Fiber push-out¶
A SiC fiber with a BN interphase is pushed out of a SiC matrix: a thin slice of the composite rests on a support ring and an indenter narrower than the fiber pushes it through the hole. The slice is a solid of revolution, so Axisymmetric models its cross section in the half plane $r \ge 0$ with nodes $(r, z)$.
Units are µm, µN and MPa, hence fracture energies are given in J/m².
In [1]:
import matplotlib.pyplot as plt
import torch
from matplotlib import animation
from torchfem import Axisymmetric
from torchfem.materials import IsotropicDamage3D
from torchfem.mesh import rect_quad
from torchfem.plot_utils import embed_animation_gif
# Set double precision
torch.set_default_dtype(torch.float64)
Geometry¶
Square elements of edge length ES give every element the same characteristic length, and the phases are element sets selected by the radius of the element centers.
In [2]:
# Fiber radius, interphase thickness, matrix thickness, outer radius and height
RF, TI, TM = 8.0, 0.5, 10.0
RO = RF + TI + TM
HEIGHT = 40.0
# Square elements of edge length ES, i.e. the characteristic length is ES
ES = 0.25
nodes, elements = rect_quad(round(RO / ES) + 1, round(HEIGHT / ES) + 1, RO, HEIGHT)
ecenters = nodes[elements].mean(dim=1)
# Element sets
fiber = ecenters[:, 0] < RF
interphase = (ecenters[:, 0] > RF) & (ecenters[:, 0] < RF + TI)
matrix = ecenters[:, 0] > RF + TI
# Node sets
axis = nodes[:, 0] < 1e-6
outer = nodes[:, 0] > RO - 1e-6
support = (nodes[:, 1] < 1e-6) & (nodes[:, 0] > RF + TI + 0.1 * RF)
push = (nodes[:, 1] > HEIGHT - 1e-6) & (nodes[:, 0] < 0.9 * RF)
Material¶
Each phase is elastic up to $\varepsilon_0 = \sigma_0 / E$ and then softens exponentially, with the fracture energy per unit area $G_\textrm{f}$ regularized against the mesh by the characteristic element length $l_\textrm{c}$.
The interphase debonds in shear under the compressing indenter. The "rankine" measure reports the principal strain largest in magnitude, which is compressive here, and $\kappa$ never decreases, so it would leave the interphase intact. The deviatoric "mises" measure is never negative and drives the debond.
In [3]:
E, nu, sigma_0, G_f = (torch.zeros(len(elements)) for _ in range(4))
# Silicon carbide (fiber and matrix)
E[fiber | matrix] = 400.0e3
nu[fiber | matrix] = 0.17
sigma_0[fiber | matrix] = 400.0
G_f[fiber | matrix] = 20.0
# Boron nitride (interphase)
E[interphase] = 30.0e3
nu[interphase] = 0.2
sigma_0[interphase] = 30.0
G_f[interphase] = 2.0
# Damage initiation strain and the cap that leaves a residual stiffness,
# so a fully damaged element never makes the tangent exactly singular
eps_0 = sigma_0 / E
D_MAX = 0.999
def eps_f(cl):
# Final strain of the softening branch, set by the fracture energy
return G_f / (E * eps_0 * cl) + eps_0 / 2.0
def d(kappa, cl):
# Damage evolution law for exponential strain softening
D = 1.0 - eps_0 / kappa * torch.exp(-(kappa - eps_0) / (eps_f(cl) - eps_0))
return D.masked_fill(kappa < eps_0, 0.0).clamp(max=D_MAX)
def d_prime(kappa, cl):
# Derivative of the damage evolution law
df = eps_f(cl) - eps_0
D = eps_0 * torch.exp(-(kappa - eps_0) / df) * (1 / kappa**2 + 1 / (kappa * df))
return D.masked_fill(kappa < eps_0, 0.0)
material = IsotropicDamage3D(E, nu, d, d_prime, "mises")
Model and boundary conditions¶
The radial displacement vanishes on the axis and on the outer radius, where the surrounding composite confines the slice. It rests on a support ring under the matrix, so fiber and interphase are free to move through the hole, and the indenter is narrower than the fiber to keep it off the interphase. Pushing by a prescribed displacement keeps the softening branch stable.
