Belytschko T. - Introduction (779635), страница 6
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The conversion of the internal nodalforce expression to the Voigt form (4.5.19) is shown in the following∂Nibσ ji dΩ = ∫ Bijbσ ji dΩ → f int = ∫ BT {σ}dΩ∂x jΩΩΩfbint = ∫1-21(4.5.23)T. Belytschko, Introduction, December 16, 1998t(a)Lagrangian Desct(b)ALE Descript1t1x, X,χx, X,χ00X1X2t (c)X3X1X4Eulerian DescrX3X4NodesMaterial Pot1x, X,χNodal TrajecMaterial Point Tra0X1X2X2X3X41-22T.
Belytschko, Chapter 2, December 16, 1998CHAPTER 2LAGRANGIAN AND EULERIAN FINITEELEMENTS IN ONE DIMENSIONby Ted BelytschkoNorthwestern University@ Copyright 19972 . 1 IntroductionIn this chapter, the equations for one-dimensional models of nonlinear continua aredescribed and the corresponding finite element equations are developed. The developmentis restricted to one dimension to simplify the mathematics so that the salient features ofLagrangian and Eulerian formulations can be demonstrated easily. These developments areapplicable to nonlinear rods and one-dimensional phenomena in continua, including fluidflow. Both Lagrangian and Eulerian meshes will be considered.
Two commonly usedtypes of Lagrangian formulations will be described: updated Lagrangian and totalLagrangian. In the former, the variables are expressed in the current (or updated)configuration, whereas in the latter the variables are expressed in terms of the initialconfiguration. It will be seen that a variety of descriptions can be developed for largedeformation problems. The appropriate description depends on the characteristics of theproblem to be solved.In addition to describing the several types of finite element formulations for nonlinearproblems, this Chapter reviews some of the concepts of finite element discretization andfinite element procedures.
These include the weak and strong forms, the operations ofassembly, gather and scatter, and the imposition of essential boundary conditions and initialconditions. Mappings between different coordinate systems are discussed along with theneed for finite element mappings to be one-to-one and onto. Continuity requirements ofsolutions and finite element approximations are also considered. While much of thismaterial is familiar to most who have studied linear finite elements, they are advised to atleast skim this Chapter to refresh their understanding.In solid mechanics, Lagrangian meshes are most popular.
Their attractiveness stemsfrom the ease with which they handle complicated boundaries and their ability to followmaterial points, so that history dependent materials can be treated accurately. In thedevelopment of Lagrangian finite elements, two approaches are commonly taken:1.
formulations in terms of the Lagrangian measures of stress and strain in whichderivatives and integrals are taken with respect to the Lagrangian (material)coordinates X, called total Lagrangian formulations2. formulations expressed in terms of Eulerian measures of stress and strain inwhich derivatives and integrals are taken with respect to the Eulerian (spatial)coordinates x, often called updated Lagrangian formulations.Both formulations employ a Lagrangian mesh, which is reflected in the term Lagrangian inthe names.Although the total and updated Lagrangian formulations are superficially quitedifferent, it will be shown that the underlying mechanics of the two formulations isidentical; furthermore, expressions in the total Lagrangian formulation can be transformedto updated Lagrangian expressions and vice versa.
The major difference between the two2-1T. Belytschko, Chapter 2, December 16, 1998formulations is in the point of view: the total Lagrangian formulation refers quantities tothe original configuration, the updated Lagrangian formulation to the current configuration,often called the deformed configuration. There are also differences in the stress anddeformation measures which are typically used in these two formulations. For example,the total Lagrangian formulation customarily uses a total measure of strain, whereas theupdated Lagrangian formulation often uses a rate measure of strain.
However these are notinherent characteristics of the formulations, for it is possible to use total measures of strainin updated Lagrangian formulations, and rate measures in total Lagrangian formulation.These attributes of the two Lagrangian formulations are discussed further in Chapter 4.Until recently, Eulerian meshes have not been used much in solid mechanics. Eulerianmeshes are most appealing in problems with very large deformations. Their advantage inthese problems is a consequence of the fact that Eulerian elements do not deform with thematerial. Therefore, regardless of the magnitudes of the deformation in a process, Eulerianelements retain their original shape. Eulerian elements are particularly useful in modelingmany manufacturing processes, where very large deformations are often encountered.For each of the formulations, a weak form of the momentum equation, which isknown as the principle of virtual work (or virtual power) will be developed.
The weakform is developed by taking the product of a test function with the governing partialdifferential equation, the momentum equation. The integration is performed over thematerial coordinates for the total Lagrangian formulation, over the spatial coordinates forthe Eulerian and updated Lagrangian formulation. It will also be shown how the tractionboundary conditions are treated so that the approximate (trial) solutions need not satisfythese boundary conditions exactly. This procedure is identical to that in linear finiteelement analysis.
