By Jens Peder Ulfkjrer" Steen Krenk,2 and Rune Brincker 3
ABSTRACT: An analytical model for load-displacement curves of concrete beams is presented. The load-displacement curve is obtained by combining two simple models. Thefracture is modeled by a fictitious crack in an elastic layer around the midsection of thebeam. Outside the elastic layer the deformations are modeled by beam theory. The stateof stress in the elastic layer is assumed to depend bilinearly on local elongation corresponding to a linear softening relation for the fictitious crack. Results from the analyticalmodel are compared with results from a more detailed model based on numerical methodsfor different beam sizes. The analytical model is shown to be in agreement with thenumerical results if the thickness of the elastic layer is taken as half the beam depth. Itis shown that the point on the load-displacement curve where the fictitious crack startsto develop and the point where the real crack starts to grow correspond to the samebending moment. Closed-form solutions for the maximum size of the fracture zone andthe minimum slope on the load-displacement curve are given.
Ever since Kaplan (1961) performed his linear elastic fracture mechanical (LEFM) investigation of notched concrete beams subjected to three- and four-point bending, much attentionhas been paid to fracture of concrete and rock. In this pioneering work and in three subsequentdiscussions (Blakey and Beresford 1962; Gliicklich 1962; Irwin 1962) the applicability of LEFMwas discussed, and the views given in these contributions (e.g., slow crack growth) are stillpopular. Today it is realized that LEFM is only applicable to large-scale initially cracked structures and ultrabrittle concrete (Bazant 1983; Planas and Elices 1989), and that it is necessaryto apply nonlinear fracture mechanics for description of fracture in ordinary concrete structures.Different models based on ,nonlinear fracture mechanical ideas describe the softening behaviorof concrete, e.g., the fictitious crack model (FC-model) by Hillerborg et al. (1976), the crackband model by Bazant (1983), and the two parameter model by Jenq and Shah (1985). In thispaper, the FC-model will be used to describe fracture in concrete.
Few researchers have considered analytical methods based on nonlinear elastic fracture mechanical models to describe fracture in concrete structures. A model has been developed byChuang and Mai (1989) based on the crack band model; however, this model predicts no sizeeffect. Also, a model based on the fictitious crack model has been developed by L10rca andElices (1990). The idea of modeling the bending failure of concrete beams by development ofa fictitious crack in an elastic layer with a thickness proportional to the beam depth was introduced by Ulfkjrer et al. (1990). In this paper, the model is presented using a linear softeningrelation and the model is validated by comparing with results from a more detailed numericalmodel.
Several general results are obtained. It is shown that the point on the load-displacement curvewhere the fictitious crack starts to develop and the point where the real crack starts to growcorrespond to the same bending moment with the points lying on either side of the peak point.Closed-form solutions for the maximum size of the fracture zone and the minimum slope onthe load-displacement curve are given. The last result is used for derivation of a general snapbackcriterion depending only on beam geometry.
The failure of a simply supported beam loaded in three-point bending is modeled by assumingdevelopment of a single fictitious crack in the midsection of the beam. In the FC-model, materialpoints on the crack extension path are assumed to be in one of three possible states: (I) Alinear elastic state; (2) a fracture state where the material is softened, caused by cohesive forcesin the fracture process zone; and (3) a state of no stress transmission. In the fracture state, thecracking process is described by a softening relation that relates stress normal to the crackedsurface (J' to the crack opening displacement, w (distance between the cracked surfaces)
'Asst. Prof.. Dept. of Build. Technol. and Struct. Engrg.. Univ. of Aalborg. DK-9000 Aalborg. Denmark.~Prof.. Dept. of Build. Technol. and Struct. Engrg.• Univ. of Aalborg. DK-9000 Aalborg. Denmark.'Assoc. Prof.. Dept. of Build. Technol. and Struct. Engrg.• Univ. of Aalborg. DK-9000 Aalborg. Denmark.Note. Discussion open until June I, 1995. To extend the closing date one month. a written request must be
where f( ) = a material function determined by uniaxial tensile tests (see Fig. 1). The areaunder the curve f( ) is termed the specific fracture energy, Gf' which is assumed to be a materialconstant (Elfgren 1989).