In [4]:
model = Axisymmetric(nodes, elements, material)
# Boundary conditions
model.constraints[axis, 0] = True
model.constraints[outer, 0] = True
model.constraints[support, 1] = True
# Indenter pushing the fiber down
U_MAX = 0.02
model.constraints[push, 1] = True
model.displacements[push, 1] = -U_MAX
phases = 1.0 * interphase + 2.0 * matrix
model.plot(element_property={"Phase": phases}, linewidth=0.0, bcs=False)
Solve¶
The indenter is pushed to an increasing fraction of U_MAX and released again, so every cycle unloads from further along the softening branch. A damage model has an unsymmetric tangent, so the solver would pick BiCGSTAB at this size, where a direct solve is considerably faster.
In [5]:
# Ramp up to each peak and back down to zero
PEAKS = [0.25, 0.5, 0.75, 1.0]
STEPS = 40
ramps = [torch.zeros(1)]
for peak in PEAKS:
ramps.append(torch.linspace(0.0, peak, STEPS + 1)[1:])
ramps.append(torch.linspace(peak, 0.0, STEPS + 1)[1:])
u, f, _, _, alpha = model.solve(
increments=torch.cat(ramps),
return_intermediate=True,
verbose=True,
method="direct",
)
────────────────────────────────────────────────────────────────────────────────────────
model Axisymmetric · 11,840 elem · 24,150 dof · float64
machine arm · 8 threads · 16 GB RAM
solver direct · scipy · cpu
newton rtol 1e-08 · atol 1e-06 · <=10 it
────────────────────────────────────────────────────────────────────────────────────────
Increment Load factor Steps Newton Krylov Residual Wall time
1 0.00625 1 1 - 7.32e-11 1.03 s
2 0.0125 1 0 - 1.46e-10 0.05 s
3 0.01875 1 0 - 2.20e-10 0.05 s
4 0.025 1 0 - 2.93e-10 0.05 s
5 0.03125 1 0 - 3.66e-10 0.05 s
6 0.0375 1 0 - 4.39e-10 0.05 s
7 0.04375 1 0 - 5.13e-10 0.05 s
8 0.05 1 0 - 5.86e-10 0.05 s
9 0.05625 1 0 - 6.59e-10 0.05 s
10 0.0625 1 0 - 7.32e-10 0.05 s
11 0.06875 1 0 - 8.05e-10 0.05 s
12 0.075 1 0 - 8.79e-10 0.05 s
13 0.08125 1 0 - 9.52e-10 0.05 s
14 0.0875 1 0 - 1.03e-09 0.05 s
15 0.09375 1 0 - 1.10e-09 0.05 s
16 0.1 1 0 - 1.17e-09 0.05 s
17 0.1063 1 0 - 1.24e-09 0.05 s
18 0.1125 1 0 - 1.32e-09 0.05 s
19 0.1188 1 0 - 1.39e-09 0.05 s
20 0.125 1 0 - 1.46e-09 0.05 s
21 0.1313 1 0 - 1.54e-09 0.05 s
22 0.1375 1 0 - 1.61e-09 0.05 s
23 0.1437 1 0 - 1.68e-09 0.05 s
24 0.15 1 4 - 1.19e-11 0.90 s
