Cumpllfen & St"tlutmtS Vol :3. So. :. ~p

Pnnted

In

ooJ5iW.(--9JY-9J9 So Yh

\63-180. 19t\6

53.00 S: 00 - .00 W

E E-- and and u u*- ?2 (J CT-. Substituting these ininequalities into eqn eqn (2.12). follows that that V’ must be equalities into L~.12). it follows V~ must be negative or zero: zero: since the the former former assumpt;on negative or since assumption would yield yield an imaginary imaginary velocity for the the discondisconwould velocity for tinuity. only V = 0 is tenable. tenable. and and it can can already already tinuity. only be concluded concluded that that ir*

Eliminating (/(I from Eliminating from eqns eqns (2.17) (7.17) and and C.IS) 12.18) yields yields E-

.= I [ J(~)

To To show show that that the the strain strain and and strain strain rates rates must must be be infinite intinite at at a point point which which strain strain softens. softens. a solution solution

be be seen seen that that II =: /1*

+

,,* l(* = - y -I'll

2

[a(t

((,, ~( tl -- $) x)~) c

2

- -.1') -

= E,,L’ H (1 E>CH(t

f’(~,


Vu (

X

Vu

(2.22a) (2.22a) E.

=

~e [H (t

- ~) - H(t - ~) e e + J, 1 + 4 (et - L) I!,) o(x) 6(r)

, (2.22b) (2.22b)

where where 5(x) 6(x) is the Dirac Dirac delta delta function. function.

ELASTIC ELASTIC

I

SOFTENING SOFTENING

I

I

~L/2-+-L/2-1

-L/2

Fig. 3. Problem Problem with strain softening at interface interface between elastic and softening domain (Ref. (Ref. [SI!. [.(]I.

I hI
W

~

• 220.

d >

~

'0

2~

4Q

------~ ~~ 70 85 '00

r:

25

R

'"r:

56.

~..J '8.

'0

(/)

2

TEf} BEL YTSCHKO ('{

F

17~

!

•5 !

43.

\oJ

~ 3t ~

..J 0-

3 ,9 .

i3

7 ~:.... _ _ _ _ _ _ _ _ _ _ _ _ _ •

8 .0

25

40

55

70

85

.00

10

25

40

55

R

R

(a)

(e)

70

85

'00

3

I

I

3

.‘

85

STRESS

tii

-

~

,

~

a

--:-_=_-=::_-=::_--::

>r

r44()l...-. _ _ _

10

25

40

55

70

85

100

380.

S270;! \

HO.

~

x

.,00~~~-=25~~4~0-~5~5--~1O=--8=5~~,00

z

V

\f - ,'10,::O~-:;25;;----:4:;;0:---;5:;;5--:1O;;--=85;--;;'00

R

43,'00 ______':T!____ ... 34.300:

1190, i

~! .Q!lOf

I

R

DISPLACEMENT

40

\oJ

tii

R 36 _ _ _ __

25

55 10 85 .00 R .7'0 • ___ ._______________

.620.

HYDROSTATIC

I~

'

a:

u

~

H.

10

100

.860 ___________.______ _

a:

u .680. HYMIOSTATIC

70

R

.BOO.

~

5> 0 . "'.,!....II I --.J),~ ,v·ut, 55

HYDROSTATIC

STRESS

t;;

e” f j

5

1.

o >

0. O. However. However. when when y'I is small small the the noise noise is exexy'I > cessive. For For one-dimensional one-dimensional planar planar waves. waves. the the cessive. value y-y ="" 0.1 gave gave noise-free noise-free response. response. Curiously. Curiously. value

the value value of of y-y for for the the cylindrical cylindrical wave wave had had to to be the higher ty (-y == 0.3) 0.25) than than the the value value for for the the spherical spherical higher wave (y (-y == 0. 0.1) in order order to to obtain obtain a noise-free noise-free rerewave I) in sponse. sponse. The values values of of the the applied applied surface surface pressure pressure are are The chosen as p. po = 0.708 0.708 for for the the spherical spherical wave wave and and p,, po chosen 0.8 for for the the cylindrical cylindrical wave. wave. For For these these boundary boundary = 0.8 outer conditions. the the wave wave propagating propagating from from the the outer conditions, surface remains remains elastic elastic until until the the wavefront wavefront reaches reaches surface

3r-----------------------,

60~----------------------'

N=10

N=10

6X

173

N=20

I(’ ,(’ _._// _*,* A ,,** .._’ /’ ,,’ ,.:.:’ jll_ _,/ I’’ ,.I’/ , ,’ ._I’ / I’’ ,.,’ / ,/ ,//I /* _:” _ / _ _** _/” _-_.-.

