Numerical analysis of catastrophic softening behaviour (snap-back instability)

0045.7w9/89 53.00 + 0.00 0 ws9krpmorlRrupk

NUMERICAL ANALYSIS OF CATASTROPHIC SOFTENING BEHAVIOUR (SNAP-BACK INSTABILITY) ALBERTO

CARPrNTtXtt and GIOVANMC~LCX#BO~

TDcpartment of Structural Engineering. Politccnico di forino, $ENEL-CRIS, 20162 Milano, Italy

10129 Torino, Italy

(Reccbecf 20 Jcmurvy 1988) Ahatract-The nature of crack and structure bchaviour can range from ductile to brittle in dependence on material properties, structure geometry, loading conditions and external constraints. The influence of variation in fracture toughness, tensile strength and geometric size scale arc investigated on the basis of the n-Theorem of dimensional analysis. Strength and toughness present in fact different physical dimensions and any consistent fracture criterion must describe mcrgy dissipation per unit vohmrc and per unit crack area respectively. A cohesive crack model is proposed, which aims to describe the size effects of fracture mechanics, i.e. the transition from ductile to brittle structure bchaviour, by increasing the sixe scale and keeping the geometric shape unchanged. For extremely brittle cases (e.g. initially untracked specimens, large and/or slender structures, low fracture toughness, high tensile strength, etc.) a snapback in the equilibrium path occurs and the load-deflection softening branch assumes a positive slope. Both load and deflection must decrease to obtain slow and controlled crak propagation (whereas in normal softening only the load must decrease). If the loading process is deflection-controlled, the loading capacity presents a discontinuity with a negative jump. It is proved that such a catastrophic event tends to reproduce the classical LEFM-instability (K, = K,,) for small fracture toughnesses and/or for large structure sizes. In these cases, neither plastic zone development nor slow crack growth occurs before unstable crack propagation.

I. INTRODU~ION

Crack growth in real structures is often stable and develops slowly with initially partial and then total stress-relaxation at the crack tip. Only in some particular cases does crack formation and/or propagation occur suddenly with a catastrophic drop in the loadcarrying capacity. The nature of crack and structure behaviour can range from ductile to brittle in dependence on material properties, structure geometry, loading condition and external constraints. The influence of variation in fracture toughness. tensile strength and geometric size scale may be investigated on the basis of the n-Theorem of dimensional analysis [l-3]. Strength and toughness parameters present, in fact, different physical dimensions and any consistent fracture criterion and numerical model must describe energy dissipation per unit volume and per unit crack area respectively [4]. When the fracture phenomenon is extremely brittle and only energy dissipation on the crack surfaces occurs, LEFM may be properly applied and describe the catastrophic structure behaviour. On the other hand, when the fracture process is extremely ductile and only energy dissipation in the ligament volume develops, Perfectly Plastic Limit Analysis may be conveniently used. The problem arises for intermediate cases, when the fracture process is initially ductile and then brittle and crack propagation is initially slow and then fast and uncontrollable. In those cases, a suitable fracture model must be adopted. In the present paper a cohesive crack model is proposed,

with the aim of describing the size effects of fracture mechanics, i.e. the transition from ductile to brittle structure behaviour, only by increasing the size scale and keeping the geometric shape unchanged. For extremely brittle cases (e.g., initially untracked specimens, large and/or slender structures, low fracture toughness, high tensile strength, etc.) a bifurcation in the equilibrium path occurs and the load P-deflection 6 softening branch assumes a positive slope (dP/db > 0). This means that both load and deflection must decrease to obtain slow and controlled crack propagation, whereas in normal softening (dPid6 < 0) only the load must decrease. If, in the former situation, the loading process is deflectioncontrolled, the P-d curve presents a discontinuity and the representative point drops on the lower branch with negative slope. It is proved that such a catastrophe tends to reproduce the classical Griffith instability for small fracture toughnesses and/or for large structure sizes. The accuracy of the numerical description of cohesive crack propagation is also investigated. It is shown that, when the finite element mesh is too coarse, i.e. when ihe cohesive forces are too far from each other, the cohesive model is unable to describe the fracture process regularly. In other words, when the structure is very large or the fracture toughness very small, the plastic or cohesive zone at the crack tip becomes relatively small and the finite element mesh must be refined, so that such a zone and the LEFM-stress-singularity can be reproduced and the equilibrium branching described. 607

608

ALDERTO CARPIHIERI and GKWANNI Coiostao

For each critical value of strain energy release rate Glc, an upper bound to the finite element size exists, beyond which the numerical representation is irregular. Vice versa, for each finite clement size a lower bound to G,c exists, below which the P-d curve is not reproducible. Such considerations are extended to the non-dimensional field and it is shown that, for low fracture toughnesses and/or large structure sizes, the loading capacity can be provided by the well-known condition K, r: Klc In these cases, it is not convenient to use a cohesive crack model with a huge number of finite elements. It is much more economical to apply a LEFM code with the stress-singularity embedded at the crack tip.

2. COHESIVE CRACK MODEL

The cohesive crack model is based on the following assumptions [S-8].

STRAIN.

(1) The cohesive fracture zone (plastic or process zone) begins to develop when the maximum principal stress achieves the ultimate tensile strength u, (Fig. la). (2) The material in the process zone is partially damaged but still able to transfer stress. Such a stress is dependent on the crack opening displacement w (Fig. lb). The real crack tip is defined as the point where the distance between the crack surfaces is qua! to the critical value of crack opening displacement w, and the normal stress vanishes (Fig. 2a). On the other hand, the fictitious crack tip is defined as the point where the normal stress attains the maximum value u, and the crack opening vanishes (Fig. 2a). The closing stresses acting on the crack surfaces (Fig. 2a) can be replaced by nodal forces (Fig. 2b). The intensity of these forces depends on the opening of the fictitious crack, w, according to the u-w constitutive law of the material (Fig. lb). When the

f

(a)

0

1 Area=louwq=

9,c

CRACK OPENING DISPLACEMENT

Fig. I. Stress-strain

IaN (a); linear stress-crack opening displacement law (b); bilinear stress-crack opening displacement law (c).

609

Numerialanalysis of catastrophic softening behaviour

On the other hand, the initial crack is stress-free and therefore: fori=1,2

F,=O,

,...,

(k-l),

(2a)

!?E

while a1 the ligament there is no displacement discontinuity:

>

w,= 0,

(2W

Equations (1) and (2) constitute a linear algebraic system of 2n equations and 2n unknowns, i.e. the elements of vectors w and F. If load P and vector F

h (b)

for i = k, (k + l), . . . , n.

