three-dimensional crack propagation

Acta Mechanica Solida Sinica,Vol.23,No.3,June,2010ISSN0894-9166 Pu b l ish e d by A M S S Press,Wu h a n,Ch i n a

FINITE ELEMENT MODELLING OF COMPLEX3D STATIC AND DYNAMIC CRACK PROPAGATION BY EMBEDDING COHESIVE ELEMENTS IN ABAQUS??

X i ang t ing Su1,2Z h e n jun Yang2?Guohua Liu1

(1College of Civil Engineering and Architecture,Zhejiang University,Hangzhou310027,China) (2Department of Engineering,the University of Liverpool,L693GQ,UK)

Received23November2009;revision received25May2010

ABSTRACT This study proposes an algorithm of embedding cohesive elements in Abaqus and

develops the computer code to model3D complex crack propagation in quasi-brittle materials in

a relatively easy and e?cient manner.The cohesive elements with softening traction-separation

relations and damage initiation and evolution laws are embedded between solid elements in re-

gions of interest in the initial mesh to model potential cracks.The initial mesh can consist of

tetrahedrons,wedges,bricks or a mixture of these elements.Neither remeshing nor objective

crack propagation criteria are needed.Four examples of concrete specimens,including a wedge-

splitting test,a notched beam under torsion,a pull-out test of an anchored cylinder and a notched

beam under impact,were modelled and analysed.The simulated crack propagation processes and

load-displacement curves agreed well with test results or other numerical simulations for all the

examples using initial meshes with reasonable densities.Making use of Abaqus’s rich pre/post-

processing functionalities and powerful standard/explicit solvers,the developed method o?ers a

practical tool for engineering analysts to model complex3D fracture problems.

KEY WORDS?nite element method,cohesive elements,three-dimensional crack propagation,

discrete crack model,concrete structures,Abaqus

I.INTRODUCTION

Crack propagation is an inherent feature in quasi-brittle materials such as concrete due to the low material tensile strength,and it is a predominant source of nonlinearity and a main culprit for ultimate failure of structures made from these materials.Accurate understanding of the crack propagation behaviour so as to assess the structural load-carrying capacity,by either experimental studies or numerical modelling,remains a worldwide challenge,especially for three-dimensional(3D)problems with complex geometries and loading conditions.

Numerical modelling of crack propagation has been an active research?eld since1960s[1,2].Nowadays there exist a large number of numerical models.In terms of how the cracks are modelled geometrically, there are discrete crack models explicitly separating crack surfaces and modelling discontinuity,smeared crack models based on continuum mechanics,and more indirectly,lattice models.In terms of numerical methods,both the traditional?nite element methods(FEM)and boundary element methods(BEM)

·272·ACTA MECHANICA SOLIDA SINICA2010 have been widely applied to crack propagation modelling.Extensive literature review of these models and methods in2D problems has been given elsewhere[3–6].

