PEAK LOAD ESTIMATION OF PRE-CRACKED PLAIN CONCRETE … · illustrate that this concise approach can provide a reasonable estimation of peak load of the pre-cracked plain concrete

Proceedings of the 6th International Conference on Mechanics and Materials in Design,

Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-30 July 2015

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PAPER REF: 5452

PEAK LOAD ESTIMATION OF PRE-CRACKED PLAIN CONCRETE

BEAMS IN MIXED-MODE FRACTURE

X. Guo1,2(*)

, R. K.L. Su3, B. Young

3

1School of Mechanical Engineering, Tianjin University, Tianjin 300072, China

2Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin 300072, China

3Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong

(*)Email: [email protected]

ABSTRACT

A major difficulty in simulating load response of a concrete structure in mixed-mode fracture

is that crack path is not known a priori. Predicting both the crack path and the associated load

response involves advanced simulation techniques and novel numerical methodologies, and

making these predictions remains a fundamental challenge to the community interested in

investigating mixed-mode fracture. Here, an intrinsic cohesive crack model is employed to

study mixed-mode fracture in a plain concrete beam with regular meshes. Simulations

illustrate that this concise approach can provide a reasonable estimation of peak load of the

pre-cracked plain concrete beams in mixed-mode fracture.

Keywords: mixed-mode fracture, cohesive crack, peak load.

INTRODUCTION

In a three-point-bending test of a concrete beam, a pre-crack, which shifts from load plane,

leads to asymmetric loading and thus a mixed-mode fracture event (John and Shah, 1990;

Guo et al., 1995). In addition, a double-edge-notched specimen in a Nooru-Mohamed test

(1992), a three-dimensional four-edge-notched specimen in a Hassanzadeh test (1991), and a

concrete plate with an embedded steel disk in a dowel disk test (1995) fracture in mixed

mode.

In contrast to simulating a concrete structure in mode-I fracture, simulating load response of a

concrete structure in mixed-mode fracture faces an extra difficulty because the crack path is

not known a priori. There have been numerous attempts to predict the load response of a

concrete structure in mixed-mode fracture. Galvez et al. (1998) found that the crack path from

linear elastic fracture mechanics (LEFM) analysis was a satisfactory approximation of the

actual crack path for concrete structures. Building on Galvez et al.’s finding, Cendon et al.

(2000) and Galvez et al. (2002a, 2002b) developed a two-step approach including numerically

predicting the crack path using the maximum tensile stress (MTS) criterion (Erdogan and Sih,

1963) and incorporating spring elements or cohesive elements.

Currently, in a framework of the finite element method (FEM), several available

methodologies to predict both the crack path and the associated load response of a concrete

structure in mix-mode fracture include the cohesive crack model, smeared crack model,

discontinuity approach, and eXtended FEM (XFEM).

The cohesive crack model was proposed to characterize phenomenology of a failure process,

based on a softening law (Dugdale, 1960; Barenblatt, 1962). Later, it was incorporated into

the FEM as an application to concrete fracture, allowing for fracture processes to be resolved

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Mechanical Behaviours of Advanced Materials and Structures at All Scales

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explicitly (Hillerborg et al., 1976; Bazant, 2002; Elices et al., 2002). Two approaches are

available in the framework of the cohesive crack model. The intrinsic cohesive crack model

usually embeds cohesive elements along the boundaries of volumetric elements as a part of

the physical model (Needleman, 1987; Guo et al., 2012a, 2012b, 2012c, 2013), whereas the

extrinsic model inserts cohesive elements into the model as fracture develops based on an

extrinsic fracture initiation criterion (Camacho and Ortiz, 1996; Ruiz et al., 2001; Yu and

Ruiz, 2006; Yu et al., 2008). Song et al. (2006) employed an intrinsic cohesive crack model

with a visco-elastic bulk material model to investigate mixed-mode fracture of asphalt

concrete. The predicted crack path was found to be in agreement with the experimental

results, but no result for the associated load response was presented. Lens et al. (2009)

investigated mixed-mode fracture of concrete and demonstrated that the cohesive model with

a “plastic” potential defined by Coulomb’s law with adherence was able to cope with the

spurious traction components. Using the extrinsic cohesive crack model, Ruiz et al. (2001)

formulated mixed-mode fracture of concrete specimens subjected to dynamic loading, and

accounted for micro-cracking, macroscopic crack development, and inertia effect. Yang and

Deeks (2007) coupled the FEM and the scaled-boundary FEM (SBFEM) for cohesive crack

propagation with a LEFM-based re-meshing procedure. The results showed that the SBFEM–

FEM coupled method was capable of predicting both the crack path and the load response

with a small number of degrees of freedom.

