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finite Deformation, finite strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids with Rheology

Applied Mathematics 2024年第1期15卷 108-168页

作者： Karan S. Surana Sri Sai Charan Mathi Department of Mechanical Engineering University of Kansas Lawrence US

This paper presents a mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and ordered rate constitutive theories in Lagrangian description derived using entropy inequality and the representation theorem for thermoviscoelastic solids (TVES) with rheology. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics and are based on contravariant deviatoric second Piola-Kirchhoff stress tensor and its work conjugate covariant Green’s strain tensor and their material derivatives of up to order m and n respectively. All published works on nonlinear dynamics of TVES with rheology are mostly based on phenomenological mathematical models. In rare instances, some aspects of CBL are used but are incorrectly altered to obtain mass, stiffness and damping matrices using space-time decoupled approaches. In the work presented in this paper, we show that this is not possible using CBL of CCM for TVES with rheology. Thus, the mathematical models used currently in the published works are not the correct description of the physics of nonlinear dynamics of TVES with rheology. The mathematical model used in the present work is strictly based on the CBL of CCM and is thermodynamically and mathematically consistent and the space-time coupled finite element methodology used in this work is unconditionally stable and provides solutions with desired accuracy and is ideally suited for nonlinear dynamics of TVES with memory. The work in this paper is the first presentation of a mathematical model strictly based on CBL of CCM and the solution of the mathematical model is obtained using unconditionally stable space-time coupled computational methodology that provides control over the errors in the evolution. Both space-time coupled and space-time decoupled finite element formulations are considered for obtaining solutions of the IVPs described by the mathematical model and are presented i

关键词： Thermoviscoelasticity Rheology Memory finite strain finite Deformation Nonlinear Dynamics Dynamic Bifurcation Ordered Rate Theories

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finite Deformation, finite strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids

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Applied Mathematics 2023年第12期14卷 773-838页

作者： Karan S. Surana Sri Sai Charan Mathi Department of Mechanical Engineering Univeristy of Kansas Lawrence KS USA

This paper presents the mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and the constitutive theories derived using entropy inequality and representation theorem for thermoviscoelastic solids (TVES) matter without memory. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics. This mathematical model is thermodynamically and mathematically consistent and is ideally suited to study nonlinear dynamics of TVES and dynamic bifurcation and is used in the work presented in this paper. The finite element formulations are constructed for obtaining the solution of the initial value problems (IVPs) described by the mathematical models. Both space-time coupled as well as space-time decoupled finite element methods are considered for obtaining solutions of the IVPs. Space-time coupled finite element formulations based on space-time residual functional (STRF) that yield space-time variationally consistent space-time integral forms are considered. This approach ensures unconditional stability of the computations during the entire evolution. In the space-time decoupled finite element method based on Galerkin method with weak form for spatial discretization, the solutions of nonlinear ODEs in time resulting from the decoupling of space and time are obtained using Newmark linear acceleration method. Newton’s linear method is used to obtain converged solution for the nonlinear system of algebraic equations at each time step in the Newmark method. The different aspects of the deformation physics leading to the factors that influence nonlinear dynamic response and dynamic bifurcation are established using the proposed mathematical model, the solution method and their validity is demonstrated through model problem studies presented in this paper. Energy methods and superposition techniques in any form including those used in obtaining solutions are neither advocated nor

关键词： Thermodynamic Consistency Dynamic Bifurcation Static Bifurcation Nonlinear Formulation finite strain finite Deformation Thermoviscoelastic Classical Continuum Mechanics Conservation and Balance Laws Nonlinear Damping

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Magnetic fabrics in fault-related fold and its relation with finite strain: an example from Mingjiang thrust structures in Western Sichuan

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Progress in Natural Science:Materials International 2007年第5期17卷 551-558页

