Belytschko T. - Introduction (779635), страница 35
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Thus a velocityboundary condition will sometimes be called a displacement boundary condition, or vice versa.The initial conditions can be applied either to the velocities and the stresses or to thedisplacements and velocities. The first set of initial conditions are more suitable for mostengineering problems, since it is usually difficult to determine the initial displacement of a body.On the other hand, initial stresses, often known as residual stresses, can sometimes be measured orestimated by equilibrium solutions. For example, it is almost impossible to determine thedisplacements of a steel part after it has been formed from an ingot.
On the other hand, goodestimates of the residual stress field in the engineering component can often be made. Similarly, ina buried tunnel, the notion of initial displacements of the soil or rock enclosing the tunnel is quitemeaningless, whereas the initial stress field can be estimated by equilibrium analysis. Therefore,initial conditions in terms of the stresses are more useful.BOX 4.1Governing Equations for Updated Lagrangian Formulationconservation of massρ(X )J (X ) = ρ 0 (X)J 0 ( X) = ρ 0 (X )(B4.1.1)conservation of linear momentum∂σ jiDvDv∇⋅ σ + ρb = ρv˙ ≡ ρor+ ρbi = ρ ˙vi ≡ ρ iDt∂x jDtconservation of angular momentum: σ = σ T4-4orσ ij =σ ji(B4.1.2)(B4.1.3)T.
Belytschko, Lagrangian Meshes, December 16, 1998conservation of energy:˙ int = D:σ −∇⋅ q + ρs or ρw˙ int = Dijσ ji −ρw∂qi+ ρs∂xiconstitutive equation: σ ∇ = S σt D(D, σ, etc.)rate-of-deformation: D = sym(∇v)boundary conditionsnj σ ji = ti on ΓtiΓti ∩ Γv i = 0initial conditionsv( x,0 ) = v 0( x)orv( x,0 ) = v 0( x)(B4.1.4)(B4.1.5)Dij =1 ∂vi ∂v j +2 ∂x j ∂xi v i = vi on ΓviΓti ∪ Γv i = Γi =1 to nSD(B4.1.6)(B4.1.7)(B4.1.8)σ ( x, 0) = σ0 (x)(B4.1.9)u (x, 0) = u0 ( x)(B4.1.10)interior continuity conditions (stationary)on Γint : n⋅σ = 0orni σ ij ≡ niA σijA + niBσ ijB = 0(B4.1.11)We have also included the interior continuity conditions on the stresses in Box 4.1as Eq.(B4.1.11).
In this equation, superscripts A and B refer to the stresses and normal on two sides ofthe discontinuity: see Section 3.5.10. These continuity conditions must be met by the tractionswherever stationary discontinuites in certain stress and strain components are possible, such as atmaterial interfaces. They must hold for bodies in equilibrium and in transient problems. Asmentioned in Chapter 2, in transient problems, moving discontinuities are also possible; however,moving discontinuities are treated in Lagrangian meshes by smearing them over several elements.Thus the moving discontinuity conditions need not be explicitly stated. Only the stationarycontinuity conditions are imposed explicitly by a finite element approximation.4.3 WEAK FORM: PRINCIPLE OF VIRTUAL POWERIn this section, the principle of virtual power, is developed for the updated Lagrangianformulation.
The principle of virtual power is the weak form for the momentum equation, thetraction boundary conditions and the interior traction continuity conditions. These three arecollectively called generalized momentum balance. The relationship of the principle of virtualpower to the momentum equations will be described in two parts:1. The principle of virtual power (weak form) will be developed from the generalizedmomentum balance (strong form), i.e. strong form to weak form.2. The principle of virtual power (weak form) will be shown to imply the generalizedmomentum balance (strong form), i.e. weak form to strong form.We first define the spaces for the test functions and trial functions.
