Belytschko T. - Introduction (779635), страница 9
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The contributions to the boundary points on the right-hand side in the aboveonly appear on the traction boundary Γ t since δu = 0 on Γ u and Γ u = Γ – Γ t (see Eqs.(2.3.6) and (2.2.30)). Combining Eqs. (2.3.10) and (2.3.2) then givesXb∫Xa(δu, X (A0 P ) dX = − ∫X δu( A0 P ), X dX + δuA0 n0 PXba) – ∑ δu A PΓt0iΓi(2.3.11)Substituting the above into Eq.
(2.3.7) gives∫XbXa[]δu ( A0 P), X + ρ 0 A0 b– ρ0 A0˙u˙ dX(0+δuA0 n 0P– t x)Γt+ ∑ δu A0 PiΓi= 0 ∀δu( X ) ∈U0(2.3.12)The conversion of the weak form to a form amenable to the use of Eq. (2.3.4-5) is nowcomplete. We can therefore deduce from the arbitrariness of the virtual displacementδu( X ) and Eqs. (2.3.4.-5) and (2.3.12) that (a more detailed derivation of this step isgiven in Chapter 4)( A0 P), X + ρ0 A0 b– ρ 0 A0˙u˙ = 0for X ∈[ X a , X b ]0(2.3.13a)n 0 P– tx = 0 on Γt(2.3.13b)A0 P = 0 on Γi(2.3.13c)These are, respectively, the momentum equation, the traction boundary conditions, and thestress jump conditions. Thus when we admit the less smooth test and trial functions, wehave an additional equation in the strong form, the jump condition (2.3.13c).2-12T. Belytschko, Chapter 2, December 16, 1998If the test functions and trial functions satisfy the classical smoothness conditions, thejump conditions do not appear.
Thus for smooth test and trial functions, the weak formimplies only the momentum equation and the traction boundary conditions.The less smooth test and trial functions are more pertinent to finite elementapproximations, where these functions are only C 0 . They are also needed to deal withdiscontinuities in the cross-sectional area and material properties. At material interfaces, theclassical strong form is not applicable, since it assumes that the second derivative isuniquely defined everywhere.
This is not true at material interfaces because the strains,and hence the derivatives of the displacement fields, are discontinuous. With the roughertest and trial functions, the conditions which hold at these interfaces. (2.3.13c) emergenaturally.In the weak form for the total Lagrangian formulation, all integrations are performedover the material coordinates, i.e.
the reference configuration, of the body, because totalLagrangian formulations involve derivatives with respect to the material coordinates X, sointegration by parts is most conveniently performed over the domain expressed in terms ofthe material coordinate X. Sometimes this is referred to as integration over theundeformed, or initial, domain. The weak form is expressed in terms of the nominalstress.Physical Names of Virtual Work Terms. For the purpose of obtaining a methodicalprocedure for obtaining the finite element equations, the virtual energies will be definedaccording to the type of work which they represent; the corresponding nodal forces willsubsequently carry identical names.Each of the terms in the weak form represents a virtual work due to the virtualdisplacement δu; this displacement δu(X) is called a often "virtual" displacement to indicatethat it is not the actual displacement; according to Webster’s dictionary, virtual means"being in essence or effect, not in fact"; this is a rather hazy meaning and we prefer thename test function.0The virtual work of the body forces b(X,t) and the prescribed tractions t x , whichcorresponds to the second and fourth terms in (2.3.4), is called the virtual external worksince it results from the external loads.
It is designated by the superscript “ext” and givenbyXbδWext(= ∫ δuρ 0 bA0 dX + δuA0 t xXa0)(2.3.16)ΓtThe first term in (2.3.4) is the called the virtual internal work, for it arises from thestresses in the material. It can be written in two equivalent forms:XbXbXaXaδWin t = ∫ δu, X PA0 dX = ∫ δFPA0dX(2.3.17)where the last form follows from (2.2.1) as follows:δu, X ( X ) = δ (φ( X )– X ), X = δφ, X =∂ (δx )=δF∂X2-13(2.3.18)T. Belytschko, Chapter 2, December 16, 1998The variation δX = 0 because the independent variable X does not change due to anincremental displacement δu(X).This definition of internal work in (2.3.17) is consistent with the internal workexpression in the energy conservation equation, Eq.
