Belytschko T. - Introduction (779635), страница 47

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This can easily be seen from Eqs. (4.9.20)or (4.9.24-25), which show that the B0 matrix depends on F , which varies with time.4-70T. Belytschko, Lagrangian Meshes, December 16, 1998The total Lagrangian equation for internal nodal forces, (4.9.22) can easily be reduced to theupdated Lagrangian form, Eq. (4.5.14) without any transformations. This is accomplished byletting the configuration at a fixed time t be the reference configuration. We then use the totalLagrangian formulation with this new reference configuration.

It is immediately clear thatF =IFij =or∂xi= δij∂X j(4.9.28)since the two coordinates systems are now coincident at time t. There are several consequences ofthis:B0 = BS= σΩ0 = ΩdΩ 0 = dΩJ=1(4.9.29)to verify this, compare (4.9.20) and (??); from Box 3.2 it follows that since F = I and S = σ .Then Eq. (4.9.22) becomesf I = ∫ BTI {σ}dΩ(4.9.30)Ωwhich agrees with Eq. (4.5.14). This process of instantenously making the current configurationthe reference configuration is a helpful trick which we will again use later.Example 4.8.

Rod in Two Dimensions. Develop the internal nodal forces for a two-noderod element in two-dimensions. The bar element is shown in Fig. 4.12. It is in a uniaxial state ofstress with the only nonzero stress along the axis of the bar.yyΩ0YΩXθX2Y211xxFig. 4.12. Rod element in rwo dimensions in total Lagrangian formulationTo simplify the formulation, we place the material coordinate system so that the X-axiscoincides with the axis of the rod, as shown in Fig.

4.12, with the origin of the materialcoordinates at node 1. The parent element coordinate is ξ, ξ ∈[ 0,1] . The material coordinates arethen related to the element coordinates by4-71T. Belytschko, Lagrangian Meshes, December 16, 1998X = X 2ξ = l 0ξ(E4.8.1)where l 0 is the initial length of the element. In this example, the coordinates X, Y are used in asomewhat different sense than before: it is no longer true that x (t = 0) = X. However, thedefinition used here corresponds to a rotation and translation of x (t = 0) .

Since neither rotationnor translation effects E or any strain measure, this choice of an X, Y coordinate system isperfectly acceptable. We could have used the element coordinates ξ as material coordinates, butthis complicates the definition of physical strain components.The spatial coordinates are given in terms of the element coordinates byx = x1(1− ξ ) + x2ξy = y1 (1−ξ ) + y2ξ x  x1 =  y   y1orx2  1− ξ y2  ξ (E4.8.2)orx (ξ,t ) = x I( t) N I (ξ )(E4.8.3)where{NI (ξ)}T = [(1− ξ) ξ] = 1− lX0Xl0 (E4.8.4)The B0 matrix as defined in (4.9.7) is given by[B ] ≡ [∂N0iII∂NT∂X i ] =  1 ∂X∂N2  1= [ −1 +1]∂X  l0where Eq.

(4.8.1) has been used to give(4.9.7):( )F = x I BI0T x1= y1(E4.8.5)∂NI 1 ∂NI=. The deformation gradient is given by∂X l0 ∂ξx2  1  −1 1  = [x − xy2  l0  1  l0 2 1y2 − y1 ] ≡1[xl 0 21y21 ](E4.8.6)The deformation gradient F is not a square matrix for the rod since there are two space dimensionsbut only one independent variable describes the motion, (E4.8.2).The only nonzero stress is along the axis of the rod. To take advantage of this, we use thenodal force formula in terms of the PK2 stress, since S11 is the only nonzero component of thisstress.

For the nominal stress, P11 is not the only nonzero component. The X axis as defined hereis corotational with the axis of the rod, so S11 is always the stress component along the axis of therod. Substituting (E4.8.5) and (E4.8.6) into Eq. (4.9.19) then gives the following expression forthe internal nodal forces:4-72T. Belytschko, Lagrangian Meshes, December 16, 1998Tfint= ∫ B0T SFT dΩ0 =Ω0T∫ N, X SF dΩ0 =Ω0∫Ω01 −11S11] [ x21[l0  +1l0y21]dΩ0(E4.8.7)Since the deformation is constant in the element, we can assume the integrand is constant, somultiplying the integrand by the volume A0l0 we havef1y int A0 S11 −x21=f2 y l0  x 21 f1xf 2x− y21 y21 (E4.8.9)This result can be transformed to the result for the corotational formulation if we use Eq.

