Research Article Damage and Failure Process of Concrete

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 631074 5 pageshttpdxdoiorg1011552013631074

Research ArticleDamage and Failure Process of Concrete Structure underUniaxial Compression Based on Peridynamics Modeling

Feng Shen Qing Zhang and Dan Huang

Department of Engineering Mechanics Hohai University Nanjing 210098 China

Correspondence should be addressed to Qing Zhang lxzhangqinghhueducn

Received 19 July 2013 Accepted 18 September 2013

Academic Editor Zhiqiang Hu

Copyright copy 2013 Feng Shen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Peridynamics is a nonlocal formulation of continuum mechanics which uses integral formulation rather than the spatial partialdifferential equations The peridynamic approach avoids using any spatial derivatives which arise naturally in the classical localtheory It has shown effectiveness and advantage in solving discontinuous problems at both macro- and microscales In this paperthe peridynamic theory is used to analyze damage and progressive failure of concrete structures A nonlocal peridynamic modelfor concrete columns under uniaxial compression is developed Numerical example illustrates that cracks in a peridynamic bodyof concrete form spontaneously The result of the example clarifies the unique advantage of modeling damage accumulationand progressive failure of concrete based on peridynamic theory This study provides a new promising alternative for analyzingcomplicated discontinuity problems Finally some open problems and future research trends in peridynamics are discussed

1 Introduction

Concrete is a complex material which is used widely in civilengineering structures such as dams buildings and bridgesConcrete is difficult to model because it is a quasibrittlematerial and precise analysis of concrete structures is elusiveIn fact concrete exhibitsmany discontinuities even before theapplication of load Expansion shrinkage and temperaturechanges often cause crackingThe formation of cracks signif-icantly affects the stress and displacement fields Even thoughconcrete has been studied for many years its progressivefailure process especially the mechanism of spontaneousformation of a crack and its subsequent growth have notbeen fully understood yet The damage of concrete is theprocess of crack initiation propagation and coalescencewhich may cause a sudden collapse suddenly Over the pastseveral decades continuum mechanics and finite elementmethod (FEM) solved many problems in solid mechanicsThe finite element method has become the most commonlyaccepted technique for the numerical solution of structuralproblems Continuum mechanics and finite element methodstart with an assumption of a spatially continuous anddifferentiable displacement field Strains are obtained from

spatial derivatives of displacement field When applied toconcrete they have limited application to model damageAs mentioned earlier problems appear when damage isinvolved The basic assumption of displacement continuityis not valid Nearly all traditional methods based on theclassical theory of continuum mechanics attempt to solvethe partial differential equations that include derivativesof the displacement components but these derivatives areundefined when the displacements are discontinuous suchas across cracks or interfaces

To overcome the shortcoming of the classical theoryof continuum mechanics Silling at Sandia Nation Labora-tories introduces the peridynamic theory [1] which doesnot assume spatial differentiability of displacement field andpermits discontinuities to arise as part of solution Theterm ldquoperidynamicrdquo comes from the Greek words ldquoperirdquoand ldquodynamicrdquo which mean ldquonearrdquo and ldquoforcerdquo respectivelyThis theory is nonlocal which can be thought of as ageneralization of classical theory of elasticity

After mathematical demonstration of the theory peridy-namics has been applied to 1D bar [2] membranes fibers[3] and so on However peridynamic theory has not beenapplied to model concrete very often Gerstle et al [4 5]

2 Mathematical Problems in Engineering

f

f

f

R

120575

x

x998400

Figure 1 Pairwise interaction of a material point with its neighbor-ing points

model the damage of concrete structures using a simpleconceptual peridynamic model and Kilic and Madenci [6]have done some research on the structural stability of aconcrete column Shen et al [7 8] and Huang et al [9 10]employed peridynamic models to simulate the damage andprogressive failure of the concrete and clarified the uniqueadvantage of peridynamics

This paper aims to investigate the capability of peridy-namic to model concrete structure under uniaxial compres-sion The remaining of the paper is organized as follows InSection 2 a brief introduction to peridynamic theory and thesolutionmethod used in the present work is given A descrip-tion of the peridynamic model for uniaxial compression ofconcrete and the simulation results are presented in Section 3Then some concluding remarks and the shortcomings ofperidynamic theory are finally provided at the end of thispaper

