Transactions of the Korean Society of Mechanical Engineers A (대한기계학회논문집A)

The Korean Society of Mechanical Engineers (대한기계학회)
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Non-Local Plasticity Constitutive Relation for Particulate Composite Material Using Combined Back-Stress Model and Shear Band Formation

비국부 이론을 이용한 입자 강화 복합재 이중후방응력 소성 구성방정식 모델 및 전단밴드 분석

  • Yun, Su-Jin (Advanced Propulsion Technology Center, The 4th R&D Institute, Agency for Defense Development) ;
  • Kim, Shin Hoe (Advanced Propulsion Technology Center, The 4th R&D Institute, Agency for Defense Development) ;
  • Park, Jae-Beom (Advanced Propulsion Technology Center, The 4th R&D Institute, Agency for Defense Development) ;
  • Jung, Gyoo Dong (Advanced Propulsion Technology Center, The 4th R&D Institute, Agency for Defense Development)
  • 윤수진 (국방과학연구소 4본부 미래추진기술센터-5실) ;
  • 김신회 (국방과학연구소 4본부 미래추진기술센터-5실) ;
  • 박재범 (국방과학연구소 4본부 미래추진기술센터-5실) ;
  • 정규동 (국방과학연구소 4본부 미래추진기술센터-5실)
  • Received : 2014.03.22
  • Accepted : 2014.07.22
  • Published : 2014.10.01
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Abstract

This paper proposes elastic-plastic constitutive relations for a composite material with two phases-inclusion and matrix phases-using a homogenization scheme. A thermodynamic framework is employed to develop non-local plasticity constitutive relations, which are specifically represented in terms of the second-order gradient terms of the internal state variables. A combined two back-stress evolution equation is also established and the degradation of the state and internal variables is expressed by continuum damage mechanics in terms of the damage factor. Then, deformation localization is analyzed; the analysis results show that the proposed model yields a wide range of shear band formation behaviors depending on the evolution of the specific internal state variables. The analysis results also show good agreement with the results of simplified Rice instability analyses.

2개의 상으로 구성된 입자 강화 복합재에 대한 균질화와 내부 상태 변수에 대해 2차 미분항이 포함된 비구역적 이론을 적용하여 탄소성 구성 방정식을 제안하였다. 열역학과 소성 포텐셜을 통해 내부 상태 변수에 대한 전개식 또한 본 논문에 포함되었다. 연속체 결함 모델을 이용, 결함 인자에 따른 물성 저하 현상도 감안되었으며 이중 후방응력이 조합된 전개식 또한 제시하였다. 일부 예에 대한 수치해석 결과, 비구역적 변수의 영향이 증가할수록 전단밴드는 감소하나 반면 특정 후방응력 전개가 지배적일수록 소성변형 집중이 증가함이 관찰되었다. 더욱이 두 개의 강소성 상으로 이루어진 복합재의 경우 강성이 높은 게재물의 비중이 증가함에 따라 전단밴드 형성이 용이한 것으로 나타났다. 그 밖에 제어변수들의 변화에 따른 전단밴드 형성에 대한 분석 결과는 Rice 소성 불안정성 분석결과와 잘 일치함 또한 밝혀졌다.

