iW.(--9JY Yh S: 00 - W C I986 Peredmon PKSS Lx.

STRAIN-SOFTENING MATERIALS AND FINITE-ELEMENT SOLUTIONS

TED BELYTSCHKO. ZDESEK P. BA~.GI-. YCL-Woosc HYL~. and T.A-PESG CHASG Department of Civil Engineering. Northwestern University. Evanston, IL 60201. U.S.A.

,\hstrsct-Closed form and tinite-element solutions are examined for \rveral problem\ \bith strain- softening materials. In the closed form solutions, strain-softening causes localization of the strain v.hich is accompanied bb an instantaneous vanishing of the stre(h. The tinite-element solutions agree closely with analytic \oIution\ in many C;LSC~ and exhibit a rate of convergence only slightly belo% that for linear problem<. The main difficulty which has been idsntitiecl in strain-xjftsning con\titutive models for damage is the absence of energy dissipation in the strain-softening dumain. and this can be corrected by II nonlocal formulation. Finite-element solution\ for the converging spherical \\ave problem exhibit multiple points of localization which change dramatically ttith ms\h retinement. With a nonlocal material formulation. thi\ pathology i\ eliminated.

I. INTRODCCTION

In materials such as concrete or rock. failure occurs by progressive damage which is manifested by phe- nomena such as microcracking and void formation. In most engineering structures. the scale of these phenomena. as compared to the scale of practical finite-element meshes. is usually too small to be modeled and their effect must be incorporated in the numerical analysis through a homogenized model which exhibits strain softening.

Strain softening. unfortunately. when incorpo- rated in a computational model. exhibits undesir- able characteristics. In static problems. finite-ele- ment solutions with strain softening often exhibit a severe dependence on element mesh size because of the inability of the mesh to adequately reproduce the localization of strain which characterizes static strain softening[ I]. Furthermore. the solutions are physically inappropriate in that with increasing

mesh refinement the energy dissipated in the strain- softening domain tends to zero[lJ.

It was first hoped. although in retrospect little practical evidence existed for this optimism. that in dynamic problems, strain softening would not be as troublesome because the inertia of the continuum would alleviate the instability. Support for this can be found in the snapthrough of an arch; in this prob- lem the load-detlection curve contains a limit point after which the force-deflection curve is negative. or softens. A static solution for the snapthrough is often very difficult. whereas a dynamic solution is relatively straightforward because the inertia of the structure alleviates some of the difficulties intro- duced by the negative slope in the force-detlection relation. The use of strain-softening models has be- come quite commonplace in dynamic concrete an- alysis. For example, in Marchertas cut tr/.[3]. strain softening appeared to reproduce the salient features of dynamic response of a concrete reactor vessel even after severe failure had developed. In the com-

munity as a whole. a certain complacency evolved and except for the work of Baiant and colleagues[l,

11. little effort was devoted to examining the basic soundness of numerical solutions with strain-soft- ening materials.

Attention was recently focused on the validity of strain-softening models by the work of Sandier and Wright[4]: they asserted that strain-softening models are basically ill posed because a small dif- ference in load results in large changes in the re- sponse. Sandler’s example. which will be described in more detail later, consists of a one-dimensional rod with the velocity prescribed at one end in which the material strain softens. By increasing the load slightly, a significantly different response was ob- tained for the problem. Sandler and Wright also noted a strong dependence of the solution on the mesh size. They concluded that “a rate-indepen- dent dynamic continuum representation of strain softening is incapable of reproducing softening be- havior in a dynamic simulation of experiments” and then proceeded to show that in this problem the introduction of viscosity eliminates the sensitivity of the response to the load. Incidentally. as will be shown later in this paper, viscous damping is not a panacea for the sensitivity observed in strain-soft- ening solutions: in certain problems which will be described herein. sensitivity to mesh size persists even after the introduction of damping.

In an effort to develop a problem with strain soft- ening in which the localization does not occur on the boundary, we investigated two problems: one, which was presented in Ref. [S]. consists of a linear elastic barjoined to a strain-softening bar. Solutions for this bar were obtained by the method of images and compared to finite-element solutions. These re- sults exhibited convergence with decreasing ele- ment length. BaZant and Belytschko[61 constructed a more interesting problem in which tensile waves are initiated at two ends of a bar so that strain soft- ening occurs at the center. It was shown that a so-

Cumpllfen & St"tlutmtS Vol :3. So. :. ~p \63-180. 19t\6 Pnnted In Gr!!at Bntain

ooJ5- -9J9 So 53.00 - .00 <: 1986 Pergamon Pre" Ltd.

STRAIN-SOFTENING MATERIALS AND FINITE-ELEMENT SOLUTIONS

TED BELYTSCHKO. ZDESEK P. BAZAsT. YCL-WOOSG Hycs. and T~-PESG CHASG Department of Civil Engineering. i'orthwestern University. Evanston. IL 6020 l. C.S.A.

Ahstract-Closed form and tinite-element solutions are e\amined for s~veral problems \\ ith '>train­softening materials. In the closed form solutions. strain-softening causes localization of the strain which is accompanied by an instantant:ous vanishing of the stress. The tlnite-element solutions agree doselv with analvtic 'iolutions in manY cases and e\hibit a rate of convergence lmlv slightlv below that f~r linear pr~blems. The main dift~culty which has been identitied in st;ain-softe~ing ~on~titutive models for damage is the absence of energy dissipation in the strain-softening domain. and this C,1n

be corrected by a non local formulation. Finite-element solutions for the converging 'ipherical wave problem exhibit multiple points of localization which change dramatically with me'ih refinement. With a non local material formulation. this rathology is eliminated.

I. J:-iTROOl'CTlO:-i

In materials such as concrete or rock. failure occurs by progressive damage which is manifested by phe­nomena such as microcracking ami void formation. In most engineering structures. the scale of these phenomena. as compared to the scale of practical finite-element meshes. is usually too small to be modeled and their effect must be incorporated in the numerical analysis through a homogenized model which exhibits strain softening.

Strain softening. unfortunately. when incorpo­rated in a computational modei. exhibits undesir­able characteristics. In static problems. finite-ele­ment solutions with strain softening often exhibit a severe dependence on element mesh size because of the inability of the mesh to adequately reproduce the localization of strain which characterizes static strain softeningf II. Furthermore. the solutions are physically inappropriate in that with increasing mesh refinement the energy dissipated in the strain­softening domain tends to zerof2J.

It was first hoped. although in retrospect little practical evidence existed for this optimism. that in dynamic problems. strain softening would not be as troublesome because the inertia of the continuum would alleviate the instability. Support for this can be found in the snapthrough of an arch; in this prob­lem the load-deflection curve contains a limit point after which the force-deflection curve is negative. or softens. A static solution for the snapthrough is often very difficult. whereas a dynamic solution is relatively straightforward because the inertia of the structure alleviates some of the difficulties intro­duced by the negative slope in the force-deflection relation. The use of strain-softening models has be­come quite commonplace in dynamic concrete an­alysis. For example. in Marchertas I!t al.f31. strain softening appeared to reproduce the salient features of dynamic response of a concrete reactor vessel even after severe failure had developed. In the com-

munity as a whole. a certain complacency evolved and except for the work of Bazant and colleagues[ I. n little effort was devoted to examining the basic soundness of numerical solutions with strain-soft­ening materials.

Attention was recently focused on the validity of strain-softening models by the work of Sandler and Wrightf4J; they asserted that strain-softening models are basically ill posed because a small dif­ference in load results in large changes in the re­sponse. Sandler's example. which will be described in more detail later, consists of a one-dimensional rod with the velocity prescribed at one end in which the material strain softens. By increasing the load slightly. a significantly different response was ob­tained for the problem. Sandler and Wright also noted a strong dependence of the solution on the mesh size. They concluded that "a rate-indepen­dent dynamic continuum representation of strain softening is incapable of reproducing softening be­havior in a dynamic simulation of experiments" and then proceeded to show that in this problem the introduction of viscosity eliminates the sensitivity of the response to the load. Incidentally. as will be shown later in this paper. viscous damping is not a panacea for the sensitivity observed in strain-soft­ening ~olutions: in certain problems which will be described herein. sensitivity to mesh size persists even after the introduction of damping.

In an effort to develop a problem with strain soft­ening in which the localization does not occur on the boundary, we investigated two problems: one, which was presented in Ref. [5 J. consists of a linear elastic bar joined to a strain-softening bar. Solutions for this bar were obtained by the method of images and compared to finite-element solutions. These re­sults exhibited convergence with decreasing ele­ment length. Bazant and Belytschko[6j constructed a more interesting problem in which tensile waves are initiated at two ends of a bar so that strain soft­ening occurs at the center. It was shown (hat a so-

lution to this problem exists but that the behavior

of the strain-softening domain is rather unusual: the

strain softening localizes to an infinitely thin do-

main. a set of measure zero. in u hich the strain

becomes instantaneously infinite and in which the

energy dissipation is zero.

In order to remedy some of these undesirable

features of strain-softening solutions. Baiant. Be-

Ivtschko. and co-workers[7.8] proposed a new non-

local formulation for treating strain softening. Non-

local theories have been introduced by Kroner[9].

Krumhansl[ IO] and Kunin[ I I], and developed fur-

ther by Eringen and co-workers[l?-l-l] and Ede-

len]lS]. The basic ingredient of a nonlocal theory

is that the stress-strain relation is not considered to

be local but relates the stress at an)- point to the

state of deformation within a finite volume about

that point. In this respect. the theory lends itself

admirably to problems of heterogeneous media.

where the representation of the microscopic detail

of strain fields and cracks is an insurmountable

task. By dealing with an average of the strain over

a finite domain about each point. the heterogeneity

can be neglected. and the dispersion which occurs

in inhomogeneous materials can be modeled vvith-

out any artifices. It should be noted that in the tran-

sition from a microstructural theory to a macro-

theory. strains and stresses are often averaged

over finite volumes. Hovvever. this averaging is

only considered in tl~~~~/r~pi/rp the macroconstitutive

equation. In the solution of the governing equa-

tions. the constitutive equations are considered to

apply only to points so the theory is local and the

localization limiting property of nonlocal theories,

i.e. the attribute of the theory which prevents lo-

calization to a set of measure zero. is then lost and

the energy dissipation associated with strain soft-

ening vanishes.

An obvious question which arises is why one

would want to introduce this complication in order

to deal with strain softening. The reason for this is

that when the constitutive equations are applied lo-

cally at points. then. as will be described here. no

dissipation of energy occurs in the strain-softening

domain because it is a set of measure zero. Thus

the material can fail without any permanent dissi-

pation of energy. which is physically quite unreal-

istic. By introducing a nonlocal character into the

constitutive law. it is possible to restrict the local-

ization to a domain of finite size just as is observed

experimentally. and to achieve a finite amount of

energy dissipation in the strain-softening domain.

However. we found we could not simply estend

the existing nonlocal models to account for strain

softeningl7. 161. The existing nonlocal laws are not

even self-adjoint. so they did not lead to symmetric

stiffness matrices. This lack ofsymmetry~ was found

to be undesirable and was corrected by introducing

the same averaging operation over the stresses as

that for the strains. Such double averaging is in fact

required by a consistent application of the v,aria-

tional principls[8]. It was ;dso found that the strxin

softening could only be introduced in the nonlocal

law in a subtle t’dshion. necessitating a split of the

constitutive equation into 2 local and nonlocal 12~.

w.ith the strain softening included only, in the non-

local portion. Numerical experiments indicated that

without this particular combination. numerical ho-

lutions were often unstable.

The nonlocal law as introduced in Kct’s. 17. 81

offers substantial promise in pro\ iding well-posed

solutions for heterogeneous materials that are sub-

jected to damage and hence strain softening. There

are hovvever. substantial breakthroughs that yet

need to be achieved: (1) efficient implementations

of nonlocal laws in the finite-element method: t2)

design of experimental methods for identifying the

local and nonlocal portions of constitutivc laws:

and (3) methods for reconciling the bifurcation be-

tween local damage. i.e. microcracking. and large-

scale fracture of a cleavage type in heterogeneous

materials. However. the work reported here has

shed light on the questions of numerical modeling

of structures in the failure regime when strain soft-

ening takes place and provides the basis for future

work.

