THE INEQUALITY OF VIRTUAL WORK FOR DISSIPATIVE SYSTEMS AND ITS APPLICATIONS

Feng Ye; Li Jie

doi:10.6052/0459-1879-23-006

Volume 55 Issue 4

Apr. 2023

Turn off MathJax

Article Contents

Abstract

References

Chinese Journal of Theoretical and Applied Mechanics > 2023 > 55(4): 895-902. > DOI: 10.6052/0459-1879-23-006 CSTR: 32045.14.0459-1879-23-006

Feng Ye, Li Jie. The inequality of virtual work for dissipative systems and its applications. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(4): 895-902. DOI: 10.6052/0459-1879-23-006

Citation:

PDF (2233 KB)

THE INEQUALITY OF VIRTUAL WORK FOR DISSIPATIVE SYSTEMS AND ITS APPLICATIONS

Feng Ye,
Li Jie^,

College of Civil Engineering, Tongji University, Shanghai 200092, China

Received Date: January 02, 2023
Accepted Date: March 10, 2023
Available Online: March 11, 2023

Graphical Abstract

Abstract

Abstract

For conservative systems, energy variational principles provide a concise way to derive the governing equations for mechanical systems. For a dissipative system, the establishment of governing equations often requires the introduction of empirical or rational assumptions, which increases the difficulty of modeling. For dissipative systems, this paper introduces the concept of local stability of mechanical systems. Based on this stability concept, we propose a variational inequality of virtual work. This inequality in fact reveals a virtual work principle for dissipative systems. The physical meaning of this principle is that the necessary condition for the system state to be stable is that the potential energy released by the system on all possible virtual paths near the state is not greater than the energy that may be dissipated in the system. This study shows that it is sufficient to only use the principle of inequality of virtual work together with the sound principle of conservation of energy to derive all the governing equations for the mechanical state variables of a quasi-static system. As an application, this paper revisits plastic mechanics and derives all the governing equations of classical plastic mechanics by combining the principle of inequality of virtual work and the principle of conservation of energy. We prove that the classical principle of maximum plastic dissipation can be derived as a corollary of the principle of inequality of virtual work. Meanwhile, this paper revisits the Mohr-Coulomb strength criterion as an example to show the application of the principle of inequality of virtual work in the strength theory. It is shown that the stress-based strength criterion can be a corollary of the energy-based criterion for system stability. The above examples illustrate the wide applicability of the principle of inequality of virtual work and verify its effectiveness in establishing the governing equations of dissipative systems.
- variational principle,
- variational inequality,
- virtual work,
- plastic mechanics,
- Mohr-Coulomb criterion

FullText(HTML)

References (31)

References

[1]	钱伟长. 变分法及有限元. 北京: 科学出版社, 1980 Qian Weichang. Variational Method and Finite Element Mehod. Beijing: Science Press, 1980 (in Chinese)
[2]	胡海昌. 弹性力学的变分原理及其应用. 北京: 科学出版社, 1981 Hu Haichang. Variational Methods in Elastic Mechanics and Its Application. Beijing: Science Press, 1981 (in Chinese))
[3]	鹫津久一郎. 弹性和塑性力学中的变分法. 北京: 科学出版社, 1984 Washizu Kyūichirō. Variational Methods in Elastic and Plasctic Mechanics. Beijing: Science Press, 1984 (in Chinese))
[4]	Zienkiewicz OC, Taylor RL, Zhu JZ. The Finite Element Method: Its Basis and Fundamentals. Elsevier, 2005
[5]	李政道. 李政道讲义: 统计力学. 上海: 上海科学技术出版社, 2006 Li Zhengdao. Lectures of Li Zhengdao: Statistical Mechanics. Shanghai: Shanghai Science and Technology Press, 2006 (in Chinese)
[6]	Kardar M. Statistical Physics of Particles. World Publishing Corporation, 2010
[7]	Hughes TJR. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Courier Corporation, 2012
[8]	Francfort G, Marigo JJ. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 1998, 46(8): 1319-1342 doi: 10.1016/S0022-5096(98)00034-9
[9]	Bourdin B, Francfort GA, Marigo JJ. Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids, 2000, 48(4): 797-826 doi: 10.1016/S0022-5096(99)00028-9
[10]	Wu JY, Nguyen VP, Nguyen CT, et al. Phase-field modeling of fracture. Advances in Applied Mechanics, 2020, 53: 1-183
[11]	吴建营. 固体结构损伤破坏统一相场理论、算法和应用. 力学学报, 2021, 53(2): 301-329 (Wu Jianying. The unified phase field theory, algorithm and application of damage to solid structures. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(2): 301-329 (in Chinese)
[12]	Wu JY, Nguyen VP, Zhou H, et al. A variationally consistent phase-field anisotropic damage model for fracture. Computer Methods in Applied Mechanics and Engineering, 2020, 358: 112629 doi: 10.1016/j.cma.2019.112629
[13]	Feng Y, Fan J, Li J. Endowing explicit cohesive laws to the phase-field fracture theory. Journal of the Mechanics and Physics of Solids, 2021, 152: 104464
[14]	Feng Y, Li J. A unified regularized variational cohesive fracture theory with directional energy decomposition. International Journal of Engineering Science, 2023, 182: 103773
[15]	Feng Y, Li J. Phase-field method with additional dissipation force for mixed-mode cohesive fracture. Journal of the Mechanics and Physics of Solids, 2022, 159: 104693 doi: 10.1016/j.jmps.2021.104693
[16]	Feng Y, Li J. Phase-field cohesive fracture theory: A unified framework for dissipative systems based on variational inequality of virtual works. Journal of the Mechanics and Physics of Solids, 2022, 159: 104737 doi: 10.1016/j.jmps.2021.104737
[17]	Lubliner J. A maximum-dissipation principle in generalized plasticity. Acta Mechanica, 1984, 52(3): 225-237
[18]	Lubliner J. Plasticity Theory. Courier Corporation, 2008
[19]	Simo JC, Hughes TJR. Computational Inelasticity. Springer Science & Business Media, 2006
[20]	Erdogan F, Sih G. On the crack extension in plates under plane loading and transverse shear. Journal of Fluids Engineeing, 1963, 85(4): 519-525
[21]	Gol'dstein RV, Salganik R. Brittle fracture of solids with arbitrary cracks. International Journal Fracture, 1974, 10: 507-523
[22]	Sih G, Macdonald B. Fracture mechanics applied to engineering problems-strain energy density fracture criterion. Engineering Fracture Mechanics, 1974, 6(2): 361-386 doi: 10.1016/0013-7944(74)90033-2
[23]	Ulloa J, Alessi R, Wambacg J, et al. On the variational modeling of non-associative plasticity. International Journal of Solids and Structures, 2021, 217: 272-296
[24]	Ortiz M, Stainier L. The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering, 1999, 171(3-4): 419-444
[25]	李杰, 吴建营, 陈建兵. 混凝土随机损伤力学. 北京: 科学出版社, 2014 Li Jie, Wu Jianying, Chen Jianbing. Stochastic Damage Mechanics of Concrete Structures. Beijing: Science Press, 2014 (in Chinese)
[26]	Simo JC, Ju JW. Strain-and stress-based continuum damage models—I. Formulation. International Journal of Solids and Structures, 1987, 23(7): 821-840 doi: 10.1016/0020-7683(87)90083-7
[27]	Maugin GA. The Thermomechanics of Plasticity and Fracture. Cambridge University Press, 1992
[28]	蒋明镜, 彭立才, 朱合华等. 珠海海积软土剪切带微观结构试验研究. 岩土力学, 2010, 31(7): 2017-2023 (Jiang Mingjing, Peng Licai, Zhu Hehua, et al. Experimental study on microstructure of shear zone of Zhuhai Marine soft soil. Rock and Soil Mechanics, 2010, 31(7): 2017-2023 (in Chinese)
[29]	Dugdale D. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids, 1960, 8(2): 100-104
[30]	Barenblatt GI. The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics, 1962, 7(1): 55-129
[31]	Camacho GT, Ortiz M. Computational modelling of impact damage in brittle materials. International Journal of Solids and Structures, 1996, 33(20-22): 2899-2938

