Research Papers

Systematic Integration of Finite Element Methods Into Multibody Dynamics Considering Hyperelasticity and Plasticity

[+] Author and Article Information
Graham Sanborn

FunctionBay, Inc.,
5F, Pangyo Seven Venture Valley 1 danji 2 dong,
625, Sampyeong-dong,
Bundang-gu, Seongnam-si,
Gyeonggi-do 463-400, South Korea
e-mail: gsanborn@functionbay.co.kr

Juhwan Choi

FunctionBay, Inc.,
5F, Pangyo Seven Venture Valley 1 danji 2 dong,
625, Sampyeong-dong,
Bundang-gu, Seongnam-si,
Gyeonggi-do 463-400, South Korea
e-mail: juhwan@functionbay.co.kr

Joon Shik Yoon

FunctionBay, Inc.,
5F, Pangyo Seven Venture Valley 1 danji 2 dong,
625, Sampyeong-dong,
Bundang-gu, Seongnam-si,
Gyeonggi-do 463-400, South Korea
e-mail: jueno@functionbay.co.kr

Sungsoo Rhim

Department of Mechanical Engineering,
KyungHee University,
1 Seochun-dong, Kihung-gu, YongIn-si,
Gyeonggi-do 449-701, South Korea
e-mail: ssrhim@khu.ac.kr

Jin Hwan Choi

Department of Mechanical Engineering,
KyungHee University,
1 Seochun-dong, Kihung-gu, YongIn-si,
Gyeonggi-do 449-701, South Korea
e-mail: jhchoi@khu.ac.kr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 31, 2013; final manuscript received April 30, 2014; published online July 11, 2014. Assoc. Editor: Javier Cuadrado.

J. Comput. Nonlinear Dynam 9(4), 041012 (Jul 11, 2014) (11 pages) Paper No: CND-13-1113; doi: 10.1115/1.4027580 History: Received May 31, 2013; Revised April 30, 2014

This study proposes a systematic extension of a multiflexible-body dynamics (MFBD) formulation that is based on a recursive formulation for rigid body dynamics. It is extended to include nonlinear plastic and hyperelastic material models for the flexible bodies. The flexible bodies in the existing MFBD formulation use a finite element formulation based on corotational elements. The rigid bodies and flexible bodies are coupled using the method of Lagrange multipliers. The extensions to add plasticity and hyperelasticity are outlined. A solid, brick-type element and a shell element are adapted from the literature for use with the plastic material, and a constant volume constraint is introduced to enforce the approximation of incompressibility with the hyperelastic materials. A brief overview of the MFBD formulation and the details required to extend the formulation to incorporate these nonlinear material models are presented. Numerical examples are presented to demonstrate the feasibility of the model.

