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mat104c_nldam_newton.F
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23!||====================================================================
24!|| mat104c_nldam_newton ../engine/source/materials/mat/mat104/mat104c_nldam_newton.F
25!||--- called by ------------------------------------------------------
26!|| sigeps104c ../engine/source/materials/mat/mat104/sigeps104c.F
27!||====================================================================
28 SUBROUTINE mat104c_nldam_newton(
29 1 NEL ,NGL ,NUPARAM ,NUVAR ,
30 2 TIME ,TIMESTEP,UPARAM ,UVAR ,JTHE ,OFF ,
31 3 GS ,RHO ,PLA ,DPLA ,EPSD ,SOUNDSP ,
32 4 DEPSXX ,DEPSYY ,DEPSXY ,DEPSYZ ,DEPSZX ,
33 5 SIGOXX ,SIGOYY ,SIGOXY ,SIGOYZ ,SIGOZX ,
34 6 SIGNXX ,SIGNYY ,SIGNXY ,SIGNYZ ,SIGNZX ,THKLY ,
35 7 THK ,SIGY ,ET ,TEMPEL ,DPLA_NL ,DMG ,
36 8 TEMP ,SEQ ,PLA_NL ,PLAP_NL )
37 !=======================================================================
38 ! Implicit types
39 !=======================================================================
40#include "implicit_f.inc"
41 !=======================================================================
42 ! Common
43 !=======================================================================
44#include "com01_c.inc"
45 !=======================================================================
46 ! Dummy arguments
47 !=======================================================================
48 INTEGER NEL,NUPARAM,NUVAR,JTHE
49 INTEGER ,DIMENSION(NEL), INTENT(IN) :: NGL
50 my_real
51 . TIME,TIMESTEP
52 my_real,DIMENSION(NUPARAM), INTENT(IN) ::
53 . UPARAM
54 my_real,DIMENSION(NEL), INTENT(IN) ::
55 . RHO,TEMPEL,
56 . depsxx,depsyy,depsxy,depsyz,depszx,
57 . sigoxx,sigoyy,sigoxy,sigoyz,sigozx,
58 . gs,thkly,pla_nl,plap_nl,dpla_nl
59c
60 my_real ,DIMENSION(NEL), INTENT(OUT) ::
61 . soundsp,sigy,et,
62 . signxx,signyy,signxy,signyz,signzx
63c
64 my_real ,DIMENSION(NEL), INTENT(INOUT) ::
65 . pla,dpla,epsd,off,thk,temp,seq
66 my_real ,DIMENSION(NEL,6), INTENT(INOUT) ::
67 . dmg
68 my_real ,DIMENSION(NEL,NUVAR), INTENT(INOUT) ::
69 . uvar
70 !=======================================================================
71 ! Local Variables
72 !=======================================================================
73 INTEGER I,II,IGURSON,ITER,NITER,NINDX,INDEX(NEL)
74c
75 my_real ::
76 . young,bulk,lam,g,g2,nu,cdr,kdr,hard,yld0,qvoce,bvoce,jcc,
77 . epsp0,mtemp,tini,tref,eta,cp,dpis,dpad,asrate,afiltr,hkhi,
78 . q1,q2,ed,an,kw,fr,fc,f0,a11,a12,nnu
79 my_real ::
80 . dti,sigvm,omega,
81 . dtemp,fcosh,fsinh,dpdt,dlam,ddep
82 my_real ::
83 . dsdrdj2,dsdrdj3,
84 . dj3dsxx,dj3dsyy,dj3dsxy,dj3dszz,
85 . normsig,
86 .
87 . sdpla,dphi_dtrsig,dfdsig2,
88 . dphi_dsig,dphi_dyld,dphi_dfdr,
89 .
