in zip(layers, lz2, lz3):
# first row
ABD[0][0] += la.Q̅11 * la.thickness # Hyer:1998, p. 290
ABD[0][1] += la.Q̅12 * la.thickness
ABD[0][2] += la.Q̅16 * la.thickness
ABD[0][3] += la.Q̅11 * z2
ABD[0][4] += la.Q̅12 * z2
ABD[0][5] += la.Q̅16 * z2
# second row
ABD[1][0] += la.Q̅12 * la.thickness
ABD[1][1] += la.Q̅22 * la.thickness
ABD[1][2] += la.Q̅26 * la.thickness
ABD[1][3] += la.Q̅12 * z2
ABD[1][4] += la.Q̅22 * z2
ABD[1][5] += la.Q̅26 * z2
# third row
ABD[2][0] += la.Q̅16 * la.thickness
ABD[2][1] += la.Q̅26 * la.thickness
ABD[2][2] += la.Q̅66 * la.thickness
ABD[2][3] += la.Q̅16 * z2
ABD[2][4] += la.Q̅26 * z2
ABD[2][5] += la.Q̅66 * z2
# fourth row
ABD[3][0] += la.Q̅11 * z2
ABD[3][1] += la.Q̅12 * z2
ABD[3][2] += la.Q̅16 * z2
ABD[3][3] += la.Q̅11 * z3
ABD[3][4] += la.Q̅12 * z3
ABD[3][5] += la.Q̅16 * z3
# fifth row
ABD[4][0] += la.Q̅12 * z2
ABD[4][1] += la.Q̅22 * z2
ABD[4][2] += la.Q̅26 * z2
ABD[4][3] += la.Q̅12 * z3
ABD[4][4] += la.Q̅22 * z3
ABD[4][5] += la.Q̅26 * z3
# sixth row
ABD[5][0] += la.Q̅16 * z2
ABD[5][1] += la.Q̅26 * z2
ABD[5][2] += la.Q̅66 * z2
ABD[5][3] += la.Q̅16 * z3
ABD[5][4] += la.Q̅26 * z3
ABD[5][5] += la.Q̅66 * z3
# Calculate unit thermal stress resultants.
# Hyer:1998, p. 445
Ntx += (la.Q̅11 * la.αx + la.Q̅12 * la.αy + la.Q̅16 * la.αxy) * la.thickness
Nty += (la.Q̅12 * la.αx + la.Q̅22 * la.αy + la.Q̅26 * la.αxy) * la.thickness
Ntxy += (la.Q̅16 * la.αx + la.Q̅26 * la.αy + la.Q̅66 * la.αxy) * la.thickness
# Calculate H matrix (derived from Barbero:2018, p. 181)
sb = 5 / 4 * (la.thickness - 4 * z3 / thickness ** 2)
H[0][0] += la.Q̅s44 * sb
H[0][1] += la.Q̅s45 * sb
H[1][0] += la.Q̅s45 * sb
H[1][1] += la.Q̅s55 * sb
# Calculate E3
c3 += la.thickness / la.E3
# Finish the matrices, discarding very small numbers in ABD and H.
ABD = lpm.clean(ABD)
H = lpm.clean(H)
abd = lpm.inv(ABD)
h = lpm.inv(H)
# Calculate the engineering properties.
# Nettles:1994, p. 34 e.v.
dABD = lpm.det(ABD)
dt1 = lpm.det(lpm.delete(ABD, 0, 0))
Ex = dABD / (dt1 * thickness)
dt2 = lpm.det(lpm.delete(ABD, 1, 1))
Ey = dABD / (dt2 * thickness)
dt3 = lpm.det(lpm.delete(ABD, 2, 2))
Gxy = dABD / (dt3 * thickness)
dt4 = lpm.det(lpm.delete(ABD, 0, 1))
dt5 = lpm.det(lpm.delete(ABD, 1, 0))
νxy = dt4 / dt1
νyx = dt5 / dt2
# See Barbero:2018, p. 197
Gyz = H[0][0] / thickness
Gxz = H[1][1] / thickness
# All layers experience the same force in Z-direction.
Ez = thickness / c3
# Calculate the coefficients of thermal expansion.
# *Technically* only valid for a symmetric laminate!
# Hyer:1998, p. 451, (11.86)
αx = abd[0][0] * Ntx + abd[0][1] * Nty + abd[0][2] * Ntxy
αy = abd[1][0] * Ntx + abd[1][1] * Nty + abd[1][2] * Ntxy
# Calculate tensor engineering properties
tEx, tEy, tEz = 1 / S[0][0], 1 / S[1][1], 1 / S[2][2]
tGxy, tGxz, tGyz = 1 / S[5][5], 1 / S[4][4], 1 / S[3][3]
tνxy, tνxz, tνyz = -S[1][0] / S[0][0], -S[2][0] / S[0][0], -S[2][1] / S[1][1]
return SimpleNamespace(
name=name,
layers=layers,
thickness=thickness,
fiber_weight=fiber_weight,
ρ=ρ,
vf=vf,
resin_weight=resin_weight,
ABD=ABD,
abd=abd,
H=H,
h=h,
Ex=Ex,
Ey=Ey,
Ez=Ez,
Gxy=Gxy,
Gyz=Gyz,
Gxz=Gxz,
νxy=νxy,
νyx=νyx,
αx=αx,
αy=αy,
wf=wf,
C=C,
S=S,
tEx=tEx,
tEy=tEy,
tEz=tEz,
tGxy=tGxy,
tGyz=tGyz,
tGxz=tGxz,
tνxy=tνxy,
tνxz=tνxz,
tνyz=tνyz,
)