It appears now that the discrete crack models,mostly based on the cohesive crack model(CCM)devel-oped in ductile materials by Barenblatt[7]and Dugdale[8]and in quasi-brittle materials by Hillerborg[9], are becoming more and more popular,because of their ability to model macroscopic cracks with strong discontinuity,the CCM’s capability of realistically representing the energy dissipation during fracture processes,and the ease of its implementation as cohesive interface elements(CIE)in the FEM and the BEM.In general,two types of approaches are used to model discrete cohesive crack propagation. The?rst type of approaches is based on sophisticated remeshing procedures that constantly change the meshes as cracks propagate[3,10–12].For problems with crack paths unknown a priori,objective crack propagation criteria are needed to judge when and in which direction a crack propagates.This usually involves calculation of stress intensity factors(SIFs)and stresses at crack tips,whose accuracy can only be ensured by?ne crack-tip meshes or using singular elements.This in turn exacerbates the complexity of the remeshing procedure.It is also very di?cult for the remeshing-based approaches to deal with complex crack propagation situations,such as multiple cracking and3D problems.Indeed,the success of the remeshing-based approaches has been largely limited to2D static fracture problems with single or a few cracks,although they are computationally e?cient because a relatively small number of nonlinear CIEs are inserted into the meshes.Another type of approaches pre-insert or pre-embed CIEs between the?nite elements in the initial meshes[6,13–15].Crack propagation is modelled by automatic opening and merging of the CIEs under applied loadings.Neither remeshing procedures nor crack prop-agation criteria are needed so complex crack propagation can be modelled.However,these approaches restrict the cracks to the?nite element edges or surfaces and the predicted crack patterns may thus be mesh-dependent.Another disadvantage is the high computational cost due to the use of a large number of nonlinear CIEs.The CIEs can also be dynamically inserted when certain crack initiation criterion is satis?ed[16,17]so that the number of CIEs changes with the loading and the total number of CIEs remains relatively low.These procedures of dynamically inserting CIEs need objective crack initiation criteria and change meshes although the remeshing operation is relatively straightforward.

In the last two decades,the partition of unity?nite element method(PUFEM)[18,19]and the extended ?nite element method(XFEM)[20–22]are becoming popular in modelling crack propagation.These methods introduce discontinuity into the?nite elements.The crack growth is described independently of the mesh,so that remeshing is not needed.Meshless methods are also receiving much attention[23,24]. These innovative methods,showing high potential in crack propagation modelling,generally need?ne crack tip meshes like the traditional FEM to calculate accurate stresses or SIFs used in evaluating the crack propagation criterion.In this sense,?ne initial meshes are needed if the crack paths are unknown in priori,which means high computational cost.The newly-developed semi-analytical scaled boundary ?nite element[25]is able to compute accurate SIFs without using?ne meshes or singular elements.It has recently been applied to model discrete cohesive crack propagation[4,5,26],but so far only2D problems were modelled.

In summary,all the above crack models and numerical methods have their own advantages and disadvantages in terms of e?ectiveness,e?ciency and applicability,and the selection largely depends upon the problems analysed and the preference and experience of the analysts.Most of the existing studies have modelled2D crack propagation.3D modelling of complex cohesive crack propagation has been sporadically conducted,and there is a long way to go before it becomes a routine exercise in research and practical design due to the inherent di?culties.In addition,most of the studies have used specialised in-house computer programs(e.g.,Cornell University’s FRANC)that may be di?cult for practical engineers to use.Most of the general-purpose commercial FEA packages,such as Ansys and Abaqus,are unable to model complicated crack propagation without user intervention or second development.

This study aims at developing a practical?nite element method for modelling complex3D crack propagation in quasi-brittle materials under static and dynamic loadings.The method makes use of a special type of CIEs called‘cohesive elements’,designed for modelling bonded interfaces in Abaqus V6.5or higher[27],to model potential cracks.An e?cient algorithm was devised to insert these CIEs into concerned regions in FE meshes consisting of various types of solid elements.The method is simple to implement.The rich pre/post-processing functionalities and powerful implicit and explicit solvers

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Vol.23,No.3Xiangting Su et al.:Complex3D Static and Dynamic Crack Propagation·273·of Abaqus can be fully exploited.Its e?ectiveness is validated by modelling various fracture problems of concrete structures.

II.THE MODELLING METHOD

The proposed method involves the following steps:

(1)Meshing the domain using Abaqus/CAE and generating an input?le.

(2)Inserting cohesive elements into the regions of interest in the initial FE mesh using an in-house computer program and generating another input?le.The initial mesh may consist of solid elements such as4-noded tetrahedrons(C3D4),6-noded wedges(C3D6)and8-noded bricks(C3D8),or a mixture of these elements.Both6-noded and8-noded cohesive elements(COH3D6and COH3D8)can be inserted. Figure1shows a mesh with solid elements only,the mesh after CIEs are inserted and the3D CIEs, respectively.

