Insttute of Structural Mechancs A comprehensve study: Boundary condtons for representatve volume elements (RVE) of compostes Srhar Kurukur A techncal report on homogenzaton technques
A comprehensve study: Boundary condtons for representatve volume elements (RVE) of compostes Abstract A comprehensve study has been carred out on the effect of dfferent types of boundary condtons mposed on the RVE to predct the effectve propertes of heterogeneous materals through the concept of homogenzaton. In ths study three types of boundary condtons were presented namely: dsplacementdfference perodc boundary condtons, homogeneous boundary condtons and prescrbed dsplacements boundary condtons. It has been realzed that, wth n numercal accuracy, the effectve propertes under perodc boundary condtons and prescrbed dsplacement boundary condtons are the same and t can easly be appled n FE analyss even when compared wth the perodc boundary condtons. It has been demonstrated that the homogeneous boundary condtons are not only over-constraned but they may also volate the tracton perodcty condtons. Further t s deduced that boundary tracton contnuty condtons can be guaranteed by the applcaton of the proposed dsplacement-dfference perodc boundary condtons and prescrbed boundary condtons. Illustratve examples are presented. 1. Introducton Composte materals are wdely used n advanced structures n astronautcs, automoble, marne, petrochemcal and many other ndustres due to ther superor propertes over conventonal engneerng materals. Consequently, predcton of the mechancal propertes of the compostes has been an actve research area for several decades. Except for the expermental studes, ether mcro- or macro mechancal methods are used to obtan the overall propertes of compostes. Mcromechancal method provdes overall behavor of the compostes from known propertes of ther consttuents (fber and matrx) through an analyss of a representatve volume element (RVE) or a unt-cell model (Aboud, 1991; Nemat-Nasser and Hor, 1993). In the macromechancal approach, on the other hand, the heterogeneous structure of the composte s replaced by a homogeneous medum wth ansotropc propertes. The advantage of the mcromechancal approach s not only the global propertes of the compostes but also varous mechansms such as damage ntaton and propagaton, can be studed through the analyss There are several mcromechancal methods used for the analyss and predcton of the overall behavor of composte materals. In partcular, upper and lower bounds for elastc modul have been derved usng energy varatonal prncples, and closed-form analytcal expressons have been obtaned (Hashn and Shtrkman, 1963; Hashn and Rosen, 1964). Unfortunately, the generalzaton of these methods to vscoelastc, elastoplastc and nonlnear compostes s very dffcult. Aboud (1991) has developed a unfed mcromechancal theory based on the study of nteractng perodc cells, and t was used to predct the overall behavor of composte materals both for elastc and nelastc consttuents. In hs work and many other references, homogeneous boundary condtons were appled to the RVE or unt cell models. In fact, the plane-remans- 1
plane s only vald for the symmetrc RVE subected to normal tractons. For a shear loadng case, many researchers, have ndcated that the plane-remans-plane boundary condtons are over-constraned boundary condtons. In ths artcle, a comprehensve study has been carred out on the effect of dfferent types of boundary condtons mposed on the RVE to predct the effectve propertes of heterogeneous materals through the concept of homogenzaton. In ths study three types of boundary condtons were presented namely: dsplacement-dfference perodc boundary condtons, homogeneous boundary condtons and prescrbed dsplacements boundary condtons. 2. Theoretcal background In ths work, three types of boundary condtons to be prescrbed on ndvdual volume element V are consdered namely: 1. Homogenous boundary condtons 2. Perodc boundary condtons 3. Prescrbed dsplacement boundary condtons. 2.1. Homogeneous boundary condtons Aboud (1991) appled on the surface of a homogeneous body wll produce a homogeneous feld there. Such boundary condtons are obtaned n the form: Knematc unform boundary condtons (KU):The dsplacement u s mposed at pont x belongng to the boundary S such that: u ( S) ε x, x S (1) ε s constant and symmetrcal second rank tensor that does not depend on x. Statc unform boundary condtons (SU): The tracton vector s prescrbed at the boundary: t ( S) σ n, x S (2) σ s a constant and symmetrcal second rank tensor ndependent of x. The vector normal to S at x s denoted by n. 2.2. Perodc boundary condtons Consder a perodc structure consstng of perodc array of repeated unt cells. The dsplacement feld for the perodc structure can be expressed as * u ( x1, x2, x3) ε x + u ( x1, x2, x3) (3) 2
In the above, s the global (average) stran tensor of the perodc structure and the frst term on the rght sde represents a lnear dstrbuted dsplacement feld. The second term on the rght sde, * u ε ( x1, x2, x3) s a perodc functon from one unt cell to another. It represents a modfcaton to the lnear dsplacement feld due to the heterogeneous structure of the compostes. Snce the perodc array of the repeated unt cells represents a contnuous physcal body, two contnutes must be satsfed at the boundares of the neghborng unt cells. One s that the dsplacements must be contnuous,.e., the adacent unt cells cannot be separated or ntrude nto each other at the boundares after the deformaton. The second condton mples that the tracton dstrbutons at the opposte parallel boundares of a unt cell must be the same. In ths manner, the ndvdual unt cell can thus be assembled as a physcally contnuous body. Obvously, the assumpton of dsplacement feld n the form of Eq. (3) meets the frst of the above requrements. Unfortunately, t cannot be drectly appled to the boundares snce the perodc part, * u ( x1, x2, x3) s generally unknown. For any unt cell, ts boundary surfaces must always appear n parallel pars, the dsplacements on a par of parallel opposte boundary surfaces can be wrtten as k + k u ε x + u (4) k + k k + k * u ε x + u (5) where ndces and dentfy the k th par of two opposte parallel boundary surfaces of a * repeated unt cell. Note that x, x, x ) s the same at the two parallel boundares (perodcty), u ( 1 2 3 therefore, the dfference between the above two equatons s u u * ε ( x x ) ε Δx (6) k + k k + k k k Snce Δx are constants for each par of the parallel boundary surfaces, wth specfed ε, the rght sde becomes constants and such equatons can be easly be appled n the fnte element analyss as nodal dsplacement constrant equatons. Eq. (6) s a specal type of dsplacement boundary condtons. Instead of gvng known values of boundary dsplacements, t specfes the dsplacement-dfferences between two opposte boundares. Obvously, the applcaton of t wll guarantee the contnuty of dsplacement feld. However, n general, such dsplacement-dfference boundary condtons, Eq. (6), may not be complete or may not guarantee the tracton contnuty condtons. The tracton contnuty condtons can be wrtten as k + k k σ σ, σ + k t σ (7) n n t k 3
σ where and are normal and shear stresses at the correspondng parallel boundary n σ t surfaces, respectvely. For general perodc boundary value problems the Eqs. (6) and (7) consttute a complete set of boundary condtons. In the followng llustratve examples, however, t has been proved that f unt cell s analyzed by usng a dsplacement-based fnte element method, the applcaton of only Eq. (6) can guarantee the unqueness of the soluton and thus Eq. (7) are automatcally satsfed. In other word, the latter boundary condtons are not necessary to be appled n the analyss. 2.3. Prescrbed dsplacement boundary condtons These boundary condtons have to be appled to the RVE n such a way that, except the strans n the drecton, n whch the effectve coeffcents have to be calculated, all other mechancal strans are zero. As a load, unform undrectonal dsplacement (prescrbed dsplacement condton) s appled. As an example: To fnd the effectve coeffcents and C the boundary condtons have to be C11 12 appled to the RVE n such a way that, except the stran n the X drecton, all other global strans are set to zero. As a load, unform undrectonal dsplacement (prescrbed dsplacement condton) s appled on the postve X-edge. The dsplacements n normal drecton on the postve and negatve Y-edges are constraned to be zero. So that the stran n the X-drecton s the only one, havng fnte value and all other strans are set to be zero. It has been observed that the effectve propertes obtaned under our prescrbed dsplacement boundary condtons and perodc boundary condtons are the same wth n numercal accuracy. 3. 3. Numercal homogenzaton usng RVE FEM has been extensvely used n the lterature to analyze unt cell, to determne the mechancal propertes and damage mechansms of compostes. In the present work the FEM mcromechancal analyss method s appled to perodc RVE. For smplcty, all the followng llustratve examples are 2 D plane stress state problems consdered. All fnte element calculatons have been carred out wth commercal FE program ANSYS. To apply the constrant equatons (6) n FEM, t s better to produce the same meshng at each two pared boundary surfaces. Then each constrant equaton n (6) contans only two dsplacement components of the pared nodes. The number of the constrant equatons s usually qute large, certan preprocessng program can be used to produce the data dependng on the ndvdual FEM code used. In all followng FEM analyses sx node plane stress elements are used wth small deformaton assumpton. The convergence of the solutons has been verfed by comparng the results wth dfferent meshng szes. 4
It s assumed that the average mechancal propertes of a RVE are equal to the average propertes of the partcular composte materal. The average stresses and strans n a RVE are defned by: In order to evaluate the effectve propertes ε 1 V σ 1 V C kl V V ε dv σ dv (8) (9), frst we should evaluate the average stress and average stran, from the Eqs. (8) and (9) wth the applcaton of dfferent boundary condtons, and then nsert them nto the consttutve relaton as follows: σ C kl (1) In numercal analyss of RVE, stress, stran have been taken from each and every element and multpled wth the volume of each element and then fnally compute the averaged stresses and strans over all elements. The stran energes predcted by the dfferent boundary condtons must satsfy the followng ε kl nequalty f the average stran ε for each case s assumed to be the same (Suquet, 1987; Hor and Nemat-Nasser, 1999; Hollster and Kkuch, 1992): t p whereu, U, U d U t p d U U (11) are the stran energy predcted by homogeneous tracton boundary condtons, perodc boundary condtons, and homogeneous dsplacement boundary condtons, respectvely. It s clear that the homogeneous dsplacement boundary condtons overestmate the effectve modul whereas the homogeneous tracton boundary condtons underestmate the effectve modul. It s also beng ponted out that the applcaton of the homogeneous dsplacement boundary condtons generally would not guarantee to produce perodc boundary tracton. Smlarly, the applcaton of the homogeneous tracton boundary condtons would not guarantee the dsplacement perodcty at the boundares. 4. Results and dscusson The results obtaned from the proposed dsplacement dfference perodc boundary condtons and prescrbed dsplacement boundary condtons, wth those obtaned by applyng homogeneous boundary condtons are compared. Consder the perodc structure as shown n Fg. 1. The volume fracton of the renforcng phase s 2 %. Assume both renforcng and matrx phases are elastc and ther materal constants are: E f 85MPa, ν. 25 and E m 28MPa, ν. 4 respectvely. f m 5
4.1. Illustratve example - I For the normal deformaton mode we apply the followng 3 dfferent sets of boundary condtons to the RVE model: (a) Perodc boundary condtons, Eq. (6): u u.5, v v, u B v B, (to elmnate the rgd body moton) (b) Homogeneous boundary condtons, Eq. (1): u ( S) ε x ε, where s the average stran. For the current example, the above equaton reduced to u.5x, v.5y u.5x, v.5y u.5x, v.5y.5y u.5x, v (c) Prescrbed dsplacement boundary condtons:.5, u v v u Where u and v are the dsplacement components along X andy, respectvely. Note that the orgn of the system s set at the pont B of the square RVE. In the case of perodc boundary condtons, one can confrm from the deformaton plot that dsplacement perodcty s satsfed, and from the stress dstrbutons that as shown n Fg.1, at the opposte parallel boundares the normal and shear stresses are the same. Thus t has been concluded that not only the dsplacements but also the stress dstrbutons along the boundares satsfy the perodcty condtons. Therefore, the average normal stress and average normal stran can be calculated from Eqs. (8) and (9) and then the resultng effectve coeffcents C & C 11 12 from Eq. (1) are: 451MPa and 167MPa, respectvely. It has also been seen from the FE smulatons that the effectve propertes under prescrbed dsplacement boundary condtons are the same, wth n numercal accuracy, as the one under perodc boundary condtons. 6
(a) Y X (b) Fg.1. FEM soluton of RVE by applyng perodc boundary condtons - Normal coeffcent: (a) deformaton ux ; (b) stress σ x (MPa); (c) stress τ xy (MPa) In the case of homogeneous dsplacement boundary condtons, t can be seen from deformaton plot that dsplacement perodcty s satsfed, but from the shear stress dstrbuton t can be seen that along the X-drecton, on postve and negatve boundares the shear stresses are not the same. Thus t can be mpled that tracton dstrbutons along the boundares does not satsfy the perodcty condtons,.e. Eq. (7) s not satsfed. Therefore, the average normal stress and average normal stran can be calculated from Eqs. (8) (c) and (9) and then the resultng effectve coeffcents 1665MPa, respectvely. C 11 & C 12 from Eq. (1) are: 4534MPa and 7
(a) Y X (b) Fg.2. FEM soluton of RVE by applyng homogeneous boundary condtons- Normal coeffcent: (a) deformaton ux ; (b) stress σ x (MPa); (c) stress τ xy (MPa) (c) 4.2. Illustratve example - II For the shear deformaton mode agan we apply the followng 3 dfferent sets of boundary condtons to the RVE model: (d) Perodc boundary condtons, Eq. (6): u u, v v. 5 u u.5, v v u B v B, (to elmnate the rgd body moton) 8
(e) Homogeneous boundary condtons, Eq. (1): u ( S) ε x ε, where s the average stran. For the current example, the above equaton reduced to u.5y, v.5x. 5 u.5y, v.5x u.5y, v. 5x u.5y, v. 5x (f) Prescrbed dsplacement boundary condtons: v v.5.5, u u.5.5 Where u and v are the dsplacement components along X andy, respectvely. Note that the orgn of the system s set at the pont B of the square RVE. The deformed shape for the case of perodc boundary condtons s shown n Fg. 3 (a). One notes that the boundares do not reman planes after the deformaton. Further assessment of the stress dstrbuton ndcates that at all opposte correspondng boundares the normal and shear stress are the same as shown n Fgs. 3 (b) and (c),.e. the RVE s subected to pure shear load. In addton, not only the dsplacements but also the stress dstrbutons along the boundares satsfy the perodcty condtons. Therefore, the average normal stress and average normal stran can be calculated from Eqs. (8) and (9) and then the resultng effectve coeffcent C 66 from Eq. (1) s: 139MPa. It has also been seen from the FE smulatons that the effectve propertes under prescrbed dsplacement boundary condtons are the same, wth n numercal accuracy, as the one under perodc boundary condtons. 9
(a) (b) (c) Fg.3. FEM soluton of RVE by applyng perodc boundary condtons - shear coeffcent: (a) deformaton ux ; (b) stress σ x (MPa); (c) stress τ xy (MPa) In contrast, Fg.4 shows the results by applyng the plane-remans-plane (homogeneous dsplacement) boundary condtons. The boundary lnes reman straght lnes. Therefore, the dsplacement perodcty s satsfed, but one can see that the normal stresses at the correspondng parallel boundares are not the same,.e. the tracton contnuty condtons, Eq. (7), s volated and therefore ths dstrbuton of stresses cannot represent the real one of physcally contnued perodc structure. Accordngly, t s clear that the homogeneous dsplacement boundary condtons are not approprate boundary condtons for the RVE of composte materals subected to a shear load. Therefore, the average normal stress and average normal stran can be calculated from Eqs. (8) and (9) and then the resultng effectve coeffcent C 66 from Eq. (1) s: 141MPa. We can see 1
that the homogeneous dsplacement boundary condton does overestmate the effectve coeffcent. (a) (b) Fg.4. FEM soluton of RVE by applyng homogeneous boundary condtons - shear coeffcent: (a) deformaton ux ; (b) stress σ x (MPa); (c) stress τ xy (MPa) (c) Conclusons: The followng conclusons have been drawn from the present study: A comprehensve study has been carred out on the effect of dfferent types of boundary condtons mposed on the RVE to predct the effectve propertes of heterogeneous materals through the concept of homogenzaton. In ths study three types of boundary condtons were presented namely: dsplacement-dfference perodc boundary condtons, homogeneous boundary condtons and prescrbed dsplacements boundary condtons. 11
The proposed explct form of dsplacement-dfference perodc boundary condtons for repettve unt cell model can easly be appled n FE analyss as a set of constrant equatons of nodal dsplacements of correspondng nodes on the opposte parallel boundary surfaces of the RVE. The applcaton of the perodc boundary condtons and prescrbed dsplacement boundary condtons can guarantee the dsplacement contnuty and tracton contnuty at the boundares of the RVE and as such t s the soluton for real perodc structure. It has also been seen from the FE smulatons that the effectve propertes under prescrbed dsplacement boundary condtons are the same, wth n numercal accuracy, as the one under dsplacement-dfference perodc boundary condtons and can easly be appled n FE analyss even when compared wth the perodc boundary condtons. The homogeneous boundary condtons (plane-reman-plane) are not only overconstraned condtons but they may also volate the stress perodcty. Ths type of boundary condtons shows a great dscrepancy n effectve propertes when applyng the shear loadng. References 1. Aboud, J., 199. Mcromechancal predcton of ntal and subsequent yeld surfaces of metal matrx compostes. Internatonal Journal of Plastcty 6, 134 141. 2. Aboud, J., 1991. Mechancs of Composte Materals, A Unfed Mcromechancal Approach. Elsever Scence Publshers, Amsterdam. 3. Hashn, Z., Shtrkman, S., 1963. A varatonal approach to the theory of elastc behavor of multphase materals. Journal of Mechancs and Physcs of Solds 11, 127 14. 4. Hashn, Z., Rosen, B.W., 1964. The elastc modul of fber-renforced materals. ASME Journal of Appled Mechancs 31, 223 232. 5. Nemat-Nasser, S., Hor, M., 1993. Mcromechancs: Overall Propertes of Heterogeneous Materals. Elsever Scence Publshers, Amsterdam. 6. Sun, C.T., Vadya, R.S., 1996. Predcton of composte propertes from a representatve volume element. Composte Scence and Technology 56, 171 179. 7. Xa, Z., Zhang, Y., Ellyn, F., 23. A unfed perodcal boundary condtons for representatve volume elements of compostes and applcatons. Internatonal Journal of Solds and Structures 4, 197 1921. 12