Belytschko T. - Introduction (779635), страница 3

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In eachof these fields a standard notation has evolved. Unfortunately, the notations arequite different, and at times contradictory or overlapping. We have tried to keepthe variety of notation to a minimum and both consistent within the book and withthe relevant literature.

To make a reasonable presentation possible, both thenotation of the finite element literature and continuum mechanics are used.Three types of notation are used: 1. indicial notation, 2. tensor notationand 3. matrix notation. Equations in continuum mechanics are written in tensorand indicial notation. The equations pertaining to the finite elementimplementation are given in indicial or matrix notation.Indicial Notation.

In indicial notation, the components of tensors or matrices areexplicitly specified. Thus a vector, which is a first order tensor, is denoted in1-6T. Belytschko, Introduction, December 16, 1998indicial notation by xi , where the range of the index is the number of dimensionsnS . Indices repeated twice in a term are summed, in conformance with the rulesof Einstein notation. For example in three dimensions, if xi is the position vectorwith magnitude rr 2 = xi xi = x1x1 + x2 x2 + x3 x3 = x 2 + y 2 + z 2(1.3.1)where the second equation indicates that x1 = x, x2 = y, x3 = z ; we will alwayswrite out the coordinates as x, y and z rather than using subscripts to avoidconfusion with nodal values.

For a vector such as the velocity vi in threedimensions, v1 = vx , v2 = vy , v3 = vz ; numerical subscripts are avoided in writingout expressions to avoid confusing components with node numbers. Indiceswhich refer to components of tensors are always lower case.Nodal indices are always indicated by upper case Latin letters, e.g.

viI isthe velocity at node I. Upper case indices repeated twice are summed over theirrange, which depends on the context. When dealing with an element, the range isover the nodes of the element, whereas when dealing with a mesh, the range isover the nodes of the mesh. Thus the velocity at a node I is written as viI , whereviI is the i-component at node I.A second order tensor is indicated by two subscripts. For example, for thesecond order tensor Eij , the components are E11 = Exx , E21 = Eyx , etc.. We willusually use indicial notation for Cartesian components but in a few of the moreadvanced sections we also use curvilinear components.Indicial notation at times leads to spaghetti-like equations, and theresulting equations are often only applicable to Cartesian coordinates.

For thosewho dislike indicial notation, it should be pointed out that it is almost unavoidablein the implementation of finite element methods, for in programming the finiteelement equations the indices must be specified.Tensor Notation. Tensor notation is frequently used in continuum mechanicsbecause tensor expression are independent of the coordinate systems. Thus whileCartesian indicial equations only apply to Cartesian coordinates, expressions intensor notation can be converted to other coordinates such as cylindricalcoordinates, curvilinear coordinates, etc.

Furthermore, equations in tensornotation are much easier to memorize. A large part of the continuum mechanicsand finite element literature employs tensor notation, so a serious student shouldbecome familiar with it.In tensor notation, we indicate tensors of order one or greater in boldface.Lower case bold-face letters are almost always used for first order tensors, whileupper case, bold-face letters are used for higher order tensors. For example, avelocity vector is indicated by v in tensor notation, while the second order tensor,such as E , is written in upper case. The major exception to this are the physicalstress tensor s , which is a second order tensor, but is denoted by a lower casesymbol.

Equation(1.3.1) is written in tensor notation as1-7T. Belytschko, Introduction, December 16, 1998r2 = x ⋅ x(1.3.2)where a dot denotes a contraction of the inner indices; in this case, the tensors onthe RHS have only one index so the contraction applies to those indices.Tensor expressions are distinguished from matrix expressions by using dots andcolons between terms, as in a ⋅ b , and A ⋅ B . The symbol ":" denotes thecontraction of a pair of repeated indices which appear in the same order, soA : B ≡ Aij Bij(1.3.3)The symbol " ⋅⋅" denotes the contraction of the outer repeated indices and theinner repeated indices, as inA ⋅⋅B = Aij Bji = A T : B(1.3.4)If one of the tensors is symmetric, the expressions in Eqs.

(1.3.3) and (1.3.4) areequivalent. This notation can also be used for contraction of higher ordermatrices. For example, the usual expression for a constitutive equation givenbelow on the left is written in tensor notation as shown on the rightσ ij = Cijkl ε klσ = C:ε(1.3.5)The functional dependence of a variable will be indicated at the beginningof a development in the standard manner by listing the independent variables. Forexample, v( x , t ) indicates that the velocity v is a function of the spacecoordinates x and the time t. In subsequent appearances of v, the identity of theindependent variables in implied. We will not hang symbols all around thevariable. This notation, which has evolved in a some of the finite elementliterature, violates esthetics, and is reminiscent of laundry hanging from thebalconies of tenements.

