Available online at www.sciencedirect.com Procedia Engineering 60 ( 2013 ) 349 – 354 6th Asia-Pacific Congress on Sports Technology (APCST) 3D-CG based stress calculation of competitive swimwear using anisotropic hyperelastic model Akihiro Matsudaa*, Hiromu Tanabeb, Takeya Nagaokab, Motomu Nakashimac, Takatsugu Shimanad, Kazuhiro Omorid aFaculty of engineering, Information and systems, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8573, Japan bGraduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba, Ibaraki, Japan cGraduate school of engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8552, Japan dResearch and development dept., Mizuno Corporation, 1-12-35 Nanko-Kita, Suminoe-Ku, Osaka, 559-8510, Japan Received 20 March 2013; revised 6 May 2013; accepted 2 June 2013 Abstract A new design method for designing competitive swimsuits using 3-dimensional stress calculation was investigated in this paper. An anisotropic hyperelastic modeling of swimwear fabric was developed from the results of tensile loading tests. 3-dimensional stress distributions of swimwear in swimming motions were calculated by the anisotropic hyperelastic model. The displacement fields of 3D-CG human model which reproduce swimming motion of human body were applied to stress calculation. For the material modeling, the uniaxial cyclic tensile loading tests were conducted to obtain the mechanical characteristics of competitive swimsuits. From loading test results, the mechanical characteristics of swimwear fabrics show strong-anisotropy and the stiffness of the fabric shows hardening along with the increase of stretch. The cyclic tensile loading test results shows reduction of stiffness which depended on the maximum deformation previously reached in the history of the material. To take the strong anisotropy and stress reduction into account for the material modeling, a stress-softening model for anisotropic hyperelastic model using stiffness ratio of warp or weft was proposed. The results of the simulation showed good agreement with that of pressure test in a design range of competitive swimsuits. Finally, 3-dimensional stress distributions of swimwear were calculated by the proposed anisotropic hyperelastic model. A polygonal model of the swimwear was prepared and deformation of swimwear was adapted to the skin of 3D-CG human model in swimming motions. From the results, stress distributions were able to be visualized on the 3D-CG swimwear model. © 2013 The Authors. Published by Elsevier Ltd. Selection© 2013 Published and peer-review by Elsevier under responsibility Ltd. Selection of theand School peer-review of Aerospace, under Mechanicalresponsibility and ofManufacturing RMIT University Engineering, RMIT University Keywords: Anisotropic hyperelasticity; swimmear; stress calculation * Corresponding author. Tel.: +81-29-853-5031; fax: +81-29-853-5207. E-mail address: [email protected]. 1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University doi: 10.1016/j.proeng.2013.07.070 350 Akihiro Matsuda et al. / Procedia Engineering 60 ( 2013 ) 349 – 354 1. Introduction Competitive swimwear was designed to compress the human body by well stretched fabrics. Therefore, the vibration reduction of compressed human body in water would be expected. Also, well stretched swimwear is possible to provide better body support of resulting in better performance. The pressure and stress given by stretch of sportswear has been investigated from the two viewpoints, the exercise performance and the reduction on the body vibration. Aoki et al. [1] suggested that a compression from 10 to 20 mmHg may have a good effect on the exercise performance over an extended period. They also suggested that a compression of 40 mmHg under continuing exercise is not good as it gives a great burden to the heart function. Therefore it is important to design the compression swimsuits so that its pressure is always under 20 mmHg. The purpose of this study is to develop a new method of designing competitive swimwear using numerical modelling. The cyclic tensile loading tests were conducted to obtain the mechanical characteristics of competitive swimwear. To reproduce the mechanical characteristics of the swimwear, an anisotropic hyperelastic model considering stress softening was proposed. The material parameters of the anisotropic hyperelastic model for the 3-dimensional stress calculation were determined from cyclic tensile loading tests results. The 3-dimenasional stress calculation was conducted to show the applicability of our method to the design of swimwear. Displacement of swimwear was calculated from the 3D-CG human model. Sub- mesh of swimwear was applied to stress calculation for reduction of calculation time and computer resources. 