A method of two-scale analysis with micro-macro decoupling scheme: application to hyperelastic composite materials

Transcription

1 Comput Mech (213) 52: DOI 1.17/s ORIGINAL PAPER A method of two-scale analyss wth mcro-macro decouplng scheme: applcaton to hyperelastc composte materals K. Terada J. Kato N. Hrayama T. Inuga K. Yamamoto Receved: 1 June 212 / Accepted: 5 May 213 / Publshed onlne: 21 May 213 Sprnger-Verlag Berln Hedelberg 213 Abstract The am of ths study s to propose a strategy for performng nonlnear two-scale analyss for composte materals wth perodc mcrostructures (unt cells), based on the assumpton that a functonal form of the macroscopc consttutve equaton s avalable. In order to solve the two-scale boundary value problems (BVP) derved wthn the framework of the homogenzaton theory, we employ a class of the mcro-macro decouplng scheme, n whch a seres of numercal materal tests (NMTs) s conducted on the unt cell model to obtan the data used for the dentfcaton of the materal parameters n the assumed consttutve model. For the NMTs wth arbtrary patterns of macro-scale loadng, we propose an extended system of the governng equatons for the mcro-scale BVP, whch s equpped wth the external materal ponts or, n the FEM, control nodes. Takng an ansotropc hyperelastc consttutve model for fber-renforced compostes as an example of the assumed macroscopc materal behavor, we ntroduce a tensor-based method of parameter dentfcaton wth the measured data n the NMTs. Once the macro-scale materal behavor s successfully ftted wth the dentfed parameters, the macro-scale analyss can be performed, and, as may be necessary, the macro-scale deformaton hstory at any pont n the macro-structure can be appled to the unt cell to evaluate the actual mcro-scale response. K. Terada J. Kato (B) T. Inuga Department of Cvl Engneerng, Tohoku Unversty, Aza-Aoba 6-6-6, Aramak, Aoba-ku, Senda , Japan e-mal: kato@rdes.tohoku.ac.jp; te@rdes.tohoku.ac.jp N. Hrayama Fukushma Research Center, Ntto Bosek Co., Ltd., 2 Sakura Ipponsug, Fukushma , Japan K. Yamamoto Cybernet Systems Co.,Ltd., Kanda-nerbecho, Chyoda-ku, Tokyo 11-22, Japan Keywords Two-scale analyss Mcro-macro decouplng scheme Numercal materal testng Homogenzaton Ansotropc hyperelastcty 1 Introducton The mathematcal theory of homogenzaton [1 3] has been recognzed as a rgorous modelng methodology for characterzng the macro-scale mechancal behavor of heterogeneous meda wth perodc mcrostructures, often called unt cells. The so-called localzaton capablty n the theory [4], whch provdes clear dstncton from the classcal or theoretcal counterparts n engneerng scence [5], s an appealng feature to researchers n the area of computatonal mechancs, snce t enables us to compute the actual mcroscopc stress and stran n a unt cell wth the help of the fnte element method (FEM). A seres of work done by Léné and hs co-workers [6,7] was probably one of the earlest developments n ths context. It s, however, no exaggeraton to say that Guedes and Kkuch [8] had a head start on the fullfledged research actvtes wth a vew to the applcatons of the homogenzaton and localzaton capabltes n engneerng practce, and ther work was followed by a strng of developments, too numerous to comprehensvely lst here. But, ths much can be safely sad the man nterest has centered on the characterzaton of the macro-scale nonlnear mechancal behavor of composte materals or heterogeneous solds by solvng the mcro-scale boundary value problem (BVP). The mcro- and macro-scale governng equatons derved for nonlnear homogenzaton problems n sold mechancs, whch defne the so-called two-scale BVP, are essentally the same as those n lnear problems. However, the major dfference between the lnear and nonlnear homogenzaton problems s that the former provdes us wth the functonal form

2 12 Comput Mech (213) 52: of the lnear macroscopc consttutve model, whose materal parameters are evaluated by a seres of numercal materal tests (NMTs), whle the latter does not. In the lnear problem, the NMTs measure the fundamental materal responses at the macro-scale so that the homogenzed materal propertes can be dentfed, and ther mcro-scale counterparts, called the characterstc functons, are obtaned by solvng the correspondng mcro-scale BVP. The ntroducton of the characterstc functons enables us to decouple the mcro- and macro-scale varables and, therefore, prevents us from solvng as many mcro-scale problems at the macroscopc materal ponts n response to the macro-scale strans there; see e.g., [2,8] for a lucd explanaton. On the other hand, nonlnear homogenzaton nether provdes an explct form of the macroscopc consttutve equaton, nor allows us to separate the mcro- and macro-scale varables. As a result, the mcro-scale BVP must be solved wth the macroscopc stran feld as datum to evaluate the correspondng macroscopc stress at a materal pont of the macro-structure. Due to ths nconvenence, many prevous studes have been entrely focused on the characterzaton of the local materal behavor by solvng a sngle mcro-scale problem under certan assumed patterns of the macroscopc loadng; see for example, References [9 12] for applcatons of elastc-plastc behavor and [13 15] for mcro-scale damage-nduced nelastc behavor of heterogeneous solds. The man purpose of these studes s to llustrate the macroscopc materal characterstcs by solvng the mcro-scale BVP for ther own unt cells, but the soluton of the macroscale BVP has receved lttle attenton. To obtan the soluton of the macro-scale BVP wthn the framework of the nonlnear homogenzaton, Terada and Kkuch [16] demonstrated the two-scale (or globallocal) computatons for elastc-plastc deformatons of fberrenforces compostes, n whch the mcro- and macro-scale BVPs are completely coupled. More specfcally, a unt cell s FE model s assgned to each ntegraton pont n the macroscale FE model, and the solutons of the macro-scale BVP, such as the macroscopc stress and stran, are the volume average of the correspondng soluton of the mcro-scale BVP. Independently, or nspred by ths attempt, several studes have been done on the development of a mcro macro couplng scheme for two-scale analyses; see, e.g. [17 21]. Ths scheme appears to be robust: t s applcable to varous types of macroscopc nonlnear materal behavor even n cases where ther macroscopc consttutve equatons are unknown. In fact, ths approach seems to be the only way to evaluate the nonlnear macroscopc materal response of a heterogeneous medum when the analytcal expresson of ts macroscopc consttutve relaton s hard to formulate; see e.g., References [22,23]. However, as readly understood from the nature of the mcro-macro couplng scheme, n whch a nonlnear mcroscale FE analyss must be conducted for a unt cell to obtan the macroscopc stress at a sngle ntegraton pont, the computatonal costs requred n executng the fully two-scale (or global-local) computatons are very large ndeed [16,21]. Thus, the applcaton of the couplng scheme to practcal problems s not feasble wthout some sort of countermeasure. In fact, there have been some attempts to mprove the computatonal effcency of the couplng algorthm; see for example, Yamada [24] who ntroduced the block Newton Raphson method to solve the two-scale BVP. A more drect approach to reduce the computatonal cost would be to utlze a dstrbuted memory parallel computer [29], wth each processor takng charge of the mcro-scale computaton. It seems, however, to be dffcult to mplement these methods for the couplng scheme nto exstng general-purpose FEM software, snce mcro- and macro-scale FE models nvolve a sngle two-scale computaton. Other approaches based on the model reducton technques are worth mentonng, whch enable us to reduce macroscopc nternal varables for nelastc homogenzaton procedures; see e.g., [25,26]. In ths context, Oskay and Fsh [28] proposed an approxmate method to reduce the model sze for the mcro-scale problems wth the help of the transformaton feld analyss. The methods are followed by Yvonnet and He [27] wth the help of the proper orthogonal decomposton, whch are utlzed to construct the macroscopc consttutve database as mentoned below. For ths reason, we expect from a vewpont of practcal use that the alternatve approach to resolve the problems mentoned above s developed. Wth our eyes set on the phlosophy of computer-aded engneerng (CAE) n ths context, the utlty value of the two-scale analyss method based on homogenzaton can be enhanced n practce by the mcro macro decouplng scheme. In ths context, Terada and Kkuch [16], who consdered the practcal applcatons whle developng ther couplng scheme, proposed the dea of a consttutve database to decouple the mcro- and macro-scale BVPs. In ther approach, the data of the dscrete macroscopc stress responses are obtaned beforehand n the macroscopc stran space by conductng a seres of NMTs, and are stored nto a database fle, whch the evaluatons of the actual macroscale stress n the macro-scale analyss use to make nterpolatons wthn the stran spaces. Ths approach has been followed by some authors to deal wth nonlnear elastc deformatons [3,31], but seems not to be feasble n the case of general nelastc deformatons. Thus, more relable and effcent means must be developed to put the homogenzatonbased method for two-scale analyses to practcal use. In ths paper, we propose a strategy of conductng nonlnear two-scale analyses of composte materals wth perodc mcrostructures (unt cells) by applyng a class of the mcromacro decouplng scheme [32] to solve the two-scale BVP, whch can be derved wthn the framework of the homoge-

