Chen Rational Mechanics II Geometrical Equations

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1、精品论文大全Chen Rational Mechanics II. Geometrical EquationsXiao JianhuaHenan Polytechnic University, Jiaozuo, Henan, P.R.C., 454000Tel: 0391-3987677, 13703916577, e-Mail: jhxiaoAbstract: Using material coordinators, deformation has two senses: distance variation and area variation. Starting from the geo

2、metrical invariant quantities in tensor theory, the macro deformation is described by the distance base vector transformation and the area base vector transformation. For pure elastic deformation, the area base vector transformation is the inverse of the distance base vector transformation. Only in

3、this special case, the deformation is equivalent with coordinator transformation. For arbitral deformation, the area base vector transformation is plastic volume variation dependent and the distance base vector transformation is the deformation gradient. Therefore, for large deformation problem, two

4、 deformation tensors should be introduced. Some remarks are made for related theoretic results which are viewed as widely accepted.PACS: 02.40.Ky, 46.05.+bKeywords: base vector, tensor, deformation, transformation, material coordinators1. IntroductionFinite deformation mechanics is the bases for man

5、y engineering problems. In finite deformation mechanics, the geometrical equation of deformation plays an essential role. For static deformation problem, the deformation gradient is measurable quantity in geometrical sense. So, the strain is defined by it. This definition equation is called geometri

6、cal equation in continuum mechanics. However, different ways of treating the deformation tensor will leads to different strain definition. In this paper, the geometrical equation is limited to be the deformation related basic equations. The strain definition will be discussed in subsequent papers.Fo

7、r infinitesimal deformation, the strain is well defined. This forms the basic geometrical equation for continuum mechanics. In engineering, this definition is used as standards for laboratory experiments or in-site experimental measurement. The most important thing is that the huge mechanical data a

8、ccumulated for various industrial applications and engineering material are obtained by the classical strain definition. Hence, this fact makes the target of this research be to save the data obtained by classical strain definition for large deformation in a rational way.Using the classical strain d

9、efinition to large deformation will leads to non-linear problem. Usually, such a kind of non-linearity is named as geometrical non-linearity. However, for large deformation, the intrinsic parameters of material do exhibit physical non-linearity. Then, in practical problems, they are happens together

10、. For distinguishing them, rational mechanics is raised. The basic concept of rational mechanics is that the physical intrinsic parameters are limited if the geometrical effects can be separated out from the phenomenology. This has no conflicts with the industrial requirements.However, the history o

11、f continuum mechanics shows that this is a very difficult problem which is to be addressed in this research. The start point of research is tensor theory for deformation. So, this paper will put focus on the tensor representation of deformation.2. Deformation Described by Covariant Base Vector Trans

12、formationFor continuum, each material point can be parameterized with continuous coordinator- 12 -x i , i = 1,2,3 . When the coordinators are fixed for each material point, the covariant gauge fieldg ij attime t will define the configuration in that time. The deformation is described by the gauge te

13、nsorgij (t )variation. The continuous coordinators endowed with the gauge field tensor define a co-movingdragging coordinator system. Usually, such a kind of material coordinators is called Lagrangian coordinators. When it is gauged, the dragging coordinator system is established. The gauge tensorgi

14、j (t )variation defines the deformation 1-4.gThe initial configuration gaugeij0 defines a distance geometric invariant:20ijds0 = g ij dx dx(1)The symmetry and positive feature of gauge tensor insures that there exist three initial covariant basegivectorsr 0 make:0r 0 r 0g ij = g i g jFor current con

15、figuration, three current covariant base vectorsgir exist which make:(2)r rg ij = g i g jThe current distance geometric invariant is:(3)2ijds = g ij dx dxFor each material point, there exists a local transformation covariant base vectors with initial covariant base vectors:(4)jF i , which relates th

