Belytschko T. - Introduction (779635), страница 15
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(2.8.2) and (2.6.5) and noting Eq.(2.8.3), the rate-of-deformation can beexpressed in terms of the shape functions byDx ( x, t) = v, x ( x,t ) = N , x ( X ( x, t ))ve (t )(2.8.12)where we have indicated the implicit dependence of the shape functions on the Euleriancoordinates. The rate-of deformation will be expressed in terms of nodal velocities via a Bmatrix bymDx = v, x = Bv =e∑B v(2.8.13)BI = NI , x(2.8.14)eI II =1whereB = N,xor2-42T. Belytschko, Chapter 2, December 16, 1998This B matrix differs from the B 0 matrix used in the total Lagrangian formulation in thatthe derivatives are taken with respect to the Eulerian coordinates..To compute the spatial derivative of the shape function, we use the chain rule−1N, ξ = N , x x, ξ so N , x = N, ξ x,ξ(2.8.15)From the above, it follows thatDx (ξ, t ) = x,ξ−1N,ξ (ξ)v e( t ) = B(ξ )ve ( t)B(ξ) = N,ξ x ,ξ−1(2.8.16)Internal and External Nodal Forces.
We now use the procedure given in Sections2.4 and 2.5 to determine nodal forces corresponding to each term of the weak form on anelement level. The assembled equations and essential boundary conditions are developedsubsequently. The internal nodal forces will be developed from the virtual internal power.Defining the element internal nodal forces so that the scalar product with the virtualvelocities gives the internal virtual power, then from (2.7.6) and (2.8.13) we can writex em ( t )δPinte≡ δv fT inte ex em ( t )∫( )δv σAdx = δv ∫( )N=T,xTex1e tσ AdxT,x(2.8.17)x`e tThe transpose is taken of the first term in the integrand even though it is a scalar so that theexpression remains consistent when δv is replaced by a matrix product.
From thearbitrariness of δve , it follows thatxme (t )fei nt=xme (t )∫N,Tx σAdxx 1e (t )∫B≡TσAdx or fein t =x1e (t )∫ B σ dΩT(2.8.18)Ω e (t )We have explicitly indicated the time dependence of the limits of integration of the integralsto emphasize that the domain of integration varies with time. The internal nodal forces canthen be evaluated in terms of element coordinates by transforming (2.8.18) to the parentdomain and using the above with dx = x ,ξ dξ , givingxme ( t )intef=∫( ) Nx1etξmξmσAdx = ∫ N x σAx,ξ dξ = ∫ NT,ξ σAdξT,xT,ξ−1,ξξ1(2.8.19)ξ1The last form in the above is nice, but this simplification can be made only in onedimension.The external nodal forces are obtained from the expression for virtual external power(2.7.7):(δPeext = δveT feext = ∫ δv T ρbdΩ + δv T At xΩ et)Γt2-43(2.8.20)T.
Belytschko, Chapter 2, December 16, 1998Substituting (2.8.11) into the right hand side of the above and using the arbitrariness ofδve givesxmefeext(= ∫ N TρbAdx + N T At x)x1eΓte=∫NTΩ e (t )(ρbdΩ+ NT At x)Γte(2.8.21)where the second term contributes only when the boundary coincides with a node of theelement.Mass Matrix. The inertial nodal forces and mass matrix are obtained from the virtualinertial power (2.7.8):xm ( t )δPinert=δv Te feinert=∫δvT ρx1 ( t )DvAdxDt(2.8.22)Substituting (2.8.11) into the above yieldsxm (t )feinert∫ ρN=TNAdx ˙v e = M e˙ve(2.8.23)x1 (t )where the inertial force has been written as the product of a mass matrix M and the nodalaccelerations.
The mass matrix is given byx m ( t)M =e∫ ρNTNA dx =x1 (t )∫ ρNTN dΩ(2.8.24)Ω e (t )The above form is inconvenient because it suggests that the mass matrix is a function oftime, since the limits of integration and the cross-sectional area are functions of time.However, if we use the mass conservation equation (2.2.10) in the form ρ 0 A0 dX = ρAdx ,we can obtain a time invariant form. Substituting the (2.2.10) into (2.8.24) givesXmM =e∫ ρ0 NTNA0dX(2.8.25)X1This formula for the mass matrix is identical to the expression developed for the totalLagrangian formulation, (2.4.11). The advantage of this expression is that it clearly showsthat the mass matrix in the updated Lagrangian formulation does not change with time andtherefore need not be recomputed during the simulation, which is not clear from (2.8.24).We will see shortly that any nodal force for a Lagrangian mesh can be computed by eitherthe total or updated Lagrangian formalism.
