Concurrent multiscale analysis of heterogeneous materials

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European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.) Vienna, Austria, September 10-14, 2012

C O N C U R R E N T M U L T I S C A L E A N A L Y S I S O F H E T E R O G E N E O U S M A T E R I A L S

Oriol Lloberas-Valls^ F r a n k E v e r d i j ^ Daniel Rixen^, Angelo Simone' and Bert Sluys'

'Faculty of Civil Engineering and Geosciences Deift University of Teclinology

P.O. Box 5048, 2600 GA Delft, The Netherlands ^Faculty 3mE

Delft University of Technology

IVIekelweg 2, 2628 CD Delft, The Netlierlands

Keywords: multiscale, domain decomposition, quasi-brittle heterogeneous materials, strain

localization

Abstract. Concurrent multiscale analysis of quasi-brittle heterogeneous materials such as con-crete and rock is conducted employing non-overlapping domain decomposition techniques. Ini-tially a coarse discretization with effective properties is considered at each domain. A zoom-in technique is performed at those domains in which non-linearities take place. The coarse mesh is replaced with a fine discretization which includes the lower scale constituents, e.g. aggre-gates or reinforcement. The present framework captures the interaction between the mesoscopic constituents and the strain/stress fields. The resulting crack path is in agreement with the one obtained with direct numerical simulations. It is shown that the interscale relations enforced at the interface between coarse andfine domains play an important role in the global response of the material.

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Oriol Lloberas-Vails, Frank Everdij, Daniel Rixen, Angelo Simone and Bert Sluys

1 I N T R O D U C T I O N

Ivlultiscale analysis o f materials used i n engineering practice has recently become a leading research topic. Several phenomena such as cracking and failure are clearly multiscale and their simulation requires sophisticated computational tecliniques. This is the case for concrete material (Figure 1) where the mesoscopic constituents such as aggregates and reinforcement play a crucial role during fracture processes.

Macroscopic concrete sample Mesoscopic concrete structure

Figure 1: Macroscopic concrete sample and mesoscopic concrete structm-e (Roels et al. [9]).

The main objective o f a general muhiscale analysis is to capture the origin and evolution o f the targeted physical processes at a fine scale and their impact at a coarse scale o f observation. This can be achieved by accounting for a refined representation o f the material which eventually requires a different simulation strategy. Examples o f such techniques can be found i n [11] where fracture i n concrete is simulated w i t h the use o f a lattice-particle model whereas standard finite elements (FE) are employed f o r the linear elastic region. The study o f localization phenomena i n granular frictional materials presented i n [10] considers a particle model at the areas affected by large deformations and standard FEs at the rest o f the sample. Other multiscale strategies are based on local material refinement and scale resolution without varying the computational approach-FEs are employed at both macro and mesoscopic scales i n [13-15].

The approach adopted i n this manuscript can be classified i n this second group and processes both macro and mesoscopic scales i n a concurrent manner. The focus is on the multiscale analy-sis o f concrete-like samples and the influence o f interscale links between macro and mesoscopic material resolutions.

2 C O N C U R R E N T M U L T I S C A L E A N A L Y S I S

I n these techniques the effect o f micro- and mesoscopic constituents is taken into account by increasing the resolution o f the material scale at certain areas o f interest. I n this scenario, the whole fine scale region is considered i n the computations as opposed to hierarchical methods where the behaviour o f a reduced fine scale area is representative f o r the one at a larger scale. For this reason, concurrent techniques are recommended f o r multiscale analyses that require a moderate j u m p i n scales due to the l i m i t i n computational resources. However, the interaction between fine scale heterogeneities and global stress and strain fields is accounted for w i t h the same level o f detail as i n a direct numerical simulation (DNS).

Most o f the concurrent multiscale methods consider a strong coupling between micro/meso and macro scales since displacement compatibility and global equilibrium are enforced across the whole structure [12,13]. When large systems need to be resolved, the use o f domain decom-position techniques is preferred. The present contribution falls i n this group and shows some

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Tf

_Ff

' I

Interface Fi

Coarse scale domain ' with effective properties Fine scale domain

X l

- ^ C j —

Aggregates ^-U^ IVIatrix ITZ Lagrange multipliers • Independent nodes o Dependent nodes

Figure 2: Concurrent multiscale analysis of the heterogeneous solid Q.

similarities w i t h the methods described i n [14-16].