25 0.1562 1 4 - 3.73e-11 0.90 s
26 0.1625 1 4 - 1.97e-11 0.95 s
27 0.1687 1 4 - 1.84e-11 1.00 s
28 0.175 1 4 - 1.10e-11 0.95 s
29 0.1812 1 4 - 2.71e-11 0.92 s
30 0.1875 1 4 - 9.69e-12 0.91 s
31 0.1938 1 3 - 4.73e-07 0.70 s
32 0.2 1 4 - 9.09e-12 0.91 s
33 0.2062 1 3 - 7.75e-07 0.70 s
34 0.2125 1 4 - 9.12e-12 1.06 s
35 0.2188 1 3 - 3.02e-07 1.00 s
36 0.225 1 3 - 2.65e-07 0.76 s
37 0.2313 1 4 - 9.36e-12 1.00 s
38 0.2375 1 4 - 9.13e-12 1.01 s
39 0.2437 1 4 - 9.18e-12 0.99 s
40 0.25 1 3 - 2.01e-07 0.71 s
41 0.2437 1 4 - 1.17e-11 1.14 s
42 0.2375 1 0 - 2.01e-07 0.07 s
43 0.2313 1 0 - 4.02e-07 0.06 s
44 0.225 1 0 - 6.03e-07 0.06 s
45 0.2188 1 0 - 8.04e-07 0.06 s
46 0.2125 1 1 - 1.17e-11 0.29 s
47 0.2062 1 0 - 8.04e-07 0.06 s
48 0.2 1 1 - 1.13e-11 0.31 s
49 0.1938 1 0 - 8.04e-07 0.06 s
50 0.1875 1 1 - 1.12e-11 0.32 s
51 0.1812 1 0 - 8.04e-07 0.08 s
52 0.175 1 1 - 1.13e-11 0.35 s
53 0.1687 1 0 - 8.04e-07 0.06 s
54 0.1625 1 1 - 1.12e-11 0.28 s
55 0.1562 1 0 - 8.04e-07 0.06 s
56 0.15 1 1 - 1.10e-11 0.31 s
57 0.1437 1 0 - 8.04e-07 0.07 s
58 0.1375 1 1 - 1.10e-11 0.31 s
59 0.1313 1 0 - 8.04e-07 0.06 s
60 0.125 1 1 - 1.11e-11 0.30 s
61 0.1188 1 0 - 8.04e-07 0.06 s
62 0.1125 1 1 - 1.10e-11 0.28 s
63 0.1063 1 0 - 8.04e-07 0.05 s
64 0.1 1 1 - 1.09e-11 0.29 s
65 0.09375 1 0 - 8.04e-07 0.06 s
66 0.0875 1 1 - 1.08e-11 0.30 s
67 0.08125 1 0 - 8.04e-07 0.07 s
68 0.075 1 1 - 1.07e-11 0.29 s
69 0.06875 1 0 - 8.04e-07 0.05 s
70 0.0625 1 1 - 1.06e-11 0.38 s
71 0.05625 1 0 - 8.04e-07 0.07 s
72 0.05 1 1 - 1.07e-11 0.30 s
73 0.04375 1 0 - 8.04e-07 0.05 s
74 0.0375 1 1 - 1.06e-11 0.33 s
75 0.03125 1 0 - 8.04e-07 0.06 s
76 0.025 1 1 - 1.07e-11 0.30 s
77 0.01875 1 0 - 8.04e-07 0.06 s
78 0.0125 1 1 - 1.07e-11 0.29 s
79 0.00625 1 0 - 8.04e-07 0.06 s
80 0 1 1 - 1.05e-11 0.28 s
81 0.0125 1 2 - 2.15e-11 0.52 s
82 0.025 1 0 - 4.40e-11 0.06 s
83 0.0375 1 0 - 6.78e-11 0.06 s
84 0.05 1 0 - 9.10e-11 0.06 s
85 0.0625 1 0 - 1.15e-10 0.06 s
86 0.075 1 0 - 1.39e-10 0.06 s
87 0.0875 1 0 - 1.62e-10 0.05 s
88 0.1 1 0 - 1.86e-10 0.05 s
89 0.1125 1 0 - 2.09e-10 0.05 s
90 0.125 1 0 - 2.33e-10 0.05 s
91 0.1375 1 0 - 2.57e-10 0.05 s
92 0.15 1 0 - 2.82e-10 0.05 s
93 0.1625 1 0 - 3.04e-10 0.05 s
94 0.175 1 0 - 3.29e-10 0.05 s
95 0.1875 1 0 - 3.52e-10 0.05 s