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Jung, S. P., Park, T. W., and Chung, W. S., 2011, “Dynamic Analysis of Rubber-Like Material Using Absolute Nodal Coordinate Formulation Based on the Non-Linear Constitutive Law,” Nonlinear Dyn., 63(1–2), pp. 149–157. [CrossRef]
Maqueda, L. G., Mohamed, A. N. A., and Shabana, A. A., 2010, “Use of General Nonlinear Material Models in Beam Problems: Application to Belts and Rubber Chains,” ASME J. Comput. Nonlinear Dyn., 5(2), p. 021003. [CrossRef]
Betsch, P., and Sänger, N., 2009, “On the Use of Geometrically Exact Shells in a Conserving Framework for Flexible Multibody Dynamics,” Comput. Methods Appl. Mech. Eng., 198(17), pp. 1609–1630. [CrossRef]
Shabana, A. A., 2008, Computational Continuum Mechanics, Cambridge University Press, Cambridge, UK.
Gerstmayr, J., and Matikainen, M. K., 2006, “Analysis of Stress and Strain in the Absolute Nodal Coordinate Formulation With Nonlinear Material Behavior,” III European Conference on Computational Mechanics, Springer, The Netherlands, pp. 615–615.
Sugiyama, H., and Shabana, A. A., 2004, “Application of Plasticity Theory and Absolute Nodal Coordinate Formulation to Flexible Multibody System Dynamics,” J. Mech. Des., 126, pp. 478–487. [CrossRef]
Ibrahimbegovic, A., Mamouri, S, Taylor, R. L., and Chen, A. J., 2000, “Finite Element Method in Dynamics of Flexible Multibody Systems: Modeling of Holonomic Constraints and Energy Conserving Integration Schemes,” Multibody Syst. Dyn., 4(2–3), pp. 195–223. [CrossRef]
Goicolea, J. M., and Orden, G., 2000, “Dynamic Analysis of Rigid and Deformable Multibody Systems With Penalty Methods and Energy-Momentum Schemes,” J. Comput. Methods Appl. Mech. Eng., 188(4), pp. 789–804. [CrossRef]
Choi, J., Ryu, H. S., and Choi, J. H., 2009, “Multi Flexible Body Dynamics Using Incremental Finite Element Formulation,” Multibody Dynamics 2009, ECCOMAS Thematic Conference.
Bae, D. S., Han, J. M., and Yoo, H. H., 1999, “A Generalized Recursive Formulation for Constrained Mechanical System Dynamics,” Mech. Struct. Mach., 27(3), pp. 293–315. [CrossRef]
Bae, D. S., Han, J. M., and Choi, J. H., 2000, “An Implementation Method for Constrained Flexible Multibody Dynamics Using a Virtual Body and Joint,” Multibody Syst. Dyn., 4, pp. 297–315. [CrossRef]
Bae, D. S., Han, J. M., Choi, J. H., and YangS. M., 2001, “A Generalized Recursive Formulation for Constrained Flexible Multibody Dynamics,” Int. J. Numer. Methods Eng., 50, pp. 1841–1859. [CrossRef]
Simo, J. C., and Hughes, T. J. R., 1998, Computational Inelasticity, Springer-Verlag, New York.
Simo, J. C., and Rifai, M. S., 1990, “A Class of Mixed Assumed Strain Methods and the Method of Incompatible Modes,” Int. J. Numer. Methods Eng., 29(8), pp. 1595–1638. [CrossRef]
Dvorkin, E. N., and Bathe, K. J., 1984, “A Continuum Mechanics Based Four-Node Shell Element for General Non-Linear Analysis,” Eng. Comput., 1(1), pp. 77–88. [CrossRef]
Ogden, R. W., 1984, Non-Linear Elastic Deformations, Ellis Horwood, Chichester.
Bonet, J., and Wood, R. D., 1997, Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University Press, Cambridge, UK.
Gent, A. N., 2001, Engineering With Rubber, Carl Hanser Verlag, Munich.
Arruda, E. M., and Boyce, M. C., 1993, “A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials,” J. Mech. Phys. Solids, 41(2), pp. 389–412. [CrossRef]
García de Jalón, D. J., UndaJ., and AvelloA., 1986, “Natural Coordinates for the Computer Analysis of Multibody Systems,” Comput. Methods Appl. Mech. Eng., 56, pp. 309–327. [CrossRef]
WittenburgJ., 1977, Dynamics of Systems of Rigid Bodies, B. G. Teubner, Stuttgart.
Shabana, A. A., 1996, “An Absolute Nodal Coordinate Formulation for the Large Rotation and Deformation Analysis of Flexible Bodies,” Technical Report No. MBS96-1-UIC, University of Illinois at Chicago, Chicago, IL.
Hughes, T. J. R., Cottrell, J. A., and Bazilevs, Y., 2005, “Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement,” Comput. Methods Appl. Mech. Eng., 194(39–41), pp. 4135–4195. [CrossRef]
Bathe, K. J., 1996, Finite Element Procedures, Vol. 2, No. 3, Prentice-Hall, Englewood Cliffs, NJ.
Belytschko, T., Moran, B., and Liu, W. K., 2000, Nonlinear Finite Element Analysis for Continua and Structures, Wiley, West Sussex, England.
Geradin, M., and Cardona, A., 2001, Flexible Multibody Dynamics: A Finite Element Approach, John Wiley, West Sussex, England.
Zienkiewicz, O. C., and Taylor, R. L., 2011, The Finite Element Method (3 Volume-Set), Butterworth-Heinemann, Oxford, England.
Belytschko, T., and Hsieh, B. J., 1973, “Non-Linear Transient Finite Element Analysis With Convected Co-Ordinates,” Int. J. Numer. Methods Eng., 7, pp. 255–271. [CrossRef]
Crisfield, M. A., and Moita, G. F., 1996, “A Unified Co-Rotational Framework for Solids, Shells and Beams,” Int. J. Solids Struct., 33, pp. 2969–2992. [CrossRef]
Wasfy, T. M., and Noor, A. K., 2003, “Computational Strategies for Flexible Multibody Systems,” ASME Appl. Mech. Rev., 56(6), pp. 553–614. [CrossRef]
Bauchau, O. A., 2011, Flexible Multibody Dynamics, Vol. 176, Springer Science + Business Media, Netherlands.
Malvern, L. E., 1969, Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Inc., NJ.
Washizu, K., 1982, Variational Methods in Elasticity and Plasticity, 3rd ed., Pergamon Press, New York.
Treloar, L. R. G., 1975, The Physics of Rubber Elasticity, Oxford University Press Oxford, UK.
Hilber, H. M., Hughes, T. J. R., and Taylor, R. L., 1977, “Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics,” Earthquake Eng. Struct. Dyn., 5(3), pp. 283–292. [CrossRef]
Jay, L. O., and Negrut, D., 2007, “Extensions of the HHT-α Method to Differential-Algebraic Equations in Mechanics,” Electron. Trans. Numer. Anal., 26, pp. 190–208.
Chung, J., and Hulbert, G. M., 1993, “A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method,” ASME J. Appl. Mech., 60(2), pp. 371–375. [CrossRef]
Jay, L. O., and Negrut, D., 2008, “A Second Order Extension of the Generalized-α Method for Constrained Systems in Mechanics,” Multibody Dynamics, Springer, The Netherlands, pp. 143–158.
Newmark, N. M., 1959, “A Method of Computation for Structural Dynamics,” ASCE J. Eng. Mech., 85(EM3), pp. 67–94.
Negrut, D., Jay, L. O., and Khude, N., 2009, “A Discussion of Low-Order Numerical Integration Formulas for Rigid and Flexible Multibody Dynamics,” ASME J. Comput. Nonlinear Dyn., 4(2), p. 021008. [CrossRef]
Crisfield, M. A., 1993, Non-Linear Finite Element Analysis of Solids and Structures, John Wiley & Sons, West Sussex, England.
abaqus 6.9, 2009, abaqus/CAE User's Manual, Dassault Systèmes.
Choi, J., Rhim, S., and Choi, J. H., 2013, “A General Purpose Contact Algorithm Using a Compliance Contact Force Model for Rigid and Flexible Bodies of Complex Geometry,” Int. J. Non-Linear Mech., 53, pp. 13–23. [CrossRef]