In the smeared crack model, infinite parallel cracks with infinitesimal openings are distributed

over the finite element, and crack propagation is simulated by a reduction in the stiffness and

strength of the material. The total strain is a sum of the strains resulting from the deformation

of the un-cracked material and the cracking process. The constitutive laws show strain

softening and introduce numerical difficulties (i.e., the system of equations may become ill-

posed) and localization instabilities and spurious mesh sensitivity may occur (Bazant and

Cedolin, 1979). Prisco et al. (2000) adopted multiple smeared crack models including

Ottosen’s model, the gradient Rankine plasticity model, and the crush-crack model based on

local and non-local approaches to investigate the mechanical behavior of plain and reinforced

concrete in mixed-mode fracture. Ozbolt and Reinhardt (2002) used a micro-plane model, a

type of the smeared crack model, for concrete in mixed-mode fracture, and predicted the

structural response and crack patterns realistically.

Sancho et al. (2007b) developed a strong discontinuity model with crack equilibrium at the

element level, cohesive cracking with central forces, and limited local crack adaptability; they

found that it could describe the crack propagation with adequate accuracy. Cohesive crack

propagation was formulated based on a truly-mixed discretization with the stresses as main

regular variables, while discontinuous displacements played the role of Lagrangian

multipliers (Bruggi and Venini, 2009). The approach handled the cohesive cracks through the

appropriate inclusion of cohesive energy terms that enrich the formulation. Generally, a major

difficulty in the discontinuity approach is that the crack propagation may lock because of

kinematical incompatibility among the cracks; therefore, special implementation should be

taken to prevent crack locking (Sancho et al., 2007a).

In the XFEM, crack propagation through a finite element is modeled by the enrichment of the

classical displacement-based finite element’s approximation by a discontinuous function

obtained through the partition of unity (Babuska and Melenk, 1997). Areias and Rabczuk

(2008) modeled set-valued softening laws in plate bending and plane crack propagation and

presented an algorithm for the softening law to determine the crack path. Generally, a major

difficulty with the XFEM lies in modeling spontaneous multiple crack initiation, branching,

and coalescence (Song et al., 2008).



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Overall, simulating both the crack path and the associated load response of a plain concrete

structure in mixed-mode fracture requires advanced simulation techniques and even novel

numerical methodologies and remains a fundamental challenge in the community interested in

investigating mixed-mode fracture. Among the above methodologies in the framework of the

FEM, the cohesive crack model is comparatively simpler in concept and easier in application

and has come to be popular in the research on non-linear fracture mechanics. Compared with

the intrinsic cohesive crack model, the extrinsic one involves some issues, both in the model

implementation and in the results interpretation (Zhang et al., 2007). Therefore, in a previous

study of Guo et al. (2012c), the intrinsic cohesive crack model was employed to investigate

mode-I fracture of three-point-bending specimens and to extract the best-fit energy and stress

ratios for both normal-strength and high-strength concrete. As an extension of Guo et al.

(2012c), the intrinsic cohesive crack model with a bilinear softening law is employed here to

study mixed-mode fracture in a plain concrete beam. Simulations with regular (not oriented)

meshes illustrate that the approach can provide a reasonable estimation on the peak load of

the pre-cracked plain concrete beam. Parametric studies are conducted to investigate effects of

the energy and stress ratios and cohesive strength on the load response and energy terms.

COHESIVE CRACK MODEL WITH A BILINEAR SOFTENING LAW

The cohesive crack model is formulated on the virtual work (Roesler et al., 2007). The

internal work done by the virtual strain tensor ( εδ ) in the domain (Ω ) and the internal work

by the virtual crack opening displacement vector ( wδ ) along the crack line ( cΓ ) are equal to

the external work by the virtual displacement vector ( uδ ) at the traction boundary (Γ ), i.e.,

∫∫∫ ΓΓΩΓ=Γ+Ω dδdδdδ TTT PuTwσε

cc

(1)

where σσσσ is the stress tensor, T is the traction vector in the cohesive zone, and P is the

external traction vector. The resulting cohesive crack model has been implemented in

ABAQUS (2011).

A bilinear softening law has been gradually accepted as a reasonable approximation to the

softening behavior in concrete. It can be fully characterized by four parameters: cohesive

strength mT , fracture energy FG , initial fracture energy fG , and kink stress kT , a stress level

corresponding to kw (a separation at the kink point). Equivalently, in line with the previous

study of Guo et al. (2012), mT , FG , an energy ratio (the ratio of the fracture energy to the

initial fracture energy FG / fG ), and a stress ratio (the ratio of the kink stress to the cohesive

strength kT / mT ) are employed in this study. To restrict the inherent mesh dependence, the

penalty stiffness is used to modify the bilinear softening law, as shown in Fig. 1. The cohesive

strength mT is the softening stress at which the separation reaches mw and damage initiates.

The fracture energy FG is the work required to create and fully separate a unit surface area of

a cohesive element and is given by the area under the softening law, i.e.,

( )∫=cw

F wwTG0

d (2)

where T is an effective traction, w is an effective separation, and cw is the critical crack

opening beyond which the traction becomes zero and the cohesive element fails completely.

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The effective traction and the effective separation can be defined as

222

sn TTT −+= η and 222

sn www η+= (3)

where nT and sT are the cohesive tractions in the normal and tangential directions,

respectively, nw and sw are the normal and tangential opening displacements across the

cohesive surfaces, respectively, and η is the tension-shear coupling constant representing the

effect of the mixed mode (Zhang et al., 2007).

