Belytschko T. - Introduction (779635), страница 72
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an important recent work is Triantifyllides and ?, whoproposed a technique for relating unit cell models to the parameters in a nonlocaltheory. deBorst et al (??) further investigated the Schreyer et al approach andshowed that that consistency (5.??) requirement then intdroduces another partialdifferential equation into the system; the boundary conditions for these partialdifferential equations are still an enigma. Hutchinson and Fleck() showed6-74T. Belytschko & B. Moran, Solution Methods, December 16, 1998expreimentally that metal plasticity depends on scale and developed a gradientplasticity theory motivated by dislocation movement.Regularization Techniques.
There are thus four regularization techniques that areunder study for unstable materials:1. gradient regularization, in which a gradient of a field variable isintroduced in the constitutive equation2. integral, or nonlocal, regularization, in which the the constitutiveequation is a function of a nonlocal variable, such as nonlocal damage,a nonlocal invariant of a strain, or a nonlocal strain.3. coupled stress regularizaztion4, regularization by introducing time dependence into the materialAll of these are except the last are still in an embryonic state of development.Little is known about the material constants and the associated material lengthscales which are required.Regularization by introducing time dependence has progressed faster than theothers because viscoplastic material laws has achieved a stat e of maturity by thetime that localization became a hot area of research.
However, viscoplasticregularization has some notable peculiarites: there is no constant length scale inthe viscoplastic maodel and the solution in the presence of matrial instability ischaracterized by exponential growth. Therefore, although a discontinuity doesnot develop in te displacement as in the rate-independent strain-softeningmaterial, the gradient in thhe displacement increases unboundedly with time.Wright and Walter have shown that this anomaly can be rectified by coupling themomentum equation to heat conduction via the energy conservation equation. thelength scales then computed agree well with observed shear band widths inmetals.The computational meodeling of localization still poses substantial difficulties.for most materials, the length scales of shear bands are much smaller than those ofthe body.
Therefore tremendous resolution is required to obtain a reasonablyaccdurate solution to these problems, see Belytschko et al for some highresolution computations. Solutions converge very slowly with mesh refinement.This behavior of numerical solutions is often called mesh sensitivity or lack ofobjectivity, though it has nothing to do with objectivity or its absence: it is simplya consequence of the inabiloity of coarse meshes to resolves high gradient inviscopladtic materials or discontinuites in rate-independent solutions.Several techniques have evolved to improve the coarse-mesh accuracy of finiteelement models for unstable materials. The first of these involve the embedmentof discontinuities in the element. Ortiz ewt al were the first to do this:theyembedded discontinuites in the strain field of the 4-node quadrilateral whenthe acoustic trensor indicated a material instability in the element.
Belytschko,Fish and Engleman attempted to embed a displacement discontinuity by enrichingthe strain field with a narrow band where the unstable material behavior occurs.In the band, the material behavior was considered homogeneous, which isridiculous since an unstable material cannot remain ina homogenous state ofstress: any perturbation will trigger a growth on the scale of the perturbation.Such is hindsight. Nevertheless these models were able to capture the evolvingdiscontinuity in displacement more effectively.
Sime and ??? invoked the theoryoof distributions to justify such techniques. They also categorized discontinuitiesas strong (in the displacements) and weak (in the strains). This categorization6-75T. Belytschko & B. Moran, Solution Methods, December 16, 1998incidentally is at odds with the widely used categorization in shocks in fluiddynamics, where discontinuites in occur in the velocity and the motion iscontinuous, see Section ??. These techniques have recently been further exploredby Armero et al () and Garipakti ad Hughes (??).Shear bands are closely related to fracture: a shear band can be viewed as adiscontinuity in the tangential displacement,a fracture as a discopntinuity in allcomponents of the displacement, see Chapter 3, Example ??.
Just as shear bandscan be viewed as the outcome of a material instability in the shear component, thedevelopment of a fracture can be viewed numerically as the outcome of a materialinstability in the directions normal (and tangential in the case of mode 2fracture)to the discontinuity. The relationship of damage and fracture has longbeen noted, see LeMaitre and Chab oche (??), where a fracture is assumed tooccur when the damage variable reaches 0.7.
the origin of the number 0,7 is quitehazy in most works on damage mechanics, but it can be seen to arise from thephase transition point based on percolation theory is 0.59275, Taylor and Francis(1985). The modeling of fracture by dmage poses some of the same difficultiesencountered in shear band modeling, since the material law becomesunstablewhen the damage excdeeds a threshold value.