90 . dyld_dpla
91c
92 my_real, DIMENSION(NEL) ::
93 . trsig,trdep,trdfds,
94 . sxx,syy,sxy,szz,sigm,j2,j3,sigdr,yld,weitemp,
95 . hardp,fhard,frate,ftherm,fdr,triax,
96 . fdam,phi,ft,fs,fg,fn,fsh,dpla_dlam,dphi_dlam,dezz,etat,
97 . normxx,normyy,normxy,normzz,sig_dfdsig,
98 . dpxx,dpyy,dpxy,dpzz,sigdr2,yld2i,dlam_nl
99c
100 !=======================================================================
101 ! DRUCKER - VOCE - JOHNSON-COOK MATERIAL LAW WITH GURSON DAMAGE
102 ! USING NON LOCAL PEERLINGS METHOD
103 !=======================================================================
104 !UVAR(1) YLD YIELD STRESS
105 !UVAR(2) EPSD PLASTIC STRAIN RATE SAVED FOR FILTERING
106 !=======================================================================
107c
108 !=======================================================================
109 ! - INITIALISATION OF COMPUTATION ON TIME STEP
110 !=======================================================================
111 ! Recovering model parameters
112 ! Elastic parameters
113 young = uparam(1)
114 bulk = uparam(2)
115 g = uparam(3)
116 g2 = uparam(4)
117 lam = uparam(5)
118 nu = uparam(6)
119 nnu = uparam(7)
120 a11 = uparam(9)
121 a12 = uparam(10)
122 ! Plastic criterion and hardening parameters [Drucker, 1948]
123 cdr = uparam(12) ! Drucker coefficient
124 kdr = uparam(13) ! Drucker 1/K coefficient
125 tini = uparam(14) ! Initial temperature
126 hard = uparam(15) ! Linear hardening
127 yld0 = uparam(16) ! Initial yield stress
128 qvoce = uparam(17) ! 1st Voce parameter
129 bvoce = uparam(18) ! 2nd Voce parameter
130 ! Strain-rate dependence parameters
131 jcc = uparam(20) ! Johnson-Cook strain rate coefficient
132 epsp0 = uparam(21) ! Johnson-Cook reference strain rate
133 ! Thermal softening and self-heating parameters
134 mtemp = uparam(22) ! Thermal softening slope
135 tref = uparam(23) ! Reference temperature
136 eta = uparam(24) ! Taylor-Quinney coefficient
137 cp = uparam(25) ! Thermal mass capacity
138 dpis = uparam(26) ! Isothermal plastic strain rate
139 dpad = uparam(27) ! Adiabatic plastic strain rate
140 ! Plastic strain-rate filtering parameters
141 asrate = uparam(28) ! Plastic strain rate filtering frequency
142 afiltr = asrate*timestep/(asrate*timestep + one)
143 dti = one / max(timestep, em20)
144c
145 ! Gurson damage model parameters parameters
146 igurson = nint(uparam(30)) ! Gurson switch flag:
147 ! = 0 => no damage model
148 ! = 1 => local damage model
149 ! = 2 => non local (Forest - micromorphic) damage model
150 ! = 3 => non local (Peerlings) damage model
151 q1 = uparam(31) ! Gurson yield criterion 1st parameter
152 q2 = uparam(32) ! Gurson yield criterion 2nd parameter
153 ed = uparam(34) ! Plastic strain trigger for nucleation
154 an = uparam(35) ! Nucleation rate
155 kw = uparam(36) ! Shear coefficient (Nahshon & Hutchinson)
156 fr = uparam(37) ! Failure void volume fraction
157 fc = uparam(38) ! Critical void volume fraction
158 f0 = uparam(39) ! Initial void volume fraction
159 hkhi = uparam(40) ! Micromorphic penalty parameter
160c
161 ! Recovering internal variables
162 DO i=1,nel
163 ! If the element is failing
164 IF (off(i) < em03) off(i) = zero
165 IF (off(i) < one) off(i) = off(i)*four_over_5
166 ! User inputs
167 yld(i) = uvar(i,1) ! Previous yield stress
168 ! Damage variables
169 fg(i) = dmg(i,2) ! Growth damage
170 fn(i) = dmg(i,3) ! Nucleation damage
171 fsh(i) = dmg(i,4) ! Shear damage
172 ft(i) = dmg(i,5) ! Total damage
173 fs(i) = dmg(i,6) ! Effective damage
174 ! Standard inputs
175 dpla(i) = zero ! Initialization of the plastic strain increment
176 et(i) = one ! Initialization of hourglass coefficient
177 hardp(i) = zero ! Initialization of hardening modulus
178 dezz(i) = zero ! Initialization of the transverse strain
179 dlam_nl(i) = zero ! Initialization of the non-local plastic multiplier
180 sig_dfdsig(i) = zero ! Initialization of the stress - normal scalar product
181 normxx(i) = zero ! Initialization of the x component of the normal tensor
182 normyy(i) = zero ! Initialization of the y component of the normal tensor
183 normzz(i) = zero ! Initialization of the z component of the normal tensor
184 ENDDO
185!