The one which is chosen is purely a matter ofconvenience. Since it is more convenient and illuminating to evaluate the mass matrix inthe total Lagrangian form, this has been done.Equivalence of Updated and Total Lagrangian Formulations. The internal andexternal nodal forces in the updated and total Lagrangian formulations can be shown to be2-44T. Belytschko, Chapter 2, December 16, 1998identical.
To show the identity for the nodal internal forces, we express the spatialderivative of the shape function in terms of the material derivative by the chain rule:N, x ( X ) = N, X ∂X = N, X F–1 = B0 F –1∂x(2.8.26)From the first equality we have N, xdx = N, X dX , and substituting this into (2.8.18) givesxm ( t )feint=∫N T, xσAdxx1 ( t )Xm= ∫ N,TX σAdX(2.8.27)X1where the limits of integration in the third expression have been changed to the materialcoordinates of the nodes since the integral has been changed to the initial configuration. Ifwe now use the identity σA= PA0 , Eq.(2.2.9), we obtain from the above thatXmfeint=∫ N , X PA0 dXT(2.8.28)X1This expression is identical to the expression for the internal nodal forces in the totalLagrangian formulation, (2.5.14).
Thus the expressions for the internal nodal forces in theupdated and total Lagrangian formulations are simply two ways of expressing the samething.The equivalence of the external nodal forces is shown by using the conservation ofmass equation, (2.2.10).
Starting with (2.8.21) and using the (2.2.10) givesxmefeext(= ∫ N ρbAdx + N At xTx1eT) Γ = ∫ NT ρ 0b A0 dX +(NT A0 t 0x ) ΓXmetetX1(2.8.29)where we have used the identity t x A = t0x A0 in the last term. The above is identical to(2.4.8), the expression in the total Lagrangian formulation.From this and the identity of the expression for the mass matrix, it can be seen that thetotal and updated Lagrangian formulations simply provide alternative expressions for thesame nodal force vectors. The formulation which is used is simply a matter ofconvenience. Moreover, it is permissible to use either of these formulations for differentnodal forces in the same calculation. For example, the internal nodal forces can beevaluated by an updated Lagrangian approach and the external nodal forces by a totalLagrangian approach in the same calculation.
Thus the total and updated Lagrangianformalisms simply reflect different ways of describing the stress and strain measures anddifferent ways of evaluating derivatives and integrals. In this Chapter, we have also useddifferent dependent variables in the two formulations, the velocity and stress in the updatedformulations, the nominal stress and the displacement in the total formulation. However,this difference is not tied to the type of Lagrangian formulation, and we have done this onlyto illustrate how different independent variables can be used in formulating the continuummechanics problem. We could have used the displacements as the dependent variables inthe updated Lagrangian formulation just as well.2-45T.
Belytschko, Chapter 2, December 16, 1998Assembly, Boundary Conditions and Initial Conditions. The assembly ofthe element matrices to obtain the global equations is identical to the procedure describedfor the total Lagrangian formulation in Section 2.5. The operations of gather are used toobtain the nodal velocities of each element, from which the strain measure, in this case therate-of-deformation, can be computed in each element.
The constitutive equation is thenused to evaluate the stresses, from which the nodal internal forces can be computed by(2.8.19). The internal and external nodal forces are assembled into the global arrays by thescatter operation. Similarly, the imposition of essential boundary conditions and initialconditions is identical and described in Section 2.4. The resulting global equations areidentical to (2.4.17) and (2.4.15).
Initial conditions are now needed on the velocities andstresses. For an unstressed body at rest, the initial conditions are given byvI = 0, I = 1to nNσ I = 0, I =1 to nQ(2.8.30)That initial conditions in terms of the stresses and velocities is more appropriate forengineering problems is discussed in Section 4.2. Nonzero initial values can be fit by anL2 projection described at the end of Section 2.4.2-46T. Belytschko, Chapter 2, December 16, 1998Box 2.3Updated Lagrangian Formulationu( X,t) = N ( X (ξ ))ue (t) = N I (X (ξ ) )u eI (t)(B2.3.1)....
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