(3)Solving the problem using Abaqus standard or explicit solvers.

This procedure can be readily automated by running a batch?le.The following brie?y discusses the key elements in the method.

Fig.1.FE meshes before and after cohesive elements are inserted and3D cohesive elements.

2.1.Cohesive Elements with Damage in Abaqus

The cohesive crack model[7–9]assumes the ex-

istence of a fracture process zone(FPZ)in front

of the real crack tip,in which energy dissipa-

tion occurs during fracture.In the FPZ,there

exist tractions in the normal direction(t n)and

the two tangential or shear directions(t s and t t)

across the crack surfaces,resulting from mech-

anisms such as material bonding,aggregate in-

terlocking and surface friction.Figure2shows a

curve relating the normal traction t n and the crack

opening displacementδn as an example.Similar

traction-separation curves can be de?ned for shear

tractions(t s and t t)-crack sliding displacements (δs andδt)relations.Before the crack initiates,a Fig.2An example of traction-separation curve with softening for cohesive elements.

linear elastic ascending phase is assumed to model the initially un-cracked material.After the crack initiates,the traction decreases monotonically as functions of the corresponding separation,which is often termed tension or strain softening.The initial tensile sti?ness k n0and the initial shear sti?ness k s0and k t0should be high enough to represent the un-cracked material,but not too high to cause numerical ill-conditioning.These initial sti?ness values are determined by a trial and error approach. Ifδn is negative during loading increments or iterations,a compressive sti?ness of magnitude equal to k n0is assigned in order to prevent penetration of crack surfaces.

·274·ACTA MECHANICA SOLIDA SINICA2010 The cohesive elements in Abaqus[27]are based on the cohesive crack model.The constitutive response of cohesive elements,de?ned in terms of traction-separation laws,assumes initially linear elastic behav-iour followed by the initiation and evolution of damage.Several damage initiation criteria are available in Abaqus.This study used the quadratic nominal stress criterion in all the examples.The damage evolution criteria can be classi?ed into three categories according to the relation between traction and separation:linear damage evolution,exponential damage evolution,and tabular damage evolution.All these criteria were used in this study.

2.2.Inserting Cohesive Elements into3D FE Meshes

Although inserting cohesive elements into2D FE meshes is relatively straightforward[6],it is not a trivial task for3D problems,especially if the FE mesh is not regular and consists of di?erent types of solid elements.One challenge is how to robustly deal with the changes in the complicated3D nodal and elemental connectivity due to the insertion of CIEs.This is tackled by an e?ective embedding algorithm implemented in a MATLAB[28]program with carefully designed data structures listed in Table1.It has the following steps:

Table1.Data structures used in the algorithm of inserting cohesive elements in3D meshes

index index index POINT elemindex

elemconn faceconn nodeconn isInDe?nedArea xcord

faceconn nodeconn elemconn ycord Members xcord type isBound zcord

ycord isInDe?nedArea newnodeindex

zcord

isInDe?nedArea

Vol.23,No.3Xiangting Su et al.:Complex 3D Static and Dynamic Crack Propagation ·275·

and CIEs are https://www.360docs.net/doc/1418636372.html,puter implementation of the above inserting algorithm must be carried out

with care,or the time spent on inserting CIEs may be unduly long.As an example,for an initial mesh

with 46800C3D8elements and 52140nodes,it takes about 30minutes for the developed MATLAB

program to insert 135,200CIEs using a PC with an Intel Xeon CPU@3.16GHz and 3GB physical

memory.

III.NUMERICAL EXAMPLES

Four 3D concrete structures subjected to static and dynamic loadings were modelled using the

developed method as examples.A PC with two Intel Xeon CPUs @3.16GHz and 3GB physical memory

was used.Both Abaqus/standard and explicit solvers were used to solve the nonlinear equation systems.

