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J ournal o f A pp Published Onli n http://dx.doi.or g Open Access De v El e ABSTRA C Most civil en g face-to-surfac extremely va r p erfect intera c tween two m a the process o f The novelty i n p les prove th e Keywords: T r 1. Introdu c Civil engine e that are syste m ious shapes a n and infrastru c p erform the m For example, p ression is q u systematic as s [1]. Under la t to horizontal compared to t though constr u computationa l ongoing rese a damentally s o analysis of s t encompassin g softening pro p gradient enha n ing damaged i N umerical p er to illustr clearly. The r e * Corresponding a p lied Mathemat i n e November 2 0 g /10.4236/jamp . v elopm e e ment U C T g ineering stru c e interaction p r iable; simplif i c tion, on the o a terials of diff e f developing a n this paper i s e stability of t h r ansient Grad i c tion e ring structure m atically and r n d forms for a e c ture. The as s m odules und e although an u ite brittle, m s embly of bri c t eral loads wa l sliding are s h t hose that fail u cted of same l ly efficient m a rch with the o und, yet si m t ructures, a r e g two material s p erties has be e n ced non-loc a i nterface inter a examples are ate the proc e e sults show th e a utho r . i cs and Ph y sics , 0 13 (http://ww w .2013.16009 e nt of a U sing D Ali Science c tures are for m p laying key ro l i ed and extre m o ther hand, pr e e rent softenin g bi-material re p s the develop m h e approach fo r i ent Non-Loca e s are formed r epetitively as e sthetically pl e s embled struc t e r similar loa d individual br i m asonry arche s c ks are relativ e l l type structu r h own to have due to diago n materials [2]. m odels are yet objective of d m plified mod e e presentative v s (Brick and M e n developed u a l homogenisa a ction of the t w provided thr o e ss of formul e stability of t h , 2013, 1, 43-4 7 w .scirp.org/jour n Bi-Ma t amage d Jelvehpour * and Engineerin g of Technolo g Email: * a.je l Receive d m ed using a n u l e on the over a m ely detailed m e dict unsafe b e g properties is d p resentative v o m ent of non-lo r a simplified R l ; Homogenis a using modul e sembled in v a e asing buildin g t ures often o u d ing conditio n i ck under co m s encompassi n e ly more duct i r es that fail d u higher ductili t n al cracking a No reliable a n available. In a d eveloping fu n e ls for reliab v olume eleme n M or t ar) of stra i u sing a transie n tion incorpor a w o materials. o ughout the p ating the R V h e approach. 7 n al / jamp ) t erial R d Hom o * , Manicka D g Faculty, Que e g y, Brisbane, A u l vehpour@qut. e d September 20 u mber of mat e a ll response o f m odels trialed e havior. In thi s developed usi n o lume elemen t cal transient d R VE and enc o a tion; Damag e e s ar - g s ut - n s. m - n g i le u e t y a l- n d a n n - le n t in n t, at - a- V E 2. Co n Qu a 2.1. E l In the isotrop i stress- s where and i C Einstei n 3). Th e damag e A d a enable [4]: where current ter wh i materi a damag e nearly Kuhn- T R eprese o genisa D hanasekar e nsland Univers i u stralia e du.au 13 e rials that are b f the structure. to date prove s paper a dam n g homogenis a t (RVE) using d amage identi f o urage applica t e Mechanics n tinuum D a a si-Brittle M l asticity Bas e classical con t i c damage qu a s train relations ij ij is the C a i jkl represents n ’s summatio n e Poisson’s ra t e . a mage loadin g the model to f eq ~ , is the eq u state of strai n i ch represents a l has experie n e growth pre s elastic. Cha n T ucker relatio n f ntative tion A p i ty b onded to ea c Unfortunatel y quite comple x age mechanic s a tion approac h damaged ho m f ication algori t t ion to other s h a ma g e Mec M aterial e d Damage M t inuum dama g a ntity ω has b hip in the foll o 1 ijkl C a uchy stress, the elasticit y n convention a t io is assume d g function is i n predict dama g eqeq ~ , ~ u ivalent strain n tensor and maximum eq u n ced. For, f s ent and mat e n ge in para m n s [4]: ,0,0 f f Volu m p proac h c h other with t y these interac x . Models tha t s based intera c h . This paper d m ogenisation a p t hm. Numeric a h apes of RVE s hanics for