作者： Jia Dong, Chen Zhuxin, Luo Liang, Hu Qianwei and Jia Qiupeng (Department of Earth Sciences, Nanjing University, Nanjing 210093, China) Department of Earth Sciences Nanjing University Nanjing 210093 China

The anisotropy of magnetic susceptibility (AMS) is a quick, effective and sensitive technique used to measure the weakly deformed sedimentary rocks, and also a reliable method to reveal the deforming mechanisms of fault-related folds. In Longmenshan front belt, a typical cross-section of fault-related folds is chosen to study the AMS. A total of 224 oriented specimens have been drilled at 23 different sampling sites which were distributed at the key structural positions of this structural section developed in the Xujiahe formation of the upper Triassic. Six elementary types of magnetic fabrics are recognized and established through this AMS study: Da sedimentary fabric; 2) an initial deformation fabric; 3) a pencil structure fabric; 4) a weak cleavage fabric; 5) a strong cleavage fabric; 6) a stretching lineation fabric. It has been found that most of magnetic fabrics are characterized by fabrics of weak deformation which belong to the pure-shear results of a pre-folding layer parallel shortening (LPS). In the fault-bend fold, almost all magnetic fabrics are the initial deformation fabrics of weak deformation, and denote that the deformation in the forelimb is stronger than that in the backlimb and no finite strain is shown in the footwall. While in the fault-propagation fold, finite strains are concentrated in the trishear zone where magnetic fabric results are approximately consistent with the estimated consequences of the kinematic model. The tectonic stress field indicated by the magnetic fabrics is basically the same along the whole structural section and shows a NW to SE compression and shortening which is accordant with the regional compressive stress field of the Longmenshan fold-thrust belt.

关键词： magnetic fabric finite strain fault-related fold trishear zone Longmenshan.

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Structural Control of the Getang Carlin-Type Gold Deposit in Southwest China

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Journal of Earth Science 2024年第2期35卷 536-552页

作者： Guoping Zeng Xinlu Hu Diwei Luo Jianzhong Liu Shuzhen Yao Yongjie Jin Wuhan Center China Geological SurveyWuhan430205China Central South China Innovation Center for Geosciences Wuhan430205China Faculty of Earth Resources China University of GeosciencesWuhan430074China Geological Party 105 Guizhou Bureau of Geology and Mineral Exploration&DevelopmentGuiyang550018China

The Getang is a representative Carlin-type gold deposit in Southwest *** has a proven reserve of about 30 tonnes at an average grade of 5.1 g/t *** orebodies occur as strat-abound lenses,and are structurally controlled by shallow NWW-and NE-trending fold-fault systems and the unconformity between the Upper and Middle *** this study,the regional-and deposit-scale structural investigations,joints and finite strain measurements,and stress and dynamic analysis were conducted with an aim to reveal the structural process of the Getang gold deposit and clarify the relationship between the gold mineralization and *** phases of deformation were identified in the deposit:(1)paleokarst was generated by a crustal uplift when the Youjiang Basin experienced extension at the end of the Middle Permian,laying the foundation for the unconformity;(2)the NW-trending structures were formed under a NNE-SSW compression during the Indochina-South China collision(Indosinian orogeny)in Triassic;(3)the NE-trending structures were generated or reactivated under a NW-SE-oriented compression during the Yanshanian intracontinental *** unconformity recorded two episodes of tectonic evolution in the NNE-SSW and NW-SE *** analyses and geochronology data suggest that the Getang gold deposit was formed as a result of tectonic transition from compression to extension during the Yanshanian intracontinental orogeny.

