International Journal of Solids and Structures 50 (2013) 3361–3374

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International Journal of Solids and Structures

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Fourier-based incremental homogenisation of coupled unilateraldamage–plasticity model for masonry structures

0020-7683/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijsolstr.2013.06.001

⇑ Corresponding author. Tel.: +852 2358 7162.E-mail address: cejkuang@ust.hk (J.S. Kuang).

Y.P. Yuen, J.S. Kuang ⇑Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 January 2013Received in revised form 30 March 2013Available online 18 June 2013

Keywords:HomogenisationRVEConstitutive model of coupled unilateraldamage–plasticity materialInelastic masonry compositesContinuum mechanics

In structural analysis of large masonry structures, nondemanding computation effort, numerical stabilityand simplified model assembly and meshing often have a higher priority over precise details of localstress or strain responses. This paper presents the development of a Fourier-based incremental homog-enisation technique, where the macro–micro transformations of mechanical variables are derived byincremental variational problems to minimise the potential energy in representative volume elements(RVEs) with respect to local fluctuating displacement fields expanded in Fourier series. In addition tothe proposed homogenisation technique, a unilateral damage–plasticity constitutive model for mortarjoints in the RVE is developed within the framework of thermomechanics, which accounts for the stiff-ness and strength degradation (or recovery) due to the transverse crack opening/closing in the mortarjoints. The numerical solution for the homogenisation problem and the performances of the proposedcoupled-damage plastic mode and Fourier-based homogenisation scheme verified by detailed case stud-ies are presented. It has been shown that the computational effort of the analysis with the proposed mod-elling technique can be considerably reduced by more than 20% as compared with that of the discretemodelling technique.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Masonry is the common construction material that has beenwidely employed for structural use, to withstand design gravityor lateral loads, or for non-structural purposes, such as partition in-fill panels. For the sake of easing construction and design work,masonry units, bricks or blocks, bonded with mortar are usuallyplaced in a periodic manner. The periodic arrangement of masonryunits allows one to extract a repetitive unit cell or RepresentativeVolume Element (RVE) from the composite masonry walls and for-mulates a homogenisation problem to determine the equivalentand macroscopic mechanical behaviour of the homogenisedcontinuum.

The continuum modelling approach has advantages over thediscrete modelling approach in saving computation effort, numer-ical stability and simplified model assembly and meshing. Differ-ent homogenisation techniques have been successfully applied toderive the effective elastic moduli and failure criteria of homoge-nised masonry materials, such as those proposed by Anthoine(1995), Luciano and Sacco (1997), Zucchini and Lourenço (2002)and Kawa et al. (2008). Despite the great success in elastic and lim-it state analysis, the homogenised continuum approach has

encountered difficulties in reproducing the full nonlinear mechan-ical response of masonry structures, particularly under non-pro-portional and reverse cyclic loading (Berto et al., 2002; Calderiniand Lagomarsino, 2008).

The primary goal of homogenisation is to link up the micro-scopic fluctuating stress or strain fields with macroscopic averagedfields, which work well for composites in the elastic or certain limitstates. However, it is not straightforward for inelastic compositesdue to the fact that equivalent macroscopic variables, such as mac-roscopic yield locus, cannot generally be obtained without evaluat-ing the whole fields of microscopic intrinsic variables in the RVE asclearly demonstrated by Suquet (1985). In other words, infiniteparameters, in which evolution can be loading-path dependent,would enter into the macroscopic inelastic constitutive relations.Therefore, given that it is virtually impractical and is of no availto derive strictly equivalent inelastic macro–micro relationsthrough rigorous homogenisation procedures, approximations orpostulations on the possible kinematic fields or microscopic distri-bution of inelastic strain and other intrinsic variables are requiredto reduce the number of intrinsic variables down to an adequateand numerically manageable level.

This paper presents a Fourier-based incremental homogenisa-tion strategy, where the macro–micro transformations of mechan-ical quantities are derived by incremental variational approachesto minimise the potential energy in the RVE with respect to local

3362 Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374

fluctuating displacement fields, which are expanded in a Fourierseries. In addition, a unilateral damage–plasticity constitutivemodel is proposed, which is strictly derived based on thermome-chanical principles and represents the fracture and frictionalbehaviour of mortar joints in the RVE. The model also accountsfor the stiffness and strength degradation (or recovery) due tothe transverse crack opening/closing in the mortar joints. Detailedcase studies for verification following the formulation of the pro-posed homogenisation strategy are conducted. It has been shownthat the computational effort of the analysis with the proposedmodelling technique can be considerably reduced as comparedwith the discrete modelling technique.

Fig. 1. Representative Volume Element (RVE) for a running-bond masonry wall.

2. Formulation of microscopic and macroscopic mechanicsfields

A homogenisation process is a kind of multi-scale analysis. It isnatural to introduce at least two different scales: macroscopic andmicroscopic scales, in which the fields of mechanical variables forthe global homogeneous structure and the local heterogeneous ele-ments or RVE are defined accordingly. In the framework of asymp-totic analysis two coordinate systems, which are referred to asslow-coordinates and fast-coordinates and commonly denoted bythe symbols x and y, respectively, are designated for the macro-and micro-scales (Manevitch et al., 2002).

2.1. Representative volume element (RVE)

The initial step in homogenisation procedures is to prescribe anRVE, which defines the domain of integration in the averagingoperations of different mechanical quantities. For a composite witha periodic structure, it is intuitive to look for a nonreducible repet-itive unit, i.e. a basic cell, to set up the geometry and material dis-tribution of the RVE. However, the basic cell may not always be thebest choice as an RVE, given that the wavelet of the local fluctua-tion fields, incurred by the presence of inclusions in which the ef-fects are to be averaged out in the homogenisation process, canbecome comparable in size with the basic cell.

It is recognised that in most situations, unlike for elastic mate-rials, it is not possible to obtain closed-form and path-independentrelations of macro- and micro-field quantities for inelastic materi-als, for instance, the transformations through localisation or con-centration tensors. Therefore, the macroscopic inelasticconstitutive relations derived by prescribing certain boundary con-ditions on the RVE boundaries, such as uniform strain, may becomeinadequate by using too small RVEs. On the other hand, the size ofthe RVE should be small enough, such that it can be satisfactorilyapproximated as a macroscopic point x on the scale of the globalstructure under consideration, for instance a masonry wall withdimensions of the order of several meters. Considering the afore-mentioned restriction, the adopted RVE for running-bond masonrywalls in this study is shown in Fig. 1. This RVE has also beenadopted by Luciano and Sacco (1997) and Calderini and Lagomar-sino (2008).

