Belytschko T. - Introduction (779635), страница 21
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The independent variables are always indicatednear the beginning of a section or chapter, and if a change of independentvariables is made, the new independent variables are noted.3-5T. Belytschko, Continuum Mechanics, December 16, 199863.3.5Displacement, Velocity and Acceleration. The displacement ofa material point is given by the difference between its current position and itsoriginal position (see Fig.
3.1), sou (X,t ) = φ( X, t) − φ(X,0) = φ(X,t ) −X ,()ui = φ i X j , t − Xi(3.2.6)where u (X,t ) = ui ei and we have used Eq. (3.2.4). The displacement is oftenwritten asu = x − X,ui = xi − Xi(3.2.7)( )where (3.2.1) has been used in (3.2.6) to replace φ X, t by x . Equation (3.2.7) issomewhat ambiguous since it expresses the displacement as the difference of twovariables, x and X, both of which are generally independent variables.
The readermust keep in mind that in expressions such as (3.2.7) the variable x represents themotion x ( X, t) ≡ φ( X,t ) .( )The velocity v X, t is the rate of change of the position vector for amaterial point, i.e. the time derivative with X held constant. Time derivativeswith X held constant are called material time derivatives; or sometimes materialderivatives.
Material time derivatives are also called total derivatives. Thevelocity can be written in the various forms shown below∂φ( X, t) ∂u ( X, t)v( X, t) = ˙u ==∂t∂t(3.2.8)In the above, the variable x is replaced by the displacement u in the fourth termby using (3.2.7) and the fact that X is independent of time. The symbol D( )/Dtand the superposed dot always denotes a material time derivative in this book,though the latter is often used for ordinary time derivatives when the variable isonly a function of time.The acceleration a( X, t) is the rate of change of velocity of a materialpoint, or in other words the material time derivative of the velocity, and can bewritten in the formsDv ˙ ∂v( X,t ) ∂ 2 u (X,t )a( X, t) =≡v ==Dt∂t∂t2(3.2.9)The above expression is called the material form of the acceleration.When the velocity is expressed in terms of the spatial coordinates and thetime, i.e.
in an Eulerian description as in v(x,t), the material time derivative isobtained as follows. The spatial coordinates in v(x,t) are first expressed as afunction of the material coordinates and time by using (3.2.3), giving v( φ( X,t ),t ) .The material time derivative is then obtained by the chain rule:3-6T. Belytschko, Continuum Mechanics, December 16, 19987Dvi ∂vi (x,t ) ∂vi (x, t ) ∂φ j (X,t ) ∂vi ∂vi=+=+v(3.2.10)Dt∂t∂x j∂t∂t ∂x j jwhere the second equality follows from (3.2.8).
The second term on the RHS of(3.2.10) is the convective term, which is also called the transport term. In(3.2.10), the first partial derivative on the RHS is taken with the spatial coordinatefixed. This is called the spatial time derivative. It is tacitly assumed throughoutthis book that when neither the independent variables nor the fixed variable areexplicitly indicated in a partial derivative with respect to time, then the spatialcoordinate is fixed and we are referring to the spatial time derivative. On theother hand, when the independent variables are specified as in (3.2.8-9), a partialderivative can specify a material time derivative.
Equation (3.2.10) is written intensor notation asDv ∂v∂v=+ v⋅∇ v =+ v⋅ grad vDt ∂t∂t(3.2.11)The material time derivative of any variable which is a function of thespatial variables x and time t can similarly be obtained by the chain rule. Thusfor a scalar function f (x, t ) and a tensor function σ ij ( x,t ) , the material timederivatives are given byDf ∂f∂f ∂f∂f= + vi= + v⋅∇ f = + v⋅grad fDt ∂t∂xi ∂t∂tDσ ij ∂σ ij∂σ ij ∂σ∂σ=+ vk=+ v⋅∇ σ =+ v⋅grad σDt∂t∂x k ∂t∂t(3.2.12)(3.2.13a)where the first term on the RHS of each equation is the spatial time derivative andthe second term is the convective term.It should be remarked that the complete description of the motion is notneeded to develop the material time derivative in an Eulerian description.
InEulerian meshes, the motion cannot be defined realistically defined as a functionof the material positions in the initial configuration; see Chapter 7. In that case,variables such as the velocity can be developed by describing the motion withrespect to a reference configuration that coincides with the configuration at a fixedtime t.For this purpose, let the configuration at time fixed time t = τ be thereference configuration and the position vector at that time, denoted by Xτ , be thereference coordinates. These reference coordinates are given byXτ = φ(X, τ )(3.2.13b)Observe we use an upper case X since we wish to clearly identify it as anindependent variable, and we add the superscript τ to indicate that these referencecoordinates are not the position vectors at the initial time.
