Belytschko T. - Introduction (779635), страница 56
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Nevertheless, the simple overstress model ( )has been very successful in capturing the strain rate dependence ofmetals over a large range of strain rates [Refs].An alternative form of rate-dependent plasticity that has been used withconsiderable success by Needleman ( ) and others is given by\begin{equation}\dot{\bar{\varepsilon}}^p = \dot{\varepsilon_0}\biggl({\bar{\sigma}\overY}\biggr)^{1/m}\end{equation}without any explicit yield surface. For plastic straining at the rate$\dot{\varepsilon_0}$,the response $\bar{\sigma}=Y$ is obtained. This response is called thereference response and can be obtained by performing a unixial stresstest with a plastic strain rate $\dot{\varepsilon}_0$. In the case ofsmall elastic strain rates, the test can be run at total strain rate of$\dot{\epsilon}_0$ without significant error (Check!).
For rates which exceed$\dot{\varepsilon}_0$ the stressis elevated above the reference stress while for lower rates the stressfalls below this value. A case of particular interest is thenear rate-independent limit when the rate-sensitivity exponent $m\to 0$.It can beseen from ( ) that, for $\bar{\sigma}<Y$,the effective plastic strain rate is negligible while for a finite plasticstrain rate the effective stress is approximately equal to the referencestress, $Y$. In this way, the model exhibits an effective yield limittogether with near elastic unloading and rate-independent response.The constitutive relations for rate-dependent plasticity in 1D aresummarized in Box 9.2T.
Belytschko & B. Moran, Solution Methods, December 16, 1998CHAPTER 6SOLUTION METHODS AND STABILITYVery Rough Draft-use equations at your own perilby Ted Belytschko and Brian MoranNorthwestern UniversityCopyright 19966.1 INTRODUCTIONThis Chapter describes solution procedures for nonlinear finite elementdiscretizations. In addition, methods for examining the physical stability ofsolutions and the stability of solution procedures are described.The first part of the chapter describes time integration, the procedures usedfor integrating the discrete momemtum equation and other time dependentequations in the system, such as the constitutive equation.
We begin with thesimplest of methods, the central difference method for explicit time integration.Next the family of Newmark β -methods, which encompass both explicit andimplicit methods, are described. Explicit and implicit methods are compared andtheir relative advantages described. As part of implicit methods, the solution ofequilibrium equations is also examined.A critical step in the solution of implicit systems and equilibrium problemsis the linearization of the governing equations. Linearization procedures for theequations of motion, and as a special case, the equilibrium equations aredescribed.6.2 EXPLICIT METHODSIn this Section the major features of explicit and implicit time integrationmethods for the discretized momentum equation and solution methods for thediscrete equilibrium equations are described.
The methods are described in thecontext of Lagrangian meshes, but can be extended to Eulerian and ALE mesheswith some techniques described in Chapter 7. The description of the solutionprocedures of equilibrium problems is combined with the description of implicitprocedures for dynamic problems, because, as we show later, the methodologiesare almost identical; the solution of a static problem by an implicit method onlyrequires that the inertial term be dropped.To illustrate the major features of explicit and implicit methods for timeintegration, the solution of the equations of motion is first considered for rateindependent materials. In this class of equations, we can avoid some of thecomplications that arise in the treatment of rate-dependent materials but stillillustrate the most important properties of explicit and implicit methods.
We willfirst describe explicit and implicit methods using only a single time integrationformula: the central difference method for explicit time integration and the6-1T. Belytschko & B. Moran, Solution Methods, December 16, 1998Newmark β-methods for implicit integration. In Section X, other time integrationformulas are considered.6.2.1.Central Difference Method.
The central difference method isamong the most popular of the explicit methods in computational mechanics andphysics. It has already been discussed in Chapter 2, where it was chosen todemonstrate some nonlinear solutions in one dimension. The central differencemethod is developed from central difference formulas for the velocity andacceleration. We consider here its application to Lagrangian meshes with rateindependent materials. Geometric and material nonlinearites are included, and infact have little effect on the time integration algorithm.For the purpose of developing this and other time integrators we will use thefollowing notation. Let the time of the simulation 0 ≤t ≤t E be subdivided intotime intervals, or time steps, ∆t n , n =1 to nTS where nTS is the number of timesteps and t E is the end-time of the simulation; ∆t n is also called the nth timeincrement.
The variables at any time step are indicated by a superscript; thus t n isthe time at time step n, t 0 = 0 is the beginning of the simulation and dn ≡ d t n isthe matrix of nodal displacements at time step n. Time increment n is given by( )(∆tn + 2 = 12 ∆tn +∆t n+11∆t n = t n − t n−1)(6.2.1)where the second equation gives the midpoint time step.The central difference formula for the velocity is˙dn+12n + 21≡v=1n+ 1∆t 2(d n +1 − dn ) ,dn +1 = dn +∆ tn + 2 v n + 211(6.2.2)where the second equation gives the corresponding integration equation which isobtained by a rearrangement of the first.