2.2. Problem formulation for local fields

As outlined by Souza and Allen (2011), even if the global struc-ture is subjected to dynamic loading, the local initial boundary va-lue problem for the RVE can be approximated to a quasi-staticproblem, given that the wavelengths of major propagating wavesare much larger than the size of RVE. This approximation is gener-ally true as long as the RVE is much smaller than the global struc-ture, which is also the case in this study and thus the formulationof the problem developed here assumes a quasi-static situation.

2.2.1. Governing equationsBy setting the origin of the micro-coordinate y 2 X at the geo-

metric centre of the RVE, where the RVE domain X ¼ Xb [Xm isa collection of sub-domains of brick units Xb and mortar jointsXm, the governing equations for the local fields in the RVE can bewritten as

@rijðyÞ@xj

þ biðyÞ ¼ 0 y 2 X ð1Þ

eijðyÞ ¼12

@ui

@yjþ @uj

@yi

!y 2 X ð2Þ

rijðyÞ ¼ q@wðeDn; TÞ

@eijy 2 X ð3Þ

rij _epij � q _wSðnÞ þ /Dð _DÞ � qkT ;k

TP 0 y 2 X ð4Þ

rijðyÞnj ¼ �rijð�yÞnj y 2 X ð5Þ

uiðyÞ � heijiyj ¼ uið�yÞ þ heijiyj y 2 @X ð6Þ

where rij, bi, eij, epij, ui, q and ni are, respectively, the Cauchy stress

tensor, the body force vector, the infinitesimal strain tensor, theinfinitesimal plastic strain tensor, the displacement vector, the massdensity and the unit outward normal vector of the RVE surfaces @X.h�i denotes an averaging operator over the RVE. Further, w, _wS, /D, D,n, qk, T are, respectively, specific Helmholtz free energy, specificstored energy rate due to plastic deformation, dissipation due todamage, damage variables controlling stiffness and yield strengthdegradation, intrinsic variables controlling plastic deformation, heatflux and temperature. Eqs. (1)–(3) are the classical equilibrium con-ditions, the infinitesimal strain-displacement relationship and theconstitutive equations under isothermal conditions. Eq. (4) is re-ferred to as the Clausius–Duhem inequality (an expression of thesecond law of thermodynamics). Eqs. (5) and (6) are the boundariesconditions: the anti-periodic condition of the traction vector andthe periodic condition of the fluctuating part of the displacementvector for periodic structures (Anthoine, 1995).

2.2.2. Constitutive relations for masonry materialsIn engineering practice of designing masonry walls, the strength

of mortar materials in the walls is often much weaker than thebrick units, and cracks will therefore likely propagate along theweak mortar joints and develop a stepping-down crack pattern.This characteristic of masonry walls leads to a common assump-tion of elastic or even rigid brick units used in many numericalor theoretical analyses (Rots, 1997). Following the common prac-tice, the brick units are assumed to behave elastically and mortar

Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374 3363

materials are modelled with the coupled damage–plasticity model,and the constitutive models are taken in the following rate forms,

_rijðyÞ ¼CB

ijkl_ekleCM

ijklðDÞð _ekl � _epklÞ ¼ LM

ijklðD; n;rÞ _ekl

(y 2 Xb

y 2 Xmð7Þ

where CBijkl and eCM

ijkl are (effective) elastic stiffness tensors of thebrick and mortar materials, respectively; LM

ijkl is an incremental ortangential stiffness tensor for mortar and it depends on damage var-iable D and the intrinsic variables n. The evolution laws and initia-tion thresholds of the damage and plastic deformation are definedin the following sections.

2.2.2.1. Damage model for mortar joints. Damage in the mortar ismainly incurred by crack formation. The effect of damage on themechanical behaviour can be modelled by introducing the conceptof effective stress ~rij and enforcing the strain equivalence principle.

One immediate effect of damage is the apparent degradation ofthe elastic stiffness. Considering the fact that the brick units wouldconstrain the crack propagation directions in the mortar joints, thecrack plane would eventually be aligned with the mortar thicknessdirection (direction-1 in Fig. 2) regardless of crack initiation. There-fore, the crack formation induces transverse anisotropy on the ini-tially isotropic stiffness tensors of the mortar. On the other hand, itis conceivable that the degraded stiffness due to the cracks formedunder tension can fully or partially be recovered when the loadingis reversed to compression. This phenomenon is called the unilat-eral or the quasi-unilateral effect (Lemaitre and Desmorat, 2005).

Therefore, it is assumed that the elastic part of the specificGibbs free energy w�E can be given by

qw�E ¼fr11g2

2ð1� D1ÞEM þf�r11g2

2ð1� hND1ÞEM þr2

22 þ r233

2ð1� D2ÞEM

� mM

EM ðr11r22 þ r22r33 þ r22r33Þ þ1þ mM� �

r223

1� D3ð ÞEM

þ1þ mM� �

r212 þ r2

13

� �1� D4ð ÞEM H r11ð Þ

þ1þ mM� �

r212 þ r2

13

� �ð1� hT D4ÞEM Hð�r11Þ ð8Þ

The symmetric property of the stress tensor is imposed in theabove expression of the Gibbs free energy, i.e. rij ¼ rji, but theoff-diagonal components should be differentiated from the in

Fig. 2. Material coordinates for mortar joints.

counterparts when a stress-dependent operator, such as one givenby Eq. (10), acts on the function. In the above equation, EM , mM arethe initial Young’s modulus and Poisson’s ratio of the mortars,respectively; D1, D2, D3 and D4 are damage variables reflectingthe degradation effect of the transverse cracks on the axial andshear stiffness along the polar or transverse directions; hN and hT

are crack closure parameters; Hð�Þ denotes Heaviside step functionwhere HðaÞ ¼ 1 if a P 0 and HðaÞ ¼ 0 if a < 0; f�g denotes Macau-lay bracket: fag ¼ a if a P 0 and fag ¼ 0 if a < 0.

