Belytschko T. - Introduction (779635), страница 19

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Substituting (A.6) into (A.5) and dividing by ∆x gives( Aσ ), x + ρAb = ρA˙u˙(A.7)The above is the momentum equation for a one-dimensional continuum of varying crosssection.To derive the momentum equation in the reference configuration, we note that theforces on the sides of the segment are given by multiplying the nominal stress by the initialarea, A0P . The net force due to the body force is ρ0 A0b∆X since ρ0 b is a force per unitinitial volume and the initial volume is A0∆X .

The mass of the segment is ρ0 A0∆X .Writing Newton's second law for the segment gives(–A0P ) X + ( A0P ) X + ∆X + ( ρ0 A0b) X + ∆X ∆X = ( ρ0 A0 bu˙˙ ) X + ∆X ∆X22-642(A.8)T. Belytschko, Chapter 2, December 16, 1998where the LHS is the sum of all forces acting on the segment and the RHS is the mass timethe acceleration. Expressing the forces due to the nominal stresses by a Taylor series as in(A.6), but in terms of the material coordinate X , substituting into (A.8) and dividing by∆X gives the momentum equation in Lagrangian form( A0 P) , X + ρ0 A0 b = ρ 0 A0˙u˙(A.9)The above can easily be transformed to the Eulerian form, Eq.

(A.7). By the stresstransformation (2.1.2), we have A0P = Aσ , so( A0 P) , x = (Aσ ), x = (Aσ ), x x, X = (Aσ ), x F(A.10)where the chain rule has been used in the third step, followed by the definition of F in Eq.(2.2.2). Substituting the (A.10) into (A.9) gives0 = ( Aσ ), x F + ρ0 A0b − ρ 0 A0˙u˙ = ( Aσ ), x F + FρAb − FρAu˙˙where the continuity equation (A.2) has been used in the last step.

Dividing by F thengives the momentum equation in Eulerian form. Note that the body force in the Lagrangianand Eulerian momentum equations is identical. Some authors distinguish the body force inthe total form by a subscript naught, i.e., Malvern (1969, p. 224), but this is superfluous ifthe body force is considered a force per unit mass so that ρb is a force per unit volume.SUMMARYThe finite element equations have been developed for one-dimensional continua ofvarying cross-section. Two mesh descriptions have been used:1.

Lagrangian meshes, where the nodes and elements move with the material;2. Eulerian meshes, in which nodes and elements are fixed in space.Two formulations have been developed for Lagrangian meshes:1. a total Lagrangian formulation, in which the strong form is expressed in spatialcoordinates, i.e.

the Eulerian coordinates;2. an updated Lagrangian formulation, where the strong form is expressed in thematerial, i.e. the Lagrangian coordinates.In both cases, the element formulation is most conveniently executed in terms of theelement coordinates. The mapping of the element coordinates from current and originalconfiguration for a valid finite element discretization is one-to-one and onto. Furthermore,the mapping to the original configuration is time invariant, so the element coordinates canserve as surrogate material coordinates.It has also been shown that the updated and total Lagrangian formulations are tworepresentations of the same mechanical behavior, and each can be transformed to the otherat both the level of partial differential equations and the level of the discrete finite elementequations. Thus the internal and external forces obtained by the total Lagrangianformulations are identical to those obtained by the updated formulation, and the choice offormulation is purely a matter of convenience.2-65T.

Belytschko, Chapter 2, December 16, 1998The equation of motion corresponds to the momentum equation and is obtained fromits weak form. As has been illustrated in the case of explicit time integration, the otherequations, measure of deformation and constitutive, are used in the course of computingthe internal nodal forces to update the displacements.

The weak form and discreteequations have been structured so that their relationship to the corresponding terms in thepartial differential equation of momentum conservation is readily apparent: the internalforces correspond to the stress terms, and the internal work (or power); the external forcescorrespond to the body forces and external work (or power); the terms Ma correspond tothe inertial terms (d'Alembert) forces and the inertial work (or power).Thiscorrespondence is summarized in Fig. 8, which shows the steps which are used to convertthe partial differential equation of momentum balance to a set of ordinary differentialequations which are called the equations of motion.

