RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅵ)—CONSERVATION LAWS OF MASS AND INERTIA

IntroductionThisworkisadirectcontinuationandasupplementofRefs .[1～8] .InRefs.[1～8]thecoupledbalancelawsandequationsofmomentum ,angularmomentumandenergyaswellasthenewHamiltonprinciple,principleofvirtualpowerandNoethertheoremhavebeenpresented .However,thecoupledconservationlawsofmassandinertiahavenotbeenreestablishedyet.Thepurposeofthispaperistoreestablishtheconservationlawsandequationsofmassandinertiaandtocombinethemwiththecoupledbalancelawsandequationsofmomentum ,angularmomentum ,energyand…     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅳ)--SURFACE COUSERVATION LAWS

The purpose is to reestablish rather complete surface conservation laws for micropolar thermomechanical continua from the translation and the rotation invariances of the general balance law. The generalized energy-momentum and energy-moment of momentum tensors are presented. The concrete forms of surface conservation laws for micropolar thermomechanical continua are derived . The existing related results are naturally derived as special cases from the results proposed in this paper . The incomplete degrees of the existing surface conservation laws are clearly seen from the process of the deduction. The surface conservation laws for nonlocal micropolar thermomechanical continua may be easily obtained via localization .     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅳ)——SURFACE COUSERVATION LAWS

The purpose is to reestablish rather complete surface conservation laws for micropolar thermomechanical continua from the translation and the rotation invariances of the general balance law. The generalized energy-momentum and energy-moment of momentum tensors are presented. The concrete forms of surface conservation laws for micropolar thermomechanical continua are derived. The existing related results are naturally derived as special cases from the results proposed in this paper. The incomplete degrees of the existing surface conservation laws are clearly seen from the process of the deduction. The surface conservation laws for nonlocal micropolar thermomechanical continua may be easily obtained via localization. Contributed by DAI Tian-min, Original Member of Editorial Committee, AMM Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931 ~)     

THE EXTREMITY LAWS OF HYDRO-THERMODYNAMICS

This paper presents the law of maximum rate of energy dissipa-tion in hydrodynamics and also in general continuum dynamicsas an addition to the classical conservation laws expressed inthe equation of continuity and the equations of motion.Thecorollary of the law is B(?)langer-B(?)ss theorem of minimum reser-ved specific energy in applied hydraulics.The mechanical energy dissipated is transformed into heatreserved in the substance.The rate of energy dissipation ata time at a given temperature gives rise to the increase in en-tropy production.Hence the maximum rate of energy dissipationsuggests itself the idea of reformulation of the second law ofthermodynamics that the rate of entropy production in mech-anical motion is always the maximum possible.The proposed extremity law in continuum dynamics has beenderived from the variational principle and the reformulatedsecond law of thermodynamics analyzed microscopically in thepaper.The two laws together form the extremity laws of hydro-thermodynamics.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅷ)—PRINCIPLE OF TOTAL WORK AND ENERGY

IntroductionRecently ,wehaverestudiedtheconventionalconservationlawofenergy ,whichhaswidelybeenusedandisusinginliteraturesofclassicalcontinuummechanicsathomeandabroad ,andpointedoutthatitistheoreticallyincompleteeveninthetraditionalframework .Thepresentsituationoftheconservationlawsofenergyforpolarcontinuaisthesameasthatforclassicalcontinuummechanics.Inordertochangethestatusquo ,itseemstobequitenecessarythatnewprinciplesoftotalworkandenergyshouldbeproposedtotaketheplaceoftheexistingconservatio…     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅷ)—PRINCIPLE OF TOTAL WORK AND ENERGY

Theoretical incompleteness of the existing conservation laws of energy for polar continuum mechanics is further clarified. For completeness, the principles of total work and energy and of total work and energy of incremental rate type are postulated. Via total variations of the former and the latter of them, the principles of virtual displacement and microrotation & stress and couple stress as well as virtual velocity and angular velocity & stress rate and couple stress rate are immediately obtained, respectively. From these principles all balance equations and boundary conditions for micropolar mechanics are naturally and simultaneously deduced. The essential differences between the nontraditional results obtained in this paper and the existing conservation laws of energy are expounded. Contributed by DAI Tian-min, Original Member of Editorial Committe, AMM Project supported by the National Natural Science Foundation of China (Nos. 10472041, 10072024) and the Science Research Foundation of Liaoning Province (No. 990111001)     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES(Ⅸ)—THERMOMECHANICS

The existing fundamental laws of thermodynamics for micropolar continuum field theories are restudied and their incompleteness is pointed out. New first and second fundamental laws for thermostatics and thermodynamics for micropolar continua are postulated. From them all equilibrium equations and the entropy inequality of thermostatics as well as all balance equations and the entropy rate inequalities are naturally and simultaneously deduced. The comparisons between the new results presented here and the corresponding results demonstrated in existing monographs and textbooks concerning micropolar continuum mechanics are made at any time. It should be emphasized to note that, the problem of why the local balance equation of energy and the local entropy inequality could not be obtained from the existing fundamental laws of thermodynamics for micropolar continua, is believed to be clarified.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅸ)— THERMOMECHANICS