The strain energy density release rates or thermodynamicforces due to damage and the elastic strain tensor are derived fromthe elastic part of the specific Gibbs free energy as

Yi ¼ q@w�E@Di

ð9Þ

eeij ¼ q

@w�E@rij

ð10Þ

Thus giving

Y1

Y2

Y3

Y4

0BBB@1CCCA ¼

1EM

fr11g2

2ð1�D1Þ2þ hN

f�r11g2

2ð1�hN D1Þ2

h i1

EM

r222þr

233

2ð1�D2Þ2

h i1

EM

ð1þmMÞr223

ð1�D3Þ2

h ið1þmMÞ r2

12þr213ð Þ

EMHðr11Þð1�D4Þ2

þ hT Hð�r11Þð1�hT D4Þ2

h i

0BBBBBBBB@

1CCCCCCCCAð11Þ

and

ee11

ee22

ee33

2ee23

2ee31

2ee12

0BBBBBBBB@

1CCCCCCCCA¼ eSðDÞh i

�

r11

r22

r33

r23

r31

r12

0BBBBBBBB@

1CCCCCCCCAð12Þ

where eSðDÞh iis the 2nd order effective compliance tensor derived

from Eq. (10) and given by

eSðDÞh iþ¼

1ð1�D1ÞEM � mM

EM � mM

EM 0 0 0

� mM

EM1

ð1�D2ÞEM � mM

EM 0 0 0

� mM

EM � mM

EM1

ð1�D2ÞEM 0 0 0

0 0 02 1þmMð Þð1�D3ÞEM 0 0

0 0 0 02 1þmMð Þð1�D4ÞEM 0

0 0 0 0 02 1þmMð Þð1�D4ÞEM

0BBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCAif r11 P 0

ð13aÞ

eSðDÞh i�¼

1ð1�hN D1ÞEM � mM

EM � mM

EM 0 0 0

� mM

EM1

ð1�D2ÞEM � mM

EM 0 0 0

� mM

EM � mM

EM1

ð1�D2ÞEM 0 0 0

0 0 02 1þmMð Þð1�D3ÞEM 0 0

0 0 0 02 1þmMð Þð1�hT D4ÞEM 0

0 0 0 0 02 1þmMð Þð1�hT D4ÞEM

0BBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCAif r11 < 0

ð13bÞ

3364 Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374

Based on the strain-equivalent principle, multiplying both sidesof Eq. (12) with the virgin stiffness tensor and time derivative ofthe equation gives

_er ¼ CM � eSðDÞ � _rþ CM � eS0ðDÞ � _D � r ð14Þ

where er and r are six-component stress vectors defined in Eq. (12).It is noticed that the increment of the effective stress consists of twoparts: the true stress increment and the damage increment.

The damage initiation and evolution are derived from the dissi-pative potential due to mixed damage f DðyÞ, which is defined by

f D ¼ Yeq � jðDÞ ¼ 0 ð15Þ

From the Ziegler’s principle of maximum dissipation, the fol-lowing normality rule is implied,

_Di ¼ _kD@f D

@Yið16Þ

where _kD is the damage multiplier, which is determined by the con-sistency condition of the damage dissipative potential, i.e._f DðY;DÞ ¼ 0, jðDÞ is a hardening or softening parameter governingthe evolution of the damage and potential, and Yeq is the effectivestrain energy density release rate. jðDÞ and Yeq are assumed inthe following forms,

jðDÞ ¼ ða � Dþ Y0Þc ð17Þ

Yeq ¼ b � Y ð18Þ

Thus, the consistency condition

_f DðY;DÞ ¼ @f D

@Yi

_Yi þ@f D

@Dk

_kD@f D

@Yk¼ 0 ð19Þ

and the damage multiplier

_kD ¼ �bi

cða � Dþ Y0Þc�1a � b_Yi ð20Þ

where the rates of the damage driving forces are

_Ykðr;DÞ ¼@Yk

@rij_rij þ

@Yk

@Dm

_Dm ð21Þ

By substituting Eqs. (20) and (21) to Eq. (16) and using Eq. (19),the rate of damage Eq. (16) can be expressed as

_Di ¼�b � @Y

@rmn_rmn

cða � Dþ Y0Þc�1a � bK�1

ik bk ¼ XD : _r ð22Þ

where b ¼ fb1;b2; b3;b4g, a ¼ fa1;a2;a3;a4g, and c and Y0 are dam-age parameters and

Kik½ ��1 ¼ dik þb � @Y

@Di

� �bk

cða � Dþ Y0Þc�1a � b

24 35�1

ð23Þ

The Kuhn–Tucker conditions for damage evolution are

_kD P 0; f D6 0; _f D ¼ 0; _kDf D ¼ 0 ð24Þ

For the special case of c ¼ 2, the damage evolution law Eq. (17)takes the same form of the evolution law adopted by Ladevèze andLe Dantec (1992). Further Y0 can be regarded as the threshold fordamage initiation.

2.2.2.2. Plasticity model for mortar joints. There are two frameworksto model the effect of the plasticity–damage coupling on the con-stitutive behaviour: the strong coupling and weak coupling mod-els. For the strong coupling model, a single dissipative potentialfunction is adopted to characterise all dissipation mechanismsincluding plastic deformation and materials damage. On the other

hand, within the framework of weak plasticity–damage dissipationcoupling models, one or more dissipative potential functions oryield surfaces, corresponding to plastic deformation, which isexplicitly independent of the state variables associated with dam-age, are introduced. More effort is virtually required to determinethe parameters of weak coupling models, but it allows more com-plicated behaviour to be modelled and thus they are widelyadopted (Bielski et al., 2005).

Some classical plasticity models can be employed in the weak-coupling model formulations but the yield surfaces are usually ex-pressed in effective stress space. The plasticity model in this studyis based on the Hsieh–Ting–Chen model (Hsie et al., 1982), whereyield surface is essentially a combination of von Mises, Drucker–Prager and Rankine surfaces with curved meridians, and it allowsnoncircular cross sections on the deviatoric plane. Both crackingstrain development and pressure-dependent frictional behaviourof mortar can be captured by this model. On the other hand, basedon the experiments by van der Pluijm (1993), dilation of mortarjoints is very small under shear action and the dilation angle issmaller than the friction angle, thus the non-associated flow ruleis adopted and the Drucker–Prager surface is employed as the plas-tic flow potential function. The yield surface f P and the flow poten-tial function gP are expressed in effective stress space as

f P eI1;eJ2; eh; �ep� �

¼ eI1 þ c1eJ2 þ

ffiffiffiffiffieJ2

qc2 cos eh þ c3

� �� 3n �epð Þ

¼ 0 ð25Þ

gP eI1;eJ2

� �¼ leI1 þ

ffiffiffiffiffieJ2

qð26Þ

where

eh ¼ 13

cos�1 3ffiffiffi3p

2

eJ3eJ3=22

!; eI1 ¼ trðerÞ; eJ2 ¼

12es : es; and

eJ3 ¼13

detðesÞ ð27Þ

are, respectively, the Lode angle with respect to the effective stresstensor, the first invariant of the effective stress tensor, the secondand third invariants of the effective deviatoric stress tensor; nð�epÞis a hardening/softening parameter, defined by

n �epð Þ ¼ n0 exp �g�epð Þ ð28Þ

where n0 is the initial hydrostatic cohesion strength, g is a materialparameter controlling the rate of evolution and �ep is the equivalentplastic strain, defined by