’

are known, it is possible to compute the beam deflection, 6 :

Fig. 2. Stress distribution across the cohesive zone (a) and quivalent nodal forces in the finite element mesh (II).

b==CrF+DpP+D,, tensile strength o, is achieved at the fictitious crack tip (Fig. Zb), the top node is opened and a cohesive force starts acting across the crack, while the fictitious crack tip moves to the next node. With reference to the three-point bending test (‘TPBT) geometry in Fig. 3, the nodes are distributed along the potential fracture line. The coefficients of influence in terms of node openings and deflection are computed by a finite element analysis, where the fictitious structure in Fig. 3 is subjected to (n + I) different loading conditions. Consider the TPBT in Fig. 4a, with initial crack of length o, and tip in node k. The crack opening displacements at the n fracture nodes may be expressed as follows: n=KF+CP+l-,

(3)

where D, is the deflection for P = 1 and D, is the deflection due to the specimen weight. After the first step, a cohesive zone forms in front of the real crack tip (Fig. 4b), say between nodes j and 1. Then qns (2) are replaced by F, = 0,

fori=l,2

Fi=F,

fori=j,(j+l)

I,

(4b)

w, = 0,

for i = I, (I + 1). . . . , n,

(44

,...,

(4a)

(j-l), ,...,

where F. is the ultimate strength nodal force (Fig. 2b): FM= ba,lm.

(1)

where

(5)

Equations (1) and (4) constitute a linear algebraical system of (2n + 1) equations and (2n + 1) unknowns, i.e. the elements of vectors w and F and the external load P. At the first step, the cohesive zone is missing (I= j = k) and the load P, producing the ultimate strength nodal force F. at the initial crack tip (node k) is computed. Such a value P,, together with the related deflection 6, computed through eqn. (3), gives the first point of the P-d curve. At the second step, the cohesive zone is between the nodes k and (k + l),

w is the vector of the crack opening displacements, K is the matrix of the coefficients of influence (nodal forces), F is the vector of the nodal forces, C is the vector of the coefficients of influence (external load), P is the external load, r is the vector of the crack opening displacements due to the specimen weight.

I

P

t

b

,-

I

Fig. 3. Finite element nodes along the potential fracture line.

ALBEWOCAIIPINYBU and GWANNI CCUXIBO

610

Fig. 4. Cohesive crack configurations at the first (a) and (I - k + 1)th (b) crack growth increment.

and the load Pz producing the force F, at the second fictitious crack tip (node k + 1) is computed. Equation (3) then provides the deflection 6,. At the third step, the fictitious crack tip is in the node (k + 2), and so on. The present numerical program simulates a loading process where the controlling parameter is the fictitious crack depth. On the other hand, real (or stress-free) crack depth. external load and deflection are obtained at each step after an iterative procedure. The program stops with the untying of the node n and, consequently, with the determination of the last pair of values F, and 6,. In this way, the complete load-deflection curve is automatically plotted by the computer.

brittleness number sh can be put also in the simpler form: SE =

WC 26

which is the ratio of the critical crack opening displacement H’,to the characteristic structure size 2b. The three-point bending beam in Fig. 3 is considered herein, with the constant geometrical proportions: span = I = 4b. thickness = I = b. The scale factor is therefore represented by the beam depth b. As is shown in Fig. 2(b), m finite elements are adjacent to the central line, whereas only n = 0.9m nodes can be untied during crack growth (Fig. 3). The finite element size h (Fig. 2b) is then connected with the beam depth 6 through the simple relation: b =mh.

3.

DUCTILE-BRITTLE

TRANSITION AND NUMERICAL

ACCURACY IN FINITE ELEMENT ANALYSIS

The accuracy of the numerical description of cohesive crack propagation is investigated in non-dimensional form. For this purpose, the brittleness number sL is introduced [8-131. G

SE =

_1_c a,b'

where G# = $,w, is the area under the U--H’curve in Fig. l(b) and represents the energy necessary to create a unit crack surface, CJ,is the ultimate tensile strength and b is the depth of the beam in Fig. 3. The

The three different finite element meshes in Fig. 5 are considered. Mesh (a) presents 20 elements and 18 fracture nodes, mesh (b) 40 elements and 36 fracture nodes, mesh (c) 80 elements and 72 fracture nodes. The load-deflection response of the three-point bending beam in Fig. 3 is represented in Fig. 6(a-c), for m = 20, 40 and 80 respectively. The initial crack depth is assumed q/b = 0.0, while the ultimate tensile strain Ey = au/E is 8.7 x 10-j and. the Poisson ratio Y = 0. I. The diagrams are plotted in non-dimensional form by varying the brittleness number sy. The simple variation in this dimensionless number reproduces all the cases related to the independent variations in G,c, b and 6,. Not the single values of Glc, b and u,, but their function s Csee eqn. (6~is responsible for the structural behaviour, which can range from ductile lo

Numerical analysis of catastrophic softening b&&our

611

m=

lb1

m x80

Fig. 5. Refinement of the finite element mesh.

brittle. Specimens with high fracture toughness are then ductile, as well as small specimens and/or specimens with low tensile strength. Vice versa, brittle behaviours are predicted for low fracture toughnesses, large specimens and/or high tensile strengths. The influence of the variation in the number sy is investigated over four orders of magnitude in Fig. 6, from 2 x lo-* to 2 x lo-‘. The results reported in Fig. 6(a-c), appear very similar. Of course, the diagrams for m = 20 (Fig. 6a) are slightly less regular than those for m = 80 (Fig. 6c), and present some weak cuspidal points, especially for low sE numbers. When ulteriorly lower sb numbers are contemplated, the P-d diagrams lose their regularity, from a mathematical point of view, and their resolution, from a graphical point of view. The influence of the variation in the sf number is further analyzed over

one order of magnitude in Fig. 7, from 2 x lo-’ to 2 x 10m6. The results reported in Fig. 7(a-c) appear much less uniform than those in Fig. 6(a-c). The diagrams for m = 20 (Fig. 7a) are lacking in mathematical regularity, graphical resolution and physical meaning. The diagrams present a slightly better regularity and resolution for m = 40 (Fig. 7b), whereas, for m = 80 (Fig. 7c), they appear s&dently regular, especially when brittleness numbers are not too low (lO-‘
612

m -

20

a,/b t/b

= 0.0 = 4

-8.7L

5

E-5

R SE z Z.OSE-6 6 SE = 4.16E-6 C SE I

6.2X-b

D SE L 6.66L-6 2 62 f

=10.462-S

SE a20.60L-6

6 SE a41 .SOE-6 Ii SE ~62~701-6 I SE rSf.bPE-6 L SE z

t .042-t

6 6E = 2.06E-3 N 6E = 4.lSE-3 0 SE 2 6.272-3 f

SE = 6.36E-6

0 SE rlO.46L-3 I

SE r20.606-3

DIMENSIONLESSDEFLECTION , 6/b X 10)