2.1 Multiscale domain decomposition modelling

Consider the equilibrium problem o f the heterogeneous solid Q w i t h boundary conditions defined i n Figure 2. The body Q is split into Ns non-overlapping domains q!^''> w i t h s e [l.Ng . The interface between domains is denoted by T i (Figure 2). Continuity o f the total displacement field u at the interface F i between non-overlapping domains .s- and p implies

u('') = u ( ^ ) o n r P n r j ' ' ) ( i )

and is satisfied by means o f linear multipoint constraints (LMPCs) as proposed i n the Finite Element Tearing and Intercomiecting (FETI) method [ 1 ] .

I n a general multiscale analysis, different resolutions, i.e. coarse, c, and fine, f, may co-exist PP P in a decomposed sample (Figure 2). I n this situation, the resulting interface T\ = Tf'^ U F j U T j , where the superscripts cc, f f and c f denote coarse-to-coarse, fine-to-fine, and coarse-to-fine mesh connection, respectively. The interfaces Fj'^'^ and Vf are conforming while the interface T f is non-conforming. I n this approach, the resulting boundary o f a fine scale mesh is obtained by subdividing the boundaiy discretization o f the initial matching mesh. I n this marmer, the non-conforming interface Vf contains a number o f matching and non-matching nodes. A l l matching nodes i n T\ are refeiTed to as independent while non-matching nodes are called dependent i n the remaining o f this text.

The continuity condition i n (1) can be expressed using a signed Boolean matrix B^'^) f o r all independent degrees o f Ji'eedom (DOFs) and a constraint matrix C^'*) for the dependent DOFs. The modified matrices B^'^) are defined by row-wise concatenation o f the Boolean matrices B^'^')

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and tlie constraint matrices C^'^^ as

; B(I) . . . B(^^') • B(I) . . . B(^^^-: ₍₂₎

A field o f Lagrange multipliers A is utilized to account f o r the LMPCs o f t h e independent DOFs while the Lagrange multipliers jU are related w i t h the LMPCs o f the dependent DOFs (Figure 2 bottom). Hence the extended field o f Lagrange multipliers

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A f t e r spatial discretization using standard finite element procedures, the discrete f o r m o f t h e equilibrium equations f o r each domain together w i t h the interface compatibility condition, read

KWuW+B(^)^A = f(^)

XBWU(^)=0,

where K^'^), and u^'') denote domain stiffness matrix, force vector and displacement vector, respectively.

As described i n [7] the main goal o f the presented framework is the multiscale analysis o f fracture processes i n quasi-brittle materials. I n this context, the fine scale domains are only utilized i n those regions where the dominant non-linear processes, i.e. damage growth and coalescence, take place. The rest o f the linear regions can be modeled w i t h a coarse scale resolution w h h effective elastic properties. During an adaptive multiscale analysis^the number o f fine domains is expected to increase and the structure o f the modified matrices B^'^) needs to be recomputed as soon as the spatial resolution o f a domain is altered. A coarse scale domain is updated to its corresponding fine scale resolution w i t h the use o f a zoom-in technique and relaxation stages as described i n detail i n [7, 8]. The moment at which a zoom-in takes place is dictated by a prediction o f an internal quantity (e.g., based on strain or stress tensors) as indicated i n [6, 7].

2.2 Micro-to-macro connection

A t a non-conforming interface Vf (top o f Figure 3) an identity constraint is enforced by the signed Boolean matrices B(*) for all independent (ind) DOFs such that

ufnd,/= <d,/> ' = 1,2, (5)

where uf^^ , and uf^^ ^ represent the displacement vectors o f two consecutive independent nodes along Fj^^. However, different interscale Wvks can be defined by constraining the dependent (dep) DOFs tlirough the constraint matrices C^'^). A master-slave relation is set between the displacements o f dependent u^^p . and independent uj^^ . nodes at the interface Fj^^. The most common choice is to enforce strong compatibility between coarse and fine scale fields using f u l l collocation (Figure 3) which can be expressed as

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Non-conforiuing interface

Coarse mesh (c) Fine mesh (f)

rf

Coarse side Fine side

^ind,2 ^ind.l 'ind,2 dep,/7/' ^dep,l ' i i i d . l

Strong compatibility Weak compatibility

Full collocation 'ind.l ••dep,/ col ind ,2 Normal collocation Average compatibility / ^ / u f ( x ) - u ^ ( x ) ) d r f = 0 Average compatibility with stillness weighting

Ej-. Young's modulus

Ei>E2

"dep./ - " = 0 / (/4(x)»f(x)-/c^»^.(x))dr[f = 0, J r,

j = LA^DOF

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Oriol Lloberas-Vails, Franlc Everdij, Daniel Rixen, Angelo Simone and Bert Sluys

n ƒ being the number o f dependent nodes between two consecutive independent nodes at Ff^^. Defining the deviation f r o m the collocation constraint as

I col

Udep,/

f u l l collocation is satisfied when

'-^ „ f . c o l _ f ~ "dep,/ "dep,/' i = \ , n f , "dep,/ col 0, i=\,nf. (7) (8) A modified version o f the collocation technique was recently introduced i n [8] where strong compatibility is only enforced along the normal direction n o f the linear segment V f . I n this view, normal collocation is satisfied when the normal component o f the deviation is nullified. The corresponding constraint equation can therefore be written as

f

"dep,/

col

• n = 0, i= l , n f . ₍₉₎

Physically, this is equivalent to state that the coarse and fine interfaces are conforming i n terms o f the resulting deformed geometiy.

The compatibility condition at the interface between coarse and fine domains can be ex-pressed i n an average sense as

| ^ ^ ^ / u f ( x ) - u ^ ( x ) ) d r f = 0. (10)

The corresponding C^'^) matrices are found using the shape functions o f the elements along the interface. The constraint i n (10) essentially applies a constant force field A ( x ) along the interface Fj'^^ delimited by two contiguous independent nodes i n order to minimize the gap between the non-matching meshes (Figure 3).

As shown i n [8], weighting functions based on nodal stiffness enforce a compatibility closer to the exact one for highly heterogeneous interfaces. Such a weak compatibility w i t h stiffness weighting reads

^^(/cf(x)i/f(x)-/c>/^^(x))dFff = 0, y = l,iVDOF, ₍₁₁₎

where A'DOF is the number o f DOFs per node and kj{x) represents a continuous function be-tween two independent nodes along T f . The values o f /cy(x) are interpolated using the fine mesh shape functions at the interface and the diagonal stiffness coefficients Kn o f the fine res-olution domain—the index / corresponds to node i (located at x) and DOF J. Conversely, kj is taken as a constant stiffiiess coefficient w h i c h corresponds to the average value kj{x) along the fine side o f the interface as

r f

d T f . (12)

The condition i n (12) guarantees consistency i n the sense that exact compatibility would also satisfy (11). Note that the force field tying two incompatible meshes increases at stiff interface segments (Figure 3).

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3 N U M E R I C A L E X A M P L E S

Several multiscale analyses o f a concrete-like material are summarized i n this section. They are designed to test the basic features o f the muhiscale approach introduced i n Section 2.

3.1 Tensile test

A concrete sample is subjected to tensile loading as depicted i n Figure 4. The body is decom-posed into three non-overlapping domains. As shown i n Figure 5, homogeneous coarse scale domains are represented by a single bilinear quadrilateral element whereas fine scale domains contain the mesoscale constituents and are meshed using linear triangular elements. Note that the mesoscopic structure is obtained by a two-dimensional translation o f a material unit cell w i t h periodic boundaries.

A gradient-enhanced damage model [2] is used i n this study to model crack growth and coalescence i n the material. A list o f material parameters f o r each phase is given i n Table 1. Initially, a set o f coarse scale domains is considered w i t h effective elastic properties consider-ing periodic boundary conditions for a representative volume element (RVE). A representative quantity o f the non-local equivalent strain ê„/ is computed f o r each domain and serves as an indicator f o r the appearance o f non-linearity (cf. [6]). The damage initiation thr-eshold Kq o f the homogeneous bulk is taken as the m i n i m u m damage initiation threshold o f the three concrete phases.

s "

I \ >

h

Figure 4: Description of the tensile test for the concrete specimen.

Material parameters Aggregates Matrix I T Z

E Young's modulus [GPa] 35.0 30.0 20.0

V Poisson's ratio [-] 0.2 0.2 0.2

£nl Non-local equivalent strain [-] Mazars Mazars Mazars

Ko Damage initiation threshold [-] dummy 0.124 X 10-4 0.1 X lO-'^

C Gradient parameter [mm^] 0.75 0.75 0.75

(OiK) Damage evolution law [-] Exponential Exponential Exponential

a Residual stress parameter [-] 0.999 0.999 0.999

Softening rate parameter [-] 500 500 500

Table 1: Material data for the concrete specimen.

Due to the initial constant strain and stress distribution i n the homogeneous bulk, a zoom-in is perforzoom-ined simultaneously for all domazoom-ins. Damage nucleates at the I T Z and propagates

60 mm

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Oriol Lloberas-Valls, Fraiil< Everdij, Daniel Rixen, Angelo Simone and Bert Sluys

Domain decomposition of the coarse discretization (1 Q4 element per domain)

Gauss point area with effective properties

Domain decomposition of the fine discretization (23112 T3 elements per domain)

X-KUJoC X-HXH- X-HJ-KC

P t y ^ :

prvr

p c p c

fmjr^-

Oo^^c? ö o c m

f^-im'C

ptRQ

p^c

Representative volume of the mesostructure (5778 T3 elements)

Figure 5: Domain decomposition ofthe coarse and fine FE discretizations.

tiirough the matrix giving rise to a series o f damage bands perpendicular to the loading direction (Figure 6). Due to the non-symmetry o f the mesoscale geometry and the applied boundaiy con-ditions strains localize at the leftmost section o f the sample w i t h the appearance o f a dominant damage band.

The applied load against the displacement registered at the right edge o f the sample is com-pared to the one obtained w i t h a D N S (Figure 7). The differences between both simulations are hardly visible except f r o m the stage i n which zoom-in is applied and a load variation is registered.

3.2 Influence of the interscale link in bending test

I n this example a bending test is performed on a concrete beam (Figure 8). The sample is decomposed into two non-overlapping domains w i t h the same geometry (Figure 9). The idea is to test different micro-to-macro connections that glue the different scale resolutions shown at the top o f Figure 10. I n this case the multiscale analysis is "static" and the resolutions do not change during the test. A linear elastic model is used f o r the homogeneous bulk and the three concrete phases considering the same moduli as i n the previous section (Table 1). Two different coarse resolutions (discretizations I and I I ) are considered. Effective elastic properties are retrieved f r o m the same RVE used i n Section 3.1. Note that for discretization I I the area o f

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Oriol Lloberas-Val Is, Frank Everdij, Daniel Rixen, Angelo Simone and Bert Sluys Stage C ^ ® © © p © ^ m@w&^ S . 0 m \ 0 din;'-) c d Stage D m® m ^®.igs©©;©e ^®:^©©@^ stage E stage F jOmoCr^G >ObOO^G O ^ O U ^ G ' © • ft)

Figure 6: Damage evolution during the adaptive multiscale analysis of the tensile test. 25 x displacement magnifi-cation.

the RVE is larger than the one o f the coarse element integration points. However, the total area in which homogenization takes places is formed by the number o f sub-areas w i t h homogeneous properties. I n this case the refinement is only employed to provide a higher flexibility to the coarse mesh and to avoid shear locking.

The bending stiffness o f the system is calculated as the relation between the vertical com-ponents o f the force and displacement at the right edge o f the sample. Table 2 contains the

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Oriol Lloberas-Valls, Fraiilt Everdij, Daniel Rixen, Angelo Simone and Bert Sluys a ' O m )/y = 0.1 mm 40 mm

Figure 8: Boundaiy conditions for the bending test.

Domain decomposition

Coarse discretization 1 (1 Q4 element per domain)

Coarse discretization 11 (256 Q4 elements per domain)

Tributaiy area with homogeneous effective properties from the RVE

Fine discretization

(23112 T3 elements per domain)

Figure 9: Domain decomposition and FE discretizations.

bending stiffness f o r each interscale constraint when using any o f the two coarse discretizations f o r domain Q^^\ I n general, the use o f discretization I I provides the closest values to the refer-ence stiffness 0.845 x 10^ N / m m obtained w i t h a fine discretization at both domains. The f u l l collocation constraint (8) turns out to perform better than the normal collocation constraint (9) when analyzing the deformation o f the sample. I n fact, the deformed configuration o f the inter-face corresponding to the reference solution is a linear segment due to the small deformations assumption. I n this view, the collocation constraints naturally provide an accurate deformation at the interface (Figure 10). Coincidently, normal collocation is able to provide a closer value to the reference one due to two counteracting effects: the too coarse domain (stiffening) and the loose compatibility (softening).

Coarse mesh Full col. Normal col. Av. comp. Stiff, weight

I 0.906 0.832 0.697 0.697

I I 0.846 0.845 0.846 0.847

Table 2: Global stiffness [xlO^N/mm] for the stadc multiscale test with different interscale constraints. The reference stiffness ofthe DNS is 0.845 x lO^N/mm.

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Oriol Lloberas-Valls, Frank Everdij, Daniel Rixen, Angelo Simone and Bert Sluys

Deformation of the multiscale analyses

Zoom area

Coarse discretization I Coarse discretization I I

Zoom of the deformed configurations

illiiil

SÊÊÊSI

Full collocation /SÉ. SlXa Normal collocation Average compatibility Stiffness weighting <

-^ — 1 i

Figure 10: Deformed configurations using different interscale constraints. 25 x displacement magnification.

Tlie average compatibility constraint (10) and average compatibility w i t h stiffness weight-ing (11) provide very similar results i n this example. Only when usweight-ing the coarse discretization I I the stiffness weighting constraint turns out to be slightly stiffer than the standard average compatibility.

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interfaces is discussed i n more detail i n [ 8 ] .

3.3 Wedge split test

A multiscale simulation o f the wedge split test, sketched i n Figure 11, is carried out. The concrete sample is decomposed into 34 non-overlapping domains and the notch F " is modeled by means o f the traction free interface F f . The tine scale discretization used f o r the different domains and the RVE sample are identical to the ones presented i n the previous examples. The coarse discretization I I employed i n Section 3.2 is adopted here f o r the coarse scale domains. A gradient-enhanced damage model is adopted again and the material parameters for each phase are listed in Table 1. The adaptive muhiscale strategy considered i n Section 3.1 w i t h a f u l l collocation interscale linlc is adopted i n this example.

o . E S o O

X

0.2 mm pn 40 mm 20 mm Interface Fj

Traction fi^ee interface F "

Figure 11: Boundary conditions (left) and domain decomposition (right) for the wedge split test.

The damage evolution after zoom-in at domains affected by non-linearity is shown i n Fig-ure 12. Despite the symmetry i n the boundary conditions and geometry o f the sample the final crack path is not aligned w i t h the notch. This is due to the heterogeneous mesostructure and its interaction w i t h the stress and strain fields at the fine resolution.

As observed i n Figure 13 the load-displacement curve is in agreement w i t h the one o f the DNS and differences are not visible after zooming at all domains involved i n the non-linear processes. The overall cost o f the multiscale analysis is clearly lower than the one o f the DNS since during the computation domains remain coarse and linear unless needed to be refined.

4 C O N C L U S I O N S

The concurrent framework presented i n this manuscript proves to be adequate for the m u l t i -scale analysis o f failure phenomena o f quasi-brittle materials such as concrete. The nucleation and evolution o f fracture processes is w e l l captured and the overall response turns out to be i n agreement w i t h the DNS. The computational cost o f the analysis is essentially linked to the extension o f the fracture process zone. I n this scenario, the domain decomposition plays an important role since larger domains w i l l trigger a larger refined area when zoom-in takes place.

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Stage A Stage B

Close-up at stage F

Figure 12: Damage evolution during the adaptive multiscale analysis of the wedge split test. lOOx displacement magnification.

Different interscale relations can be supported by the cuixent framework. I t is observed that collocation teclmiques generally provide a stiffer link than average compatibility constraints.

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0 i ' ' ' '

0 0.04 0.08 0.12 0.16 0.2 Displacement [mm]

Figure 13: Load-displacement curves for the wedge split test DNS and multiscale analysis. The inset shows the close-up of the load-displacement curve around a zoom-in event.

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