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Fig. 1

Two contiguous rigid bodies

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Fig. 2

Example shell element reference frame rl and Al=[xl yl zl]

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Fig. 3

The interpolation points for the out-of-plane shear strains of the MITC4

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Fig. 8

The model used for verification of the hyperelastic material behavior

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Fig. 9

Total force applied to Force applied to the free end of the hyperelastic model

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Fig. 4

Cantilever model with plastic material and hexahedral element formulation. Traction load applied to free end. Displacement compared at point marked . Von Mises stress compared at point marked . (Model with 10 mm elements shown).

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Fig. 5

Distributed edge pressure load applied in the vertical direction to free end of cantilever plate model

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Fig. 6

Displacement of node at free end of cantilever, (a) time 0 to −2 sec, and (b) time 1.9 to −2.0 sec, for element sizes 2.5 and 5 mm, for abaqus (Ab) and the proposed formulation (Pr), for hexa-type solid elements

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Fig. 10

(a) x-displacement and (b) von Mises stress of proposed versus. abaqus model for a Mooney–Rivlin material

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Fig. 11

A test model including rigid bodies modeled with the recursive formulation and flexible bodies that use plasticity and hyperelasticity

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Fig. 7

von Mises stress of node near base of cantilever, (a) time 0 to −2 sec, and (b) time 1.9 to −2.0 sec, for element sizes 2.5 and 5 mm, for abaqus (Ab) and the proposed formulation (Pr), for hexa-type solid elements

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Fig. 12

Configuration of the mechanism model between times 0.0 and 3.0 sec

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Fig. 13

Magnitude of contact forces between bodies B and E and between bodies C and E

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Fig. 14

Equivalent von Mises strain for total and plastic strain components on the top (inside of curvature after deformation) and bottom (outside of curvature after plastic deformation) of body E at the point of maximum strain

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Fig. 15

von Mises stress at a node near the midpoint of body F (hyperelastic body)




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