Usually, the FC-model is combined with numerical methods like the finite-element method(Hillerborg et al. 1976), or a boundary-element method like the substructure method introducedby Petersson (1981), and no simple methods of analysis are directly available. In this paper weinvestigate the simplest possible version of the FC-model for a crack in a beam in bending. Themodel is based on two assumptions: (1) That the elastic response of the beam is approximatedby two contributions (a local flexibility, due to the crack, represented by a thin layer of springs.and a global beam-type flexibility); and (2) that the softening relation is assumed to be linear.The aim of the model is to demonstrate that many of the characteristic features associated withcracking of a concrete beam, such as change of maximum moment with size and softeningbehavior, can be captured by this simple model that rests solely on bending equilibrium, acracking criterion, and the relative stiffness of crack softening and the local stiffness of thebeam.
The assumption of linear softening might be changed to a Dugdale relation or another softeningrelation. However, using the assumption of a linear softening relation, the fracture energy isgiven by G f = (1/2)(1"w" where (1" = ultimate tensile stress; and We = critical crack openingdisplacement (see Fig. 1). Assuming a linear softening relation, the constitutive relation of thelayer becomes a bilinear relation between the axial stress, (1, and the elongation, v (Fig. 2). Onthe ascending branch, the elongation is linear elastic v = Ve' and no crack opening is present.The linear response is given by V e = (1hIE, where h = thickness of the layer; and E = Young'smodulus. On the descending branch, the total deformation v consists of two contributions v =
V e + w, where W = crack opening displacement. The peak point corresponds to the deformation11 = v,,; and total fracture corresponds to v = We'
To have a model that allows developing of a finite-size fracture zone the elastic layer mustbe stable in displacement-controlled loading corresponding to
(2)
In concrete fracture, the material parameters and the beam size I are usually described by oneparameter, the brittleness number, sometimes defined as (1;J!G F E (Elfgren 1989). In the presentcase, the brittleness is represented most directly by the parameter
B = (1,,!!.. = (1?,hWe E 2GF E
The stability condition (2) can be written in terms of B
B<1
(3)
(4)
Thus, the brittleness number B varies between zero, corresponding to ideal ductile behavior,and one, corresponding to ideal brittle behavior. When B ~ 1, the fictitious crack will notdevelop and brittle fracture will occur.
The thickness h of the equivalent elastic layer, representing the local stiffness of the beam,is an important parameter of the model. The results predicted by the analytical model are ingood agreement with finite-element analysis by assuming h to be proportional to the beam depthh = kb independent of crack length. Computations of cohesive joints in Krenk et al. (1994)
FIG. 1. (a) Concrete Rod Subjected to UniaxialDisplacement Controlled Loading; (b) Division ofDisplacement Into Stress-Strain Relation andStress Crack Opening Displacement Relation
FIG. 2. Constitutive Relation for Midsection ofLayer: (a) When Layer Is Stable (8 < 1); (b) WhenLayer is Unstable (8 = 1)
(b)
8<1.0u
0u
8=1.0uu
Y-~~~~-'o--V V
~ ~=~v.(y) Vu v(y)
(a)
u(y) ~
!IR I" / "h-f(W)
uffillll L~ ~wL1L-w we
~=-L--
L+L1L
b)
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have shown that the local flexibility associated with a crack penetrating half the beam depth isequivalent to an elastic layer with k = hlb = 0.5.
SOLUTIONS FOR LOAD-DISPLACEMENT CURVE
As mentioned previously, the flexibility of the beam is divided into two contributions: (I) Alocal layer of bilinear springs; and (2) global linear beam flexibility. First, the deformation ofthe springs is considered separately (Fig. 3).
The fracture process is divided into three phases.
Phase I: before the tensile strength is reached in the tensile side of the beamPhase II: development of a fictitious crack in the layerPhase III: crack propagation
The stress distribution in each phase of the fracture process is illustrated in Fig. 4.In phase I, a linear elastic constitutive relation is used for all parts of the layer V e = ah IE.
By simple geometric considerations it is seen that V e = q:>(b - 2y), where q:> = rotation; b =beam depth; and y = vertical coordinate (Fig. 2). The neutral axis is at the midpoint of thebeam corresponding to y = b12. Instead of the bending moment M and the rotation q:>, it isconvenient to introduce the nondimensional bending moment
6fL = M--,
a"b-t
and the corresponding dimensionless displacement
bE E6=q:>-=q:>-
ha" ka"
giving the simple load-displacement relation corresponding to the elastic spring layer
(5)
(6)
for 0 < 6 < 1 (7)
In the limit situation of phase I, the stress for y = 0 equals the tensile strength; and thenondimensional bending moment equals one. Thus in phase I the load-displacement curve is astraight line between the origin and (6, fL) = (1, 1) (see Fig. 5).
