-
Notifications
You must be signed in to change notification settings - Fork 6
/
Copy pathUHYPER_Lefevre_Lopez-Pamies.for
330 lines (330 loc) · 12.9 KB
/
UHYPER_Lefevre_Lopez-Pamies.for
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
!**********************************************************************
! Legal notice: UHYPER_Lefevre_Lopez-Pamies.for (Windows)
!
! Copyright (C) 2018 Victor Lefèvre ([email protected])
! Oscar Lopez-Pamies ([email protected])
!
! This ABAQUS UHYPER subroutine implements the hyperelastic energy
! density derived in [1] for the macroscopic elastic response of
! isotropic and incompressible filled elastomers. The results applies
! to general non-percolative isotropic distributions of stiff
! inclusions, stiff interphases, and stiff occluded rubber. This result
! is valid for any choice of I1-based incompressible energy density
! characterizing the non-Gaussian isotropic elastic response of the
! underlying elastomer. The present subroutine is implemented for the
! choice of strain energy density proposed in [2].
!
! For the special case of equiaxed particles surrounded by interphases
! and occluded rubber, the model reduces to the result derived in [3].
! For the special case of equiaxed particles without interphases
! nor occluded rubber, both models reduce to the result obtained in
! [4].
!
! This program is free software: you can redistribute it and/or modify
! it under the terms of the GNU General Public License as published by
! the Free Software Foundation, either version 3 of the License, or
! (at your option) any later version.
!
! This program is distributed in the hope that it will be useful,
! but WITHOUT ANY WARRANTY; without even the implied warranty of
! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
! GNU General Public License for more details.
!
! You should have received a copy of the GNU General Public License
! along with this program. If not, see https://www.gnu.org/licenses/
!
!**********************************************************************
! Usage:
!
! The subroutine is to be used as an incompressible USER hyperelastic
! model with either 8 (arbitrary microstructures) or 7 (equiaxed
! particles) material properties, e.g.,
! *HYPERELASTIC, USER, TYPE=INCOMPRESSIBLE, PROPERTIES=8
! or
! *HYPERELASTIC, USER, TYPE=INCOMPRESSIBLE, PROPERTIES=7
! in the input (.inp) file.
!
! 1/ For arbitrary microstructures, the 8 materials properties required
! by the model to be input to the subroutine via the PROPS array are
! listed in the table below:
!
! AMU1 = PROPS(1) ! PARAMETER #1 OF THE ELASTOMER
! ALPHA1 = PROPS(2) ! EXPONENT #1 OF THE ELASTOMER
! AMU2 = PROPS(3) ! PARAMETER #2 OF THE ELASTOMER
! ALPHA2 = PROPS(4) ! EXPONENT #2 OF THE ELASTOMER
! ACP = PROPS(5) ! VOLUME FRACTION OF PARTICLES
! ACI = PROPS(6) ! VOLUME FRACTION OF INTERPHASE
! ACO = PROPS(7) ! VOLUME FRACTION OF OCCLUDED RUBBER
! AMUT = PROPS(8) ! INITIAL SHEAR MODULUS OF FILLED ELASTOMER
!
! The two material parameters AMU1, AMU2 characterizing the elastic
! behavior of the underlying elastomer are non-negative real numbers
! (AMU1 >= 0, AMU2 >= 0) with strictly positive sum (AMU1 + AMU2 > 0).
! The two exponents ALPHA1, ALPHA2 are non-zero real numbers
! (ALPHA1 ≠ 0, ALPHA2 ≠ 0) leading to a strongly elliptic strain
! energy (see eq. (22) in [2]). This is left to the user to check.
!
! The volume fractions of particles (ACP), interphases (ACI), and
! and occluded rubber (ACO) must satisfy 0 <= ACP + ACI + ACO <= 1.
!
! The initial shear modulus of the filler elastomer AMUT must be a
! non-negative real number (AMUT >= 0).
!
! 2/ For microstructures comprising stiff equiaxed particles,
! interphases, and occluded rubber, the 7 materials properties required
! by the model to be input to the subroutine via the PROPS array are
! listed in the table below:
!