2 Peridynamic Theory and Solution Method

The peridynamic theory was first introduced by Silling [1] in2000 which makes no assumption of continuity of displace-ments In peridynamics the material domain is discretizedinto an infinite number of infinitesimal interacting particlesParticles separated by a finite distance can interact with eachother The maximum interaction distance between particlesin peridynamics is called the ldquomaterial horizonrdquo When thedistance between two particles is less than the materialhorizon they interact with each other otherwise they donot interact The interacting forces between particles arecalculated through empirical particle interaction relationsParticles move in accordance with Newtonrsquos second law

Referring to Figure 1 the peridynamic formulation pro-posed by Silling [1] is defined by the following equation

119871u (x 119905) = int119877

119891 (u (x1015840 119905) minus u (x 119905) x1015840 minus x) 119889119881x1015840

forallx isin 119877 119905 ge 0(1)

In this formulation at any point 119909 in the reference configura-tion119877 at any time 119905 each pair of particles interacts through aforce function 119891 119871 is the pair wise force acting upon x due toall particles x1015840 within its horizon 119906 is the displacement fieldand119889119881x1015840 is the integration variable that indicates infinitesimaldomain located at point x1015840 According toNewtonrsquos second lawthe peridynamic equation of motion is given by

1205881205972u1205971199052

= int119867

119891 (u (x1015840 119905) minus u (x 119905) x1015840 minus x) 119889119881x1015840 + b (x 119905) (2)

where b is a prescribed body-force density field whichrepresents the external force per unit reference volume 120588is mass density in the reference configuration and 119867 is theneighborhood of material points x within the horizon size of120575 as follows

119867 = 119867(x 120575) = x1015840 isin 119877 10038171003817100381710038171003817x1015840minus x10038171003817100381710038171003817 le 120575 (3)

120575 is a positive value called the material horizon Whenthe distance between two particles is less than or equalto 120575 they will interact with each other through the forcefunction 119891 otherwise if the distance exceeds horizon theinteraction may vanish In calculation the value of horizonsize 120575 can be confirmed according to the requirement ofsimulation efficiency and precision It can be found thata much larger horizon size will lead to a higher accuracybut lower efficiency Referring to the literatures [11 12] andpractices for macroscale modeling we choose the horizonwhich is three times the lattice constant And119891 is the pairwiseforce between particles x and x1015840 which depends on therelative position and the relative displacement between thetwo particles Silling uses 120585 120578 for the relative position and therelative displacement vectors respectively as follows

120585 = x1015840 minus x 120578 = u1015840 minus u (4)

Therefore the vector 120578 + 120585 = (x1015840 + u1015840) minus (x + u) denotesthe current relative position between points x1015840 and x in thedeformed configuration This means that the force vectorbetween two particles is parallel to their relative currentposition vector The pairwise function 119891 contains all of theconstitutive information of the material Because the interac-tion of particles is at a finite distance peridynamic is in thecategory of nonlocal models Peridynamics is not defined inthe form of stress and strain and it eliminates the assumptionon the continuity or differentiability of the displacement fieldor small-deformation behavior thoroughly Many studies[13ndash15] proposed various peridynamic constitutive modelscombined with the constitutive information of the materialin the force function119891 Any function satisfying the linear and

Mathematical Problems in Engineering 3

the angular admissibility condition is a valid force functionFor a bond-based peridynamic linear elastic material thepairwise force function 119891 is given by [11]

119891 (120578 120585) = 119888s (5)

where119891 has units of force per unit volume squared 119888 is calledthe peridynamic microelastic constant and s is the bondstretch as follows

s =1003816100381610038161003816120585 + 120578

1003816100381610038161003816 minus |120585|

|120585| (6)

Regardless of the distance among interacting points the forcefunction 119891 assumes that the interaction force remains thesame for a constant stretch Considering the effect of thermalloading (5) can be modified as

119891 (120578 120585) = 119888 (s minus 120572120579) (7)