3 Comput Mech (213) 52: nzaton theory. The suggested scheme strongly reles on the method of numercal materal testng, whch corresponds to the homogenzaton process for the unt cells, just lke the computatonal homogenzaton method for lnear problems. To be more specfc, assumng the concrete functonal form of the macro-scopc consttutve model, we conduct a seres of NMTs on the numercal specmen,.e., the unt cell s FE model, to obtan the nonlnear macro-scale materal behavor. By means of the measured data n the NMTs, the materal parameters n the assumed consttutve model are dentfed wth an approprate method of parameter dentfcaton. Once the macro-scale materal behavor s successfully ftted wth the dentfed parameters, the macro-scale analyss can be performed, and, as may be necessary, the macro-scale deformaton hstory at any pont n the macro-structure can be appled to the unt cell to evaluate the actual mcro-scale response. In ths paper, an ansotropc hyperelastc consttutve model for fber-renforced compostes s taken as a smple and lucd example to demonstrate the proposed method. A smlar, but essentally dfferent study has been reported by Ren et al. [33], whch proposed a contnuum damage model that accounts for the mcro-crack evolutons wth the help of numercal materal testng concept. Although the nelastc and brttle damage behavor due to mcro-crackng can successfully be represented by the proposed model, nether the mcro macro consstency wthn the framework of mathematcal homogenzaton has been examned by comparng the macroscopc consttutve responses wth those obtaned by the mcroscopc analyses, nor the extended system for the mcroscopc problem s utlzed wth an ntenton to mplement the model nto the general-purpose software. An outlne of ths paper s as follows. In Sect. 2, westart out by provdng the two-scale BVP for general fnte deformaton problems and then ntroduce a decouplng scheme to solve the mcro- and macro-scale BVPs. Secton 3 s devoted to detaled explanatons of the numercal materal testng suggested as a process n our scheme. Here, the extended system for the mcro-scale BVP wth the perodc boundary condton s formulated by ntroducng external materal ponts, whose counterparts n the FEM are referred to as the control nodes n ths study, so that any pattern of macroscopc stress and deformaton can be appled to the unt cell models. It s to be noted that, thanks to the ntroducton of the control nodes located outsde the unt cell model, the correspondng mcroscale analyses can be conducted by general-purpose FEM software avalable n the market. In Sect. 4, employng an ansotropc hyperelastc consttutve model to represent the macroscopc materal behavor of fber-renforced compostes, a tensor-based method of parameter dentfcaton for the model s provded. In Sect. 5, we llustrate the tran of numercal analyses nvolved n the proposed strategy of two-scale analyses for fber-renforced compostes. The numercal examples demonstrate that the proposed approach s expected to be elgble for both the mcro- and macro-scale CAE systems, snce the mcro- and macro-scale numercal analyses are completely decoupled, yet are related to each other wth regard to the adequacy of the assumed consttutve model. 2 Two-scale analyss based on homogenzaton theory In the multscale mathematcal modelng for composte materals wth perodc mcrostructures (unt cells) by means of the homogenzaton theory [1 3], mcro- and macroscopc boundary-value problems (BVP) are separately derved, and the resultng set of BVPs s referred to as a two-scale BVP [21,34]. In ths secton, after presentng the ndvdual sets of mcro- and macroscopc governng equatons that defne the two-scale BVP, we descrbe the mcro-macro decouplng scheme to perform the correspondng two-scale analyss. 2.1 Two-scale boundary-value problem Wth reference to Fg. 1, we provde the two-scale BVP for a composte materal wth unt cells. The formulaton here s made consstently wthn the framework of fnte stran theory [34]. To measure the mcroscopc mechancal behavor of a unt cell, the spatal poston Y n the mcro-scale ntal or reference confguraton Y of the unt cell doman and the spatal poston y n the mcro-scale current confguraton Y are ntroduced. They are nter-related by the mcro-scale moton as y = ϕ(y) = Y + w(y), where w s the mcro-scale dsplacement of the unt cell. Then the mcro-scale deformaton gradent s defned as F = Y ϕ(x; Y) = Y w(x; Y) + 1 = H(X) + Y u (X; Y) + 1, (1) where X denotes the macroscopc materal pont n the macro-scale reference confguraton, but s not an ndependent varable n the mcro-scale knematcs. Here, Y s the gradent operator wth respect to the mcro-scale Y, H s the macroscopc dsplacement gradent that s ndependent of Y, 1 s the second-order dentty tensor, and u s the Y- perodc dsplacement feld that represents a fluctuaton due to mcro-scale heterogenety. The fluctuaton dsplacement u n (1) s assumed to be subjected to the perodc boundary condton on the unt cell s external boundary Y as follows: u Y [J] = u Y [ J] (J = 1, 2, 3), (2) where Y [±J] ndcates a par of opposte external boundares of the unt cell [4]. Ths condton s referred to as the Y-perodcty n the theory. It s assumed that a unt cell s a rectangular parallelepped-shape, and ts external boundares are arranged parallel to the three mcro-scale coord-