When the quasi-static explicit solver was used to model static problems,the analysis time was carefully

chosen to avoid dynamic e?ects.Because the failure mode of all the examples is mainly tensile fracture

and there is no experimental data about shear fracture resistance,the strength and the initial sti?ness

in shear are assumed as 10times those in tension for the cohesive elements so that shear fracture failure

does not occur.

3.1.Wedge-Splitting Test

The ?rst example is the wedge-splitting test of a concrete specimen CP250carried out by Trunk [29].

The test was modelled in 2D by Trunk [29]and Feist [30],and in 3D by Areias and Belytschko [22].The

geometry and the loading condition are shown in Fig.3.The concrete has a Young’s modulus E =28300

MPa,the Poisson ratio ν=0.2and mass density ρ=2.5×10?6kg/mm 3.A bilinear softening curve

shown in Fig.4with parameters obtained from the experiment was used to model the normal traction-

separation relation of the CIEs.The area under the curve is the fracture energy G f =0.49N/mm.The

initial sti?ness k n 0was assumed as 1×105MPa ·mm.The Abaqus/explicit quasi-static solver used an

analysis time of 0.1

s.

Fig.3.Wedge-splitting test specimen (unit:

mm).

Fig. 4.Bilinear softening law used for the wedge-

splitting test.

Figure 5shows a deformed FE mesh with 774C3D6and 1392C3D8R solid elements,1854CIEs

and 6852nodes in total.It can be seen that the crack path was vertical as expected because of the

symmetry of the mesh and loading conditions.Finer meshes were also modelled but they led to little

di?erence from the predicted crack path and the load-crack mouth opening displacement (CMOD)

curves.Figure 6compares the load-CMOD curves from the experiment and modelling using di?erent

solvers and solid elements.The standard solver always failed by divergence at the post-peak stage,e.g.,

it failed at CMOD=3.2mm for C3D8as shown in Fig.6.This also often occurred in 2D simulations [6].

In contrast,the explicit solver always managed to model the whole fracture process when small enough

time increments were used.The explicit solver using C3D8R with reduced integration and hourglass

control (HC)resulted in nearly the same pre-peak responses as the standard solver using C3D8,while

the explicit solver using C3D8R but without hourglass control led to too soft pre-peak responses.C3D8

was also tried with the explicit solver but the computational time nearly doubled when C3D8R was

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·276·ACTA MECHANICA SOLIDA SINICA 2010

used.Therefore,the explicit solver was used in all the following simulations and C3D8R elements with

hourglass control were chosen when 8-noded brick elements were

used.

Fig.5.Deformed mesh (CMOD=4mm,

scale=100).Fig.6.Load-CMOD curves:e?ect of solid element types

and solvers.

Figure 7compares the load-CMOD curves from 3D and 2D simulations.In the 2D simula-tion,a similar mesh to the in-plane mesh of the 3D mesh (Fig.5)was used assuming a plane stress condition.It can be seen that the peak load pre-dicted from the 3D simulation is higher than that from the 2D,which re?ects the constraining e?ect of the specimen thickness.A very similar mesh to Fig.5but with a non-symmetric distribution of elements along the vertical central line was also modelled.The deformed mesh is shown in Fig.8.The crack path deviated from the central line,indicating the e?ect of the initial mesh.The

load-Fig.7Load-CMOD curves from 2D and 3D.CMOD curves predicted from the two meshes are shown in

Fig.9.

Fig.8.Deformed mesh from a non-symmetric mesh

(CMOD=4mm,

scale=100).Fig.9.Load-CMOD curves:e?ect of mesh arrangement.

3.2.Torsion Fracture Test

The second example is a concrete bar under torsion tested by Brokenshire [31].The geometry and the

loading condition are illustrated in Fig.10.3D modelling was carried out by Je?erson et al.[32]and Gasser

and Holzapfel [33].The three support points were assumed as ?xed and the loading incremental procedure

was controlled by the vertical displacement at the point C .The concrete has E =35000MPa,ν=0.2and

Vol.23,No.3Xiangting Su et al.:Complex3D Static and Dynamic Crack Propagation·277·

Fig.10.A concrete bar under torsion(unit:mm).