In phase II, the size of the elastic tensile zone is determined by simple geometrical considerations. When the fictitious crack develops, it is necessary to determine the crack openingdisplacement. By assuming that the stress in the fictitious crack is equal to the stress in theelastic layer, the crack opening displacement becomes
26W = 1 _ B (ar - Y) (8)
where af = length of fictitious crack. This corresponds to a linear crack profile. The linearsoftening relation is expressed as
a = a" (1 - ~)W"~
(9)
b)
FIG. 3. (a) Beam where Hatched Area Is ElasticLayer; (b) Deformed Beam where only Rigid-BodyDisplacements are Considered
\
::\ :~u=uui I a=aUI at
J : at : a FOu<uu I I
Phase I I Phase II Phase III
F :III
'-------------9
FIG. 4. Stress Distributions of Each Phase:(a) Phase I with Elastic Stress Distribution; (b)Phase II with Fictitious Crack; (c) Phase III withReal Crack; (d) Load-Displacement Curve
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L-....-L-----Lo-c
------0
FIG. 5. Moment Rotation Curve of Beam whenonly Rigid Body Displacements are Considered
200
" Scale=O.25
"\
\
\
\
\\1.0
I
\\
\
"
0.50 1.00 1.50
Rotation, rp (Thousands)
Slenderness= 6.0- - Numerical~- Analytical
0.00 -D-TTTTrl"nTTTTTrl"nTTTTTTTTTl"nTTTl0.00
0.75
0.25
0.50
'0Q)
.~ 1.00<0E...oZ
2.00
::l.. 1.75
~ 1.50
Eo
:::i! 1.25
FIG. 6. Comparison Between Analytical Model andDirect Substructure Method Using Standard Beamat Four Different Size Scales
IIIII
(10)
The length of the fictitious crack can be determined by combining (9) with the equilibriumcondition (resultant axial force equal to zero), In nondimensional form the result reduces to
atCa) J (1)af(a) = -b- = 1 - B - (1 - B) 0 - B
The equivalent moment is determined by integrating the axial stresses
(2a(a)J )
fL(a) = a 1 ~ B - 6af (a) + 4 - 3; for 1 < a < a, (II)
where 6e marks the transition to phase Ill.In order to stay in phase II, the crack opening displacement at the bottom of the beam must
be smaller than the critical crack opening w(O) < w'" which by use of (8), (10), and (11) canbe formulated as
fL(a) > 1 or 6 < ac (12)
where
ac =1 + VB
2B(13)
(14)
Thus, during the development of the fictitious crack the moment increases from 1 to its ultimatevalue and then decreases. When the moment reaches the value 1 on the descending branchcorresponding to 6 = 6", the real crack starts to grow (see Fig. 5). Thus, the real crack willonly propagate on the descending branch of the load-displacement curve as found by Harder(1991).
In phase III, the real crack starts to grow. The real crack length is termed a (see Fig. 4). Thesize of the elastic tensile zone is determined by the condition that w(a + af ) = "/I' The size ofthe fictitious crack at is obtained by the conditon w(a) = Wco giving in nondimensional form
1 1 - Baf = 2a-B-
The crack length a is determined through the equilibrium condition that the resultant axial forceis equal to zero
aa=-=. b (15)
As in phase II, the nondimensional bending moment is determined by integrating the axialstresses. The result is
(16)for ac < afL(a) = (~rThe results for the moment-rotation curve, including only rigid-body displacement of the beamparts, are shown in Fig. 5.
The descending moment-rotation relation in phase Ill, crack propagation, is of the form M
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ex 'P - 2 ex a-2, independent of the details of the stress-elongation behavior of the layer. Thisfollows from the following argument. There is a linear distribution of the elongation in theelastic layer; thus, the stress distribution in the layer is similar to that of the material shown inFig. 2(a). There is no axial force, and therefore the negative and positive parts always have thesame force resultant. The moment at the center of the beam is proportional to a" and the squareof the depth of the uncracked part (b - a)l. Thus, in plase III M ex a,,(b - a)2. The elongationat the top and the bottom of the uncracked part of the section remains constant during crackpropagation, and thus 'P ex (b - a) -1. This establishes the result M ex 'P - 2 in the crack propagationphase, independent of the details of the constitutive relation of the layer.