N=40

N 40

iv=40

0

M

,."

600

r

N=160

~~~>, - _'1..

10

x

01J ol’ J) )J 10

11f7

i j!

x

-100

Fig. 14. 14. Comparison Comparison of numerical solutions for elastic. elastic. linear spherical geometry geometry with analytic analytic solutions. solutions.

17~

TED BEL YTSCHKO

309c 30% of of the thickness thickness h h - ({. (I. The dimensions dimensions are ({(I == = 10. 10. hh == = 100. 100. L == = hh - ({(I == = 90. In order to ascertain the mesh refinement order ascertain refinement necessary essary for this class of of problems. problems. elastic elastic local local so· solutions lutions were were obtained obtained first. first. The The convergence convergence with with

1'{

ul.

increasing increasing numbers numbers of of elements elements (N (ic’ == 10. IO. 20. 20. 40. 80. 80. 160) 160) is is shown shown in Figs. 14 I-l and 15. 15. The results results converge converge to the exact exact solution solution given given by eqn (5.9). Subsequently. Subsequently, the problem problem was solved solved for aa nonlocal local continuum continuum (f" (E, = = I. €.r Ej = 5) 5) with with characteristic characteristic

6\,--------------------, 1 i N=10 N=10 1

a-

“y------

I

r

t 0 3

r i’

-1

I

/

I

N=20

/ /

/

/ /

:k r

/

I

I

/

,

4g~/~~/='~~~==~======~

N=40 N=40

N=40 ,/; i 1, ; /\

/ )/

,‘d’

i’ii-,

N=160

N=160

4g~,.,~~=u~,I~~,,=,====~====~ ! 11

::: ,..,.»

.......

11_.

S.haUoQ

,

fI_

.... "'.10

"'e.•

II I

10

.... ., ...-

AtJ,al711c.1

""IUaD

x

100

10

I

I I I I I. I

x

Fig. 15. 15. Numerical Numerical solutions solutions for for elastic, elastic, linear linear cylindrical cylindrical geometry. geometry. Fig.

100

175 175

Strain-softening materials materials and and finite-element finite-element solutions solutions Strain-softening

length 1I == length

LlIO. The The results results for for progressively progressively finer finer L/IO. meshes IN (N == IO. 10. 20.10.80. :!O. 40. 80. 160) are are shown shown in in Figs, Figs. meshes 16-19. right columns. columns. For For comparison. comparison. the the solution solution 1619. right was also run for a local continuum (I '= Ir = L/I%’ LIN was also run for a local continuum ii = Ir = variable with with A’) iV) having having the the same same stress-strain stress-strain = variable

relation with with strain strain softening: softening: see see Figs. Figs. 16-19. 16-19. left left relation columns. ItIt is seen seen (Figs. (Figs. 16-19) 16-19) that that the the nonlocal non local columns. solutions with with strain strain softening softening converge converge well well with with solutions N. On On the the other other hand, hand. the the corresponding corresponding increasing N. increasing do not not converge converge at at all. all. local solutions solutions do local

I'~nlocal

N=lO

N=10

~~"i l /

/

'

0

/

,... ,

,

t"

...

-

,

!

~

/

•• !t

3

N=40

(I)

till

~ C'd

.c.:

u

(I)

E

;:I

,. ,.

0

>

3,

.-

.......... ~ ..

.'

-_L

Iv=80 1'1 N N=80 N;"80 = 80

I

I

!

"

--'-'--------"""

N=160

10 10

xX

100 loo

10

xX

100 loo

Fig. 16. local materials 16. Comparison Comparison of of numerical numerical results results for for local and non nonlocal materials with with spherical spherical geometrygeometryvolume volume change. change.

I~h

TEO BEl.YTSUlr;" ('(

lit.

------

- - - - - 50.-

80~----

nonlocal nonlocal

local local

____*. ,’

/ "

_-+-_-

N= 10

N=10

N=20

----'

N-40

N=40 G=

/I

1:I.;: 1. ,.-. ,___-*

/

_;’

,;’

,;’ ,,,’

,,1’

.:’

__=

I

. J

O~.

80,

l N= 160 I

f

r f

I

O~rL-L-~~ 10 10

______~~~~o __~~__~~__=-_~~~ xX

100 100 10 JO

xX

100 10