�ep ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi23ep

ijepij

rð29Þ

The plastic flow rate is determined by the normality conditionwith respect to the flow potential,

_epij ¼ _kP

@gP eI1;eJ2

� �@ erij

ð30Þ

The evolution of n �epð Þ is thus given by

_n ¼ � _kPgn0 exp g�epð Þ2ep

ij

3�ep

@gP eI1;eJ2

� �@ erij

ð31Þ

where epij is integrated over the loading history. As a weak-coupling

model, the plastic multiplier _kP , a Lagrange multiplier is introducedindependently of the damage potential and is determined by theconsistency condition of the yield surface,

_f P er; n� �¼ @f P

@ erij

_erij þ _kP@f P

@n@n

@epmn

@gP

@ ermn¼ 0 ð32Þ

Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374 3365

which gives

_kP ¼ �@f P

@er ij

@f P

@n@n@ep

mn

@gP

@ermn

_erij ð33Þ

The partial derivatives of the yield surface and flow potentialwith respect to effective stress and intrinsic variables are thenobtained,

@f P

@ erij¼ dij þ c1esij þ c2 cos eh esij

2ffiffiffiffiffieJ2

q � sin eh NffiffiffiffiffieJ2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� N2

p esij

2eJ2

�evij

3eJ3

!264375

þ c3

esij

2ffiffiffiffiffieJ2

q ¼ 0 ð34Þ

@gP

@ erij¼ ldij þ

esij

2ffiffiffiffiffieJ2

q ð35Þ

@f P

@n¼ �3 ð36Þ

@n

@epij

¼ �n0g exp �g�epð Þ2ep

ij

3�epð37Þ

where N ¼ 3ffiffi3p

2eJ3eJ3=22

� �. Using Eqs. (14) and (30) with the Voigt nota-

tion, the plastic strain rate can be expressed in terms of the truestress rate,

_epi ¼

1Kp

@gP

@ eri

@f P

@ erjCM

jkeSkm _rm þ CM

jn@eSno

@DqXD

ql_rlro

!¼ XDP

ij_rj ð38Þ

where

XDPij ¼

1Kp

@gP

@ eri

@f P

@ ermCM

mkeSkj þ CM

jn@eSno

@DqXD

qjro

!ð39Þ

Kp ¼ �@f P

@n@n

@epmn

@gP

@ ermnð40Þ

Same as for the damage evolution, the Kuhn–Tucker conditionis also defined for plastic deformation,

_kP P 0; f P6 0; _f P ¼ 0; _kPf P ¼ 0 ð41Þ

2.2.3. Summary of model formulation for local fieldsBoth local kinetics and kinematics of the masonry materials,

bricks and mortar due to the elasto-plastic deformation and dam-age have been rigorously formulated within the framework of thethermodynamic laws and the weak damage–plastic coupling mod-els has been presented in previous sections.

Brick materials are assumed to behave elastically, given that thestrength of mortar materials in the walls is often much weakerthan brick units, therefore the damage and inelastic deformationare likely localised in the mortar joints while the bricks suffer noor very little damage. The developed damage model accounts forthe stiffness and strength degradation of mortar due to the trans-verse cracks formation that renders the damaged mortar trans-versely with isotropic mechanical properties, with the polardirection aligned with the thickness direction. The plasticity repre-sentation is based on the Hsieh–Ting–Chen model (Hsie et al.,1982) with the non-associated flow rule to model the pressure-dependent frictional behaviour of mortar and cracking straindevelopment.

Using the plastic strain rate Eq. (38) and the rate form ofthe constitutive laws Eq. (7) for the mortar materials, thetrue stress rate in terms of total strain rate can be resolvedas

_ri ¼ dim þ eCMmnX

DPni

� ��1eC Mmj

_ej ¼ LMij ðD; n;rÞ _ej ð42Þ

where the tangential stiffness tensor Lmij ðD; n;rÞ depends on the

current stress and damage states and the plastic deformationhistory. The model formulation for the local fields is summarisedin Table 1.

The total number of the model parameters is 22, out of which 4are elastic constants and 6 are for the plasticity model and 12 arefor the damage model. The four elastic constants, the first threeyield surface parameters c1, c2 and c3 and dilatation parameterg in the plasticity model can be determined by following a stan-dard procedure proposed by Chen (1982) with monotonic loadingtest data, including uniaxial and biaxial compression and tensiontests and direct shear tests of masonry unit specimens under var-ious levels of compression.

As the damage and plastic evolutions are coupled, the last yieldsurface parameter n0 controlling the evolution of the yield surfaceand the parameters of the proposed damage model are calibratedtogether. The plastic evolution parameter n0, and mode 1 and mode3 damage parameters a1, b1, a3 and b3, which are the major dam-age mechanisms of typical masonry units, can be determined fromthe mode I and II fracture tests respectively, while the damagethreshold parameter Y0 is related to the strain energy release rateat the crack imitation.

To precisely calibrate the mode 2 and mode 4 damage parame-ters, information about the transverse strain or Poisson’s effect ofthe damaged specimens is required but, unfortunately, such datais generally not available, so in this study the damage parametersof relevant deformation modes are calibrated together throughextensive scatterplots analysis. Furthermore, the calibration of uni-lateral parameters hN and hT requires reverse cyclic loading tests,which again generally not available, hence the recommended val-ues for quasi-brittle materials by Lemaitre and Desmorat (2005)are adopted in this study.

2.3. Macroscopic averaging process

The macroscopic state variables at a macroscopic point x are de-rived from averaging the local fields, which depend on both macro-and micro-coordinates of x and y, respectively, with correspondingvariables in the RVE. In other words, the originally heterogeneousmasonry composite of bricks and mortar is now smeared to givea single homogenised masonry material.

However, for most of the cases, there is no one-off homogenisa-tion process to obtain closed-form macro–micro relations, in whichtransformation tensors are commonly known as localisation ten-sors, for evaluating the fluctuating displacement field of inelasticcomposites in contrast to elastic composites, due to the fact thatsome intrinsic variables, such as the fluctuating fields, exhibit load-ing-path dependent characteristics as mentioned in the previoussections. Therefore, homogenisation problems for inelastic com-posites should be resolved for each increment, and this approachhas been adopted by Miehe et al. (2002), Dascalu et al. (2008)and Souza and Allen (2011). In view of the complexity of the localcoupled damage–plasticity constitutive model for mortars andpossible loading patterns and paths that would be encounteredby masonry structures under seismic loading, the incrementalhomogenisation approach is found to be more suitable in thisstudy.

Table 1Model formulation for local fields in RVE.