Elastic deformations in the beam parts outside the elastic layer are taken into account bysubtracting the elastic deformation J..l(a) from the elastic layer, leaving only deformations dueto crack growth, and then adding the elastic deformations of the whole beam using a solutionfor an elastic beam with a concentrated force given by Timoshenko (1955). The central elasticbeam deflection is
Mf20,. = 12EI ~(A) (17)
where EI = bending stiffness of the beam; 13 = a factor describing the influence of the concentrated load 13 = 1 + 2.85/~2 - 0.84/~3; ~ = slenderness ratio ~ = lib. The elastic rotationsimilar to equation (6) is
a = 2 ~ bE,. I hu"
The relation (17) can be written in nondimensional form
(18)
a,. = 'YJ..l;~Awhere'Y = 3k
(19,20)
MODEL VALIDATION
The effect of the beam flexibility is introduced by adding a,. and subtracting the deformation ofthe elastic springs. Thus, the total deformation at is
af = a + a,. - J..l = a + (oy - 1)J..l (21)
In this relation, the moment J..l(a) = J..l is given by (7), (11), and (16) for the phases I-III.Hence, the complete moment-rotation curve is fully determined by the brittleness number B,and the slenderness ~.
In this section, results from the analytical model are compared with results from a moredetailed numerical model. The numerical model is based on the fictitious crack model and alinear softening relation, and the results are obtained by the direct substructure (DSS) method(Brincker and Dahl 1989). Four-node elements and an element mesh with 21 nodes in themidsection were used.
Results for one beam geometry (slenderness ratio ~ = 8) are compared at different brittlenesslevels in order to see how well the model predicts the load-displacement curve. It was foundthat the best agreement was obtained if the size of the elastic layer is proportional to the beamdepth h = kb with k = 0.5. Therefore k was assumed equal to 0.5. A beam geometry similarto the Reunion Internationale des Laboratoires d'Essais et des Recherches sur les Materiauxet les Constructions (RILEM) beam with material parameters corresponding to a normal strengthconcrete is chosen as standard beam (see Table 1). With the chosen material parameters, themaximum beam depth according to (4) is 888 mm corresponding to a scale factor of 8.88.
In Fig. 6, a comparison is shown between the analytical model and the numerical results forthe standard beam on four different size scales (0.25, 0.5, 1.0, and 3.0). It is observed that the
TABLE 1. Geometry and Materials Parameters for Standard Beam
Property Symbol Unit of measurement Value(1 ) (2) (3) (4)
beam depth b mm 100.0beam width t mm 100.0beam length L mm XOO.Onotch depth Ui mm 0.0specific fracture energy GF Nmm/mmc 0.100tensile strength u" N/mmc 3.0modulus of elasticity E N/mmc 20.000brittleness number B - " 0.1125
"Nondimensional.
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shape of the moment-rotation curves is almost identical and that the model predicts the ultimateload quite well. However, in the analytical model the snapback effect is slightly more pronounced. This could be adjusted by introducing a slight dependence of h on crack depth.However, in view of the simplicity of the model with k = 0.5, the agreement is quite satisfactory.
In Fig. 7, results for the size of the fictitious crack are compared. It is seen that the size ofthe fictitious crack calculated by the analytical model is slightly smaller than that calculated bythe numerical method before the real crack starts growing (the ascending branch of the curves),and larger at the descending branch. The small kinks in the numerical results are due to thediscretization in the numerical model. With a larger number of nodes in the midsection thesekinks would disappear. In Fig. 8, the real crack lengths for the two models are compared. It isseen that the real crack grows faster in the numerical model.
Comparing numerical results with results for the analytical model, it can be concluded thatdeviations are relatively small. The errors introduced by the elastic layer and the assumptionof wedge-like crack-opening are typically smaller than errors due to the simple linear softeningrelation (Brincker and Dahl 1989).
SIZE EFFECTS PREDICTED BY MODEL
The peak loads j.Lmax predicted by the analytical model are shown in log-log scale in Fig. 9.The model has a limit beyond which no size effects are predicted; size effects are only predictedin the small-scale region. This might seem surprising, but as will be explained, the behavior isa direct consequence of the basic model assumptions.
In linear fracture mechanics there is only one state parameter, the stress-intensity factor. Ifgeometrically similar specimens are considered (an initial crack is scaled with the size), thestress-intensity factor is proportional to the square root of the specimen size. Therefore, inlinear fracture mechanics, size effects on the strength are always described by the well-knownsquare root law (Bazant and Oh 1983). In nonlinear fracture mechanics, however, the statecannot be described solely by one parameter. The size of the fracture zone has to be taken intoconsideration. If the fracture zone is large compared to the specimen size, the situation is closeto ideal plastic behavior in which no size effects are predicted. On the other hand, if the fracturezone is small compared to the specimen size, the influence from the fracture zone vanishes, andthe situation is described well by linear fracture mechanics. Thus, two size effects are usuallypresent: one caused by the influence of scaled defects, and one caused by the influence of varyingfracture-zone size.