We have organized the material as follows: in

Sec. 2 we describe several of the generic onc-di-

mensional problems which can be used to examine

the mathematical character of dynamic strain-soft-

ening solutions. In Sec. 3, the nonlocal continuum

law will be described. In Sec. 4. some finite-element

solutions are presented for planar problems that in-

dicate that the solutions converge to the analytic

solution: however. only for a nonlocal law can finite

energy dissipation be achieved in a strain-softening

domain. In Sec. 5. local and nonlocal solutions will

be given for the converging spherical and cylindri-

cal wave. followed by conclusions in Sec. 6.

The problem by Sandier and Wright[4] is shown

in Fig. I. The essence of their argument was that

the solutions are very sensitive to the constant qi.

which represents the maximum prescribed velocity

at the left-hand boundary. and that the solution

changes markedly and so does not appear to con-

verge as the mesh is refined. Although the Sandkr-

Wright stress-strain law is nice from the viewpoint

“0 w 1-x

lution to thi, probkm eXi'its but that the behavior of the strain-softening domain is rather unusual: the ,train softening localize~ to an infinitely thin do­main. a set of measure zero. in which the strain becomes instantaneuusly infinite aOll in which the energy dissipation is zero.

In urder to remedy some of these undesirable features of strain-softening solutions. Bazant. Be­lytschko. and co-workers[7. 8] proposed a new non­local formulation for treating strain softening. Non­local theories have been introduced by Krbner[9]. Krumhansl[lO] and Kunin[ll], and developed fur­ther by Eringen and co-workers[ 12-14] and Ede­len[ 151. The basic ingredient uf a nonlucal theory is that the stress-strain relatiun is not considered to be local but relates the stress at any point to the state of deformatiun within a finite vulume about that point. In this respect. the theury lends itself admirably to problems of heterogeneou'i media. where the representatiun of the microscupic detail of strain fields and cracks is an insurmountabk task. By dealing with an average of the strain uver a finite dumain abuut each puint. the heterogeneity can be neglected. and the dispersiun which uccurs in inhomugeneuus materials can be modeled with­uut any artifices. It should be nuted that in the tran­sition from a microstructural theury tu a macro­theury. strains and stresses are often averaged over finite vulumes. However. this averaging is unly cunsidered in del'el()pillg the macrocunstitutive equation. In the sulutiun uf the governing equa­tiuns. the constitutive equations are considered to apply only to puints su the theory is local and the localization limiting property uf nunlocal theories. i.e. the attribute uf the theory which prevents lo­calization to a set of measure zero. is then lost and the energy dissipation assuciated with strain suft­ening vanishes.

An ubvious question which arises is why une would want tu introduce this cumplication in order to deal w'ith strain suftening. The reason for this is that when the constitutive equatiuns are applied lo­cally at points. then. as will be described here. no dissipatiun of energy occurs in the strain-softening domain because it is a set of measure zero. Thus the material can fail without any permanent dissi­pation of energy. which is physically quite unreal­istic. By introducing a nonlucal character into the constitutive law. it is possible tu restrict the lucal­ization to a domain of tinite size just as is ubserved experimentally. and to achieve a finite amount of energy dissipatiun in the strain-softening domain.

However. we found we could not simply extend the existing nonlocal models to account for strain softeningl7. 161. The existing nonlocallaws are not even self-adjuint. su they did not lead to symmetric stiffness matrices. This lack ofsymmt:try was found to be undesirable and was corrected by introducing the same averaging operation over the stresses as that for the strains. Such double averaging is in fact required by a consistent application of the varia-

tional principle[81. It \\a-; ,ti,o fLlund that the ,tl',lin softening could only be introduced in the nllOiL1cai law in a subtle fashion. neces,itating a split llf the constitutive equation into a local and nonlocal la\\. with the strain suftening included only in the non­lucal portion. Numerical experiments indicated that without this particular combinatiLln. numerical so­lutiuns were often unstable.

The nonlucal law as introduced in Refs. 17. 81 offers substantial promise in providing well-posed sulutions for heterogeneous material-; that are sub­jected to damage and hence strain softening. There are however. substantial breakthmughs that yet need to be achieved: (I) efficient implementatilln, of nonlocal laws in the finite-element method: 12) design of experimental method, for identifying the local and non local portions of conqitutive I,m,: and (3) methods for reconciling the bifurcation be­tween local damage. i.e. microcracking. and large­scale fracture of a cleavage type in heterogeneous materials. However. the work reported here has shed light un the questions of numel'ical mmkling of structures in the failure regime when strain 'ioft­ening takes place and provides the basis for future wurk.

We have organized the material a'i follows: in Sec. 2 we describe several of the generic one-di­mensional problems which can be used to examine the mathematical character of dynamic strain-soft­ening solutions. In Sec. 3. the non local continuum law will be described. In Sec. 4. some finite-element solutions are presented for planar problems that in­dicate that the solutions converge to the analytic solution; however. only fora nunlocallaw can tinite energy dissipatiun be achieved in a strain-suftening domain. In Sec. 5. local and nonlocal solutions will be given for the converging spherical and cylindri­cal wave. followed by conclusions in Sec. 6.

2. A"iALYTlCAL SOLlTIOl'i FOR STR.\I'·SOFTE"iI:\G

:\IATERIALS

The problem by Sandler and Wright[4] is shown in Fig. I. The essence of their argument was that the solutiuns are very sensitive to the cunstant "". which represents the maximum prescribed velocity at the left-hand boundary. and that the solution changes markedly and so does not appear to con­verge as the mesh is refined. Although the Sandler­Wright stress-strain law is nice from the viewpoint

Fig. I. Samller-Wrightl-ll problem: the ,Ire,,-,lrain 1.,\\ is IT = E" E exp( - E!E"I. E" = O.O(j2. 11 ... 111. II = ~ ;',,11 -cos(;,{/{"iI for { < {". lI.riO. {I = ;'" t'ur { < {". {" = 0.2

m,.

that it provid2s a continuous relationship between stress and strain in the loading domain. it is not amenable to any attempts at a closed form solution by d’.Ilembert mzthods because of the dispersive charactsr of the wave solution even in the loading range. For this r2ason. w2 have limited our studies to piecewise linear prescribed velocities or stresses and stress-strain laws of the type shown in Fig. 2: this will be called th2 modified Sandler-Wright problem. Note that the stress goes to zero as the strain becomes large on the strain-softening sides of the law tusually~ the tensile side).

The analytical solutions to this problem are de- velop2d next. The salient characteristic of the ana- lytical solution is the appearance ofan infinite strain on the boundary once the strain E,, is exceeded. The construction of the solution for this case will follow the approach presented by Bafant and Be- lytschko]6] for a similar problem. As will be seen. when strain softening occurs. then the strain im- mediatrly localizes and reaches infinity within a time interval that approaches zero. Therefore. the solution can be generated by adding an image wave which cancels the incident wave so that the strain- softening point is instantaneously converted to a free boundary.

The governing equations can be stated as fol- lows:

0.8 = pi/.,,. (‘.I)

rJ.r = E(E)//.,, = E(Ek.,. (7.‘) - -

where u and E are the stress and strain. II the dis- placement. and subscripts preceded by commas denote differentiation: P is the density and E the tangent modulus. We will consider two types of boundary conditions on the left-hand side, .r = 0:

velocity condition: I/.,(0. 1) = -V,,(1). (2.3)

traction condition: ato. t) = U,)(f). (2.4)

where (f) designates fH(f). H being the Heaviside step function. The velocity boundary condition will be considered first. The right-hand boundary is as-

Fig. 2. Stress-strain law nith strain softening showing no- menclature.

sumed to be sufficiently far so that the rod can be considered semi-infinite.

Not2 that prior to the onset of strain softening. the problem is governed by the standard one-di- mensional vvave equation

I II.,, = 7 II.,,. 12.5)

c’-

where

(2 = t‘ P .

(2.6)

Once the strain-softening regime of the material is attained. then at those points the governing equa- tion is

_T

(‘- II.,, + II.,, = 0. (2.7)

E (-2 = _ -

P (2.8)

and T: vanishes once E,, is attained. Equation (2.7) is elliptic in space-time, which is peculiar in that information can be transmitted at infinite speeds. Hadamard[ 171 commented on this in 1903 and he claimed that the negative character of the square of the wave speed precluded its applicability to real materials since the wave speed would then be im- aginary. However, the case of c = 0 has been treated extensively by Taylor[l8], who noted that for perfectly plastic solids the deformation is lo- calized at the point of impact. Wu and Freund[l9] have recently presented a lucid description of these localization phenomena and investigated the effects of strain-rat2 sensitivity and heat transfer on the localization. However. the analyses were limited to the case where in the limit c = 7 = 0.

We will here consider the strain-softening situ- ation using the concepts developed in 161. The present situation differs from [6] in that the stress wave is a ramp rather than a step wave. but it will be found that all of the singularities associated with a step input remain.

The procedure of constructing a solution con- sists of three steps:

(I)

12)

(3)

It is shown that the boundary between the strain-softening and elastic domain cannot move, so the strain-softening domain is limited to a point. This is shown to imply that the strain and strain rate in the strain-softening points must be in- finite. Since the strain rate is infinite for the class of materials considered here. in which u - 0 as E - x. the stress can be considered to vanish instantaneously at the points which strain soften.

The last conclusion enables the solution to be easily constructed by the d’Alembert method by

Strain·,oftening materials and linite·element ,olutilln'i 1('5

that it provides a continuous relationship between stress and strain in the loading domain. it is not amenable to any attempts at a closed form solution by d' :1.lembert methods because of the dispersive character of the wave solution even in the loading range. For this reason. we have limited our studies to piecewise linear prescribed velocities or stresses and stress-strain laws of the type shown in Fig. 2: this will be called the moditied Sandler-Wright problem. Note that the stress goes to zero as the strain becomes large on the strain-softening sides of the law (usually the tensile side!.

The analytical solutions to this problem are de· veloped next. The salient characteristic of the ana­lytical solution is the appearance of an intinite strain on the boundary once the strain E" is exceeded. The construction of the solution for this case will follow' the approach presented by Bazant and Be­lytschkof6] for a similar problem. As will be seen. when strain softening occurs. then the strain im· mediately localizes and reaches intinity within a time interval that approaches zero. Therefore. the solution can be generated by adding an image wave which cancels the incident wave so that the strain· softening point is instantaneously converted to a free boundary.

The governing equations can be stated as fol­lows:

u .. plI.". 12.1 )

12.2)

where a and E are the stress and strain. II the dis­placement. and subscripts preceded by commas denote differentiation: p is the density and E the tangent modulus. We will consider two types of boundary conditions on the left-hand side . .r = 0:

velocity condition:

traction condition:

11.,(0. t) = -1',,(1). (2.3)

a(O. t) = a,,(r}. (2.4)

where <f) designates f HU). H being the Heaviside step function. The velocity boundary condition will be considered first. The right-hand boundary is as-

(J

Fig. 2. Stress·strain law with strain softening showing no· menclature.

sumed to be sufticiently far so that the rod can be considered semi-intinite.

Note that prior to the onset of strain softening. the problem is governed by the standard one-di­mensional wave equation

H' II

where

E p

12.5)

(2.6)

Once the strain-softening regime of the material is attained. then at those points the governing equa­tion is

E Z~ = -

p

(2.7)

(2.8)

and Z vanishes once E" is attained. Equation 12.7) is elliptic in space·time. which is peculiar in that information can be transmitted at infinite speeds. Hadamard! 17] commented on this in 1903 and he claimed that the negative character of the square of the wave speed precluded its applicability to real materials since the wave speed would then be im­aginary. However. the case of c = 0 has been treated extensively by Taylor[ 18], who noted that for perfectly plastic solids the deformation is lo­calized at the point of impact. Wu and Freund[19] have recently presented a lucid description of these localization phenomena and investigated the effects of strain-rate sensitivity and heat transfer on the localization. However. the analyses were limited to the case where in the limit c = Z = O.

We will here consider the strain-softening situ­ation using the concepts developed in [6]. The present situation differs from [6J in that the stress wave is a ramp rather than a step wave. but it wili be found that ali of the singularities associated with a step input remain.

The procedure of constructing a solution con­sists of three steps: 1 I) It is shown that the boundary between the

strain-softening and elastic domain cannot move. so the strain-softening domain is limited to a point.