关键词： Carlin-type gold deposit structural features finite strain Getang Southwest China economy geology geochemistry

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Deformation characteristics of granitic rocks in Erguna ductile shear zone,NE China

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Global Geology 2024年第2期27卷 63-75页

作者： NWABUEZE Ekene Chibuike LIANG Chenyue SONG Zhiwei ZHAO jiaqi ENEZE Florence Ego ZHENG Changqing College of Earth Sciences Jilin UniversityChangchun 130061China Key Laboratory of Mineral Resources Evaluation in Northeast Asia Ministry of Natural ResourcesChangchun 130026China School of International Studies Jinan UniversityGuangzhuo 510632China

The Erguna ductile shear zone is situated in the Erguna Massif,which has been exposed along the eastern bank of the Erguna River in northeastern *** authors present comprehensive study results on the macro-and micro-structures,finite strain and kinematic vorticity,quartz electron backscatter diffraction(EBSD)fabrics,and geochronology of granitic rocks in the Erguna ductile shear *** deformed granitic rocks have experienced significant SE-trending dextral strike-slip *** strain and kinematic vorticity in all deformed granitic rocks indicate that the deformation is characterized by simple sheardominated general shearing with S-L *** deformation behaviors and quartz C-axis textures demonstrate that the deformed granitic rocks developed under greenschist to amphibolite facies conditions at deformation temperatures ranging from 450 to 550℃.New LA-ICP-MS zircon U-Pb ages indicate that these granitic rocks were formed in Early Triassic(~248.6 Ma)and Early Cretaceous(~136.7 Ma).All the evidence indicates that this deformation may have occurred in Early Cretaceous and was related to the compression resulting from the final closure of the Mongol-Okhotsk Ocean.

关键词： zircon U-Pb dating EBSD analysis finite strain kinematic vorticity Erguna ductile shear zone

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有限应变问题基于聚类的方法之比较研究

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Acta Mechanica Sinica 2022年第4期38卷 104-115,I0003页

作者：杨洋张磊唐少强 MoE Key Laboratory of High Energy Density Physics Simulations(HEDPS)and State Key Laboratory of Turbulence and Complex Systems(LTCS) College of EngineeringPeking UniversityBeijing 100871China

虚拟聚类分析(VCA)是一种数值均匀化的降阶方法.我们给出了有限应变问题的VCA算法.说明了它的实现,并提供了免费下载和阅读的代码.四个测试实例的比较表明,VCA和自洽聚类分析(SCA)都很好地逼近了基于FFT的直接数值模拟(DNS)结果.且计算... 详细信息

虚拟聚类分析(VCA)是一种数值均匀化的降阶方法.我们给出了有限应变问题的VCA算法.说明了它的实现,并提供了免费下载和阅读的代码.四个测试实例的比较表明,VCA和自洽聚类分析(SCA)都很好地逼近了基于FFT的直接数值模拟(DNS)结果.且计算代价大大降低.与SCA相比,VCA在线上阶段更为快捷.

关键词： Virtual clustering analysis finite strain Homogenization Machine learning

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An Extended Chaboche's Viscoplastic Law at finite strains: Theoretical and Numerical Aspects

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Journal of Materials Science & Technology 2005年第2期21卷 145-147页

作者： R.C.Lin W.Brocks Institute for Materials Research GKSS Research Centre Max-Planck-Street D-21502 Geesthacht GermanyInstitute for Materials Research GKSS Research Centre Max-Planck-Street D-21502 Geesthacht Germany

This paper presents a newly extended Chaboche’s viscoplastic law at finite strains, so that the classical Chaboche’s theories can be applied to the physical and numerical simulation of metals processing and behavior description of spatial metal structures. The extension is based on a new dissipation inequality at finite strains. The evolution equations are formulated in terms of the corotational rates of the logarithmic elastic strain and the strain-like internal variable conjugate to the back stress as well as the material time derivative of the accumulated plastic strain. The stress equation is expressed on the hyperelastic theory. Therefore, the possible inconsistency with elasticity, caused by the hypoelastic equations, is completely removed. A set of numerical examples with finite deformations are presented to prove the effectivities of the new model and numerical algorithms.