For the analyical model presented here, however, no initial defects are included. Furthermore,if a defect was assumed to be present, the model would not be able to trace the linear fracture
o. 00 -h-TTTTTT-ri-rTTTTTT-rn"CTTTTTTT:IITTTTTi--'--l~
0.00 0.50 1,00 1.50 2.00
Rotation, 'P (Thousands)
FIG. 7. Size of Fictitious Crack for AnalyticalModel and Direct Substructure Method (Loops aredue to Snapback)
FIG. 9. Peak Load at Different Size Scales Predicted by Model and by Direct Substructure Method
FIG. 10. Moment Displacement Curve for Beamon Critical Size Scale
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mechanical behavior. This lacking capability of the model is due to the simple local form of theconstitutive equation of the elastic layer. The complicated elastic interaction between the different points around a crack tip cannot be described by this simple model. Thus, the model canonly predict the weak size effects caused by varying fracture zone size-the fact that the fracturezone does not scale like other geometrical quantities.
When the brittleness number B approaches the value 1 (approaching ideal brittle behavior),the fracture zone approaches zero. When the fracture zone vanishes, then its influence does,too. Therefore, the model can never predict any size effects beyond the limit corresponding toB = 1. Thus the size effects predicted by the model agree perfectly with what should be expectedfrom the basic model assumptions.
The size effects predicted by the numerical solution are also shown in Fig. 9. As it appears,the analytical and the numerical results agree quite well. Since no initial crack is present, thenumerical model shows the same lack of size effect prediction beyond the limit B = 1. For thenumerical solution, however, since it is based on the finite-element method and thereforeincludes any elastic response, the numerical solution would have approached the square rootlaw in the large-scale region, if an initial crack had been present.
When the size of the beam changes, the stress distribution in the partially fractured midsectionchanges, as does the shape of the load-displacement curve. A few closed-form solutions aregiven for these size effects predicted by the analytical model.
An important parameter describing the stress distribution in the partially fractured midsectionis the maximum size ar,max of the fictitious crack. Since aar/aS > 0 in phase II, and oar/aS < 0in phase Ill, at is largest at the end of phase II. Thus, the maximum size of the fictitious crackis found by combining (10) and (12)
ar'max = b(l - VB) (22)
(23)
Thus, for small ductile beams the relative size of the fictitious crack is large, and for large brittlebeams the relative size of the fictitious crack approaches zero.
The way in which the load-displacement curve changes with size is more difficult to describe.One important parameter of the load-displacement curve is the peak load, fJ.max. The peak loadmight be obtained from the condition afL/aS, = O. However, no simple expression has beenderived for this case.
Another important key-parameter for the load-displacement curve is the maximum slope Son the descending branch. The slope is found by taking the derivative of (21)
as ofL afL= - - + ('1 - 1)-
afL as, ae,
from which
OfLae, (
-Iae
'1 - 1 + oJ (24)
(25)
(27)
(26)
The steepest point on the descending branch of the load-displacement curve is at the transitionfrom phase II to phase III, i.e., for S = Se- Thus, the minimum value of as/afL is found from(16) and (13), which together with (24) yields the result
S = 4B1 + VB - 4c"y - l)B
The quantity S = brittleness of the structure; the larger maximum slope on the descendingbranch, the more brittle the behavior of the beam will be. The brittleness number varies betweenzero, corresponding to ideal ductile behavior, and infinity, corresponding to the case where themaximum slope becomes infinite. If the point of infinity slope is exceeded, snapback occurs,and the brittleness number S becomes meaningless. Thus S only describes the brittleness ofstructures without snapback on the load-displacement curve.
The maximum slope on the descending branch becomes infinite when the denominator in(25) vanishes; i.e., when
1 + VB - 4c"y - l)B = 0
The solution to this equation defines a critical brittleness number for the elastic layer
= [1 + \11 + 16c"y - 1)]2Ber 8c"y - 1)
If the brittleness number B of the layer is larger than the critical brittleness number Bn , thenthere is snapback on the load-displacement curve. Otherwise there is no snapback. For thestandard beam the critical brittleness number is found as Ber = 0.069, corresponding to a scale
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factor of 0.615. The case is illustrated in Fig. 10, where the results for the analytical model areshown for B = B,,.