4 122 Comput Mech (213) 52: Fg. 1 Concept of numercal materal testng based on homogenzaton method (n 2D). nate planes Y J so that the bass vector E [J] s an outward unt normal vector on Y [J] n the ntal confguraton. The mcro-scale self-equlbrum equaton for the unt cell s gven as Y P = n Y, (3) where P s the mcro-scale 1st-Pola Krchhoff (PK) or nomnal stress. The mcro-scale governng equaton s completed by the ntroducton of a relevant consttutve model as a functon of the mcro-scale deformaton gradent F defned by (1), and possbly other mcro-scale nternal state varables n the case of nelastc materals. Although arbtrary consttutve models are acceptable for the mcro-scale stress response n ths framework, we take a class of hyperelastc models n ths study so that the correspondng macroscopc consttutve model could be an ansotropc hyperelastc one. Owng to the Y-perodcty, the Pola tracton vector T (N) = P N, wth N beng the outward unt normal vector on the correspondng surface, satsfes the followng antperodcty condtons on the unt cell boundary Y n the ntal confguraton: T [J] + T [ J] =, (4) where we have defned T [±J] := T (±E[J]) wth E [J] beng the bass vector of the Y J -axs. On the other hand, denotng the macro-scale reference and current confguratons by B and B, respectvely, and the macro-scale ntal poston by X B, we have ts current poston by the macro-scale moton x = ϕ(x) B, and we can defne the macro-scale deformaton gradent as F = X ϕ wth X beng the gradent operator wth respect to the macro-scale X. At the same tme, F s defned as the volume average of the correspondng mcro-scale deformaton gradent over the unt cell as F = 1 FdY = H + 1, (5) Y Y where Y s the ntal volume of the unt cell. Here, ths relatonshp s dervable from (1) along wth the Y-perodcty of the fluctuaton dsplacement u, and the macro-scale dsplacement gradent can be dentfed wth H = X ũ(x), wth ũ beng the macro-scale dsplacement feld. Smlarly, the macro-scale 1st PK stress can be defned as the volume average of the correspondng mcro-scale stress over the unt cell as. P = 1 PdY, (6) Y Y whch satsfes the followng macro-scale equlbrum equaton: X P + b = n B, (7) where b s the body force. It s well known that the nonlnear homogenzaton theory does not have a logc that accommodates the explct form of the macroscopc consttutve equaton, but t allows us to use (6) to evaluate the macroscopc stress P after solvng the mcro-scale problem for the equlbrated mcro-scale stress P. In summary, the mcro-scale BVP s to be solved for the set of solutons w, F, P that satsfes the mcro-scale equ-

5 Comput Mech (213) 52: lbrum equaton (3) along wth the knematc condton (1) and a relevant consttutve equaton, whle the macro-scale BVP s for ũ, F, P that satsfes (5), (6) and (7). It s noted that the mcro-scale BVP can be solved only f the macroscale soluton s gven and vce versa. The BVP composed of the mcro- and macro-scale BVPs s called the two-scale BVP n the mathematcal homogenzaton theory. 2.2 Mcro macro couplng and decouplng schemes for the two-scale BVP In the two-scale BVP, the macroscopc consttutve equaton s an mplct functon of the solutons of the mcro-scale BVP and, thus, the mcro-scale BVP ndrectly represents the macroscopc materal response. That s, t s not untl the mcro-scale equlbrated stress s determned that the macroscopc stress can be calculated n vew of (6). Therefore, f the two-scale couplng analyss s performed by the FEM, the mcro-scale BVP must be assocated wth an ntegraton pont located n a macro-scale fnte element model and solved for the mcro-scale equlbrated stress to evaluate the macro-scale stress by the averagng relaton (6), whch must satsfy the macro-scale BVP at the same tme. In partcular, when an mplct and ncremental soluton method wth a Newton-Raphson type teratve procedure s employed to solve the two-scale BVP, the mcro-scale BVP s to be solved n every teraton to attan the macro-scale equlbrum state at every loadng step. Needless to say, the mcro-scale BVP s also nonlnear and therefore requres the teratve method. Ths type of soluton scheme to solve the two-scale BVP s referred to as the mcro-macro (or global-local) couplng scheme and s typfed n [16,21,34]. The mcro-macro couplng scheme s promsng n the sense that almost all of the varous types of macroscopc materal behavor can be captured wthout knowng ther explct functonal forms of materal models f the unt cell s elgble for a RVE. However, the nature of the method means t requres a sgnfcant amount of computatonal cost. In fact, the model sze of the macro-scale BVP rases the number of mcro-scale BVPs to the second power, snce each macroscale ntegraton pont s assocated wth ts own mcroscale BVP. Although some parallel algorthms can reduce the cost to some extent [29], we are bound to say that the couplng scheme s all but useless n most practcal applcatons. Therefore, the decouplng of mcro- and macro-scale BVPs s ndspensable for applyng the two-scale approach based on homogenzaton to varous problems encountered n practce [32]. The precondton of decouplng s that we are able to pck up a consttutve model to properly characterze the macroscopc materal behavor that would be obtaned from the numercal analyss on the mcro-scale BVP. It s noted that, from a practcal pont of vew, approxmated consttutve models allow alternatves, snce there mght not be a rgorous model avalable dependng on the type of composte materals. Once the functonal form of an approprate macroscopc consttutve equaton s assumed, several mcro-scale numercal analyses are performed on the unt cell to obtan ts materal parameters. The set of mcro-scale analyses for ths purpose can be referred to as numercal materal testng (NMT), an essental process of the mcro-macro decouplng scheme [32]. The concrete procedure of the method s descrbed as follows: () An approprate consttutve model relevant for the macroscopc materal behavor under consderaton s assumed. () A seres of NMTs s conducted on a unt cell model (FE mesh), whch s regarded as a numercal specmen, to obtan the homogenzed or macroscopc materal behavor. Note that the loadng patterns here hnge on the selected consttutve model. () Materal parameters of the assumed consttutve model are dentfed by means of the emprcal data obtaned from the NMTs and an approprate curve fttng scheme. (v) FE analyses are carred out to solve the macro-scale BVP usng the assumed consttutve model wth dentfed materal parameters. (v) If necessary, after extractng the tme-seres of macroscopc deformaton hstory from the macroscopc analyss result and applyng t as a seres of boundary condtons, the localzaton analyses are performed to evaluate what has actually been happenng nsde the unt cell durng the macroscopc deformaton process. Snce the materal models used n unt cells are supposed to be gven n the computatonal homogenzaton method, the homogenzed or macroscopc materal model to be assumed n Step () s expected to partally nhert the mcro-scale materal behavor. For example, f the unt cell model of a fber-renforced plastc s assumed to be composed of sotropc hyperelastc materals, the correspondng macroscopc materal behavor can be ansotropc hyperelastc. Lkewse, f the consttuents are elastc-plastc materals, the macroscopc consttutve model should be wthn the scope of ansotropc plastcty. Even though mcro-scale crackng s taken nto account as n [23], the correspondng macroscopc materal behavor may be represented by the ansotropc damage model approxmately. However, assumed macroscopc consttutve models do not always properly represent the macroscopc materal behavor properly, the decouplng scheme s just an approxmate scheme. Thus, two-scale analysts are responsble for the degree of approxmaton, but the couplng scheme can be used rather than the decouplng one, f the hghest level of accuracy s desred rrespectve of computatonal costs.