186 epsd(1:nel) = uvar(1:nel,2)
187!
188 ! Initialization of damage, temperature and self-heating weight factor
189 IF (time == zero) THEN
190 temp(1:nel) = tini
191 IF (isigi == 0) THEN
192 dmg(1:nel,5) = f0
193 ft(1:nel) = f0
194 dmg(1:nel,1) = f0/fr
195 IF (f0<fc) THEN
196 dmg(1:nel,6) = f0
197 ELSE
198 dmg(1:nel,6) = fc + (one/q1-fc)*(f0-fc)/(fr-fc)
199 ENDIF
200 fs(1:nel) = dmg(1:nel,6)
201 ENDIF
202 ENDIF
203 IF (cp > zero) THEN
204 IF (jthe == 0) THEN
205 DO i=1,nel
206 ! Computation of the weight factor
207 IF (plap_nl(i) < dpis) THEN
208 weitemp(i) = zero
209 ELSEIF (plap_nl(i) > dpad) THEN
210 weitemp(i) = one
211 ELSE
212 weitemp(i) = ((plap_nl(i)-dpis)**2 )
213 . * (three*dpad - two*plap_nl(i) - dpis)
214 . / ((dpad-dpis)**3)
215 ENDIF
216 ENDDO
217 ELSE
218 temp(1:nel) = tempel(1:nel)
219 ENDIF
220 ENDIF
221c
222 !========================================================================
223 ! NON-LOCAL VARIABLES UPDATE
224 !========================================================================
225 DO i=1,nel
226c
227 ! Previous value of Drucker equivalent stress
228 trsig(i) = sigoxx(i) + sigoyy(i)
229 sigm(i) = -trsig(i) * third
230 sxx(i) = sigoxx(i) + sigm(i)
231 syy(i) = sigoyy(i) + sigm(i)
232 szz(i) = sigm(i)
233 sxy(i) = sigoxy(i)
234 j2(i) = half*(sxx(i)**2 + syy(i)**2 + szz(i)**2 ) + sxy(i)**2
235 j3(i) = sxx(i) * syy(i) * szz(i) - szz(i) * sxy(i)**2
236 fdr(i) = j2(i)**3 - cdr*(j3(i)**2)
237 IF (fdr(i) > zero) THEN
238 sigdr(i) = kdr * exp((one/six)*log(fdr(i)))
239 ELSE
240 sigdr(i) = zero
241 ENDIF
242c
243 ! Computation of the stress triaxiality and the etaT factor
244 IF (sigdr(i)>zero) THEN
245 triax(i) = (trsig(i)*third)/sigdr(i)
246 ELSE
247 triax(i) = zero
248 ENDIF
249 IF (trsig(i)<zero) THEN
250 etat(i) = zero
251 ELSE
252 etat(i) = one
253 ENDIF
254c
255 ! Normal to the previous yield surface
256 IF (yld(i)>zero) THEN
257 yld2i(i) = one / yld(i)**2
258 dphi_dsig = two * sigdr(i) * yld2i(i)
259 fsinh = sinh(q2*etat(i)*trsig(i)/(yld(i)*two))
260 dphi_dtrsig = q1*q2*etat(i)*fs(i)*fsinh/yld(i)
261 ELSE
262 yld2i(i) = zero
263 dphi_dsig = zero
264 fsinh = zero
265 dphi_dtrsig = zero
266 ENDIF
267c
268 ! Computation of the Eulerian norm of the stress tensor
269 normsig = sqrt(sigoxx(i)*sigoxx(i)
270 . + sigoyy(i)*sigoyy(i)
271 . + two*sigoxy(i)*sigoxy(i))
272 normsig = max(normsig,one)
273c
274 ! Computation of the normal to the yield surface
275 fdr(i) = (j2(i)/(normsig**2))**3 - cdr*((j3(i)/(normsig**3))**2)
276 IF (fdr(i) > zero) THEN
277 dphi_dfdr = dphi_dsig*kdr*(one/six)*exp(-(five/six)*log(fdr(i)))
278 ELSE
279 dphi_dfdr = zero
280 ENDIF
281 dsdrdj2 = dphi_dfdr*three*(j2(i)/(normsig**2))**2
282 dsdrdj3 = -dphi_dfdr*two*cdr*(j3(i)/(normsig**3))
283 !dJ3/dS
284 dj3dsxx = two_third*(syy(i)*szz(i))/(normsig**2)