120572 and 120579 stand for the coefficient of thermal expansion andthe temperature difference between current and referencetemperatures between material points x1015840 and x According tothe prototype microelastic brittle (PMB) model introducedby Silling and Askari [12] we have

119888 =18119870

1205871205754 (8)

where 119870 is the bulk modulus of the material expressed interms of elastic modulus 119864 and Poissonrsquos ratio ] as

119896 =119864

3 (1 minus 2]) (9)

So the peridynamic equation of motion (2) can be rewrittenas

1205881205972u1205971199052

= int119867x

f (120578 120585) 119889119881120585+ b (x 119905) (10)

with

f (120578 120585) = 119891 (120578 120585) 120585 + 1205781003816100381610038161003816120585 + 1205781003816100381610038161003816

(11)

where119867x is the neighborhood of the particle 119909 it is assumedto be a spherical region centered at x with the radius of thematerial horizon and the horizon is analogous to the cutoffradius used in molecular dynamics

The break of the bond is described by the factor 120583 asfollows

120583 (119905 120585) = 1 119904 (119905

1015840 120585) minus 120572120579 lt 119904

0 0 le 119905

1015840le 119905

0 otherwise(12)

where 1199040is a critical stretch of the material for failure

Although 1199040is expressed as a property of a particle bond

breaking must be a symmetric operation for all particle pairssharing a bond [12] Therefore we can define the damage at apoint x as

120601 (x 119905) = 1 minusint119867x120583 (120578 120585) 119889119881x1015840

int119867x119889119881x1015840

(13)

To solve the peridynamic equation of motion the regiondefining a peridynamic material is discretized into particles(material points) forming a simple cubic lattice with thelattice constant Δx and a numerical approximation is usuallyimplemented by reducing the spatial integration into finitesum So the integral in (2) is replaced by

120588u119899i = sum

119901isin119876

f (u119899119901minus u119899119894 x119901minus x119894)119881119901+ b119899119894 (14)

where 119899 is the number of time steps subscripts denote theparticle number and 119876 represents the family of particles forwhich particle x shares a bond in the reference configurationwithin the horizon That is

119876 = 119901 |10038171003817100381710038171003817x1015840 minus x10038171003817100381710038171003817 le 120575 (15)

Through calculation test considering both efficiency and sta-bility the appropriate time step size for different simulationtask can be confirmed [12]

3 Numerical Implementation and Results

In the following a concrete column with the diameter of015m is presented The height of the column is 03m Theconcrete material considered is assumed to be homogeneousisotropic and brittle elastic and the elastic constants areidentical to those in classical continuummodelThe concretehas density 120588 = 2200 kgm3 The bulk modulus 119870 = 20GPaand a critical bond stretch for failure of a bond is set ass0= 0005m In the peridynamic simulation the column is

discretized into particles forming a simple cubic lattice withlattice constant Δx = 0005m and horizon 0015m (Thehorizon is three times the lattice constant) The temperatureof the whole system is kept at 300K

The position and velocity of each particle are calculatedas a solution to the equation of motion using the velocity-velert time integration algorithm and the time step is set to1 times 10

minus7 s based on the combined optimization test of bothintegration stability and computation efficiency [12]

The numerical experiment was performed to simulatethe damage and progressive failure of the concrete columnunder uniaxial compression In the simulation the velocityof all particles was initially set to zero along the threeorthogonal directions in Cartesian coordinates When theuniaxial deformation simulation is implemented the tenlayers of particles at the bottom of column are constrained tobe fixed with a zero velocity as the top ten layers of particleswere set to have a downward velocity of 20ms Figure 2shows the damage and failure process of the concrete columnunder uniaxial compression It can be found that the break ofbonds appears symmetrically The top ten layers of particleshave moved down during the simulation and the specimenhas formed the diagonal shear bands as well as the dynamicejection of crushed material from the middle of the columnEvenmore the fracture tracks along the direction of 45∘ abovehorizon which is consistent with the experimental resultsof concrete structures under uniaxial compression Figure 3displays the failure mode of concrete column section underuniaxial compression using peridynamic model


(a) (b) (c)