! AMU1 = PROPS(1) ! PARAMETER #1 OF THE ELASTOMER
! ALPHA1 = PROPS(2) ! EXPONENT #1 OF THE ELASTOMER
! AMU2 = PROPS(3) ! PARAMETER #2 OF THE ELASTOMER
! ALPHA2 = PROPS(4) ! EXPONENT #2 OF THE ELASTOMER
! ACP = PROPS(5) ! VOLUME FRACTION OF PARTICLES
! ACI = PROPS(6) ! VOLUME FRACTION OF INTERPHASE
! ACO = PROPS(7) ! VOLUME FRACTION OF OCCLUDED RUBBER
!
! These 7 material properties are subjected to the same restrictions
! listed above.
!
!**********************************************************************
! Additional information:
!
! This subroutine does not create solution-dependent state variables
! nor predefined field variables.
!
! Please consult the ABAQUS Documentation for additional references
! regarding the use of incompressible USER hyperelastic models with
! the UHYPER subroutine.
!
! Due the incompressible nature of this model, use of hybrid elements
! is strongly recommended.
!
!**********************************************************************
! References:
!
! [1] Lefèvre, V., Lopez-Pamies, O. 2017. Nonlinear electroelastic
! deformations of dielectric elastomer composites: II — Non-Gaus-
! sian elastic dielectrics. J. Mech. Phys. Solids 99, 438--470.
! [2] Lopez-Pamies, O., 2010. A new I1-based hyperelastic model for
! rubber elastic materials. C. R. Mec. 338, 3--11.
! [3] Goudarzi, T., Spring, D.W., Paulino, G.H., Lopez-Pamies, O., 2015
! . Filled elastomers: a theory of filler reinforcement based on
! hydrodynamic and interphasial effects. J. Mech. Phys. Solids
! 80, 37--67.
! [4] Lopez-Pamies, O., Goudarzi, T., Danas, K., 2013. The nonlinear
! elastic response of suspensions of rigid inclusions inr ubber:
! II—a simple explicit approximation for finite-concentration
! suspensions. J. Mech. Phys. Solids 61, 19--37.
!
!**********************************************************************
!
SUBROUTINE UHYPER(BI1,BI2,AJ,U,UI1,UI2,UI3,TEMP,NOEL,
1 CMNAME,INCMPFLAG,NUMSTATEV,STATEV,NUMFIELDV,FIELDV,
2 FIELDVINC,NUMPROPS,PROPS)
!
INCLUDE 'ABA_PARAM.INC'
#INCLUDE <SMAASPUSERSUBROUTINES.HDR>
!
CHARACTER*80 CMNAME
DIMENSION U(2),UI1(3),UI2(6),UI3(6),STATEV(*),FIELDV(*),
1 FIELDVINC(*),PROPS(*)
!
! STDB_ABQERR AND GET_THREAD_ID INITIALIZATION
!
DIMENSION INTV(1),REALV(4)
CHARACTER*8 CHARV(1)
CHARACTER*100 STRING1, STRING2, STRING3, STRING4
CHARACTER*300 STRING
!
INTEGER MYTHREADID
!
INTV(1)=0
REALV(1)=0.
REALV(2)=0.
REALV(3)=0.
REALV(4)=0.
CHARV(1)=''
!
MYTHREADID = GET_THREAD_ID()
!
! INPUT CHECKS
!
IF (MYTHREADID.EQ.0) THEN
IF (INCMPFLAG.EQ.0) THEN
STRING1='INCOMPRESSIBILITY FLAG IS 0. THE MODEL IS INCOMPRES'
STRING2='SIBLE. SET USER TYPE=INCOMPRESSIBLE.'
STRING = TRIM(STRING1) // TRIM(STRING2)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
ELSE IF (NUMSTATEV.NE.0) THEN
INTV(1)=NUMSTATEV
STRING1='RECEIVED REQUEST FOR %I SOLUTION-DEPENDENT STATE'
STRING2=' VARIABLES. THE SUBROUTINE DOES NOT CREATE SOLUTION'
STRING3='-DEPENDENT STATE VARIABLES.'