Fig.6(a).

r: d

In=40 a,/b ~0.0 t/b =I Eu =a.74 E-5 A SE = 2 .OOE-6

* =, g 5" . c1

E SE :10.46E-S f

? 0

SE :20.90E-5

0 SE =41.60E-S 8 SE =62.?OE-5

5 n

2;

1 SE 163.6SE-6 L SE t

z zf! g

4.18E-6 b.P’IE-S

0 SE = 6.36E-6

z ::

8 SE : C SE f

a

SE

=

a

0”

l.OlE-3

!I SE z 2.06E-3

%c b

N 6E i

4.lSE-3

0 SE i

6.271-3

P SE z 6.36E-3 0 SE =lO.lbE-3

,~~~~,

.

,

,

,

).

r,

*

II SE :20sSOE-3

*

8 GO.0

I .2

2.4

6.6

DIHENSIO1YLESS DEFLECTION , 6/b X 10' Fig. 6(b).

4.6

Nw&oal

analysis of atuvophic

8o!hinp

613

bchviour

= 0.0

a,/b

=L cu - 0.74 t/b

A SE *

2.O@E-6

D BE *

4.IBE-6

E-5

C SE s 6.27E-6 D 82

= I.SBE-6

E SE 810.4SE-6 F IE

rOO.#OE-6

0 SC =41.8OE-6 II SE =82.7OE-6 1 SE x13.691-6 L IE

s l.OIE-3

.?I IL

= 2.08E-3

N EL = 4.102-3 0 IL

= 6.27E-3

? IL

= 8.36E-5

0 SE =10.46E-3 R SE rZO.BOE-3

DIMENSIONLESSDEFLECTION . 6/b X 10' Fig. 6(c). Fig. 6. Dimensionless load-deflection response of an initially uncrackcd specimen, by varying the brittleness number, sE = G,,/aJ = w,/2b, between 2 x IO-’ and 2 x IO-‘. (a) m = 20, (b) m = 40, (c) m = 80

q = 20 a,/b = 0.0 L/b = L Eu = a.7.i~-5

R SE

:

B SE

=

.42E-S

C SE =

.6?iE-6

0 SE

=

.84E-6

E SE

s 1 .OlE-6

F SE : 0 SE

.21E-5

1.2bE-6

E 1.46E-5

H BE L I.b’lE-S I

SE

s

l.WE-6

L BE x 2.0@E-6

1.

i

DIMENSIONLESSDEFLECTION

,

6/b X lo* Fig. 7(a).

614

m =LO a,/b t/b

= 0.0

=L

A SE I

.2lE-5

B SE E

.42E-6

C SE E

.63E-S

0 6E =

.64E-S

E SE = l.O4E-S F SE = 1.2SE-6 0 SE t 1.4bE-6 Ii If s l.VE-S I SE = l.OE-S L SE c 2.0BE-6

I’ DIHEKSIONLESS

i DEFLECTION

b/b X 10’

,

Fig. 7(b).

m

= 80

a,/b = 0.0 L/b = 1 = a.74 E”

~-5

R SE =

.ZlE-5

6 SE =

.42E-5

C SE E

.63E-5

0 SE :

.64E-5

E SE s l.O4E-5 F SE = 1.2SE-5 C SE = 1.4SE-5 Ii SE o 1.67E-5 I SE E l.IBE-5 L SE : Z.OSE-5

DIMENSIOKLESS

DEFLECTION

,

6/b X lo4

Fig. 7(c). Fig. 7. Dimensionless load-deflection response of an initially untracked specimen, by varying the brittleness number, sL = G,,,uJ = w,,‘Zb, between 2 x 10m6 and 2 x IO-‘. (a) m = 20, (b) m = 40, (c) m = 80.

Nun&cd

rndysis of atastrophk

softening bchaviour

m I

615

20

l,/b =O.l t/b =I .8.74 E-5 E" 0 SE = 4.16L-6 C SE = 6.27E-6 0 SE = 6.36C-6 E SE xlO.4SE-6 F 6E xZO.BOE-6 0 SE rll.OOE-6 II SE r62.7OE-6 I SE rlS.bBE-6 L 6E z

l.OIE-3

II SE I

2.06E-3

II 6E 5 4.I6E-3 0 SE t

6.27E-3

? SE I

6.36E-9

0 6E I

.lO.lbE-3

SE x20.6OE-3

DIMSNSIOKLESSDEFLECTION, 6/b X 10'

Fig. 8(a).

R MiO.OI 6 OFzO.02 c of=o.o3 0 oF=O.Ol E OfsO. F of=0.10 D ofs0.20 Ii w+o.30

DIMENSIONLESSDEFLECTION , d/b X 10'

Fig. E(b).

ALDERTO CARPINYEFU and GKWANNI coLoMe0

616

A SE s 2.06E-6 6 SE x 4.16E-S C 6E

x 6.27E-6

0 6E L 6.66E-s E 62

=10.4SE-6

F 6E s20.60E-6 0 6E %41.6OE-S Ii 6E =62.7OE-6 I

6E =6S.SOE-s

L 6E

:

1 .OIE-3

Ii SE :

2.06E-9

N SE = 4.16E-3 0 SE = 6.27E-3 P 6E = 6.36E-3 0 SE :10.45E-I R SE r20.60E-3

dd.0

1’.2

2’.4

i.6

DIKENSIONLZSS DEFLECTION , 6/b

4.6

X 10’

Fig. 8(c) Fig. 8. Dimensionless load-deflection response of an initially cracked specimen (a,:b = 0. I). by varying the brittleness number, sL = G,,/a,b = ~,,2b, between 2 x IO-’ and 2 x IO-‘. (a) WI= 20, (b) m = 40, (c) M = 80.