3366 Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374

2.3.1. Link between micro- and macro-variablesDefine the averaging operator over the RVE as

h�i ¼ 1VRVE

ZVRVE

�dVy y 2 X ð43Þ

The macroscopic mechanical quantities, as averaged quantitiesof the corresponding macroscopic counterparts, can be expressedby

RðxÞ ¼ hrix 2Y

ð44Þ

EðxÞ ¼ heix 2Y

ð45Þ

�qFðxÞ ¼ hqwix 2Y

ð46Þ

DðxÞ ¼ h/ix 2Y

ð47Þ

whereQ

is the domain of the global structure; R, E, �q, F and D are,respectively, the macroscopic stress tensor, the macroscopic straintensor, the averaged mass density, the macroscopic specific Helm-holtz free energy and the macroscopic dissipation as the average

of the microscopic dissipation / due to plasticity /P and damage/D. The displacement vector and strain tensor can be decomposedin two parts: average and fluctuating, given by

uðyÞ ¼ E � y þ u ð48Þ

eðyÞ ¼ Eþ eðuÞ ð49Þ

The average of the fluctuating part of the strain tensor is zeroi.e. hei ¼ 0. Following the usual practice and ignoring the effect ofthe body force, i.e. r � rðyÞ ¼ 0 in the homogenisation problem,Hill’s lemma, which is one of the most important results in homog-enisation theory, states that

R : E ¼ hr : ei ð50Þ

Hill’s lemma holds the true for any admissible stress and strainfield that satisfies the field equations and the boundary conditions.

2.3.2. Evaluation of fluctuating displacement fieldFor the standard displacement-based FEM, the increment of the

macroscopic strain tensor E at each integration point of the globalstructure is computed by solving the global boundary value prob-

Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374 3367

lem in each increment or iteration. Based on the given macroscopicstrain tensor increment, the goal is to resolve the fluctuation dis-placement field, which is influenced by the mechanical properties,

eðyÞ ¼

e1;1

e2;2

e3;3

2e2;3

2e3;1

2e1;2

0BBBBBBBB@

1CCCCCCCCA¼

Xm;n

Amnmk cos mky1 þ h1m

� �sin nly2 þ h2

n

� �Xm;n

Bmnnl sin mky1 þx1m

� �cos nly2 þx2

n

� �000X

m;n

Amnnl sin mky1 þ h1m

� �cos nly2 þ h2

n

� �þ Bmnmk cos mky1 þx1

m

� �sin nly2 þx2

n

� �� �

0BBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCAð54Þ

Fig. 3. Partition of RVE for numerical integration.

geometry and distribution of the composite materials in the RVE.After obtaining the fluctuating displacement field, it is straight-

forward to evaluate other microscopic mechanics quantities andtheir averages, which are the corresponding macroscopic quanti-ties to be updated at the integration point. The fluctuating dis-placement field u can be evaluated by minimising the totalpotential energy with the following variational problem,

dW ¼Z

VRVE

r : dedV �Z@VRVE

n � r � dudS�Z

VRVE

b � dudV ¼ 0 ð51Þ

The variation of the displacement and strain fields are equal totheir fluctuating fields as the variations of the macroscopic straintensor vanish in the local RVE scale i.e. du ¼ du and de ¼ de. It isnoted that the second and the last integrals vanish, since the sur-face traction and fluctuating displacement are anti-periodic andperiodic respectively [see Eqs. (5) and (6)] and the body force is ig-nored. Therefore, Eq. (64) is reduced toZ

VRVE

r : dedV ¼ 0 ð52Þ

The variational problem can be solved numerically by the stan-dard FEM method through discretising the RVE domain into a finiteset of elements. However, it requires the enforcement of the peri-odic boundary condition for the fluctuating displacement by, forexample, the Lagrange multiplier or penalty method (Anthoine,1995; Luciano and Sacco, 1997; Miehe et al., 2002). Here a more di-rect approach is adopted by expanding the fluctuating displace-ment by a Fourier series,

uðyÞ ¼u1

u2

u3

0B@1CA ¼

Xm;n

Amn sin mky1 þ h1m

� �sin nly2 þ h2

n

� �Xm;n

Bmn sin mky1 þx1m

� �sin nly2 þx2

n

� �0

0BBBB@1CCCCA ð53Þ

where Amn and Bmn are Fourier coefficients for the two in-plane fluc-tuating displacement fields u1 and u2, respectively. h1

m, h2n, x1

m, x2n

are phase angles, which can be determined by the boundary condi-tions or symmetry conditions (if any) on the RVE. On the otherhand, for most of masonry wall structures, the ratio of thicknessto width is much less than unity (on the scale of 0.03–0.1). Thus,it is reasonable to assume that the in-plane displacement fieldsare independent of the thickness coordinate y3 and the out-of-planenormal strain is uniform along the thickness direction, i.e. e33 ¼ E33,as long as the out-of-plane action is negligible. Nevertheless, theformulation can be easily extended to incorporate the possible

dependency of the thickness coordinate y3 following the sameconcept.

The fluctuating strain fields are

By setting the angular frequencies k and l as

k ¼ 2pa; l ¼ 2p

bð55Þ

where a and b are the width and height of the RVE, as shown inFig. 3, the periodic boundary condition for the fluctuating displace-ment fields Eq. (6) and the average of the fluctuating strain fieldbeing vanished, as strictly required by Eq. (45), are automaticallysatisfied. Therefore, both of the necessary boundary conditionsand mean field requirements are naturally admitted by the Fourierseries expansion of the fluctuating displacement field. Furthermore,the orthogonal planes of symmetry on the RVE allow determinationof the phase angles as h1

m ¼ 0, h2n ¼ 90�, x1

m ¼ 90� and x2n ¼ 0. The

remaining unknowns in the Fourier series, Eq. (54), are determinedby solving the variation problem Eq. (52).

When the fluctuating displacement field is obtained, the macro-scopic stress tensor R and tangential moduli L can be calculated by

R ¼ 1VRVE

@Wmin

@Eð56Þ

L ¼ 1VRVE

@2Wmin

@E� @Eð57Þ

where Wmin ¼ infu

W is the minimum potential energy with respectto the fluctuating displacement field.

Table 2Averaging process formulation.

3368 Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374

2.3.3. Summary of the average processThe general principle in homogenisation theory and the linkage

between microscopic and macroscopic mechanics field quantitiesare systematically outlined in the previous sections. The fluctuat-ing displacement field in the RVE is expanded by a Fourier series,and the associated boundary conditions and vanishing require-ment of its average are naturally satisfied by specifying the angularfrequencies. The formulation and governing equations for the aver-age process are given in Table 2.