This process is called a spatialdiscretization or semidiscretization.The discretization has been carried out for the general case when inertial forces are notnegligible. If the inertial forces can be neglected, the term Ma is omitted from the discreteequations. The resulting equations are either nonlinear algebraic equations or ordinarydifferential equations, depending on the character of the constitutive equation.The governing equations have been developed for a one-dimensional rod of varyingcross-section and from these a weak form has been developed by integrating over thedomain.

When the equations are given in terms of partial derivatives with respect to thematerial derivatives, it is natural to develop the weak form by integrating over theundeformed domain. This leads to the total Lagrangian formulation where all nodalforces are obtained by integrating over the material coordinates. When the partialderivatives are with respect to the spatial coordinates, it is natural to integrate over thecurrent configuration, which leads to the updated Lagrangian formulation.The process of discretization for multidimensional problems is very similar.However, in multi-dimensional problems we will have to deal with the major consequenceof geometric nonlinearities, large rotations, which are completely absent in one-dimensionalproblems.ExercisesExercise: Repeat Example 2.8.3 for spherical symmetry, where η r h = ηθ ηφ  σ r s = σ θ σ φ give B, feint , feext , M e2-66T.

Belytschko, Continuum Mechanics, December 16, 1998CHAPTER 3CONTINUUM MECHANICSby Ted BelytschkoNorthwestern UniversityCopyright 1996DRAFT3.1 INTRODUCTIONContinuum mechanics is an essential building block of nonlinear finiteelement analysis, and a mastery of continuum mechanics is essential for a goodunderstanding of nonlinear finite elements.

This chapter summarizes thefundamentals of nonlinear continuum mechanics which are needed for adevelopment of nonlinear finite element methods. It is, however, insufficient forthoroughly learning continuum mechanics. Instead, it provides a review of thetopics that are particularly relevant to nonlinear finite element analysis. Thecontent of this chapter is limited to topics that are needed for the remainder of thebook.Readers who have little or no familiarity with continuum mechanicsshould consult texts such as Hodge (1970), Mase and Mase (1992), Fung (1994),Malvern (1969), or Chandrasekharaiah and Debnath (1994).

The first three arethe most elementary. Hodge (1970) is particularly useful for learning indicialnotation and the fundamental topics. Mase and Mase (1992) gives a conciseintroduction with notation almost identical to that used here. Fung (1994) is aninteresting book with many discussions of how continuum mechanics is applied.The text by Malvern (1969) has become a classic in this field for it provides avery lucid and comprehensive description of the field. Chandrasekharaiah andDebnath (1994) gives a thorough introduction with an emphasis on tensornotation. The only topic treated here which is not presented in greater depth in allof these texts is the topic of objective stress rates, which is only covered inMalvern. Monographs of a more advanced character are Marsden and Hughes(1983), Ogden (1984) and Gurtin (). Prager (1961), while an older book, stillprovides a useful description of continuum mechanics for the reader with anintermediate background.

The classic treatise on continuum mechanics isTruesdell and Noll (1965) which discusses the fundamental issues from a verygeneral viewpoint. The work of Eringen (1962) also provides a comprehensivedescription of the topic.This Chapter begins with a description of deformation and motion,including some useful equations for characterizing deformation and the timederivatives of variables. Rigid body motion is described with an emphasis onrigid body rotation.

Rigid body rotation plays a central role in nonlinearcontinuum mechanics, and many of the more difficult and complicated aspects ofnonlinear continuum mechanics stem from rigid body rotation. The materialconcerning rigid body rotation should be carefully studied.Next, the concepts of stress and strain in nonlinear continuum mechanicsare described. Stress and strain can be defined in many ways in nonlinearcontinuum mechanics.

We will confine our attention to the strain and stress3-11T. Belytschko, Continuum Mechanics, December 16, 1998measures which are most frequently employed in nonlinear finite elementprograms. We cover the following kinematic measures in detail: the Green straintensor and the rate-of-deformation. The second is actually a measure of strainrate, but these two are used in the majority of software. The stress measurestreated are: the physical (Cauchy) stress, the nominal stress and the second PiolaKirchhoff stress, which we call PK2 for brevity.

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