(E3.9.8)xyand note that cos θ = 21 and sinθ = 21 .llIn Voigt notation, the nonzero entries of the B0 matrix are the first row of (4.9.24), soB0 I = [ x , X NI , Xy, X N I , X ] = [ cosθ N I , X sin θ N I , X ]Noting that N1, X = −1 l0 , N2 , X = 1 l0 , we have that[B0 = B10]B02 =1[−cos θl0− sinθ cos θsinθ ]The expression for the nodal forces, (4.5.19) then becomesintf int fx1  f y1 ≡  fx2  fy2 −cos θ 1 − sinθ = ∫ B0T {S}dΩ0 = ∫{S11}dΩ0cosθl0Ω0Ω0 sinθ Example 4.9.Triangular Element. Develop expressions for the deformation gradient,nodal internal forces and nodal external forces for the 3-node, linear displacement triangle.

Theelement was developed in the updated Lagrangian formulation in Example 4.1; the element isshown in Fig. 4.2.The motion of the element is given by the same linear map as in Example 4.1, Eq. (E4.1.2)in terms of the triangular coordinates ξ I . The B0 matrix is given by (4.9.7):4-73T. Belytschko, Lagrangian Meshes, December 16, 1998[ ] []B0 I = B 0jI = ∂N I ∂X j , B0 = [B01 B02 ∂N1B03 ] =  ∂X∂N 1 ∂Y∂N2∂X∂N2∂Y∂N3 ∂X ∂N3 ∂X 1  Y23 Y31 Y12 2 A0  X32 X13 X21 1A0 = ( X32Y12 − X12Y32 )2=(E4.9.1)where A0 is the area of the undeformed element and X IJ = X I − X J ,Y IJ = Y I − YJ . These equationsare identical to those given in the updated Lagrangian formulation except that the initial nodalcoordinates and initial area are used. The internal forces are then given by (4.9.11b):Tfint f1x= [ fiI ] =  f2 x f3 x Y231 = ∫ Y312A0A0Y12intf1y f2 y  = ∫ B T0 PdΩ0Ω0f3 y X32  Y23P11 P12 a0 X13 a0 dA0 =  Y312P21 P22 X 21 Y12X32 PX13  11PX 21  21(E4.9.2)P12 P22 Voigt Notation.

The expression for the internal nodal forces in Voigt notation requires the B0matrix. Using Eq. (4.9.24) and the derivatives of the shape functions in Eq. (E4.9.1) givesY23 x,XB0 = X32 x, YY x, +X x,23 Y32XY23 y, XX32 y,YY23 y,Y + X32 y,XY31 x,XY31 y, XX13x, YX13y, YY31x,Y + X13 x, X Y31y, Y +X13 y, XY12 x, XX21x,YY12 x,Y +X21 x, XY12 y, XX21 y,YY12 y,Y + X21y, X (E4.9.3)The terms of the F matrix, x,X , y, X , etc., are evaluated by Eq. (4.9.6); for example:x,X = NI , X x I =1(Y x + Y x + Y x )2A0 23 1 31 2 12 3(E4.9.4)Note that the F matrix is constant in the element, and so is B0 . The nodal forces are then given byEq. (4.9.22):4-74T.

Belytschko, Lagrangian Meshes, December 16, 1998intfint f1x f  1y  f2 x = { fa } =   f2 y  f3x   f3y =∫ S11 TB0 S22  dΩ 0S 12Ω0(E4.9.5)Example 4.10.Two-Dimensional Isoparametric Element. Construct the discreteequations for two- and three-dimensional isoparametric elements in indicial matrix notation andVoigt notation. The element is shown in Fig. 4.4; the same element in the updated Lagrangianform was considered in Example 4.2.The motion of the element is given in Eq. (E4.2.1), followed by the shape functions andtheir derivatives with respect to the spatial coordinates.

