Chen Rational Mechanics II Geometrical Equations.doc
《Chen Rational Mechanics II Geometrical Equations.doc》由会员分享,可在线阅读,更多相关《Chen Rational Mechanics II Geometrical Equations.doc(12页珍藏版)》请在三一文库上搜索。
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: Abstract: Using material coordinators, deformation has two senses: distance variation and area variation. Starting from the geometric
2、al 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 this s
3、pecial 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 defor
4、mation 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 many engi
5、neering 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 geometrical eq
6、uation 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.For infi
7、nitesimal 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 accumul
8、ated 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 definit
9、ion 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. For
10、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 of cont
11、inuum 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 Transformat
12、ionFor 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 tensorgi
13、j (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 tensorgij (t )
14、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 configura
15、tion, 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 the curr
16、entiiggr = 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 be in
17、troduced, 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-variant ba
18、se 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 j = g
19、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 more mat
20、hematic 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 gr j(1
21、2)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 indexgreprese
22、nts 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 the mi
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- Chen Rational Mechanics II Geometrical Equations
链接地址:https://www.31doc.com/p-3618738.html