(2) This is shown to imply that the strain and strain rate in the strain-softening points must be in­tinite.

(3) Since the strain rate is intinite for the class of materials considered here. in which a ~ 0 as E -- :c. the stress can be considered to vanish instantaneously at the points which strain soften.

The last conclusion enables the solution to be easily constructed by the d'Alembert method by

simply adding a wave to satisfy the zero stress con- the displacements and stress conditions across the

dition. interface. From eqn (2. IJal. it follows that

For the prescribed velocity problem. let I, be the

time when the left-hand end. .V = 0. reaches E,, and

begin5 to strain soften: I, is given by

it0 e- = z = (itI - f,) f E,,. (2.17,

If the two displacement solutions. eqns (2.1-k)

and Q.15). are now matched across the interface

.Y = s. then

hi,, I, = -

i’,l

and the solution prior to the onset ofstrain softening

is given b),

Strain softening first occurs at .V = 0. We now

show that the boL]nd~tr~ between the elastic and the

softening interface cannot move. For this purpose.

the usual formula for velocity V ot’ discontinuities

is used tn development is given in [6l):

U- - <r- = p L%* - EC). (2.1’)

where the superscripts + and - designate the state

variables to the right and left of the discontinuity,

respectively. If the material is strain softening be-

hind the interface and not yet before it, it follows

that E- > E- and u* 2 CT-. Substituting these in-

equalities into eqn (2.12). it follows that V’ must be

negative or zero: since the former assumpt;on

would yield an imaginary velocity for the discon-

tinuity. only V = 0 is tenable. and it can already

be concluded that

ir* = CT-. (2.13)

To show that the strain and strain rates must be

infinite at a point which strain softens. a solution

is constructed in the strain-softening domain which

is considered to be U 4 A G s where .s - 0. It can

be seen that

,,* = - y ((,, - $) (2.14b)

satisfies the governing equation in the strain-soft-

ening domain. 12.7). This solution. t2.14). is now

matched to a solution in the elastic domain

where the second term is a wave emanating in the

strain-softening region w,hich w?ll be used to match

II” + It/( f - II ) + E,,j.T

= - T ((t - $) f f(E). i7.i&l)

(2.18b)

Eliminating (I from eqns (7.17) and 12.18) yields

It can be seen that ass-+ 0. E- -+ 0 instantaneously.

which through eqn (2.13) implies uc = 0. The func-

tion f(Q is then found from this condition. Using

the displacement field of eqn f2.15) and letting tr+.

and hence E + , vanish. we find

f’(~, = E,,L’ H (1 - I, - ;)

f LI~(~ - ti - $, (2.20af

f(S) = E,‘(’ (r - II - ;>

Hence the complete solution is

This solution will subsequently be compared to fi-

nite-element solutions.

The solution for the stress boundary condition.

eqn (2.4). can be found by replacing z’(, by u,,ciE.

However. in the prescribed stress form of this prob-

lem. eqn (7.4). the introduction of the image at the

strain-softening point poses a difficulty since the

first point to strain soften is initially on the bound-

ary. Thus. in one sense it can be said that this

1f1f1

simply adding a waVe to satisfy the zero stress con­dition.

For the prescribed velocity problem. let (I be the time when the left-hand end . .r = n. reaches fir' and begin-; to strain soften: II is given by

(2.9) I.'p

and the solution prior to the onset of strain softening

the displacements and stress conditions across the interface. From eqn 12.14a). it follows that

E (2.17)

If the two displacement solutions. eqns (2.14a) and (2.15). are now matched across the interface :c = s. then

is given by 1/* + [a( t tl) + Er,jS

(2.10)

(2.11 )

Strain softening first occurs at.r = n. We now show that the boundary between the elastic and the softening interface cannot move. For this purpose. the lIslIal formula for vdocity V of discontinuities is lIsed (a development is given in [61):

(2.12)

where the superscripts + and designate the state variables to the right and left of the discontinuity. respectively. If the material is strain softening be­hind the interface and not yet before it. it follows that E - > E - and u - ? (J - • Substituting these in­equalities into eqn L~.12). it follows that V~ must be negative or zero: since the former assumption would yield an imaginary velocity for the discon­tinuity. only V = 0 is tenable. and it can already be concluded that

(2.13)

To show that the strain and strain rates must be intinite at a point which strain softens. a solution is constructed in the strain-softening domain which is considered to be 0 ~ x ~ s where 5 -- O. It can be seen that

II =: /1* + [a(t (2.14a)

l(* I'll ~( x)~) - - tl - -2 c

(2.14b)

satisfies the governing equation in the strain-soft­ening domain. (2.7l. This solution. 12.14). is now matched to a solution in the elastic domain

X I:.=I---tl' - C

(2.15)

(2.16)

where the second term is a wave emanating in the strain-softening region which will be lIsed to match

~ ~(t n) + f(~). (2.ISa)

s (2.ISb)

('

Eliminating (/ from eqns (2.17) and C.IS) yields

I [ . - ;'" ~ .1') < J E- = J(~) - - t - - - fI~ . s 2 c

(2.19)

It can be seen that as J -- O. E - ~ 0 instantaneously. which through eqn (2.13) implies u" = O. The func­tion f(~) is then found from this condition. Using the displacement field of eqn (2.15) and letting u" .

and hence E + • vanish. we find

E>CH(t - tl -::) I ('

+ Vo ~t - tl - ~). (2.20a)

E,C/t - tl -::) I \ ('

"0 ~« +- t ., tl - D). 12.20b)

Hence the complete solution is

II = - ¥ ~t - ~) + EpC ~t - 11 - ~)

+~~(t tl-D) (2.2Ia)

-~) + EpC H (t - tl - D + ;'u ~t - tl - ~). (2.2Ib)

This solution will subsequently be compared to fi­nite-element solutions.

The solution for the stress boundary condition. eqn (2.4). can be found by replacing 1'0 by UnclE. However. in the prescribed stress form of this prob­lem. eqn (2.4). the introduction of the image at the strain-softening point poses a difticulty since the first point to strain soften is initially on the bound­ary. Thus. in one sense it can be said that this

Strain-softening material5 and finite-element wIution4 167

boundary must satisfy two different boundary con- softening instantaneously and no wave should

ditions: eqn (3.4) and cr = 0. reach the interior.

This contradiction can only be reconciled by re-

quiring the second boundary condition (that the stress vanishes) to take precedence. In fact. the first boundary condition would be impossible to enforce in an experiment. The notion of a boundary con- dition depending on the result of the solution is not totally unexpected in an analysis of a continuum which fails. For example. in a buckling problem with unstable postbuckling behavior, the pre- scribed force would also be limited by the capacity of the structure. Yet the situation in the buckling problem is not completely analogous: in a dynamic buckling problem. any force may be prescribed and the excess force will generate accelerations. which depend on the magnitude of the force, whereas in this problem, the solution is completely indepen- dent of the value of the prescribed stress once the failure stress is exceeded. Nevertheless, this model does appear to represent a physically meaningful situation: the behavior of a rod in which the material can sustain a limited tensile strain before it fails,

and the solution appears reasonable.

(2) The solution does not depend on the specific functional dependence of stress on strain in the strain-softening portion if the stress vanishes as the strain becomes large.

One conceptual difficulty of the Sandler-Wright problem is that strain softening occurs only at the boundary, which confuses the role of the boundary condition and the strain softening. For this reason we have attempted to construct problems in which strain softening occurs within the domain of the problem.

From a mathematical viewpoint, the character

of the solution presents some other dilemmas. First of all, we consider 0 < x < s to be a segment in developments of eqn (2.19), but it is only a point. Second, since the strain softening portion localizes to a point at which the displacement is discontin- uous, within conventional theories for partial dif- ferential equations, the body would no longer be considered to be a single body: strain softening sub- divides the initial body into two by introducing in- terior boundaries. Similarly in three-dimensional problems, the strain-softening domain is a surface. In both cases, these are sets of measure zero. Al- though mathematical theories for such partial dif- ferential equations are not known to us, there is no reason to arbitrarily exclude such phenomenolog- ical models.

The strictly one-dimensional problems of this type are shown in Figs. 3 and 4. The first consists of an elastic rod joined to a rod with a strain-soft- ening material[S]. We will not give the closed form solution but only explain its major features. If the applied stress is sufficiently large. then strain soft- ening is initiated at the interface between the two materials. The strain localizes at this point. and as in the previous problem. the stress vanishes in- stantaneously at the interface. The solution can thus be viewed as a case in which a body separates into two.

The second problem, given in (61. consists en- tirely of a strain-softening material. Equal and op- posite velocities I’,) (or forces) are applied to the two ends of the bar. so that tensile waves are generated at the two ends. These propagate to the center; when they meet at the center. the stress doubles for elastic behavior, so strain softening is possible at this midpoint even though it did not occur at the boundaries.

The solution is given in [6] for prescribed ve- locities that are step functions in time. As in the previous case, localization occurs at the midpoint where the strain becomes infinite. The solution is symmetric about the midpoint x = 0; for the left half (x < 0 and 0 C t c 2Llc)

Another difficulty posed by this model is that the energy dissipated in the formulation of the strain-softening region is not finite but instead van- ishes. This can be seen from the fact that the only irreversible energy loss in the material shown in Fig. 2 occurs in the strain-softening domain. Be- cause the strain-softening domain in a one-dimen- sional problem becomes a point, and since the en- ergy dissipated per unit length is finite, the total energy dissipated vanishes. This in fact is a more serious difficulty than the mathematical difficulties, for the strain-softening constitutive equation is often intended to represent microcracking. which is a dissipative process. It will be seen that, in spite of the mathematical questions, the behavior of fi- nite-element solutions is not pathological.

II = Ll,,(f - +> - <!,,(, - +).

(2.22a)

+ 4 (et - I!,) 6(r) , (2.22b) 1 where 6(x) is the Dirac delta function.

Remarks: (1) The solution is puzzling when the ramp loads, eqns (2.3) and (2.4). are replaced by step functions. According to the present analysis, if u,,

> E,,E, then the boundary point should reach strain

ELASTIC SOFTENING

-L/2

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

Strain-'ioftening materials am! tinite-elemc:nt ,olutions 167

boundary must satisfy two different boundary con­ditions: eqn (2.4) and a = O.

This contradiction can only be reconciled by re­quiring the second boundary condition (that the stress vanishes) to take precedence. In fact. the tirst boundary condition would be impossible to enforce in an experiment. The notion of a boundary con­dition depending on the result of the solution is not totally unexpected in an analysis of a continuum which fails. For example. in a buckling problem with unstable postbuckling behavior. the pre­scribed force would also be limited by the capacity of the structure. Yet the situation in the buckling problem is not completely analogous: in a dynamic buckling problem. any force may be prescribed and the excess force will generate accelerations. which depend on the magnitude of the force. whereas in this problem, the solution is completely indepen­dent of the value of the prescribed stress once the failure stress is exceeded. Nevertheless. this model does appear to represent a physically meaningful situation: the behavior of a rod in which the material can sustain a limited tensile strain before it fails. and the solution appears reasonable.

From a mathematical viewpoint. the character of the solution presents some other dilemmas. First of all. we consider 0 < x < s to be a segment in developments of eqn (2.19), but it is only a point. Second. since the strain softening portion localizes to a point at which the displacement is discontin­uous, within conventional theories for partial dif­ferential equations, the body would no longer be considered to be a single body: strain softening sub­divides the initial body into two by introducing in­terior boundaries. Similarly in three-dimensional problems, the strain-softening domain is a surface. In both cases, these are sets of measure zero. Al­though mathematical theories for such partial dif­ferential equations are not known to us. there is no reason to arbitrarily exclude such phenomenolog­ical models.

Another difficulty posed by this model is that the energy dissipated in the formulation of the strain-softening region is not finite but instead van­ishes. This can be seen from the fact that the only irreversible energy loss in the material shown in Fig. 2 occurs in the strain-softening domain. Be­cause the strain-softening domain in a one-dimen­sional problem becomes a point, and since the en­ergy dissipated per unit length is finite. the total energy dissipated vanishes. This in fact is a more serious difficulty than the mathematical difficulties, for the strain-softening constitutive equation is often intended to represent microcracking. which is a dissipative process. It will be seen that, in spite of the mathematical questions, the behavior of fi­nite-element solutions is not pathological.