关键词： Viscoplastic law Dissipation inequality finite strain

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Algorithmic tangent modulus at finite strains based on multiplicative decomposition

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Applied Mathematics and Mechanics(English Edition) 2014年第3期35卷 345-358页

作者：李朝君冯吉利 State Key Laboratory for Geomechanics and Deep Underground Engineering School of Mechanics and Civil Engineering China University of Mining and Technology (Beijing)

The algorithmic tangent modulus at finite strains in current configuration plays an important role in the nonlinear finite element method. In this work, the exact tensorial forms of the algorithmic tangent modulus at finite strains are derived in the principal space and their corresponding matrix expressions are also presented. The algorithmic tangent modulus consists of two terms. The first term depends on a specific yield surface, while the second term is independent of the specific yield surface. The elastoplastic matrix in the principal space associated with the specific yield surface is derived by the logarithmic strains in terms of the local multiplicative decomposition. The Drucker-Prager yield function of elastoplastic material is used as a numerical example to verify the present algorithmic tangent modulus at finite strains.

关键词： algorithmic tangent modulus matrix expression finite strain multiplicative decomposition

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Deformation Kinematics of Main Central Thrust Zone(MCTZ)in the Western Himalayas

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Journal of Earth Science 2022年第2期33卷 452-461页

作者： Mohsin Ahmad Ahanger Ghulam Jeelani Department of Earth Sciences University of KashmirSrinagar 190006India

The main central thrust(MCT)is one of the major thrusts in *** central Himalaya,MCT was defined as a contact between underlying Lesser Himalayan Sequence(LHS)and overlying higher Himalayan crystallines(HHC).However,in the Kashmir Himalayas,the main central thrust zone(MCTZ),shear zone associated with MCT,is overlain by Kashmir Tethyan Sequence suggesting that the MCTZ has been deformed through a mechanism different than the mechanism responsible for MCTZ evolution in other parts of the *** the present study we used structural,microfabric and kinematic analyses to investigate the deformation kinematics of *** investigation revealed that the quartz in orthogneiss mylonites of MCTZ was dynamically recrystallized by grain boundary migration(GBM)and sub-grain rotation recrystallisation(SGR)with top-toSW sense of *** mean kinematic vorticity number(W;)just above the thrust ranges from 0.72to 0.84(40%–52%pure shear component)decreasing upwards to 0.65–0.71(35%–50%pure shear component).Deformation in the MCTZ is characterized by Rxzstrain ratio varying from 2.7 to *** present study suggested that the MCTZ suffered 3%–40%vertical shortening and 3%–66%transportparallel *** results suggested that the HHC’s were not completely exhumed to the topographic surfaces in the Kashmir *** the basal decollement,i.e.,the main Himalayan thrust(MHT),the deformation continued until MCTZ reached the brittle-ductile transition where deformation mechanism changed to the brittle and the MCTZ rocks were transported to the surface through slip on brittle MCT.

关键词： kinematics mylonites finite strain mean kinematic vorticity number(Wm) general shear tectonics

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Theoretical Assessment of Burst Failure of Internally Pressurized Defect-Free Pipelines with Plastic Anisotropy

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China Ocean Engineering 2013年第4期27卷 427-436页

作者：张永强王立忠金伟良 College of Civil Engineering and Architecture Zhejiang University

In the framework of finite deformation theory, the burst failure analysis of end-opened defect-free pipes with plastic anisotropy under internal pressure is carried out. The analytical solutions of burst pressure and the corresponding equivalent stress and strain are obtained for thin-walled pipes, which can take into account the effects of material plastic anisotropy and strain hardening exponent. The influences of plastic anisotropy on the burst pressure and the corresponding equivalent stress and strain are discussed. It is shown that the burst pressure and the corresponding equivalent stress and strain are dependent upon the plastic anisotropy of material, and the degree of dependence is related to the strain hardening exponent of material. In addition, the effects of the strain hardening exponent on burst failure are investigated.

关键词： burst pressure plastic anisotropy end-opened pipe finite strain

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