(d) (e) (f)

Figure 2 Peridynamic simulation of concrete column subjected by uniaxial compression where 119905 = 1000 2000 3000 4000 5000 6000 timesteps for (a) to (f) respectively

Figure 3 Peridynamic simulation of the failure mode of concretecolumn under uniaxial compression

From the simulation results it is observed that thedamage and failure of the column is just spontaneous andno special failure criteria are required in order to predict thepath along which the crack develops

4 Concluding Remarks

This paper takes initial attempt to solve the concrete col-umn under uniaxial compression based on the peridynamictheory The result of the study clarifies the unique advan-tage of peridynamics in modeling concrete structures As areformation of the classical continuum mechanics theoriesperidynamics is a powerful tool to solve discontinuitiesproblems The peridynamic theory can avoid singularity insolving discontinuity problems by integral equations ratherthan differential equations The basic equations can beapplied anywhere in the model including at the cracksso no additional technique is necessary for the study ofdamage and cracking The problem of concrete columnunder uniaxial compression shows that a relatively simpleperidynamicmodel can simulate the entire damage evolutionand fracture process in concrete structures effectively withgreat potential over both mesh free method and moleculardynamics In other words the peridynamic model can beregarded as a coarse-grain molecular dynamics model The


pairwise function contains all of the constitutive informationof the material and the limitations such as remeshing andmapping can be removed thoroughly

As a new powerful tool the peridynamic theory hassome shortcomings itself Dealing with an infinite number ofmaterial particles amuch greater calculation time is requiredAs each particle will interact with its neighbors within thehorizon besides the nearest ones the peridynamic simulationat a nondamaged region is computationally expensive com-pared with Finite Element Method In addition to this theoriginal peridynamic constitutive models oversimplify theinteraction between the particles in the sense that the mutualforce between two particles is assumed to be independentof the positions of other particles within the horizon Therotation of the particles is not considered which influencethe exactitude of the constitutive relations of thematerialTheplasticity and viscosity in the classical theory of continuummechanics based on peridynamics are still under researchwhich limits the application And more efficient solutionalgorithms and appropriate peridynamic constitutive modelsneed to be developed Furthermore a multiscale researchapproach combining peridynamics with molecular dynamicsand finite element method will be a promising way to solvediscontinuous problems more accurately and effectively Andthe boundary and the size effects arising in the peridynamicmodel need to be investigated in the future

Conflict of Interests

The authors of the paper declare that there is no conflict ofinterests regarding the publication of this paper The authorsdo not have a direct financial relation with the commercialidentity that might lead to a conflict of interests for any of theauthors

Acknowledgment

This work was supported by the National Nature Sci-ence Foundation of China (Grant nos 11372099 5117906411132003 and 51209079) and the Graduate Students Researchand Innovation Plan of Jiangsu Province (no CXZZ11 0425)

References

[1] S A Silling ldquoReformulation of elasticity theory for discontinu-ities and long-range forcesrdquo Journal of theMechanics and Physicsof Solids vol 48 no 1 pp 175ndash209 2000

[2] S A Silling M Zimmermann and R Abeyaratne ldquoDeforma-tion of a peridynamic barrdquo Journal of Elasticity vol 73 no 1ndash3pp 173ndash190 2003

[3] S A Silling and F Bobaru ldquoPeridynamic modeling of mem-branes and fibersrdquo International Journal of Non-Linear Mechan-ics vol 40 no 2-3 pp 395ndash409 2005

[4] W Gerstle N Sau and E Aguilera ldquoPeridynamic modeling ofplain and reinforced concrete structuresrdquo in Proceedings of the18th International Conference on StructuralMechanics in ReactorTechnology (SMiRT rsquo05) Beijing China 2005

[5] W Gerstle N Sau and S Silling ldquoPeridynamic modeling ofconcrete structuresrdquo Nuclear Engineering and Design vol 237no 12-13 pp 1250ndash1258 2007

[6] B Kilic and EMadenci ldquoStructural stability and failure analysisusing peridynamic theoryrdquo International Journal of Non-LinearMechanics vol 44 no 8 pp 845ndash854 2009