The motion can bedescribed in terms of these reference coordinates by3-7T. Belytschko, Continuum Mechanics, December 16, 1998()x = φτ Xτ , tfor t ≥τ8(3.2.13c)Now the arguments used to develop (3.2.10) can be repeated; noting thatv( x, t ) = v φτ (X, t ), t()Dvi ∂vi( x, t) ∂v (x, t ) ∂φiτ=+Dt∂t∂xi∂t(3.2.13d)with t = τ . Reference configurations coincident with a configuration other thanthe initial configuration will also be employed in the development of finiteelement equations.3.2.6DeformationGradient. The description of deformation and themeasure of strain are essential parts of nonlinear continuum mechanics. Animportant variable in the characterization of deformation is the deformationgradient.
The deformation gradient is defined byFij =∂φi ∂xi≡∂X j ∂X jor F =∂φ ∂xT≡≡ (∇ X φ)∂X ∂X(3.2.14)Note in the above that the first index of Fij refers to the component of thedeformation, the second to the partial derivative. The order can be rememberedby noting that the indices appear in the same order in Fij as in the expression forthe partial derivative if it is written horizontally as ∂φi ∂X j . The operator ∇X isthe left gradient with respect to the material coordinates. We will only use theleft gradient in this book, but to maintain consistency with the notation of otherssuch as Malvern, we follow his convention exactly.
Therefore, the transpose of∇X φ appears in the above because of the convention on subscripts: for the leftgradient, the first subscript is the pertains to the gradient, but in Fij the gradient isassociated with the second index. The distinction between left and right gradientsis not of importance in this book because we will always use the left gradient, butwe adhere to the convention so that our equations are consistent with thecontinuum mechanics literature. In the terminology of mathematics, thedeformation gradient is the Jacobian matrix of the vector function φ( X,t ) .If we consider an infinitesmal line segment dX in the referenceconfiguration, then it follows from (3.2.14) that the corresponding line segmentdx in the current configuration is given bydx = F ⋅dX or dx i = Fij dX j(3.2.15)In the above expression, the dot could have been omitted between the F and dX ,since the expression is also valid as a matrix expression.
We have retained it toconform to our conventionof always explicitly indicating contractions in tensorexpressions.3-8T. Belytschko, Continuum Mechanics, December 16, 19989In two dimensions, the deformation gradient in a rectangular coordinatesystem is given by ∂x1∂XF = ∂x 1 2 ∂X 1∂x1 ∂x∂X 2 = ∂X∂y∂x2∂X 2 ∂X∂x ∂Y ∂y∂Y (3.2.16)As can be seen in the above, in writing a second-order tensor in matrix form, weuse the first index for the row number, the second index for the column number.The determinant of F is denoted by J and called the Jacobian determinantor the determinant of the deformation gradientJ = det(F )(3.2.17)The Jacobian determinant can be used to relate integrals in the current andreference configurations by∫ f dΩ= ∫ f J dΩ0Ωor in 2D:Ω0∫ f ( x, y) d xdy = ∫ f ( X,Y ) J dXdYΩ(3.2.18)Ω0The material derivative of the Jacobian determinant is given byDJ ˙∂v≡ J = Jdivv ≡ J iDt∂xi(3.2.19)The derivation of this formula is left as an exercise.3.2.6 Conditions on Motion.
The mapping φ( X,t ) which describes themotion and deformation of the body is assumed to satisfy the followingconditions:1. the function φ( X,t ) is continuous and continuously differentiableexcept on a finite number of sets of measure zero;2. the function φ( X,t ) is one-to-one and onto;3. the Jacobian determinant satisfies the condition J>0.These conditions ensure that φ( X,t ) is sufficiently smooth so that compatibility issatisfied, i.e. so there are no gaps or overlaps in the deformed body.
The motionand its derivatives can be discontinuous or posses dicontinuous derivatives on setsof measure zero; see Section 1.5, so it is characterized as piecewise continuouslydifferentiable. Sets of measure zero are points in one dimension, lines in twodimensions and planes in three dimensions because a point has zero length, a linehas zero area, and a surface has zero volume.The deformation gradient, i.e. the derivatives of the motion, is generallydiscontinuous on interfaces between materials. Discontinuities in the motionitself characterize phenomena such as a growing crack.
We require the number ofdiscontinuities in a motion and its derivatives to be finite. In fact, in some3-9T. Belytschko, Continuum Mechanics, December 16, 199810nonlinear problems, it has been found that the solutions posses an infinite numberof discontinuities, see for example James () and Belytschko, et al (1986).However, these solutions are quite unusual and cannot be treated effectively byfinite element methods, so we will not concern ourselves with these types ofproblems.The second condition in the above list requires that for each point in thereference configuration Ω 0 , there is a unique point in Ω and vice versa. This is asufficient and necessary condition for the regularity of F, i.e.
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