96 0.2 1 0 - 3.76e-10 0.06 s
97 0.2125 1 0 - 4.00e-10 0.06 s
98 0.225 1 0 - 4.23e-10 0.06 s
99 0.2375 1 0 - 4.47e-10 0.06 s
100 0.25 1 0 - 5.04e-08 0.05 s
101 0.2625 3 17 - 1.53e-07 4.17 s ↓1 ↑2
102 0.275 2 8 - 4.33e-10 1.91 s ↑2
103 0.2875 2 9 - 5.38e-06 2.17 s ↑2
104 0.3 2 9 - 1.60e-05 2.17 s ↑2
105 0.3125 1 5 - 2.13e-11 1.21 s
106 0.325 1 4 - 1.51e-10 0.97 s
107 0.3375 1 4 - 1.99e-10 0.95 s
108 0.35 1 4 - 1.48e-11 0.98 s
109 0.3625 1 4 - 3.19e-11 0.98 s
110 0.375 1 4 - 1.99e-11 0.95 s
111 0.3875 1 4 - 1.49e-11 0.95 s
112 0.4 1 4 - 3.24e-11 0.96 s
113 0.4125 1 4 - 2.54e-07 0.92 s
114 0.425 1 4 - 1.49e-11 0.91 s
115 0.4375 1 4 - 1.62e-11 0.99 s
116 0.45 1 4 - 4.14e-07 0.94 s
117 0.4625 1 4 - 2.45e-09 0.94 s
118 0.475 1 4 - 1.63e-11 0.93 s
119 0.4875 1 5 - 1.61e-11 1.16 s
120 0.5 1 5 - 1.60e-11 1.17 s
121 0.4875 1 3 - 3.97e-11 0.78 s
122 0.475 1 0 - 8.00e-11 0.05 s
123 0.4625 1 0 - 1.36e-10 0.05 s
124 0.45 1 0 - 1.74e-10 0.05 s
125 0.4375 1 0 - 2.13e-10 0.05 s
126 0.425 1 0 - 2.53e-10 0.05 s
127 0.4125 1 0 - 2.94e-10 0.05 s
128 0.4 1 0 - 3.51e-10 0.06 s
129 0.3875 1 0 - 3.91e-10 0.05 s
130 0.375 1 0 - 4.30e-10 0.06 s
131 0.3625 1 0 - 4.70e-10 0.06 s
132 0.35 1 0 - 5.11e-10 0.05 s
133 0.3375 1 0 - 5.51e-10 0.05 s
134 0.325 1 0 - 6.07e-10 0.06 s
135 0.3125 1 0 - 6.47e-10 0.05 s
136 0.3 1 0 - 6.87e-10 0.05 s
137 0.2875 1 0 - 7.28e-10 0.05 s
138 0.275 1 0 - 7.68e-10 0.05 s
139 0.2625 1 0 - 8.25e-10 0.05 s
140 0.25 1 0 - 8.64e-10 0.05 s
141 0.2375 1 0 - 9.13e-10 0.05 s
142 0.225 1 0 - 9.53e-10 0.05 s
143 0.2125 1 0 - 1.00e-09 0.05 s
144 0.2 1 0 - 1.04e-09 0.05 s
145 0.1875 1 0 - 1.08e-09 0.05 s
146 0.175 1 0 - 1.13e-09 0.05 s
147 0.1625 1 0 - 1.17e-09 0.05 s
148 0.15 1 0 - 1.22e-09 0.05 s
149 0.1375 1 0 - 1.26e-09 0.05 s
150 0.125 1 0 - 1.30e-09 0.06 s
151 0.1125 1 0 - 1.34e-09 0.06 s
152 0.1 1 0 - 1.39e-09 0.06 s
153 0.0875 1 0 - 1.43e-09 0.06 s
154 0.075 1 0 - 1.48e-09 0.05 s
155 0.0625 1 0 - 1.52e-09 0.06 s
156 0.05 1 0 - 1.56e-09 0.06 s
157 0.0375 1 0 - 1.60e-09 0.05 s
158 0.025 1 0 - 1.65e-09 0.05 s
159 0.0125 1 0 - 1.69e-09 0.05 s
160 0 1 0 - 1.73e-09 0.06 s
161 0.01875 1 2 - 2.61e-11 0.48 s
162 0.0375 1 0 - 1.73e-09 0.06 s
163 0.05625 1 0 - 3.47e-09 0.05 s
164 0.075 1 0 - 5.20e-09 0.06 s
165 0.09375 1 0 - 6.94e-09 0.05 s