Fig. local materials Fig. 17. 17. Comparison Comparison of of numerical numerical results results for for local local and and non nonlocal materials with with spherical spherical geometrygeometryradial radial displacement. displacement.

In In Sec. Sec. 3. it was was shown shown that that for for the the uniaxial uniaxial planar solutions converge to planar wave, wave, finite-element finite-element solutions converge to the with mathe exact exact solution solution with aa local local strain-softening strain-softening material. terial. For For spherical spherical waves. waves. no no exact exact local local material material solution with has been solution with strain strain softening softening been found. found. As As

in the the exact exact solution solution for for the the planar planar wave. wave. the the strainstrainsoftening sinsoftening region region appears appears to to localize. localize. producing producing singular gular strains, strains, as can can be seen seen from from the the spikes spikes in Fig. Fig. 16 I6 on on the the left. left. These These singular singular strains strains begin begin to apappear. when pear. as expected. expected. when the the strain strain at the the wavefront wavefront

177

Strain·softening materials materials and and finite-element tinite·element solutions sQlutions Strain-softening

nonlocal \

local

-- --

/

\

-+-,' 0

YJJ .'

211l-~-

I

N=80

N

~\i ,!,:~ J

j

I

=8O

:1

. .-

.........

a 2

IO

xX

100

10

xX

too 100

Fig. 18. 18. Comparison Comparison of of numerical numerical results results for for local local and and nonlocal nonlocal materials materials with with cylindrical cylindrical geometrygeometryFig. volume volume change. change.

reaches reaches the the strain strain at at which which strain strain softening softening begins. begins. In In contrast contrast to to the the planar planar wave, wave. there there is is not not one one point point of of localization localization but but many. many, and and they they appear appear at at difdifferent ferent locations locations for for different different N N (i.e. [i.e. for for different different mesh mesh refinements), re~nements~. and and the the number number of of localization localization

points increases increases with with increasing increasing mesh mesh refinement. refinement. points The impression impression isis that that of of chaos. chaos. ItIt isis anticipated anticipated The that ifif the the loading loading isis aa ramp ramp function function in in time. time, the the that set of of localization localization points points would would be be an an infinitely infinitely set dense set set of of discrete discrete points. points. i.e. i.e. aa Cantor Cantor set. set. The The dense

liX

TED BELYTKHKO BELYTSCHKO ('r £'1 ol. (/1. TED

__.

6 0 - , - - - -______ 60, local local

N = N=

10

10

,

T

nonlocal nonlocal >,,,’

i:,,’

/,’

~

_/’ ,/: ,/:

’

,/’

I

I i --G--

Jr

*. ..2.

N=20

,’ ,’ ,I ;....’ ,,/‘, ,’ ;’ , ,/ ,, ,a .Y’ , >’ .,,I’ // *, / ,,,’ ; - --. _,

N=40

N=80

N=80

N=160

N= 160

/'

Fig. of Fig. 19. 19. Comparison Comparison of numerical numerical results results for for local local and and nonlocal nonlocal materials materials with with cylindrical cylindrical geometrygeometryradial radial displacement. displacement.

179 179

Strain-softening Strain-softening materials materials and and finite-element finire-element solutions solutions