=20

m

a,/b l/b E”

= 3.1 = 1 D 6.71

R SE -_

.21E-5

6 SE =

.42E-S

C SE =

.63E-S

D SE o

.S4E-S

SE :

I .OIE-S

F SE z

1.2SE-S

E

0 SE i I.lbE-S ” SE z I

l.S’)E-S

SE : 1.66E-S

L SE : 2.0SE-S

DItiEKSIONLESS

DEFLECTION

,

6/b

X 10’

Fig. 9(a)

E-5

NUIIXIM pnolysi~ of ~tastrophic softening b&&our

617

m = LO

a,/b

=

t/b

= L

EU

= 0.12

0.1 E-5

R W~O.001 I of~0.002 c Of~0.003 0 Ofco.004 t of~0.005 f Of~O.006 0 Dfr0.007 Ii OfrO.OO1 I Ofr0.009 L Of~0.010

DIHENSIOfiLESS DEFLECTION , 6/b X 10’

Fig. 9(b).

m * 80 a,/b

= 0.1

t/b E"

= L 5 0.7.d

R SE

=

E-5

.21E-S

I SE =

.42E-6

C SE =

.6X-6

0 SE =

.84E-5

E SE = l.O4E-6 f SE : 1.2SE-5 0 SE = 1.46E-5 Ii SE = 1.67E-6 ! SE = l.ME-5 L SE : 2.09E-5

g d

.a

DIMENSIONLESSDEFLECTION , 6/b X 10' Fig. 9(c). Fig. 9. Dimensionless load-deflection response of an initially cracked specimen (a,$ = 0.1). by varying the brittleness number, sE = G&b = u,/tb, between 2 x 10e6 and 2 x 10-j. (a) WI= 20, (b) m = 40, (c) m = 80.

618

ALBERTO CMP:NTERI and

GIOVANNI COLOMWJ

R SE = 2.09E-6 B SE s 4.lOE-6 C SE z S.27E-8 0 SE a S.SbE-S E SE o10.4SE-8 F SE *20.@OE-6 0 SE z41.IOE-8 Ii SE rS2.7OE-6 I SE zSS.SSE-s L SE :

1 .OlE-3

n SE i

Z.OSE-3

N SE 5 4.18E-9 0 SE L: 6.27E-3 P SE D S.SbE-3 0 SE :10.45E-3 R SE =20.OOE-3

DIKEt!SIONiESS

DE’LECTIOC

,

6/b

:!

10’

Fig. IO(a)

=LO

I?.

a,/b

=

t/b

=L .a.71

c”

0.3 E-5

R 5L z Z.OSE-5 3 C 0 E r 0 n I L )I w 0 ? 0 I

;II~E.'ISIOI;L2SS DPPLECTION ,

6/b

Fig. IO(b)

X

10’

SE SE SE SL SE SE IL SE SE IL SE SC SE 3E SE

i 4.13E-5 i 6.27E-5 i 3.36E-5 :,0.4tx-s n2O.SOE-5 i‘l.wE-5 sS?..70E-5 d3.SSL-S = l.OdE-3 c 2.o*c-3 i l.l8f-3 L I.Z’IE-3 : 1.36E-3 s,O.,SE-3 =ZO.SOE-3

Num&al

analysis of ataatrophic

a&nine

II 8 C D L F 0 n 1 L ” Y 0 r 2 n Q.0

6.6 i.2 i.0 DIflENSIONLESSDEFLECTIOC , 6/b

619

bchaviour

St SE IL IL 82 52 52 52 8E SC SE 12 SC SE SE SE

s Z.OIE-6 s 4.lOE-6 s 5.27E-6 * 0.552-6 .lO.lSE-6 ~20.502-5 sbt.ME-6 sSZ.‘)Oi--6 dS.LSE-6 = 1.04E-5 s Z.OSE-5 * 4.152-5 s 5.27E-5 z D.SSE-2 *10.4SE-5 r20.5OE-5

i.4 X

10’

Fig. IO(c). Fig. 10. Dimensionless load-deflection response of an initially cracked specimen (o,/b = 0.3). by varying the brittleness number, sE = G&r& = w,/26, between 2 x 10m5and 2 x lo-‘. (a) m = 20, (b) m = 40, (c) m = 80.

m = 20 a.,/b= 0.3 l/b C” R 0 C 0 E F 0 W 1 L

.6

DIf4EXSIONLESSDEFLECiIOii,

6/b

X

10’

Fig. II(a)

=L a 0.71

SE i SE = 52 s

E-5

.ZlE-6

.GE-6 .52E-6

SE E .84L-6 SE SE SE SE 62 SE

= E i i i E

l.O4L-5 1.2SE-5 ,.46E-s 1.672-S L.OSE-6 2.0x-6

ALBERTO CARPMEN and OIOVANNCOLOMBO

6 DIMENSIOHLESS

DEFLECTION

,

6/b

X 10’

Fig. 1l(b).

p15 89 a,/b

- 0.3

t/b YJ R II c 0 E F 0 n I L

DiKZ,‘SIO’;LTSS

DE?tECTIO?I

,

6/b

.:i p 8.7:

SE SE SE SE SE SE SE SE SE SE

: : I I i i f i F I

E-5

.21E-S .42E-S .6X-S .S4L-5 1.042-5 l.ZSE-5 i .462-S i.6?E-5 1 .SSE-5 t.OSE-5

X 10’

Fig. 1I(c). Fig. 11. Dimensionless load-deflection response of an initially cracked specimen (rr,/b = 0.3), by varying the brittleness number. sE = G&a& = ss,/2b, between 2 x 10Vb and 2 x lo-‘. (a) m = 20, (b) m = 40, (c) m = 80.

Numericalanalysis of catastrophicsoftening bchaviour

. I20

r,/b - 0.5 L/b = 4

A IL * 2.OBE-6 1 IL C IL 0 SE E CC f 8E 0 8E PISE 1 SE L SE PISE N SE 0 If f SE 0 SE R 8E

DIN':iS?ONLSSS

DE'LECTICS

,

d/b X 10

= 4.IIE-6 m S.t'lE-L a I.IIE-6 slQ.ICE-6 *20.8OE-5 s4I.IOE-6 dZ.'IOE-6 a#l.SBt-S s l.O4E-2 = Z.OIE-2 = 4.llE-3 L 6.27E-3 x a-ME-3 rlO.lSE-3 rZO.SOE-2

3

Fig. 12(a).

D

=

4.,/b t/b

= 0.5 = L

I C 0 E f

SE SE SE SE SE SE

I 2.08E-5 * 4.tIE-6 L 6.27E-5 i I.ME-5 tlO.lSE-5 r20.005-5

0 n I L H N 0 ? 0 I

SE SE SE SE SE SE SE SE SE SE

dl.IOE-5 =62.?OE-S =I3.S#E-6 L l.O4E-3 D 2.OIE-3 i 4.tIE-5 s 6.27E-3 L I.YIE-3 rlO.4SE-3 =20.90E-5

R

Fig. 12(b).