3. Numerical solution of the homogenisation problem

The governing equations for the microscopic mechanical fieldsin the RVE and their linkage to the corresponding macroscopicmechanical fields are thoroughly defined and discussed in the pre-vious sections. The last step is to establish a numerical approxima-tion scheme to resolve the incremental homogenisation problem.Increments of different field quantities in the RVE at time t þ Dtare approximated in the following linearised forms,

rðt þ DtÞ ¼ rðtÞ þ Dr ð58Þ

eðt þ DtÞ ¼ eðtÞ þ De ð59Þ

uðt þ DtÞ ¼ uðtÞ þ Du ð60Þ

where rðtÞ, eðtÞ and uðtÞ are known quantities and the stress tensorincrement is calculated by the incremental constitutive relationsEqs. (7), (42)

DrðyÞ ¼ Cb : DeðyÞLmðD; n;rÞ : DeðyÞ

(y 2 Xb

y 2 Xmð61Þ

The total strain and displacement increments can be decom-posed into two parts

De ¼ DEþ De ð62Þ

DuðyÞ ¼ DE � y þ Du ð63Þ

where the macroscopic strain increment DE at a macroscopic pointx is determined in a global iteration step i, thus it is also a knownquantity for the local field problem in the RVE.

3.1. Incremental variational problem

Substituting the incremental forms into Eq. (52) givesZVRVE

ðrðtÞ þ DrÞ : dDedV ¼ 0 ð64Þ

Using Hill’s lemma and the fact thathDei ¼ heðt þ DtÞi � heðtÞi ¼ 0, it can be shown thatZ

VRVE

r : dDedV ¼ 0 ð65Þ

thusZVRVE

Dr : dDedV ¼ 0 ð66Þ

Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374 3369

The above incremental variational problem Eq. (66) has a simi-lar form to that adopted by Souza and Allen (2011) to evaluate thelocal displacement field. Using Eqs (42), (45), and (54) with Voigtnotation, Eq. (66) can be expressed byZ

VRVE

dDei � Lij � DEj þ Dej� �

dV ¼ 0 ð67Þ

The increments of the nonvanishing fluctuating strain compo-nents are approximated as

De1;1 ¼Xm;n

DAmnmkcos mky1 þ h1m

� �sin nly2 þ h2

n

� � ð68Þ

De2;2 ¼Xm;n

DBmnnl sin mky1 þx1m

� �cos nly2 þx2

n

� � ð69Þ

2De1;2 ¼Xm;n

DAmnnl sinðmky1 þ h1mÞ cosðnly2 þ h2

nÞþDBmnmk cos mky1 þx1

m

� �sin nly2 þx2

n

� �" #ð70Þ

The increments of the fluctuating strain can be expressed in thefollowing concise form

Dek ¼Xm;n

Deðm;nÞk ¼Xm;n

Cðm;nÞkl DQ ðm;nÞl ð71Þ

where fDQgT ¼ fDA DB g and ½C� is given by

½C� ¼

mkcos ey1þh1m

� �sin ey2þh2

n

� �0

0 nlsin ey1þx1m

� �cos ey2þx2

n

� �0 00 00 0

nlsin ey1þh1m

� �cos ey2þh2

n

� �mkcos ey1þx1

m

� �sin ey2þx2

n

� �

26666666664

37777777775ð72Þ

where ey1 ¼ mky1 and ey2 ¼ nly2 are angular coordinates.Noted that the variations of the parameters in the Fourier series

of the fluctuating displacement fdDQ ðm;nÞg are arbitrary; then bysubstituting Eq. (71) into Eq. (67) results in

dDQ ðm;nÞl

ZVRVE

Cðm;nÞil � Lij � DEj þ Cðm;nÞjo DQ ðm;nÞo

� �dV ¼ 0 ð73Þ

or

Fig. 4. Schematic diagram of increme

fDQg ¼ �½G��1fDSg ð74Þ

where

DSðm;nÞl ¼

ZVRVE

CT � L � DEdV ¼Z

VRVE

Cðm;nÞil LijDEjdV ð75Þ

Gðm;nÞlo ¼

ZVRVE

CT � L � CdV ¼Z

VRVE

Cðm;nÞil LijCðm;nÞjo dV ð76Þ

In the above equations, fDSg and ½G� can be regarded as the gen-eralised macroscopic strain increment and generalised globalisa-tion tensor, i.e. its inverse can be regarded as the generalisedlocalisation tensor. The parameters in the Fourier series fQg canbe iteratively solved by the standard Newton–Raphson (NR)Scheme with Eqs (61). Fig. 4 shows the schematic diagram of theincremental homogenisation procedures.

4. Verification of the model

The proposed coupled-damage plasticity model and Fourier-based homogenisation scheme were successfully implemented inABAQUS through the user-subroutine for constitutive models. Inte-grations of local constitutive relations are performed with themodified Euler scheme (Sloan, Abbo & Sheng, 2001) and Eqs. (75)and (76) are evaluated using the Euler–Maclaurin formula (Davisand Rabinowitz, 1984). The model performances are tested withseveral case studies presented in the following sections.

4.1. Performance of the coupled-damage plasticity model

To verify the coupled damage–plasticity model performance,numerical loading tests including monotonic and cyclic shear andtension tests are performed to simulate the following experimentalstudies: direct shear tests by Van der Pluijm (1993); cyclic sheartest by Atkinson et al. (1988); monotonic tension tests by Ver-mltfoort and Van der Pluijm (1991). The numerical models wereestablished to simulate the realistic boundary conditions, speci-men sizes and layouts of the experiments used in the verificationstudies, which can be found in the corresponding literature.

ntal homogenisation procedures.

3370 Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374

4.1.1. Monotonic shear tests with varying normal stressThe mortar joints exhibit pressure dependent frictional behav-

iour under shear, so that the peak and residual shear strengths varywith the normal compressive stress. To verify the shear behaviourof the model, simulations of the shear tests of the mortar bed joints(Joosten clay brick with mortar 1:2:9), which were conducted atTNO (Van der Pluijm, 1993), are performed. The model parametersare presented in Table 3. The simulated responses of nominal shearstress against shear displacement under different normal compres-sive stress levels are plotted together with the test results inFig. 5(a). It can be seen that the results of numerical analysis agreewell with the experimental data, where the simulated peak andresidual shear strengths vary linearly with the normal stress levels.

4.1.2. Cyclic shear testIn addition to monotonic shear behaviour, the cyclic shear

behaviour based on the coupled-damage plasticity model is stud-ied by simulating the cyclic direct shear tests (Atkinson et al.,1988). The masonry specimen is subjected to a constant nominalnormal stress of 36 kN/m2. The simulated responses are plotted to-gether with the test results in Fig. 5(b). The shear strength of thespecimen gradually deteriorates to the residual strength after thepeak response, followed by box-shaped hysteresis loops under

Table 3Model parameters for test simulations.