166 0.1125 1 0 - 8.67e-09 0.05 s
167 0.1313 1 0 - 1.04e-08 0.05 s
168 0.15 1 0 - 1.21e-08 0.05 s
169 0.1687 1 0 - 1.39e-08 0.05 s
170 0.1875 1 0 - 1.56e-08 0.05 s
171 0.2062 1 0 - 1.73e-08 0.05 s
172 0.225 1 0 - 1.91e-08 0.05 s
173 0.2437 1 0 - 2.08e-08 0.06 s
174 0.2625 1 0 - 2.25e-08 0.06 s
175 0.2812 1 0 - 2.43e-08 0.05 s
176 0.3 1 0 - 2.60e-08 0.05 s
177 0.3187 1 0 - 2.77e-08 0.05 s
178 0.3375 1 0 - 2.95e-08 0.05 s
179 0.3563 1 0 - 3.12e-08 0.05 s
180 0.375 1 0 - 3.29e-08 0.05 s
181 0.3937 1 0 - 3.47e-08 0.05 s
182 0.4125 1 0 - 3.64e-08 0.05 s
183 0.4313 1 0 - 3.81e-08 0.05 s
184 0.45 1 0 - 3.99e-08 0.05 s
185 0.4688 1 0 - 4.16e-08 0.05 s
186 0.4875 1 0 - 4.34e-08 0.05 s
187 0.5062 1 5 - 3.93e-11 1.10 s
188 0.525 3 19 - 8.26e-07 4.49 s ↓1 ↑2
189 0.5437 2 7 - 3.71e-05 1.64 s ↑2
190 0.5625 2 9 - 1.35e-06 2.04 s ↑2
191 0.5813 2 9 - 3.32e-08 2.47 s ↑2
192 0.6 1 6 - 1.19e-09 1.34 s
193 0.6188 1 5 - 3.56e-11 1.11 s
194 0.6375 1 4 - 4.76e-11 1.01 s
195 0.6562 1 4 - 3.56e-11 0.91 s
196 0.675 1 5 - 3.43e-11 1.10 s
197 0.6937 1 4 - 3.55e-11 0.89 s
198 0.7125 1 4 - 3.52e-11 0.90 s
199 0.7312 1 3 - 4.79e-07 0.71 s
200 0.75 1 4 - 3.49e-11 0.89 s
201 0.7312 1 4 - 2.68e-11 0.90 s
202 0.7125 1 0 - 1.43e-10 0.05 s
203 0.6937 1 0 - 1.65e-10 0.05 s
204 0.675 1 0 - 2.95e-10 0.05 s
205 0.6562 1 0 - 3.19e-10 0.05 s
206 0.6375 1 0 - 3.46e-10 0.05 s
207 0.6188 1 0 - 4.73e-10 0.05 s
208 0.6 1 0 - 4.99e-10 0.05 s
209 0.5813 1 0 - 6.28e-10 0.05 s
210 0.5625 1 0 - 6.54e-10 0.05 s
211 0.5437 1 0 - 6.81e-10 0.05 s
212 0.525 1 0 - 8.08e-10 0.05 s
213 0.5062 1 0 - 8.35e-10 0.05 s
214 0.4875 1 0 - 9.10e-10 0.05 s
215 0.4688 1 0 - 9.86e-10 0.05 s
216 0.45 1 0 - 1.06e-09 0.05 s
217 0.4313 1 0 - 1.14e-09 0.05 s
218 0.4125 1 0 - 1.21e-09 0.05 s
219 0.3937 1 0 - 1.24e-09 0.05 s
220 0.375 1 0 - 1.32e-09 0.05 s
221 0.3563 1 0 - 1.39e-09 0.05 s
222 0.3375 1 0 - 1.42e-09 0.05 s
223 0.3187 1 0 - 1.50e-09 0.05 s
224 0.3 1 0 - 1.57e-09 0.05 s
225 0.2812 1 0 - 1.65e-09 0.05 s
226 0.2625 1 0 - 1.72e-09 0.05 s
227 0.2437 1 0 - 1.77e-09 0.05 s
228 0.225 1 0 - 1.83e-09 0.05 s
229 0.2062 1 0 - 1.90e-09 0.05 s
230 0.1875 1 0 - 1.98e-09 0.05 s
231 0.1687 1 0 - 2.03e-09 0.05 s
232 0.15 1 0 - 2.10e-09 0.05 s
233 0.1313 1 0 - 2.18e-09 0.05 s