Remarks: (1) The solution is puzzling when the ramp loads, eqns (2.3) and (2.4), are replaced by step functions. According to the present analysis. if au > E.pE. then the boundary point should reach strain

softening instantaneously and no wave should reach the interior.

(2) The solution does not depend on the specific functional dependence of stress on strain in the strain-softening portion if the stress vanishes as the strain becomes large.

One conceptual difficulty of the Sandler-Wright problem is that strain softening occurs only at the boundary. which confuses the role of the boundary condition and the strain softening. For this reason we have attempted to construct problems in which strain softening occurs within the domain of the problem.

The strictly one-dimensional problems of this type are shown in Figs. 3 and 4. The first consists of an elastic rod joined to a rod with a strain-soft­ening material[5J. We will not give the closed form solution but only explain its major features. If the applied stress is sufficiently large. then strain soft­ening is initiated at the interface between the two materials. The strain localizes at this point. and as in the previous problem. the stress vanishes in­stantaneously at the interface. The solution can thus be viewed as a case in which a body separates into two.

The second problem. given in [6J. consists en­tirely of a strain-softening material. Equal and op­posite velocities "u (or forces) are applied to the two ends of the bar, so that tensile waves are generated at the two ends. These propagate to the center: when they meet at the center. the stress doubles for elastic behavior, so strain softening is possible at this midpoint even though it did not occur at the boundaries.

The solution is given in [6J for prescribed ve­locities that are step functions in time. As in the previous case, localization occurs at the midpoint where the strain becomes infinite. The solution is symmetric about the midpoint x = 0; for the left half (x < 0 and 0 .;; t .;; 2L1c)

(X + L) (_~). II = Vu t - -c-' - - Vu t

e

(2.22a)

E. = ~ [H (t -~) - H (t -~) e e e

+ 4 (et - L) o(x) J, (2.22b)

where 5(x) is the Dirac delta function.

ELASTIC SOFTENING

I I I

~L/2-+-L/2-1 Fig. 3. Problem with strain softening at interface between

elastic and softening domain (Ref. [SI!.

“0 I V, bX

I------+-----A Fig. 1. One-dimensional problem in uhich strain wtiening

occurs at v = 0 (Ref. 161).

Fig. 5. Spherical converging wave problem

Another problem we have considered is a sphere

loaded on its exterior surface (see Fig. 0. This

problem is not easily physically realizable with a

tensile load: however. it is physically meaningful

with a compressive load, and strain softening can

occur in some materials in compression (although

the stress usually will not vanish as the dilatation

becomes large).

This problem has an intriguing feature. Consider

a load which is a ramp function in time. Prior to the

onset of strain softening at an interior surface, part

of the wave can have passed through this surface.

Since the stresses in the wave which are inside the

initial surface of strain softening are amplified as

the wave passes to the center. the formation of ad-

ditional strain-softening surfaces is possible. As a

result. this problem has considerably more struc-

ture than the one-dimensional problems.

3. NONLOCAL CONTINLWI FOR STRAIN SO+TENISG

The major shortcoming of strain-softening

models for representing local damage is that the lo-

calization phenomenon associated with strain soft-

ening results in no dissipation of energy. For ex-

ample, in one-dimensional problems. the strain-

softening process is limited to a single point and

since the rate of work per unit length is finite, the

amount of work dissipated in the strain-softening

domain vanishes. Analogously. in three-dimen-

sional problems, strain softening localizes to a sur-

face. and since the work per unit volume is again

finite. the total energy dissipation due to the sep-

aration across the surface is finite. Thus. no energy

is required to separate the material along the surface

of strain softening, which is physically quite un-

realistic. It should be stressed that all of these com-

ments apply only to materials in which the stress

across the surface tend\ to zero ;IS the strain be-

comes large: if the stress tends to some nonzrro

value. then the response of the material may be

quite different and the dissipation no longer van-

ishes. However, any dissipation in excess of that

which corresponds to the perfectly. plastic dissi-

pation associated with the final stress vanishes.

To achieve finite energy dissipation during fail-

ure by strain softening. the artifice of imposing ;I

certain minimum element size which depends on

the aggregate size (crack band model) has been pro-

posed[Z]. However, this artifice may be inconven-

ient in practical analysis. since it requires the ele-

ment size to be dictated by a material constant

rather than by the size of the structure.

In order to avoid this difficulty. we have inves-

tigated the possibility of using nonlocal constitutive

laws in which the stress at a point is related to the

weighted average of the strain in a neighborhood ot

that point. This is a special case of the existing

(or classical) nonlocal continuum theories[%-14).

However, it was found to be necessary to make two

modifications of this theory in order to obtain re-

alistic results in the strain-softening regime: ( I) the

existing nonlocal theory is not self-adjoint and

hence possesses certain spurious zero-energy

modes of deformation: (2) in order to achieve stable

solutions even in one dimension in the presence of

strain softening with a constant weight. ;I material

law consisting of a combination of a local law vrith-

out strain softening and a nonlocal law with strain

softening was required.

We will now sketch the essential features of this

nonlocal theory for one-dimensional problems. De-

tails may be found in Refs. 17. 81. The fundamental

assumption in a nonlocal theory is that the nonlocal

strain r at a point is a weighted average of the strains

in a neighborhood of that point. Thus

i

t-12

E(s) = ct.\’ + .S)II’l.S) d.s t-12 (3. I)

-I -I 2 i),,

= J , _, ~ z (x + .S)~l~(.S) d.s.

where n’(.s) is a given weighting function. The

stress-strain law is then written in terms of Z. and

its rate form is

u.,(x) = L-(E)E.,lx). (3.2)

Although the classical nonlocal theory directly

uses the stress CT in the momentum equation. eqn

(2. I ). the resulting form is not self-adjoint[ In]. This

leads to the existence of spurious. zero-energy’

modes of deformation for certain weighting func-

tions II.(.V): deformations which are associated with

vanishing strains E and hence do not generate any’

stresses. These spurious modes have been found fc71

uniform (constant) weighting functions 1i.t~).

To remedy this difficulty. the stress 17 is pro-

I hI<

Va Va

<9C=====~===~~3~$-'" X I.. L-~~\

Fig. -I. One-dimensional problem in "hich strain softening occurs at.r = 0 iRd. 16j). -

uniform

traction uo

Fig. 5. Spherical converging wave problem.

Another problem we have considered is a sphere loaded on its exterior surface (see Fig. 5). This problem is not easily physically realizable with a tensile load: however. it is physically meaningful with a compressive load. and strain softening can occur in some materials in compression (although the stress usually will not vanish as the dilatation becomes large).

This problem has an intriguing feature. Consider a load which is a ramp function in time. Prior to the onset of strain softening at an interior surface. part of the wave can have passed through this surface. Since the stresses in the wave which are inside the initial surface of strain softening are amplified as the wave passes to the center. the formation of ad­ditional strain-softening surfaces is possible. As a result. this problem has considerably more struc­ture than the one-dimensional problems.

3. NON LOCAL CO!llTINUU:<'1 t'OR STRAIN SOtTENING

The major shortcoming of strain-softening models for representing local damage is that the lo­calization phenomenon associated with strain soft­ening results in no dissipation of energy. For ex­ample. in one-dimensional problems. the strain­softening process is limited to a single point and since the rate of work per unit length is finite. the amount of work dissipated in the strain-softening domain vanishes. Analogously. in three-dimen­sional problems. strain softening localizes to a sur­face. and since the work per unit volume is again finite. the total energy dissipation due to the sep­aration across the surface is finite. Thus. no energy is required to separate the material along the surface of strain softening. which is physically quite un­realistic. It should be stressed that all of these com­ments apply only to materials in which the stress

across the surface tend~ to zero as the ~train be­comes large: if the stress tends to ,ome nonzero value. then the re'iponse of the material may be quite different and the dissipation no longer van­ishes. However. any dissipation in exce,s of that which corresponds to the perfectly plastic dissi­pation associated with the tinal stress vanishes.

To achieve finite energy di'isipation during fail­ure by strain softening. the artitice of imposing a certain minimum element size which depends on the aggregate size (crack baml model) has been pro­posed[2J. However. this artifice may be inconven­ient in practical analysis. since it requires the ele­ment size to be dictated by a material constant rather than by the size of the structure.

In order to avoid this difficulty. we have inves­tigated the possibility of using non local constituti ve laws in which the stress at a point is related to the weighted average of the strain in a neighborhood of that point. This is a special case of the existing (or classical) nonlocal continuum theories[9-141. However. it was found to be necessary to make two modifications of this theory in order tLl obtain re­alistic results in the strain-softening regime: (I) the existing nonlocal theory is not self-adjoint and hence possesses certain spurious zero-energy modes of deformation: (2) in order to achieve stable solutions even in one dimension in the presence of strain softening with a constant weight. a material law consisting of a combination of a local law with­out strain softening and a non local law with strain softening was required.

We will now sketch the essential features of this nonlocal theory for one-dimensional problems. De­tails may be found in Refs. [7.81. The fundamental assumption in a nonlocal theory is that the non local strain E at a point is a weighted average of the strains in a neighborhood of that point. Thus

E(X) f '-':: ELr + S)II'(S) lis \ -/ 2

j-' -/:: (Ill ~ (.r + .I')II'(S) ds.

,-I: (IX

(3.1 )

where \1'(.1') is a given weighting function. The stress-strain law is then written in terms of E. and its rate form is

Although the classical non local theory directly uses the stress a in the momentum equation. eqn (2.1). the resulting form is not self-adjoint[161. This leads to the existence of spurious, zero-energy modes of deformation for certain weighting func­tions 1\'(.1'): deformations which are associated with vanishing strains E and hence do not generate any stresses. These spurious modes have been found for uniform (constant) weighting functions 11(.1').

To remedy this difficulty. the stress IT is prLl-

Strain-\oftenlng material\ and finite-element wlurion~

cessed through an operator identical to (3.1).

I

I -/I i3.r) = a(.r + .s)u~(.v) ds. (3.3)

t-12

and the resulting stress is used in the equation of

motion. eqn (2.1). Once eqn (3.3) is added to the

system. spurious modes are eliminated even for

constant weighting functions nG). It was also

shown by Bafant[S] that the averaging of a(x) is

required within a consistent application of a vari-

ational principle such as the principle of virtual

work. A nonlocal continuum which is characterized

by double averaging. once on the strains. as in 0. I) and then on the stresses. as in (3.3). is the limiting

case of a series of imbricated (overlapping) finite

elements. Therefore. this type of nonlocal contin-

uum was termed an imbricated continuum by Ba-

iant[8].

Even with a self-adjoint form of the nonlocal

laws. solutions for strain-softening materials are un-

stable for constant weights ~cG). So far, only by combining a local and nonlocal law has stability

been achieved[l6, 71. By superimposing two dis-

tinct field systems, one local and wirhour strain soft-

ening. and a nonlocal law ~vith strain softening,

stability is achieved in a model which exhibits a

negative slope for a finite segment.

The governing equations for this model can be summarized as follows:

o.r = EC., Ecan be negative, (3.4)

T., = EE,~, E>O. (3.5)

eqn (3.3): u--+ ij’, (3.6)

S=Cl -y)cJ+yT, O<y<l. (3.7)

5-C = Pll.rr. (3.8)

Finite element solutions for this nonlocal law are given in Sec. 5 for cylindrical and spherical geo-

metries. These results were obtained with an im- bricate finite-element model, and with various fi- nite-element meshes. It will be seen that the strain distributions computed by the nonlocal law are quite well-behaved for various meshes, whereas the local formulation predicts strain that vary errati- cally with mesh refinement.

1. FISITE-EI.EMENT SOLUTIONS FOR PLA\NAR

PROBLE\lS

Finite-element solutions for the moditied San-

dler-Wright problem. Fig. I, obtained with the

local material law given in Fig. 2. are shown in Fig.

6. Solutions were obtained with meshes of 50. 100.

and 200 elements. Linear displacement. constant

strain elements. and lumped mass matrices were

used. Time integration was performed with the cen-

tral difference method.

g -3.700

0 -e- AN.4 soi

-6.700 + 100 ELE

- 10.000 ,000 2.000 4.000 6.000 8.000 10.000

TIME - SEC (xl O-‘)

Fig. 6. Velocity-time history at .v = L/4 for geometry shown in Fig. I. stress-strain law given in Fig. 2: t,, =

O.OI. E, = 0.05. L = 100. C‘ = IO’.