[7] F Shen Q Zhang D Huang and J Zhao ldquoPeridynamicsmodeling of failure process of concrete structure subjected toimpact loadingrdquo Engineering Mechanics vol 29 no s1 pp 12ndash15 2012

[8] F Shen Q Zhang D Huang and J Zhao ldquoDamage and failureprocess of concrete structure under uni-axial tension basedon peridynamics modelingrdquo Chinese Journal of ComputationalMechanics vol 30 pp 79ndash83 2013

[9] D Huang Q Zhang and P Z Qiao ldquoDamage and progressivefailure of concrete structures using non-local peridynamicmodelingrdquo Science China Technological Sciences vol 54 no 3pp 591ndash596 2011

[10] D Huang Q Zhang P Qiao and F Shen ldquoA review on peri-dynamics method and its applicationsrdquo Advances in Mechanicsvol 40 no 4 pp 448ndash459 2010

[11] M L Parks R B Lehoucq S J Plimpton and S A SillingldquoImplementing peridynamics within a molecular dynamicscoderdquo Computer Physics Communications vol 179 no 11 pp777ndash783 2008

[12] S A Silling and E Askari ldquoA meshfree method based onthe peridynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[13] Q Du andK Zhou ldquoMathematical analysis for the peridynamicnonlocal continuum theoryrdquo ESAIM Mathematical Modellingand Numerical Analysis vol 45 no 2 pp 217ndash234 2011

[14] B K Tuniki Peridynamic constitutive model of concrete [MSthesis] University of New Mexico 2012

[15] P Seleson andM L Parks ldquoOn the role of the influence functionin the peridynamic theoryrdquo International Journal for MultiscaleComputational Engineering vol 9 no 6 pp 689ndash706 2011

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Stochastic AnalysisInternational Journal of


f

f

f

R

120575

x

x998400

Figure 1 Pairwise interaction of a material point with its neighbor-ing points

model the damage of concrete structures using a simpleconceptual peridynamic model and Kilic and Madenci [6]have done some research on the structural stability of aconcrete column Shen et al [7 8] and Huang et al [9 10]employed peridynamic models to simulate the damage andprogressive failure of the concrete and clarified the uniqueadvantage of peridynamics

This paper aims to investigate the capability of peridy-namic to model concrete structure under uniaxial compres-sion The remaining of the paper is organized as follows InSection 2 a brief introduction to peridynamic theory and thesolutionmethod used in the present work is given A descrip-tion of the peridynamic model for uniaxial compression ofconcrete and the simulation results are presented in Section 3Then some concluding remarks and the shortcomings ofperidynamic theory are finally provided at the end of thispaper

2 Peridynamic Theory and Solution Method

The peridynamic theory was first introduced by Silling [1] in2000 which makes no assumption of continuity of displace-ments In peridynamics the material domain is discretizedinto an infinite number of infinitesimal interacting particlesParticles separated by a finite distance can interact with eachother The maximum interaction distance between particlesin peridynamics is called the ldquomaterial horizonrdquo When thedistance between two particles is less than the materialhorizon they interact with each other otherwise they donot interact The interacting forces between particles arecalculated through empirical particle interaction relationsParticles move in accordance with Newtonrsquos second law

Referring to Figure 1 the peridynamic formulation pro-posed by Silling [1] is defined by the following equation

119871u (x 119905) = int119877

119891 (u (x1015840 119905) minus u (x 119905) x1015840 minus x) 119889119881x1015840

forallx isin 119877 119905 ge 0(1)

In this formulation at any point 119909 in the reference configura-tion119877 at any time 119905 each pair of particles interacts through aforce function 119891 119871 is the pair wise force acting upon x due toall particles x1015840 within its horizon 119906 is the displacement fieldand119889119881x1015840 is the integration variable that indicates infinitesimaldomain located at point x1015840 According toNewtonrsquos second lawthe peridynamic equation of motion is given by

1205881205972u1205971199052

= int119867

119891 (u (x1015840 119905) minus u (x 119905) x1015840 minus x) 119889119881x1015840 + b (x 119905) (2)

where b is a prescribed body-force density field whichrepresents the external force per unit reference volume 120588is mass density in the reference configuration and 119867 is theneighborhood of material points x within the horizon size of120575 as follows

