Belytschko T. - Introduction (779635), страница 20
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There are many others, butfrankly even these are too many for most beginning students. The profusion ofstress and strain measures is one of the obstacles to understanding nonlinearcontinuum mechanics. Once one understands the field, one realizes that the largevariety of measures adds nothing fundamental, and is perhaps just a manifestationof academic excess. Nonlinear continuum mechanics could be taught with justone measure of stress and strain, but additional ones need to be covered so that theliterature and software can be understood.The conservation equations, which are often called the balance equations,are derived next. These equations are common to both solid and fluid mechanics.They consist of the conservation of mass, momentum and energy.
Theequilibrium equation is a special case of the momentum equation which applieswhen there are no accelerations in the body. The conservation equations arederived both in the spatial and the material domains. In a first reading orintroductory course, the derivations can be skipped, but the equations should bethoroughly known in at least one form.The Chapter concludes with further study of the role of rotations in largedeformation continuum mechanics. The polar decomposition theorem is derivedand explained.
Then objective rates, also called frame-invariant rates, of theCauchy stress tensor are examined. It is shown why rate type constitutiveequations in large rotation problems require objective rates and several objectiverates frequently used in nonlinear finite elements are presented. Differencesbetween objective rates are examined and some examples of the application ofobjective rates are illustrated.3.2 DEFORMATION AND MOTION3.2.1 Definitions.Continuum mechanics is concerned with models of solidsand fluids in which the properties and response can be characterized by smoothfunctions of spatial variables, with at most a limited number of discontinuities.
Itignores inhomogeneities such as molecular, grain or crystal structures. Featuressuch as crystal structure sometimes appear in continuum models through theconstitutive equations, and an example of this kind of model will be given inChapter 5, but in all cases the response and properties are assumed to be smoothwith a countable number of discontinuities. The objective of continuummechanics is to provide a description to model the macroscopic behavior of fluids,solids and structures.Consider a body in an initial state at a time t=0 as shown in Fig.
3.1; thedomain of the body in the initial state is denoted by Ω 0 and called the initialconfiguration. In describing the motion of the body and deformation, we alsoneed a configuration to which various equations are referred; this is called thereference configuration.
Unless we specify otherwise, the initial configuration isused as the reference configuration. However, other configurations can also beused as the reference configuration and we will do so in some derivations. The3-22T. Belytschko, Continuum Mechanics, December 16, 19983significance of the reference configuration lies in the fact that motion is definedwith respect to this configuration.φ ( X, t)y, YuXΩ0xΓ0ΩΓx, XFig. 3.1.
Deformed (current) and undeformed (initial) configurations of a body.In many cases, we will also need to specify a configuration which isconsidered to be an undeformed configuration. The notion of an "undeformed"configuration should be viewed as an idealization, since undeformed objectsseldom exist in reality. Most objects previously had a different configuration andwere changed by deformations: a metal pipe was once a steel ingot, a cellulartelephone housing was once a vat of liquid plastic, an airport runway was once atruckload of concrete.
So the term undeformed configuration is only relative anddesignates the configuration with respect to which we measure deformation. Inthis Chapter, the undeformed configuration is considered to be the initialconfiguration unless we specifically say otherwise, so it is tacitly assumed that inmost cases the initial, reference, and undeformed configurations are identical .The current configuration of the body is denoted by Ω ; this will often alsobe called the deformed configuration. The domain currently occupied by the bodywill also be denoted by Ω . The domain can be one, two or three dimensional; Ωthen refers to a line, an area, or a volume, respectively.
The boundary of thedomain is denoted by Γ , and corresponds to the two end-points of a segment inone dimension, a curve in two dimensions, and a surface in three dimensions. Thedevelopments which follow hold for a model of any dimension from one to three.The dimension of a model is denoted by nSD , where “SD” denotes spacedimensions.For a Lagrangian finite element mesh, the initial mesh is a discrete modelof the initial, undeformed configuration, which is also the reference configuration.The configurations of the solution meshes are the current, deformedconfigurations. In an Eulerian mesh, the correspondence is more difficult topicture and is deferred until later.3.2.2 Eulerian and Lagrangian Coordinates.