The existing fundamental laws of thermodynamics for micropolar continuum field theories are restudied and their incompleteness is pointed out. New first and second fundamental laws for thermostatics and thermodynamics for micropolar continua are postulated. From them all equilibrium equations and the entropy inequality of thermostatics as well as all balance equations and the entropy rate inequalities are naturally and simultaneously deduced. The comparisons between the new results presented here and the corresponding results demonstrated in existing monographs and textbooks concerning micropolar continuum mechanics are made at any time. It should be emphasized to note that, the problem of why the local balance equation of energy and the local entropy inequality could not be obtained from the existing fundamental laws of thermodynamics for micropolar continua, is believed to be clarified.     

BCKLUND TRANSFORMATION, NONLINEAR SUPERPOSITION FORMULAE AND INFINITE CONSERVED LAWS OF BENJAMIN EQUATION

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＲｅｆ.[1 ]ｐｏｉｎｔｅｄｏｕｔｔｈａｔｓｏｍｅｔｙｐｉｃａｌｎｏｎｌｉｎｅａｒｗａｖｅｅｑｕａｔｉｏｎｓｐｏｓｓｅｓｓｅｄｆｉｖｅｃｏｍｍｏｎｆａｃｔｏｒｓ,ｗｈｉｃｈｗｅｒｅｗｉｔｈｓｏｌｉｔａｒｙｓｏｌｕｔｉｏｎｓ,ｉｎｆｉｎｉｔｅｃｏｎｓｅｒｖｅｄｌａｗｓ,ｅｘａｃｔｌｙｓｏｌｖａｂｌｅｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓｂｙｔｈｅｕｓｅｏｆｉｎｖｅｒｓｅｓｃａｔｔｅｒｉｎｇｍｅｔｈｏｄ .Ｂ ｃｋｌｕｎｄｔｒａｎｓｆｏｒｍａｔｉｏｎｓ,ｂｅｉｎｇｒｅｄｕｃｅｄｔｏ…     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅰ)—MICROPOLAR CONTINUA

Based on the restudies of existing polar continuum theories rather complete systems of basic balance laws and equations for micropolar continuum theory are presented. In these new systems not only the additional angular momentum, surface moment and body moment produced by the linear momentum, surface force and body force, respectively, but also the additional velocity produced by the angular velocity are considered. The new coupled balance laws of linear momentum, angular momentum and energy are reestablished. From them the new coupled local and nonlocal balance equations are naturally derived. Via contrast it can be clearly seen that the new results are believed to be rather general and complete. Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: Dupai Tian-min (1931≈)     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅴ)--POLAR THERMOMECHANICAL CONTINUA

The purpose is to reestablish rather complete basic balance equations and boundary conditions for polar thermomechanical continua based on the restudy of the traditional theories of micropolar thermoelasticity and thermopiezoelectricity . The equations of motion and the local balance equation of energy rate for micropolar thermoelasticity are derived from the rather complete principle of virtual power. The equations of motion, the balance equation of entropy and all boundary conditions are derived from the rather complete Hamilton principle . The new balance equations of momentum and energy rate which are essentially different from the existing results are presented. The corresponding results of micromorphic thermoelasticity and couple stress elastodynamics may be naturally obtained by the transition and the reduction from the micropolar case , respectively . Finally , the results of micropolar thermopiezoelectricity are directly given .     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CON-TINUUM THEORIES (Ⅱ)—MICROMORPHIC CONTINUUM THEORY AND COUPLE STRESS THEORY

The purpose is to reestablish the balance laws of momentum, angular momentum and energy and to derive the corresponding local and nonlocal balance equations for micromorphic continuum mechanics and couple stress theory. The desired results for micromorphic continuum mechanics and couple stress theory are naturally obtained via direct transitions and reductions from the coupled conservation law of energy for micropolar continuum theory, respectively. The basic balance laws and equations for micromorphic continuum mechanics and couple stress theory are constituted by combining these results derived here and the traditional conservation laws and equations of mass and microinertia and the entropy inequality. The incomplete degrees of the former related continuum theories are clarified. Finally, some special cases are conveniently derived. Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931≈)     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅲ)—NOETHER''''S THEOREM

