Belytschko T. - Introduction (779635), страница 77
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Unlike the Lagrangian formulations, the spatialdomain changes continuously throughout the computation. The mesh update procedure willbe described in section ????.In the subsequent development of this chapter, we shall divide the discussion of ALEformulations into four parts:1) Updated ALE formulations for continuum material models without memory, that is, theevaluation of constitutive laws is independent of the strain history. A simple example ofthis kind of materials is a slightly compressible Newtonian fluid which will be discussed insection ???2) Updated ALE formulations of continuum material models with memory, that is, theevaluation of constitutive laws is strain history dependent.
This will be described in section???.3) The Petrov-Galerkin finite element method, of which the discretization of the transportterm requires special treatment. For high velocities, if the mesh is not sufficiently refined,the Galerkin method gives rise to oscillatory solutions. To overcome this difficulty, variousschemes, collectively known as upwinding, will be described in section ????.4) The mesh update procedure, which is described in section ????.The reason for the distinction between ALE formulations with and without memory is dueto the difference between ALE material update procedures. Recall that the material timeW.K.Liu, Chapter 728derivative of a given function f can be related to the referential time derivative byEq.(7.2.20), that isDff˙ == ˆf ,t[ χ ] + c j f , jDtIt is noted that f can represent density ρ , velocity v , energy E, Cauchy stress σ , etc., asappearing in the governing equations Box 7.1.The next section will illustrate the difference between Lagrangian and ALE material updateprocedures.7.3 ALE Material UpdatesIn the Lagrangian description, the updating of any material-related state variable issimple.
Since the Lagrangian coordinate is always associated with the same material point,a Taylor series expansion in time may be used. Using first order accuracy gives:F(X,t + dt) = F(X,t) + dt∂F(X,t) + K∂tX(7.3.1)However, in a referential description, updating of a material state variable introducescomplicates.
To illustrate this, we expand a state variable fˆ (χ,t) in a Taylor series:ˆˆf ( χ, t+ dt) = ˆf ( χ, t) + dt ∂f ( χ, t) + K∂tχ(7.3.2)or, by referring everything to the particle X at time t (see Eq. (7.2.10)), as:∂fˆfˆ[ ψ (X,t + dt),t + dt ] = fˆ[ ψ(X,t),t ] + dt [ ψ (X,t ),t ]+K∂tχ=ψ (X,t )(7.3.3)which is equivalent toF(X,t + dt) = F(X,t) + dt∂F(X,t)+K∂tX= ψ −1 (χ ,t)(7.3.4a)whereX = ψ −1( χ, t + dt) and X = ψ −1( χ, t)(7.3.4b)Comparing Eq. (7.3.4a) with Eq.
(7.3.1), even though the terms in the right hand side ofthe equations are the same , shows that X and X are two different material particles, whichW.K.Liu, Chapter 729at t and t + dt, respectively, have the same referential coordinates. Therefore, in areferential description, a simple updating technique, such as Eq. (7.3.2) cannot be used formaterial point-related variables, such as state variables in path-dependent materials;however, for homogeneous materials with no memory, such as the generalized Newtonianfluids, Eq. (7.3.2) can be implemented with no further complications. Further detail willbe given on this subject later.7.16 ALE Finite Element Method for Path-Dependent MaterialsThe purpose of this section is to provide a general formulation and an explicitcomputational procedure for nonlinear ALE finite element analyses.
Emphasis is placed onthe stress update procedure for path-dependent materials. First, after the generalformulations for the ALE description are reviewed, according to strong form, weak formand finite element form, the most important part of the ALE application for path-dependentmaterials, the stress update procedure, is studied in detail. Formulations for regularGalerkin method, Streamline-upwind/Petrov-Galerkin(SUPG) method and operator splitmethod are derived respectively.
All the path-dependent state variables are updated with asimilar procedure. Further, the stress update procedures are specified in 1-D case. And thematrices corresponding to these three methods are listed. Then, an explicit computationalmethod and a flowchart are presented. Finally, elastic and elastic-plastic wave propagationexamples are given.7.16.1 Formulations for Updated ALE:7.16.1.1 Strong Form Formulations for ALE:In addition to continuity and momentum equations given in section ????, for the purpose ofintroducing path-dependent material model, the Cauchy stress may be decomposed into thedeviatoric stress tensor sij and the hydrostatic pressure p such thatσ ij = sij − pδij(7.16.1c)and the components of the deviatoric stress term are given by the Jaumann rate constitutiveequationsij,t[ χ ] + ck sij,k = Cijkl Dkl + skj Wik + ski W jk(7.16.1d)Similarly the rate from of the equation of state is given by:p,t[ χ ] + ci p,i = p(ρ )(7.16.1e)In the above equations, Dij and Wij are of deformation tensor and the spin tensor,respectively; such thatDij =1(v + v j,i )2 i, j(7.16.1f)W.K.Liu, Chapter 71Wij = (vi, j − v j,i )230(7.16.1g)and Cijkl is the material response tensor which relates any frame-invariant rate of theCauchy stress to the velocity strain.