LO

621

ALBERmc-

md GIOVANNICoLorw,

.I30

0

a./b t/b cu R s C 0 E F 0 n I L n n 0 P 0 R

DIMENSIONLESS DEFLECTION ,

6/b

=0.5 -4

.0.7L

SE SE SE SE SE SE SE SE SE SE SE SE SE SE SE SE

E-5

* 2.0SE-6 = l.lSE-6 s S.l’lE-6 i S.SSE-6 *10.46E-6 r2O.SOE-6 dI.SOE-6 zS2.?OE-s =Sl.SSE-6 = I .04E-1 = Z.OSE-3 = 4.lSE-3 : S.Z7E-I = S.SSE-6 .10.4SE-2 rZO.SOE-6

X 10'

Fig. 12(c). Fig. 12. Dimensionless load-deflection response of an initially cracked specimen (a,lb = 0.5), by varying the brittleness number, sL = G,c,a,,b = w,,‘Zb, between 2 x 10eJ and 2 x lo-?. (a) m = 20, (b) m = 40, (c) m = 80.

m = 20 a,/b

=

t/b

= L

=u

0.5

I a.74

E-5

R SE = .212-s S SE = .t2E-5 c SE s .SSE-6 0 E F 0 )I I L

DIMEGSIONLESS DEFLECTION , 6/b X 104

Fig. 13(a).

SE SE SE SE SE SE SE

s . z s z z L

.S4E-6 l.O4E-S I.ZSE-6 1.4SE-5 I.E7E-6 I.SSE-6 2.08E-5

Numerical analycis of ut86trophic

6okning

bchaviour

a =

LO

l,/b

. 0.5

t/b - 4 cu * 0.74 II BE . I :t a C 0 E f 0 II I I.

0

I

DIKEKSIONLESS

2

6

H 6L 6E 6L 6L 6C 6E 6E

. s s s s s s s

E-5

.6lL-s .42f-6 .66E-6 .64E-6 1.04E.6 1.26t-6 1.46L-6 ,.67L-6 1.661-‘6 I.OOL-6

4

DEFLECTION . 6/b

x LO4

Fig 13(b).

R SLE 6 C D E P

0

I

DIf4ENSlOi;iESS

2

DETLECTIOh’

2

,

6/b

6E 66 SE 6L SE

.2lE-6 z .42E-6 : .62E-6 5 .64E-6 i ,.O.E-6 5 ,.26E-6

1 X

lo*

Fig. 13(c). Fig. 13. Dimensionless load4eflection response of an initially cracked specimen &/b = 0.5). by varying the brittleness number sE = G,,!aJ = w,,/2b, between 2 x 10e6 and 2 x IO-‘. (a) m = 20, (b) m = 40, (c) m = 80.

624

AwEKm c-

and GKIVANNI Coiosmo

From the cases shown in Figs 6 and 7, the sE threshold below which the results are unacceptable is approximately w, 80 x 10-S. W=G-m --

(8)

The lower bound to s, can be regarded as an upper bound to the finite element size h: h d 600~~.

(9)

For a normal concrete with maximum aggregrate sixe of 2 cm, w, 2 0.1 mm and then eqn. (9) gives h d6cm. The loaddeflection response shows the same trends even when an initial crack is present in the lower edge of the three-point bending beam. The following initial crack depths are considered: aJb = 0.1 (Figs 8 and 9), oo/b = 0.3 (Figs 10 and 1l), q,/b = 0.5 (Figs 12 and 13). The deeper the initial crack is, the more ductile the resulting heam behaviour is, as will be further discussed in the next section. 4. INFLUENCE

OF

THE INITIAL CRACK

DEPTH

A three-point bending test specimen of depth b=lScm, thickness I =b=lScm and span l46 = 60 cm is considered. The finite element mesh in Fig. 5(b) (m = 40) is utilized in the numerical simulation of a concrete-like material with the following mechanical properties: Young’s modulus E = 365,000 kg/cm*, tensile strength u, = 3 1.90 kg/cm?, Poisson ratio v = 0.10. For two different values of fracture toughness, G,c = 0.05 and 0.01 kg/cm respec-

tively, the initial crack depth q,/b is varied from 0.0 (initially uncracked specimen) to 0.5. For all the cases considered, the load-deflection (Pd) and load-crack mouth opening displacement (P-w,) curves are obtained. The P-d curves in Fig. 14(a) are related to different initial crack depths. Obviously, stiffness and maximum loading capacity of the specimen decrease by increasing the initial crack depth a,Jb. On the other hand, even the slope of the softening branch decreases, so that the untracked specimen reveals considerable instability and a nearly vertical drop in its loading capacity, whereas the cracked specimens appear much more ductile and controllable in the P-d descending stage. The terminal softening branch ap pears as totally independent of the initial crack depth. This is a direct consequence of the assumption of a damage zone collinear to the crack and concentrated on a line of zero thickness. The Pd curves in Fig. 14(b) describe the specimen behaviour when Glc= 0.01 kg/cm. For aJb Q 0.25 a snap-back instability occurs [14-181, namely, a softening branch with positive slope (dP/dB > 0) is revealed. If the loading process is controlled by the deflection, the loading capacity will show a discontinuity with a negative jump and the representative point will drop on the fourth branch of the P-d curve (Fig. 15). The part BD of the P-d curve in Fig. 15 appears as a virtual path, which can be revealed only by decreasing both load and deflection while the crack grows and opens. The four stages of the P-d diagram in Fig. 15 can be defined as follows: (OA)P>O,

6>0,

(430,

+,>o;

UOa)

(AB) t
d >O,

H 20,

6, >O;

(lob)

E

IO

DEFLECTION,

6 km1

Fig. 14(a).

1,/b

=

0.20

Numerial andyis

DEFLECTION.

ofatutrophic

625

roAminp beluviour

0.00 0.a 0.10

A

8,/b

-

5

rob

-

C

8,/b

=

D

a,,/b =

0.15

E

a,,/b =

0.20 0.25

F

qb

=

G

a,/b

-

0.30

H I

a@

-

0.36

=

0.40

L

r,/b

-

0.45

M

rob

-

aso

aOb

6 kml

Fig. 14(b). Fig.