2. Material testing and anisotropic modeling of swimsuits 2.1. Anisotropic hyperelastic model The second Piola-Kirchhoff stress tensor S of hyperelasticity is given by the partial differentiation of strain energy function W(C ) with respect to the right Cauchy-Green deformation tensor C as follows: ∂W ()C S = 2 (1) ∂C Where, the right Cauchy-Green deformation tensor C is calculated by CFF=t ⋅ (2) Here, F is the deformation gradient tensor and F t is the transpose of the deformation gradient tensor. A hyperelastic body is characterized by the strain energy function W(C ) . In the case of anisotropic material behavior, W(C ) reduces to an anisotropic function of C and the following structural tensor M )1( and M )2( [2]. (1)= (1) ⊗ (1) (2)= (2) ⊗ (2) M n n, M n n (3) Where, n()a is the unit vector which is oriented parallel to each fiber. In this study, a = 1 shows warp direction and a = 2 shows weft direction. W(C ) is represented as a function of the three principal invariants of C I CC:ICC:ICC= I = cof I = (4) 1(),()(),()2 3 det( ) and the first invariants of CM⋅ ()a and CM2⋅ (a ) , respectively. )1( =[] ⋅ )1( )1( = [2 ⋅ (1) ])2( = [] ⋅ )2( )2( = [2 ⋅ (2) ] I4 tr CM , I5 tr CM , I4 tr CM , I5 tr CM (5) Akihiro Matsuda et al. / Procedia Engineering 60 ( 2013 ) 349 – 354 351 where I4 and I5 are the anisotropic invariants that correspond to the second power of each fiber length. )1( )2( K1 and K1 are other invariants needed to satisfy the polyconvex condition which is required for stable iteration of numerical simulation[3]. K ()()a (,)CM a=IIII() a −()a + (6) 1 5 1 4 2 In this study, the strain energy function of anisotropic hyperelastic body was represented as the sum of the isotropic term W and anisotropic term W as follows: iso ani WW()=C + SI ()1( )W(,CM )1( )+ SI ()1( )W(,CM )2( ) (7) iso 4 max ani 4 max ani Here, SI( )1( ) and SI( )2( ) are softening function, I and I are the first and second invariants of 4 max 4 max 1 2 the volume preserve right Cauchy-Green deformation tensor as follows: = = − 3/1 = = − 3/1 I1 ()CC:IC:ICC:III3 ,()()()2 cof cof I3 C:I (8) − Here, cof ()C is given by CCT det( ) . The Mooney-Rivlin model was introduced to the isotropic part[4]: W()C =c ( I − 3) + c ( I − 3) + p ( J − 1) (9) iso 1 1 2 2 where c1 and c2 are material parameter and p is hydrostatic pressure, respectively. J is calculated as J = det(F) . For anisotropic part, following functions were introduced [5]: 2 ()a ()a ª ª c ()a º C ()a ()a 1 n3 ()a dn 4 k ()a dk ()a()() a a() a (a ) 2 W ani = {}(I )− 1+ {} (K )− 1 −{} (c + c − C)( I−)1 + Cln( J ) ] «¦« ()a 4 » ()a 1 13 23 k 4 k 4 ¬« n=1 ¬ dn4 ¼ d k (10) ()()n a ≥ ()()n a ≥ Here, ca3 , ck 0 and da4 , dk 1 . The relationship between nominal stress and nominal stretch is calculated theoretically using equations (7), (9) and (10). The mechanical characteristics of swimwear-fabrics show the effect of dependence on maximum elongation recorded during stretching [3]. Therefore, the materials lose the stiffness depending on the maximum elongation. We proposed a new softening model expressed by recorded maximum value of the anisotropic invariants as follows: )1( = − α[] −{} − γ())1( − )2( = − α[] −{} − γ())2( − S(I 4 max ) 11 1 exp 1I 4 max 1 ,SI (4 max ) 12 1 exp 2I 4 max 1 (11) α α ≥ γ γ where, 1, 2 0 1 and 2 are the softening parameters for fabric. 2.2. Cyclic loading tests Cyclic loading tests were conducted to evaluate the mechanical characteristics under the cyclic deformation. The test specimens were plane weave fiber made from the fabric which was same to competition swimwear developed by MIZUNO corp. The specimens are 120mm in length, 30mm in width and 0.2mm in thick (in Fig 1 (a)). The both ends of the specimen were fixed to the uniaxial loading machine (in Fig 1 (b)). The specimens, the fiber orientation angle θ which were 0°, 45° and 90° were prepared for the test. The maximum tensile displacements are 168mm, 180mm, 192mm, 204mm and 216mm which are correspond to the stretch of 140%, 150%, 160% and 180% along the specimen. 5- cyclic tensile deformation were applied to the specimens for each elongation. Speed of fixing jigs was set 352 Akihiro Matsuda et al. / Procedia Engineering 60 ( 2013 ) 349 – 354 as 1mm/sec. Fig 2 shows cyclic loading test and identification result of material parameters. Theoretical stress-strain relationships were calculated by original computer program which was developed by FORTRAN compiler. From Fig 2, the theoretical calculations are in good agreement with experimental data. Tensile direction Tensile direction direction Tensile 30mm 80mm 120mm 80mm Fig.1 (a) Geometry of specimen for cyclic loading test; (b) Fixture and specimen of cycling loading test 3 10 Experiment Theory Experiment Theory 2.5 8 2 6 1.5 4 1 Nominal Stress [MPa] Stress Nominal [MPa] Stress Nominal 2 0.5 0 0 1 1.2 1.4 1.6 1.8 1 1.2 1.4 1.6 1.8 Nominal Stretch Nominal Stretch Fig.