We will attach short words to some of the symbols. Thisis intended to help a reader who delves into the middle of the book. It is notintended that such complex symbols be used working through equations.Mathematical symbols and equations should be kept as simple as possible.Matrix Notation. In implementation of finite element methods, we will often usematrix notation. We will use the same notations for matrices as for tensors and butwill not use connective symbols.

Thus Eq. (2) in matrix notation is written asr 2 = xT x(1.3.6)All first order matrices will be denoted by lower case boldface letters, such as v,and will be considered column matrices. Examples of column matrices arex  x = yz   v1  v = v2 v  3(1.3.7)1-8T. Belytschko, Introduction, December 16, 1998Usually rectangular matrices, of which second tensors are a special case, will bedenoted by upper case boldface, such as A. The transpose of a matrix is denotedby a superscript “T” , and the first index always refers to a row number, thesecond to a column number.

Thus a 2x2 matrix A and a 2x3 matrix B are writtenout as (the order of a matrix is also written with number of rows by number ofcolumns, with rows always first): A11A= A21A12 A22  B11B=  B21B12B22B13 B23 (1.3.8)In summary, we show the quadratic form associated with A in threenotationsx T Ax = x ⋅ A ⋅ x = xi Aij x j(1.3.9)The above are all equivalent: the first is matrix notation, the second in tensornotation, the third in indicial notation.Second-order tensors are often converted to Voigt.

Voigt notation isdescribed in the Glossary.1.4. MESH DESCRIPTIONSOne of the themes of this book is partially the different descriptions thatare used in the formulation of the governing equations and the discretization ofthe continuum mechanics. We will classify the finite element model in threeparts, Belytschko (1977):1. the mesh description;2. the kinetic description, which is determined by the choice of the stresstensor and the form of the momentum equation;3. the kinematic description, which is determined by the choice of thestrain measure.In this Section, we will introduce the types of mesh descriptions. For thispurpose, it is useful to introduce some definitions and concepts which will be usedthroughout this book. The first are the definitions of material coordinates andspatial coordinates.

Spatial coordinates are denoted by x; spatial coordinates arealso called Eulerian coordinates. A spatial (or Eulerian coordinate) specifies thelocation of a point in space. The material coordinates, also called Lagrangiancoordinates, are denoted by X. The material coordinate labels a material point:each material point has a unique material coordinate, which is usually taken to beits spatial coordinate in the initial configuration of the body, so at t=0, X =x.Since in a deforming body, the positions of the material points change with time,the spatial coordinates of material points will be functions of time.The motion or deformation of a body can be described by a functionφ(X, t ) , with the material coordinates X and the time t as the independent1-9T.

Belytschko, Introduction, December 16, 1998variables. This function gives the spatial positions of the material points as afunction of time through:x = φ(X, t )(1.4.1)This is also called a map between the initial and current configurations.

Thedisplacement of a material point is the difference between its current position andits original position, so it is given byu( X , t ) = φ( X , t ) − X(1.4.2)To illustrate these definitions, consider the following motion in onedimension:x = φ( X , t ) = ( 1 − X ) t + 12 Xt 2 + X(1.4.3)This motion is shown in Fig. 1.1; the motion of several points are plotted in spacetime to exhibit their trajectories; we obviously cannot plot the motion of allmaterial points since there are an infinite number.

The velocity of a material pointis the time derivative of the motion with the material coordinate fixed, i.e. thevelocity is given byv( X , t ) =∂φ( X , t )= 1 + X (t − 1)∂t(1.4.4)The mesh description is based on the choice of independent variables. Forpurposes of illustration, let us consider the velocity field. We can describe thevelocity field as a function of the Lagrangian (material) coordinates, as in Eq.(1.4.4) or we can describe the velocity as a function of the Eulerian (spatial)coordinatesv( x , t ) = v(φ −1 ( x , t ), t )(1.4.5)In the above we have placed a bar over the velocity symbol to indicate that thevelocity field when expressed in terms of the spatial coordinate x and the time twill not be the same function as that given in Eq.

(1.4.4), although in theremainder of the book we will usually not distinguish different functions whichare used to represent the same field. We have also used the concept of an inversemap which to express the material coordinates in terms of the spatial coordinatesX = ϕ −1( x , t )(1.4.6)Such inverse mappings can generally not be expressed in closed form for arbitrarymotions, but for the simple motion given in Eq. (1.4.3) it is given byX=x−t− t +1(1.4.7)1 t22Substituting the above into (3) gives1-10T. Belytschko, Introduction, December 16, 1998v( x , t ) = 1 +( x − t )(t − 1) 1 − x + xt − 12 t 21 t22− t +1=1 t22− t +1(1.4.8)Equations (1.4.4) and (1.4.8) give the same physical velocity fields, but expressthem in terms of different independent variables, so that they are differentfunctions.

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