The position vector of amaterial point in the reference configuration is given by X, where3-3T. Belytschko, Continuum Mechanics, December 16, 1998X = X iei ≡4nSD∑ Xie i(3.2.1)i =1where Xi are the components of the position vector in the reference configurationand e i are the unit base vectors of a rectangular Cartesian coordinate system;indicial notation as described in Section 1.3 has been used in the secondexpression and will be used throughout this book. Some authors, such as Malvern(1969), also define material particles and carefully distinguish between materialpoints and particles in a continuum. The notion of particles in a continuum issomewhat confusing, for the concept of particles to most of us is discrete ratherthan continuous.
Therefore we will refer only to material points of the continuum.The vector variable X for a given material point does not change withtime; the variables X are called material coordinates or Lagrangian coordinatesand provide labels for material points. Thus if we want to track the functionf ( X,t ) at a given material point, we simply track that function at a constant valueof X. The position of a point in the current configuration is given byx = xi ei ≡nSD∑ xie i(3.2.2)i =1where xi are the components of the position vector in the current configuration.3.2.3 Motion. The motion of the body is described byx = φ ( X,t )orxi = φi ( X, t)(3.2.3)where x = xi ei is the position at time t of the material point X. The coordinates xgive the spatial position of a particle, and are called spatial, or Euleriancoordinates. The function φ( X,t ) maps the reference configuration into thecurrent configuration at time t., and is often called a mapping or map.When the reference configuration is identical to the initial configuration,as assumed in this Chapter, the position vector x of any point at time t=0coincides with the material coordinates, soX = x( X,0 ) ≡ φ( X, 0)Xi = xi( X, 0) = φi (X, 0)or(3.2.4)Thus the mapping φ( X,0) is the identity mapping.Lines of constant Xi , when etched into the material, behave just like aLagrangian mesh; when viewed in the deformed configuration, these lines are nolonger Cartesian.
Viewed in this way, the material coordinates are often calledconvected coordinates. In pure shear for example, they become skewedcoordinates, just like a Lagrangian mesh becomes skewed, see Fig. 1.2. However,when we view the material coordinates in the reference configuration, they areinvariant with time. In the equations to be developed here, the material3-4T. Belytschko, Continuum Mechanics, December 16, 19985coordinates are viewed in the reference configuration, so they are treated as aCartesian coordinate system.
The spatial coordinates, on the other hand, do notchange with time regardless of how they are viewed.3.2.4 Eulerian and Lagrangian Descriptions.Two approaches areused to describe the deformation and response of a continuum. In the firstapproach, the independent variables are the material coordinates X and the time t,as in Eq. (3.2.3); this description is called a material description or Lagrangiandescription.
In the second approach, the independent variables are the spatialcoordinates x and the time t. This is called a spatial or Eulerian description. Theduality is similar to that in mesh descriptions, but as we have already seen in finiteelement formulations, not all aspects of a single formulation are exclusivelyEulerian or Lagrangian; instead some finite element formulations combineEulerian and Lagrangian descriptions as needed.In fluid mechanics, it is often impossible and unnecessary to describe themotion with respect to a reference configuration.
For example, if we consider theflow around an airfoil, a reference configuration is usually not needed for thebehavior of the fluid is independent of its history. On the other hand, in solids,the stresses generally depend on the history of deformation and an undeformedconfiguration must be specified to define the strain. Because of the historydependence of most solids, Lagrangian descriptions are prevalent in solidmechanics.In the mathematics and continuum mechanics literature, cf. Marsden andHughes (1983), different symbols are often used for the same field when it isexpressed in terms of different independent variables, i.e.
when the description isEulerian or Lagrangian. In this convention, the function which in an Euleriandescription is f(x,t) is denoted by F(X,t) in a Lagrangian description. The twofunctions are related byF( X, t ) = f ( φ( X, t), t), or F = f o φ(3.2.5)This is called a composition of functions; the notation on the right is frequentlyused in the mathematics literature; see for example Spivak(1965, p.11). Thenotation for the composition of functions will be used infrequently in this bookbecause it is unfamiliar to most engineers.The convention of referring to different functions by different symbols isattractive and often adds clarity. However in finite element methods, because ofthe need to refer to three or more sets of independent variables, this conventionbecomes quite awkward. Therefore in this book, we associate a symbol with afield, and the specific function is defined by specifying the independent variables.Thus f(x,t) is the function which describes the field f for the independent variablesx and t, whereas f(X,t) is a different function which describes the same field interms of the material coordinates.
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