285 . - third*(sxx(i)*szz(i))/(normsig**2)
286 . - third*(sxx(i)*syy(i)-sxy(i)**2)/(normsig**2)
287 dj3dsyy = - third*(syy(i)*szz(i))/(normsig**2)
288 . + two_third*(sxx(i)*szz(i))/(normsig**2)
289 . - third*(sxx(i)*syy(i)-sxy(i)**2)/(normsig**2)
290 dj3dszz = - third*(syy(i)*szz(i))/(normsig**2)
291 . - third*(sxx(i)*szz(i))/(normsig**2)
292 . + two_third*(sxx(i)*syy(i)-sxy(i)**2)/(normsig**2)
293 dj3dsxy = two*(sxx(i)*sxy(i) + sxy(i)*syy(i))/(normsig**2)
294c
295 ! dPhi/dSig
296 normxx(i) = dsdrdj2*sxx(i)/normsig + dsdrdj3*dj3dsxx + dphi_dtrsig
297 normyy(i) = dsdrdj2*syy(i)/normsig + dsdrdj3*dj3dsyy + dphi_dtrsig
298 normzz(i) = dsdrdj2*szz(i)/normsig + dsdrdj3*dj3dszz + dphi_dtrsig
299 normxy(i) = two*dsdrdj2*sxy(i)/normsig + dsdrdj3*dj3dsxy
300 trdfds(i) = normxx(i) + normyy(i) + normzz(i)
301 sig_dfdsig(i) = sigoxx(i)*normxx(i) + sigoyy(i)*normyy(i)
302 . + sigoxy(i)*normxy(i)
303c
304 ! Computation of the non-local plastic multiplier
305 IF (sig_dfdsig(i)>zero) THEN
306 dlam_nl(i) = ((one - ft(i))*yld(i)*dpla_nl(i))/max(sig_dfdsig(i),em20)
307 ELSE
308 dlam_nl(i) = zero
309 ENDIF
310c
311 ! Damage growth update
312 IF ((trdfds(i)>zero).AND.(ft(i)>zero).AND.(ft(i)<fr)) THEN
313 fg(i) = fg(i) + (one-ft(i))*dlam_nl(i)*trdfds(i)
314 ENDIF
315 fg(i) = max(fg(i),zero)
316c
317 ! Nucleation damage update
318 IF ((pla_nl(i) >= ed).AND.(ft(i)<fr)) THEN
319 ! Case for positive stress triaxiality
320 IF (triax(i)>=zero) THEN
321 fn(i) = fn(i) + an*dpla_nl(i)
322 ! Case for negative stress triaxiality
323 ELSEIF ((triax(i)<zero).AND.(triax(i)>=-third)) THEN
324 fn(i) = fn(i) + an*max(one + three*triax(i),zero)*dpla_nl(i)
325 ENDIF
326 ENDIF
327 fn(i) = max(fn(i),zero)
328c
329 ! Shear damage update
330 IF ((sigdr(i) > zero).AND.(ft(i)>zero).AND.(ft(i)<fr)) THEN
331 sigvm = sqrt(max(em20,three*(j2(i)/(normsig**2))))
332 omega = one - ((twenty7 *(j3(i)/(normsig**3)))/(two*(sigvm**3)))**2
333 omega = max(omega,zero)
334 omega = min(omega,one)
335 sdpla = (sxx(i)*normxx(i)+syy(i)*normyy(i)+ szz(i)*normzz(i)
336 . + sxy(i)*normxy(i))
337 . * dlam_nl(i)
338 fsh(i) = fsh(i) + kw*omega*ft(i)*(sdpla/sigdr(i))
339 ENDIF
340 fsh(i) = max(fsh(i),zero)
341c
342 ! Total damage update
343 ft(i) = f0 + fg(i) + fn(i) + fsh(i)
344 ft(i) = min(ft(i), fr)
345 IF (ft(i) >= fr) THEN
346 IF (off(i)==one) off(i) = four_over_5
347 ENDIF
348c
349 ! Effective update
350 IF (ft(i) < fc)THEN
351 fs(i) = ft(i)
352 ELSE
353 fs(i) = fc + (one/q1 - fc) * (ft(i)-fc)/(fr-fc)
354 ENDIF
355 fs(i) = min(fs(i),one/q1)
356c
357 ! Temperature update
358 IF (cp > zero ) THEN
359 IF (jthe == 0) THEN
360 dtemp = weitemp(i)*(one-ft(i))*yld(i)*dpla_nl(i)*eta/(rho(i)*cp)
361 temp(i) = temp(i) + dtemp
362 ENDIF
363 ENDIF
364 ENDDO
365c
