Development of a Bi Material Representative Volume Element Using Damaged Homogenisation Approach

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Journal of App Published Onlin http://dx.doi.org Open Access

Dev

Ele

ABSTRAC

Most civil eng face-to-surfac extremely var perfect interac tween two ma the process of The novelty in ples prove the

Keywords: Tr

1. Introduc

Civil enginee that are system ious shapes an and infrastruc perform the m For example, pression is qu systematic ass [1]. Under lat to horizontal compared to t though constru computational ongoing resea damentally so analysis of st encompassing softening prop gradient enhan ing damaged i

Numerical per to illustr clearly. The re

*_{Corresponding a}

plied Mathemati ne November 20 g/10.4236/jamp.

velopme

ement U

CT

gineering struc e interaction p riable; simplifi ction, on the o aterials of diffe f developing a n this paper is e stability of th

ransient Gradi

ction

ering structure

matically and r nd forms for ae cture. The ass

modules unde although an uite brittle, m sembly of bric teral loads wal sliding are sh those that fail ucted of same lly efficient m arch with the ound, yet sim tructures, a re g two materials

perties has bee nced non-loca interface intera examples are ate the proce esults show the

author.

ics and Physics, 013 (http://www .2013.16009

ent of a

Using D

Ali Science

ctures are form playing key rol

ied and extrem other hand, pre erent softening bi-material rep s the developm he approach for

ient Non-Loca

es are formed repetitively as esthetically ple sembled struct er similar load individual bri masonry arches cks are relative ll type structur hown to have

due to diagon materials [2]. models are yet objective of d mplified mode epresentative v s (Brick and M en developed u al homogenisa action of the tw

provided thro ess of formul

e stability of th

, 2013, 1, 43-47 w.scirp.org/journ

Bi-Mat

amaged

Jelvehpour* and Engineering of Technolog Email: *_a.jel Received

med using a nu le on the overa mely detailed m

edict unsafe be g properties is d

presentative vo ment of non-lo

r a simplified R

l; Homogenisa

using module sembled in va easing building tures often ou ding condition ick under com s encompassin ely more ducti

res that fail du higher ductilit nal cracking a No reliable an available. In a developing fun els for reliab volume elemen Mortar) of strai using a transien tion incorpora wo materials. oughout the p

ating the RV he approach.

7 nal/jamp)

terial R

d Homo

*

, Manicka D g Faculty, Quee gy, Brisbane, Au [email protected] d September 20

umber of mate all response of models trialed ehavior. In this developed usin olume element

cal transient d RVE and enco

ation; Damage es ar- gs ut- ns. m- ng ile ue ty al- nd an n- le nt in nt, at- a- VE

2. Con

Qua

2.1. El In the isotropi stress-s where and Ci

Einstein 3). The damage A da enable [4]: where current ter whi materia damage nearly Kuhn-T

Represe

ogenisa

Dhanasekar ensland Universi ustralia edu.au 13

erials that are b f the structure.

to date prove s paper a dam ng homogenisa t (RVE) using damage identif ourage applicat e Mechanics

ntinuum Da

asi-Brittle M

lasticity Base classical cont ic damage qua strain relations

ij



ij

 is the Ca

ijkl represents

n’s summation e Poisson’s rat

e.

amage loading the model to



f

eq

~ , is the equ

state of strain ich represents al has experien

e growth pres elastic. Chan Tucker relation f

ntative

tion Ap

ity

bonded to eac Unfortunately quite complex age mechanics ation approach damaged hom fication algorit tion to other sh

amage Mec

Material

ed Damage M

tinuum damag antity ω has b hip in the follo



1 



C_ijkl  

auchy stress,  the elasticity n convention a tio is assumed

g function is in predict damag



~_eq,



~_eq  uivalent strain n tensor and  maximum equ nced. For, f sent and mate nge in param ns [4]:

, 0 ,

0 

  f

f

Volum

pproach

ch other with t y these interac x. Models that s based interac h. This paper d mogenisation ap

thm. Numerica hapes of RVEs

hanics for

Mechanics

ge mechanics been introduce owing way [3]

kl



kl

 is the line y matrix com applies (i, j, k, d not to be aff

ntroduced, in ge growth, as





which depend

, is a history uivalent strain

0

 , there wi erial would be meter  , foll

0   JAMP

me

h

their sur- tions are t assume ction be- describes pproach. al exam- s. a scalar ed in the

: (1)

ear strain mponents. l = 1, 2, fected by

order to follows

(2)

ds on the parame- n that the ill be no ehave li- ows the

(2)

which implies that damage growth will happen when

0 

f , 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 ~ , which maps the _eq 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 compressive 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

1 2

2 2

1 2 1 2

1

1 12

2 1 2 1

eq k

I k

k k

I J

k 



 

 

 

 

 



(4)

where I₁, is the first invariant of strain tensor and J₂, 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 damage 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 developed 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





 



_e i



i _ _  

 

_₁_ ₁_ _   ₍₅₎

Due to crack bridging the experimentally obtained load displacement data has a long tail. Using this expres- sion when , stress approaches



1



Ei which

can represent this long tail. Parameter  controls the damage growth rate which depends on the tensile fracture energy of the material. When  is higher，model will show a faster crack growth and a more brittle response. This type of damage evolution lawwas used for both constituents.