The softening law in Fig. 1 has two stages. In stage O-A, the cohesive element deforms

elastically. Damage initiates at the critical point A. Note that mw should be much smaller than

kw so that the penalty stiffness can be sufficiently large. During stage A-B-C, the separation

of the cohesive element increases, and the cohesive element is softened, i.e., damage evolves.

The softening stress can be formulated as

( )

( )( ) k c

for (A B);

for w w w (B C).

m k

m m k

k

k k

k

c k

T T wT w w w

wT

T w wT

w w

−− < ≤ →

=

− − < ≤ → −

(4)

Same with Song et al. (2006), the tension-shear coupling constant η is taken as 1, i.e., the

quadratic strain criterion is employed for damage initiation and evolution in the cohesive

elements.

Fig. 1 - Softening law with a penalty stiffness to restrict mesh dependence.

MIXED-MODE FRACTURE SIMULATION OF PRE-CRACKED PLAIN

CONCRETE BEAMS

Here, the four-point, single-edge notched shear beams tested by Arrea and Ingraffea (1982)

are studied. Their testing results, especially load versus crack mouth sliding displacement

(CMSD) curves, have become a benchmark for mixed-mode crack propagation analysis. The

geometry and boundary conditions of the series B beams in their test are shown in Fig. 2. The

analytical configuration adopted has a span of 914 mm, a depth of 308.5 mm, a thickness of

Separation w

SofteningStressT

Gf

GF

O

(wm, T

m)

B (wk, T

k)

Penalty stiffness

A

C (wc, 0)


Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores,

152 mm, and a pre-crack length

prevailed. The specimen is discretized into the volumetric and cohesive elements. Uniform

linear cross-triangular meshes

the experiments, displacement control

Fig. 2 - An analytical configuration

A linear elastic constitutive law

constitutive parameters in Arrea and Ingraffea

Poisson’s ratio of 0.18, and compressive strength of 45.5 MPa, which

the present simulations. The fracture behavior predicted by the cohesive cr

heavily on the cohesive strength and the fracture energy employed

et al., 2008). Generally, although

by the work-of-fracture method,

measuring the mode-II fracture energy

crack model, the tensile strength has generally been taken as the cohesive strength

and Ortiz, 1996; Ruiz et al., 2001;

intrinsic cohesive crack model, t

example, from a direct tensile test

tensile tests of concrete are difficult to perform and the results always scatter extensively.

Generally, the result of a splitting

(Guinea et al., 1994). In line with prior studies employing the intrinsic cohesive crack model

for concrete (Roesler et al., 2007

taken here to be its mode-I cohesive strength

employing the extrinsic cohesive crack model for concrete, the tensile strength of

is also taken here to be its mixed

Neither the fracture energy FG

(1982). A wide range of values were used for

listed in Table 1, where mIIT

=3.7 MPa, which are the same as

taken to be the same as FIG

Unless otherwise stated, the

shear stress when shear failure occurs

compressive strength cuf was measured to be 45.5

internal friction angle φ can be



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crack length 0a of 82 mm. It is assumed that the condition of plain strain

s discretized into the volumetric and cohesive elements. Uniform

triangular meshes (3 mm in size) are used in the potential fractured zone.

the experiments, displacement control is used in the simulations.

An analytical configuration for a four-point single-edge notched shear beam

constitutive law is applied to the concrete itself. The available material

parameters in Arrea and Ingraffea (1982) include Young’s modulus of 24.8 GPa,

Poisson’s ratio of 0.18, and compressive strength of 45.5 MPa, which are

The fracture behavior predicted by the cohesive cr

heavily on the cohesive strength and the fracture energy employed (Roesler et al., 2007; Park

although the mode-I fracture energy FIG of concrete

fracture method, proposed by RILEM TC 50-FMC (1985)

II fracture energy FIIG is still under the way. In the extrinsic cohesive

crack model, the tensile strength has generally been taken as the cohesive strength

2001; Yu and Ruiz, 2006; Yu et al., 2008). Theoretically, in the

intrinsic cohesive crack model, the cohesive strength is unknown and should be calibrated, for

example, from a direct tensile test (Cornec et al., 2003; Guo et al., 2010, 2012c)


splitting-tension test can provide an estimate of the tensile strength

In line with prior studies employing the intrinsic cohesive crack model

et al., 2007; Guo et al., 2012b), the tensile strength of concrete,

I cohesive strength mIT . Furthermore, in line with prior studies

ic cohesive crack model for concrete, the tensile strength of

mixed-mode cohesive strength mT .

F nor the tensile strength tf was reported in Arrea and Ingraffea

. A wide range of values were used for the series B beams by different analysts, as

is the mode-II cohesive strength. Here, FIG

ame as those in Galvez et al. (2002a, 2002b), are

such that the total fracture energy is FG =Unless otherwise stated, the mode-II cohesive strength is evaluated as the critical internal

shear stress when shear failure occurs in a simple compression, i.e., c =

was measured to be 45.5 MPa (Arrea and Ingraffea

can be taken as 030 , and thus, 7.19=c MPa. The

s assumed that the condition of plain strain

s discretized into the volumetric and cohesive elements. Uniform

used in the potential fractured zone. As in

edge notched shear beam.