(2.2.16-2.2.17): if we change the ratesin (2.2.11) to virtual increments, then ρ 0 δwint = δFP . The virtual internal work δWin t isdefined over the entire domain, so we haveXbδWint=int∫ δw ρ0 A0dX =Xa∫XbXaδFPA0 dX(2.3.19)which is the same term that appears in the weak form in (2.2.18).The term ρ 0 A0˙u˙ can be considered a body force which acts in the direction opposite tothe acceleration, i.e. in a d'Alembert sense. We will designate the corresponding virtualwork by δW inert and call it the virtual inertial work, soXbδW inert = ∫ δuρ0 A0˙u˙ dX(2.3.20)XaThis is the work by the inertial forces on the body.Principle of Virtual Work.
The principle of virtual work is now stated using thesephysically motivated names. By using Eqs. (2.3.16-2.3.20), Eq. (2.3.4) can then bewritten asδW (δu, u) ≡δW int −δW ext +δW inert = 0∀δu ∈U 0(2.3.21)The above equation, with the definitions in Eqs. (2.3.16-2.3.20), is the weak formcorresponding to the strong form which consists of the momentum equation, the tractionboundary conditions and the stress jump conditions. The weak form implies the strongform and that the strong form implies the weak form. Thus the weak form and the strongform are equivalent.
This equivalence of the strong and weak forms for the momentumequation is called the principle of virtual work.All of the terms in the principle of virtual work δW are energies or virtual work terms,which is why it is called a virtual work principle. That the terms are energies isimmediately apparent from δWext : since ρ 0b is a force per unit volume, its product with avirtual displacement δu gives a virtual work per unit volume, and the integral over thedomain gives the total virtual work of the body force.
Since the other terms in the weakform must be dimensionally consistent with the external work term, they must also bevirtual energies. This view of the weak form as consisting of virtual work or energy termsprovides a unifying perspective which is quite useful for constructing weak forms for othercoordinate systems and other types of problems: it is only necessary to write an equationfor the virtual energies to obtain the weak form, so the procedure we have just gonethrough can be avoided. The virtual work schema is also useful in memorizing the weakform. However, from a mathematical viewpoint it is not necessary to think of the testfunctions δu(X) as virtual displacements: they are simply test functions which satisfycontinuity conditions and vanish on the boundaries as specified by (2.3.6).
This second2-14T. Belytschko, Chapter 2, December 16, 1998viewpoint becomes useful when a finite element discretization is applied to equations wherethe product with a test function does not have a physical meaning. The principle of virtualwork is summarized in Box 2.1.Box 2.1.
Principle of Virtual Work forOne Dimensional Total Lagrangian FormulationIf the trial functions u( X, t) ∈U , then(Weak Form) δW = 0∀ δu ∈ U0(B2.1.1)is equivalent to(Strong Form)the momentum equation (2.2.12): ( A0 P ), X + ρ 0 A0 b = ρ 0 A0˙u˙ ,the traction boundary conditions (2.2.28): n0 P =and the jump conditions (2.2.33): 〈A 0P〉 = 0.0txon Γt ,(B2.1.2)(B2.1.3)(B2.1.4)Weak form definitions:δW ≡δW int −δW ext +δW inertXbXbXaXa(B2.1.5)δWint = ∫ δu, X PA0 dX = ∫ δFPA0dX ,XbδW inert = ∫ δuρ0 A0˙u˙ dXXa(B2.1.6)Xb(δW ext = ∫ δuρ 0 bA0 dX + δuA0 t xXa0)(B2.1.7)Γt2.4 Finite Element Discretization In Total Lagrangian FormulationFinite Element Approximations.