234 0.1125 1 0 - 2.23e-09 0.05 s
235 0.09375 1 0 - 2.31e-09 0.05 s
236 0.075 1 0 - 2.37e-09 0.05 s
237 0.05625 1 0 - 2.43e-09 0.05 s
238 0.0375 1 0 - 2.51e-09 0.05 s
239 0.01875 1 0 - 2.57e-09 0.05 s
240 0 1 0 - 2.64e-09 0.05 s
241 0.025 1 2 - 3.22e-11 0.47 s
242 0.05 1 0 - 2.64e-09 0.05 s
243 0.075 1 0 - 5.26e-09 0.05 s
244 0.1 1 0 - 7.91e-09 0.05 s
245 0.125 1 0 - 1.05e-08 0.05 s
246 0.15 1 0 - 1.32e-08 0.05 s
247 0.175 1 0 - 1.58e-08 0.05 s
248 0.2 1 0 - 1.85e-08 0.05 s
249 0.225 1 0 - 2.11e-08 0.05 s
250 0.25 1 0 - 2.37e-08 0.05 s
251 0.275 1 0 - 2.64e-08 0.05 s
252 0.3 1 0 - 2.90e-08 0.05 s
253 0.325 1 0 - 3.16e-08 0.05 s
254 0.35 1 0 - 3.43e-08 0.05 s
255 0.375 1 0 - 3.69e-08 0.05 s
256 0.4 1 0 - 3.95e-08 0.06 s
257 0.425 1 0 - 4.22e-08 0.05 s
258 0.45 1 0 - 4.48e-08 0.05 s
259 0.475 1 0 - 4.74e-08 0.06 s
260 0.5 1 0 - 5.01e-08 0.05 s
261 0.525 1 0 - 5.27e-08 0.06 s
262 0.55 1 0 - 5.54e-08 0.06 s
263 0.575 1 0 - 5.81e-08 0.06 s
264 0.6 1 0 - 6.07e-08 0.05 s
265 0.625 1 0 - 6.33e-08 0.06 s
266 0.65 1 0 - 6.59e-08 0.05 s
267 0.675 1 0 - 6.86e-08 0.06 s
268 0.7 1 0 - 7.12e-08 0.05 s
269 0.725 1 0 - 7.39e-08 0.06 s
270 0.75 1 0 - 7.65e-08 0.06 s
271 0.775 1 5 - 1.53e-09 1.17 s
272 0.8 1 3 - 1.30e-10 0.71 s
273 0.825 1 3 - 8.04e-11 0.70 s
274 0.85 1 3 - 6.45e-11 0.70 s
275 0.875 1 2 - 1.81e-07 0.48 s
276 0.9 1 2 - 7.68e-09 0.48 s
277 0.925 1 2 - 1.14e-08 0.48 s
278 0.95 1 2 - 1.49e-08 0.50 s
279 0.975 1 2 - 1.70e-08 0.65 s
280 1 1 2 - 1.80e-08 0.47 s
281 0.975 3 12 - 1.80e-08 2.91 s ↓1 ↑2
282 0.95 2 3 - 1.74e-11 0.72 s ↑2
283 0.925 2 4 - 1.39e-11 0.93 s ↑2
284 0.9 2 4 - 1.35e-11 0.93 s ↑2
285 0.875 1 2 - 3.76e-11 0.46 s
286 0.85 1 0 - 7.74e-11 0.05 s
287 0.825 1 0 - 1.18e-10 0.05 s
288 0.8 1 0 - 2.03e-10 0.05 s
289 0.775 1 0 - 2.36e-10 0.05 s
290 0.75 1 0 - 2.71e-10 0.05 s
291 0.725 1 0 - 3.08e-10 0.05 s
292 0.7 1 0 - 3.46e-10 0.05 s
293 0.675 1 0 - 3.84e-10 0.05 s
294 0.65 1 0 - 4.76e-10 0.05 s
295 0.625 1 0 - 5.11e-10 0.05 s
296 0.6 1 0 - 5.47e-10 0.05 s
297 0.575 1 0 - 5.83e-10 0.05 s
298 0.55 1 0 - 6.20e-10 0.05 s
299 0.525 1 0 - 7.16e-10 0.05 s
300 0.5 1 0 - 7.51e-10 0.05 s
301 0.475 1 0 - 8.19e-10 0.05 s
302 0.45 1 0 - 8.53e-10 0.05 s
303 0.425 1 0 - 9.22e-10 0.05 s
304 0.4 1 0 - 9.56e-10 0.05 s