366 ! Computation of the initial yield stress
367 DO i=1,nel
368 ! a) - Hardening law
369 fhard(i) = yld0 + hard*pla(i) + qvoce*(one-exp(-bvoce*pla(i)))
370 ! b) - Correction factor for strain-rate dependence (Johnson-Cook)
371 frate(i) = one
372 IF (epsd(i) > epsp0) frate(i) = frate(i) + jcc*log(epsd(i)/epsp0)
373 ! c) - Correction factor for thermal effects
374 ftherm(i) = one
375 IF (cp > zero) ftherm(i) = one - mtemp*(temp(i) - tref)
376 ! d) - Computation of the yield function
377 yld(i) = fhard(i)*frate(i)*ftherm(i)
378 ! e) - Checking values
379 yld(i) = max(em10, yld(i))
380 ENDDO
381c
382 !========================================================================
383 ! - COMPUTATION OF TRIAL VALUES
384 !========================================================================
385 DO i=1,nel
386c
387 ! Computation of the trial stress tensor
388 signxx(i) = sigoxx(i) + (a11*depsxx(i) + a12*depsyy(i))
389 signyy(i) = sigoyy(i) + (a11*depsyy(i) + a12*depsxx(i))
390 signxy(i) = sigoxy(i) + (depsxy(i)*g)
391 signyz(i) = sigoyz(i) + (depsyz(i)*gs(i))
392 signzx(i) = sigozx(i) + (depszx(i)*gs(i))
393 ! Computation of the trace of the trial stress tensor
394 trsig(i) = signxx(i) + signyy(i)
395 sigm(i) = -trsig(i) * third
396 ! Computation of the deviatoric trial stress tensor
397 sxx(i) = signxx(i) + sigm(i)
398 syy(i) = signyy(i) + sigm(i)
399 szz(i) = sigm(i)
400 sxy(i) = signxy(i)
401 ! Second deviatoric invariant
402 j2(i) = half*(sxx(i)**2 + syy(i)**2 + szz(i)**2 ) + sxy(i)**2
403 ! Third deviatoric invariant
404 j3(i) = sxx(i)*syy(i)*szz(i) - szz(i)*sxy(i)**2
405 ! Drucker equivalent stress
406 fdr(i) = j2(i)**3 - cdr*(j3(i)**2)
407 ! Checking equivalent stress values
408 IF (fdr(i) > zero) THEN
409 sigdr(i) = kdr * exp((one/six)*log(fdr(i))) ! FDR(I)**(1/6)
410 ELSE
411 sigdr(i) = zero
412 ENDIF
413 ! Computation of the stress triaxiality and the etaT factor
414 IF (sigdr(i)>zero) THEN
415 triax(i) = (trsig(i)*third)/sigdr(i)
416 ELSE
417 triax(i) = zero
418 ENDIF
419 IF (trsig(i)<zero) THEN
420 etat(i) = zero
421 ELSE
422 etat(i) = one
423 ENDIF
424c
425 ! Update out of plane strain increment
426 IF (off(i) == one) THEN
427 dezz(i) = -nnu*depsxx(i)-nnu*depsyy(i)
428 IF (sig_dfdsig(i) > em01) THEN
429 dezz(i) = dezz(i) + nnu*(dlam_nl(i)*normxx(i))
430 . + nnu*(dlam_nl(i)*normyy(i))
431 . + dlam_nl(i)*normzz(i)
432 ENDIF
433 ENDIF
434 ENDDO
435c
436 !========================================================================
437 ! - COMPUTATION OF YIELD FONCTION
438 !========================================================================
439 DO i=1,nel
440 fdam(i) = two*q1*fs(i)*cosh(q2*etat(i)*trsig(i)/yld(i)/two) - (q1*fs(i))**2
441 phi(i) = (sigdr(i) / yld(i))**2 - one + fdam(i)
442 ENDDO
443c
444 ! Checking plastic behavior for all elements