The finite-element solutions are compared with

the analytic solution given in the Sec. 2. It can be

seen that the agreement is quite good and improves

with mesh refinement. although the instantaneous

drop in the velocity. which is a result of the strain

localization, cannot be reproduced even with the

finest mesh.

The rate of convergence for the case when E, =

er = 0.01 is shown in Fig. 7. Here the error (3 is

defined by

- I,‘~,‘)’ d.v dt. (4. I)

where v with superscripts FEM and ANA are the

finite-element and analytic velocities. respectively.

As can be seen from Fig. 7. the rate of convergence

is approximately proportional to I/‘.‘” where 11 is the

element length. This is not much less than the the-

ij

( (i=&

I i

In h 1

1

1 Fig. 7. Rate of convergence of finite-element solution for velocity to analytic solution for modified Sandier-Wright

problem with e, = e, = 0.01.

Strain-,oftening malerial-- and tinile-ekmenl ,ululion,

cessed through an operator identical to (3.1).

f' -/;

a(x) '" cr(x + s)"'(s) ds. \ -/ ~

(3.3)

and the resulting stress is used in the equation of motion. eqn (2.1). Once eqn (3.3) is added to the system. spurious modes are eliminated even for constant weighting functions II-(.\"}. It was also shown by Bazant[8] that the averaging of cr(x) is required within a consistent application of a vari­ational principle such as the principle of virtual work. A non local continuum which is characterized by double averaging. once on the strains. as in (3.1) and then on the stresses. as in (3.3). is the limiting case of a series of imbricated (overlapping) finite elements. Therefore. this type of non local contin­uum was termed an imbricated continuum by 8a­zant[8].

Even with a self-adjoint form of the non local laws. solutions for strain-softening materials are un­stable for constant weights I\"(x). So far, only by combining a local and non local law has stability been achieved[ 16. 7]. By superimposing two dis­tinct field systems. one local and without strain soft­ening. and a non local law with strain softening. stability is achieved in a model which exhibits a negative slope for a fmite segment.

The governing equations for this model can be summarized as follows:

cr. r = E'E.. r

'.r = EE.r.

eqn (3.3): cr-+ a. 5 = (I - 'Y)a + 'Y'.

5., = pll.rr.

E can be negative.

E>O.

0<'Y<1.

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

Finite element solutions for this non local law are given in Sec. 5 for cylindrical and spherical geo­metries. These results were obtained with an im­bricate finite-element model, and with various fi­nite-element meshes. It will be seen that the strain distributions computed by the non local law are quite well-behaved for various meshes, whereas the local formulation predicts strain that vary errati­cally with mesh refinement.

~. FI:-;ITE·ELDIE:"IT SOLt.:TlONS FOR PLA:"IAR

PROBLDIS

Finite-element solutions for the modi tied San­dier-Wright problem. Fig. I. obtained with the local material law given in Fig. 2. are shown in Fig. 6. Solutions were obtained with meshes of 50. 100. and 200 elements. Linear displacement. constant strain elements. and lumped mass matrices were used. Time integration was performed with the cen­tral difference method.

-.700 If

i: -J.700

~·I <,qo; i I 't, tc ' TI ...... ANA SOL u

0 ....J

II/ -u- 50 ElE w

> -6.700 -&- 100 ELE

--.!r- 200 ELE

-10.000 +------+-----+-4"---+----+---4 .000 2.000 '.000 6.000 B.OOO 10.000

TIME - SEC (x 1 0- 2)

Fig. 6. Velocity-time history at x = LI~ for geometry shown in Fig. I, stress-strain law given in Fig. ;: t,.

0.01, tr = O,OS. L = 100. (" = 10'.

The finite-element solutions are compared with the analytic solution given in the Sec. 2. It can be seen that the agreement is quite good and improves with mesh refinement. although the instantaneous drop in the velocity. which is a result of the strain localization. cannot be reproduced even with the finest mesh.

The rate of convergence for the case when E, = E, = 0.01 is shown in Fig. 7. Here the error (' is defined by

where " with superscripts FEM and ANA are the finite-element and analytic velocities. respectively. As can be seen from Fig. 7. the rate of convergence is approximately proportional to 1114h where II is the element length. This is not much less than the the-

In lI~iie r--------,

Convergence: of Velocity

Ey = Er = 0.0 I

t = loLe

146

--------7~---------lnh -I

o

-I

Fig, 7. Rate of converg.ence of finite-element solution for velocity to analytic solution for modified Sandler-Wright

problem with t, = £/ = (Ull.

TED BELYTKHKOC~ trl

i0 -1.900 -1.450 -1.000 - 550

In h

Fig. 8. Rate of convergence of finite-element solution for velocity to analytic solution for modified Sandler-Wright

problem with t, = ZEN = 0.02.

-.lLm I=--- & -.900

s Y

-1.700 I

*so0 i ma 2.cal 4.m 6.000 8.000 10.000

TIME - SEC (x10-')

- AhA SOL

-a- 50 ELE

+3- 100 ELE

Fig. 9. Finite-element solution to modified Sandler- Wright problem with p = p,,H(f). H being the Heaviside

step function.

oretical value of 11’ expected for linear solutions by

these methods, so the sensitivity to meshing which

Sandier and Wright pointed out is not evident. The

rate of convergence for a material with E, = Ze,. =

0.02 is shown in Fig. 8; as E( increases, the rate of

convergence deteriorates. Figure 9 shows the finite-

element solution when the input is a step function.

Theoretically. there should be no waves inside the

rod. but because of the finite size of the elements.

a small pulse penetrates the mesh in a numerical

solution.

5. FINITE-ELEMENT SOLUTIONS FOR SPHERICAL AND CYLINDRICAL PROBLEMS

As mentioned in Sec. 2. the converging spherical

wave problem is particularly intriguing because it

offers the possibility of strain softening being ini-

tiated at many points by a wave. Although we have

not been able to find a closed form solution for this

problem. we have examined both local and nonlocal

finite-element solutions. The local finite-element

solutions exhibit radical differences as the mesh is

refined. and this ostensible absence of convergence

is not alleviated by the addition of damping. Non-

local solutions are free of these difficulties. These

solutions will be summarized in this section.

In spherical or cylindrical coordinates. the

strain-displacement relations are

Ex=u,,, c,=;, t; = l y (spherical),

eL = 0 (cylindrical), (5.1)

E = E, + 3,. PI = i (E, - E,).

e, = 6 (E, - E,) (spherical). (5.2)

E = E, + 2EV, (‘, = i (ZE, - E, ).

e,. = i (EV - E,) (cylindrical). (5.3)

We denote by F,. E,.. E. 7,. 7,. the means of e,.

ey. e, e,, e,, respectively; defined as in eqn (3.1).

The stress-strain law is

o.r.1 = i& + z?Z,,,,

o,., = KE., f 2?%,.,,.

79.1 = KE,, + ZGr,,,,

‘Tb .I = KE.~ + ZGr,,,.

and the total stress S is obtained by

1 .I * /:2

v, Lr) = a,(~ + S)II~S) ds. r_,,?

s,, = (1 - y)Z, + yir,

s,. = (I - y)aV + -yTY.

The equation of motion is

s,,., + f (5, - S,.) = PlJ.,, (spherical),

(5.4)

(5.5)

(5.6)

(5.7)

ST., + ; (S, - S,.) = ptt,,, (cylindrical). (5.8)

Here u = radial displacement (Fig. 10~): e.<, E, =

radial and circumferential normal strains (local): E

= volumetric strain; e,. e,. = deviatoric strains

(local): T.,, TV = local radial and circumferential nor-

mal stresses: o.,, u,. = broad-range radial and cir-

cumferential normal stresses: S,, S,. = total radial -- and circumferential normal stresses: K. G. K, G = local and broad-range (nonlocal) bulk and shear

moduli. In eqns (5.4) and (5.5) isotropy of the ma-

terial is assumed. The shear moduli G and c are

assumed to be constant. while K and ?? depend on

17() TED BELYTSCHKO f[ al.

2500

1.700

~ .900 c:

0< .97

1.

100

-.700

-1.500 I-----I-----+----+---_+_---' -2.350 -1.900 -1.450 -1.000 -.550

.2.n h

Fig. 8. Rate of convergence of finite-element solution for velocity to analytic solution for modified Sandler-Wright

problem with E, = 2Er = 0.02.

~ U 0 -.J W >

-.100

-.900

-1.700

2.500 .000

Er =0.05

t = 0

...... ANA SOL

-e!-- 50 ELE

-e- 100 ELE

-tr- 200 ELE

2.000 ~.OOO 6.000 8.000 10.000

TIME - SEC (x10-2)

Fig. 9. Finite-element solution to modified Sandler­Wright problem with p = p"H(t). H being the Heaviside

step function.

oretical value of 112 expected for linear solutions by these methods. so the sensitivity to meshing which Sandler and Wright pointed out is not evident. The rate of convergence for a material with Er = 2E" = 0.02 is shown in Fig. 8; as E( increases. the rate of convergence deteriorates. Figure 9 shows the finite­element solution when the input is a step function. Theoretically. there should be no waves inside the rod. but because of the finite size of the elements. a small pulse penetrates the mesh in a numerical solution.

5. FINITE-ELEMENT SOLUTIONS fOR SPHERICAL AND

CYLINDRICAL PROBLE.\IS

As mentioned in Sec. 2. the converging spherical wave problem is particularly intriguing because it offers the possibility of strain softening being ini-

tiated at many points by a wave. Although we have not been able to find a closed form solution for this problem. we have examined both local and nonlocal finite-element solutions. The local finite-element solutions exhibit radical differences as the mesh is refined. and this ostensible absence of convergence is not alleviated by the addition of damping. Non­local solutions are free of these difficulties. These solutions will be summarized in this section.

In spherical or cylindrical coordinates. the strain-displacement relations are

Ex = U,x, _ ~ {EZ = Ey (spherical).

Ey - X· Ez = 0 (cylindrical). (5.1)

E = E, + ~E\ .• e.\" = ~ (E,. - E,.).

e,. :\ (E, - Ex) (spherical) . (5.2)

E = Ex + 2E\ .• e,\' = :1 (2E, - E, ).

e,. k (E\. - Ex) (cylindricalJ. (5.3)

We denote by ex. E,. E. ex. e, the means of E,.

Ey. E. ex. ey• respectively; defined as in eqn (3.1). The stress-strain law is

ax.! KE., + 2Ge",.

U\".I KE., + 2Ge, .. ,. (5.4)

T\.l KE., + 2Ge,.,.

'\',f KE., + 2Ge,.,. (5.5)

and the total stress 5 is obtained by

1X~/!2 a,(x) (Tx(x + 5)11'(5) d5.

r-II~

1.1'+1/2 aJrJ (T,.(X + 5)11'(5) ds.

.r - /I~ . (5.6)

5.1' (I 'I fax + 'ITx.

5,. (I 'I)a, + 'IT)' . (5.7)

The equation of motion is

')

S'" + ~ (S, - 5,) x . PII," (spherical).

I Su + - (S, - 5,) = PII," (cylindrical). (5.8)

x

Here II = radial displacement (Fig. 10c): Ex. E.' = radial and circumferential normal strains (iocal); E

= volumetric strain; e,. e,. = deviatoric strains (local); Tx. T., = local radial and circumferential nor­mal stresses; (Tx. (T,. = broad-range radial and cir­cumferential normal stresses; S,. S,. = total radial and circumferential normal stresses; K. G. K. G = local and broad-range (nonlocal) bulk and shear moduli. In eqns (5.4) and (5.5) isotropy of the ma­terial is assumed. The shear moduli G and G are assumed to be constant. while K and K depend on

Strain-softening materials and finite-element solulions lil

b) used: bulk modulus K = I .O. density p = I. shear modulus G = I x IOwh. eV = 1.0. e, = 5.0 (see

bottom of Fig. I I ). Damping was added by a viscous

stress defined by

d) local

-%,\I1 - _ -- _.. e-7:

-al nonlocal ;-l i itl -

I

r--

Fig. IO. Notation for spherical and cylindrical wave prob- lems.