reason of strain-softreason for for the the appearance appearance of multiple multiple strain-softening ening points points (the (the spikes spikes in in the the left left columns columns of of Figs. Figs. 16 and 18) is that part of the strain step at the 16 and 18) is that part of the strain step at the wavewavefront across front is is transmitted transmitted across the the surface surface of of strain strain softsoftening ening before before the the stress stress is is reduced reduced to to zero. zero. The The part part of then of the the wave wave which which has has been been transmitted transmitted then grows grows as converges toward as the the wavefront wavefront converges toward the the center center until until the the elastic elastic limit limit Ep ep is is reached reached again. again. so so the the situation situation repeats repeats itself. itself.

6. 6. CO:"lCLl"SIO:--;S COSCLLWOSS

The are the of The following following the major major conclusions conclusions of the the work: work: I.I. Analytic solutions for Analytic solutions can can be established established for cercertain simple problems which include strain-softening tain simple problems which include strain-softening materials. The materials. The solutions solutions exhibit exhibit singular singular strain strain disdistributions but of tributions but the the rate rate of of convergence convergence of finite-eletinite-element is ment solutions solutions is quite quite rapid. rapid. 2. In numerical In the the spherical spherical wave wave problem. problem, numerical sosolutions models lutions of of strain-softening strain-softening models exhibit exhibit severe severe dedependence pendence on on element element mesh mesh size. size. This This is is particularly particularly true inside true of of field field variables variables inside the the surface surface of of initial initial strain Nonlocal models rapidly strain softening. softening. Nonlocal models provide provide rapidly converging solutions to converging solutions to this this problem. problem. 3. A of A major major difficulty difficulty of local local laws laws with \vith strain strain softening is vanishes. softening is that that the the energy energy dissipation dissipation vanishes. Thus. process is by Thus, the the failure failure process is not not accompanied accompanied by enenwhich physically unrealistic. ergy ergy dissipation. dissipation. which is is physically unrealistic. Nonlocal laws provide aa means 4. 4. Nonlocal laws provide means for for obtaining obtaining rapid and in rapid convergence convergence and finite finite energy energy dissipation dissipation in failure. the for imfailure. However. However, the technology technology for efficiently efficiently implementing these in large-scale. multiplementing these techniques techniques large-scale, multidimensional problems problems remains remains to to be be developed. developed. dimensional Ackl/(}l\"ledgmenlS-We gratefully gratefully acknowledge acknowledge the the supsupAck,ro,c,/rd.~fnrnrs-We port of of Air Air Force Force AFOSR AFOSR Grant Grant 83-0009 lo to Northwestern Northwestern port University. University.

REFERENCES REFERENCES

I. Z. P. Batant Ba:!ant and and L. Cedolin, Cedolin, Fracture Fracture mechanics mechanics of of 1.

2.

3.

4. 4.

5. 5.

6. 6.

reinforced concrete. concrete. J. 1. Engng Engng Mech. Mech. Div. Div. ASCE ASCE 106, reinforced 1287-1306 (1980): (1980); Discussion Discussion and and Closure Closure in 108.464108,4641287-1306 471 (1982): (1982). Z. P. Bafant, Batant, Instability, Instability, ductility ductility and and size size effect effect in Z. strain-softening concrete. concrete. J. 1. Engng Engng Mech. Mech. Div. ASCE ASCE strain-softening 102, EM2. EM2, 331-334 331-334 (1976): (1976); discussion discussion 103. 103, 357-358. 357-358, 775-777 (1977); 104,‘501-502 104,501-502 (1978). 