ρ=2.5×10?6kg/mm3.The same linear soften-

ing curve(Fig.11)as used in Ref.[32]was adopted

to model the normal traction-separation relation

with G f=0.08N/mm.The initial sti?ness was

set as k n0=7.5×106MPa·mm.An analysis time

of0.3s was used in the explicit quasi-static pro-

cedure.CMOD is de?ned as the relative displace-

ment between the points A and B normal to the

notch surface(ref.Fig.10).Fig.11Linear softening law used for the torsion test.

Two FE meshes with di?erent solid element types were modelled to investigate the e?ect on the prediction of crack surfaces.Mesh A in Fig.12(a)has10651C3D4solid elements and9606CIEs,and Mesh B in Fig.12(b)has7027C3D8R and C3D6elements and8055CIEs.The predicted crack surfaces are compared with the test result in Fig.13.It is clear that Mesh A with C3D4led to very realistic, tortuous fracture surfaces whereas Mesh B with C3D8R predicted an unrealistically?at and smooth crack path.The predicted load-CMOD curves are presented in Fig.14.It can be seen that both meshes led to curves close to the experimental data.Mesh A predicted a peak load slightly higher than Mesh B.This may be explained by the predicted crack surfaces in Fig.13,where the tortuous and unsmooth crack surfaces from Mesh A o?er higher fracture resistance than the?at and smooth ones from Mesh B. Je?erson et al.[32]also modelled the same test using a similar mesh to Mesh B and the‘Craft’concrete

Fig.12.Two FE meshes used for the torsion test.

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·278·ACTA MECHANICA SOLIDA SINICA

2010

Fig.13.Crack surfaces after failure for the torsion

test.

Fig.14.Load-CMOD curves for the torsion

test.Fig.15.Pull-out test of a steel bar embedded in concrete

(unit:mm).

model.The predicted load-CMOD curve is also shown in Fig.14and it is very close to that from Mesh

B in this study.

3.3.Pull-out Test of a Steel-Embedded Concrete Anchor

This example models the pull-out behaviour of a steel anchor embedded in concrete.The detailed

geometry,applied boundary and loading conditions of one-quarter specimen are illustrated in Fig.15.

This example was also modelled in Ref.[19,22,23]using other methods.The anchor bolt,excluded from

the FE model,was simulated by a boundary condition of vertical displacement imposed on the

top

Fig.16.Meshes used for the pull-out test.

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Vol.23,No.3Xiangting Su et al.:Complex3D Static and Dynamic Crack Propagation·279·surface of the groove.The concrete properties are:E=30000MPa,ν=0.2andρ=2.5×10?6kg/mm3.

A linear softening curve with G f=0.1N/mm,tensile strength=3MPa and k n0=6×104MPa·mm was used.An analysis time of0.1s was used in the explicit solver.

To investigate the mesh-dependence of results,two meshes were modelled.The coarse mesh in Fig.16(a)has31889C3D4solid elements and60164CIEs,and the numbers for the?ne mesh in Fig.16(b) are69798and138169,respectively.Figure17shows the load(F)-displacement(d)curves from the two meshes and other studies.It can be seen that the two meshes predicted very close responses,especially the same failure point when the load suddenly dropped at d=0.25mm.

Fig.17.Load-displacement curves for the pull-out test.

Similar crack propagation processes and?nal crack patterns were predicted from the two meshes. Figure18illustrates four stages of the predicted crack propagation using the?ne mesh,where the cracks are represented by grey areas consisting of highly-damaged cohesive elements(with the damage Fig.18.Predicted crack propagation process of the pull-out test.