E and Z, respectively: K > 0 but ?? may become negative, which represents strain softening.

We consider waves generated by a sudden ap- plication of a uniform normal traction at the exterior surface of a hollow sphere or a hollow infinite cyl- inder. The traction is a Heaviside step function of time. so the boundary condition at .r = h (Fig. IOa) is u., = p. H(r), and the interior surface is load- free, i.e. ux = 0 at x = o (a, h = internal and external surface radii). Initially, the body is at rest. The elastic solution[20], consists of a step wave with a strain which grows as the wave propagates toward the center.

The closed form solution is

for 7 2 0, (5.9)

as may be checked by substitution in eqns (5. I )- (5.8) with y = I. K = constant; 7 = t - (h - v)/

c,, 5 = 2 cj/(bC,), w = .$+z)’ - I I”‘, (‘, = [3K( I

- u)/(l + u) p]“‘, cz = (G/p)“‘, v = (3K - 2G)/

(6K + 2G) = Poisson’s ratio.

For the numerical solutions, one-dimensional meshes with (Fig. lob) two-node elements with lin- ear displacements and a single quadrature point (at the element center) were used. The local elements are of length /I and the imbricate elements of length I = n/r. The mass matrix is lumped.

The first group of numerical solutions was made with the aim of examining the character of the spherical wave solution and determining whether damping is sufficient to achieve well posedness, as in the planar one-dimensional problem[4].

Shear moduli G and ?? are assumed to be neg- ligibly small (IO-‘). The following constants were

g;;‘ = 21 (pK)’ ’ E8,,. (5.10)

with n = 0. I. This provided enough viscosity to damp the cutoff frequency of the mesh by 44% and 6% of critical damping in the fine and coarse meshes, respectively. In some solutions. the damp- ing was turned off whenever er, the strain at which the stress vanishes. was reached. This is called a

damping cutoff. Comparing the results in Fig. I2 for the various

meshes, one can see a striking difference in the re- sults within the strain-softening domain. which is the domain to the left of what is marked as the tran- sition point. As the mesh is refined, several points of localization develop and the points of localiza- tion change arbitrarily with mesh refinement even with damping. However, the displacement outside the transition surface appears to converge with mesh refinement.

Figure I3 shows a similar solution except that the damping was not cut off when localization de- veloped. Again, the solution does not exhibit any pattern of convergence in the strain-softening do- main. The stresses in the region near the transition point are quite different from those in Fig. 12, which employed a damping cutoff.

The next group of solutions examines the effect of the nonlocal formulation. In these solutions. the bulk behavior follows the bilinear total stress-strain diagram in Fig. I la, characterized by elastic bulk moduli K = K,, = I. strain E,, = I at peak stress and er = 5 at the end of strain softening. This dia- gram is obtained by assuming that the local behav- ior is elastic-ideal plastic (Fig. I lb), and that the

broad-range behavior (Fig. I la) exhibits oversoft-

6”

t s Total

Fig. I I. Stress-strain law for spherical and cylindrical wave problems: only the lowest figure applies to local

laws.

Strain-softening materials and finite-element ,olutions 171

b)

c) d)

tk tm k (ij";r------,-. -, --~+ m (i)

[oca[

Fig. 10. Notation for spherical and cylindrical wave prob­lems.

E and E. respectively: K > 0 but K may become negative. which represents strain softening.

We consider waves generated by a sudden ap­plication of a uniform normal traction at the exterior surface of a hollow sphere or a hollow infinite cyl­inder. The traction is a Heaviside step function of time. so the boundary condition at x = b (Fig. lOa) is 0'" = p() H( 0. and the interior surface is load­free. i.e. O'x = 0 at x = a (a. b = internal and external surface radii). Initially. the body is at rest. The elastic solution[20J. consists of a step wave with a strain which grows as the wave propagates toward the center.

The closed form solution is

b)p() { T [£ (2X ). ] 1I=4Gx~ I-e~ ;;; /;-1 SIOWT+COSWT

for T ~ O. (5.9)

as may be checked by substitution in eqns (5.1)­(5.8) with -y = I. K = constant; T = t - (b - 1')1 c,.£ = 2d/(bC,).w = Wc,/c~)~ - 1J'f~.(', = [3K(1 - v)/( I + v) pJ"~, C2 = (G/p)1I2, V = (3K - 2G)1 (6K + 2G) = Poisson's ratio.

For the numerical solutions. one-dimensional meshes with (Fig. lOb) two-node elements with lin­ear displacements and a single quadrature point (at the element center) were used. The local elements are of length It and the imbricate elements of length { = "fr. The mass matrix is lumped.

The first group of numerical solutions was made with the aim of examining the character of the spherical wave solution and determining whether ?amping is sufficient to achieve well posedness. as 10 the planar one-dimensional problem[4].

Shear moduli G and G are assumed to be neg­ligibly small (10- 6

). The following constants were

used: bulk modulus K = 1.0. density p = I. shear modulus G = I x 10- 10

• E, = 1.0. E, = 5.0 (see bottom of Fig. II). Damping was added by a viscous stress defined by

(5.10)

with 11 = 0.1. This provided enough viscosity to damp the cutoff frequency of the mesh by 44% and 6% of critical damping in the fine and coarse meshes. respectively. In some solutions. the damp­ing was turned off whenever E,. the strain at which the stress vanishes. was reached. This is called a damping cutoff.

Comparing the results in Fig. 12 for the various meshes, one can see a striking difference in the re­sults within the strain-softening domain. which is the domain to the left of what is marked as the tran­sition point. As the mesh is refined. several points of localization develop and the points of localiza­tion change arbitrarily with mesh refinement even with damping. However. the displacement outside the transition surface appears to converge with mesh refinement.

Figure 13 shows a similar solution except that the damping was not cut off when localization de­veloped. Again, the solution does not exhibit any pattern of convergence in the strain-softening do­main. The stresses in the region near the transition point are quite different from those in Fig. 12. which employed a damping cutoff. -

The next group of solutions examines the effect of the nonlocal formulation. In these solutions. the bulk behavior follows the bilinear total stress-strain diagram In F~. II a. characterized by elastic bulk moduli K = Ku = I. strain E,. = I at peak stress and EJ = 5 at the end of strain softening. This dia­gram is obtained by assuming that the local behav­ior is elastic-ideal plastic (Fig. II b). and that the broad-range behavior (Fig. Ila) exhibits oversoft-

(1-1)0' broad -range

Kt

t "Y T local

I ~-------S Total

Fig. II. Stress-strain law for spherical and cylindrical wave problems; only the lowest figure applies to local

laws.

DIS

PL

AC

EM

EN

?

DIS

PL

AC

EM

EN

T

HY

MIO

STA

TIC

S

TR

ES

S

HY

DR

OS

TA

TIC

S

TR

ES

S

VO

LUM

ETR

IC

STR

AIN

VO

LUM

ETR

IC

STR

AIN

5 ‘I!

“m

,.

__

-i---

1

5 j

e”

f j

.‘

I

3 I

HY

DR

OS

TA

TIC

S

TR

ES

S

VO

LUM

ETR

IC

STR

AIN

DIS

PL

AC

EM

EN

T

DIS

PL

AC

EM

EN

T

HY

DR

OS

TA

TIC

S

TR

ES

S

HY

DR

OS

TA

TIC

S

TR

ES

S

VO

LUM

ETR

IC

STR

AIN

VO

LUM

ETR

IC

STR

AIN

F

2

17~

z <i ~ 2.660. _f"_ .. 1\0011 """., iII.UZI

'" ~ f940. >-W

~ • 220.

d > ~ ------~

(/)

'" ILl

'0 2~ 4Q ~~ 70 85 '00 R

920-.----_______ . ____ _

r: .800. (/)

u .680. >= ~ <Jl 0 .560 •

l5 >-r 44()

'0 25 40 55 70 85 '00 R

36

>- 29. , , Z '. -----W

~ 22,

~ ! ..J

• 5 ! CL <Il i3

8 .0 25 40 55 70 85 .00

R (a)

TEf} BEL YTSCHKO ('{ al.

'0 25 4()

.1- ,,_ ........... .. 'I I

55 R

70 '00

z <i r: 56.

'" u ~ 37. .... W

~ '8. ..J o

1-- "-""" "',., ..... I I

> II ., tfr-...~ ... -;·':";_"'·=4~~~55:c=="70:'" 9~ 100

R

'" .730~--_----·--·_--

& ~ ;420.

u >= :! .tiO_

'" o ~ '.200 . >­r

.5.0."=0---c2"'5c--4-::0o--c:55=-- 70 85 '00

R 55 ____ . ____ -. ______ · ___ __

>-z 43. w ::I! \oJ ~ 3t ~ ..J 0-

3 ,9 .

7 ~:.... _____________ •

10 25 40 55 70 85 '00 R (e)

Fig. 12. Solutions to spherical wave problem for various meshes with damping for meshes of (a) 25 elements, (b) 100 elements, and (c) 200 elements.

3.380, ____ -;-_.-_____ __

u ~ '940.

w ::I!

d • 220.

> ~~. -----------~~=--

10 25 40 55 10 85 .00 R

<Il .920r-----, .------ .

'" w a: .BOO. t;; u .680.

I~ r44()l...-. ___ --:-_=_-=::_-=::_--::

10 25 40 55 70 85 100

36 ____ _

'0 ZS

R

55 R (0)

,00

z

~ 2': >- • <Il

\- ,. .... ,,, ...... , ..... U a: 14. I-\oJ ::I! 3 1 .

o >

!

55 70 85 R

<Jl .860 ___________ . ______ _ <Jl - , \oJ a: tii .620.

u ~ 380.

~ a >­r

HO. I

100

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

43,'00 ______ ':T! ___ _

... 34.300: z

r; ,I "

!2~~: J~I: '" 16.1'00. is \

7.100"" ,I I 'OZS40"

R (b)

70

z <i r: 33, <Jl

U a: 22. .... ' w ~ H.

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

10 25 40 55 10 85 .00 R

~ .7'0 • ___ . ______________ _

\oJ a: tii 1190, i

S270;! \ ~! V ~ .Q!lOf x

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

R

43r,-------~~-----------'

Fig. 13. Solutions to spherical wave problem for various meshes with damping without damping cutoff for meshes of (a) 25 elements. (b) (00 elements, and (c) 200 elements.

Strain-Softening mLlterials and finite-element solutions 173

wing for the purpose of achieving total behavior in which the strain softening terminates with a zero stress value[?j. The introduction of the local stresses through the coefficient y. see eqn (3.7) is required for stability. For one-dimensional planar bvaves. it was shown[7] that stability is assurred for y > 0. However. when y is small the noise is ex- cessive. For one-dimensional planar waves. the value y = 0.1 gave noise-free response. Curiously.

6X

600 r

N=20 I(’ ,(’ _._-

jll_

// _*,* A ,,** /’ ,,’ .._’ .:’ / ’ _, I’ ,.:

/ / , ,’ ,.I’-

/ ’ I’ ._I’ ,/ ,//I /* _:”

,.,’

_ / _ _** _/” _-_.-.

the value of y for the cylindrical wave had to be higher ty = 0.3) than the value for the spherical

wave (y = 0. I) in order to obtain a noise-free re-

sponse. The values of the applied surface pressure are

chosen as p. = 0.708 for the spherical wave and p,, = 0.8 for the cylindrical wave. For these boundary conditions, the wave propagating from the outer surface remains elastic until the wavefront reaches

0 iv=40 M

ol’ J J !

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

Strain-'ioftening materials ami finite-element solutions

ening for the purpose of achieving total behavior in \vhich the strain softening terminates with a zero stress value[7J. The introduction of the local stresses through the coefficient -yo see eqn (3.7) is required for stability. For one-dimensional planar waves. it was shown[7J that stability is assurred for 'I > O. However. when 'I is small the noise is ex­cessive. For one-dimensional planar waves. the value -y "" 0.1 gave noise-free response. Curiously.

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

N=10

N=40

,."

10 x

the value of -y for the cylindrical wave had to be higher (-y = 0.25) than the value for the spherical wave (-y = 0.1) in order to obtain a noise-free re­sponse.