[1]	Wan Zheng, Meng Da. ANISOTROPIC STRENGTH CRITERION BASED ON T CRITERION AND THE TRANSFORMATION STRESS METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(5): 1519-1537. DOI: 10.6052/0459-1879-20-138
[2]	Wan Zheng, Song Chenchen, Meng Da. A NONLINEAR STRENGTH CRITERION AND TRANSFORMATION STRESS METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(4): 1210-1222. DOI: 10.6052/0459-1879-19-039
[3]	Mingqing You. STUDY ON SHEAR STRENGTH OF ROCKS USING THE EXPONENTIAL CRITERION IN MOHR'S STRESS SPACE[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(2): 607-619. DOI: 10.6052/0459-1879-18-291
[4]	Wan Zheng, Qiu Rendong, Guo Jinxue. A KIND OF STRENGTH AND YIELD CRITERION FOR GEOMATERIALS AND ITS TRANSFORMATION STRESS METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(3): 726-740. DOI: 10.6052/0459-1879-16-297
[5]	Wang Guosheng, Lu Dechun, Du Xiuli, Li Meng. DYNAMIC MULTIAXIAL STRENGTH CRITERION FOR CONCRETE DEVELOPED BASED ON THE S CRITEIRON[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(3): 636-653. DOI: 10.6052/0459-1879-15-433
[6]	Li Chunguang, Zhu Yufei, Liu Feng, Deng Qin, Zheng Hong. A NEW LINEARIZATION METHOD OF MOHR-COULOMB YIELD SURFACE FOR LOWER BOUND PROBLEMS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(2): 245-250. DOI: 10.6052/0459-1879-12-187
[7]	Yueqiang Qian, Xinming QIU, Xiong Zhang. Analysis of yield surface and crack tip plastic zone of Kagome lattice structure[J]. Chinese Journal of Theoretical and Applied Mechanics, 2008, 40(6): 826-833. DOI: 10.6052/0459-1879-2008-6-2007-109
[8]	The representation of failure criterions in generalized mohr space and its applications[J]. Chinese Journal of Theoretical and Applied Mechanics, 2005, 37(3): 368-373. DOI: 10.6052/0459-1879-2005-3-2004-251
[9]	SMOOTH RIDGE MODEL OF GENERALIZED TWIN SHEAR STRESS CRITERION[J]. Chinese Journal of Theoretical and Applied Mechanics, 1990, 22(2): 213-216. DOI: 10.6052/0459-1879-1990-2-1995-935

Cited By

Cited by

Periodical cited type(0)

Other cited types(1)

Get Citation

PDF

XML

Article Metrics

Article views (778) PDF downloads (141) Cited by(1)

THE INEQUALITY OF VIRTUAL WORK FOR DISSIPATIVE SYSTEMS AND ITS APPLICATIONS

Abstract

References

Related Articles

Cited by

Periodical cited type(0)

Other cited types(1)

Catalog

Article Metrics

Related

Download Center

Author Center

THE INEQUALITY OF VIRTUAL WORK FOR DISSIPATIVE SYSTEMS AND ITS APPLICATIONS

Abstract

References

Related Articles

Cited by

Periodical cited type(0)

Other cited types(1)

Catalog

Article Metrics

Related

Download Center

Author Center

Export File

Citation

Format

Content