Both geometric and material nonlinearities are includedin the setting of Eqs. (7.16.1a-g).REMARK 7.16.1.1 The right-hand sides of Eqs. (7.16.1a,b,d and e) remain the same forall descriptions.REMARK 7.16.1.2. Eqs. (7.16.1a,b,d and e) are referred to as the “quasi-Eulerian”description (Belytschko and Kennedy, 1978) because these equations have a strongresemblance to the Eulerian equations. In particular, the Eulerian equations can be readilyobtained by choosing c = v , i.e., χ = x .REMARK 7.16.1.3. With the help of integration by part, Eq.(7.16.1d) is equivalent to thefollowing equations:sij,t[ χ ] + yijk,k − ck ,k sij = Cijkl Dkl + skj Wik + skiW jk(7.16.1h)yijk = sij ck(7.16.1i)andwhere yijk is the stress-velocity product.
In the following finite element computation, thesetwo equations will replace Eq.(7.16.1d) in the weak form.7.16.1.2 Weak Form of the ALE Equations:Similar to the case for continuity and momentum equations, we may obtain the weak formof the constitutive equations:∫Ω δsijsij,t[χ ]dΩ + ∫Ω δsij yijk,k dΩ− ∫Ω δsijc k,k sij dΩ= ∫ δsij Cijkl Dkl dΩ + ∫ δsij {skj Wik + ski W jk}dΩΩΩ(7.16.2c)and∫Ω δyijk yijkdΩ = ∫Ω δyijksijck dΩ(7.16.2d)Equation of state∫Ω δppt,[χ ]dΩ + ∫Ω δpci p,idΩ = ∫Ω δpp(ρ)dΩ(7.16.2e)Eq.(7.16.2a) represents the control volume form of material conservation. Eq.(7.16.2b) isa generalization of the principle of virtual work to the control volume form with the firstintegral brought in as d’Alembert forces.7.16.1.3 Finite Element Discretization:Similar to the finite element discretization for continuity and momentum equations, we canobtain the finite element form for constitutive equations by employing N s , N y and N p asW.K.Liu, Chapter 731sets of shape functions, and N s , N y and N p as corresponding sets of test functions tointerpolate the deviatoric stress, stress-velocity product, and hydrostatic pressurerespectively.
Note that the test functions and the shape functions for deviatoric stresses areused only in the constitutive equations.Constitutive Equations:M s s,t[ χ ] + GT y − Ds = z(7.16.5a)M y y = Lys(7.16.5b)where the superscript T denotes matrix transpose, M s and D are the generalized mass anddiffusion matrices for deviatoric stress respectively; G T y corresponds to the generalizedconvective term; M y and Ly are the generalized mass and convective matrices for stressvelocity product respectively; and z is the generalized deviatoric stress vector such thatM s = [M IJs ] = ∫ N Is NJs dΩ(7.16.5c)yG T = [GIJT ] = ∫ N Is N J,xdΩ(7.16.5d)D = [DIJ ] = ∫ N Is ck ,k NJs dΩ(7.16.5e)ΩΩΩz = [zijI] = ∫ NIsCijkl Dkl dΩ+Ω∫Ω NI {skjWik + skiW jk}dΩs(7.16.5f)M y = [M IJy ] = ∫ NIy N JydΩ(7.16.5g)Ly = [LyIJ ] = ∫ N Is cN Js dΩ(7.16.5h)ΩΩEquation of State:M p p,t[χ ] + Lp p = u(7.16.6a)where M p and Lp are the generalized mass and convective matrices for pressurerespectively; and u is the generalized pressure vector, such thatM p = [M IJp ] = ∫ NIp N Jp dΩ(7.16.6b)Lp = [LpIJ ] = ∫ N Ip ci N Jp,i dΩ(7.16.6c)u = [uI ] = ∫ N Ip p(ρ)dΩ(7.16.6d)ΩΩΩW.K.Liu, Chapter 732REMARK 7.16.1.6 The stress-velocity product y is stored at each node as a vector with adimension of (number of space dimensions) × (number of stress components).
Theydiagonal form for M is considered by location the numerical integration points at nodes.REMARK 7.16.1.7 The numerical integration of (7.16.3) and (7.16.4) has been discussedby Liu & Belytschko(1983), Liu(1981) and Liu & Ma(1982). A procedure for the stressupdate Eqs. (7.16.5a) and (7.16.5b) is presented in the next section to clarify the temporalintegration for path-dependent materials. All the path-dependent quantities are updatedanalogous to Eqs(7.16.5a) and (7.16.5b). The Petrov-Galerkin formulation of thecontinuity and momentum equation derived in section 7.14 can be adopted here.7.16.2 Stress Update Procedures:7.16.2.1 Stress Update Procedure for Galerkin MethodThe stress state in a path-dependent material depends on the stress history of the materialpoint.
A stress history can be readily treated in Lagrangian description because elementscontain the same material points regardless of the deformation of the continuum; similarly,quadrature points at which stresses are computed in Lagrangian elements coincide withmaterial points throughout the deformation. On the other hand, in an ALE description, amesh point does not coincide with a material point so that the stress history needs to beconvected by the relative velocity c, as indicated in Eq.(7.16.1d).
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