Belytschko T. - Introduction (779635), страница 87
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Loworder elements, when applied to incompressible materials, tend to exhibit volumetriclocking. In volumetric locking, the displacements are underpredicted by large factors, 5 to10 is not uncommon for otherwise reasonable meshes. Although incompressible materialsare quite rare in linear stress analysis, in the nonlinear regime many materials behave in anearly incompressible manner. For example, Mises elastic-plastic materials areincompressible in their plastic behavior. Though the elastic behavior may be compressible,the overall behavior is nearly incompressible, and an element that locks volumetrically willnot perform well for Mises elastic-plastic materials.
Rubbers are also incompressible inlarge deformations. To be applicable to a large class of nonlinear materials, an elementmust be able to treat incompressible materials effectively. However, most elements haveshortcomings in their performance when applied to incompressible or nearlyincompressible materials. An understanding of these shortcomings are crucial in theselection of elements for nonlinear analysis.To eliminate volumetric locking, two classes of techniques have evolved:1. multi-field elements in which the pressures or complete stress and strain fieldsare also considered as dependent variables;2. reduced integration procedures in which certain terms of the weak form for theinternal forces are underintegrated.Multi-field elements are based on multi-field weak forms or variational principles; these arealso known as mixed variational principles.
In multi-field elements, additional variables,such as the stresses or strains, are considered as dependent, at least on the element level,and interpolated independently of the displacements. This enables the strain or stress fieldsto be designed so as to avoid volumetric locking. In many cases, the strain or stress fieldsare also designed to achieve better accuracy for beam bending problems. These methodscannot improve the performance of an element in general when there are no constraints6such as incompressibility.
In fact, for a 4-node quadrilateral, only a 3 parameter family ofelements is convergent and the rate of convergence can never exceed that of the 4-nodequadrilateral. Thus the only goals that can be achieved by mixed elements is to avoidlocking and to improve behavior in a selected class of problems, such as beam bending.The unfortunate byproduct of using multi-field variational principles is that in manycases the resulting elements posses instabilities in the additonal fields.
Thus most 4-nodequadrilaterals based on multi-field weak forms are subject to a pressure instability. Thisrequires another fix, so that the resulting element can be quite complex. The developemntof truly robust elements is not easy, particularly for low order elements. For this reason,an understanding of element technology is useful to anyone engaged in finite elementanalysis.Elements developed by means of underintegration in its various forms are quite similarfrom a fundamental and practical viewpoint to elements based on multi-field variationalprinciples, and the equivalence was proven by Malkus and Hughes() for certain classes ofelements. Therefore, while underintegration is more easily understood than multi-fieldapproaches, the methods suffer from the same shortcomings as multi-field elements:pressure instabilities.
Nevertheless, they provide a straightforward way to overcomelocking in certain classes of elements.We will begin the chapter with an overview of element performance in Section 8.1.This Section describes the characteristics of many of the most widely used elements forcontinuum analysis. The description is limited to elements which are based on polynomialsof quadratic order or lower, since elements of higher order are seldom used in nonlinearanalysis at this time.
This will set the stage for the material that follows. Many readersmay want to skip the remainder of the Chapter or only read selected parts based on whatthey have learned from this Section.Although the techniques introduced in this Chapter are primarily useful for controllingvolumetric locking for incompressible and nearly incompressible materials, they applymore generally to what can collectively be called constrained media problems.
Anotherimportant class of such problems are structural problems, such as thin-walled shells andbeams. The same techniques described in this Chapter will be used in Chapter 9 to developbeam and shell elements.Section 8.3 describes the patch tests. These are important, useful tests for theperformance of an element.
Patch tests can be used to examine whether an element isconvergent, whether it avoids locking and whether it is stable. Various forms of the patchtest are described which are applicable to both static programs and programs with explicittime integration. They test both the underlying soundness of the approximations used inthe elements and the correctness of the implementation.Section 8.4 describes some of the major multi-field weak forms and their application toelement development. Although the first major multi-field variational principle to bediscovered for elasticity was the Hellinger-Reissner variational principle, it is notconsidered because it can not be readily used with strain-driven constitutive equations innonlinear analysis.
Therefore, we will confine ourselves to various forms of the HuWashizu principles and some simplifications that are useful in the design of new elements.7We will also describe some limitation principles and stability issues which pertain to mixedelements.To illustrate the application of element technology, we will focus on the 4-nodeisoparametric quadrilateral element (QUAD4).
This element is convergent for compressiblematerial without any modifications, so none of the techniques described in this Chapter areneeded if this element is to be used for compressible materials. On the other hand, forincompressible or nearly incompressible materials, this element locks. We will illustratetwo classes of techniques to eliminate volumetric locking: reduced integration and multifield elements. We then show that reduced integration by one-point quadrature is rankdeficient, which leads to spurious singular modes.
To stabilize these modes, we firstconsider perturbation hourglass stabilization of Flanagan and Belytschko (1981). We thenderive mixed methods for stabilization of Belytschko and Bachrach (1986), and assumedstrain stabilization of Belytschko and Bindeman (1991). We show that assumed strainstabilization can be used with multiple-point quadrature to obtain better results when thematerial response is nonlinear without great increases in cost. The elements of Pian andSumihara() and Simo and Rifai() are also described and compared. Numerical results arealso presented to demonstrate the performance of various implementations of this element.Finally, the extension of these results to the 8-node hexhedron is sketched.8.2.
Overview of Element PerformanceIn this Section, we will provide an overview of characteristics of various widely-usedelements with the aim of giving the reader a general idea of how these elements perform,their advantages and their major difficulties. This will provide the reader with anunderstanding of the consequences of the theoretical results and procedures which aredescribed later in this Chapter. We will concentrate on elements in two dimensions, sincethe properties of these elements parallel those in three dimensions; the correspondingelements in three dimension will be specified and briefly discussed. The overview islimited to continuum elements; the properties of shell elements are described in Chapter 9.In choosing elements, the ease of mesh generation for a particular element should beborne in mind.
Triangles and tetrahedral elements are very attractive because the mostpowerful mesh generators today are only applicable to these elements. Mesh generators forquadrilateral elements tend to be less robust and more time consuming. Therefore,triangular and tetrahedral elements are preferable when all other performance characteristicsare the same for general purpose analysis.The most frequently used low-order elements are the three-node triangle and the fournode quadrilateral. The corresponding three dimensional elements are the 4-nodetetrahedron and the 8-node hexahedron.
The detailed displacement and strain fields aregiven later, but as is well-known to anyone familiar with linear finite element theory, thedisplacement fields of the triangle and tetrahedron are linear and the strains are constant.The displacement fields of the quadrilateral and hexahedron are bilinear and trilinear,respectively.
All of these elements can represent a linear displacement field and constantstrain field exactly. Consequently they satisfy the standard patch test, which is described inSection 8.3. The satisfaction of the standard patch test insures that the elements converge8in linear analysis, and provide a good guarantee for convergent behavior in nonlinearproblems also, although there are no theoretical proofs of this statement.We will first discuss the simplest elements, the three-node triangle in two dimensions,the four-node tetrahedron in three dimensions. These are also known as simplex elementsbecause a simplex is a set of n+1 points in n dimensions. Neither simplex elementperforms very well for incompressible materials.
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