14. Load-deflection

plots by varying the initial crack depth (b) Glc = 0.01 kg/cm.

a>o,

(BC)P
8<0,

(CD)P
d>O, a>o, 3,>0;

G,>O;

(1~)

u&

(a)

G,=

0.05 kg/m;

5. INFLUENCE OF SLENDERNESS AND ULTIMATE TENSILE

STRMN

(104

In addition to the slenderness l/b = 4 considered in Sec. 3, the ratios I/b = 8 and 16 are herein contemwhere the dot indicates derivative with respect to a plated. For initially untracked specimens (u,/b = quantity increasing monotonically with time (e.g. O.O), Fig. 6(b) (I/b =4) is to be compared with time itself or crack length). For the cases investigated Fig. 17(a) (I/b = 8) and Fig. 17(b) (I/b = 16). Increase numerically, points A and B tend to coincide, so that in brittleness with the slab slenderness is manifest a sharp bifurcation in the P-d path is revealed, as also for initially cracked specimens in Fig. 18 shown in Fig 14(b). (uo/b = 0.3). The catastrophic drop in the loading capacity may Two different ultimate tensile strains are then taken be avoided and the P-d snap-back behaviour experinto consideration: E, = 17.48 x lo-’ and 34.96 x imentally shown by controlling the crack mouth 10e5, in addition to the value e, P 8.74 x lo-” of Sec. opening displacement w,, instead of the beam 3. For a,Jb = 0.0, the results are reported in Fig. 19, deflection d (141. In fact, as shown in Fig. 16(a,b), H’, whereas the case 4/b = 0.3 is plotted in Fig. 20. increases monotonically in the softening stage during A brittleness increase appears by increasing the ultithe crack propagation. mate tensile strain. Both the trends described in this section are due to the variation in the elastic compliance of the nondamaged zone. An increase of such a compliance produces an increase of brittleness in the system. In the softening stage, in fact, the elastic recovery prevails over the localized increase of deformation, so that a snap-back instability occurs.

& SIZE-SCALE EFFECTS DECREASE OF APPARENT

STRENGTH AND INCREASE OF PKlTI’lOlJS PRAClWRE TOUCHNE!%S DECLECTION.

6

Fig. IS. Catastrophic softening bchaviour.

The maximum loading capacity P!.& of initially untracked specimens with I = 46 is obtained from

Tr-L Y,c = 0.05 kg/cm

I

0.00

A

8,/b

=

B

a,/b

=

0.05

C D

r,lb r,/b

= =

0.10 0.15

E F

a,/b

=

0.20

8,/b

=

0.25

G

$,fb

=

0.30

H

+,/b

=

0.35

I

8,/b

=

0.40

L

a,lb

-

0.45

M

8,/b

=

0.50

.

5

0

1.10-‘1

7

IO

CRACK MOUTH OPENING

I5 DISPLACEMENT,

w,k.m)

Fig. 16(a)

A

a,lb

=

0.00

B

so/b I0 Ib

= =

0.05 0.10 0.15

C

a,/b

=

a,ib

=

0.20

a,lb

-

0.25

a,lb

=

0.30

I

a,/b co/b

= =

0.35 0.40

L

qb

-

0.45

M

a,lb

=

0.50

D E F G H

7

2

1

CRACK MOUTH OPENING DISPLACEMENT.

lolo-‘)

6

6 I,

k-n)

Fig. 16(b) Fig. 16. Load-crack

mouth opening displacement plots by varying the initial crack depth a,,% (a) G,, = 0.05 kg, cm; (b) G,, = 0.01 kg ‘cm.

627

Numerical analysis of atutrophic sohning bchaviour

40

m =

= 0.0

6,/b L/b

E” II 6 C 0 E r 0 R I L I( N 6 r 0 I

= I

6E 6E SE 6E 6E 6E 6E 6E 6E 6E 6E 6E 6E SE SE SE

0

8.74

E-5

z t.wE-6 : 4.11-6 = 6.27E-6 = 6.36E-6 =IO.lSE-6 rlO.OOE-6 *4t.6OE-6 =6Z.?OE-6 =63.66E-6 = I.O(E-S : 2.06E-1 = r.l6E-S i 6.27E-S = 6.S6E-1 :10.46E-S rZO.OOE-S

DIKSNSIOKLESS DEFLECT:% , 6/b X 10'

Fig. 17(a).

D = 40 a,/b = 0.0 L/b = 16 .0.74 E-5 c" R SE : 2.06E-5 6 SE z ,.,6E-S C SE s 6.27E-S

sz-

0 E f 0 ”

SE SE SE SE SE

x 6.16E-5 :lO.GE-5 =ZO.OOE-S rdl.IOE-S r62.76E-S

I L ” W

SE SE SE SE

16S.S6E-5 i I.O4E-S . Z.OIE-3 i ‘.,6E-S

0 P 0 I

SE SE SE SE

s 6.27E-S : 6.16E-3 :lO.ISE-1 =?O.OOE-,

DIMENSIONLESS DEFLECTION , 6/b X 10'

Fig. 17(b) Fig. 17. Dimensionless load-deflection response of an initially uncrackcd specimen, by varying the brittleness number, sE = G,,.‘ob = H.
p1

=

40

a,/b C/b tu

= 0.3 = 8 I 0.74

E-5

II Sf * P.OSE-6 I c 0 L P 0 W 1 L n

EE 6C SE SE SE IE SE SE SE

* 4.m-6 = S.Z?E-6 . S.ML-6 iIO.4SE-6 aZO.SOE-6 ~41 .SOL-6 &2.701-S *83.bst-s = 1.04E-3

0 C P R

SE 5 S.rK-3 St = S.SIE-S SE 110.46E-3 SE c?O.SOE-f

SE i 2.06E-3 w SE : I.ISE-3

DIHEBSIONLESS DEFLECTION , 6/b X 10’

Fig. 18(a).

c1= LC a,/b = 0.3 C/b = 16 = 6.71 E-5

E"

R SE i Z.QSE-5 6 SE C SE 0 SE E SE

i

i

sZO.SOE-S

SE

4.lSE-S

: 6.2?E-5 i S.,SE-5 =lO.rSC-5

0 SE rl,.SDE-5

n

SE =EP.‘)OE-S

I

SE

,,

SE t 1.O,E-3

=SS.SSE-S

tl SE

= 2.OW-3

N SC

i

‘.,SE-3

: S.t?E-1 P SE i *.16E-3 0 SE =tO.GE-3 D SE

R

SE ~20.602-3

D 0

3 6 9 DIHENSIONLESS DEFLECTION , 6/b X 10'