The available material

Young’s modulus of 24.8 GPa,

are also employed in

The fracture behavior predicted by the cohesive crack model depends

(Roesler et al., 2007; Park

of concrete can be measured

(1985), an approach to

In the extrinsic cohesive

crack model, the tensile strength has generally been taken as the cohesive strength (Camacho

Theoretically, in the

ohesive strength is unknown and should be calibrated, for

al., 2010, 2012c). Direct


test can provide an estimate of the tensile strength

In line with prior studies employing the intrinsic cohesive crack model

, the tensile strength of concrete, tf , is

n line with prior studies

ic cohesive crack model for concrete, the tensile strength of concrete tf ,

was reported in Arrea and Ingraffea

series B beams by different analysts, as

FI =107 N/m and tf

are used, and FIIG is

FIFIIFI GGG 2=+= .

the critical internal

φφ cossincuf . The

Arrea and Ingraffea, 1982), and the

The mode-I cohesive

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strength mIT and the mode-II cohesive strength mIIT are used in the penalty and damage

initiation parts in Fig. 1 while the mixed-mode cohesive strength mT is used in the damage

evolution part in Fig. 1.

Table 1 Available approaches to simulate experiments in Arrea and Ingraffea (1982) and the fracture energy GF

and tensile strength tf employed.

Reference Approach Fracture energy FG (N/m) Tensile strength

tf (MPa)

Rots & De Borst (1987) Smeared crack model 75 2.7

Xie & Gerstle (1995) Energy-based extrinsic

cohesive crack model

150 4.0

Saleh & Aliabadi (1995) Boundary element

model

100 2.8

Cendon et al. (2000) LEFM + spring model 125 4.0

Galvez et al. (2002a,

2002b)

LEFM + intrinsic

cohesive crack model FIG =107 3.7

Yang & Deeks (2007) Scaled boundary FEM

+ extrinsic cohesive

crack model + LEFM-

based re-meshing

FIG =100 N/m, FIIG =0.1

FIG

=10 N/m mIT =

tf =3.0 MPa,

0.13/ == tmII fT MPa

Yu et al. (2008) Extrinsic cohesive

crack model

99.4 2.8

The present authors Intrinsic cohesive crack

model FIG =107 N/m,

FIIG =FIG

=107 N/m mIT =

tf =3.7 MPa, mIIT

=c=19.7 MPa

As a feature and also a limitation due to its basic assumption, in the cohesive crack model,

crack path is confined to bulk-element boundaries, which are not necessarily coincident with

the actual crack path. Fig. 3 shows a typical crack path in the present simulation with an

energy ratio of 2.0 and a stress ratio of 0.25. It can be found that the main crack propagation

included two line segments, namely, the first one starting from the pre-crack tip and oriented 045 and the second one kinked 045 and pointing upward. The crack path is very close to that

in Oliver et al. (2002) via the strong discontinuity approach. There is an obvious local mixed

mode when the crack propagates from the pre-crack, as the kink between the pre-crack and

the first line segment shows. Upon careful observation of the kink region between the two line

segments, multiple micro-cracks initiate there and actually formed a transitional region. The

deviation of the crack path having two kinked line segments from the actual crack path being

a smooth curve has a potential effect on energy release rate and the associated load response.

Fig. 3 - Crack path in the simulation with an energy ratio of 2.0 and a stress ratio of 0.25 when the CMSD is

0.133 mm (a deformation scale factor of 100 is employed for clear illustration).



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EFFECTS OF THE ENERGY RATIO ON THE LOAD RESPONSE AND

ENERGY TERMS

Bazant and Becq-Giraudon (2002) confirmed a value of 2.5 as the optimal energy ratio, with a

variation of approximate 40% from a database involving 238 test series. Bazant (2002)

reviewed the available research on concrete fracture and found that the energy ratio could be

taken as 2.5 and that the stress ratio varied from 0.15 to 0.33. In this section, three levels of

energy ratio (1.5, 2.0, and 2.5) and a stress ratio of 0.25 are chosen. Load-CMSD curves in the

present simulations, together with the lower and upper bounds of experimental results in

Arrea and Ingraffea (1982), are illustrated in Fig. 4. As the energy ratio increases, the initial

fracture energy decreases, and the concrete is damaged more easily; thus, both the peak load

and the peak-load CMSD decrease. By comparing the simulation results with the

experimental results in Fig. 4, it can be found that the width of the peak region larger than that

in the experimental results. After the peak region, the simulation results are larger than the

experimental ones, due to the difference between the crack path in the simulations and the

actual crack path, as shown in Fig. 3. The maximum energy release rate criterion states that

the actual crack propagates in a direction that maximizes the release of stored energy in the

structure. Therefore, the crack path in the simulation releases stored energy at a slower pace.