Keywords

References

  1. Liu, X. and Hu, G., 2005, "A Continuum Micromechanical Theory of Overall Plasticity for Particulate Composites Including Particle Size Effect," Int. J. of Plas., Vol. 21, pp. 777-799. https://doi.org/10.1016/j.ijplas.2004.04.014
  2. Li, J. and Weng, G. J., 1998, "A Unified Approach from Elasticity to Viscoelasticity to Viscoplascity of Particle-Reinforced Solids," Int. J. Plas. Vol. 14, No. 1-3, pp. 193-208. https://doi.org/10.1016/S0749-6419(97)00048-X
  3. Coulibaly, M. and Sabar, H., 2011, "New Integral Formulation and Self-Consistent Modeling of Elastic-Viscoplastic Heterogeneous Materials," Int. J. Sol. Struc., Vol. 48, pp. 753-763. https://doi.org/10.1016/j.ijsolstr.2010.11.012
  4. Bardella, L., 2003, "An Extension of the Secant Method for the Homogenization of the Nonlinear Behavior of Composite Materials," Int. J. Eng. Sci., Vol. 41, pp. 741-768. https://doi.org/10.1016/S0020-7225(02)00276-8
  5. Pierard, O., Gonzalez, C., Segurado, J., LLorca, J. and Doghri, I., 2007 "Micromechanics of Elasto-Plastic Materials Reinforced with Ellipsoidal Inclusions," Int. J. Sol. Struc., Vol. 44, pp. 6945-6962. https://doi.org/10.1016/j.ijsolstr.2007.03.019
  6. Shen, L. and Yi, S., 2001, "An Effective Inclusion Model for Effective Moduli of Heterogeneous Materials with Ellipsoidal Inhomogeneities," Int. J. Sol. Struc., Vol. 38 pp. 5789-5805. https://doi.org/10.1016/S0020-7683(00)00370-X
  7. Xun, F., Hu, G. and Huang, Z., 2004, "Size-Dependence of Overall In-Plane Plasticity for Fiber Composites," Int. J. Sol. Struc., Vol. 41, pp. 4713-4730. https://doi.org/10.1016/j.ijsolstr.2004.02.063
  8. Mercier, S. and Molinari, A., 2009, "Homogenization of Elastic-Viscoplastic Heterogeneous Materials: Self-Consistent and Mori-Tanaka Schemes," Int. J. Plas., Vol. 25, pp. 1024-1048. https://doi.org/10.1016/j.ijplas.2008.08.006
  9. Mori, T. and Tanaka, K., 1973, "Average Sress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions," Acta Metal., Vol. 21, May, pp. 571-574. https://doi.org/10.1016/0001-6160(73)90064-3
  10. Sabar, H., Berveiller, M., Favier, V. and Berbenni, S., 2002, "A New Class of Micro-Macro Models for Elastic-Viscoplastic Heterogeneous Materials," Int. J. Sol. Struc., Vol. 39, pp. 3257-3276. https://doi.org/10.1016/S0020-7683(02)00256-1
  11. Mercier, S., Jacques, N. and Molinari, A., 2005, "Validation of an Interaction Law for the Eshelby Inclusion Problem in Elasto-Viscoplasticity," Int. J. Sol. Struc., Vol. 42, pp. 1923-1941. https://doi.org/10.1016/j.ijsolstr.2004.08.016
  12. Pensee, V. and He, Q.-C., 2007, "Generalized Self-Consistent Estimation of the Apparent Isotropic Elastic Moduli and Minimum Representative Volume Element Size of Heterogeneous Media," Int. J. Sol. Struc., Vol. 44, pp. 2225-2243. https://doi.org/10.1016/j.ijsolstr.2006.07.003
  13. Ramtani, S., Bui, H. Q. and Dirras, G., 2009, "A Revisited Generalized Self-Consistent Polycrystal Model Following an Incremental Small Strain Formulation and Including Grain-Size Distribution Effect," Int. J. Eng. Sci., Vol. 47, pp. 537-553. https://doi.org/10.1016/j.ijengsci.2008.09.005
  14. Berbenni, S., Favier, V. and Berveiller, M., 2007, "Impact of the Grain Size Distribution on the Yield Stress of Heterogeneous Materials," Int. J. Plas. Vol. 23, pp. 114-142. https://doi.org/10.1016/j.ijplas.2006.03.004
  15. Ponte Castaneda P., 1991, "The Effective Mechanical Properties of Nonlinear Isotropic Composites," J. Mech. Phys. Solids, Vol. 39. No. 1, pp. 45-71. https://doi.org/10.1016/0022-5096(91)90030-R
  16. Hutter, G., Linse, T., Muhlich, U. and Kuna, M., 2013, "Simulation of Ductile Crack Initiation and Propagation by Means of a Non-Local Gurson-Model," Int. J. Eng. Sci., Vol. 50, pp 662-671.
  17. Aravas, N. and Ponte Castaneda P., 2004, "Numerical Methods for Porous Metals with Deformation-Induced Anisotropy," Comput. Methods Appl. Mech. Engrg. Vol. 193, pp. 3767-3805. https://doi.org/10.1016/j.cma.2004.02.009
  18. Xu, F., Sofronis, P., Aravas, N. and Meyer, S., 2007, "Constitutive Modeling of Porous Viscoelastic Materials," European J. Mech. A/Solids, Vol. 26, pp. 936-955. https://doi.org/10.1016/j.euromechsol.2007.05.008
  19. Tohgo, K. and Itoh, T., 2005, "Elastic and Elastic-Plastic Singular Fields Around a Crack-Tip in Particulate-Reinforced Composites with Progressive Debonding Damage," Int. J. Sol. Struc., Vol. 42, pp. 6566-6585. https://doi.org/10.1016/j.ijsolstr.2005.04.013
  20. Li, C. and Ellyin, F., 2000, "A Mesomechanical Approach to Inhomogeneous Particulate Composite Undergoing Localized Damage: Part II Theory And Application," Int. J. Sol. Struc., Vol. 37, pp. 1389-1401. https://doi.org/10.1016/S0020-7683(98)00297-2
