Belytschko T. - Introduction (779635), страница 14
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Belytschko, Chapter 2, December 16, 1998This weak form is often called the principle of virtual power (or principle of virtualvelocities, see Malvern (1969), p. 241). If the test function is considered a velocity, theneach term in the above corresponds to a variation of power, or rate of work; for exampleρAbdx is a force, and when multiplied by δv(X) gives a variation in power. Therefore, theterms in the above weak form will be distinguished form the principle of virtual work inSection (2.3) by designating each term by P with the appropriate superscript. However, itshould be stressed that this physical interpretation of the weak form is entirely a matter ofchoice; the test function δv(X) need not be attributed any of the properties of a velocity; itcan be any function which satisfies Eq.
(2.7.2).We define the virtual internal power byδPintxbxbxaxa= ∫ δv,xσAdx = ∫δD σAdx = ∫ δDσdΩ(2.7.6)Ωwhere the second equality is obtained by taking a variation of (2.6.5), i.e., δDx = δv, x ,while the third equality results from the relation dΩ = Adx which parallels (2.5.20). Theintegral in Eq. (2.7.6) corresponds to the internal energy rate in the energy conservationequation (2.6.7) except that the rate-of-deformation D is replaced by δD, so designatingthis term as a virtual internal power is consistent with the energy equation.The virtual powers due to external and inertial forces are defined similarly:xbδPext= ∫ δvρbAdx + (δvAt x ) Γ = ∫ δvρbdΩ + ( δvAt x ) ΓtxaxbδPinert= ∫ δvρxa(2.7.7)tΩDvDvAdx = ∫ δvρdΩDtDtΩ(2.7.8)Using Eqs.
(2.7.6-2.7.8). the weak form (2.7.5) can then be written asδ P =δ P int − δ P ext + δ P inert = 0(2.7.9)where the terms are defined above. In summary, the principle of virtual power states thatif v( X, t) ∈U and δP= 0∀ δv( X) ∈U0(2.7.10)then the momentum equation (2.6.4), the traction boundary conditions (2.6.13) and thejump conditions are satisfied. The validity of this principle can be established by simplyreversing the steps used to obtain Eq. (2.7.5).
All of the steps are reversible so we candeduce the strong form from the weak form.The key difference of this weak form, as compared to the weak form for the totalLagrangian formulation, is that all integrals are over the current domain and are expressedin terms of variables which have a spatial character.
However, the two weak forms are justdifferent forms of the same principle; it is left as an Exercise to show that the principle ofvirtual work can be transformed to the principle of virtual power by using transformationson the integrals and the variables.2-38T. Belytschko, Chapter 2, December 16, 1998Exercise. Replace the virtual displacement in the principle of virtual work by avelocity and use the relations to show that it can be transformed into the principle of virtualpower.2.8.
Element Equations for Updated Lagrangian FormulationWe will now develop the updated Lagrangian formulation. As will become clear, theupdated Lagrangian formulation is simply a transformation of the total Lagrangianformulation. Numerically, the discrete equations are identical, and in fact, as we shall see,we can use the total Lagrangian formulation for some of the nodal forces and the updatedfor others in the same program. Students often ask why both methods are presented whenthey are basically identical.
We must confess that the major reason for presenting bothformulations today is that both are widely used, so to understand today's software andliterature, a familiarity with both formulations is essential. However, in a first course, it isoften useful to skip one of these Lagrangian formulations.The domain is subdivided into elements Ω e, so that Ω =∪Ω e . The coordinates of thenodes in the initial configuration are given by X1 , X2 ,KX nN and the positions of the nodesare given by x1( t), x2 (t ),Kx m( t ) . The m nodes of element e in the initial configuration bedenoted by X1e , X2e , KXme , and the positions of these nodes in the current configuration begiven by x1e (t ), x2e(t ),Kx me ( t) . The spatial coordinates of the nodes are given by the finiteelement approximation to the motionx I (t ) = x ( XI ,t )(2.8.1)Thus each node of the mesh remains coincident with a material point.We will develop the equations on an element level and then obtain the global equationsby assembly using the scheme described in Section 2.5.
As before, the relationshipsbetween the terms of the virtual power expression and the corresponding nodal forcesalong with the physically motivated names will be employed to systematize the procedure.The dependent variables in this development will be the velocity and the stress. Theconstitutive equation, combined with the expression for the velocity-strain, and the massconservation equation are treated in strong form, the momentum equation in weak form.The mass conservation equation can be used to easily compute the density at any pointsince it is a simple algebraic equation. We develop the equations as if there were noessential boundary conditions and then impose these subsequently.The velocity field in each element is approximated byv( X, t) =m∑ NI ( X ) vIe (t ) = N ( X ) v e( t)I =12-39(2.8.2)T.