775-777 H. Marchertas, Marchertas, S. H. H. Fistedis, Fistedis, Z. P. Barant, Batant, and and S. H. T. Belytschko, Belytschko, Analysis Analysis and and application application of of prepreT. stressed concrete concrete reactor reactor vessels vessels for for LMFBR LMFBR conconstressed tainment. Nucl. Nucl. Engng Engng Des. Des. 49, 49, 155-174 155-174 (1978). (1978). tainment. Sandler and and J. 1. Wright, Wright, Summary Summary of of strain-softening. strain-softening. I.I. Sandier In Theoretical Theoretical Foundations Foundations for for Large-Scale Large-Scale CompuCompuIn tations ofNonlinearMateria1 ofNonlinear Material Behavior, Behavior, DARPA-NSF DARPA-NSF tations Workshop (Edited (Edited by by S. S. Nemat-Nasser). Nemat-Nasser). KorthwestNorthwestWorkshop ern University University (1984). (1984). em T. Belytschko, Belytschko. Discussion Discussion of of I.1. Sandier Sandler in in Theoretical Theoretical T. Foundations for for Large-Scale Large-Scale Computations Computalions of of NonNonFoundations linear Material Material Behavior, Behavior. DARPA-NSF DARPA-NSF Workshop Workshop linear (Edited by by S. S. Nemat-Nasser), Nemat-Nasser). pp. pp. 285-315. 285-315. NorthNorth(Edited western University University (1984). (1984). western Z. P. P. Bafant Ba:!ant and and T. T. B. B. Belytschko, Belytschko. Wave Wave Propagation Propagation Z. Exact Solution. Solution. Retort Report No. No. in Strain-Softenine Strain-Softening Bar: Bar: Exact in _.

83-IOi401w 83-lOi401w to to D:-OA. DNA. Center Center for for Concrete Concrete and and GeoGeomaterials. materials, Northwestern Northwestern University. University, Evanston. Evanston, llIiIllinois. nois, Oct. Oct. 1983. 1983. Also. Also, 1. J. Engng Engng Mech. .Mech. Div. Div. ASCE. ASCE, lll, 111, 381-389 381-389 (1985). (1985). 7. 7. Z. Z. P. P. Ba:!ant. Batant. T. T. B. B. Belytschko. Belytschko. and and T.-P. T.-P. Chang. Chang. Continuum Continuum theory theory for for strain strain softening. softening. 1. 1. Engng Engng ,Hech. Mech. ASCE ASCE 1l0. 110, 1666-169:! 1666-1692 (1984). (1984). 8. Z. P. P. Bazant. Baiant. Imbricate Imbricate continuum continuum and and its its variational variational 8. Z. derivation. derivation. 1. J. Engng EnanP J[ech. Jfech. ASCE ASCE 1l0, 110. 1593-1712 1593-1712 (1984). (1984). 9. E. Kroner. Kraner, Elasticity Elasticity theory theory of of materials materials with with long long 9. E. range range cohesive cohesive forces. forces. Inl. Int. 1. J. Solids Solids Slruct. Srruct. 3,7313, 731742 742 (1967). (1967). 10. 10. 1. J. A. A. Krumhansl. Krumhansl. Some Some Considerations Considerations of of the the Relation Relation Between Between Solid Solid State State Physics Physics and and Generalized Generalized ContinContinuum Mechanics. of Continua Mechanics. Mechanics Mechanics of Generalized Generalized Conrinua (Edited (Edited by by E. E. Kroner). Krbner), pp. pp. 198-311. 198-31 I. Springer-Verlag. Springer-Verlag, Berlin Berlin (1968). (1968). II. I.I. A. Kunin. Kunin, The Theory Theory of of Elastic Elastic Media LMediawith Microstructure crostructure and and the Theory Theory of of Dislocations. Dislocations. MeMechanics chanics of of Generalized Generalized Continua Continua (Edited (Edited by E. Kroner). Kriiner), pp. 321-328. 321-328. Springer-Verlag. Springer-Verlag, Berlin Berlin (1968). (1968). 12. 12. A. C. Eringen. Eringen, Nonlocal Nonlocal polar polar elastic elastic continua. continua. Int. 1. J. Engng Engng Sci. 10, 10, 1-16 I-16 (1972). (1972). 13. 13. A. A. C. C. Eringen Eringen and and D. D. G. G. B. B. Edelen. Edelen, On On nonlocal nonlocal elasticity. ticity. Int. 1. J. Engng Engng Sci. Sci. 10,233-248 10, 233-248 (1972). (1972). 14. 