STRING = TRIM(STRING1) // TRIM(STRING2) // TRIM(STRING3)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
ELSE IF (NUMFIELDV.NE.0) THEN
INTV(1)=NUMFIELDV
STRING1='RECEIVED REQUEST FOR %I PREDEFINED FIELD VARI'
STRING2='ABLES. THE SUBROUTINE DOES NOT CREATE PREDEFINED'
STRING3=' FIELD VARIABLES.'
STRING = TRIM(STRING1) // TRIM(STRING2) // TRIM(STRING3)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
ELSE IF ((NUMPROPS.NE.7).AND.(NUMPROPS.NE.8)) THEN
INTV(1)=NUMPROPS
STRING1='RECEIVED %I MATERIAL PROPERTIES. THE SUBROUTINE'
STRING2=' REQUIRES EITHER 7 OR 8 MATERIAL PROPERTIES.'
STRING = TRIM(STRING1) // TRIM(STRING2)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
END IF
END IF
!
! MATERIAL PARAMETERS
!
AMU1 = PROPS(1) ! PARAMETER #1 OF THE ELASTOMER
ALPHA1 = PROPS(2) ! I1 EXPONENT #1 OF THE ELASTOMER
AMU2 = PROPS(3) ! PARAMETER #2 OF THE ELASTOMER
ALPHA2 = PROPS(4) ! I1 EXPONENT #2 OF THE ELASTOMER
ACP = PROPS(5) ! VOLUME FRACTION OF PARTICLES
ACI = PROPS(6) ! VOLUME FRACTION OF INTERPHASE
ACO = PROPS(7) ! VOLUME FRACTION OF OCCLUDED RUBBER
IF (NUMPROPS.EQ.8) THEN
AMUT = PROPS(8) ! INITIAL SHEAR MODULUS OF FILLED ELASTOMER
END IF
!
! PARTIAL MATERIAL PARAMETERS CHECKS
!
IF (((AMU1.LT.0.).OR.(AMU2.LT.0.).OR.(AMU1+AMU2.LE.0.))
1 .AND.(MYTHREADID.EQ.0)) THEN
REALV(1)=AMU1
REALV(2)=AMU2
REALV(3)=AMU1+AMU2
STRING1='RECEIVED AMU1 = %R AND AMU2 = %R, AMU1 + AMU2 = %R.'
STRING2=' THE PARAMETERS AMU1 AND AMU2 MUST BE NON-NEGATIVE'
STRING3=' AND AMU1 + AMU2, MUST BE GREATER THAT ZERO.'
STRING = TRIM(STRING1) // TRIM(STRING2) // TRIM(STRING3)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
END IF
!
IF (((ALPHA1.EQ.0.).OR.(ALPHA2.EQ.0.)).AND.
1 (MYTHREADID.EQ.0)) THEN
REALV(1)=ALPHA1
REALV(2)=ALPHA2
STRING1='RECEIVED ALPHA1 = %R AND ALPHA2 = %R.'
STRING2=' THE EXPONENTS ALPHA1 AND ALPHA2 MUST BE NON-ZERO.'
STRING = TRIM(STRING1) // TRIM(STRING2)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
END IF
!
AC = ACP + ACI + ACO
!
IF (AC.LT.0) THEN
IF (MYTHREADID.EQ.0) THEN
REALV(1)=ACP
REALV(2)=ACI
REALV(3)=ACO
REALV(4)=AC
STRING1='RECEIVED ACP = %R, ACI = %R, ACO = %R.'
STRING2=' "TOTAL" VOLUME FRACTION AC = ACP+ACI+ACO = %R'
STRING3=' IS NEGATIVE.'
STRING = TRIM(STRING1) // TRIM(STRING2) // TRIM(STRING3)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
ELSE
CALL XIT
END IF
ELSE IF (AC.GE.1) THEN
IF (MYTHREADID.EQ.0) THEN
REALV(1)=ACP
REALV(2)=ACI
REALV(3)=ACO
REALV(4)=AC
STRING1='RECEIVED ACP = %R, ACI = %R, ACO = %R.'