We will consider theminimum smoothness required for the functions to be defined in the sense of distributions, i.e. weallow Dirac delta functions to be derivatives of functions. Thus, the derivatives will not be defined4-5T. Belytschko, Lagrangian Meshes, December 16, 1998according to classical definitions of derivatives; instead, we will admit derivatives of piecewisecontinuous functions, where the derivatives include Dirac delta functions; this generalization wasdiscussed in Chapter 2.The space of test functions is defined by:δv j ( X) ∈U0{U0 = δvi δvi ∈C 0 (X ),δvi = 0 on Γvi}(4.3.1)This selection of the space for the test functions δv is dictated by foresight from what will ensue inthe development of the weak form; with this construction, only prescribed tractions are left in thefinal expression of the weak form.
The test functions δv are sometimes called the virtualvelocities.The velocity trial functions live in the space given byvi ( X, t) ∈U{U = vi vi ∈C 0 (X), vi = vi on Γvi}(4.3.2)The space of displacements in U is often called kinematically admissible displacements orcompatible displacements; they satisfy the continuity conditions required for compatibility and thevelocity boundary conditions. Note that the space of test functions is identical to the space of trialfunctions except that the virtual velocities vanish wherever the trial velocities are prescribed.
Wehave selected a specific class of test and trial spaces that are applicable to finite elements; the weakform holds also for more general spaces, which is the space of functions with square integrablederivatives, called a Hilbert space.Since the displacement ui ( X, t ) is the time integral of the velocity, the displacement fieldcan also be considered to be the trial function. We shall see that the constitutive equation can beexpressed in terms of the displacements or velocities. Whether the displacements or velocities areconsidered the trial functions is a matter of taste.4.3.1 Strong Form to Weak Form.
As we have already noted, the strong form, orgeneralized momentum balance, consists of the momentum equation, the traction boundaryconditions and the traction continuity conditions, which are respectively:∂σ ji∂x j+ ρbi = ρ˙vi in Ω(4.3.3a)nj σ ji = ti on Γti(4.3.3b)nj σ ji = 0 on Γint(4.3.3c)where Γint is the union of all surfaces (lines in two dimensions) on which the stresses arediscontinuous in the body.Since the velocities are C 0 (X) , the displacements are similarly C 0 (X) ; the rate-ofdeformation and the rate of Green strain will then be C −1 (X) since they are related to spatialderivatives of the velocity.
The stress σ is a function of the velocities via the constitutive equation(B4.1.4relates the rate-of-deformation to the velocities) and Eq. (B4.1.5), which or the Green4-6T. Belytschko, Lagrangian Meshes, December 16, 1998strain to the displacement. It is assumed that the constitutive equation leads to a stress that is awell-behaved function of the Green strain tensor, so that the stresses will also be C −1 (X) .Note that the stress rate is often not a continuous function of the rate-of-deformation; for example,it is discontinuous at the transition between plastic behavior and elastic unloading.The first step in the development of the weak form, as in the one-dimensional case inChapter 2, consists of taking the product of a test function δvi with the momentum equation andintegrating over the current configuration: ∂σ ji˙δv+ρb−ρvii dΩ = 0∫ i ∂x jΩ(4.3.4)In the intergral, all variables must be implicitly transformed to be functions of the Euleriancoordinates by (???).
However, this transformation is never needed in the implementation. Thefirst term in (4.3.4) is next expanded by the product rule, which gives∫ δviΩ ∂∂ (δvi ) dΩ = ∫ δviσ ji −σ ji dΩ∂x j∂x∂xjjΩ∂σ ji()(4.3.5)Since the velocities are C 0 and the stresses are C −1 , the termδviσ ji on the RHS of the above isC −1 . We assume that the discontinuities occur over a finite set of surfaces Γ int , so Gauss'stheorem, Eq. (3.5.4) gives∂∫ ∂x j (δviσ ji )dΩ = ∫ δviΩΓintn jσ ji dΓ+ ∫ δvi n jσ ji dΓ(4.3.6)ΓFrom the strong form (4.3.3c), the first integral on the RHS vanishes.