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index D ≥0.99).At F =310kN (60%of the peak load F u =503kN),tensile cracks initiated at

the top edge of the groove,forming a cone around the groove (Fig.18(a)).As the load approached

the peak,a splitting-mode longitudinal crack initiated at the top surface and propagated in the radial

direction (Fig.18(b)).Beyond the peak load (Fig.18(c)),the longitudinal crack propagated rapidly,

leading to a drastic decline on the load-displacement curve (Fig.17).After the complete propagation

of the longitudinal crack (Fig.18(d)),the conical crack continued to advance with the load decreasing

slowly.The simultaneous propagation of the longitudinal crack and the conical crack,observed in the

experiments by Rots [34],was well reproduced.This may explain the di?erence between the predicted

load-displacement curves (Fig.17)by the present study and other studies [22,23]which only modelled

the conical crack.

3.4.Impact Test of a Concrete Beam

The fourth example is a concrete beam under

impact tested and modelled in 2D by Du et al.[35].

3D modelling was reported by Belytschko et al.[36].

The geometry and the boundary condition are

shown in Fig.19.The impact load history obtained

from the test (Fig.20)was input as the loading

condition.The concrete properties are:E =34480MPa,ν=0.2and ρ=2.5×10?6kg/mm 3.

An

Fig.19Impact test of a concrete beam (unit:mm).exponential softening curve (Fig.21)with the exponential law parameter α=1.5and the equivalent

dynamic fracture energy G f =0.152N/mm,suggested in Ref.[35],was adopted to model the CIEs.

The initial sti?ness k n 0=1×105MPa ·mm was used.

Figure 22shows a deformed mesh at the post-peak stage when t =1ms.The mesh consists of

3645solid elements and 606cohesive elements.Figure 23compares favourably the predicted load-

de?ection curve with the test data.Figure 24compares the measured and computed strain histories

at three locations (SG01,SG02and SG03).The strain history computed for SG03agreed well with

the measured.For the other two locations,the strain histories were underestimated,probably due to Fig.20.Load history used for the impact

test.

Fig.21.Exponential softening law for the impact

test.

Fig.22.Deformed mesh for the impact

test.

Fig.23.Load-displacement curves for the impact test.

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Vol.23,No.3Xiangting Su et al.:Complex 3D Static and Dynamic Crack Propagation ·281·

the negligence of shrinkage e?ects in the numerical modelling [37].The crack-tip extension histories are

compared in

Fig.25.

Fig.24.Strain history for the impact

test.Fig.25.Crack-tip extension history for the impact test.

IV.CONCLUSIONS

A simple yet e?ective ?nite element methodology has been developed to simulate 3D complex cohesive

crack propagation in quasi-brittle materials,using Abaqus’s cohesive elements.An e?cient algorithm

to insert the cohesive elements into initial ?nite element meshes was devised and implemented by an

in-house computer program.Several concrete examples were modelled to demonstrate the e?ectiveness

of the method.It is concluded that,making use of the rich pre/post-processing functionalities and

powerful standard/explicit solvers in Abaqus,the developed methodology seems promising to provide

a practical simulation tool for modelling realistic 3D fracture in various industries.

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FINITE ELEMENT MODELLING OF COMPLEX 3D STATIC AND DYNAMIC CRACK

PROPAGATION BY EMBEDDING COHESIVE ELEMENTS IN ABAQUS

作者：Xiangting Su， Zhenjun Yang， Guohua Liu

作者单位：Xiangting Su(College of Civil Engineering and Architecture,Zhejiang University,Hangzhou

310027,China;Department of Engineering,the University of Liverpool,L69 3GQ,UK)， Zhenjun

Yang(Department of Engineering,the University of Liverpool,L69 3GQ,UK)， Guohua Liu(College of

Civil Engineering and Architecture,Zhejiang University,Hangzhou 310027,China)

刊名：

固体力学学报（英文版）

英文刊名：ACTA MECHANICA SOLIDA SINICA

年，卷(期)：2010,23(3)

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