The acceleration is given by(˙d˙ n ≡ a n = 1 vn + 21 − vn − 21∆t n)v n + 2 = vn − 2 +∆ t n a n11(6.2.3a)As can be seen from the above, the velocities are defined at the midpoints of thetime intervals, or at half-steps. By substituting (6.2.2a) and its counterpart for theprevious time step into (6.2.3), the acceleration can be expressed directly in termsof the displacements˙d˙ n ≡ a n =∆tn− 12(dn +1 − dn ) − ∆t n + ( dn − d n−1 )12n∆t ∆tn − 21n + 21∆tFor the case of equal time steps the above reduces to6-2(6.2.3b)T. Belytschko & B.
Moran, Solution Methods, December 16, 1998˙d˙ n ≡ a n =(dn+1 − 2d n + dn− 1)2( ∆t n )(6.2.3c)This is the well known central difference formula for the second derivative of afunction.We now consider the time integration of the undamped equations ofmotion for rate-independent materials, Eq. (4.x.x.), which at time step n are givenby()(Man = f n = f ext dn , t n − f int dn , t n)( )subject to g I dn = 0, I =1to nc(6.2.4a)(6.2.4b)where (6.2.4b) is a generalized representation of the nc displacement boundaryconditions; constraints may also arise from other conditions on the model. Themass matrix in this expression is considered constant because as noted in SectionX, it is time independent for a Lagrangian mesh.
Methods for Eulerian meshesare discussed in Chapter 7. The internal and external nodal forces are functions ofthe nodal displacements and the time. The external loads are usually prescribed asfunctions of time; they may also be functions of the nodal displacements becausethey may depend on the configuration of the structure, as when pressures areapplied to the surfaces which undergo large deformations. The dependence of theinternal nodal forces on displacements is quite obvious: the nodal displacementsdetermine the strains, which in turn determine the stresses and hence the nodalinternal forces by Eq. (4.4.11).
Internal nodal forces are generally not directlydependent on time, but there are situations of engineering relevance when this isthe case; for example, when the temperature is prescribed as a function of time,the stresses and hence the internal nodal forces depend directly on time.The equations for updating the nodal velocities and displacements areobtained as follows. Substituting Eq. (6.2.4a) into (6.2.3b) givesvn + 21= ∆t nM −1f n + vn − 21(6.2.5)which provides an update for the nodal velocities; the displacements are thenupdated by (6.2.2).At any time step n, the displacements dn will be known. The nodalforces f n can be determined by using in sequence the strain-displacementequations, the constitutive equation and the relation for the nodal internal forces.n+1Thus the entire right hand side of (6.2.5) can be evaluated, which gives υ 2 ,and the displacements dn+1 at time step n+1 can be determined by (6.2.2b).
Theentire update can be accomplished without solving any system equations providedthat the mass matrix M is diagonal. This is the salient characteristic of an explicitmethod:6-3T. Belytschko & B. Moran, Solution Methods, December 16, 1998in an explicit method, the time integration of the discrete momentum equations fora finite element model does not require the solution of any equations.In numerical analysis, integration methods are classified according to thestructure of the time difference equation. The difference equations for first andsecond derivatives are written in the general formsnS∑(αn dnS−nn=0nS)∑(αn dn−∆ tβn˙d n = 0S−nn=0)−∆ t 2βn˙d˙ n = 0(6.2.6)where nS is the number of steps in the difference equation.
The differenceformula for the first or second derivatives is called explicit if β0 = 0 or β0 = 0 ,respectively. From (6.2.3c) it can be seen that β0 = 0, β1 = 1, β2 = 0 , so theformula is explicit. Thus the difference formula is called explicit if the equationfor the function at time step n only involves the derivatives at previous time steps.Difference equations which are explicit according to this classification generallylead to solution schemes which require no solution of equations. In most casesthere is no benefit in using explicit schemes which involve the solution ofequations, so the use of such explicit schemes is rare.
There are a few exceptions.For example, if the consistent mass is used with the central difference method,even though the difference equation is classified as explicit, system equations stillneed to be solved in the update.6.2.2.Implementation. A flow chart for explicit time integration of a finiteelement model with rate-independent materials is shown in Box 6.1. Thisflowchart generalizes the flowchart given in Chapter 2 by considering nonzeroinitial conditions, a variable time step and including elements which require morethan one-point quadrature. The primary dependent variables in this flowchart arethe velocities and the Cauchy stresses.
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