445 nindx = 0
446 DO i=1,nel
447 IF ((phi(i) >= zero).AND.(off(i) == one).AND.(ft(i)<fr)) THEN
448 nindx=nindx+1
449 index(nindx)=i
450 ENDIF
451 ENDDO
452c
453 !====================================================================
454 ! - PLASTIC CORRECTION WITH CUTTING PLANE METHOD (SEMI-IMPLICIT)
455 !====================================================================
456c
457 ! Number of iterations
458 niter = 3
459c
460 ! Loop over yielding elements
461#include "vectorize.inc"
462 DO ii=1,nindx
463c
464 ! Number of the element with plastic behaviour
465 i = index(ii)
466c
467 ! Initialization of the iterative Newton procedure
468 sigdr2(i) = sigdr(i)**2
469 yld2i(i) = one / yld(i)**2
470 dpxx(i) = zero
471 dpyy(i) = zero
472 dpzz(i) = zero
473 dpxy(i) = zero
474 ENDDO
475c
476 ! Loop over the iterations
477 DO iter = 1, niter
478#include "vectorize.inc"
479 DO ii=1,nindx
480 i = index(ii)
481c
482 ! Note: in this part, the purpose is to compute for each iteration
483 ! a plastic multiplier allowing to update internal variables to satisfy
484 ! the consistency condition using the cutting plane semi-implicit
485 ! iterative procedure.
486 ! Its expression at each iteration is : DLAMBDA = - PHI/DPHI_DLAMBDA
487 ! -> PHI : current value of yield function (known)
488 ! -> DPHI_DLAMBDA : derivative of PHI with respect to DLAMBDA by taking
489 ! into account of internal variables kinetic :
490 ! plasticity, temperature and damage(to compute)
491c
492 ! 1 - Computation of DPHI_DSIG the normal to the yield surface
493 !-------------------------------------------------------------
494c
495 ! Derivative with respect to the equivalent stress and trace
496 yld2i(i) = one/(yld(i)**2)
497 dphi_dsig = two*sigdr(i)*yld2i(i)
498 fsinh = sinh(q2*etat(i)*trsig(i)/(yld(i)*two))
499 dphi_dtrsig = q1*q2*etat(i)*fs(i)*fsinh/yld(i)
500c
501 ! Computation of the Eulerian norm of the stress tensor
502 normsig = sqrt(signxx(i)*signxx(i)
503 . + signyy(i)*signyy(i)
504 . + two*signxy(i)*signxy(i))
505 normsig = max(normsig,one)
506c
507 ! Derivative with respect to Fdr
508 fdr(i) = (j2(i)/(normsig**2))**3 - cdr*((j3(i)/(normsig**3))**2)
509 dphi_dfdr = dphi_dsig*kdr*(one/six)*exp(-(five/six)*log(fdr(i)))
510 dsdrdj2 = dphi_dfdr*three*(j2(i)/(normsig**2))**2
511 dsdrdj3 = -dphi_dfdr*two*cdr*(j3(i)/(normsig**3))
512 ! dJ3/dSig
513 dj3dsxx = two_third*(syy(i)*szz(i))/(normsig**2)
514 . - third*(sxx(i)*szz(i))/(normsig**2)
515 . - third*(sxx(i)*syy(i)-sxy(i)**2)/(normsig**2)
516 dj3dsyy = - third*(syy(i)*szz(i))/(normsig**2)
517 . + two_third*(sxx(i)*szz(i))/(normsig**2)
518 . - third*(sxx(i)*syy(i)-sxy(i)**2)/(normsig**2)
519 dj3dszz = - third*(syy(i)*szz(i))/(normsig**2)
520 . - third*(sxx(i)*szz(i))/(normsig**2)