  21. Voyiadjis, G. Z. and Thiagarajan, G., 1997, "Micro and Macro Anisotropic Cyclic Damage-Plasticity Models for MMCS," Int. J. Eng. Sci., Vol. 35, No. 5, pp. 467-184. https://doi.org/10.1016/S0020-7225(96)00125-5
  22. Voyiadjis, G. Z. and Park, T., 1995, "Local and Interfacial Damage Analysis of Metal Matrix Composite," Int. J. Eng. Sci., Vol. 33, No. 11, pp. 1595-1621. https://doi.org/10.1016/0020-7225(95)00029-W
  23. Voyiadjis, G. Z. and Deliktas, B., 2009, "Mechanics of Strain Gradient Plasticity with Particular Reference to Decomposition of the State Variables into Energetic and Dissipative Components," Int. J. Eng. Sci., Vol. 47, pp. 1405-1423. https://doi.org/10.1016/j.ijengsci.2009.05.013
  24. Ganghoffer, J. F. and de Borst, R., 2000, "A New Framework in Nonlocal Mechanics," Int. J. Eng. Sci., Vol. 38, pp. 453-486. https://doi.org/10.1016/S0020-7225(99)00030-0
  25. Makowski, J., Stumpf, H. and Hackl, K., 2006, "The Fundamental Role of Nonlocal and Local Balance Laws of Material Forces in Finite Elastoplasticity and Damage Mechanics," Int. J. Sol. Struc., Vol. 43, pp. 3940-3959. https://doi.org/10.1016/j.ijsolstr.2005.04.038
  26. Buryachenko, V. A., 2011, "On Thermoelastostatics of Composites with Nonlocal Properties of Constituents I. General Representation for Effective Material and Field Parameters," Int. J. Sol. Struc., Vol. 48, pp. 1818-1828. https://doi.org/10.1016/j.ijsolstr.2011.02.023
  27. Polizzotto, C., Fuschi, P. and Pisano, A. A., 2004, "A Strain-Difference-Based Nonlocal Elasticity Model," Int. J. Sol. Struc., Vol. 41, 2383-2401 https://doi.org/10.1016/j.ijsolstr.2003.12.013
  28. Chung, P. W. and Namburu, R. R., 2003, "On a Formulation for a Multiscale Atomistic-Continuum Homogenization Method," Int. J. Sol. Struc., Vol. 40, pp. 2563-2588. https://doi.org/10.1016/S0020-7683(03)00058-1
  29. Stromberg, L., 2008, "A Special Case of Equivalence Between Nonlocal Plasticity and Gradient Plasticity in a One-Dimensional Formulation," Int. J. Eng. Sci., Vol. 46, pp. 835-841. https://doi.org/10.1016/j.ijengsci.2008.01.019
  30. Jirasek, M. and Rolshoven, S., 2003, "Comparison of Integral-Type Nonlocal Plasticity Models for Strain-Softening Materials," Int. J. Eng. Sci., Vol. 41, pp. 1553-1602. https://doi.org/10.1016/S0020-7225(03)00027-2
  31. Voyiadjis, G. Z. and Park, T., 1999, "Kinematics Description of Damage for Finite Strain Plasticity," Int. J. Eng. Sci., Vol. 56, Nos 4, pp. 483-511.
  32. Bonora, N., 1997, "A Nonlinear CDM Model for Ductile Failure," Engng. Fracture Mech., Vol. 58(1/2): pp. 11-28. https://doi.org/10.1016/S0013-7944(97)00074-X
  33. Engelen, R. A. B., Geers, M. G. D. and Baaijen, F. P. T., 2003, "Nonlocal Implicit Gradient-Enhanced Elasto-Plasticity for the Modelling of Softening Behavior, Int. J. of Plas., Vol. 19, pp. 403-433. https://doi.org/10.1016/S0749-6419(01)00042-0
  34. Kouznetsova, V. G., Geers, M. G. D. and Brekelmans, W. A. M., 2004, "Multi-Scale Second-Order Computational Homogenization of Multi-Phase Materials: A Nested Finite Element Solution Strategy," Compt. Meth. Appl. Mech. Engrg., Vol. 193, pp. 5525-5550. https://doi.org/10.1016/j.cma.2003.12.073
  35. Aifantis, E. C., 1992, "On the Role of Gradients in the Localization of Deformation and Fracture," Int. J. Engr. Sci., Vol. 30, pp. 1279-1299. https://doi.org/10.1016/0020-7225(92)90141-3
  36. Voyiadjis, G. Z., Al-Rub, R. A. and Palazotto, A. N., 2004, "Thermodynamic Framework for Coupling of Anisotropic Viscodamage for Dynamic Localiztion Problems Using Gradient Theory," Int. J. Plast., Vol. 20, pp. 981-1038. https://doi.org/10.1016/j.ijplas.2003.10.002
  37. Mroz, Z., Shrivastava, H. P. and Dubey, R. N., 1976, "A Non-Linear Hardening Model and Its Application to Cyclic Loading," Acta Mech., Vol. 25, pp. 51-61. https://doi.org/10.1007/BF01176929
  38. Phillips, A., Tang, J. L. and Ricciuti, M., 1974, "Some New Observation on Yield Surfaces," Acta Mech, Vol. 20 pp. 23-39. https://doi.org/10.1007/BF01374960
  39. Lee, E. H., 1969, "Elastic-Plastic Deformation at Finite Strains," J. Appl. Mech., Vol. 36, pp. 1-6. https://doi.org/10.1115/1.3564580
  40. Prawoto, Y., 2012, "How to Compute Plastic Zones of Heterogeneous Materials: A Simple Approach Using Classical Continuum and Fracture Mechanics," Int. J. Sol. Struc., Vol. 49, pp. 2195-2201. https://doi.org/10.1016/j.ijsolstr.2012.04.037
  41. Hui, T. and Oskay, C., 2013, "A Nonlocal Homogenization Model for Wave Dispersion in Dissipative Composite Materials," Int. J. Sol. Struc., Vol. 50, pp. 38-48. https://doi.org/10.1016/j.ijsolstr.2012.09.007