6 124 Comput Mech (213) 52: A method of numercal materal testng Assumng that an approprate consttutve model s found for the macroscopc nonlnear materal behavor, we are concerned wth the accuracy and valdty of the NMT as part of the two-scale analyss. Gven the macroscopc stress or deformaton as a datum, the mcro-scale BVP has to be solved for the mcro-scale stress. Then, usng (5) and (6), we obtan dscrete macroscopc stress stran curves, whch can be regarded as expermental results. In ths secton, we frst ntroduce an extended system of the mcro-scale BVP so that the arbtrary patterns of macroscopc loadng are utlzable n the NMTs, and provde the concrete procedure of the NMT by usng the standard FEM. 3.1 Alternatve form of the mcro-scale BVP The ntegraton of (1) wth respect to Y yelds the followng form of the mcro-scale dsplacement w of the unt cell: w(x; Y) = H(X) Y + u (X; Y) + c(x), (8) where c s a constant vector ndependent of Y. By substtutng ths expresson nto (2) that mpose the Y-perdocty of the fluctuaton dsplacement feld, we have the constrant condton as w [J] w [ J] = H L [J], (9) where w [±k] := w Y [±k]. Here, we have defned the vector connectng the materal ponts of a Y-perodcty par as follows: L [J] := Y Y [J] Y Y [ J], (1) whch can be called the sde vector of a unt cell. Owng to the ant-perodcty of the Pola tracton vector (4), the followng relatohshp can be derved: T [J] = P E [J] = 1 P E [J] dy Y Y 1 = T [J] ds, (11) Y [J] Y [J] where Y [J] s the area of the unt cell boundary Y[J]. Also, denotng the spatal bass vector by e [], the components of P n (11) can be expressed as T [J] = P J = e [] ( P E [J] )= 1 Y [J] T [J] ds. (12) Y [J] That s, the J-component of the macro-scale 1st PK stress, P J, s the area average of the correspondng mcro-scale Pola tracton vector, T [J], at the unt cell boundary Y [J]. Therefore, the area average of the ant-perodcty condton (4) for the mcro-scale Pola tracton vector s nothng but the acton and reacton law of the macro-scale Pola tracton vector on the macro-scale surface whose outward unt normal vector concdes wth E [J]. The proof of Eq. (11), or equvalently (12), s provded n Appendx A. When the response functon of the mcro-scale 1st PK stress F(F) s assumed for a materal model used n the unt cell, the mcro-scale BVP s gven anew as Y P = F = Y w + 1 P = F(F) n Y, (13) w [J] w [ J] = H L [J] and T [J] 1 = Y [J] P E [J] ds Y [J] on Y [J]. (14) The data to be prepared for ths BVP are the macro-scale deformaton (e.g. H) and/or the macroscopc stress (e.g. P) n addton to the nformaton about the unt cell s geometry and the materal models for consttuents wth relevant propertes. It s realzed that ths BVP s a standard qaus-statc equlbrum problem except that the dsplacement constrants and the loadng condtons at the boundares are somewhat specal. As wll be seen later, snce the 9 components of the macro-scale dsplacement gradent H and the 9 components of the mcro-scale 1st PK stress P are dual, they cannot be gven ndependently. 3.2 Extended system of the mcro-scale BVP wth external materal ponts We here ntroduce an extended system of the mcro-scale BVP by ntroducng fcttous materal ponts located outsde the unt cell doman. Although these addtonal ponts can be nsde the unt cell, we call them external materal ponts n ths study. The constrant condton (9), or equvalently the frst equaton n (14), can be re-wrtten as w [J] w [ J] = q [J], (15) where we have defned the relatve dsplacement vectors as q [J] := H L [J]. (16) Correspondng to the three pars of the unt cell boundary surfaces Y [±J], we ntroduce three external materal ponts, to whch the degrees-of-freedom (DOFs) of the three relatve dsplacement vectors q [J] n (16) are assgned as depcted n Fg. 2. Thus, the resultng extended system of the governng equatons for a unt cell has nne more DOFs than the organl one (13) wth (14) n a 3D settng. Owng to the ntroducton of the external materal ponts, Eq. (15) s regarded as a three-pont constrant equaton that

7 Comput Mech (213) 52: (a) (b) (c) Fg. 2 External materal ponts for controllng relatve dsplacements and reacton forces on unt cell boundary surfaces: a boundary surfaces; b relatve dsplacements; c reacton forces. relates the dsplacement vectors of an arbtrary par of materal ponts on the boundary surface Y [±J] to the relatve dsplacement vector of the correspondng sngle external pont, whle eqn. (9) s regarded as a two-pont constrant condton wth q [J] as a constant vector. As wll seen later, ths feature s not only of partcular convenence when evaluatng the macroscopc stress and stran, but also the only way to mpose the macroscopc stress components drectly to the unt cell wthout solvng the macroscopc BVP (7). For nstance, f we solve the mcro-scale equlbrum equaton (3) for a unt cell by specfyng the -th component of the dsplacement q [J] at an external pont, not only the mcro-scale stress and stran felds, but also the -th component of the reacton force R [J] at the external pont should be obtaned. We then note that ths reacton force R [J] can be dentfed wth the area ntegral of the mcro-scale Pola tracton vector T [J] over Y [J], whch s assocated wth the constrant condton (15) for the relatve dsplacement w [J] w [ J].Thats,wehave R [J] = Y [J] T [J] ds, (17) from whch the correspondng component of the macro-scale Pola tracton vector can be obtaned as P J = T [J] = R [J] Y [J]. (18) Therefore, once all the resultant forces actng on the three external materal ponts are evaluated, all the components of the macro-scale stress can be obtaned. On another front, the specfcaton of the components of to an external pont s possble, and mples that the components of the macroscopc stress P J can be mposed on a unt cell rrespectve of the correspondng macro-scale problem. In ths case, the component of a resultant force R [J] the dsplacement q [J] s unknown, but can be translated to the component of the macro-scale dsplacement H K L [J] by means of (16). In the mathematcal theory of homogenzaton, a unt cell doman s dentfed wth a sngle macroscopc materal pont. Ths means that each macroscopc feld varable s determned from a sngle unt cell. In ths context, the three external ponts have nne DOFs n total, whch are the same n number as the ndependent components of the macro-scale 1st PK stress or of the macro-scale dsplacement gradent. K

8 126 Comput Mech (213) 52: Therefore, the set of components of the macroscopc stress P s unquely assocated wth the set of components of the reacton force R at the three external materal ponts. Also, the dsplacement gradent H s unquely related to the relatve dsplacement q [J]. Thus, these external materal ponts enable us to evaluate the macroscopc quanttes wthout usng the correspondng mcroscopc quanttes. Ths feature s of partcular advantage especally when the numercal materal testng s conducted wth commercal FEMsoftware, as explaned below. 3.3 Fnte element analyss for a unt cell wth external materal ponts We here explan the usage of the external materal ponts to solve the mcro-scale BVP wth a vew to utlzng a generalpurpose FEM code avalable on the market. For the sake of smplcty, only a rectangular paralleleppe-shaped unt cell s consdered, and ts sdes are assumed to be parallel to one of the mcro-scale coordnate planes so that only the J-th component of the sde vector L [J] s non-zero. For preparaton of the fnte element analyss (FEA) for the mcro-scale BVP, the spatal doman of the unt cell s dscretzed to generate ts FE mesh. At the same tme, each external materal pont s also dscretzed to an element wth a sngle node whch has three DOFs and no mass. Snce the external materal ponts enables us to control the components of the macro-scale stress and deformaton, as explaned above, the node correspondng to an external materal pont s referred to as a control node n ths study. Thus, we obtan an extended system of FE-dscretzed equatons nvolvng nne addtonal DOFs of three control nodes. In the followng, we ntroduce some specfc usages of the three control nodes to solve the extended system. Frst, the macro-scale deformaton s assumed to be known; that s, all the components of the macroscopc dsplacement gradent H are gven as data. Usng (16), we obtan all the components of the nodal dsplacement vector q [J] at the three control nodes located on the unt cell boundary Y [J] (J = 1,,2, 3). Then, gven all the components q [J], we solve the extended system of FE equatons for the mcroscale BVP (13) wth the nne sets of two-pont constrants realzed by (15). The results of the FEA contan not only the mcro-scale dsplacement, stran and stress, but also the reacton force R [J]. Therefore, the macro-scale 1st PK stress P can be computed from (18), wthout performng a numercal ntegraton on (6). Also, snce the macro-scale dsplacement gradent H has been gven as a datum, the macro-scale deformaton gradent can be computed as F = 1+ H and n turn ts determnant J = det F so that the macro-scale true (Cauchy) stress s computed as σ = P F T / J Secondly, let us suppose that the macro-scale stress s known; that s, all the components of the macro-scale 1st PK stress P are gven as data. All the components of the resultant force vector R [J] at the three control nodes are determned by means of (18). In ths case, all the components of the nodal dsplacement vector q [J] are unknown n the extended system of FE equatons for the unt cell. Once the soluton of the system s obtaned, we can obtan the followng relatonshp: Q = H L, (19) from whch the macro-scale dsplacement gradent can be evaluated as H = Q L 1. Here, we have defned Q as the matrx composed of three sets of the dsplacement vectors q [J] at the control nodes and L [J] (J = 1,,2, 3) as the matrx composed of the sde vectors. To be more presce, they are respectvely defned as Q = [ q [1] q [2] q [3]] and L = [ L [1] L [2] L [3]]. (2) The computaton of the macro-scale deformaton gradent F = 1 + H s straghtforward and s followed by the evaluaton of the macro-scale rght- and left-cauchy-green (CG) deformaton tensors as, respectvely, C = F T F and b = F F T. Furthermore, we denote the egenvalues of the macro-scale rght- and left-stretch tensors Ũ and Ṽ by λ α, and the correspondng egenvectors by N α and n α, respectvely. Then, the spectral decomposton of C and b are respectvely gven as follows: C = 3 λ 2 α N α N α, and b = α=1 3 λ 2 α n α n α, (21) α=1 from whch the materal and spatal logarthmc strans can be computed. If ether H J or P J are gven as data for the control nodes n actual computatons, the macros-scale quanttes are evaluated n the same way as outlned above. It s, however, noted that both the dsplacement q [J] and the resultant force R [J] cannot be specfed to the same component number due to the nature of the extended system. Also, the localzaton procedure can be performed n exactly the same manner by usng the tme-seres data of the macro-scale dsplacement gradent, whch are supposed to be obtaned n a macro-scale analyss. 4 Parameter dentfcaton for ansotropc hyperelastc consttutve law After the measurements wth the numercal materal testng, the materal parameters n the assumed macroscopc consttutve model can be dentfed. In ths secton, takng a class of ansotropc hyperelastc consttutve models as an exam-