16、e currentiiggr = Fj r 0j(5)So, the current covariant gauge tensor can be expressed as:k l 0g ij = FiF j g kl(6)FjThe local transformationirepresents the local length transformation.3. Deformation Described by Contra-variant Base Vectors TransformationIn Riemann geometry, contra-variant gaugeg 0ijcan

17、 be introduced, which meets condition:gg0il 0jlj= i(7)Similarly, contra-variant base vectorsgr i ,gr 0ican be introduced for current configuration and initialconfiguration respectively. Mathematically, there are:g ij = gr i gr j ,g 0ij = gr 0i gr 0 j(8)There exists a local transformation contra-vari

18、ant base vectors:jgr i = G i gr 0 jjG i (represents area transformation), which relates the(9)So, the current contra-variant gauge tensor (area gauge tensor) can be expressed as:g ij = G i G j g 0 kl(10)k lBy Equations (5) and (9), the current volume can be calculated as:i ligjThe mixed tensoriGl F

19、j = g jGdescribes the current volume distortion. The local transformation(11)i jrepresents the local area transformation. The essential difference between deformation mechanics andmathematic tensor theory is thatg i ifor deformation mechanics while in mathematical tensorjjtheoryg i = i . For much mo

20、re mathematic discussion about tensor in deformation mechanics, pleasejjsee reference 5.4. Deformation Described by Mixed Base VectorsFjBased on above research, the transformationirelates the initial contra-variant base vectorswith current contra-variant base vectors in such a way that:jgr 0i = F i

21、gr j(12)FjTherefore, the transformationiis a mixture tensor. Its lower index represents covariant componentincurrentgijconfiguration,itsupperindexrepresentscontra-variantcomponentinijinitial g 0 configuration.GjSimilar discussion shows that the local transformationiis a mixture tensor, lower indexgr

22、epresents covariant component in initialij0 configuration, upper index represents contra-variantcomponent in currentg ijconfiguration. It is easy to find that:gijir 0 = G j gr(13)Other two important equations are:Fji = gr 0iir i grjr 0(14)G j = g g j(15)FjBy these equations, theican be explained as

23、the mixed tensor as that it is the dot product ofcontra-variant base vector in initial configuration and covariant base vector in current configuration. When the two configurations are the same, it becomes the standard Kronecker-delta. For thejgjG i transformation, similar interpretation can be made

24、. Theiis the extended Kronecker-delta atgjcurrent configuration. To clarify the true mixed tensoriFjin mathematic sense, the mixed tensoriGjandiwill be referred as transformation tensor. Chen Zhida points out that they represent point-settransformation which belongs to one-parameter group transforma

25、tion 1.5. Interpretation about the Related StressFjFrom above equations to see, geometrically, it is found that theimeasures the currentcovariant base vector referring with the initial contra-variant base vector as reference. For simple linear elastic deformation, when stress tensor is defined as:ii

26、k ( ll ) j = E jlFk k(16)Its mechanic meaning can be explained as the lower index represents component of surface force in current direction, and the upper index represents the initial surface normal where surface force acts on.It takes the initial surface as surface force reference.ThejG i measures

27、 the initial covariant base vector with the current contra-variant base vector asreference. The corresponding stress can be explained as the lower index represents component of surface force in initial direction, and the upper index represents the current surface normal where surface force acts on.F

28、or such a kind of mixture tensor, it is defined on the same point with different configurations (that is initial and current). So, the name of two-point tensor 4 given by C Truessdell is not correct. This disguise interpretation has caused many doubts cast on the feature of transformation tensor. So

29、me even said that the mixture tensor is meaningless.However, during treating the non-symmetric field theory, Einstein believes that the use of mixturetensor is more reasonable that the pure covariant or pure contra-variant tensor 6. So, we have sound reason to use mixture tensor 7 in continuum mecha

30、nics, as it can give clear physical meaning for thedefinition of stress.6. Mathematic RemarksFjNote that the transformationiis completely determined by the deformation measured in initialijconfiguration with gauge tensor g 0 . Mathematically, the covariant differentiation is taken in the initialFjge