Fig. 4 - Simulated load-CMSD curves with three levels of energy ratio and a stress ratio of 0.25, together with

the lower and upper bounds of experimental results.

Energy terms are extracted from the simulations for further analysis. For the specimen here,

the damage dissipation, external work, and strain energy versus the CMSD are illustrated in

Fig. 5. Both the damage dissipation and the external work increase monotonically with the

CMSD, i.e., with the fracture process. As shown in Fig. 5a, the damage dissipation increases

with an increasing energy ratio. After damage initiation, as the energy ratio increases, the

initial fracture energy decreases, i.e., the softening stress in this region also decreases.

Therefore, the concrete is damaged more easily, and a comparatively larger fracture process

zone (FPZ) occurs, leading to larger cumulative damage dissipation. Fig. 5b shows that, after

damage initiation, the external work decreases with the energy ratio because less work is

required to overcome the initial fracture energy for the case of the larger energy ratio. Before

the peak-load occurs, the strain energy increases in a parabolic manner, as shown in Fig. 5c.

After the strain-energy peak, as the FPZ increases, the unloading in the undamaged zone

occurs around the FPZ, which explains why the strain energy decreases when extensive

CMSD (mm)

LoadP(KN)

0 0.02 0.04 0.06 0.080

50

100

150

200

Stress ratio=0.25

Lower bound of exp. resultsUpper bound of exp. results

Energy ratio=1.5Energy ratio=2.0

Energy ratio=2.5

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damage is present.

Fig. 5 - (a) Damage dissipation, (b) external work, and (c) strain energy versus the CMSD with three levels of

energy ratio and a stress ratio of 0.25.

EFFECTS OF THE STRESS RATIO ON THE LOAD RESPONSE AND THE

ENERGY TERMS

In this section, an energy ratio of 2.5 and three levels of stress ratio (0.15, 0.25, and 0.35) are

chosen. The predicted load-CMSD curves are illustrated in Fig. 6. For a specimen with a

median size, when the load is at the peak position, the FPZ around the pre-crack tip is small;

thus, the softening stress at the crack tip is almost defined by the initial linear segment of the

bilinear softening law. Therefore, the peak load is dominated by the initial fracture energy Gf.

When the energy ratio is fixed, the peak of the load-CMSD curves remains almost the same,

as shown in Fig. 6, and is in agreement with many available numerical simulations, as

reviewed in Bazant (2002). Because the peak of the load-CMSD curve is mainly dominated

by the initial fracture energy and depends less on the stress ratio, the stress ratio of 0.15

recommended by the CEB-FIP Model Code (1993) predicts the same peak load as the other

stress ratios. After the peak, with the FPZ around the pre-crack developing further, the crack

tip displacement (the vector sum of crack tip opening and sliding displacements) is larger than

kw , and thus, the softening stress in the FPZ is larger for a larger stress ratio. As the stress

ratio increases from 0.15 to 0.25, the load level in the post-peak region remains almost

identical; as the stress ratio further increases from 0.25 to 0.35, the load level in the post-peak

region increases significantly.

Fig. 6 - Simulated load-CMSD curves with an energy ratio of 2.5 and three levels of stress ratio.

CMSD (mm)

Damagedissipation(J)

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

Stress ratio=0.25

Energy ratio

1.52.02.5

CMSD (mm)

Externalwork(J)

0 0.02 0.04 0.06 0.08 0.10

5

10

15

20

Stress ratio=0.25

Energy ratio1.52.02.5

CMSD (mm)

Strainenergy(J)

0 0.02 0.04 0.06 0.08 0.10

5

10

15

Stress ratio=0.25

Energy ratio1.52.02.5

CMSD (mm)

LoadP(KN)

0 0.02 0.04 0.06 0.080

50

100

150

200

Energy ratio=2.5

Stress ratio=0.15Stress ratio=0.25Stress ratio=0.35

a b c



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Next, the effect of the stress ratio on the energy terms is studied. For the specimen here, the

damage dissipation, the external work, and the strain energy versus the CMSD are illustrated

in Figs. 7. All three energy terms are almost independent of the stress ratio before the peak-

load CMSD. Both the damage dissipation and the external work increase monotonically with

the fracture process. As shown in Fig. 7a, the damage dissipation has negligible dependence

on the stress ratio. Fig. 7b shows that the external work increases with the stress ratio after the

peak-load CMSD because, when extensive damage occurs, more work is required to

overcome the larger softening stress in the case of the larger stress ratio. Before the peak-load

CMSD, the strain energy increases in a parabolic manner, as shown in Fig. 7c. After the peak-

load CMSD, for the case of the stress ratios 0.15 and 0.25, the strain energy decreases because

of the unloading in the undamaged zone; for the stress ratio 0.35, the strain energy remains

constant and then increases.

Fig. 7 - (a) Damage dissipation, (b) external work, and (c) strain energy versus the CMSD with an energy ratio of

2.5 and three levels of stress ratio.