M echanics g e mechanics b een introduc e o wing way [3] kl kl is the lin e y matrix co m a pplies (i, j, k, d not to be af f n t r oduced, in g e growth, as which depen d , is a history u ivalent strai n 0, there w i e rial would b e m eter , foll 0 JAMP m e h t heir su r - tions are t assume c tion be- d escribes p proach. a l exam- s . a scalar e d in the : (1) e ar strain m ponents. l = 1, 2, f ected by order to follows (2) d s on the parame- n that the i ll be no e have li- ows the (3) A. JELVEHPOUR, M. DHANASEKAR Open Access JAMP 44 which implies that damage growth will happen when 0f, where history parameter satisfies eq ~ . 2.2. Definition of Equivalent Strain Shape and size of the loading surface depends on the definition of equivalent strain eq ~, which maps the strain tensor into a scalar value by weighting its compo- nents considering their different effects on cracking. In this paper, a strain based modified von Mises definition which is able to differentiate between tensile and com- pressive strains. This differentiation is needed in order to predict the behavior of quasi-brittle materials such as concrete and brick. This definition reads as follows [5]: 1 2 2 12 22 1 212 1 112 212 1 eq kI k kk IJ k (4) where 1 I, is the first invariant of strain tensor and 2 J, is the second invariant of deviatoric strain tensor. Pa- rameter k controls the sensitivity of the equivalent strain to tension and compression which is usually set to the ratio of compressive strength of material to its tensile strength. In this equation, Poisson’s ratio has been repre- sented by . 2.3. Damage Evolution Law for Quasi-Brittle Material Fracture in quasi-brittle material is not a result of growth of one dominant defect but a collective process of dam- age growth and nucleation in its microstructure. Quasi- brittle materials like concrete, brick and mortar, demon- strate a gradual loss of strength, instead of a sudden loss of deformation resistance like brittle fracture. Several scalar damage evolution laws have been de- veloped in the literature [6,7]. In engineering material softening is nonlinear which has a relatively steep stress drop when cracking starts and a moderate decrease af- terwards. In this work, an exponential softening law for concrete has been used in the form of i e i 11 (5) Due to crack bridging the experimentally obtained load displacement data has a long tail. Using this expres- sion when , stress approaches i E 1 which can represent this long tail. Parameter controls the damage growth rate which depends on the tensile frac- ture energy of the material. When is higher，model will show a faster crack growth and a more brittle re- sponse. This type of damage evolution law was used for both constituents. 3. Non-Local Model for Strain Softening Material Damage growth is highly dependent on the microstruc- ture of the material. In quasi-brittle material such as con- crete, brick and mortar, cracks are bridged by aggregates. Therefore, the fracture process is directly related to the aggregates size and distribution. However, in classical damage mechanics models the scale of microstructure has not been included. This unrealistic shortcoming of classical damage models results in the damage localisa- tion [8]. To overcome this localisation problem in simu- lation of strain softening material, we can introduce non-locality to the constitutive relation so that the growth of damage variable depends on the average deformation of the material in a certain region. Addition of this non- local concept to the damage model will result in a smooth damage growth depending on the length scale [9]. 3.1. Gradient Enhanced Non-Local Model Non-local strain can be introduced as the solution of the following partial differential equation eqeqeq c ~ 2 (6) This means that the damage field variable should de- pend on a non-local equivalent strain eq , instead of local equivalent strain eq ~ . Gradient parameter c, is a constant related to the squared of the internal length pa- rameter. eq , can now be implicitly calculated in terms of eq ~ , using a 0 C-continues finite element domain. To solve this Helmholtz partial deferential equation a natural boundary condition has been considered as proposed in [9]. 