All of the phenomena found inshear banding then occur: localization to a set of measure zero for rateindependent models, exponential growth for simple rate-dependent models, zerodissipation in failure and absence of a length scale.These difficulties were grasped and resolved in a novel way early in the evolutionof fintie elements by Hillerborg et al (??), Basant (??) and Willam(??) have alsocontributed to this approach. The idea is to match the energy of fracture to theenergy dissipated by the element in which the localization occurs.[??] H.M. Hiller, T.J.R.
Hughes, and R.L. Taylor, "Improved NumericalDissipation for Time Integration Algorithms in Structural Dynamics," EarthquakeEngineering and Structural Dyanmics, Vol. 5, 282-292, 1977.The tangent moduli are denoted by C SE and a general constitutive equation canbe written as˙S =C SE : E˙ or S˙ = C SE E˙irirkl klPij = Cirkl E˙ kl FrjT + SirF˙ rjT˙ in terms of F˙ and noting the minor symmetry ofNow using (3.3.20) to express Ethe tangent modulus marix (see Section 5.?) gives˙ TPij = Cirkl Fkm F˙ lm FrjT +Sir Frj6-76T.
Belytschko & B. Moran, Solution Methods, December 16, 1998(6.4.3)Norms.Norms are used in this book primarily for simplifying the notation. No proofs aregiven that rely on the properties of normed spaces so the student need only learnthe definitions of the norms as given below. It is also worthwhile to learn aninterpretation of a norm as a distance. This is easily grasped by first learning thenorms in the space l n , which is a norm in the space of vectors of real numbers.The extension to function spaces such as the Hilbert spaces and the space ofLebesque integrable functions, L 2 , (often named el-two) is then straightforward.The norms on l n are defined by the following.
We begin with the norm l 2 ,which is simply Euclidan distance. If we consider an n-dimensional vector a ,often written as a ∈ Rn , then the l 2 norm is given by1a2 n 2= ∑ ai2 i=1 In the above, the symbol ⋅indicates a norm and the subscript 2 in combinationwith the fact that the enclosed variable is a vector indicates that we are referring tothe l 2 norm. For n = 2 or 3 , respectively, the l 2 norm is simply the length ofthe enclosed vector. The distance between two points, or the difference betweentwo vectors, is written asa−b2n= ∑ ai − bi i=1(122) Fundamental properties of the l 2 norm are that:1.
it is positive,2. it satisfies the triangle inequality3. it is linearThe l k norms are generalizations for the above definition to arbitrary k >1asfollows:6-77T. Belytschko & B. Moran, Solution Methods, December 16, 1998akn= ∑ ai i =11k kNorms for k ≠ 2 are seldom used except for k = ∞ , which is called the infinitynorm. The infinity norm gives the component of the vector with the maximumabsolute value, which can easily be figured out by thinking about (??) a little bit.Thus we can write thata∞= max aiiOne of the principal applications of these norms is to define the error in a vector.Thus if we have a approximate solution to a set of discrete equations dapp and theexact solution is dexact , then a measure of the error iserror = dapp − d exact2If you are concerned with the maximum error in any component of the solution,then you should select the infinity norm.
When the concern is with the error overa selected number of components, then the norm can be restricted to thosecomponents. The idea is that you use norms to achieve what you need: they arenot immutable. In using norms to asses errors in solutions, it is recommended thatthe error be normalized, e.g.dapp − d exacterror =2dappbecause absolute errors are very difficult to interprete and are meaningless unlessthe approximate magnitude of the solution is reported.Norms of functions are defined analogously to the above. The relationshipbetween functions and vectors is that a function can be thought of as an infintedimensional vector. Thus the norm in function space that corresponds to l 2 isgiven by()a xL211 n2 12= ∑ a2 xi ∆x = ∫0 a 2 x dx i =1()()6-78T.
Belytschko & B. Moran, Solution Methods, December 16, 1998This norm is called the L 2 , and the space of functions for which this norm iswell-defined and bounded is called the L 2 space; usually just the number isindicated This space is the set of all functions which are square integrable, and itincludes the space of all functions which are piecewise continuous.( )The Dirac delta function δ x − y is defined by()f x =+∞∫−∞()( )f y δ x − y dyis not square integrable. It can be thought of as a function which is infinite at x=ybut vanishes everywhere else. The mathematical definition of this function is thetopic of the theory of Schwartz distributions, which is needed for a goodunderstanding of convergence theory but not for nonlinear finite element analysis.The exact delineation of the space L 2 can get quite technical, sincemathematicians are concerned with questions such as whether the function()()f x = 1 when x is rational, f x = 0 otherwise, is square integrable (it is not).But for engineers concerned with the finite element method, it is sufficient toknow that any function mentioned in this book except the Dirac delta functionposseses an L 2 norm.The space of functions L 2 is a special case of a more general group of spacescalled Hilbert spaces.
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