EFFECTS OF THE COHESIVE STRENGTH ON THE LOAD RESPONSE AND THE

ENERGY TERMS

Fig. 8 illustrates the simulated load-CMSD curves with an energy ratio of 2.5, a stress ratio of

0.25, and three levels of the mode-I cohesive strength mIT , i.e., tf80.0 (2.96 MPa), tf90.0

(3.33 MPa), and tf (3.70 MPa). Fig. 8 shows that the peak load increases clearly with mIT

while the increasing rate of the peak load is smaller than that of mIT . Furthermore, the peak-

load CMSD decreases as mIT increases. The associated damage dissipation, the external work,

and the strain energy versus the CMSD are illustrated in Fig. 9. The damage dissipation

increases negligibly with mIT , as shown in Fig. 9a. Fig. 9b shows that the external work

always increases with the CMSD. Fig. 9c shows that the strain energy increases with the

CMSD before the peak-load CMSD and then decreases after the peak-load CMSD due to

unloading in the undamaged zone around the FPZ. Figs. 9b and c show that the external work

and the strain energy increase with mIT .

CMSD (mm)


0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

Energy ratio=2.5

Stress ratio

0.150.250.35

CMSD (mm)

Externalwork(J)

0 0.02 0.04 0.06 0.08 0.10

5

10

15

Energy ratio=2.5

Stress ratio

0.150.250.35

CMSD (mm)

Strainenergy(J)

0 0.02 0.04 0.06 0.08 0.10

5

10

15

Energy ratio=2.5

Stress ratio

0.150.250.35

a b c

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Fig. 8 - Load-CMSD curves with an energy ratio of 2.5, a stress ratio of 0.25, and three levels of mIT .

Fig. 9 - (a) Damage dissipation, (b) external work, and (c) strain energy versus the CMSD with an energy ratio of

2.5, a stress ratio of 0.25, and three levels of mIT .

Fig. 10 illustrates load-CMSD curves with an energy ratio of 2.0, a stress ratio of 0.25, and

three levels of the mode-II cohesive strength mIIT , namely, c50.0 (9.85 MPa), c75.0 (14.775

MPa), and c (19.70 MPa). Fig. 10 shows that the load curves essentially coincide and show

nearly no dependence on mIIT . The associated damage dissipation, the external work, and the

strain energy versus the CMSD are illustrated in Fig. 11. After the peak-load CMSD, the

damage dissipation in the case of c50.0 is larger than that in the case of c75.0 and c by

almost a constant, as shown in Fig. 11a. On careful observation of Fig. 11a, the constant

amount of damage dissipation difference is accumulated during the initiation stage of the

crack propagation from the pre-crack. Figs. 11b and c show that the external work and the

strain energy are essentially independent of the mIIT . These observations are in agreement

with the current understanding that the crack grows mainly in a stable manner in local mode-I

under a global mixed-mode loading, as implied by the MTS criterion (2002), and that a

mixed-mode fracture mechanism dominates at crack initiation, though the specimens finally

fracture in mode-I manner (Ozbolt and Reinhardt, 2002). Furthermore, Figs. 8-11 show that

mIT had much larger effects on the load response and the energy terms than mIIT .

CMSD (mm)

Load(KN)

0 0.02 0.04 0.06 0.080

50

100

150

Energy ratio=2.5Stress ratio=0.25

TmI=0.80f

t

TmI=0.90f

t

TmI=1.00f

t

CMSD (mm)


0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5


TmI=0.80f

tTmI=0.90f

tTmI=1.00f

t

CMSD (mm)

Externalwork(J)

0 0.02 0.04 0.06 0.08 0.10

5

10

15


TmI=0.80f

tTmI=0.90f

t

TmI=1.00f

t

CMSD (mm)

Strainenergy(J)

0 0.02 0.04 0.06 0.08 0.10

5

10

15


TmI=0.80f

tTmI=0.90f

tTmI=1.00f

t

a b c



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Fig. 10 - Load-CMSD curves with an energy ratio of 2.0, a stress ratio of 0.25, and three levels of mIIT .

Fig. 11 - (a) Damage dissipation, (b) external work, and (c) strain energy versus the CMSD with an energy ratio

of 2.0, a stress ratio of 0.25, and three levels of mIIT .

CONCLUSIONS

In this study, the intrinsic cohesive crack model with a bilinear softening law is employed to

investigate mixed-mode fracture in a plain concrete beam. In contrast to the available

research, the present approach requires neither preliminary results from LEFM simulations

nor a re-meshing procedure or special implementation to prevent crack locking. The

simulations with regular meshes show that the approach can provide a reasonable estimation

of the peak load of the plain concrete structure. The parametric studies illustrate that the

energy ratio in the bilinear softening law have larger effects on the load response and energy

terms than the stress ratio. The simulations also confirm that the crack initiates in a mixed-

mode manner but grows in local mode-I under a global mixed-mode loading and that the

mode-I cohesive strength has much larger effects on the load response and the energy terms

than the mode-II cohesive strength.

ACKNOWLEDGMENTS

X. Guo acknowledges financial support from National Natural Science Foundation of China

(Project nos. 11372214 and 11102128) and Postdoctoral Fellowship from Faculty of

Engineering and UDF-Computational Science and Engineering (CSE) at The University of

Hong Kong.