0. n eq (7) in which n, is the unit normal to boundary . 3.2. The Transient-Gradient Damage Model Using a constant parameter c, in the gradient model leads to an increase of damage growth in and outside the localisation zone. This issue can be resolved by consid- ering a transient value instead of a constant for the gra- dient parameter [10]. This modification transforms Equa- tion (6) as follows eqeqeq ~ 2 (8) in which , represents the transient gradient parameter and is defined as follows c c n (9) A. JELVEHPOUR, M. DHANASEKAR Open Access JAMP 45 In order to solve this new partial differential equation, an extra set of continuity equation needs to be added to the original gradient enhanced model. To avoid adding this extra continuity equation, Equation (9) can be di- vided by 0 , which leads to the diffusion equation [11] eq eq eq ~ 2 (10) which requires the same number of continuity equations as the original gradient enhanced model. The transient gradient parameter needs to be slightly changed to avoid division by zero into [11] c ccc n )( 00 (9) in which 0 c, is considered to be an arbitrary positive value so that non-local interaction is prevented at the beginning of the analysis. 4. Computational Homogenisation To derive an enhanced constitutive material model for a complex composite like masonry computational homog- enisation can be used so that we can derive the global behaviour of the masonry from its constituents such as concrete block and mortar. Uniform loading and periodic geometry for masonry has been assumed and thus, ho- mogenisation theory for periodic media which was adopted in [12,13] seems suitable to use. Computations have been performed on a single representative volume element (RVE) which contains the information of the entire mesostructure. A boundary value problem has been solved on the RVE using finite element method. Based on homogenisation theory for periodic media, strains should be compatible and the stresses should be anti-periodic on two opposite sides of the RVE. This will ensure that two neighbouring RVEs fit together. 4.1. Strain-Periodic Displacement Field The strain-periodic displacement field has the form )(.)( xwxxu (11) where , is the macroscopic strain tensor, x , is the position vector and )(xw , is a mesoscopic displacement fluctuation field which distinguishes the real meso- structural displacement field from the linear x . , field [13]. The fluctuation field is assumed to be periodic. The volume average of the mesoscopic strain field resulting from equation (11) is given by RVERVE dwxdu RVERVE . 1 )( 1 (12) which shows the volume average of mesoscopic strain field is equal to macroscopic strain . By using the Hill-Mandel work equivalence the total macro-stress can be determined as RVE d m RVE 1 (13) in which , and m , represent macro and meso stress, respectively. 4.2. Mesoscopic Representative Volume Element (RVE) In order to minimize the computational cost at the mesoscopic scale and also capture all possible failure mechanisms, the RVE should be chosen carefully. It is important to note that if the average behaviour remains unique any periodic RVE predicts equivalent results, i.e. in an infinitesimal strain setting and before localisation happens. For masonry, due to periodicity of the initial mesostructure, the RVE is chosen as the smallest peri- odic element. Based on the assumption that arrangement of the constituent materials is the main cause of average stiffness degradation, the initial and damage induced anisotropy will be captured correctly using this RVE [13]. 