521 . + two_third*(sxx(i)*syy(i)-sxy(i)**2)/(normsig**2)
522 dj3dsxy = two*(sxx(i)*sxy(i) + sxy(i)*syy(i))/(normsig**2)
523 ! dPhi/dSig
524 normxx(i) = dsdrdj2*sxx(i)/normsig + dsdrdj3*dj3dsxx + dphi_dtrsig
525 normyy(i) = dsdrdj2*syy(i)/normsig + dsdrdj3*dj3dsyy + dphi_dtrsig
526 normzz(i) = dsdrdj2*szz(i)/normsig + dsdrdj3*dj3dszz + dphi_dtrsig
527 normxy(i) = two*dsdrdj2*sxy(i)/normsig + dsdrdj3*dj3dsxy
528c
529 ! 2 - Computation of DPHI_DLAMBDA
530 !---------------------------------------------------------
531c
532 ! a) Derivative with respect stress increments tensor DSIG
533 ! --------------------------------------------------------
534 trdfds(i) = normxx(i) + normyy(i) + normzz(i)
535 dfdsig2 = normxx(i) * (a11*normxx(i) + a12*normyy(i))
536 . + normyy(i) * (a11*normyy(i) + a12*normxx(i))
537 . + normxy(i) * normxy(i) * g
538c
539 ! b) Derivatives with respect to plastic strain P
540 ! ------------------------------------------------
541c
542 ! i) Derivative of the yield stress with respect to plastic strain dYLD / dPLA
543 ! ----------------------------------------------------------------------------
544 hardp(i) = hard + qvoce*bvoce*exp(-bvoce*pla(i))
545 dyld_dpla = hardp(i)*frate(i)*ftherm(i)
546c
547 ! ii) Derivative of dPLA with respect to DLAM
548 ! -------------------------------------------
549 sig_dfdsig(i) = signxx(i) * normxx(i)
550 . + signyy(i) * normyy(i)
551 . + signxy(i) * normxy(i)
552 dpla_dlam(i) = sig_dfdsig(i) / ((one - ft(i))*yld(i))
553c
554 ! c) Derivative with respect to the yield stress
555 ! ----------------------------------------------
556 sigdr2(i) = sigdr(i)**2
557 dphi_dyld = -two*sigdr2(i)/yld(i)**3-dphi_dtrsig*trsig(i)/yld(i)
558c
559 ! d) Derivative of PHI with respect to DLAM
560 dphi_dlam(i) = - dfdsig2 + (dphi_dyld*dyld_dpla*dpla_dlam(i))
561 dphi_dlam(i) = sign(max(abs(dphi_dlam(i)),em20) ,dphi_dlam(i))
562c
563 ! Computation of the plastic multiplier
564 dlam = -phi(i)/dphi_dlam(i)
565c
566 ! Plastic strains tensor update
567 dpxx(i) = dlam * normxx(i)
568 dpyy(i) = dlam * normyy(i)
569 dpzz(i) = dlam * normzz(i)
570 dpxy(i) = dlam * normxy(i)
571 trdep(i) = dpxx(i) + dpyy(i) + dpzz(i)
572c
573 ! Elasto-plastic stresses update
574 signxx(i) = signxx(i) - (a11*dpxx(i) + a12*dpyy(i))
575 signyy(i) = signyy(i) - (a11*dpyy(i) + a12*dpxx(i))
576 signxy(i) = signxy(i) - dpxy(i)*g
577 trsig(i) = signxx(i) + signyy(i)
578 sigm(i) = -trsig(i) * third
579 sxx(i) = signxx(i) + sigm(i)
580 syy(i) = signyy(i) + sigm(i)
581 szz(i) = sigm(i)
582 sxy(i) = signxy(i)
583c
584 ! Cumulated plastic strain and strain rate update
585 ddep = (dlam/((one - ft(i))*yld(i)))*sig_dfdsig(i)
586 dpla(i) = max(zero, dpla(i) + ddep)
587 pla(i) = pla(i) + ddep
588c
589 ! Drucker equivalent stress update