The major difference in geometrically nonlinear formulations is the needto define the coordinates over which the integrals are evaluated and to specify the choice ofstress and strain measures.The discrete equations for a finite element approximation will then be derived. Forproblems in which the accelerations are important (often called dynamic problems) or thoseinvolving rate-dependent materials, the resulting discrete finite element equations areordinary differential equations (ODEs). The process of discretizing in space is called asemidiscretization since the finite element procedure only converts the spatial differentialoperators to discrete form; the derivatives in time are not discretized.
For static problemswith rate-independent materials, the discrete equations are independent of time, so the finiteelement discretization results in a set of nonlinear algebraic equations.Examples of the total and updated Lagrangian formulations are given for the 2-node,linear displacement and 3-node, quadratic displacement elements. Finally, to enable thestudent to solve some nonlinear problems, a central difference explicit time-integrationprocedures is described.2 .
2 Governing Equations For Total Lagrangian FormulationNomenclature. Consider the rod shown in Fig. 1. The initial configuration, alsocalled the undeformed configuration of the rod, is shown in the top of the figure. Thisconfiguration plays an important role in the large deformation analysis of solids.
It is alsocalled the reference configuration, since all equations in the total Lagrangian formulationare referred to this configuration. The current or deformed configuration is shown at thebottom of the figure. The spatial (Eulerian) coordinate is denoted by x and the coordinatesin the reference configuration, or material (Lagrangian) coordinates, by X . The initialcross-sectional area of the rod is denoted by A0( X ) and its initial density by ρ0 ( X ) ;2-2T. Belytschko, Chapter 2, December 16, 1998variables pertaining to the reference (initial, undeformed) configuration will always beidentified by a subscript or superscript nought. In this convention, we could indicate thematerial coordinates by x0 since they correspond to the initial coordinates of the materialpoints, but this is not consistent with most of the continuum mechanics literature, so wewill always use X for the material coordinates.The cross-sectional area in the deformed state is denoted by A( X, t ) ; as indicated, it isa function of space and time.
The spatial dependence of this variable and all others isexpressed in terms of the material coordinates. The density is denoted by ρ( X,t ) and thedisplacement by u( X,t ) . The boundary points in the reference configuration are X a andXb .A o(X)Tx,XXbXa_A(X) = A(x)TxXaxb = ( Xb, t )Fig. 1.1. The undeformed (reference) configuration and deformed (current) configurations for a onedimensional rod loaded at the left end; this is the model problem for Sections 2.2 to 2.8.Deformation and Strain Measure. The variables which specify the deformation andthe stress in the body will first be described.
The motion of the body is described by afunction of the Lagrangian coordinates and time which specifies the position of eachmaterial point as a function of time:x = φ ( X,t )X ∈[ Xa , X b](2.2.1)where φ( X, t) is called a deformation function. This function is often called a map betweenthe initial and current domains. The material coordinates are given by the deformationfunction at time t = 0, soX =φ ( X, 0)(2.2.2)As can be seen from the above, the deformation function at t = 0 is the identity map.The displacement u(X ,t ) is given by the difference between the current position andthe original position of a material point, sou(X ,t ) = φ (X, t ) − X or u = x − X(2.2.3)The deformation gradient is defined by2-3T.
Belytschko, Chapter 2, December 16, 1998F=∂φ ∂x=∂X ∂X(2.2.4)The second definitions in Eq. (2.2.3) and (2.2.4) can at times be ambiguous. Forexample, Eq. (2.2.4) appears to involve the partial derivative of an independent variable xwith respect to another independent variable X , which is meaningless. Therefore, itshould be understood that whenever x appears in a context that implies it is a function, thedefinition x = φ ( X,t ) is implied.Let J be the Jacobian between the current and reference configurations. The Jacobianis usually defined by J( x( X) ) = ∂x / ∂X for one-dimensional mappings; however, tomaintain consistency with multi-dimensional formulations of continuum mechanics, wewill define the Jacobian as the ratio of an infinitesimal volume in the deformed body, A∆x ,to the corresponding volume of the segment in the undeformed body A0∆X , so it is givenbyAFAJ = ∂x=∂X A0A0(2.2.5)The deformation gradient F is an unusual measure of strain since its value is one whenthe body is undeformed.
We will therefore define the measure of strain byε ( X, t ) = F ( X, t ) –1≡ ∂x –1= ∂u∂X∂X(2.2.6)so that it vanishes in the undeformed configuration. There are many other measures ofstrain, but this is the most convenient for this presentation. This measure of straincorresponds to what is known as the stretch tensor in multi-dimensional problems. In onedimension, it is equivalent to the engineering strain.Stress Measure. The stress measure which is used in total Lagrangian formulationsdoes not correspond to the well known physical stress. To explain the measure of stress tobe used, we will first define the physical stress, which is also known as the Cauchy stress.Let the total force across a given section be denoted by T and assume that the stress isconstant across the cross-section.
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