31、ometry also, although the physical meaning ofiis that it relates two configurations.It is valuable to point out that, if the initial coordinator system is taken as Cartesian system, theFji can be transformed into pure covariant form:r 0 rGjTheiFij = g i g jcan be transformed into pure contra-variant

32、 form:G ij = gr i gr 0 j(17)(18)Although it is acceptable in form for the special case of taking Cartesian system as the initial coordinator system, the intrinsic meaning of deformation tensor is completely destroyed by such a formulation. That may be the main reason for the rational mechanics const

33、ructed by C Truessdell et al in 1960s.Mathematically, once the initial gauge field is selected, the current gauge field must be obtained by the given physical deformation. In this sense, the current gauge field is viewed as the physical field. So, the covariant differentiation is taken respect with

34、the initial configuration.C. Truessdell argues that the covariant differentiation should be taken one index in initial configuration, another index in current configuration 4,8. This concept has strongly effects on the development of finite deformation mechanics. Such a kind of misunderstanding inde

35、ed is caused by the equations (17-18).Historically, Chen Zhida is the first one to clear the ambiguity 1 caused by equations (17) and(18) systematically. His monograph “Rational Mechanics”(1987) treats this topic in depth.There are many critics about the mathematics used in Chens rational mechanics

36、theory. The most comment criticism is based on following reasoning. For a coordinator transformationjjdx i = Ai dX j ,dX i = ( Ai ) 1 dx j(19)jThe covariant and contra-variant components of tensor is defined by its transformation from newAjcoordinator system to old coordinator system (or from old to

37、 new), represented byiformulations. However, such a tensor definition is to make:or ( Ai ) 1ijds 2= g dx i dx j= Gij dXi dX j(20)be invariant. Such a tensor describes a continuum without any deformation. This is the main topic in mathematics theory of tensors. In fact, such a tensor feature is only

38、to say the objective indifference forcoordinator system selection (the geometrical invariantsds2 ).However, many physicists and mechanists treat the deformation of continuum as theAjtransformationior ( Ai ) 1 . Mathematically, the equation (20) can be rewritten as:j2ijk lijijor:ds = g ij dx dx= g kl

39、 Ai A j dX dX= Gij dX dX(21)ds 2= Gij dXklii dX j= G ( A k ) 1( Al ) 1jijdx i dx j= g dx i dx j(22)In the mathematic tensor theory, it is clear that the covariant invariant feature must be maintained bythe coordinator system choice. It has no any meaning of deformation.In Chens geometry, for time pa

40、rameter t ,2ijklijds (t ) = g ij (t )dx dx= g kl (0)Fi(t )F j (t )dx dx(23)The gauge field is time dependent, while the coordinator is fixed (called intrinsic coordinator, orLagrangian coordinators).The equation (23) can be rewritten as:ds 2 (t ) = g(t )dx i dx j = g(0)F k (t )F l (t )dx i dx j = g(

41、0)dX k (t )dX l (t )(24)ijklijklIt is similar in form with equation (21). But their mechanical interpretation is strikingly different.7. Displacement Field as the Macro Phenomenon of DeformationSure, one can view the following transformation represent the deformation:jdX i (t ) = F i (t )dx j(25)It

42、can be argued that the initial coordinator is transformed into current coordinator. This understanding is true if both the coordinators are defined in the same coordinator system. In this case, it says that the initial position is changed into current position. Hence, it describes the displacement f

43、ield measured in the initial coordinator system.This point is very important. In standard rectangular measurement coordinator system, for amaterial point initially at coordinator:material point is:xi , its current position isX i (t) , the displacement field of theui (t) = X i (t ) xi(26)Taking its d

44、ifferentiation about the initial coordinators (usually the initial coordinator system is rectangular, and the initial rectangular coordinator is taken as the Lagrangian coordinator), one has:iiu (t) = X (t) i(27)x jx jjHence, comparing with Equation (25) one finds that:iiF i (t ) = X (t) = u (t) + i