119867 = 119867(x 120575) = x1015840 isin 119877 10038171003817100381710038171003817x1015840minus x10038171003817100381710038171003817 le 120575 (3)

120575 is a positive value called the material horizon Whenthe distance between two particles is less than or equalto 120575 they will interact with each other through the forcefunction 119891 otherwise if the distance exceeds horizon theinteraction may vanish In calculation the value of horizonsize 120575 can be confirmed according to the requirement ofsimulation efficiency and precision It can be found thata much larger horizon size will lead to a higher accuracybut lower efficiency Referring to the literatures [11 12] andpractices for macroscale modeling we choose the horizonwhich is three times the lattice constant And119891 is the pairwiseforce between particles x and x1015840 which depends on therelative position and the relative displacement between thetwo particles Silling uses 120585 120578 for the relative position and therelative displacement vectors respectively as follows

120585 = x1015840 minus x 120578 = u1015840 minus u (4)

Therefore the vector 120578 + 120585 = (x1015840 + u1015840) minus (x + u) denotesthe current relative position between points x1015840 and x in thedeformed configuration This means that the force vectorbetween two particles is parallel to their relative currentposition vector The pairwise function 119891 contains all of theconstitutive information of the material Because the interac-tion of particles is at a finite distance peridynamic is in thecategory of nonlocal models Peridynamics is not defined inthe form of stress and strain and it eliminates the assumptionon the continuity or differentiability of the displacement fieldor small-deformation behavior thoroughly Many studies[13ndash15] proposed various peridynamic constitutive modelscombined with the constitutive information of the materialin the force function119891 Any function satisfying the linear and



119891 (120578 120585) = 119888s (5)


s =1003816100381610038161003816120585 + 120578

1003816100381610038161003816 minus |120585|

|120585| (6)


119891 (120578 120585) = 119888 (s minus 120572120579) (7)


119888 =18119870

1205871205754 (8)


119896 =119864

3 (1 minus 2]) (9)


1205881205972u1205971199052

= int119867x

f (120578 120585) 119889119881120585+ b (x 119905) (10)

with

f (120578 120585) = 119891 (120578 120585) 120585 + 1205781003816100381610038161003816120585 + 1205781003816100381610038161003816

(11)



120583 (119905 120585) = 1 119904 (119905

1015840 120585) minus 120572120579 lt 119904

0 0 le 119905

1015840le 119905

0 otherwise(12)




120601 (x 119905) = 1 minusint119867x120583 (120578 120585) 119889119881x1015840

int119867x119889119881x1015840

(13)


120588u119899i = sum

119901isin119876



119876 = 119901 |10038171003817100381710038171003817x1015840 minus x10038171003817100381710038171003817 le 120575 (15)









(a) (b) (c)

(d) (e) (f)











Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891 (120578 120585) = 119888s (5)


s =1003816100381610038161003816120585 + 120578

1003816100381610038161003816 minus |120585|

|120585| (6)


119891 (120578 120585) = 119888 (s minus 120572120579) (7)


119888 =18119870

1205871205754 (8)


119896 =119864

3 (1 minus 2]) (9)


1205881205972u1205971199052

= int119867x

f (120578 120585) 119889119881120585+ b (x 119905) (10)

with

f (120578 120585) = 119891 (120578 120585) 120585 + 1205781003816100381610038161003816120585 + 1205781003816100381610038161003816

(11)



120583 (119905 120585) = 1 119904 (119905

1015840 120585) minus 120572120579 lt 119904

0 0 le 119905

1015840le 119905

0 otherwise(12)




120601 (x 119905) = 1 minusint119867x120583 (120578 120585) 119889119881x1015840

int119867x119889119881x1015840

(13)


120588u119899i = sum

119901isin119876



119876 = 119901 |10038171003817100381710038171003817x1015840 minus x10038171003817100381710038171003817 le 120575 (15)









(a) (b) (c)

(d) (e) (f)











Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

(d) (e) (f)











Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Damage and Failure Process of Concrete