The key difference in the formulation ofthe isoparametric element in the total Lagrangian formulation is that the matrix of derivatives of theshape functions with respect to the material coordinates must be found. By implicit differentiationN NI ,X −1  I ,ξ 0 = X ,ξ  N  = Fξ N I ,Y  I, η ( )−1 N I,ξ N  I ,η (E4.10.1)whereX, ξ = X I N I ,ξ∂Xi∂N I= XiI∂ξ j∂ξ jor(E4.10.2)Writing out the above givesX,ξY ,ξX, η  X I =  NY ,η   YI  I , ξ[NI ,η](E4.10.3)which can be evaluated from the shape functions and nodal coordinates; details are given for the 4node quadrilateral in Eqs. (E4.2.7-8) in terms of the updated coordinates and the formulas for thematerial coordinates can be obtained by replacing ( x I , y I ) by ( X I ,YI ) .

The inverse of X, ξ is thengiven byX,−1ξ X,ξ=Y, ξX, η −1 1  Y, η= ξY,η J 0 −Y,ξ− X,η −1 ξ, X η, X =X,ξ  ξ,Y η,Y where the determinant of the Jacobian between the parent and reference configurations is given byJ0ξ = X ,ξ Y,η −Y ,ξ X,ηThe B0 I matrices are given by4-75T. Belytschko, Lagrangian Meshes, December 16, 1998[B0TI = [ N I , XNI , Y ] = NI , ξ][−1NI ,η X,ξ= NI ,ξξ, XN I ,η ξ, Y]η, X η, Y (E4.10.4)The gradient of the displacement field H is given by u xI H = u IB0TI =  [ NI , Xu yI NI ,Y ](E4.10.5)The deformation gradient is then given byF=I+H(E4.10.6)The Green strain E is obtained from (B4.7.4) and the the stress S is evaluated by the constitutiveequation; the nominal stress P can then be computed by P = SFT ; see Box 3.2.The internal nodal forces are given by Eq. (4.9.11b):(f )int TI1 1= ∫ B PdΩ 0 = ∫ ∫ [ NI , XT0IΩ0−1 −1P11NI ,Y ] P21P12  ξJ dξdηP22  0(E4.10.7)where( )( )J0ξ =det X,ξ = det Fξ0(E4.10.8)If the Voigt form is used, the internal forces are computed by Eq.

(4.9.22) in terms of S .The external nodal forces, particularly those due to pressure, are usually best computed in theupdated form. The mass matrix was computed in the total Lagrangian form in Example 4.2.Example 4.12. Three-Dimensional Element. Develop the strain and nodal force equationsfor a general three-dimensional element in the total Lagrangian format. The element is shown inFigure 4.5. The parent element coordinates are ξ = (ξ1, ξ2 , ξ3 ) ≡ ( ξ,η,ζ ) for an isoparametricelement, ξ = (ξ1, ξ2 , ξ3 ) for a tetrahedral element, where for the latter ξi are the volume(barycentric) coordinates.Matrix Form.

The standard expressions for the motion, Eqs. (4.9.1-5) are used. Thedeformation gradient is given by Eq. (4.9.6). The Jacobian matrix relating the referenceconfiguration to the parent is X,ξX,ξ =  Y,ξ Z,ξX,ηY,ηZ,ηX X,ξ  ITY,ξ  = X I B0 I = {X I } ∂N I ∂ξ j =  YI  NI , ξZ Z,ξ  I[The deformation gradient is given by4-76][NI , η]NI , ξ (E4.12.1)T. Belytschko, Lagrangian Meshes, December 16, 1998x , ..., x N   N I, X  ∂N I   1Fij = [ x iI ]  =  y1 , ..., yN   N I, Y  ∂X J   z1, ..., z N   N I, Z [ ](E4.12.2) ∂N   NI , X   ∂N   ∂ξ   ∂N  -1 I  =  NI , Y  =  I   k  =  I X,ξ ∂X j   N   ∂ξk  ∂Xj   ∂ξk  I, Z (E4.12.3)wherewhere X,ξ-1 is evaluated numerically from Eq.

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