IntroductionThispaperisadirectcontinuationofthepreviouspapers [1 ,2 ] .ThefieldequationsoflinearcouplestresselasticityweredevelopedbyMindlinandTiersten[3].AgeneralcouplestresstheorywaspresentedbyToupin[4 ].Eringen[5 ,6 ]haspointedoutthatthecouplestresstheoriesofelasticmediafluidsmaybededucedfromthemicropolarelasticityandfluidmechanicswhenthemicrorotationsiandψiareconstrainedbytherotationvectorωiofclassicalelasticityandvorticityvectorrioffluidmechanics,respectively .AnexpressionoftheJ_int…     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅰ) —MICROPOLAR CONTINUA

IntroductionThemicropolarcontinuumtheoryisatypicalandwideapplicatedtheoryinthegeneralizedcontinuumtheoriesandwasgenerallyrecognizedlongago .Manymonographsandalargenumberofscientificpapersconcerningthisfieldarepublished .Wehaverestudiedtheexistingpolarcontinuumtheoriesandfoundthatsomebasicbalancelawsandprinciplesofthemareincompleteandthereexistsometheoreticaldefectswhichshouldberemoved .Forcontrastandclaritywenowlistthetraditionalbasicbalancelawsandequationsformicropolarcontinuumtheoryasfollows…     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES(Ⅶ)—INCREMENTAL RATE TYPE

The purpose is to establish the rather complete equations of motion, boundary conditions and equation of energy rate of incremental rate type for micropolar continua. To this end the rather complete definitions for rates of deformation gradient and its inverse are made. The new relations between various stress and couple stress rate tensors are derived. Finally, the coupled equations of motion, boundary conditions and equation of energy rate of incremental rate type for continuum mechanics are obtained as a special case. Contributed by Dai Tian-min, Original Member of Editorial Committee, AMM Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: Dai Tian-min (1931≈)     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅴ)——POLAR THERMOMECHANICAL CONTINUA

The purpose is to reestablish rather complete basic balance equations and boundary conditions for polar thermomechanical continua based on the restudy of the traditional theories of micropolar thermoelasticity and thermopiezoelectricity. The equations of motion and the local balance equation of energy rate for micropolar thermoelasticity are derived from the rather complete principle of virtual power. The equations of motion, the balance equation of entropy and all boundary conditions are derived from the rather complete Hamilton principle. The new balance equations of momentum and energy rate which are essentially different from the existing results are presented. The corresponding results of micromorphic thermoelasticity and couple stress elastodynamics may be naturally obtained by the transition and the reduction from the micropolar case, respectively. Finally, the results of micropolar thermopiezoelectricity are directly given. Contributed by DAI Tian-min, Original Member of Editorial Committee, AMM Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931-)     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CON-TINUUM THEORIES (Ⅱ)—MICROMORPHIC CONTINUUM THEORY AND COUPLE STRESS THEORY

IntroductionThispaperisadirectcontinuationofRef.[1 ] .InitthecoupledconservationlawofenergypresentedinRef.[2 ]wasextendedandtherathercompletesystemsofbasicbalancelawsandequationsformicropolarcontinuumtheoryhavebeenconstitutedbycombiningtherenewedresultsandthetraditionalconservationlawsofmassandmicroinertiaandtheentropyinequality .Thepurposeofthispaperistorestablishthesystemsofbasicbalancelawsandequationsformicromorphiccontinuumtheoryandcouplestresstheoryviadirecttransitionsandreductionsfromth…     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES(Ⅶ)—INCREMENTAL RATE TYPE

IntroductionTherelationsbetweenvariousstresstensors,theequationofmomentumandtheboundaryconditionsofincrementalratetypeforclassicalcontinuummechanicshavebeensystematicallyderivedbyKuanginRef.[1 ] .InRef.[2 ]wehavederivedtherelationsbetweenvariouscouplestresstensorsandtheirratesandpresentedtheequationsofmotionandtheboundaryconditionsofincrementalratetypeofCauchyform ,PiolaformandKirchhoffformforpolarcontinua .InRef.[3 ]wehavepresentednewprinciplesofpowerandenergyrateofincrementalratetypeformi…     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅹ) —MASTER BALANCE LAW

Through a comparison between the expressions of master balance laws and the conservation laws derived by Noether's theorem, a unified master balance law and six physically possible balance equations for micropolar continuum mechanics are naturally deduced. Among them, by extending the well-known conventional concept of energymomentum tensor, the rather general conservation laws and balance equations named after energy-momentum, energy-angular momentum and energy-energy are obtained. It is clear that the forms of the physical field quantities in the master balance law for the last three cases could not be assumed directly by perceiving through the intuition. Finally, some existing results are reduced immediately as special cases.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅲ) —NOETHER’S THEOREM

The existing various couple stress theories have been carefully restudied.The purpose is to propose a coupled Noether’s theorem and to reestablish rather complete conservation laws and balance equations for couple stress elastodynamics. The new concrete forms of various conservation laws of couple stress elasticity are derived. The precise nature of these conservation laws which result from the given invariance requirements are established. Various special cases are reduced and the results of micropolar continua may be naturally transited from the results presented in this paper.