STRING2=' "TOTAL" VOLUME FRACTION AC = ACP+ACI+ACO = %R'
STRING3=' IS GREATER OR EQUAL THAN 1.'
STRING = TRIM(STRING1) // TRIM(STRING2) // TRIM(STRING3)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
ELSE
CALL XIT
END IF
END IF
!
IF ((NUMPROPS.EQ.8).AND.(MYTHREADID.EQ.0)) THEN
IF (AMUT.LT.0.) THEN
REALV(1)=AMUT
STRING1='RECEIVED AMUT = %R.'
STRING2=' THE INITIAL SHEAR MODULUS OF THE FILLED ELASTOMER'
STRING3=' MUST BE NON-NEGATIVE.'
STRING = TRIM(STRING1) // TRIM(STRING2) // TRIM(STRING3)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
ELSE IF (AMUT.LT.(AMU1+AMU2)) THEN
REALV(1)=AMUT
REALV(2)=AMU1+AMU2
STRING1='RECEIVED AMUT = %R. AND AMU1+AMU2 = %R.'
STRING2=' THE INITIAL SHEAR MODULUS OF THE FILLED ELASTOMER'
STRING3=' MUST BE GREATER THAN THAT OF THE'
STRING4=' UNDERLYING ELASTOMER.'
STRING = TRIM(STRING1) // TRIM(STRING2)
STRING = TRIM(STRING) // TRIM(STRING3) // TRIM(STRING4)
CALL STDB_ABQERR(-3,STRING,INTV,REALV,CHARV)
END IF
END IF
!
IF (NUMPROPS.EQ.8) THEN
! ARBITRARY MICROSTRUCTURE [1]
RC = AMUT/((AMU1+AMU2)*(1.-AC))
ELSEIF (NUMPROPS.EQ.7) THEN
! STIFF EQUIAXED INCLUSIONS, INTERPHASES, OCCLUDED RUBBER [3,4]
RC = (1.-AC)**(-3.5)
END IF
!
! ‘‘AMPLIFIED’’ STRAIN MEASURE
!
AII1 = RC*(BI1-3.)+3.
!
! NON-GAUSSIAN HYPERELASTIC MODEL FOR THE ELASTOMER [2]
! AND DERIVATIVES
!
AP1 = AMU1*3.**(1.-ALPHA1)*0.5
AP2 = AMU2*3.**(1.-ALPHA2)*0.5
!
PSI = AP1/ALPHA1*(AII1**ALPHA1-3.**ALPHA1)+
1 AP2/ALPHA2*(AII1**ALPHA2-3.**ALPHA2)
!
DPSI = AP1*AII1**(ALPHA1-1.)+AP2*AII1**(ALPHA2-1.)
!
DDPSI = AP1*(ALPHA1-1.)*AII1**(ALPHA1-2.)+
1 AP2*(ALPHA2-1.)*AII1**(ALPHA2-2.)
!
! MACROSCOPIC STRAIN ENERGY DENSITY FUNCTION FOR
! FILLED ELASTOMER [1,3,4]
!
U(1) = (1.-AC)*PSI
U(2) = 0.
!
! FIRST PARTIAL DERIVATIVES
!
UI1(1) = (1.-AC)*RC*DPSI
UI1(2) = 0.
UI1(3) = 0.
!
! SECOND PARTIAL DERIVATIVES
!
UI2(1) = (1.-AC)*RC**2.*DDPSI
UI2(2) = 0.
UI2(3) = 0.
UI2(4) = 0.
UI2(5) = 0.
UI2(6) = 0.
!
! THIRD PARTIAL DERIVATIVES
!
UI3(1) = 0.
UI3(2) = 0.
UI3(3) = 0.
UI3(4) = 0.
UI3(5) = 0.
UI3(6) = 0.
!
RETURN
END
!
!**********************************************************************