CMSD (mm)

LoadP(KN)

0 0.02 0.04 0.06 0.080

50

100

150

200


TmII=0.50c

TmII=0.75c

TmII=1.00c

CMSD (mm)


0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5


TmII=0.50c

TmII=0.75c

TmII=1.00c

CMSD (mm)

Externalwork(J)

0 0.02 0.04 0.06 0.08 0.10

5

10

15


TmII=0.50c

TmII=0.75c

TmII=1.00c

CMSD (mm)

Strainenergy(J)

0 0.02 0.04 0.06 0.08 0.10

5

10

15


TmII=0.50c

TmII=0.75c

TmII=1.00c

a b c

Symposium_17


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REFERENCES

[1]-ABAQUS. ABAQUS theory manual and user's manual, version 6.8, Dassault, 2011.

[2]-Areias PMA, Rabczuk T. Quasi-static crack propagation in plane and plate structures

using set-valued traction-separation laws. Int J Numer Methods Engng, 2008, 74, p. 475-505.

[3]-Arrea M, Ingraffea AR. Mixed-mode crack propagation in mortar and concrete. Report

No. 81-13, Dept Struct Engng, Cornel University, Ithaca, New York, 1982.

[4]-Babuska I, Melenk J. The partition of unity method. Int J Numer Methods Engng, 1997,

40, p. 727-758.

[5]-Barenblatt GI. The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl

Mech, 1962, 7, p. 55-129.

[6]-Bazant ZP. Concrete fracture models: testing and practice. Engng Fract Mech, 2002, 69, p.

165-205.

[7]-Bazant ZP, Becq-Giraudon E. Statistical prediction of fracture parameters of concrete and

implications for choice of testing standard. Cement Concrete Res, 2002, 32, p. 529-556.

[8]-Bazant ZP, Cedolin L. Blunt crack band propagation in finite element analysis. J Eng

Mech, 1979, 111, p. 381-389.

[9]-Bruggi M, Venini P. Modeling cohesive crack growth via a truly-mixed formulation.

Comput Methods Appl Mech Engng, 2009, 198, p. 3836-3851.

[10]-Camacho GT, Ortiz M. Computational modelling of impact damage in brittle materials.

Int J Solids Struct, 1996, 33, p. 2899-2938.

[11]-CEB-FIP. Model Code 90. London, UK: Thomas Telford Ltd. 1993.

[12]-Cendon DA, Galvez JC, Elices M, Planas J. Modelling the fracture of concrete under

mixed loading. Int J Fract, 2000, 103, p. 293-310.

[13]-Cornec A, Scheider I, Schwalbe KH. On the practical application of the cohesive model.

Eng Fract Mech, 2003, 70, p. 1963-1987.

[14]-Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Solids, 1960, 8, p.

100-104.

[15]-Elices M, Guinea GV, Gomez J, Planas J. The cohesive zone model: advantages,

limitations and challenges. Eng Fract Mech, 2002, 69, p. 137-163.

[16]-Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse

shear. J Basic Eng, 1963, 85, p. 519-527.

[17]-Ferrara L. On the validity of non-local models for concrete: analysis of stress states

characterized by high strain-gradients. MSc Thesis, Politecnico di Milano, 1995.

[18]-Galvez JC, Cendon DA, Planas J. Influence of shear parameters on mixed-mode fracture

of concrete. Int J Fract, 2002a, 118, p. 163-189.

[19]-Galvez JC, Cervenka J, Cendon DA, Saouma V. A discrete crack approach to

normal/shear cracking of concrete. Cement Concrete Res, 2002b, 32, p. 1567-1585.

[20]-Galvez JC, Elices M, Guinea GV, Planas J. Mixed mode fracture of concrete under

proportional and nonproportional loading. Int J Fract, 1998, 94, p. 267-284.

[21]-Guinea GV, Planas J, Elices M. A general bilinear fit for the softening curve of concrete.



-1735-

Mater Struct, 1994, 27, p. 99-105.

[22]-Guo X, Chang K, Chen LQ, Zhou M. Determination of fracture toughness of AZ31 Mg

alloy using the cohesive finite element method. Eng Fract Mech, 2012a, 96, p. 401-415.

[23]-Guo X, Leung AYT, Chen AY, Ruan HH, Lu J. Investigation of non-local cracking in

layered stainless steel with nanostructured interface. Scripta Mater, 2010, 63, p. 403-406.

[24]-Guo X, Su RKL, Young B. Numerical investigation of the bilinear softening law in the

cohesive crack model for normal-strength and high-strength concrete. Adv Struct Eng, 2012b,

15, p. 373-387.

[25]-Guo X, Weng GJ, Soh AK. Ductility enhancement of layered stainless steel with

nanograined interface layers. Comput Mater Sci, 2012c, 55, p. 350-355.

[26]-Guo X, Zhang WJ, Zhu LL, Lu J. Mesh dependence of transverse cracking in laminated

metals with nanograined interface layers. Eng Fract Mech, 2013, 105, p. 211-220.