590 j2(i) = half*(sxx(i)**2 + syy(i)**2 + szz(i)**2 ) + sxy(i)**2
591 j3(i) = sxx(i) * syy(i) * szz(i) - szz(i) * sxy(i)**2
592 fdr(i) = j2(i)**3 - cdr*(j3(i)**2)
593 sigdr(i) = kdr * exp((one/six)*log(fdr(i)))
594 ! Computation of the stress triaxiality and the etaT factor
595 triax(i) = (trsig(i)*third)/sigdr(i)
596 IF (trsig(i)<zero) THEN
597 etat(i) = zero
598 ELSE
599 etat(i) = one
600 ENDIF
601c
602 ! Hardening law update
603 fhard(i) = yld0 + hard * pla(i) + qvoce*(one-exp(-bvoce*pla(i)))
604c
605 ! Yield stress update
606 yld(i) = fhard(i) * frate(i) * ftherm(i)
607 yld(i) = max(yld(i), em10)
608c
609 ! Yield function value update
610 sigdr2(i) = sigdr(i)**2
611 yld2i(i) = one / yld(i)**2
612 fcosh = cosh(q2*etat(i)*trsig(i)/(yld(i)*two))
613 fdam(i) = two*q1*fs(i)*fcosh - (q1*fs(i))**2
614 phi(i) = sigdr2(i) * yld2i(i) - one + fdam(i)
615c
616 ENDDO
617 ENDDO
618 ! End of the loop over the iterations
619 !===================================================================
620 ! - END OF PLASTIC CORRECTION WITH CUTTING PLANE ITERATIVE METHOD
621 !===================================================================
622c
623 ! Storing new values
624 DO i=1,nel
625 ! USR Outputs
626 uvar(i,1) = yld(i) ! Yield stress
627 ! Standard outputs
628 dmg(i,1) = ft(i)/fr ! Normalized total damage
629 dmg(i,2) = fg(i) ! Growth damage
630 dmg(i,3) = fn(i) ! Nucleation damage
631 dmg(i,4) = fsh(i) ! Shear damage
632 dmg(i,5) = min(ft(i),fr) ! Total damage
633 dmg(i,6) = min(fs(i),one/q1) ! Effective damage
634 seq(i) = sigdr(i) ! Equivalent stress
635 ! If element is broken
636 IF (ft(i) >= fr) THEN
637 dpla(i) = zero
638 signxx(i) = zero
639 signyy(i) = zero
640 signxy(i) = zero
641 signyz(i) = zero
642 signzx(i) = zero
643 seq(i) = zero
644 ENDIF
645 ! Plastic strain-rate (filtered)
646 dpdt = dpla(i) / max(em20,timestep)
647 epsd(i) = afiltr * dpdt + (one - afiltr) * epsd(i)
648 uvar(i,2) = epsd(i)
649 ! Coefficient for hourglass
650 IF (dpla(i) > zero) THEN
651 et(i) = hardp(i)*frate(i) / (hardp(i)*frate(i) + young)
652 ELSE
653 et(i) = one
654 ENDIF
655 ! Computation of the sound speed
656 soundsp(i) = sqrt((a11)/rho(i))
657 ! Storing the yield stress
658 sigy(i) = yld(i)
659 ! Computation of the thickness variation
660 thk(i) = thk(i) + dezz(i)*thkly(i)*off(i)
661 ENDDO
662c
663 END
min
#define min(a, b)
Definition macros.h:20
max
#define max(a, b)
Definition macros.h:21
mat104c_nldam_newton
subroutine mat104c_nldam_newton(nel, ngl, nuparam, nuvar, time, timestep, uparam, uvar, jthe, off, gs, rho, pla, dpla, epsd, soundsp, depsxx, depsyy, depsxy, depsyz, depszx, sigoxx, sigoyy, sigoxy, sigoyz, sigozx, signxx, signyy, signxy, signyz, signzx, thkly, thk, sigy, et, tempel, dpla_nl, dmg, temp, seq, pla_nl, plap_nl)
Definition mat104c_nldam_newton.F:37