Belytschko, Chapter 2, December 16, 1998Although the shape functions are functions of the material coordinates X, they can beexpressed in terms of spatial coordinates. For this purpose, the mapping x = φ ( X,t ) isinverted to give X =φ −1( x,t ) so the velocity field is()v( x, t) = N φ −1 ( x, t) ve ( t)(2.8.3)Although developing the inverse mapping is often impossible, partial derivatives withrespect to the spatial coordinates can be obtained by implicit differentiation, so the inversemapping need never be calculated.The acceleration field is given by taking the material time derivative of (2.8.2), whichgivesv˙ (X ,t ) = N( X )v˙ (t ) ≡ N( X)a(t )(2.8.4)It can be seen from this step that it is crucial that the shape functions be expressed asfunctions of the material coordinates.
If the shape functions are expressed in terms of theEulerian coordinates byv( x, t) = N( x ) ve ( t) = N( φ( X, t)) ve (t )(2.8.5)then material time derivative of the shape functions does not vanish and the accelerationscannot be expressed as a product of the same shape functions and nodal accelerations.Therefore, the shape functions are considered to be functions of the material coordinates inthe updated Lagrangian method.
In fact, expressing the shape functions in terms of spatialcoordinates is incompatible with a Lagrangian mesh, since we need to approximate thevelocity in an element, which is a material subdomain.2-40T. Belytschko, Chapter 2, December 16, 1998Current configurationt12ξ=0x(ξ, t )ξ=1ξParentx1 (t )x2 (t )X( ξ)x, XX1X2Reference configurationFig. 2.5. Role of parent configuration, showing mappings to the initial, undeformed configuration and thecurrent, deformed configuration in a Lagrangian mesh.Element Coordinates. Calculations in the updated Lagrangian formulation are usuallyperformed in the element coordinate system ξ in the parent domain. This is in fact simplerthan working in the spatial domain.
We have already used element coordinates to simplifythe evaluation of element nodal forces in the examples. Element coordinates, such astriangular coordinates and isoparametric coordinates, are particularly convenient for multidimensional elements.Consider Fig. 2.5, which shows a two-node element in the initial and currentconfigurations and the parent domain, which is the interval 0 ≤ξ ≤ 1.
The parent domaincan be mapped onto the initial and current configurations as shown. For example, in thetwo-node element, the mapping between the element coordinates and the Euleriancoordinates is given byx (ξ,t ) = x1(t )(1−ξ ) + x2 (t )ξ(2.8.6)or for a general one dimensional element asx (ξ,t ) = N(ξ )xe (t )(2.8.7)Specializing the above to the initial time gives the map between the parent domain and theinitial configurationmX (ξ) = ∑ NI (ξ ) XIe = N(ξ)Xe(2.8.8)I =12-41T.
Belytschko, Chapter 2, December 16, 1998which for the two-node element isX (ξ) = X1 (1− ξ ) + X 2ξ(2.8.9)The mapping between the Eulerian coordinates and the element coordinates, (2.8.6),changes with time, while the map between the initial configuration and the element domainis time invariant in a Lagrangian mesh. Therefore shape functions expressed in terms ofthe element coordinates by (2.8.8) will be independent of time. If the initial map is suchthat every point in the parent element ξ maps onto a unique point of the initialconfiguration, and for every point X there exists a point ξ , then the parent elementcoordinates can serve as material labels. Such a map is called one-to-one and onto.
Themap between the parent domain and the current configuration must be one-to-one and ontofor all time; this is discussed further in Example 2.8.3 and Chapter 3.As shown in Fig. 2.5, at any time the shape functions can be used to map between thecurrent and parent element configurations. Thus the element coordinates provide a linkbetween the initial configuration and the current configuration of the element which can beused in the evaluation of derivatives and integrals.It follows from Eqs. (2.8.7) and (2.8.8) that the displacements can also be interpolatedby the same shape functions since()u(ξ, t ) = x(ξ, t) − X (ξ ) = N(ξ ) x e( t) − Xe = N (ξ )ue (t )(2.8.10)The velocities and accelerations are also given by material derivatives of the displacement,while the test function is given by the same shape functions, sov (ξ,t ) = N(ξ )ve( t)a(ξ, t) = N(ξ )˙u˙ e (t )δv(ξ, t) = N(ξ )δve(2.8.11)since the shape functions are independent of time.Using Eqs.
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