12

Fig. 18(b). Fig. 18. Dimensionless load-deflection response of an initially cracked specimen (4,/b = 0.3), by varying = ~,/26, between 2 x 10e5 and 2 x 10m2.(a) I/b = 8; (b) I/b = 16. the brittleness number. sE = G&J

Numerical analysis of cPtastrophic sohning bchaviour

629

m - LO

a&b

90.0

Lib = 6 tU =l?.bB

E-5

a SE . P.ODE-6 I c D E r 0 I) I L II w 0 ? E I

SE SE SC SE SE SC SE SE I SE SL SE SE SE SE

s 4(.l(iE-6 s 6.171-S . S.%E-6 mto.4sc-6 sto.soE-6 dl .soc-6 ~Sz.?oE-s lS3*SSE-3 s l.O4E-3 a 2.osc-s * I.lSE-S s 6.27E-J i S.SBE-3 .10.8SE-3 rZO*SOE-f

DIKENSIONLBSS DEFLECTION , 6/b X 10'

Fig. 19(a).

0 SE L O.SSE-5 E SE rlP.4SE-S F SE *ZO.SOL-s

DIMENSIONLESS DEFLECTION , d/b X 10'

J

Fig. 19(b). Fig. 19. Dimensionless load~efl~tion response of an initially amcracked pmen, by varying the brittleness number, s, = G,Ju,b = H;/Zb, &wan 2 x IO-’ and 2 x IO- . (a) p;= 17.48 x IO-$ (b) E,= 34.96 x IO-’

ALBERTO CARPINIEIU and GIOVANNI (3xmf~0

m = LO

no/b = 0.3 t/b = L = 17.a E-5 YJ R SE = Z.OBE-5 I SC I 4.lEE-6 c SE

= 6.271-S

0 SE

i

E SE

rlO.46E-5

F SE

G?O.OOE-5

D SE

=41.8OE-6

”

SE

=62.7OE-6

I

SE

rW.WE-6

l.l8L-6

L SE

:

I.O4E-3

”

SE

:

2.01E-3

”

SE

i

..lBE-3

0 SE

:

6.27E-3

P SE

i

O.llE-3

D SE

zIO.tSE-3

R SE

=20.9OE-3

DItCNSIONLESS DEPLECTIJN , 6/b X 10'

Fig. 20(a).

m = LO a,/b = 0.3 Lfb = L = 34.96 c"

E-5

R SC E *.09c-s 0 SE : 4.IBE-5 t

SE

= 6.2-/E-6

0 SE

i

E SE

~lO.lSE-S

F SE

=ZO.SOE-5

0 SE

:4l.OOE-6

n IL

~62.7oc-5

I

8.3BE-5

SC :83.69C-6

L SE

i

l.O4E-3

n SE

i

?.O)E-3

H SE

z 4.IEE-S

0 SE

i

r

= 8.96E-3

SE

6.27E-3

0 SE

:lO.(SE-S

I

rZO.SOE-3

SE

DIMENSIONLESS DIFLEC:I3N , 6/b x 10'

Fig. 20(b). Fig. 20. Dimensionless load-deflection response of an initially cracked specimen (a,/b = 0.3), by varying the brittleness number. sE= G&J = w,!Zb, between 2 x lo-’ and 2 x IO-*. (a) E, = 17.48 x lo-‘; (b) E, = 34.96 x 10-5.

Nurnerbl adyais

ofcptuttophic

Figs 6 and 7. On the other hand, the maximum load P& of ultimate strength is given by

(11) The values of the ratio P(l) /P”) may also be regarded as the ratio of the apTare:; tensile strength C/ wven by the maximum load Pa?,and applying eqn. (ll)] to the true tensile strength a, (considered as a material constant). It is evident from Fig. 21 that the results of the cohesive crack model tend to those of the ultimate strength analysis for low So values: lim Pz

SE-.0

= Pz.

(12)

Therefore, only for comparatively large specimen sizes can the tensile strength u, be obtained as u, = 0,. With the usual laboratory specimens, an apparent strength higher than the true one is always found. The maximum loading capacity P& of initially cracked specimens according to the cohesive crack model is obtained from the P-d diagrams in Figs 8-13. On the other hand, the maximum loading capacity PC& according to LEFM can be derived from the following formula [19]: K,tb 3’

p(2)

,x-m

with the shape-function

and

the

critical

(13)

g given by

value

of stress-intensity

factor

K,,computed according to the well-known relationship

K,, =

m.

(14)

Eventually, a simple ultimate strength analysis on the center-line with the assumption of a butterfly stress variation through the ligament, gives pwma1= -2 u,t(b -ao)’ 3 I .

(19

The values of the ratios PC’) m.r/P”) mmand P&/P& are reported as functions of the inverse of the brittleness number s, in Fig. 22. The values E, = 8.7 x lo-‘, t = b, 1 = 46 are assumed. The ratio PzJP& may also be regarded as the ratio of the fictitious fracture

ndhing

631

bhaviour

toughness (given by the nonlinear maximum load) to the true fracture toughness (considered as a material constant). It is evident that, for high s, numbers, the ultimate strength collapse results in a more critical condition than that of LEFM (Ps < Ps); also, the results of the cohesive crack model tend to those of LEFM for low sE values:

(16) The fictitious crack depth at the maximum load is reported as a function of the inverse of the brittleness niunber sL in Fig. 23. The increase in the structure brittleness for s,+O is evident also from this diagram, the process zone at dP/dc5 = 0 tending to disappear, whereas it tends to cover the whole ligament for sE+co (ductile collapse). The real (or stress-free) crack depth at the maximum load is nearly coincident with the initial crack depth for each value of s,. This means that the slow crack growth does not start before the softening stage.

7. INFLUENCE OF THE SHAPE OF THE COHFSWE DIAGRAM e-w Let us consider a three-point bending test specimen with geometrical ratios I = 46 and t = b (Fig. 3) and a central crack of depth 4/b = 0.1, and a material with ultimate strain E, = 8.7 x lo-’ and Poisson ratio v = 0.1. A bilinear u-w diagram is assumed (Fig. lc) with the knee at the point of coordinates w = 2/9 w_,, a = 1/3c7,. The dimensionless load-deflection diagrams are represented in Fig. 24 by varying the brittleness number So The ratio of the maximum loading capacity obtained by the cohesive crack model to the maximum loading according to LEFM is equal to the ratio of the fictitious fracture toughness to the true one, K,c. Such a ratio is demonstrated to tend to unity when ~~-0, for the linear as well as for the bilinear u-w constitutive softening relations (Fig. 25). This means that, for large structure sizes and/or low fracture energies, the bifurcation of the load-deflection curve tends to describe the classical LEFM-instability. Therefore, it is possible to conclude that, although the u-w shape affects the theoretical value of the maximum load, such a dependence is reduced by decreasing the number So As a -limit case, for ~~-0, the maximum load derives from the wellknown LEFM relation: K, = K,, =&%%, and is dependent only on the area Glc under the assumed u-w diagram, and not on the shape of this diagram [ 13,201.