The values of the applied surface pressure are chosen as po = 0.708 for the spherical wave and po = 0.8 for the cylindrical wave. For these boundary conditions. the wave propagating from the outer surface remains elastic until the wavefront reaches

3r-----------------------, N=10

N 40

N=160

~~~>, - _'1.. 11f7 --

i 01J )) j

10 x 100

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

30% of the thickness h - (I. The dimensions are (I = 10. h = 100. L = h - (I = 90.

In order to ascertain the mesh refinement nec- essary for this class of problems. elastic local so- lutions were obtained first. The convergence with

a-

t

i 0 ’ 3 r I I

: k r N=40

increasing numbers of elements (ic’ = IO. 20. 40. 80. 160) is shown in Figs. I-l and 15. The results converge to the exact solution given by eqn (5.9).

Subsequently, the problem was solved for a non- local continuum (E, = I. E j = 5) with characteristic

“y------ 1 1 N=10

I

-1 N=20

N=40

,/; i 1,

/ ; /\ )/ ,‘d’ i’ ii-,

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

17~ TED BEL YTSCHKO 1'{ ul.

309c of the thickness h ({. The dimensions are ({ == 10. h == 100. L == h - ({ == 90.

In order to ascertain the mesh refinement nec­essary for this class of problems. elastic local so· lutions were obtained first. The convergence with

/ /

I I

/ /

/ /

/

/ ,

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

N=40

N=160

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

10

! ::: ""IUaD 11 .,.» ,. ... ....

x 100

increasing numbers of elements (N = 10. 20. 40. 80. 160) is shown in Figs. 14 and 15. The results converge to the exact solution given by eqn (5.9).

Subsequently. the problem was solved for a non­local continuum (f" = I. €.r = 5) with characteristic

6\,--------------------, i r

II I

I I I I I I. I

10

AtJ,al711c.1

S.haUoQ

N=10

x

N=160

11_. fI_ , .... . ...-, "'.10

"'e .• ....

100

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

Strain-softening materials and finite-element solutions 175

length 1 = L/IO. The results for progressively finer relation with strain softening: see Figs. 16-19. left

meshes IN = IO. 20.10.80. 160) are shown in Figs, columns. It is seen (Figs. 16-19) that the nonlocal

16- 19. right columns. For comparison. the solution solutions with strain softening converge well with

was also run for a local continuum ii = Ir = L/I%’ increasing N. On the other hand, the corresponding

= variable with A’) having the same stress-strain local solutions do not converge at all.

N=lO

10 X

-_L

Iv=80

N=160

loo 10

N=40

N=80

X loo

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

Strain-softening materials and finite-element solutions 175

length I = LlIO. The results for progressively finer meshes (N = 10. :!O. 40. 80. 160) are shown in Figs. 16-19. right columns. For comparison. the solution was also run for a local continuum (I '= Ir = LIN

variable with iV) having the same stress-strain

/ , 0 ,... , •• !t

3 (I)

till ~ C'd

.c.: u (I)

E ;:I -0 >

. -

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

I'~nlocal N=10

~~"i / / t" , !

' ... - / ~

.' .......... ~ ..

,. ,.

l

3,

I I !

N;"80 1'1 N = 80

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

10 x 100 x 100

Fig. 16. Comparison of numerical results for local and non local materials with spherical geometry­volume change.

local

N-40

nonlocal ____*. / ,’ _-+-_-

N=40 G= /I

/ ,.-. ,___-*

1. _;’ 1:I.;: ,;’

,;’ ,,,’

,,1’ .:’

__=

10 X 100 JO X 10

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

In Sec. 3. it was shown that for the uniaxial in the exact solution for the planar wave. the strain-

planar wave, finite-element solutions converge to softening region appears to localize. producing sin-

the exact solution with a local strain-softening ma- gular strains, as can be seen from the spikes in Fig.

terial. For spherical waves. no exact local material I6 on the left. These singular strains begin to ap-

solution with strain softening has been found. As pear. as expected. when the strain at the wavefront

I~h TEO BEl.YTSUlr;" ('( lit.

80~---- ----- 50.- ------"

local nonlocal

N=10 N= 10

N=20 ----'

I . J O~.

80,

l N= 160 I

f

r

f I

O~rL-L-~~ ______ ~~~~o __ ~~ __ ~~ __ =-_~~~ 10 x 100 10 x 100

Fig. 17. Comparison of numerical results for local and non local materials with spherical geometry­radial displacement.

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

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

Strain-softening materials and finite-element solutions

IO X

100 10 X too

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

reaches the strain at which strain softening begins. points increases with increasing mesh refinement. In contrast to the planar wave. there is not one point The impression is that of chaos. It is anticipated of localization but many, and they appear at dif- that if the loading is a ramp function in time, the ferent locations for different N [i.e. for different set of localization points would be an infinitely mesh re~nements~. and the number of localization dense set of discrete points. i.e. a Cantor set. The

Strain·softening materials and tinite·element sQlutions 177

local

----/ " /,~ .... :::-...

-+-,'<"- - ~-I ..

I :' /, :'

/ I ..

I '. !

I " ..-

I I

/ /

-+­I

I

I -'

/ "

nonlocal \

\

.. :; I - ~ - ~;-C-l

l ' " i

~~"='=/~=""~'==~~~====~ I

/'" /

-"

,

J I . .

: I : , . .-a / , •• a' a -2

N=40 2~ N=40 Q,) QI)

~ c.I:S

.c: u Q,) -i' E

YJJ .' ;:j -0 >

211l-~- N=80 IN=8O

J I ~\i ,!,:~ j :1

.-. ......... a

2

x x 100

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

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

points increases with increasing mesh refinement. The impression is that of chaos. It is anticipated that if the loading is a ramp function in time. the set of localization points would be an infinitely dense set of discrete points. i.e. a Cantor set. The

TED BELYTKHKO ('r ol.

60, ______ __. T

local nonlocal

N = 10 , i:,,’ >,,,’ _/’

/,’ ~

,/: ’

,/: ,/’ I

I i

Jr --G-- *. ..2.

N=20

,’ ,’ ,I ,,/‘, ,’ ;’

;.. ..’

,/ ,a , ,, .Y’

// , >’ .,,I’

*, / ,,,’ _, ; - --.

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

liX TED BELYTSCHKO £'1 (/1.

60-,----

local nonlocal

N= 10

N=40

N=80 N=80

N=160 N= 160

/'

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

Strain-softening materials and finire-element solutions 179

reason for the appearance of multiple strain-soft-

ening points (the spikes in the left columns of Figs.

16 and 18) is that part of the strain step at the wave-

front is transmitted across the surface of strain soft-

ening before the stress is reduced to zero. The part

of the wave which has been transmitted then grows

as the wavefront converges toward the center until

the elastic limit ep is reached again. so the situation

repeats itself.

6. COSCLLWOSS

The following are the major conclusions of the

work:

I. Analytic solutions can be established for cer-

tain simple problems which include strain-softening

materials. The solutions exhibit singular strain dis-

tributions but the rate of convergence of tinite-ele-

ment solutions is quite rapid.

2. In the spherical wave problem, numerical so-

lutions of strain-softening models exhibit severe de-

pendence on element mesh size. This is particularly

true of field variables inside the surface of initial

strain softening. Nonlocal models provide rapidly

converging solutions to this problem.

3. A major difficulty of local laws \vith strain

softening is that the energy dissipation vanishes.

Thus, the failure process is not accompanied by en-

ergy dissipation. which is physically unrealistic.

4. Nonlocal laws provide a means for obtaining

rapid convergence and finite energy dissipation in

failure. However, the technology for efficiently im-

plementing these techniques in large-scale, multi- dimensional problems remains to be developed.

Ack,ro,c,/rd.~fnrnrs-We gratefully acknowledge the sup- port of Air Force AFOSR Grant 83-0009 lo Northwestern University.

1.

2.

3.

4.

5.

6.

REFERENCES

Z. P. Batant and L. Cedolin, Fracture mechanics of reinforced concrete. J. Engng Mech. Div. ASCE 106, 1287-1306 (1980): Discussion and Closure in 108.464- 471 (1982): Z. P. Bafant, Instability, ductility and size effect in strain-softening concrete. J. Engng Mech. Div. ASCE 102, EM2. 331-334 (1976): discussion 103. 357-358. 775-777 (1977); 104,‘501-502 (1978). S. H. Marchertas, S. H. Fistedis, Z. P. Barant, and T. Belytschko, Analysis and application of pre- stressed concrete reactor vessels for LMFBR con- tainment. Nucl. Engng Des. 49, 155-174 (1978). I. Sandier and J. Wright, Summary of strain-softening. In Theoretical Foundations for Large-Scale Compu- tations ofNonlinearMateria1 Behavior, DARPA-NSF Workshop (Edited by S. Nemat-Nasser). Korthwest- em University (1984). T. Belytschko, Discussion of I. Sandier in Theoretical Foundations for Large-Scale Computations of Non- linear Material Behavior, DARPA-NSF Workshop (Edited by S. Nemat-Nasser), pp. 285-315. North- western University (1984).

Ntrmrricrrl rrlgorithm for sphc~ricul an~c,vlindricctl NY,~‘~.s

(I) Read (I. h. 1. n. It (time step). N,.. N,.. NI. N,(number of local elements, imbricate elements. nodes, and time steps, respectively), y. and po. Generate arrays l(i). k(i).

I,(i). x,.(i). I(i). Z(i). i,.(i), T,.(i) giving the number of the left and right nodes of the ith local or imbricate element, its length. and its coordinate at the center ofelement. Also generate externally applied nodal forces ft.::. Initialize as zero the values (for r = I) of ill. kI. ;rl. T,~. v,~. max Q. max fl. T,,.~ *.A. urn.,< c.l forall k = I. . . N,. (local

Z. P. Bafant and T. B. Belytschko, Wave Propagation in Strain-Softenine Bar: Exact Solution. Retort No.

elements) and k = I. . Iv,. (imbricate elements). (2) DO 8. r = 2. . N,. (3) Initialize nodal forces FL = 0. fl = 0 (for all nodes, !i = I. , Nk). (4) DO 5 i =_ I. , N,. (local elements). (5) Set k = k(i), m = G(i). x = x,.(i) and evaluate 1c,i = (v.,, - I%)~frill. E,, 6 E,, + AE,, AE,, = l&,, + 1’1) ~rtl2.r. Ed, +-r,.; + SE,.. For spherical wave l\e, = AC,, + XE,~. and for cylindrical wave AC, +- AE,, + 16,;: e, + ei + ALL;. Then call a subroutine which determines the incre- mental (tangential) moduli K,. G, for the local elements from their strainse,,. E,,. E, and also decides whether virgin loading. unloading. or reloading applies. Then calculate

_. AT,, = K; AC, + ZG, e,,. AT,, = Ki 1~; + 2G, e,,. T,~ +

7.

8.

9.

10.

II.

12.

13.

14.

15.

16.

17.

18.

19.

20.

83-lOi401w to DNA. Center for Concrete and Geo- materials, Northwestern University, Evanston, Illi- nois, Oct. 1983. Also, J. Engng .Mech. Div. ASCE, 111, 381-389 (1985). Z. P. Batant. T. B. Belytschko. and T.-P. Chang. Continuum theory for strain softening. 1. Engng

Mech. ASCE 110, 1666-1692 (1984). Z. P. Baiant. Imbricate continuum and its variational derivation. J. EnanP Jfech. ASCE 110. 1593-1712

I I

(1984). E. Kraner, Elasticity theory of materials with long range cohesive forces. Int. J. Solids Srruct. 3, 731-

742 (1967). J. A. Krumhansl. Some Considerations of the Relation Between Solid State Physics and Generalized Contin- uum Mechanics. Mechanics of Generalized Conrinua

(Edited by E. Krbner), pp. 198-31 I. Springer-Verlag, Berlin (1968). I. A. Kunin, The Theory of Elastic LMedia with Mi- crostructure and the Theory of Dislocations. Me- chanics of Generalized Continua (Edited by E. Kriiner), pp. 321-328. Springer-Verlag, Berlin (1968). A. C. Eringen, Nonlocal polar elastic continua. Int. J. Engng Sci. 10, I-16 (1972). A. C. Eringen and D. G. B. Edelen, On nonlocal elas- ticity. Int. J. Engng Sci. 10, 233-248 (1972). A. C. Eringen and N. Ari, Nonlocal stress field at Griffith crack. Cyst. Latt. Def. Amorph. Mat. 10,33- 38 (1983). D. C. B. Edelen. Non-local variational mechanics I- Stationarity conditions with one unknown. Int. J. Engng Sci. 7, 269-285 (1969). Z. P. Barant and T.-P. Chang, Nonlocal Continuum and Strain-Averaging Rules, Report No. 83-I 1/4OIi. Center for Concrete and Geomaterials, Northwestern University, Evanston, Illinois, Nov. 1983; also J. Engng Mech. ASCE (in press). J. Hadamard, Leqons sur la propagation des ondes, Chapt. VI. Hermann et tie, Paris (1903). G. I. Taylor, The testing of materials at high rates of loading. J. Insr. Civ. Engng 26, 486-519 (1946).

F. H. Wu and L. B. Freund, Deformation trapping due to thermoplastic instability in one-dimensional wave propagation. J. Mech. Phys. Solids 32(2), 119- 132 (1984). J. D. Achenbach, Wave Propagation in Elastic Solids. North-Holland, Amsterdam (1973).

Strain-softening materials and finite-element solutions 179

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

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

The following are the major conclusions of the work:

I. Analytic solutions can be established for cer­tain simple problems which include strain-softening materials. The solutions exhibit singular strain dis­tributions but the rate of convergence of finite-ele­ment solutions is quite rapid.

2. In the spherical wave problem. numerical so­lutions of strain-softening models exhibit severe de­pendence on element mesh size. This is particularly true of field variables inside the surface of initial strain softening. Nonlocal models provide rapidly converging solutions to this problem.

3. A major difficulty of local laws with strain softening is that the energy dissipation vanishes. Thus. the failure process is not accompanied by en­ergy dissipation. which is physically unrealistic.

4. Nonlocal laws provide a means for obtaining rapid convergence and finite energy dissipation in failure. However. the technology for efficiently im­plementing these techniques in large-scale. multi­dimensional problems remains to be developed.

Ackl/(}l\"ledgmenlS-We gratefully acknowledge the sup­port of Air Force AFOSR Grant 83-0009 to Northwestern University.

REFERENCES

I. Z. P. Ba:!ant and L. Cedolin, Fracture mechanics of reinforced concrete. 1. Engng Mech. Div. ASCE 106, 1287-1306 (1980); Discussion and Closure in 108,464-471 (1982).

2. Z. P. Batant, Instability, ductility and size effect in strain-softening concrete. 1. Engng Mech. Div. ASCE 102, EM2, 331-334 (1976); discussion 103, 357-358, 775-777 (1977); 104,501-502 (1978).

3. S. H. Marchertas, S. H. Fistedis, Z. P. Batant, and T. Belytschko, Analysis and application of pre­stressed concrete reactor vessels for LMFBR con­tainment. Nucl. Engng Des. 49, 155-174 (1978).

4. I. Sandler and 1. Wright, Summary of strain-softening. In Theoretical Foundations for Large-Scale Compu­tations of Nonlinear Material Behavior, DARPA-NSF Workshop (Edited by S. Nemat-Nasser). Northwest­ern University (1984).

5. T. Belytschko. Discussion of 1. Sandler in Theoretical Foundations for Large-Scale Computalions of Non­linear Material Behavior. DARPA-NSF Workshop (Edited by S. Nemat-Nasser). pp. 285-315. North­western University (1984).

6. Z. P. Ba:!ant and T. B. Belytschko. Wave Propagation in Strain-Softening Bar: Exact Solution. Report No.

83-IOi401w to D:-OA. Center for Concrete and Geo­materials. Northwestern University. Evanston. llIi­nois. Oct. 1983. Also. 1. Engng Mech. Div. ASCE. lll, 381-389 (1985).

7. Z. P. Ba:!ant. T. B. Belytschko. and T.-P. Chang. Continuum theory for strain softening. 1. Engng ,Hech. ASCE 1l0. 1666-169:! (1984).

8. Z. P. Bazant. Imbricate continuum and its variational derivation. 1. Engng J[ech. ASCE 1l0, 1593-1712 (1984).

9. E. Kroner. Elasticity theory of materials with long range cohesive forces. Inl. 1. Solids Slruct. 3,731-742 (1967).

10. 1. A. Krumhansl. Some Considerations of the Relation Between Solid State Physics and Generalized Contin­uum Mechanics. Mechanics of Generalized Continua (Edited by E. Kroner). pp. 198-311. Springer-Verlag. Berlin (1968).

II. I. A. Kunin. The Theory of Elastic Media with Mi­crostructure and the Theory of Dislocations. Me­chanics of Generalized Continua (Edited by E. Kroner). pp. 321-328. Springer-Verlag. Berlin (1968).

12. A. C. Eringen. Nonlocal polar elastic continua. Int. 1. Engng Sci. 10, 1-16 (1972).

13. A. C. Eringen and D. G. B. Edelen. On nonlocal elas­ticity. Int. 1. Engng Sci. 10,233-248 (1972).

14. A. C. Eringen and N. Ari. Nonlocal stress field at Griffith crack. CrySI. Latt. Def. Amorph. Mal. 10,33-38 (1983).

15. D. C. B. Edelen. Non-local variational mechanics 1-Stationarity conditions with one unknown. Inl. 1. Engng Sci. 7, 269-285 (1969).

16. Z. P. Ba:!ant and T.-P. Chang. Nonlocal Continuum and Strain-Averaging Rules. Report No. 83-11140Ii. Center for Concrete and Geomaterials. Northwestern University. Evanston. Illinois. Nov. 1983; also 1. Engng Mech. ASCE (in press).

17. 1. Hadamard, Le~ons sur Ja propagalion des andes, Chapt. VI. Hermann et cie. Paris (1903).

18. G. 1. Taylor. The testing of materials at high rates of loading. 1. Insl. Civ. Engng 26, 486-519 (1946).

19. F. H. Wu and L. B. Freund. Deformation trapping due to thermoplastic instability in one-dimensional wave propagation. 1. Mech. Phys. Solids 32(2), 119-132 (1984).

20. 1. D. Achenbach. Wave Propagalion in Elastic Solids. North-Holland. Amsterdam (1973).

APPENOIX A

Numerical (lIRorilhm for spherical lIns{cylindric(l1 "'l/I'es (I) Read (I. h. J. n. ~I (time step), N c • N, .. N,. N, (number of local elements. imbricate elements. nodes. and time steps. respectively). 'Y. and Po. Generate arrays k(i). Iil(i).

le(i). x,.(i). k(i). Iii(i). I,.(i). x,.(i) giving the number of the left and right nodes of the ith local or imbricate element. its length. and its coordinate at the center of element. Also generate externally applied nodal forces fe:. Initialize as zero the values (for r = I) of 1',. fl.,. ;". ;". a". max E,. max t,. Tmax d. am .. , d for all k = I. .... N,. (local elements) and k = I ..... Nc (imbricate elements). (2) DO 8. r = 2 ..... N,. (3) Initialize nodal forces F, = O. f, = 0 (for all nodes. k = I ..... Nd. (4) DO 5 i = I ..... N .. (local elements). (5) Set k = k(i). m = Ifl(i). x = x,.(i) and evaluate ~E,; =

(i'm - t'd~llh. Exi -- Exi + ~E'i :lE,·i = (t'm + t'd ~1/2x. E,., -- E,.; + :lE, .. For spherical wave ~E, = ~E.'i + 2~E,;. and for cylindrical wave ~Ei <- ~En + ~E'·i: Ei <- E, + :lEi' Then call a subroutine which determines the incre­mental (tangentiall moduli K i • Gi for the local elements from their strains E,i' E". Ei and also decides whether virgin loading. unloading. or reloading applies. Then calculate ~Txi = Ki j,£; + 2G; eri • .l.T\"j = K; .lEi + 2G{ e,"j. Ixi ~

IX0 TM BELI.TSC.HKO (21 cd.

78, L A;,,. :,, = T,, - AT,,. Then calculate all the local Then calculate all the nonlocal nodal forces at element nodal forces a[ element nodes /, and VI f Fig. t&c). nodes b and m (Fig. IOd).

= - y [Tc, x2 - T,, (I - ?h)hl.

(spherical)

AFL= --

(Al) (spherical) (MI

AF,,, = + [u.;s’ + u,, (.r + ;j h] . Icylindricall (AZ)

Af,,, = y (T,, .I- + I/r r,,)J. AF& = - + (o,,.r - ; UXi) .

These forces are accumulated at each node, fk + fk +

Af4. f”, +-- fm + Afrn.L (6) DO 7. i 5 I, , E, (imbricate elements). (7) Set k = k(i). m = Xi). I = i,fi). x = f,(i) and evaluate . _._ AZ,; + (v, - vm)Arll. ,i +- ?,i + A&* AZ.“/ - (u, + L’~)A rl2.r. Fyi + Z,; + AZ,., For spherical wave AZ; = A,, + 217,;. and for cylindrical wave Zi = AZ.,, + AZ,+; E = 7; + AZ;. Then call a subroutine w&h determines the incremental (tangential) moduli K,. Gi for the imbricate elements from their mean strains &. Z.Vi, Zi and also de- cides whether virgin loading, unloading, or reloading ap- plies. Then calculate Au,, = Ki AC, + ZGiAeri. Au,; =

z,Aei + ZG,Ae,.i, u.1; + u,, + Au,;. cruj + cr,., + AuO,.~.

(cylindrical) (A4)

I--Y AF, = -

n These forces are then accumulated at each node: FL + FL + AFL. F.,, + Fe,, f AF,,,. (8) DO 8. k = I. . . . NA (all nodes). (9) Calculate 1’~ = r’l + (FL + FL + fC”)~/i(plr.v’). 11~ = ,,I, + l.llV~.

For more detailed explanations. a similar algorithm for planar stress wave given in Ref. 171 may be consulted. It may be checked that the sum of the nodal forces given by eqns (A I )-(A41 on one node yields a second-order discrete approximation of the conrinuum equation of motion (5.X).

I XII TED BELYTSCH"O ('I III.

T" ~ .IT,,. T" = T" - .IT,,. Then calculate all the lo~al nodal forcc:s at c:lement nodes J.: and 1/1 (Fig. lOcI.

.If, = - 'I [."x1 - T,,(X - !h)h].

(spherical) (A I)

'I [T.,.r + TVi (x + ih)h],

.lI, = 'I (T,i.r - til T,,1.

(cylindrical) (A::!)

.lI", = 'I (T".r + til T,,)I.

These forces are accumulated at each node. fk <- fk + .lfb fm <- f", + .If,,,. (6) DO 7. i = 1. .... N, (imbricate elements). (7) Set k = k(i). m = m(i).! = jeW. x = x,(i) and evaluate olE,; <- {v", - vdt:l.111. Ex; <- E." + .lE.". t:l.Ey; ....- (v", + Vk).l1/2x. Ev; <- EVi + .lEvi. For spherical wave t:l.E; = t:l.Ex; + 2.lE.,;. and for cylindncal wave Ei = t:l.Ex; + t:l.Ey;; E; =

Ei + .lE;. Then call a subroutine which determines the incremental (tangential) moduli Ki • G; for the imbricate elements from their mean strains E.". Eyi. E; and also de­cides whether virgin loading. unloading. or reloading ap­plies. Then calculate .l O'xi = K;.l E; + 2Gi .lex;. t:l. 0',.; = Ki.lEi + 2G j .le v;, ax; ~ ax; + ~(J.Ti. Clvi +- a,vi + .;lO\'j.

Then calculate all the nonlocal nodal forces at element nodes band m (Fig. IOd) .

.l F, = - 1 ~ 'I [ 0' "x~ - rr" (.r - ~) il J . (spheri(;al) I AJ)

(cylindrical) (A4)

1 - 'I ( h) t:l.F", = -- O'x;X + - O'v; . n 2 . These forces are then accumulated at each node: F, ..... F, + .!IF,. F", <- F,,, + ::'F",. (8) DO 8. J.: = I ..... N, (all nodesl. (9) Calculate!', = 1', + (F, + F, + If" ).l I/(pil.r1I. II, = II, + .!lll'"

For more detailed explanations. a similar algorithm for planar stress wave given in Ref. [71 may be consulted. It may be checked that the sum of the nodal forces given by eqns (A 1 )-(A4) on one node yields a second-order discrete approximation of the continuum equation of motion (5.8).