The discrete equations for a finite element model areobtained from the principle of virtual work by using finite element interpolants for the testand trial functions. For the purpose of a finite element discretization, the interval [X a,X b]is subdivided into elements e=1 to ne with nN nodes. The nodes are denoted by X I, I = 1to nN, and the nodes of a generic element by X eI , I = 1 to m, where m is the number ofnodes per element.
The domain of each element is [ X1e , Xme ], which is denoted by Ω e . Forsimplicity, we consider a model problem in which node 1 is a prescribed displacementboundary and node nN a prescribed traction boundary. However, to derive the governingequations we first treat the model as if there were no prescribed displacement boundariesand impose the displacement boundary conditions in the last step.The finite element trial function u(X,t) is written as2-15T. Belytschko, Chapter 2, December 16, 1998u(X ,t ) =nN∑ N I ( X )uI (t )(2.4.1)I =1In the above, NI ( X ) are C0 interpolants, they are often called shape functions in the finiteelementliterature; uI (t ), I =1to nN , are the nodal displacements, which are functions oftime, and are to be determined in the solution of the equations.
The nodal displacementsare considered functions of time even in static, equilibrium problems, since in nonlinearproblems we must follow the evolution of the load; in many cases, t may simply be amonotonically increasing parameter. The shape functions, like all interpolants, satisfy theconditionNI ( X J ) = δ IJ(2.4.2)where δ IJ is the Kronecker delta or unit matrix: δ IJ =1 if I = J , δ IJ = 0 if I ≠ J . We notehere that if we set u1(t ) = u (0, t ) then the trial function u( X,t ) ∈U , i.e.
it is kinematicallyadmissible since it has the requisite continuity and satisfies the essential boundaryconditions. Equation (2.4.1) represents a separation of variables: the spatial dependence ofthe solution is entirely represented by the shape functions, whereas the time dependence isascribed to the nodal variables. This characteristic of the finite element approximation willhave important ramifications in finite element solutions of wave propagation problems.The test functions (or virtual displacements) depend only on the material coordinatesnNδu( X ) = ∑ N I ( X)δuI(2.4.3)I =1where δuI are the nodal values of the test function; they are not functions of time.Nodal Forces.
To provide a systematic procedure for developing the finite elementequations, nodal forces are developed for each of the virtual work terms. These nodalforces are given names which correspond to the names of the virtual work terms. ThusδWin t=nN∑ δuI fIint = δuT fin t(2.4.4a)I =1δWext=nN∑ δuI fIext =δuT f ext(2.4.4b)I =1δWinertnN= ∑δu I fIinert = δu T f inert(2.4.4c)I =1[δuT = δu1 δu2 ...δu nN][f T = f1f2 ...fn N](2.4.4d)where f int are the internal nodal forces, f ext are the external nodal forces, and f inert are theinertial, or d'Alembert, nodal forces. These names give a physical meaning to the nodal2-16T.
Belytschko, Chapter 2, December 16, 1998forces : the internal nodal forces correspond to the stresses “in” the material, the externalnodal forces correspond to the externally applied loads, while the inertial nodal forcescorrespond to the inertia term due to the accelerations.Nodal forces are always defined so that they are conjugate to the nodal displacementsin the sense of work, i.e. so the scalar product of an increment of nodal displacements withthe nodal forces gives an increment of work. This rule should be observed in theconstruction of the discrete equations, for when it is violated many of the importantsymmetries, such as that of the mass and stiffness matrices, are lost.Next we develop expressions for the various nodal forces in terms of the continuousvariables in the partial differential equation by using (2.3.16-2.3.20).
In developing thenodal force expressions, we continue to ignore the displacement boundary conditions andconsider δuI arbitrary at all nodes. The expressions for the nodal forces are then obtainedby combining Eqs. (2.3.16) to (2.3.20) with the definitions given in Eqs. (2.4.4) and thefinite element approximations for the trial and test functions. Thus to define the internalnodal forces in terms of the nominal stress, we use (2.4.4a) and Eq.
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