4.3. RVE’s Boundary Conditions Periodic boundary conditions have been applied on three controlling nodes (see Figure 1) of the RVE as indicated in [13] and justified in [12]. The periodicity conditions for edges can then be for- mulated in terms of the controlling nodes as ,, ,, ,, 32 31 21 eq Feq F eq Eeq B eq Deq A FC EB DA uuuu uuuu uuuu (14) Figure 1. Controlling nodes and periodicity conditions on a typical masonry RVE. Open Access 46 This mean s riodic in the p eriodic stres is applied on p oints as illu s and L, are wi tively. More o allel to bed j shear stress, r e 5. Numeri c After imple m sections, a R length, 60 m m 110 mm. The 5 mm mortar critisation w h lustrated in F have been c o experimental obtained in [1 Mesoscopi c can be found chanics has b e Quadratic i ered for the d functions ha v Parameters f o model are sh o method is e m b een obtaine d Two loadi n compression p p endicular to have been ap p Evolution of dicular and p a and 5 , respec t to bed joint, d brick-mortar i In the second allel to bed j o spot and cont i Figure s both displac e RVE. These p ses and strain the RVE by m s trated in Fi gu dth, height a n o ver, xx , y oint, stress p e e spectively. c al Exampl e m enting the p r R VE has bee n m height and RVE consist s around all its h ich was used F i g ure 3 . Th e o nsidered, in o results on c 4]. c material pr o in Table 1 . e en considere d i nterpolation f d isplacement f v e been consi d o r the transien t o wn in Table m ployed since t d under load c o n g cases have p arallel to be d bed joint. T p lied on the R V damage und e a rallel to bed j t ively. In case d amage growt h i nterface and c case which i s o int, damage g i nues to grow i 2. Loading mo d e ment and da m p eriodicity co n s mesoscopic m eans of the t h u re 2 . In thes e n d length of t h y y , and, xy , e rpendicular t o e s r ocess explai n n constructed an ou t -of- p l a s of a 230*50 c sides. The fi n for the anal y e se geometri c o rder to sim u onventional m o perties used f Elasticity bas e d for both mat e functions ha v f ields and lin e d ered for the t gradient enh a 2 . Conventio n t he experime n o ntrol t ests. been conside r d joint and c o T hese loading V E as illustra t e r compressi o oint can be s e of compressi o h initiates on t h c ontinues into m s under compr e g rowth initiate s i nto the perpe n d es applied on A. JELVEHPO U m age will be p n ditions lead t fields. Loadi n h ree controlli n e figures W, H h e RVE, respe , are stress p a o bed joint a n n ed in previo u with 240 m m ne thickness o c lay brick an d n ite element d i sis has been i c configuratio n u late the biax i m asonry pan e f or the analy s e d damage m e e rials. v e been consi d e ar interpolati o non-local fiel a nced non-loc n al load contr o n tal results ha v r ed for analys i o mpression p e configuratio n t ed in Fi g ure o n load perpe n e en in Fi g ures o n perpendicul h e corner of t h m ortar bed joi n e ssion load p a s from the sa m n d joints. the RVE. U R, M. DHAN A e- t o n g n g H , c- ar - n d u s m o f d a i s- i l- n s i al e ls s is e- d - o n d. c al ol v e i s; er - n s 2 . n - 4 ar h e n t. ar - m e Figure Tab Mate r Bri c Mor t T a Mate r Bri c Mor t Figure perpen d Da m have a p erime n sient g r 6. Co n In this als wit h gated u ous da m dient e n the mo A SEKAR 3. Typical fini t le 1. Material p r ial E ck 14000 t a r 3000 a ble 2. Parame t r ial c ck 2 t a r 2 4. Evolution o f d icular to bed j m age evolution good agreem e n tal investiga t r adient model h n clusion contribution, h different so u sing a homo g m age growth a n hanced non- l del. Influenc e t e element mes h p arameters for 0.2 1.0 0.2 1.0 t ers for tran sie n 0 c 0.01 0.01 f damage in th e j oint. patterns obta i e nt with failu r t ions. Mesh s h as been inve s the interactio n ftening prope r g enisation app r a nd mesh sens l ocal model h e of the choic e h used in comp RVE comput a 800 10 100 10 n t gradient m o n 1.0 1.0 e RVE for co m i ned from the r e patterns se e ensitivity of t s tigated in [10 , n between tw o r ties has bee n r oach. To avo itivity, a trans h as been intro d e of this mod e JAMP utations. a tions. i 0.0002 0.0003 o del. e 0.002 0.003 m pression analysis e n in ex- t he tran- , 11]. o materi- n investi- id spuri- ient gra- d uced to e l on the A. JELVEHPOUR, M. DHANASEKAR Open Access JAMP 47 Figure 5. Evolution of damage in the RVE for compression parallel to bed joint. evolution of damage under two loading cases has been investigated. The overall RVE behaviour using the tran- sient gradient model indicates that calibration of the tran- sient parameter under different loading conditions needs to be investigated. REFERENCES [1] A. Cavicchi and L. 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