AcknowleQemenrs-The numerical results reported in the present paper were obtained in a joint research program between ENEL-CRlS-Milan0 Bologna.

and

the

University

of

632

03 0

1

2

3

4

5

lh,

DIMENSIONLESS SIZE. bo,/ %,c (lo”) Fig. 21. Decrease of the apparent ultimate tensile strength o,, by increasing the specimen size.

LEFM

%

I

1

I

1

2

3

I

4

DIMENSIONLESS SIZE, bo,/ Y,, (10’) Fig. 22(a)

5

*

l/s,

analysis of atastrophic

Jhm&al

%

100 80

633

rofkning behwiour

LEFM

-

--/ Of

0

Cohesiw Crack Model Ultimate Strength

I

I

I

I

I

1

2

3

4

5

DIMENSIONLESS SIZE, bo,l

glc

(103)

Fig. 22(b).

KfiC’. IC

“;c” i3I 4 x

I

I

%

a,/b=0.5

LEFM

‘W-

Cohesive Crack Model Ultimate Strength

-

--1 2

I 3

DIMENSIONLESS SIZE, bo,/

I 4 Y,,

I 5

Ifs,

W

(10’)

Fig. 22(c). Fig. 22. Increase of the fictitious fracture toughness K,,jr’ , by increasing the specimen size. (a) %/b = 0.1; (b) a,,/b = 0.3; (c) %/b = 0.5.

ALDERIDCAR~NTENand GKIVANNICoLorw,

(a)

I

a,/b=O.l

0.2 -

0.0

1

0

initial

crack depth I

I

1

2

L

3

DIMENSIONLESSSIZE, bo,/9,c

4

5

l/s,

(10’)

(b)

0.0

0

I

I

1

2

w

3

DIMENSIONLESSSIZE. bo,/+

4

5

l/s,

(103)

Cc)

0.4 ‘\

initial crack depth

0.2

i 1

2

3

DIMENSIONLESSSIZE, bo,/qc

4

5

l/SE

(lo31

Fig. 23. Fictitious crack depth at the maximum load vs specimen size. (a) a,/b (c) a,,/6 = 0.5.

= 0.1;

(b) 4/b

= 0.3;

Ntmerial analysisof aHastrophic mfhing

635

behaviour

BILIllLAR0-w LAW m -LO a-/b = 0.1 l/b

- 4

cy 9 8.74

E-5

A SE - I.OW-s # C 0 c F 0

SE SE SE SE SE SE

= 4.lDE-S - &.27E-S = O.NE-s .10.4SE-s =2O.SOE-S ..1.DoE-s

II I 1 II Y 0 P 0 I)

I SE SE SE SE SE S2 SE SE

[email protected] -m.s1-s - ,.OIE-3 . I.ODE-3 . 4.!@L-2 . 6.27E-2 l I.ME-3 .10.4SE-3 lZO.ML-2

I

DIKENSIONLESS DEFLECTION , 6/b X 10'

Fig. 24. Dimensionless IoadAeflcction response of an initially cracked specimen &lb = 0.1) by varying = wJ26, between 2 x lo-’ and 2 x 10ez, when the bilinear the brittleness number, sE = G&J stress-crack opening displacement law in Fig. l(c) is utilized.

loon%

LEFM

80 -

0 Linear u - w Irw o Bilinear u.

DIMENSIONLESSSIZE. bo,/9,,,

w law

llo’j

Fig. 25. Size-scale transition toward LEFM instability: linear vs bilinear cohesive law.

REFERENCES

I. A. Carpinteri. Size effect in fracture toughness testing: a dimensional analysis approach. In Proc. ht. Con/. Analyrical and Experimental Fracture Mechanics, June 23-27, 1980, Rome (Edited by G. C. Sih and M. Mirabilc). DD. 785-797. Siithoff Br Noordhoff (1981). 2. A. Carpinteri, Notch se&ivity in fracture testing’of aggregative materials. Engng Fract. Mech. 16, 467481 I.

s.

(1982). 3. A. Carpinteti,

C. Marega and A. Savadori, Ductile brittle transition by varying structural size. Engng. - Fract. Mech. 21, 26~3-271.(1985). 4. A. Carninteri and G. C. Sih. Damaae accumulation and crack growth in bilinear materials v&h softening Theor. appl. Fract. Mach. 1, 145-159 (1984).

5. A. Hillerborg, M. Modeer and P. E. Petersson, Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement Concrete Res. 6, 773-782 (1976).

6. P. E. Petersson, Crack growth and development of fracture zones in plain concrete and similar materials. Rep. TVBM-1006, Division of Building Materials, Lund Institute of Technology (1981). 7. G. Colombo and E. Limido, A numerical method for the analysis of stable TPBT tests: comparison with some experimental data (in Italian) XI Conuegno Narionale &ll’Associazione Italiana per I’Analisi delle Sollecitazioni, Torino, pp. 233-243 (1983).

8. A. Carpinteri, Interpretation of the Griffith instability as a bifurcation of the global equilibrium. In NATO Advanced Research Workshop on Application of Fracture Mechanics to Cementitious Composites, Evanston, IL,

4-7 September 1984 (Edited by S.P. Shah), pp. 287-316. Martinus NijholI (1985). 9. A. Carpinteri, A. Di Tommaso and M. Fanelli, Influence of material parameters and geometry on cohesive crack propagation. In Proc. Int. Con/. Fracture Mechanics of Concrete, Lausanne, l-3 October 1985 (Edited by F. H. Wittmann). Fracture Toughness and Fracrure Energy of Concrete, pp. 117-135. Elsevier, Amsterdam (1986). 10. A. Carpinteri, G. Colombo and G. Giuseppetti, Accuracy of the numerical description of cohesive crack propagation. In Proc. Int. Con/. Fracture Mechanics of Concrete, Lausanne, l-3 October 1985 (Edited by F. H. Wittmann), Fracture Toughness and Fracture Energy of Concrete, pp. 189-195. Elsevier, Amsterdam (1986).

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