[27]-Guo ZK, Kobayashi AS, Hawkins NM. Dynamic mixed mode fracture of concrete. Int J

Solids Struct, 1995, 32, p. 2591-2607.

[28]-Hassanzadeh M. Behaviour of fracture process zones in concrete influenced by

simultaneously applied normal and shear displacements. PhD Thesis, Lund Institute of

Technol, 1991, 1-104.

[29]-Hillerborg A, Modeer M, Petersson PE. Analysis of crack formation and crack growth in

concrete by means of fracture mechanics and finite elements. Cement Concrete Res, 1976, 6,

p. 773-782.

[30]-John R, Shah SP. Mixed-mode fracture of concrete subjected to impact loading. J Struct

Eng, 1990, 116, p. 585-602.

[31]-Lens LN, Bittencourt E, d’Avila VMR. Constitutive models for cohesive zones in

mixed-mode fracture of plain concrete. Eng Fract Mech, 2009, 76, p. 2281-2297.

[32]-Needleman A. A continuum model for void nucleation by inclusion debonding. J Appl

Mech, 1987, 54, p. 525-531.

[33]-Nooru-Mohamed MB. Mixed mode fracture of concrete: an experimental approach. PhD

Thesis, Delft Univ. Technol 1992, 1-151.

[34]-Oliver J, Huespe AE, Pulido MDG, Chaves E. From continuum mechanics to fracture

mechanics: the strong discontinuity approach. Eng Fract Mech, 2002, 69, p. 113-136.

[35]-Ozbolt J, Reinhardt HW. Numerical study of mixed-mode fracture in concrete. Int J

Fract, 2002, 118, p. 145-161.

[36]-Park K, Paulino GH, Roesler JR. Determination of the kink point in the bilinear

softening model for concrete. Eng Fract Mech, 2008, 75, p. 3806-3818.

[37]-Prisco MD, Ferrara L, Meftah F, Pamin J, De Borst R, Mazars J, Reynouard JM. Mixed

mode fracture in plain and reinforced concrete: some results on benchmark tests. Int J Fract,

2000, 103, p. 127-148.

[38]-RILEM TC 50-FMC. Determination of the fracture energy of mortar and concrete by

means of three-point bend tests on notched beams. Mater Struct, 1985, 18, p. 287-290.

[39]-Roesler J, Paulino GH, Park K, Gaedicke C. Concrete fracture prediction using bilinear

softening. Cement & Concrete Compos, 2007, 29, p. 300-312.

Symposium_17


-1736-

[40]-Rots J, De Borst R. Analysis of mixed-mode fracture in concrete. J Eng Mech, 1987,

113, p. 1739-1758.

[41]-Ruiz G, Pandolfi A, Ortiz M. Three-dimensional cohesive modeling of dynamic mixed-

mode fracture. Int J Numer Methods Engng, 2001, 52, p. 97-120.

[42]-Saleh AL, Aliabadi MH. Crack growth analysis in concrete using boundary element

method. Eng Fract Mech, 1995, 51, p. 533-545.

[43]-Sancho JM, Planas J, Cendon DA, Reyes E, Galvez JC. An embedded crack model for

finite element analysis of concrete fracture. Eng Fract Mech, 2007a, 74, p. 75-86.

[44]-Sancho JM, Planas J, Fathy AM, Galvez JC, Cendon DA. Three-dimensional simulation

of concrete fracture using embedded crack elements without enforcing crack path continuity.

Int J Numer Analytical Methods Geomech, 2007b, 31, p. 173-187.

[45]-Song JH, Wang H, Belytschko T. A comparative study on finite element methods for

dynamic fracture. Comput Mech, 2008, 42, p. 239-250.

[46]-Song SH, Paulino GH, Buttlar WG. A bilinear cohesive zone model tailored for fracture

of asphalt concrete considering viscoelastic bulk material. Eng Fract Mech, 2006, 73, p. 2829-

2848.

[47]-Xie M, Gerstle WH. Energy-based cohesive crack propagation modeling. J Eng Mech,

1995, 121, p. 1349-1358.

[48]-Yang ZJ, Deeks AJ. Fully-automatic modelling of cohesive crack growth using a finite

element-scaled boundary finite element coupled method. Eng Fract Mech, 2007, 74, p. 2547-

2573.

[49]-Yu RC, Ruiz G. Explicit finite element modeling of static crack propagation in

reinforced concrete. Int J Fract, 2006, 141, p. 357-372.

[50]-Yu RC, Ruiz G, Chaves EWV. A comparative study between discrete and continuum

models to simulate concrete fracture. Eng Fract Mech, 2008, 75, p. 117-127.

[51]-Zhang Z, Paulino GH, Celes W. Extrinsic cohesive modelling of dynamic fracture and

microbranching instability in brittle materials. Int J Numer Methods Engng, 2007, 72, p. 893-

923.

Documents

PEAK LOAD ESTIMATION OF PRE-CRACKED PLAIN CONCRETE … · illustrate that this concise approach can provide a reasonable estimation of peak load of the pre-cracked plain concrete