3D-CG Based Stress Calculation of Competitive Swimwear Using Anisotropic Hyperelastic Model

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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 C)(W 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 t ⋅= FFC (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 C)(W . In the case of anisotropic material behavior, C)(W reduces to an anisotropic function of C and the following structural tensor M )1( and M )2( [2]. ⊗= ⊗= )2()2()2()1()1()1( , nnMnnM (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. C)(W is represented as a function of the three principal invariants of C I = I = cof I = CCI:CCI:CC (4) 1 2 3 )(,)()(,)( det( ) and the first invariants of ⋅ MC a)( and ⋅ MC a)(2 , respectively. )1( []⋅= )1( )1( [⋅= )1(2 ])2( []⋅= )2( )2( [⋅= )2(2 ] I4 tr MC , I5 tr MC , I4 tr MC , I5 tr MC (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 MC )()( ),( aa )( a)( +−= IIII (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 C += IS )1( MC )1( + IS )1( MC )2( ),(W)(),(W)()(WW (7) iso 4 max ani 4 max ani Here, (IS )1( ) and (IS )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 )( 3 II 2 cof I:CCI:CI:CC cof I3 )()()(, I:C (8) − Here, cof C)( is given by T det(CC ) . The Mooney-Rivlin model was introduced to the isotropic part[4]: −+−+−= (9) iso C 11 22 JpIcIc )1()3()3()(W

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 )( aa )()( aa )( a 2)( W ani = {}I − + {}K {}(1)(1)( −+−− )( )1 +− ln(JCICcc ) ] «¦« a)( 4 » a)( 1 13 23 k 4 k 4 ¬« n=1 ¬ dn4 ¼ d k (10) n a)()( ≥ n a)()( ≥ Here, a3 cc k 0, and a4 dd k 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( []{}())1( −γ−−α−= )2( []{}())2( −γ−−α−= S I 4 max 1 11)( exp I 41 max IS 4 max 2 11)(,1 exp I 42 max 1 (11)

αα ≥ γ γ where, 21 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 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. 2. Comparison of cyclic loading test and theoretical calculation by nominal stress, (a) ș =0°;(b) ș =90°

2.3. 3-Dimensional stress calculation of swimwear

3-dimentional stress calculation of swimwear was conducted to calculate the effect of mechanical characteristics of swimwear on the human body. We prepare the polygonal model of swimwear and adopt all of node position of sub-mesh to the 3D-CG of human body. The polygonal model of swimwear and 3D-CG human body is shown in Fig. 3. The 3D-CG of human body was developed by MIZUNO Corporation using the 3D-CG software Autodesk Maya. 57400 nodes are applied to the 3D-CG of human body so that surface elongation of CG corresponds to some subjects who were selected from engineering staffs. 680 nodes and 481 elements were applied to the polygonal model of swimwear. Also swimwear was supposed to be stretched to 120% in vertical direction and 150% in horizontal direction by wearing the swimwear, initially. The swimwear was supposed to be fixed to the human body, because the swimwear was supposed to be very tight to human body. Therefore, the friction between swimwear and the human body was not considered in this paper. Deformations of swimwear were calculated from Akihiro Matsuda et al. / Procedia Engineering 60 ( 2013 ) 349 – 354 353 swimming motion of 3D-CG human model which was pretend the four swimming motions (crawl, breaststroke, backstroke and butterfly). A snapshot of 3D-CG model of crawl and deformation of swimwear were shown in Fig. 4. The Mises-stress distributions of swimwear to swim crawl were shown in Fig. 5. The Mises stresses were calculated using displacement of polygonal model of swimwear. High stress areas on buttocks and front thigh were observed.

3. Conclusion

3-dimentional stress calculation of polygonal model of swimwear using 3D-CG of human model was investigated in this paper. Stress response of swimwear was calculated by anisotropic hyperelastic model considering stress softening of the swimwear fabric. The anisotropic hyperelastic model considering stress softening was newly proposed in this paper. The stress softening function was formulated to decrease as the recorded maximum elongation increased. Material parameters of the anisotropic hyperelastic model and the stress softening model were approximated from the cyclic tensile loading test results. The swimming motions of 3D-CG model which represent human body motions were applied to stress calculation of polygonal model of swimwear. From the calculated results, stress distributions of swimwear were possible to be visualized. From the result of stress calculation, relationships between stress in the swimwear and swimming motion were possible to investigate.

Fig.3 (a) Polygonal model of swimwear; (b) 3D-CG human model. 354 Akihiro Matsuda et al. / Procedia Engineering 60 ( 2013 ) 349 – 354

Fig.4 (a) 3D-CG of crawl motion with polygonal model of swimwear; (b) Deformation of polygonal model.

3.5 Right half Left half 3 Base

2.5

1.5

Cauchy Stress[MPa] 1

0.5

0 5 1015202530354045 Frame No.

Fig. 5 (a) Snapshot of the Mises stress distribution of polygonal swimwear model in crawl; (b) Relationships between maximum Cauchy stress and frame number of 3D-CG calculation

References

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