305 0.375 1 0 - 9.91e-10 0.05 s
306 0.35 1 0 - 1.06e-09 0.05 s
307 0.325 1 0 - 1.09e-09 0.05 s
308 0.3 1 0 - 1.16e-09 0.05 s
309 0.275 1 0 - 1.20e-09 0.11 s
310 0.25 1 0 - 1.23e-09 0.05 s
311 0.225 1 0 - 1.28e-09 0.05 s
312 0.2 1 0 - 1.33e-09 0.05 s
313 0.175 1 0 - 1.39e-09 0.05 s
314 0.15 1 0 - 1.44e-09 0.05 s
315 0.125 1 0 - 1.47e-09 0.05 s
316 0.1 1 0 - 1.52e-09 0.05 s
317 0.075 1 0 - 1.57e-09 0.05 s
318 0.05 1 0 - 1.62e-09 0.05 s
319 0.025 1 0 - 1.66e-09 0.05 s
320 0 1 0 - 1.71e-09 0.05 s
────────────────────────────────────────────────────────────────────────────────────────
converged · 320 increments · 348 Newton iterations · 97.55 s
Postprocessing¶
The load peaks once the debond has run most of the way along the fiber, then eases off as it reaches the far end and the damaged interphase is left carrying the sliding resistance.
Each cycle unloads along a straight secant to the origin and reloads along it, since scalar damage degrades the stiffness without leaving a permanent strain.
In [6]:
disp = u[:, push, 1].mean(dim=1).abs()
load = f[:, push, 1].sum(dim=1).abs()
fig, ax = plt.subplots(figsize=(7, 5))
ax.plot(disp, load, ".-", color="deeppink")
ax.set_xlabel("Indenter displacement in µm")
ax.set_ylabel("Load in µN")
ax.grid()
plt.show()
Damage evolution¶
The debond starts at the indented end and runs down the fiber, localizing in the element layer on the fiber side of the interphase.
The displacement is magnified by a factor 50. The damage holds still through the unloading and reloading of each cycle, where the damage does not change.
In [7]:
fig, ax = plt.subplots(figsize=(4, 6))
style = {"cmap": "Blues", "vmin": 0.0, "vmax": 1.0, "linewidth": 0.1, "bcs": False}
# Drawn once, as the colorbar survives the clearing of the axes in each frame
model.plot(element_property={"Damage": alpha[0, :, 1]}, colorbar=True, ax=ax, **style)
def animate(i):
ax.clear()
model.plot(u=50 * u[i], element_property={"Damage": alpha[i, :, 1]}, ax=ax, **style)
return () # we have to return a (empty) tuple of artists
anim = animation.FuncAnimation(
fig, animate, frames=range(len(alpha))[::8], interval=50, blit=False, repeat=False
)
plt.close(fig) # prevent static plot
embed_animation_gif(anim, fps=10)
Out[7]:
