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AIAA JOURNAL V ol. 38, No. 12, December 2000 Micromechanics-Based Predictive Model for Compressively Loaded Angle-Ply Composite Laminates Jung Hyun Ahn, ¤ and Anthony M. Waas † University of Mic higan, Ann Ar bor , Mic higan 48109-2140 A micromechanics-based analysis to predict damage initiation in compressively loaded symmetric angle-ply laminates is described. The nite element method in conjunction with the commercial code Abaqus is used to solve the governing system of equations. The results obtained for the predictions are compared against a set of experime ntal r esults previously made availablefor AS4/3502 symmetr ic angle- ply laminates.A unie d model that captures damage initiation and that describes failure mode transition as a function of ply angle is reported. The predi ction of the model is fou nd to compare fa vorably against the exper imental data. I. Introducti on T HE response of composite laminates when subjected to me- chanical loads is inuenced by the material type and stacking sequence adopted. In order to exploit the multitude of benecial fact orsthat compos itelami nates have to off er, it is importantto gain a thorough underst andingof how the respon se to mec hanical loads and ultimate failure stress are related to the lami nate microst ruc- ture. The compressiv e strengt h of laminates is an impor tant design parame terfor aerospac estructuresmadeof compositelaminat es.In- deed, the probl em of predicti ng compr essi ve stre ngth has recei ved considera ble attention in the recent literature , as evidenced by the large number of research papersdiscussedin, for exampl e, the sur- vey papers by Waas and Schultheisz 1 and Schultheisz and Waas. 2 Pre vious experimentaland analyticalresults pertai ning to the static compre ssive strengt h of a lamin ate based on a unidire ctionalpoly- mer matr ix haveestablishedthatber micr obuckli ngin the presence of a nonline ar matrix is the domina nt mode of compressive failure in these laminates . Thes e result s have been thoroughly discuss ed by Budiansky and Fleck, 3 Kyriakides et al., 4 Sun and Jun, 5 and Schapery. 6,7 Similar investigationsthat account for time-dependent effe cts have been presente d by Sun and Thir uppukuzhi , 8 Weeks and Sun, 9 Woldesenbet and Vinson, 10 Hsi ao andDaniel, 11 Hsiao et al. , 12 Oguni and Ravichandr an, 13 and Lee and Waas. 14 The importanc e of establ ishingthe connecti onbetween the unidir ectionalcomp ressi ve strengt h and the compressive strengt h of a lami nate that has mul ti- directionalplies (wi th zer o plie s incl uded ) has recei vedlesser atten- tion, although Drapier et al., 15 Swanson, 16 Swain et al., 17 Lesko et al., 18 andXuetal., 19 haveintroducedmodelsthatcapturesomeofthe effe ctsof stacking .The situatio nbecomesless satisf actorywhe n one exami nes the compressiv e strength of angle-pl y lami nates(no zero plies) . Rotemand Has hin 20 andKim 21 conductedexperimentalstud- ies of angle-ply laminates and found that shear failure mechanisms were signi cant. Shua rt 22 conducteda carefuland systematicexper- imental study on the compression failure of (+h / ¡h ) ns laminates made of AS4/3502 epoxy . Shuar t was able to ident ify the tran sition of failur e betwe en ber micro buckling / kink banding (w hi ch he also call ed ber broo ming, bec ausesomet imes the laminat e brok e along the band thatwas formed nearthe boundaryof the loaded edge ), in- plane matr ix shearing, and matr ix compr ession.No uni fyingmodel was introduced to capture the different r egimes of failur e. Instead, different simplied analytical models were constructed to explain the different failure mechanisms, which change as a function of ply angle. However, Shuart present ed a complet e set of e xperimen- Received 18 January 2000; revision received 31 March 2000; accepted for publication 31 March 2000. Copyright c ° 2000 by Jung Hyun Ahn and Anthony M. Waas . Pub lished by the American Institute of Aeronautics and Astronautics, Inc., with permission. ¤ Res earch Fellow , Department of Aeros pace Engine ering; doolyii@ umich .edu. Member AIAA. † Professor, Department of Aerospace Engineering; [email protected]. As- sociate Fellow AIAA. tal data to show the different mechanisms of failure as well as the dependence of laminat e compre ssivestrength on the ply angle. In thepresentpaper,we repo rtthe resultsfroma mic romechanic s- based ni te ele ment anal ysi s thatwas usedto model the mic rost ruc- tural aspects of the response and failure of an angle-ply lamina te subject ed to compre ssionloads. This analysi s incorpor atesthe non- line ar cons tituti veresponseof the mat rix,which play s a cruc ialrole in the failure mechanism. Shuart’s experimental data for the lami- nate failure hav e been used here as a benchmark for the trends to be expected when the compressive failure of angle-ply laminates is invest igated,as well as for comparison against t he analysis pre- dictions. Good agreement between the model predictions and the experi mental results is found. II. Mechanical Model for Compres sive Response of an Angle-Ply Lamina The congurati on studied is as shown in Fig. 1, where an angle- ply laminate, (§h ) ns , is subjecte d to unifor m compressionloading. The ( x , y ) axes denote the lamina orthot ropic ax es, and the (1, 2) axes denote the ort hotropicmaterial axes. The approa ch take n here mirrors the earlier work of the authors, 23 in which a microregion situat ed in an area of intens e stress is mode led as an array of alter - nating laye rs of ber and matrix. In the present context, there is no preferntial location for the microregion, because each lamina is in a state of homogeneousdeformation. A microregion with dimension H £ W is of interes t for t he ni te element analysis. First, the compliances of a lamina in the x – y coordinate frame are related to the orthotropic compliances in the 1– 2 coordinateframe by S x x =m 4 S 11 + n 4 S 22 + 2 m 2 n 2 S 12 + m 2 n 2 S 66 S yy =n 4 S 11 + m 4 S 22 + 2 m 2 n 2 S 12 + m 2 n 2 S 66 S x y =m 2 n 2 S 11 + m 2 n 2 S 22 + (m 4 + n 4 ) S 12 ¡ m 2 n 2 S 66 (1) where m =cos(h ) and n =sin(h ). The orthotropic compliances S 11 , S 22 , and S 66 are computed by using the la mina proper tieslisted in Table 1, and the relations among lamina properties and com- pliances are as given in Hyer. 24 From Eq. (1) and with the use of classical lamination theory , the laminate engineering properties E x x , E yy , G x y , and m x y are computed. These values are shown in Table 2. When these values and a loadi ng condition of unit com- pressive stress in the x direction are used, the laminate strains are found from ² x x ² yy c x y = 1 / E x x ¡m x y / E x x 0 ¡m x y / E x x 1 / E yy 0 0 0 1 / G x y r xx =¡1 r yy =0 s x y =0 (2) Once the laminate strains in the x – y coordinate frame are ob- tained, it is necessary to transform the lami nate strains to the 1 – 2 2299
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