Download - Cohesive cracks versus nonlocal models: Closing the gap

International Journal of Fracture 63: 173-187, 1993. © 1993 Kluwer Academic Publishers. Printed in the Nethdrlands. 173

Cohesive cracks versus nonlocal models: Closing the gap

J. PLANAS, M. ELICES and G.V. GUINEA Departamento Ciencia de Materiales, Universidad Polit~cnica de Madrid, ETS de Ingenieros de Caminos, Ciudad Universitaria, 28040 Madrid, Spain

Received 15 November 1992; accepted in revised form 13 July 1993

Abstract. Fracture of quasi-brittle materials, particularly concrete and cement-based material, has been treated in the past using a number of mildly related models. Hillerborg's fictitious crack model (also called cohesive crack model and Dugdale-Barenblatt model), Ba~ant's crack band model, and nonlocal models, are three of the most used for theoretical as well as for applied analysis. Using a uniaxial formulation and a Rankine-type model, the present work shows that the cohesive crack may be obtained as a particular case of a fully nonlocal formulation. The discussion of the generalization of the uniaxial formulation to triaxial behavior suggests that a directional averaging, rather than an isotropic one, may be necessary.

1. Introduction

The cohesive crack model, called fictitious crack model by Hillerborg and co-workers, has been one of the essential tools in the analysis of the fracture of concrete and cement based materials since its first application to structural analysis in the mid seventies [1].

The smeared cracking counterpart - the crack band model introduced by Ba~ant - was based on the idea of using classical continuum mechanics equations - stress-strain formulation - with softening, together with an extra condition limiting the minimum size of the localization zone [2]. The localization limitation is required in order to avoid failure modes with zero energy dissipation, and in this approach it is introduced by postulating the shape (band), size (fixed width) and internal field distribution (uniform) of the localization zone.

Although over a long period there has been a permanent debate regarding which of the two models, the cohesive crack or the crack band, is the more suitable, the analysis performed by the authors suggested that as far as tensile fracture is concerned, the two approaches are computationally equivalent [3].

To introduce a localization limiter in a more systematic way, BaZant extended the concepts of nonlocal elasticity developed by Eringen and others (cf. [4]) to nonlinear situations. The early developments [5, 6] relied on defining nonlocal stresses and strains by spatially averaging the local fields, and defining a suitable constitutive equation relating the nonlocal variables rather than the classical, local ones. However, this turned out to be a very difficult formulation to handle numerically, and in the late eighties, Ba~,ant and co-workers introduced the idea of averaging only internal variables, such as damage variables or cumulated plastic strain [7, 8, 9].

The basic idea of using nonlocal counterparts of only the internal variables has been recently applied by de Borst et al. [10] to build a higher order gradient theory in tensile fracture, the analysis of which has largely inspired the present work.

In essence, this paper shows that in a uniaxial approximation, a cohesive crack model is the solution of a well formulated nonlocal model in the sense of Ba~,ant and Pijaudier-Cabot [7], (see also Simo [11] for a formalization and discussion of this kind of model).

174 J. Planas et al.

Section 2 presents the essentials of the cohesive crack model as it is used in this paper. Section 3 presents the underlying local softening model, which is taken to be a plastic model

with softening, such as those considered by BaZant and Pijaudier-Cabot [7], de Borst et al. [10] and Simo [11]. The so called localization into a step - a single surface of discontinuity in plastic strain is produced: half the specimen unloads while the other half uniformly strains - is first analyzed. Then the localization into a band is studied (all the specimen unloads except for a band that is uniformly strained) to recover the well known result of the strain being able to localize into a zone of extension zero with zero resultant elongation and zero energy

dissipation. In Section 4 the model is made nonlocal by forcing the instantaneous strength to depend on a

nonlocal variable as the mean inelastic strain over a segment, which is the simplest formulation. Then the analytical solution for step localization of the nonlocal variable is elaborated, which in terms of the local strain comes to consist of an array of equally spaced cohesive cracks over half the specimen. The cohesive cracks do lead to a non-zero extension and to a non-zero energy dissipation. The band localization is then analyzed and it is found that the nonlocal variable can localize into a band whose width is a multiple N of the averaging length. In terms of the local strain, this corresponds to N cohesive cracks. The preferred localization configuration is that in which the required external work supply is a minimum, hence, the uniaxial model will tend to fail through a single cohesive crack, as postulated in the straight approach to cohesive cracks. However, the nonlocal theory of cohesive cracks also shows that the minimum spacing between cohesive cracks is fixed, and equal to the averaging length. The need for the existence of a minimum cohesive crack spacing has been repeatedly emphasized by BaZant [12], and in the present approach this limitation is clear.

In Section 5 the nonlocal model is generalized to non-uniform averaging. It is found that a cohesive crack is indeed an exact solution of such a model, and a general method is provided to find the minimum spacing of an array of cracks. It is found that such minimum spacing is related to the averaging distance, but in general it is not equal to it. However, some exotic weighting distributions are found that do not provide a lower bound for the crack spacing.

In Section 6 the first steps toward a triaxial generalization are given. It is shown that for a uniform field, consistency with the cohesive crack approximation requires the averaging to be directional, and that the averaging direction must coincide with the minimum principal stress direction. Section 7 draws the main conclusions.

2. Outline of the fictitious or cohesive crack model

For the uniaxial formulation of the cohesive model, one may follow HiUerborg [13] or Elices and Planas [3]. Consider a bar of uniform cross-section subjected to monotonic extension as shown in Fig. 1, where a sketch of the stress-elongation curve is also shown. Along the hardening portion of the curve OP the strain is uniform. At point P the tensile strength f , is reached and bifurcation occurs. The model assumes that a cohesive crack forms somewhere in the bar and that the bulk of the material unloads. Henceforth the bulk unloads following the curve PB, while the crack concentrates the displacement as a crack opening w, which adds to the uniform strain of the bulk to give the total bar elongation (point A).

The characteristics of the fictitious crack are contained in its stress-crack opening relationship (the softening curve) which in the simplest approximation is assumed to be unique, or more

Cohesive cracks vs. nonlocal models 175

stress, G

/p crack

o// elongation, AL

Fig. I. Stress-elongation curve for a cohesive crack model.

specifically, independent of triaxiality, so one can write

a = f ( w ) . (1)

The hardening behavior need not be linear, but in our specific problem this point is largely irrelevant and a linear elastic pre-peak behavior is assumed.

The specific form of the softening curve is also of little relevance and need not be linear. The only essential features are the following

(1) it is non-negative and non-increasing, (2) for zero crack opening its value equals the tensile strength, (3) it tends to zero for large crack openings (complete failure, zero strength) and (4) it is integrable over (0, ~).

This integral is the work of fracture per unit surface of complete crack, and is denoted as Ge

Gv = f ( w ) dw. (2)

For future use, we may view the crack opening w as a displacement jump corresponding to an inelastic strain distribution which is mathematically described as a Dirac's 6-function. Hence- forth, if we take the x-axis to coincide with the axis of the specimen and let the crack be at position xc, we may write the inelastic strain el as

ei = w6(x - xc). (3)

3. UniaXial local model

3.1. Formulat ion

We consider a uniaxial elastic-plastic-softening defined by the following equations, to be explained below

o = ~ + 8P, (4)


F(a) =-- a - at <. O, (5)

'de i~>O if a = a , and d a = d a t , deP=de i with [ d e i = O otherwise, (6)

at = ~b(ei). (7)

The first equation just states the classical splitting of the strain into elastic and plastic parts, and E represents the elastic modulus. Equation (5) states that the loading surface, F(a) = 0, is assumed to be of the Rankine type (failure only in tension) where at/> 0 is the instantaneous tensile strength, whose evolution will be defined in (7). The flow rule is given in (6) which simply forces the plastic strain increment to take place in the tensile direction and only when the stress satisfies the loading condition.

The introduction of the effective uniaxial elastic strain ei is not strictly necessary in this uni.axial formulation. However, it is introduced to ease the generalization to triaxial states, where e p is a tensor, while e i is a scalar.

Finally, (7) is the hardening-softening function stating that the instantaneous local strength is a function of the effective uniaxial inelastic strain. In this work we assume that the initial value of this function is the tensile strength f t and that once the initial tensile strength has been reached, softening begins. This behavior is illustrated in Fig. 2. The points representative of the strength state of a material point always lie on that curve and move from point P downwards, never going backwards.

The plastic work per unit volume 7 is easily seen to be integral under the softening curve, so that work density required for complete fracture is 7F

f c~

Ye : (b(e i) de i. (8) o

3.2. Bi furcat ion and strain localization

Consider a uniform bar of the material just defined subjected to a monotonically increasing elongation by pulling its ends apart. The equilibrium condition requires the stress to be

f, - e o

o I equivalent uniaxial inelastic strain

Fig. 2. Softening relationship for the uniaxial local model: Instantaneous tensile strength vs. equivalent uniaxial inelastic strain.

f, (a)

.p

- t -


equivalent uniaxial inelastic strain, E i

(b)

B

(c)

El

V

[- x - - ~

D B I

Fig. 3. Bifurcation and localization for the local model. (a) Loading path OPA and loading-unloading path OPB. (b) Localization into a step. (c) Localization into a band.

uniform. Henceforth, the elastic strain alE is also uniform and only the plastic strain may be variable. We may draw the path of a material point as a curve in a a - el plot shown in

Fig. 3a. From the equations set in the preceding paragraph, the only points that may be reached on this graph are those in the area limited by the vertical (a) axis and the softening curve. A material point following the path OPA is subjected to monotonic loading (strictly increasing ~i).

A possible solution of the problem is that in which all the points of the bar follow the loading branch along OP, and upon reaching P bifurcation occurs and some of the points of the bar follow PA, while the remaining points unload following PO. The equilibrium condition requires that the unloaded portions of the bar be in the state defined by point B in which the stress is the

same as that at point A. The set of material points in state A is any non-void set. One of the infinitely many solutions

is a localization into a step, in which all the material points of the bar on the right of a given cross-section load, while all those on the left unload (Fig. 3b). Another possibility is a localization into a band, where the points which are monotonically loading are those within a band of given thickness h, as shown in Fig. 3c. For this latter case, the required work supply for complete failure is

W p = ShTv, (9)

where S is the cross-section of the bar. The fundamental issue is that of all the possible (admissible) modes of failure, the actual one is

that in which the required work supply W p is minimum. And from (9) it follows that this minimum is zero and corresponds to failure through a vanishingly thin band, i.e. failure at a single 'mathematical' cross-section.

This mode of failure is prevented in BaZant's crack band model by postulating that the band width h cannot be less than some minimum width hc which is a material property.

In the following section this zero-measure failure mode is prevented by introducing a nonlocal variable into the constitutive equations.


4. A simple nonlocai model: Uniform averaging

4.1. Nonlocal modification o f the local model

The uniaxial model of the previous section is made nonlocal by introducing a nonlocal variable as the spatial average of the effective uniaxial inelastic strain. In this section we use the

simplest approximation, the uniform averaging, which was used by Ba2ant in the early developments of the nonlocal theory; see [12]. The effect of using weighted averaging will be

explored in the next section. Let the material points of the model be defined by their coordinate x, and let )~ be an

averaging distance or internal length, which is a material property. We define the nonlocal

effective inelastic strain at a point as

1 ('x + 2/2 f~(x) = - I e i (x ' )dx '. (10)

Now, of the four equations, (4)-(7), defining the local model, the first three are kept unchanged and the last one, defining the softening function, is modified by stating that the instantaneous

tensile strength at is a function of the nonlocal variable f~

~t = ~(K2). (11)

With these modifications, the nonlocal model is complete and ready to be analyzed.

4.2. Localization into a step

Before going any further, we notice that since the instantaneous strength is defined by (11) in terms of the nonlocal variable, a a - e i plot is now meaningless and must be substituted by a

- f2 plot as shown in Fig. 4a. An analysis strictly parallel to that in Section 3.2 shows that the possible equilibrium situations are those in which some points of the bar are in state A, wih

= f~a, while all the rest are in state B, with ~q = 0. One of the possibilities is the localization into a step, with I'~ = 0 on the left of the step and

f~ = f2 A on the right of it, as depicted in Fig. 4b. To simplify the formulation, we take, without loss of generality, the step to be located at the origin, and we use the Heaviside step function H(x) to write the distribution of f2 in the step localization as

f~(x) = f lai l (x) . (12)

The problem now is to find which distribution of plastic strains and effective uniaxial inelastic strains (which are identical to each other in this uniaxial problem) correspond to the above distribution of fL This is done by solving the integral equation that results from substituting (12) into (10)

f[ + a/2 d(x ' ) dx ' = AllAH(X). (13) - ~ . 1 2


t~

(a) -p

(a)

nonlocal equiv, uniax, inelastic strain, X"2

(b) A f~ V

~ - x ~

t B I t

(d) u i F

I

I

Fig. 4. Bifurcation and localizaton for the nonlocal model. (a) Loading path OPA and loading-unloading path OPB. (b) Localization into a step of the nonlocal variable. (c) The local inelastic strain localizes into an array of equally spaced 6-functions (cohesive cracks). (d) Inelastic displacement distribution.

The solution of this integral equation may be sought as a the sum of a particular solution, say e~(x), plus the general solution of the homogeneous integral equation. One of the particular solutions turns out to be a discrete, but indefinite, array of Dirac's f-distribution over the positive part of the x-axis

e~,(X)=2f~a,~16 X ~ 2 . (14)

The homogeneous integral equation obviously accepts as a solution any alternating function of period (main wavelength) equal to 2, say Aa(x). To find the actual solution we need to impose a further restriction on it. This restriction is that e i must be everywhere positive or zero. Since the particular solution is zero almost everywhere (in particular it is identically zero on the negative portion of the x-axis) and any non-zero alternating function contains, by definition, negative values, it follows that A~(x) must be identically zero. Therefore, the solution to the inelastic strain distribution is given by the array of 6-distributions of (14), as depicted in Fig 4c.

The above distribution is equivalent to an array of equally spaced cohesive cracks with spacing 2. Indeed, if we compute the inelastic displacement associated with the inelastic strain as

f x

ui(x) = ei(x') dx', (15) - o o

180 J. P l a n a s et al.

we find the stepped distribution shown in Fig. 4d. One may call the displacement jump at each

step 'crack opening', represent it as w, and easily obtain its value as

W = /~'-~A" ( 1 6 )

The same conclusion is reached by directly comparing the solution (14) with the inelastic strain distribution (3) for a single cohesive crack.

To obtain the softening function f ( w ) of the cohesive crack, one determines the stress transferred through the cracks from the softening relation (11), the condition that at point A cr = a,, and the relationship (16) between w and ~. The result is

a = ~)(w/2) - f ( w ) . (17)

This completes the cohesive model to which the nonlocal model is equivalent. The only difference from the straight approach to the fictitious crack given in Section 2 is that there a single crack was postulated, and up to now we have an array of infinite cracks. We turn to the single crack in the next subsection.

4.3. L o c a l i z a t i o n into a band

Consider now the case in which the nonlocal variable f~ localizes into a band. Let us assume that the band is over the interval (0, h) where, for the moment, h is an arbitrary length. Following the same steps as in the section before, we are bound to find the solution of the integral equation

x+,V2 d ( x ' ) d x ' = 2 D A [ H ( x ) -- H ( x -- h)], - 4/2

(18)

and, because of the linearity of the equation, we may use a superposition procedure to find that the solution is

2n'lE 2n11} d ( x ) = 2 ~ , , y ~ 6 x ~ ~. - 6 x - h ~ ,~ , n = l

(19)

However, this solution cannot be valid in general, because, as already stated, d cannot be negative. Because of the negative 6's appearing in the sum in the above equation, the only valid solutions are those in which the positive and negative 6's cancel each other from one on. This only happens if h is an integer multiple, say N, of the internal length 2. This is in fact the loca l i za t ion l imi t ing cond i t ion

h = N 2 N = 1 , 2 , 3 . . . . (20)

The corresponding inelastic strain distribution is a set of N cohesive cracks

2nll gi(x) = 2f~a 6 x ~ 2 . (21) n = l


The energy consumed in creating these cracks is N times that for a single crack. Hence the preferred mode of failure is that in which a single crack forms. With this, the proof that the uniaxial cohesive crack model is the solution of a particular nonlocal model is complete.

Remark 4.1. A surprising feature of the foregoing solution is that, in some sense, it displays degradation at a distance. Indeed, while all the inelastic strain is localized at a point (where all the inelastic work is concentrated), the strength of the material decreases (degrades) at points a finite distance apart from the inelastic point. This will be a more obvious feature in the general case discussed in the next section.

5. Nonlocal models with weighted averaging

When a weighting function is introduced, a methodology slightly different from that in the previous section is more convenient. After introducing the weighting function we show that localization into a single cohesive crack (a Dirac's delta function in e i, as before) is a valid solution of the nonlocal model. Next we examine the possibility of an array of cohesive cracks evolving simultaneously, and develop a graphical method to determine the minimum crack spacing for any weighting distribution.

5.1. The weighting distribution

A weighted average is introduced defining the nonlocal variable ~ as

lff ~(x - x')ei(x ') dx', (22) ~ ( x ) = ~ _

where ~(s) is a dimensionless weight distribution defined on ( - 2, 2). For practical purposes, the existence interval of this function may be extended to ( - oo, @) by setting

~(s )=0 for [s [>2. (23)

In this paper we restrict ourselves to weight functions satisfying the following conditions:

(1) It is even: ~(s) = ~(-s); (2) It is non-negative: ~(s) >~ 0; (3) It is bounded; (4) It has only a maximum at s = 0; (5) The value of its maximum is 1:

~(0) = 1 (24)

A number of weight distributions satisfying this condition have been plotted in Fig. 5. Note that the weight distribution used in the preceding section is a particular case (Fig. 5b).


(fl) Ct (b)

?-

(c) c~

(d) a

- 5 s 2 1

°c=e) 1

(e) o:

--$2 ~

(t3 a

-Isl 1

Fig. 5. Weight distributions. (a) General. (b) Uniform averaging. (c~(f) Some analytically defined cases.

Remark 5.1. In this work we use (24) as the normalization condition rather than forcing the integral of ~(s) to be unity, which is more usual. The reason for this will become clear in the following. For the time being, just note that the only modification introduced by the normalization is a scale shift in ~, which may be absorbed by adequate re-scaling of function q~(~) in (11). Therefore, no loss of generality is involved in adopting (24), except that the weight function is necessarily bounded.

5.2. The cohesive crack: an exact solution

We now examine whether a cohesive crack can or cannot be a solution of the problem of the monotonic stretching of the bar. Taking the origin of coordinates to coincide with the crack site, Fig. 6a, we write the inelastic strain distribution as

ei(x) = w6(x) (25)

where w is, as before, the crack opening. From this expression, the nonlocal parameter ~ is found from (22)

w ~2(x)=~-~(x). (26)

This ~) distribution has been sketched in Fig. 6b. To be consistent with the model defined by (4)-(6) and (11), the following two conditions must be met:

1. The stress must coincide with the instantaneous tensile strength at at the points where inelastic strain takes place, i.e. at point x = 0. Therefore, according to (5) and (11) we must have

a = ~b[n(0)] = e(0 = , (27)


(a)

~ crack

wA

A° l A2 A1

~g O

(c) -p

o,=,(a)

D

Fig. 6. Cohesive crack as a solution of a nonlocal model: (a) Inelastic strain distribution. (b) f~ distribution. (c) tr - ~q trajectories.

where the second equality follows from (26) and the last one from the normalization condition (24). Point Ao in Fig. 6c represents the condition (27).

2. At any other point, where d = 0, inequality (5) must be satisfied, i.e. a ~< a,. According to (27) and (11), this is satisfied if

f~(x) ~ f~(O) or ~(x) ~< ~(0) for x # O,

which, in view of the restrictions imposed to 7(s), is automatically satisfied, which is obvious from Fig. 6b.

Thus, for a bounded weight function, a cohesive crack is an exact solution of the nonlocal problem as formulated here. It is not unique, but it is a good candidate to be the solution requiring minimum work supply.

Remark 5.2. In this solution, the effect of degradation at a distance is fully noticeable. While only one point (that at the crack site) undergoes inelastic strain, all the dashed region in Fig. 6b experiences loss of strength. Note that only one point (Ao) follows the a - t2 softening curve. The other points follow ~ - f~ trajectories which lie below the limiting softening curve, as depicted in Fig. 6c by thin lines P - A1 and P - A2.

5.3. Arrays o f cracks

We now examine how close cohesive cracks may be placed in order to be a possible solution. Consider two identical cracks of opening w located, respectively, at x = 0 and x = h. The inelastic strain distribution is, thus,

~i = w[6(x) + 6(x -- h)], (28)


(b) ~ ~ .......

?_ t _ , _ h .

~:~!, ~ ~ ~7'~ ....... ii I . . . . .

~ h m .

@ (f)

Fig. 7. (a) Right hand member of (32) forbids points into the shaded region. (b) Left hand member of (32) represents function ~(s) with origin at the heavy dot. For the distribution in Fig. 5e: (c) valid situation, (d) non-valid situation, and (e) limit situation. (f) For the weight distribution with a central cusp of Fig. 5f there is no restriction on h.

with the corresponding ~ distribution being, from (22)

f~ = ~ [a(x) + ~(x -- h)]. (29)

The steps to follow to see what values of h are acceptable are exactly the same as before:

(1) At the crack location sites one must have a = a~; (2) a ~< at everywhere else.

These conditions are rewritten, using (11), (29) and (24):

a = qSEn(O)] = 4~Ef~(h)] = q$[z[1 + a ( -h ) ] ] , (30)

c~(x) + c~(x -- h) ~< 1 + c~(-h). (31)

The set of values of h satisfying inequality (31) are the physically admissible spacings for an array of equally spaced cracks. The minimum spacing hm, depends on the specific shape of the function ~(x). A very easy graphical analysis may be performed by writing the equation in the form

e(x - h) - ~(-h) ~< 1 - e(x). (32)

The right hand member defines the fixed shaded region shown in Fig. 7a. According to the inequality, no point of the left hand member can lie inside the dashed region. The left hand side is a curve depending on parameter h. Its graphic representation is a parallel displacement of the curve ~(x) in such a way that the origin is translated to ( - h , ~(-h)) as shown by the heavy dot in Fig. 7b. The valid solutions are those in which, when Fig. 7b is superimposed on Fig. 7a with the heavy dot at the origin, no point of the curve falls onto the dashed region.


To illustrate the method, we use the weight function depicted in Fig. 5e. Then, Figs. 7c, 7d and

7e respectively show a valid situation, a non-valid situation, and the limiting situation for which h takes the minimum acceptable value h,,. Note that for this weighting distribution, h,. ~. 1.32. For the distribution in Figs. 5b-5d it is easily shown using the graphic procedure that h,, = 2.

A very special case is that in which the weight function has a cusp point at the origin and is concave on the sides (Fig. 5f). In such a case any spacing is possible as shown in Fig. 71". Such a kind of weighting function should, thus, be used with care.

6. T o w a r d s a tr iax ia l genera l izat ion

The triaxial generalization of the uniaxial formulation is not simple, but one may go some steps towards it. The simplest generalization is to keep a scalar softening model with a Rankine-type of loading function. To take into account the plastic strain directionality (i.e. the fact that cracking occurs essentially perpendicularly to the maximum principal stress direction), one may use a unit eigenvector associated with the principal stress ab say p~. Then, by definition,

t rp l=a lp l and p l ' p l = l , (33)

where a is the stress tensor and the dot indicates scalar (internal) product. For directional tensors, we use the projector tensor P~ associated to the eigenstress cr~

PI = Pl (~ Pl and Pl"o = Pl'(~Pl) = o ' i (34)

where ® indicates tensor product and the dot, again, scalar product.

With this definition, (4)-(6), common to local and nonlocal formulations, and the softening rule (11), are rewritten as

---- Co" + e p (35)

F(~r) -- th - o't ~ 0 (36)

~dg i ~> 0 if a = at and da = d 6 t

dgP- -P lde I with [ d e i = o otherwise (37)

~ , = ~ ( f l ) (38)

where the e and ~P represent strain and plastic strain tensors respectively; C represents the fourth order tensor of elastic compliance; f~ is a nonlocal variable to be defined in the sequel; and the other variables were already defined in Section 3.1 and in (34).

The next step is to try to adequately define the nonlocal scalar variable ft. In order to keep the cohesive crack solution in at least one situation, namely, proportional uniform loading of a large plate, as depicted in Fig. 8, it turns out that the averaging must be a line averaging rather than a volume averaging. This must be so because otherwise the cohesive crack would be itself nonlocal, i.e. the transferred stress at a point would depend on the crack openings in a


Fio. 8. Proportional uniform stressing of a plate. Inelastic strain localizes into a crack normal to the maximum principal stress direction, defined by vector Pv

neighborhood of that point. Remark that we do not state that this could not be the case. We simply do not want to look at such a complex case right now. Note also that even if the usual formulations of nonlocal models take a non-directional volume average (isotropic average), in BaZant's bar, td formulation, what is of primary importance is the average across the band, not along it. Hence, our findings do not seem to lie far from the usual understanding.

Going to a particular formulation, the simplest one is to choose the principal stress direction as the averaging direction, and then define the nonlocal average as

= c~(s)ei[x - spl(x)] ds, (39) n(x) ~-~

where now x is the position vector of a given point, and s the arc-length taken along a straight line in the direction of the maximum principal stress at that point. This equation reduces exactly to (22) for uniform states when the x-axis is taken to coincide with the direction of p~, with the obvious change of variables s = x - x'.

Obviously, this generalization is simple, but by no means the only one at hand. Many line averages may be defined in place of that of (39) that give the same result for uniform states. The most obvious and most appealing is the one in which the integration line coincides with the maximum tension isostatic line through the point at which the average is taken. Work is in progress to ascertain which of these generalizations is more suitable.

7. Conclusions

The essential conclusions of this work are the following:

1. The simple nonlocal model presented in Section 4 and the more sophisticated models with weighted averaging of Section 4 are able to prevent the development of failure modes with


zero measure. They accept a cohesive crack as an exact solution, thus providing a sound

basis for con t inuum-based approaches to cohesive crack models. 2. The nonlocal variable f~ defined by (22) localizes into a band in a manner similar to Ba2ant 's

crack band model. The f~-distribution has the same shape as the averaging weight function

~(s) (Fig. 5 and 6). 3. The local inelastic strain localizes into a Dirac 's f-distr ibution. This is physically equivalent

to a cohesive crack because a l though the geometrical width of the localization is zero - it

provides a finite displacement j u m p and requires a finite work supply. 4. The nonlocal model natural ly introduces the result that if multiple parallel cracking is to

occur, the cohesive cracks must keep a spacing not less than some distance h,,. The value of

hm is related to the averaging distance ), but, in general, is not equal to it. A graphical

me thod is provided for a rapid determinat ion of h,,. 5. The first steps towards a triaxial generalization show that in order to keep cohesive cracks

local (stress at a point related to crack opening at this point only) the averaging must be

per formed along a line rather than over a volume. This is perfectly in accordance with the

general feeling that in a band approx imat ion what is essential is the averaging across the

band, not the averaging along it. 6. The nonlocal averaging direction must be initially controlled by the m a x i m u m principal

stress direction. Averaging along a straight line as defined in (39) is the simplest approxi-

mat ion, but averaging along isostatic lines may be another reasonable issue which is

currently being analyzed.

Acknowldgements

The authors gratefully acknowledge financial suppor t for this research provided by the

Comis i6n Interminister ial de Ciencia y Tecnologia D I G I C Y T , Spain, under grants PB90-0276

and MAT92-0031.

References

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