3. Non-Local Model for Strain Softening

Material

Damage growth is highly dependent on the microstructure of the material. In quasi-brittle material such as concrete, 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 localisation [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

eq eq

eq c  

 _ _2 _~ ₍₆₎

This means that the damage field variable should depend on a non-local equivalent strain eq, instead of

local equivalent strain ~ . Gradient parameter eq c, is a

constant related to the squared of the internal length parameter. eq, can now be implicitly calculated in terms

of ~ , using a eq C0-continues finite element domain. To

solve this Helmholtz partial deferential equation a natural boundary condition has been considered as proposed in [9].

0 . 

_eqn (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 considering a transient value instead of a constant for the gradient parameter [10]. This modification transforms Equa- tion (6) as follows

eq eq

eq   

 _ _2 _~ ₍₈₎

in which , represents the transient gradient parameter and is defined as follows

     

  

       

  

 

  

 



c c

n

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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 c c c

n

)

( ₀

0 ₍₉₎

in which c₀, 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, homogenisation 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

) ( . )

(x x w x

u      (11) where , is the macroscopic strain tensor, x, is the position vector and w(x), 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









_



 

    

  

RVE RVE

d w x d

u

RVE RVE

  

. 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 periodic 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

, ,

3 2

3 1

2 1

eq F eq F

eq E eq B

eq D eq A

F C

E B

D A

u u u u

 

  

  

   

[image:3.595.316.538.494.705.2]

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This means riodic in the periodic stres is applied on points as illus and L, are wi tively. Moreo allel to bed j shear stress, re

5. Numeric

After implem sections, a R length, 60 mm 110 mm. The 5 mm mortar critisation wh lustrated in F

have been co experimental obtained in [1 Mesoscopic can be found chanics has be Quadratic i ered for the d functions hav Parameters fo model are sho method is em been obtained Two loadin compression p pendicular to have been app Evolution of dicular and pa and 5, respect

to bed joint, d brick-mortar i In the second allel to bed jo spot and conti

Figure

s both displace RVE. These p ses and strain

the RVE by m strated in Figu

dth, height an over, xx, y

oint, stress pe espectively.

cal Example

menting the pr RVE has been m height and RVE consists around all its hich was used

Figure 3. The

onsidered, in o results on c 4].

c material pro in Table 1.

een considered interpolation f displacement f ve been consid or the transient

own in Table

mployed since t d under load co ng cases have parallel to bed

bed joint. T plied on the RV

damage unde arallel to bed j tively. In case damage growth

interface and c case which is oint, damage g inues to grow i

2. Loading mod

ement and dam periodicity con s mesoscopic means of the th

ure 2. In these

nd length of th

yy, and, xy,

erpendicular to

es

rocess explain n constructed

an out-of-pla s of a 230*50 c

sides. The fin for the analy ese geometric order to simu onventional m

operties used f Elasticity base d for both mate functions hav fields and line dered for the t gradient enha

2. Convention

the experimen ontrol tests.

been consider d joint and co These loading

VE as illustrat er compressio oint can be se of compressio h initiates on th continues into m s under compre growth initiates into the perpen

des applied on

mage will be p nditions lead t

fields. Loadin hree controllin e figures W, H he RVE, respe

, are stress pa o bed joint an

ned in previou with 240 mm ne thickness o clay brick and nite element di sis has been i c configuration ulate the biaxi

masonry pane

for the analys ed damage me erials.

ve been consid ear interpolatio non-local fiel anced non-loc nal load contro ntal results hav

red for analysi ompression pe configuration ted in Figure

on load perpen een in Figures

on perpendicul he corner of th mortar bed join ession load pa s from the sam nd joints.

the RVE.

e- to ng ng H, c- ar- nd

us m of d a is- il- ns ial els

sis e-

d- on d. cal ol ve

is; er- ns

2.

n-

4

ar he nt. ar- me

Figure

Tab

Mater Bric Mort

Ta

Mater Bric Mort

Figure perpend

Dam have a perimen sient gr

6. Con

In this als with gated u ous dam dient en the mo

3. Typical finit

le 1. Material p

rial E ck 14000

tar 3000

able 2. Paramet

rial c ck 2

tar 2

4. Evolution of dicular to bed j

mage evolution good agreeme ntal investigat radient model h

nclusion

contribution, h different so using a homog mage growth a nhanced non-l del. Influence

te element mesh

parameters for

 

0.2 1.0 0.2 1.0

ters for transien

0

c

0.01 0.01

f damage in the joint.

patterns obtai ent with failur tions. Mesh s has been inves

the interaction ftening proper genisation appr and mesh sens local model h e of the choice

h used in comp

RVE computa

 

800 10 100 10

nt gradient mo

n 1.0 1.0

e RVE for com

ined from the re patterns see ensitivity of t stigated in [10,

n between two rties has been roach. To avo itivity, a trans has been introd

e of this mode

utations.

ations.

i



0.0002 0.0003

odel.

e 0.002 0.003

mpression

analysis en in ex- the tran- ,11].

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A. JELVEHPOUR, M. DHANASEKAR

[image:5.595.65.282.86.331.2]

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 transient gradient model indicates that calibration of the transient parameter under different loading conditions needs to be investigated.

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