CONCLUSIONS
A simple model for calculation of load-displacement curves of concrete beams in three-pointbending is presented. The results from the analytical model are compared to results from anumerical model based on the direct substructure method. Using the simple relationh = kb, where k = 0.5 and b = depth of the midsection, the analytical model appears to givegood results.
The analytical model is described by a set of simple equations and, therefore, the calculationtime is considerably less than the calculation time using a numerical model. Therefore, if a linearsoftening relation is acceptable, the analytical model is well suited for estimation of materialparameters from test results by regression.
Since the analytical model takes both elastic and fracture energies into consideration, it isable to predict both size effects and snapback. However, the model is restricted to brittlenessnumbers B < 1.0 and is therefore not applicable to large brittle beams.
ACKNOWLEDGMENTS
Financial support from the Danish Technical Research Council is gratefully acknowledged.
APPEND~I. REFERENCES
Bazant. Z. P., and Oh, B. H. (1983). "Crack band theory for fracture of concrete." Mat. and Struct., Paris,France, Vol. 16, 155-177.
Blakey, F. A., and Beresford, F. D. (1962). "Discussion of 'Crack propagation and the fracture of concrete' byM. E. Kaplan." J. Am. Concrete Inst., 59, 919-923.
Brincker, R., and Dahl, H. (1989). "On the fictitious crack model of concrete fracture." Mag. of Concrete Res.,41(147), 79-86.
Chuang, T., and Mai, Y. W. (1989). "Flexural behavior of strain-softening solids." Int. 1. Solids and Struct..25(12),1427-1443.
Elfgren, L. (1989). "Fracture mechanics of concrete structures-from theory to application." Rep., ReunionInternationale des Laboratoires d'Essais et de Recherchais sur les Materiaux et les Constructions (RILEM),Chapman & Hall, Ltd., London, England.
Ghicklich, J. (1962). "Discussion of 'Crack propagation and the fracture of concrete,' by M. F. Kaplan." 1. Am.Concrete InH., Vol. 59, 919-923.
Harder, N. A. (1991). "A theorem on brittleness." Rep. No. R9126, University of Aalborg, Aalborg, Denmark,1-17.
Hillerborg, A., Modcer, M., and Petersson, P. E. (1976). "Analysis of crack formation and crack growth inconcrete by means of fracture mechanics and finite elements." Cement and Concrete Res., Vol. 6, 773- 782.
Irwin, G. R. (1962). "Discussion of 'Crack propagation and the fracture of concrete,' by M. F. Kaplan." J. Am.Concrete Inst., 59(929).
Jenq, Y. S., and Shah, S. P. (1985). ''Two parameter fracture model for concrete." J. Engrg. Mech., ASCE,111(No. 10), 1227-1240.
Kaplan, M. F. (1961). "Crack propagation and the fracture of concrete." J. Am. Concrete Inst., 58(11), 591610.
Krenk, S., Jonsson, J., and Hansen, L. P. (1994). "Fatigue analysis and testing of adhesive joints." Engrg. Mech.Paper No. 23, Aalborg University, Denmark.
Petersson, P. E. (1981). "Crack growth and development of fracture zones in plain concrete and similar materials."Rep. TVBM-I006, Div. of Build. Mat., Lund Inst. of Techno!.. Sweden.
Planas, J., Elices, M. (1989). "Asymptotic analysis of the development of a cohesive crack zone in mode I loadingfor arbitrary softening curves." Proc., of Fracture of Concrete and Rock, S. P. Shah and S. E. Swartz, cds.,SEM-RILEM Conf., Society of Experimental Mechanics, Houston, Tex., 384.
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Fracture-ECF8, D. Finao, ed., Engrg. Mat. Advisory Services LTD, Vol. II, 612-617.
APPENDIX II. NOTATION
The following symbols are used in this paper:
a crack length;at fictitious crack length;B brittleness modulus;b beam depth;E modulus of elasticity;
f( ) material function;GF specific fracture energy;
h thickness of elastic layer;I moment of inertia;
14 JOURNAL OF ENGINEERING MECHANICS
J. Eng. Mech. 1995.121:7-15.
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k elasticity coefficient, which determines thickness of elastic layer;I beam length;
M cross-sectional moment;t beam thickness;tI = elongation of layer;
tlu elongation of layer corresponding to the tensile strength;W crack opening displacement;
We critical crack opening displacement;0. normalized crack length;