14. A. C. Eringen Eringen and N. Ari. Ari, Nonlocal Nonlocal stress stress field at Griffith Mal. Griffith crack. crack. CrySI. Cyst. Latt. Latt. Def. Def. Amorph. Amorph. Mat. 10,3310,3338 38 (1983). (1983). 15. 15. D. C. B. Edelen. Edelen. Non-local Non-local variational variational mechanics mechanics 1IStationarity Stationarity conditions conditions with one unknown. unknown. Inl. Int. 1. J. Engng Engng Sci. Sci. 7, 7, 269-285 269-285 (1969). (1969). 16. 16. Z. P. Ba:!ant Barant and T.-P. T.-P. Chang. Chang, Nonlocal Nonlocal Continuum Continuum and Strain-Averaging Strain-Averaging Rules. Rules, Report Report No. 83-11140Ii. 83-I 1/4OIi. Center Northwestern Center for Concrete Concrete and Geomaterials. Geomaterials, Northwestern University. Nov. 1983; University, Evanston. Evanston, Illinois. Illinois, Nov. 1983; also also 1. J. Engng Mech. Mech. ASCE ASCE (in press). press). Engng 17. Le~ons sur sur la Ja propagation propagalion des 17. 1. J. Hadamard, Hadamard, Leqons des andes, ondes, Chapt. Chapt. VI. Hermann Hermann et cie. tie, Paris Paris (1903). (1903). 18. 18. G. 1. I. Taylor. Taylor, The The testing testing of of materials materials at high rates rates of of loading. Insl. Civ. Engng Engng 26, (1946). loading. 1. J. Insr. 26, 486-519 486-519 (1946). 19. and L. L. B. Freund, Freund. Deformation Deformation trapping trapping 19. F. H. Wu and due to thermoplastic thermoplastic instability instability in one-dimensional one-dimensional due wave propagation. propagation. 1. Mech. Phys. Phys. Solids Solids 32(2), 32(2), 119wave J. Mech. 119132 (1984). (1984). Achenbach. Wave Propagation Propagalion in in Elastic Elastic Solids. Solids. 20. 1. J. D. Achenbach, North-Holland. Amsterdam Amsterdam (1973). North-Holland, I

I

APPENOIX A

Numerical rrlgorithm (lIRorilhm for for sphc~ricul spherical an~c,vlindricctl lIns{cylindric(l1 NY,~‘~.s "'l/I'es Ntrmrricrrl (I) Read Read (I. (I. h. h. 1. J. n. n. It ~I (time (time step). step), N,.. N c • N,.. N, .. NI. N,. N,(number N, (number (I) of local local elements, elements. imbricate imbricate elements. elements. nodes, nodes. and and time time of steps. respectively), respectively). y. 'Y. and and po. Po. Generate Generate arrays arrays l(i). k(i). k(i). Iil(i). steps, le(i). x,.(i). x,.(i). I(i). k(i). Z(i). Iii(i). i,.(i), I,.(i). T,.(i) x,.(i) giving giving the the number number of of the the I,(i). of the the ith ith local local or or imbricate imbricate element, element. left and and right right nodes nodes of left its length. length. and and its its coordinate coordinate at the the center center ofelement. of element. Also Also its generate externally externally applied applied nodal nodal forces forces ft.::. Initialize Initialize as as generate zero the the values values (for (for r = I) of of ill. 1',. kI. fl.,. ;rl. ;". T,~. ;". v,~. a". max max Q. E,. zero max fl. t,. T,,.~ Tmax *.A. d . urn.,< am .. , c.l d forall for all k == I. I. .... (local . . N,. (local max elements) and and k = I. I ...... Iv,. Nc (imbricate (imbricate elements). elements). elements) (2) DO DO 8. 8. rr == 2. 2 . . . . .. N,. N,. (2) (3) Initialize Initialize nodal nodal forces forces FL F, == 0. O. flf, == 00 (for (for all all nodes, nodes. (3) Nd. !ik == I.I ....., Nk). (4) DO DO 55 ii =_ = I. I . . . . ., N,. N .. (local (local elements). elements). (4) Set kk == k(i), k(i). m m == G(i). Ifl(i). xx == x,.(i) x,.(i) and and evaluate evaluate 1c,i ~E,; == (5) Set (5) (i'm -- I%)~frill. t'd~llh. E,, Exi 6 - - E,, Exi + + AE,, ~E'i AE,, :lE,·i == l&,, (t'm + + 1’1) t'd ~rtl2.r. ~1/2x. (v.,, E,., +-r,.; -- E,.; ++ SE,.. :lE,.. For For spherical spherical wave wave l\e, ~E, = = AC,, ~E.'i + + XE,~. 2~E,;. Ed, and for for cylindrical cylindrical wave wave AC, ~Ei +