9 Comput Mech (213) 52: ple, we ntroduce a method of parameter dentfcaton wth all the tensor components measured n the NMTs. 4.1 Strategy for parameter dentfcaton There are some establshed methods of parameter dentfcaton for most sotropc hyperelastcty models avalable n general-purpose FEM-software. After the selecton of a consttutve model that s expected to properly characterze the mechancal behavor of the rubber-lke or polymerc materals under consderaton, t s a common practce to perform unaxal tenson, pure shear and equ-baxal tenson tests on sheet materals [35,36]. Then, by usng the measured data as nput, a lnear or nonlnear least square method s appled to dentfy the parameters used n the selected model. Although such multple patterns of loadng are prepared, t s dffcult to correctly measure all the necessary components of the stress and deformaton. For nstance, when the membrane specmen s subjected to the unaxal or equ-baxal tensle loadng, ts out-of-plane deformaton s generally not measured. Therefore, ncompressblty s usually assumed n the establshed method of dentfcaton so that the deformaton n a certan drecton, whch has not been measured, can be estmated from the measured data of deformaton n other drectons. However, t cannot be expected that the same strategy can be appled to ansotropc hyperelastc models. In fact, deformaton that s not measured cannot be reasonably estmated n ansotropc models, as a general rule. Also, snce there are sgnfcantly more parameters for ansotropc models than for many sotropc models, t s necessary for the measured data to be more relable and contan more nformaton about the tensor components of the stress and stran than n the case of sotropc materals. In ths regard, t s fortunate that we can utlze the emprcal data obtaned from the NMT on a sngle numercal specmen, whch s actually an FE model of a unt cell. That s, snce the NMT enables us to evaluate all the components of the stress and stran along the deformaton hstory of the numercal specmen, a bare mnmum of data can be obtaned for parameter dentfcaton of ansotropc hyperelastc consttutve functons. Of course, multple patterns of loadng are to be appled to the numercal specmen to acqure suffcent data, but the concrete patterns and ther number have not yet been dscussed. Thus, the present study s lkely to be the frst trals on the determnaton of the loadng patterns n the NMT for parameter dentfcaton. 4.2 Loadng patterns n NMT A consttutve model s a functonal representaton of materal behavor and essentally provdes the relatonshp between the stress and stran tensors. That s, a consttutve model s a tensor-valued tensor functon and s regarded as a devce to output all the components of the stress tensor by nputtng all the components of the stran tensor. In ths context, we remember that the NMTs for homogenzaton n 3D lnear elastcty are conducted on the numercal specmen, namely the unt cell model, wth sx-ndependent patterns of the macro-scale stran whch have sx correspondng sets of macroscopc stffness, whch s equvalent to the macroscopc stress. That s, twenty-one components of the macroscopc elastc coeffcent matrx can be determned wth only sx NMTs. To be more specfc, we apply the sx-patterns of macroscopc unt strans 1 (1) ={1,,,,, } T 1 (6) = {,,,,, 1} T to the unt cell separately, obtan the mcroscale stress σ and stran ε, whch satsfes the mcro-scale BVP, and take ther volume average over the unt cell to evaluate the followng ansotropc elastcty matrx D H n the macroscopc consttutve equaton σ =Σ = D H E = D H ε wthn the lnear elastcty framework: [ ] D H = D H (1) DH (2) DH (3) DH (4) DH (5) DH (6) (22) where each column vector D H () contans the macroscopc stress components n response to -th test case wth nput data 1 (). Although the macroscopc stress responses are dfferent dependng on the macroscopc stran levels for nonlnear problems, the macroscopc stress stran curves are unquely determned n hyperelastcty once the macroscopc deformaton patterns are gven. It s therefore reasonable that sxndependent patterns of macroscopc deformaton are gven to the numercal specmen separately to obtan the sx sets of response curves of sx (or nne) components of the macroscopc dsplacement gradent tensor and 1st PK stress tensor. Usngthese6 6 stress stran curves, we are able to formulate the mnmzaton problem to determne the unknown materal parameters of an assumed consttutve functon for ansotropc hyperelastcty, as detaled n the next subsecton. An example of macroscopc deformaton and stress patterns n NMTs s provded n Table 1. Here, Ĥ and are the specfed and unspecfed components of the macro-scale dsplacement gradent H [α] (α = 1,, 6), respectvely, and mples the value of the component s fxed to zero durng the NMT. Thus, the components of the macroscopc 1st PK stress that corresponds to n H [α] are zero. Also, and are the components of the macroscopc stress caused by the specfcaton of Ĥ and zero n H [α], respectvely. Although we admt of arbtrarness n selectng loadng patterns, the materal parameters are unquely dentfed by the method proposed below. 4.3 Tensor-based method for parameter dentfcaton On the presumpton that the sx sets of sx response curves of all the components of the macroscopc dsplacement gradent tensor and 1st PK stress tensor have been obtaned by the

10 128 Comput Mech (213) 52: Table 1 Loadng patterns for numercal materal tests Case-1: Tenson n the X 1 -drecton Case-2: Tenson n the X 2 -drecton H [1] Ĥ =, P [1] = H [2] = Ĥ, P [2] = Case-3: Tenson n the X 3 -drecton Case-4: Shear n the X 1 X 2, X 2 X 1 -plane H [3] =, P [3] = H [4] = Ĥ Ĥ, P [4] = Ĥ Case-5: Shear n the X 2 X 3, X 3 X 2 -plane Case-6: Shear n the X 3 X 1, X 1 X 3 -plane H [5] = Ĥ, P [5] = H [6] = Ĥ, P [6] = Ĥ Ĥ sx sets of the NMTs, we ntroduce a method of parameter dentfcaton for ansotropc hyperelastc consttutve models. Although the method can be appled for arbtrary forms of ansotropc consttutve models, we confne ourselves to a certan class among them, n whch the functonal form s lnear wth respect to materal parameters. A typcal example of ths class of models s presented n Appendx B, and we employ t n the followng sectons. Denotng the materal parameters by p [k] and the number of them by n para, an ansotropc hyperelastc consttutve model whch s lnear wth respect to p [k] can be wrtten as S( p) = n para k p [k] g [k] (23) where p = { p [1],, p [n para] }, S s the macroscopc 2nd Pola Krchhoff (PK) stress tensor and g [k] (k = 1,, para) are tensor-valued functons that are nonlnear functons of the macroscopc deformaton gradent or the alternatves. Then now, the number of deformaton patterns of the NMTs s fxed to n test = 6, and the number of measured data ponts obtaned by a NMT for loadng pattern α s denoted by n [α] step. More concretely, we perform mcro-scale analyses n test = 6 tmes for a sngle unt cell of the hyperelastc composte materal and, for a loadng pattern α, store n [α] step sets of data, each of whch contans the sx components of the macroscopc rght Cauchy-Green (CG) deformaton tnsor ˆ C [n,α] C and the sx components of the macroscopc 2nd PK stress tensor ˆ S [n,α] for each selected step n among the loadng steps. Here, ˆ C [n,α] and ˆ S [n,α] can respectvely be computed by the macroscopc dsplacement gradent ˆ H [n,α] and the 1st PK stress tensor ˆ P [n,α], both of them are ether the nput data or the measured data n the NMTs as explaned n the prevous subsecton. On the other hand, gven the data of the macroscopc dsplacement gradent ˆ H [n,α], the consttutve equaton (23) can be evaluated to compute the correspondng macroscopc stress. We denote ths consttutve response by n para S [n,α] ( p) = k p [k] g [k,n,α] (24) where g [k,n,α] are supposed to be computed by ˆ H [n,α],or equvalently, ˆ C [n,α]. Then, the followng scalar-valued functon can be defned to sum up the errors measured by the norms of the stress tensors: χ(p) = 1 2 n test α=1 or equvalently, 1 n [α] step χ(p) = 1 n test n [α] step 1 2 α=1 n [α] step n=1 n [α] step S [n,α] ( p) ˆ S [n,α] 2 n=1 ˆ S [n,α] 2 ( S [n,α] ( p) ˆ S [n,α] )( S [n,α] ( p) ˆ S [n,α] KL ˆ S [n,α] KL ˆ S [n,α] ) (25) (26) n whch the summaton conventon s employed for the ndces on the macroscopc 2nd PK stress tensor. It s to be noted that, n ths error functon, all the stress components are used to defne the error between the consttutve response and the stress response n the NMT for the same macroscopc deformaton. Note also that the role of the denomnator n the error functon s to normalze the error n each step n by the norm of the stress tensor obtaned at the same step of the NMT and n turn to mtgate the loss of sgnfcant dgts n the numercal treatments of the parameter dentfcaton. Snce the assumed consttutve equaton (24) s lnear wth respect to the materal parameters p, the dfferentaton of the error functon (25) wth respect to these parameters yelds the followng system of lnear equatons to be solved for p: χ(p) p [l] = (l = 1,, n para ) (27)

11 Comput Mech (213) 52: To be more specfc, the obtaned algebrac equatons are as follows: n test α=1 1 n [α] step n para k=1 = n step [α] n test α=1 n=1 1 n [α] step g[l,n,α] ˆ S [n,α] KL n [α] step n=1 g [k,n,α] ˆ S [n,α] KL g[l,n,α] ˆ S [n,α] KL p [k] ˆ S [n,α] ˆ S [n,α] KL (28) whch can be dentfed wth Gp= b and solved for p,fthe coeffcent matrx G s nvertble. In fact, unless the assumed consttutve equaton has some nadequaces, the regularty of the matrx should be guaranteed snce the selected loadng patterns provde mutually ndependent stress responses. In summary the entre procedure of the proposed method s presented n the followng box. I. Select a macroscopc consttutve materal model II. Conduct NMTs on a unt cell model usng FE mesh () Gve the macroscopc dsplacement gradent H and then the relatve dsplacement vector q [J] of the external ponts q [J] := H L [J] () Buld the extended mcroscopc BVP Eqs. (13), (14) by mposng w [J] w [ J] = q [J] () Obtan 1st PK stress P J at each ncremental step n for all loadng patterns α from the reacton force vector drectly obtaned by solvng the extended mcro-scale BVP [J] ˆ P J = ˆ R / Y [J] III. Identfy macroscopc materal parameters () Usng the NMT data calculate the macro-scale 2nd PK stress ˆ S [n,α] ( ( = F [n,α]) ) 1 ˆ P [n,α] and the 5 Numercal examples Numercal analyses are conducted to demonstrate the feasblty of the two-scale couplng analyss wth the mcromacro decouplng scheme and to assess the valdty of the present method of parameter dentfcaton by means of the NMT. The ansotropc hyperelastc consttutve model [38,39] gven n Appendx B s employed as a macro-scale materal model for the macro-scale BVP, and a generalpurpose FEM software, ANSYS [37], s used for both mcro- and macro-scale analyses. 5.1 Condtons for NMTs The unt cell models for the numercal verfcaton are shown n Fgs. 3 and 4, whch are referred to as UC-1 and UC- 2, respectvely. UC-1 s a perodc mcrostructure of a undrectonal fber-renforced composte (UD-FRC) wth A = {,, 1} T, and UC-2 s that of a 3 -crossed fber-renforced composte (3CR-FRC) wth A ={1/2,, 3/2} T and B ={ 1/2,, 3/2} T. The volume fractons of UC-1 and UC-2 are 28.3 % and 36.1 %, respectvely. Also, ten-node tetrahedral elements (SOLID 187) n the ANSYS s element lbrary s used for ther FE meshes. For the matrx materal n both of the unt cells, an sotropc hyperelastc model of Ogden [36] s assumed, and ts materal parameters are set at μ 1 = ,μ 2 =.14 [MPa],α 1 = 1.3, α 2 = 5. Fg. 3 UC-1: un-drectonal fber-renforced composte. rght-cg deformaton tensors ˆ C [n,α] by makng use of Eq. (43) and store all sets of data over n [α] step () Buld a functon of Eq. (23) wth the materal parameters p () Identfy the macroscopc materal parameters p by solvng the obtaned algebrac equatons Gp= b IV. Macroscopc FE-analyss () Solve the macro-scale BVP usng the assumed consttutve model wth dentfed materal parameters Fg. 4 UC-2: 3 -crossed fber-renforced composte.

12 121 Comput Mech (213) 52: and d O = n the Ogden s energy functonal 2 μ W O = ( λ α 1 α + λ α 2 + λ α ) (J 1) 2 (29) d O =1 For the fbers, we take materal parameters μ = 7 [MPa] and d H = 1 3 [MPa] n the followng sotropc neo- Hookean energy functonal: W H = μ (Ī1 1 ) + 1 (J 1) 2 (3) 2 d H Remark 1 We have been aware that a suffcent number of dgts after the decmal pont s essental n recodng measured data n the NMTs. To be more specfc, 15 or 16 dgts after the decmal pont are necessary to mtgate the effect of roundng errors, when we use double precson real numbers for computatons. The number of dgts seems to be somewhat excessve from the vewpont of effectve dgts for engneerng judgment, but s requred for the present tensor-based method of parameter dentfcaton. Remark 2 Smlarly to Remark 1, the qualty of FE meshes for unt cells s also nfluental on the accuracy of the NMTs and n turn that of the parameter dentfcaton demonstrated n Subsect For nstance, we need the maxmum degree of conformty n the coordnates of the nodes on the opposed sdes of a unt cell, at whch the two-pont or three-pont constrants (15) s mposed. In addton, the maxmum degree of geometrcal symmetry of the FE model of a unt cell s desred to obtan the symmetrc moton of a geometrcally symmetrc unt cell n response to a macroscopcally symmetrc loadng as n Table 1. Otherwse, the parameter dentfcaton wth the present method suffers from the superfluous reacton forces at the control nodes. Remark 3 The relablty of the data measured n the NMT depends on the accuracy of the mcro-scale analyss for the unt cell models. It s, therefore, preferable that the most fundamental response, such as a stress stran curve of the un-axal tenson test, s calbrated so that we can elmnate any analyss errors due to the defcences n both the FE approxmaton and the nput data for the unt cell models. That s, some sort of emprcal valdaton s ndspensable f the accuracy of the NMTs s essental. Nonetheless, the method of parameter dentfcaton can be verfed on the assumpton that the results of the NMTs are relable enough. 5.2 NMTs and parameter dentfcaton In Step () as explaned n Sect. 2.1, we conduct a seres of NMTs on the two unt cell models separately by applyng the sx macroscopc loadng patterns descrbed n Subsect. 4.2 by means of the control ponts ntroduced n Subsect The number of loadng steps n step = 2 s taken for each loadng pattern, and the same number of sets of the macro-scale 2nd PK stress and rght-cg deformaton tensors, S [n,α] and C [n,α], are obtaned n a sngle NMT wth loadng pattern α, as explaned n Subsect We denote the components of the macro-scale 2nd PK stress and rght-cg deformaton tensors [n,α] [n,α] by ˆ C and ˆ S, respectvely, to dstngush them from the values determned by the assumed consttutve equaton. The results of the NMTs are shown n Fgs. 5 and 6, n whch only one curve s depcted for each loadng pattern, although all the tensor components were obtaned and stored nto a fle. The curves characterze the ansotropc materal responses as expected. In ths study, we assume the followng functonal form of the 2nd PK to ft these curves [38,39]: S = S vol + S so (31) where S vol = 2 D J (J 1) C 1 (32) ( S so = I 1/3 3 I 1 ) 3 C 1 [ C : γ 1 1+ γ 2 C + γ 4 (A A) + γ 5 (A CA+ CA ) A + γ 6 (B B) + γ 7 (B CB+ CB ) ] B + γ 8 (A B) (33) Here, γ ( = 1,, 8) and D are descrbed n Appendx B. Ths expresson of the stress corresponds to UC-2, whereas UC-1 should provde the transversely sotoropc behavor that can be realzed by settng B = n (33); see [38]. Now, let us go on to the Step () as descrbed n Sect That s, we apply the method of parameter dentfcaton, whch s proposed n Subsect. 4.3, to dentfy the materal parameters of the assumed consttutve equaton. To be more specfc, the materal parameters to be determned are a, b, c ( = 1, 2, 3) and d, e, f, g ( = 2,, 6) n (51) aswellasd n (32) or(5), though the parameters e, f, g are unnecessary for UC-1 as mentoned before. The value of the error functon (25) for each loadng pattern s presented n Table 2. Usng the dentfed parameters n (24) ˆ C [n,α] for the assumed consttutve model and the same used n the NMTs for the arguments C [n,α], we provde the functonal responses of S [n,α] n Fgs. 7, 9 and 8, where the results of the NMTs are also shown for comparson. The macroscale stress stran curves n Fgs. 7 and 8 are the responses to the macro-scale deformaton patterns consstent wth the NMTs, whle those n Fg. 9 are not realzed n the NMTs. In partcular, Fg. 9a shows the macroscopc response of UC- 1 when tensle loadng s appled n the Y 1 -drecton wth the Y 2 -drecton stress free and wth the Y 3 -drecton fxed, and Fg. 9b shows the relatonshps between the macroscopc axal stress components and the rght CG deformaton tensor components of UC-2 when tensle loadng s appled n each

13 Comput Mech (213) 52: Fg. 5 Numercal materal test results for unt cell of un-drectonal fber-renforced composte: a three normal normal components; b three shear components. (a) (b) one of the normal drectons wth the sde lengths n the other two drectons fxed. It can be seen from Fgs. 7a, b for UC-1 that the curves obtaned wth the dentfed parameters show farly good agreement wth those of the NMTs, although conformty may not be satsfactory, especally n the case of the shear deformaton patterns. These levels of accuracy are also seen n Table 2. As can be seen from Fg. 9a, some level of accuracy s observed n response to the loadng patterns not contaned n the sx patterns n the NMTs for parameter dentfcaton. A note s appended wth regard havng obtaned equvalent results for other knds of unt cells of UD-FRC of dfferent volume fractons of fbers. Smlar studes can be made on the results shown n Fg. 8 for UC-2, whch compares the consttutve responses wth the dentfed parameters and the data obtaned n NMTs for the macroscopcally orthotropc behavor of the 3CR-FRC. [n,α] [n,α] Although the dsagreement n the response ˆ C 33 ˆ S 33 s by no means small and s confrmable n Table 2, the parameter dentfcaton seems to be largely successful. The reason why the error calculated for the 2nd pattern wth H 22 beng controlled n the NMT s relatvely large n Table 2, even though farly good agreement s obtaned for the response n S 22 C 22, s that the functonal responses of some components other than S 22 C 22 to ths loadng pattern devate from those of the NMTs. We tred other error functons besdes (25), but faled to obtan better results. The assumed consttutve model for orthotropc hyperelastcty s not capable of accurately reproducng the materal behavor expected from the prepared unt cell models,

14 1212 Comput Mech (213) 52: Fg. 6 Numercal materal test results unt cell of 3 -crossed fber-renforced composte: a three normal normal components; b three shear components. (a) (b) Table 2 Square errors n parameter dentfcaton Loadng patterns n NMT UC UC though the level of accuracy n the parameter dentfcaton s assured to some extent. It was, however, not untl the parameter dentfcaton was realzed by means of the NMTs that the performance of the consttutve model was examned n ths study. If there were more approprate consttutve models avalable, the present approach for homogenzaton n the two-scale analyses could be advocated. 5.3 Parameter dentfcaton for non-proportonal loadng In ths numercal example, non-proportonal loadng condton s consdered n terms of the unt cell model UC- 1 to verfy the further qualty of the parameter dentfcaton. We use the materal parameters obtaned at Subsect. 5.2 and compare the analyss results wth the dentfed

15 Comput Mech (213) 52: (a) (a) (b) (b) Fg. 8 Fttng results for unt cell of 3 -crossed fber-renforced composte: a three normal normal components; b three shear components. Fg. 7 Fttng results for unt cell of un-drectonal fber-renforced composte. Fundamental responses as n NMTs: a three normal normal components; b three shear components. materal parameters and by the mcro-scale analyss. The assumed non-proportonal loadng s unform tenson n the Y 1 -drecton followed by unloadng and transverse shear n the Y 12 -drecton. To be more specfc, the followng two macroscopc dsplacement gradents, expressed as H (1) and H (2), are gven to the three external materal ponts n turn as constrants, H (1) =, H (2) =.15 (34) where the superscrpt denotes smply the order of appled loadng condton. Fgure 1 shows the results of the analyss wth the dentfed materal parameters and of the mcro-scale analyss usng the UC-1. Here, the relaton of the equvalent 2nd PK stress and the equvalent dsplacement gradent s dsplayed to examne the response wth respect to the non-proportonal loadng. It s observed that the response shows almost lnear behavor durng the deformaton by H (1), then turns out to be complex nonlnear behavor durng the deformaton by H (2). As can be seen, the result wth dentfed materal parameters shows good agreement wth the mcro-scale analyss, although dscrepancy s observed between two responses. In general, ths knd of errors tend to be accumulated and ncreased as the loadng drecton s changed frequently. However, the present errors occur durng the deformaton by H (1) and thereafter stll preserve almost constant even under the complex response by H (2). Ths means that the parameter dentfcaton was successfully mplemented and the dentfed parameters provde suffcent qualty for the macro-scale analyss wthn the scope of errors n Table Macro-scale and mcro-scale analyses The next step, namely Step (v), s to conduct the macroscopc analyss wth the assumed macroscopc consttutve

16 1214 Comput Mech (213) 52: (a) Fg. 1 Comparson of analyss result applyng dentfed materal parameters wth result of macro-scale analyss. (b) Fg. 9 Responses to the macro-scale deformaton patterns not realzed n NMTs: a macroscopc response of UC-1 when tensle loadng s appled n the Y 1 -drecton wth the Y 2 -drecton stress free and wth the Y 3 -drecton fxed; b relatonshps between the macroscopc axal stress components and the rght CG deformaton tensor components of UC-2 when tensle loadng s appled n each one of the normal axal drectons wth the sde lengths n the other two drectons fxed. model usng the materal parameters dentfed above. Below, a demonstraton s made only for the UC-1 of Fg. 3. We consder the macro-structure as shown n Fg. 11, whch also llustrates the support and loadng condtons. Here, the mcro-scale coordnate system O-Y 1 Y 2 Y 3 s dfferent from that used n the NMTs n Step () n ths example. More specfcally, the fber drecton parallel to the Y 3 - axs s rotated by 6 n the counterclockwse drecton wth respect to the X 2 axs that s dentcal wth the Y 2 axs, whle the Y 2 and Y 2 -axes are dentcal. Knowng that the macroscale coordnate system n Steps () and () concdes wth the mcro-scale one, we have to set the fber drecton at A ={1/2,, 3/2} T along wth B = for (33). After the macro-scale analyss, the localzaton analyss n Step (v) can be performed, f necessary, by usng the macroscopc deformaton hstores obtaned n the macroscale analyss n Step (v) at certan ponts of nterest n the macro-structure. In ths example, after selectng the center ponts of two representatve elements, Ponts A and B, ndcated n Fg. 11, we extract the tme-seres data of the macroscale dsplacement gradent tensors at these ponts. Before applyng the values to each unt cell to carry out the correspondng mcro-scale analyss for localzaton, the tensor components of the macro-scale dsplacement gradents have to be transformed to the values n the rotated macro-scale coordnate system O-X 1 X 2 X 3 whose axes conform wth the mcro-scale system O-Y 1 Y 2 Y 3. That s, we need to transform the tme seres data of H pq at Ponts A and B by the followng coordnate transformaton rule: H j = 3 p=1 q=1 3 T p H pq T jq (35) where T j are the components of the coordnate transformaton matrx defned for ths partcular example as cos θ snθ [T ]= 1 (36) sn θ cos θ wth θ beng set at 6. By applyng the components H j at Ponts A and B to the control nodes wth reference to (15) and (16), we carry out the mcro-scale analyses to evaluate the actual mcro-scale stress and stran n the correspondng unt cells. Fgure 12 shows the results of both the macro-scale analyss and the localzaton analyses assocated wth Ponts A and B for the loadng step 2/4 and 4/4. As can be seen from the fgure, each mcro-scale moton properly reflects

17 Comput Mech (213) 52: Fg. 11 Macroscopc problem: a FE model wth boundary condtons; b fber algnment. (a) (b) the correspondng macro-scale dsplacement gradent at the selected ponts of nterest n the macro-structure. Concretely speakng, the unt cell located at Pont A s domnated by a rgd-body rotaton wth shear deformaton manly due to the nclned fber, whle the unt cell at Pont B exhbts stretchng n the X 1 -drecton wth a relatvely hgh stress value but wthout severe a-rgd-body rotaton. The results carry the mplcaton of the so-called two-scale knematcs ntroduced n [34], and thus llustrate that the macro-scale materal behavor s consstent wth the mcro-scale structural response n the proposed method of two-scale analyss. By comparng the macro-scale stresses obtaned from the mcro- and macro-scale analyses, we can conduct the verfcaton of the results. In fact, the localzaton analyss for each selected pont enables us to compute the reacton forces at the control nodes (17) and n turn the macroscopc stress wth (18). It was confrmed that the macroscopc stress computed ths way was almost the same as that obtaned by the macroscale analyss, though the overlapped two stress stran curves are not shown here. Accordngly, t can be concluded that the proposed method of two-scale analyss wth the mcro macro decouplng scheme s relable enough to the extent of the adequacy of the assumed macroscopc consttutve model. 6 Concludng remarks Intendng to develop a multscale CAE system for composte materals, we have ntroduced a method of two-scale analyss by applyng the mcro-macro decouplng scheme under the assumpton that a functonal form of the macroscopc consttutve equaton s avalable. The key ngredent of the method s the numercal materal testng, whch corresponds to the homogenzaton process realzed by carryng out mcro-scale numercal analyses for perodc mcrostructures (unt cells) of composte materals. To be more specfc, assumng that the concrete functonal form of the macroscopc consttutve model s known, a seres of numercal materals tests (NMTs) s conducted on the numercal specmen,.e., the unt cell s FE model, to obtan the nonlnear macro-scale materal behavor. Ths has been successfully conducted thanks to the ntroducton of the extended system for the two-scale BVP, and the correspondng mcro-scale analyses have been realzed by a general-purpose FEM code by vrtue of the utlzaton of the control nodes located outsde the unt cell model. Then, the proposed tensor-based method of parameter dentfcaton enables us to determne the materal parameters n the assumed model by means of the measured data n the NMTs. Once the macro-scale materal behavor ftted wth the dentfed parameters s satsfactory to us, the macro-scale analyss can be performed, whch must be conducve n the macro-scale CAE. Moreover, as may be necessary, we are able to carry out the mcro-scale analyss to evaluate mcro-scale mechancal behavor of the unt cell assocated wth a macro-scale pont, by applyng the macroscale deformaton hstory at that pont to the control nodes. Ths fnal process suggests the possblty of the mcro-scale CAE. Takng an ansotropc hyperelastc consttutve model of fber-renforced compostes as an example of the assumed macroscopc materal behavor, we have demonstrated the potental and promse of the method. A foreseeable extenson of ths study would be to develop a CAE system that enables us to deal wth a varety of materal behavor arsng from arbtrary knds of mcrostructures. The bottleneck n development must come n the form of a relable macro-scale consttutve equaton, whch should be, deally, represent the actual macro-scale materal behavor. Therefore, the development of relevant consttutve mod-

18 1216 Comput Mech (213) 52: Fg. 12 Macro- and mcroscopc analyss results.