Parameters

Young’s modulus of brickPoisson’s ratio of brickYoung’s modulus of mortarPoisson’s ratio of mortarHTC yield surface parameter 1HTC yield surface parameter 2HTC yield surface parameter 3HTC yield surface parameter 4HTC yield surface hardening parameterDilatation parameter for DP flow surfaceDamage potential’s hardening parameter (mode 1)Damage potential’s hardening parameter (mode 2)Damage potential’s hardening parameter (mode 3)Damage potential’s hardening parameter (mode 4)Coefficient for effective damage forces (mode 1)Coefficient for effective damage forces (mode 2)Coefficient for effective damage forces (mode 3)Coefficient for effective damage forces (mode 4)Exponent for damage potentialDamage thresholdUnilateral parameter (normal)Unilateral parameter (tangential)

Fig. 5. Shear behaviour of mortar joints. (

cycles of reversing shear, which is again captured by the proposedmodel.

4.1.3. Monotonic tension testsThe tension behaviour of the model is examined by simulation

of monotonic tension tests (series T.CS) by Vermltfoort and Van derPluijm (1991). Young’s modulus and Poisson’s ratio of the mortarjoints and bricks are 13400 MPa and 0.17, and 2630 MPa and0.16, respectively. The tension cohesion strength is 2.5 MPa. Bycomparing with shear tests, a much more rapid drop in the post-peak strength of the mortar joints under tension is observed inFig. 6(a), due to the fact that the loading plane coincides with theprincipal damaged bed plane.

4.1.4. Cyclic tension simulationFurther to the study of monotonic tension behaviour, the cycle

tension behaviour is also examined. Given that there are veryscarce cyclic tension tests for mortar joints that can be found inthe literature, the cycle tension simulation is performed with thesame model of the monotonic tension simulation but subjectedto reversed cyclic loading. The simulated cyclic and monotonic ten-sion responses are both plotted in Fig. 6(b). It can be seen that themodel reveals the stiffness degradation as well as the recovery

Symbols (units) Values

Eb (N/mm2) 16700mb 0.16Em (N/mm2) 2500mm 0.2c1 0.13619c2(mm2/N) 2.9108c3 1.291n(N/mm2) 1.956g 300l 0.001a1 0.002a2 0.1a3 0.1a4 0.0062b1 1b2 0.01b3 0.02b4 0.6c 1.5Y0 (N/mm2) 0.0057hN 0.2hT 0.1

a) Monotonic loads; (b) cyclic loads.

Fig. 6. Uniaxial behaviour of mortar joints. (a) Monotonic loads; (b) cyclic loads.

Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374 3371

(unilateral effect) associated with tensile cracks openings and clo-sures in the bed joints.

It can be seen from Fig. 6(b) that inelastic residual strains aredeveloped in the mortar joints even when the specimen is com-pletely unloaded from tension. This development of tensile inelas-tic strains in the proposed material model is to capture the effect ofmicro-cracking, which constitute the macroscopic irrecoverableinelastic deformation, in the quasi-brittle materials formed undertension. Hence, in order to recover the original length of the spec-imens from tension loading, further compression is required inaddition to merely unload the specimens, of which phenomenonhas been reported for concrete and different quasi-brittle materials(Richard and Ragueneau, 2013; Bazant and Planas, 1998; Yankelev-sky and Reinhardt, 1989). Nevertheless, it is usually assumed thattensile cracks in mortar joints are completely closed at the com-plete unloading since the induced inelastic strain would be rela-tively small and negligible. Actually, the presented model canalso replicate such origin-oriented unloading behaviour of thedamaged mortar joints by adjusting the model parameters to makethe total energy loss of the damaged materials be solely contrib-uted by the damage dissipation.

4.2. Performance of the Fourier-based homogenisation scheme

4.2.1. Verification for numerical schemeThe fluctuating strain fields Eq. (54) in the RVE must vanish for

homogeneous materials. Although this condition is apparently

Fig. 7. Uniaxial behaviour of RVE with homogeneous elastic properties.

ensured by Eq. (75), in which the integration vanishes for homoge-neous materials given the periodicity of the fluctuating displace-ment, the numerical integration schemes may pose errors on theresults. To examine the soundness of the numerical analysisschemes, a verification case study is performed by comparing theanalysis results of the FBH model with the homogenous materialproperties of the reference model using ABAQUS’ internal material.The uniaxial force–displacement responses of the two models areplotted in Fig. 7. In can be seen that this case study clearly demon-strates the reliability and accuracy of the numerical schemes usedin the FBH model and the relative error is very small at the order of1�5.

4.2.2. Orthotropic mechanical propertiesThe orthotropic geometry of the RVE, as shown in Fig. 3, would

render the virgin homogenised masonry materials with orthotropicmechanical properties, though both the brick and mortar are re-garded as isotropic materials. To verify the orthotropic propertiesof the FBH model, the load-deformation behaviour of the RVE invarious directions is studied. The results are compared with thoseof the discrete analysis where the brick units and mortar joints inthe RVE are discretely modelled, as shown in Fig. 8(a).

Indeed, it is ideal to prescribe periodic boundary conditions onthe finite RVEs but uniform boundary condition is applied in thisstudy due to the fact that the volumetric ratio of the mortar joints,as the second phase material, is only about 13% of the gross volumeof the RVE, so the uniform boundary condition can already give avery good approximation to this particular elastic problem. Thejustification of the use of uniform boundary condition was outlinedby Suquet (1985) and it has been applied in many studies, i.e. Bou-chart et al. (2008), Wang et al. (2007), Hashin (1983). Hence, con-sidering the nature of this problem, the use of uniform boundarycondition is satisfactory and thus only negligible discrepancy inthe responses of the discrete and homogenised models are seenin Fig. 8a. It is also expected that the discrete model would revealstiffer responses due to the kinematic constraints on theboundaries.

The elastic uniaxial and shear responses of the two models invarious (in-plane) directions are plotted in Fig. 8(a) and the corre-sponding von Mises stress distributions are shown in Fig. 8(b). Theloading curves shown in Fig. 8a are the responses integratedthrough the whole volume or boundary, where the mean responsesare equivalent to the macroscopic stress–strain responses of thehomogenised RVE. Fig. 8(b) reveals the microscopic stress fieldswith local fluctuation in the discrete RVE under various modes ofdeformation and the averages of the stress fields are equivalent

Fig. 8. Load–deformation behaviour of RVE based on discrete and continuum (FBH) modelling approaches with heterogeneous elastic properties. (a) von Mises stressdistributions; (b) loading curves.

