Elastic waves in heterogeneous materials as in multiscale-multifield continua

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1 Prc. Estnian Acad. Sci. Phys. Math., 007, 56,, Elastic waves in hetergeneus materials as in multiscale-multifield cntinua Patrizia Trvalusci and Giuseppe Rega Department f Structural Engineering and Getechnics, University f Rme La Sapienza, Via Gramsci 53, Rme, Italy; patrizia.trvalusci@unirma1.it, giuseppe.rega@unirma1.it Received 15 January 007 Abstract. A multifield cntinuum t describe grssly the dynamic behaviur f cmpsite materials (fibre reinfrced, plymers, masnry-like, etc.) is prpsed using a multiscale mdelling based n the hyptheses f the classical mlecular thery f elasticity. Referring t a nedimensinal sample, the pssibility f revealing the presence f internal hetergeneities is investigated. Key wrds: cmpsite materials, micrcracked materials, cntinua with micrstructure, multiscale mdels, wave prpagatin. 1. INTRODUCTION The mechanical behaviur f cmplex materials, characterized by the presence f hetergeneities f significant size and texture at finer scales, strngly depends n their micrstructural features. Lacking in material internal scale parameters, the classical cntinuus mdel seems nt always apprpriate t describe the macrscpic behaviur f such materials taking int accunt the size, rientatin, and dispsitin f the micr-hetergeneities [ 1 ]. Mrever, the basic hypthesis f lcal unifrmity f the classic stress/strain fields is inapprpriate in critical regins with high gradients, e.g. in the case f gemetrical r lading singularities. These difficulties, while leading t ill-psed prblems in the case f nnmntnic cnstitutive laws [,3 ], claim fr develping cntinuus mdels different frm the simple Cauchy ne (grade 1). In this wrk the effectiveness f nnstandard cntinuum mdelling fr these materials is investigated making recurse t the thery f multifield cntinua [ 4 ] and addressing the relevant wave prpagatin prperties. Attentin is fcused n 100

2 cmpsite media made f shrt, stiff, and strng fibres embedded in a mre defrmable matrix that, due t manufacturing defects r lack f chesin, presents distributed micrcracks. Starting frm the kinematics f a cmplex lattice mdel (micrmdel), the multifield cntinuum (macrmdel) is built up using a strategy based n an energy equivalence principle linking different material scales (multiscale mdelling) [ 5,6 ]. This cntinuum has additinal field descriptrs accunting fr the presence f the micrstructure which must satisfy additinal balance equatins, as in the s-called mdels with cnfiguratinal frces [ 7,8 ]. The capability f such a multiscale-multifield cntinuum t reveal the presence f material micrstructure is investigated by studying the wave prpagatin in a ne-dimensinal system describing a material with a micrcracked elastic matrix. In particular, it is shwn that the additinal micrstructural field descriptrs make the equatins f mtin dispersive, with phase-velcities changing with frequency. This is the peculiar feature f mst mdels prpsed t vercme the intrinsic drawbacks f the simple Cauchy cntinuum, such as ratedependent, nnlcal, strain gradient mdels [,3,9,10 ]. In all f these mdels, nnstandard strain measures implying spatial r time derivatives f an rder ther than the secnd rder in the equatins f mtin, are intrduced. In the multifield mdels additinal stress measures are als intrduced, crrespnding in the sense f the virtual wrk t the nnstandard strain measures invlved, and this circumstance makes the prblem thermdynamically cnsistent [ 11 ]. Due t the dispersin prperties, the multifield mdel shws t be able t describe changes in the shape f travelling waves generally assciated with scattering. This seems a necessary feature t study the strain lcalizatin phenmena [ 3 ] that ften characterize the mechanical behaviur f brittle cmpsite materials.. THE MULTISCALE-MULTIFIELD MODEL The cntinuus mdel f the generalized hmgeneus material (macrmdel) is built up based n the linearized kinematics f a prper lattice mdel (micrmdel) f the kind prpsed in [ 1,1 ]. Our reference material is a fibrereinfrced micrcracked cmpsite, which culd be a ductile plymer cmpsite with lng carbn fibres as well as a masnry-like material with stne/bricks embedded in a mrtar matrix, bth with a distributin f slit micrcavities in the matrix. At the micrscpic level such a material is described by tw interacting lattice systems: ne lattice made f interacting rigid particles f given shape, representing the fibres, and the ther lattice made f interacting slits f arbitrary shape with a predminant dimensin, representing the micrcracks. T identify the cnstitutive functins fr the cntinuum internal and external (inertial) actins, the prcedure prpsed in [ 5 ] is adpted. This prcedure allws us t link tw different scale mdels based n tw hyptheses: (i) macrscpic hmgeneus defrmatins are impsed t a representative vlume element (mdule) f the material peridic micrstructure; (ii) the vlume average f the stred energy functin f the mdule is equated, thrugh the lcalizatin 101

3 therem, t the strain energy density f the macrmdel. These hyptheses are standard in the classical mlecular thery f elasticity [ 13 ], as riginally delineated in the basic wrk f Vigt [ 14 ]. Briefly, the linearized strain measures f the mdule are as fllws: (a) the a b relative displacement between tw pints p and p, belnging t tw particles ab A and B, represented by the vectr u ; (b) the relative rtatin between A and ab B, represented by the skew-symmetric tensr W ; (c) the pening-displacement n a slit H ( K ), centred at the psitin hk ( ), represented by the vectr d h k h k ( d ); (d) the relative displacement d d ; (e) the relative displacement a h between tw pints p and p, f a particle A and a slit H, als accunting fr h ah d, represented by the vectr ω. The generalized frces assciated with the abve kinematic quantities are: (a) the frce and (b) the cuple between A and B, represented by the vectr t ab ab and the skew-symmetric tensr C, h k respectively; (c) the pening frce n H ( K ), represented by the vectr z ( z ); hk (d) the slit interactin frce, represented by the vectr z ; (e) the particle-slit ah interactin frce, represented by the vectr q. The mean strain energy f the mdule, f vlume V, reads 1 ab ab a a b 1 ab a b ε = t ( u W ( p p )) + C ( W W ) ab V h h hk h k ah ah a a h + z d + ( ) + ( ( )), h z d d hk q ω W p p ah (1) where, t a first apprximatin, all f the generalized frces are assumed linear ab ab ab ab elastic functins f the strain measures: t ( u ); C ( W ); h ( h z d ); hk ( h k ah ah a z d d ); q ( ω ), with W the rtatin f the reference particle. Based n hypthesis (i), all f the kinematic descriptrs in Eq. (1) are expressed in terms f regular fields defined n the current cnfiguratin f the cntinuum (multifield), namely: the standard displacement vectr field, u ; the micrrtatin tensr field (skew-symmetric), W ; and the micrdisplacement vectr field, d. The nnstandard micrscpic fields accunt fr the rtatins f the individual fibres and fr the distributed displacement jump due t the presence f micrcracks in the matrix. Then (hypthesis (ii)), the strain energy density fr the cntinuum can be derived as 1 1 ε = S ( u W) + S W + z d + Z d. () The ensuing quantities { S, S, z, Z} have the meaning f generalized stress fields: S is a nnsymmetric stress tensr; S is the cuple-stress tensr; z is the internal vlume frce related t the presence f the micrcracks, playing the rle f the frce respnsible fr the internal changes f the system cnfiguratin [ 7,8 ], and Z is the micrstress tensr. The stress strain relatins read 10

4 S = A ( u W) + B W + C d + D d, S = E ( u W) + F W + G d + H d, z = I ( u W) + L W + M d + N d, Z = O ( u W) + P W + Q d + R d, (3) where A R are elastic tensrs f different rder, with cmpnents depending n the size, shape, arrangement, and rientatin f the internal phases, besides the elastic cnstants f the matrix. The material hyperelasticity entails symmetry relatins between the cmpnents f the pairs f tensrs ( BE, ), ( CI, ), ( D, O ), ( GL, ), ( HP, ), ( NQ, ). If the material is centrally symmetric, the tensrs BCEHINPQ,,,,,,, are null. Nte that if the micrcracks are nt present, the identified cntinuum reduces t a Csserat cntinuum. The energetic equivalence criterin can als be pstulated fr the external actins, and the cntinuum, macr and micr, inertial actins can be identified as functins f u, W, and d, where the dt indicates time derivative. 3. WAVE PROPAGATION IN A MICROCRACKED CONTINUUM a b A simplified mdel in which particle rtatins, W ( W ), are neglected is cnsidered. This physically crrespnds t assumed pint-size fibres cinciding with matrix particles. In this case the identificatin prcedure yields null tensrs FGHLP,,,,. If als cnsidering micrcracks arranged accrding t the central symmetry, the cnstitutive equatins fr the stress measures becme S = A u+ D d, S = 0, (4) z = Md, Z = O u+ R d. Accrding t the aximatic descriptin in [ 15 ], the equatins f mtin and micrmtin are derived by assuming the equivalence f the internal and external wrks fr any u, d, d, and an additinal balance equatin ensues frm the vanishing f the internal wrk under macr and micr rigid mtins [ 1 ] div S + b= ρ u, div Z z = µ d, T T T S S + z d d z+ Z d d Z = 0, where b is the bdy density frce, ρ and µ are the identified macr and micr mass densities [ 16 ]. In rder t verify the capability f the multifield cntinuum t reveal the presence f the micrstructure, wave prpagatin is analysed. As a sample test, a ne-dimensinal bar characterized by a distributin f micrcracks f length l m, (5) 103

5 arranged accrding t the transverse istrpic symmetry, is cnsidered. Denting with u and d the lngitudinal cmpnents f the macrdisplacement and micrdisplacement fields, respectively, the equatins f macr- and micrmtin fr this prblem read u α u βd = 0, d εu ϕ d + ηd = 0, (6) where α = A ρ, β= D ρ, ε = O µ, ϕ =R µ, and η = M µ, with A, D, O, R, and M being the sle independent cmpnents f the cnstitutive tensrs in Eqs (3). The apex indicates spatial derivative. In particular, A = Y, the axial stiffness; R = ny ρmπlm and M = myρm πl m, where ρ m is the micrcrack density per unit length, and n and m are cnstants depending n the number and arrangement f the slits in the mdule. The cupling term D= O als depends n the slit size and arrangement and n the elastic cnstants f the matrix. As the micrmass density is the mass relevant t micrcracks, µ = ρ and thus ε = β. Denting with x the crdinate f the bar axis and t the time variable, let us cnsider waves which prpagate in the x-directin with the wave number k and angular frequency ω. A general slutin fr u and d f the frm u = u exp[ i(kx ω t)] d = d exp[ i(k x ω t)] (7) is assumed, with d and u cnstant. Substitutin f Eqs (7) int Eqs (6) gives where ( c ) =, Q I v 0 (8) T {} v = { u d}, c = ω /k, [ Q ] = α β β η/k ϕ + and I is the identity tensr. A nntrivial slutin f system (8) exists if the characteristic equatin (α c)( ϕ c + η /k ) β = 0 is satisfied. Tensr Q plays the rle f the acustic tensr f the multifield system; the psitive square rts f its eigenvalues, c,c, u d are the macr and micr wave velcities. In general, bth these velcities depend n the wave number, k, and the system is dispersive. Nte that if the cupling term, β, is null, Eq. (6a) crrespnds t the standard wave equatin, satisfied by a macrwave prpagating with cnstant macrvelcity, cu = α, while Eq. (6b) remains dispersive with velcity depending n the wave number r frequency: cd = ϕ + η/k =ω ϕ / ω η (dispersin relatin). This circumstance ccurs when n interactins between particles and slits are cnsidered in the lattice mdel. The slutin fr the multifield prblem can be searched fr as a superpsitin f waves prpagating with different velcities depending n frequencies and material parameters. By way f example, let us cnsider the superpsitin f tw linear harmnic waves, u and d, f equal amplitude, u, with different wave 104

6 numbers, k u and k d, and different phase-velcities, c u and c. d Their superpsitin has the frm u+ d = usin k( x c t)cs k( x c gt), with k = (ku + k d), k = ku k d, c = (cu + c d), cg = cu c d. Due t the dispersin prperties, the grup velcity, c, g generally differs frm the average velcity, c, and the shape f the resulting wave is altered. Physically, this seems t be a cnsequence f partial reflectins f waves ccurring in encuntering micrcracks (scattering). The presence f micrcracks in the multifield mdel, represented by the additinal field d, can then be interpreted as a disturbance spread alng the bar that, different frm the classical cntinuum, alters the shape f travelling waves, depending n the micrcrack density. In particular, fr lwdamaged materials the disturbance is lcalized as in a beating-like phenmenn (Fig. 1a), while in high-damaged materials it spreads alng the bar carried by the elastic wave (Fig. 1b). Finally, accrding t experimental results, bth the velcity and the average amplitude, u, f the resulting wave decrease with the increase in micrcrack density (Figs, 3). Fig. 1. Travelling waves alng the bar: (a) lw micrcrack density, (b) high micrcrack density. The elastic wave is shwn by the thin line, the resulting wave by the thick line. Fig.. Phase-velcities vs micrcrack density. The elastic bar is shwn by the thin line, the micrcracked bar by the thick line. Fig. 3. Average amplitude rati vs micrcrack density. The elastic bar is shwn by the thin line, the micrcracked bar by the thick line. 105

7 4. FINAL REMARKS This wrk prvides a multifield cntinuum fr fibre-reinfrced micrcracked bdies accunting fr the presence f fibres and micrcracks by means f additinal kinematic and dynamic fields. Main features f the mdel are the presence f internal length scales and the capability f exhibiting dispersin prperties. Analysis f wave prpagatin in the 1D prblem shws the influence f micrcrack density and the pssibility f describing changes in the shape f the travelling waves generally assciated with scattering. Based n dispersin prperties, which can result in a well-psed set f PDEs, further analysis will be cncerned with describing fracture phenmena taking material nnlinearities int accunt. REFERENCES 1. Trvalusci, P. and Masiani, R. Nn-linear micrplar and classical cntinua fr anistrpic discntinuus materials. Int. J. Slids Struct., 003, 40, Aifantis, E. C. On the rle f gradients in the lcalizatin f defrmatin and fracture. Int. J. Eng. Sci., 199, 30, Sluys, L. J., de Brst, R. and Mühlhaus, H.-B. Wave prpagatin, lcalizatin and dispersin in a gradient-dependent medium. Int. J. Slids Struct., 1993, 30, Capriz, G. Cntinua with Micrstructure. Springer-Verlag, Berlin, Trvalusci, P. and Masiani, R. A multifield mdel fr blcky materials based n multiscale descriptin. Int. J. Slids Struct., 005, 4, Sansalne, V., Trvalusci, P. and Cleri, F. Multiscale mdelling f cmpsite materials by a multifield finite element methd. Int. J. Multiscale Cmput. Eng., 005, 3, Maugin, G. Material Inhmgeneities in Elasticity. Chapman & Hall, Lndn, Gurtin, M. E. Cnfiguratinal Frces as Basic Cncepts f Cntinuum Physics. Springer- Verlag, Berlin, Needleman, A. Material rate dependence and mesh sensitivity n lcalizatin prblems. Cmput. Methds Appl. Mech. Eng., 1988, 67, Pijaudier-Cabt, G. and Bažant, Z. P. Nnlcal damage thery. J. Eng. Mech., ASCE, 1987, 113, Gurtin, M. E. Thermdynamics and the pssibility f spatial interactin in elastic materials. Arch. Rat. Mech. Anal., 1965, 19, Trvalusci, P. and Augusti, G. A cntinuum mdel with micrstructure fr materials with flaws and inclusins. J. Physique IV, 1998, Pr8, Ericksen, J. L. Special tpics in elaststatics. Adv. App. Mech., 1977, 17, Vigt, W. Lehrbuch der Kristallphysik. Math. Wissenschaften, 1910, XXXIV, Di Carl, A. A nn-standard frmat fr cntinuum mechanics. In Cntemprary Research in the Mechanics and Mathematics f Materials (Batra, R. C. and Beatty, M. F., eds). CIMNE, Barcelna, 1996, Trvalusci, P. and Rega, G. A cntinuum mdel fr the analysis f prpagating elastic waves in micrcracked materials. In Prceedings ICM9, Geneve, 003 (CD ROM). 106

8 Elastsed lained hetergeensetes materjalides kui multiskalaarsetes mitmekmpnentsetes pidevates keskkndades Patrizia Trvalusci ja Giuseppe Rega Kasutades multiskalaarset mdelleerimist, mis baseerub klassikalise mlekulaarse elastsusteria hüpteesidel, n esitatud mitmekmpnentse pideva keskknna mudel kmpsiitmaterjalide (kiududega armeeritud materjalide, plümeeride, müüritise tüüpi materjalide jne) dünaamika ligilähedaseks kirjeldamiseks. Lainelevi ühemõõtmelisele prbleemile tuginedes n uuritud võimalust määratleda sisemiste hetergeensuste lemaslu materjalides. 107