45、(28)jx jx jjThis equation makes the covariant base vector be calculated by displacement gradient. It forms the basic stone for rational mechanics.jNote that theF i (t )is named as deformation gradient tensor 4. It is defined in initialconfiguration. To emphasis this feature, the commoving dragging c

46、oordinator system 1 is introduced for identify the invariance of material coordinators for a given material point.8. Infinitesimal DeformationTaking differentiation of Equation (26) about the current coordinators (that is the current coordinator system is rectangular and is taken as the Lagrangian c

47、oordinator, and the initial rectangularcoordinator is arbitral), one has: ui (t )ixiThat is:= j X j (t )X j (t )(29) xiiui (t )Mathematically:= j X j (t )X j (t )(30)ililX (t) x= i = ( i + u (t ) )( l u (t) )(31)That is equivalent with:xlX j (t )jlxljX j (t )ui (t )ui (t )ui (t) ul (t)=(32)x jX j (t

48、 )xlX j (t)For infinitesimal deformation, the higher order infinitesimal can be omitted. Then the equation becomes:ui (t )x jui (t )X j (t )(33)Therefore, for infinitesimal deformation, the difference between the reference configurations can be omitted.9. Large DeformationFor large deformation, the

49、inverse deformation may exist or not. If the deformation is pure elastic,jby applying an inverse deformation(F i )1the material has no any kind of volume variation. Letiig j = j , by Equation (11), one has:lGi =j l i (Fj )1 = (lF i )1(34)GjTherefore, the transformationiis the inverse of the local tr

50、ansformationjF i . In pure elastic case,the deformation can be purely described by the local transformation deformation, one has:jF i . Hence, for pure elastici liGl Fj = j(35)It says that the pure elastic deformation is equivalent with coordinator transformation. For pure elastic deformation, the R

51、iemann tensor theory is an equivalence. Just is this special case which fools many researchers on finite deformation mechanics. In fact, this equation can be viewed as the definition ofreversible deformation.jHowever, for plastic deformation, if the plastic volume variation isgi , then followed by t

52、heinverse deformationj(F i )1 , one has:i = (i + i )(j )1Gl jg j Fl(35)GjThe transformationiis not the inverse of the local transformationjF i . In this case, theGjtransformationiis plastic dependent. In fact, this equation can be viewed as the definition forgjirreversible deformation. Only when the

53、 plastic volume variation igjis measured, the transformationGji can be calculated. Hence, it is viewed as a measurable quantity.Therefore, for large deformation problem, two deformation tensors should be introduced. TheFjdirect measurable quantities are:ijandgi . Geometrically, the iis the volume ca

54、lculated byarea base vector (upper index i ) and length base vector (lower index j ) on plastic configuration. Generally, it has nine independent parameters. For isotropic plastic deformation, it has only onegjparameter. For symmetric plastic deformation, it has three parameters. In fact, the equiva

55、lence of iin classical deformation theory is the plastic strain.10. Remarks on Stress in Classical MechanicsReasoning from this equation, the area variation transformation is the key to understand theGjintrinsic meaning of plasticity. So, the transformationior plastic volume mixed tensor imustgjbe i

56、ntroduced for large deformation. Usually, the volume mixed tensor igjis introduced in classicalgjplasticity theory in an implied form. As the volume mixed tensor iis determined by physicaljfeature of materials, the stress tensorican be introduced to replace the contra-variant base vectortransformati

57、onjGi .as:To see this, it is helpful to view the deformation energy W for static deformation 3. It is definedk ( ll )W = l Fk k(36)jWhere,iis deformation stress in usual senses. By Equation (11), the volume deformation energycan be defined as: k ( ll )W = l gk k(37)jWhere, iis the stress related with the volume deformation. It will be named as intrinsic stress later.Physically, it represents the required stress for volume deformation. The ratio of the volume deformation energy over the deformation energy is a physical parameter which depends on the ma