3372 Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374

to the mean stress in the homogenised RVE, which would be ob-served in the macroscopic scale.

As expected, the discrete model shows slightly stiffer responses,particularly for shear action applied on the vertical faces of theRVE, i.e. shear 12. As aforementioned, such stiffening effect is in-duced by the kinematic constraints on the boundaries and this ef-fect becomes more significant when the amplitudes of thefluctuating displacement fields become stronger on the con-strained boundaries. Obviously, the fluctuating fields on the verti-cal faces have higher strength than those on the horizontal face,which is promoted by a higher degree of heterogeneous materialdistribution on the vertical faces, and therefore higher stiffness isobtained in the discrete model subjected to uniformly imposedloading on the vertical faces. Furthermore, to maintain a ratherhigh quality elements mesh, which is necessary to attain sufficientaccuracy and robustness of the analyses and with an aspect ratioless than 1.5 for the discrete model, considerably large numbersof elements is required, as shown in Fig. 8(b). As a result, the com-putational time of the discrete models is almost 3 times as long asthat of the FBH models.

4.2.3. Nonlinear responses of masonry piersThe last simulation investigates the nonlinear global or macro-

scopic mechanical behaviour of a prototype masonry structuresubjected to in-plane lateral loading. The nonlinear performanceof the homogenisation model is verified against original heteroge-neous discrete model by comparing the macroscopic load-defor-mation behaviour, the stress distributions and the developedmacroscopic damage patterns. In this study, the simulations areperformed in a deterministic manner and hence real tests are not

adopted in this study to evade many uncertain factors in large-scale specimen tests, of which experimental data are often muchscatter than pure material tests and one should apply probabilityfunctions to describe random distributions of defects in the struc-tures. It is clear that the homogenisation model can no way per-fectly replicate the local behaviour but it should adequatelymatch the global behaviour observed in the original heterogeneousstructure. The masonry wall of size 1.5 1.6 0.09 m(width height thickness) is composed of 240 115 90 mmmasonry bricks and 10 mm thick mortar joints. The mechanicalproperties of the masonry materials are presented in Table 3.

Two finite element models, a discrete model and a continuummodel with FBH technique (m ¼ n ¼ 6), are constructed. Their non-linear load–deformation behaviour is studied by pushover analysisunder a vertical load of 0.1 N/mm2 acting on the top. The elementmeshes of the two models are shown in Fig. 9(a). The element sizeof the discrete model is much smaller than that of the continuummodel to prevent elongated elements.

The resulting damage patterns (plots of equivalent plastic straindistributions) and the force transfer mechanisms (plots of principalcompressive stress distributions) of the prototype structures areshown in Fig. 9(c) and (d), respectively. The tensile flexural cracksare initiated in the mortar bed joints at the top and bottom of thewalls, as shown in Fig. 9(c). Further loading on the walls leads tothe formation of the shear cracks, as a typical stepping-down pat-tern, which is often observed in real loading tests of masonry struc-tures (Rots, 1997), for the discrete model and a diagonal band forthe continuum model, initiated in the mortar joints at the centralparts of the walls. The cracks propagate towards the loading cornerand the opposite corner. Before formation of the shear cracks, the

Fig. 9. Nonlinear pushover behaviour of masonry piers. (a) Element meshes of prototype structures; (b) equivalent plastic strain distributions [drift ratio = 0.05% (above),=0.4% (below)]; (c) load–deformation curves; (d) principal compressive stress distributions [drift ratio = 0.05% (above), =0.4% (below)].

Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374 3373

lateral loading is mainly transferred through a diagonal compres-sive strut, as shown in Fig. 9(d), and then the shear cracks cutthe strut into halves, where the post-peak stress transfer through.

The load–deformation curves of the two models are plotted inFig. 9(b). It can be seen that the continuum model result satisfacto-rily agrees with the discrete model result, nevertheless, the contin-uum model shows stiffer response in the loading curve (solid line)than the discrete model (dashed line), particularly in the post-peakcurve. This is because the deformation of the elements in the con-tinuum is retrained by the Fourier series approximation of thekinematic modes. The accuracy of the continuum simulation canbe improved by increasing the order, i.e. m and n, of the Fourierseries. Similar to the previous simulation, computational effort issignificantly saved by the continuum approach that the analysistime is 804 s for the continuum model but it is 4703 s for the dis-crete model.

5. Concluding remarks

It is recognised that the high computational effort and compli-cated model assembly of the discrete modelling approach hinderfrom practical design utilisation for high-rise infilled frame struc-tures. Therefore a continuum modelling approach is developed

on the basis of a Fourier-based incremental homogenisation strat-egy. The continuum model aims to minimise the computationaltime and the model assembly effort, yet be able to retain sufficientaccuracy in simulating global structural responses of masonry orinfilled structures. In addition to proposing the homogenisationstrategy, a unilateral damage–plasticity constitutive model formortar joints in the representative volume element (RVE) is alsodeveloped.

The local constitutive model for mortar joints is strictly devel-oped within the framework of thermomechanics. The unilateraldamage model accounts for the stiffness and strength degradation(or recovery) due to the transverse crack opening/closing in mortarjoints that render the damaged mortars transversely isotropicmechanical properties. The plasticity model is based on theHsieh–Ting–Chen model (Hsie et al., 1982) with the non-associatedflow rule to model the pressure-dependent frictional behaviour ofmortar. The coupling effect of the damage and plasticity models isaccounted for through the strain-equivalence principle. Thehomogenisation problem for the RVE is solved as an incrementalvariational problem to minimise the total potential energy with re-spect to the fluctuating displacement field, which is expanded by aFourier series. After evaluating the fluctuating fields, the equiva-lent macroscopic mechanical quantities for the RVE are obtainedby averaging the corresponding microscopic quantities.

3374 Y.P. Yuen, J.S. Kuang / International Journal of Solids and Structures 50 (2013) 3361–3374

The performances of the proposed local constitutive model andthe continuum approach are verified by case studies in comparisonwith experimental results. As shown in the numerical studies, themodels satisfactorily simulate the mechanical behaviour of ma-sonry structures. Furthermore, the computation time of the contin-uum model is also greatly reduced to as much as 5 times less thanthat of the discrete approach for the same structural model. Thus,the proposed continuum approach provides an effective, yet accu-rate, means of analysis and design as well as further research onhigh-rise infilled frame structures, if the local mechanical re-sponses of the infill components are deemed to be insignificantor not of interest.

Acknowledgement

The support of the Hong Kong Research Grand Council undergrant number 614011 is gratefully acknowledged.

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