For the second integral onthe RHS we can use another part of the strong form, the traction boundary conditions (4.3.3b) onthe prescribed traction boundaries. Since the test function vanishes on the complement of thetraction boundaries, (4.3.6) givesnSD∂δvσdΩ=∑ ∫ δvitidΓ∫ ∂x j i jii=1 ΓΩ()(4.3.7)tiThe summation sign is included on the RHS to avoid any confusion arising from the presence of athird index i in Γti ; if this index is ignored in the summation convention then there is no need for asummation sign.If (4.3.7) is substituted into (4.3.4) we obtain∫ δviΩ∂σ ji∂x jnSDdΩ = ∑ ∫ δvi ti dΓ −i=1 Γti∂ (δvi )σ ji dΩ∂xjΩ∫4-7(4.3.8)T.
Belytschko, Lagrangian Meshes, December 16, 1998The process of obtaining the above is called integration by parts. If Eq. (4.3.8) is then substitutedinto (4.3.4), we obtainnSD∂ (δvi )σdΩ−δvρbdΩ−δ vi tidΓ + ∫ δvi ρv˙ i dΩ= 0∫ ∂x j ji ∫ i i ∑∫i =1 ΓΩΩΩ(4.3.9)tiThe above is the weak form for the momentum equation, the traction boundary conditions and theinterior continuity conditions.
It is known as the principle of virtual power, see Malvern (1969),for each of the terms in the weak form is a virtual power; see Section 2.5.4.3.2. Weak Form to Strong Form. It will now be shown that the weak form (4.3.9)implies the strong form or generalized momentum balance: the momentum equation, the tractionboundary conditions and the interior continuity conditions, Eqs. (4.3.3). To obtain the strongform, the derivative of the test function must be eliminated from (4.3.9). This is accomplished byusing the derivative product rule on the first term, which gives()∂ δviσ ji∂σ ji∂ (δvi )σdΩ=dΩ−δv∫ ∂x j ji∫ ∂x j∫ i ∂x j dΩΩΩΩ(4.3.10)We now apply Gauss’s theorem, see Section 3.5.2, to the first term on the RHS of the above∫(∂ δviσ ji∂x jΩ)dΩ=∫ δvi n jσ ji dΓ+ ∫ δviΓΓintnSD∑ ∫ δvi n jσ jidΓ + ∫ δvii =1 Γtn jσ ji dΓ=(4.3.11)n jσ ji dΓΓintiwhere the second equality follows because δvi = 0 on Γvi , (see Eq.
(4.3.1) and Eq. (B4.1.7)).Substituting Eq. (4.3.11) into Eq. (4.3.10) and in turn to (4.3.9), we obtain ∂σ ji˙δv+ρb−ρvii dΩ−∫ i ∂x jΩnSD∑ ∫Γi=1ti()δvi n jσ ji − t i dΓ− ∫Γintδvi n j σ ji dΓ = 0(4.3.12)We will now prove that the coefficients of the test functions in the above integrals mustvanish. For this purpose, we prove the following theoremif αi (X), β i (X ),γ i (X) ∈C−1 and δvi (X) ∈U 0nSDand∫ δviα idΩ + ∑ ∫ δviβ idΓ +Ωi=1 Γti∫ δviγ idΓ = 0Γint∀δvi (X)(4.3.13)then αi (X) = 0 in Ω , βi (X) = 0 on Γt i ,γ i ( X) = 0 on Γintwhere the integral is either transformed to the reference configuration or the variables are expressedin terms of the Eulerian coordinates by the inverse map prior to evaluation of the integrals.4-8T. Belytschko, Lagrangian Meshes, December 16, 1998In functional analysis, the statement in (4.3.13) is called the density theorem, Oden andReddy (1976, p.19).
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