Belytschko T. - Introduction (779635), страница 41

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Belytschko, Lagrangian Meshes, December 16, 1998where A0 is the initial area of the element.Voigt Notation.We first develop the element equations in Voigt notation, which should befamiliar to those who have studied linear finite elements. Those who like more condensed matrixnotation can skip directly to that form. In Voigt notation, the displacement field is often written interms of triangular coordinates asux  ξ1 0 ξ 2 0 = uy   0 ξ1 0 ξ2ξ3 0 d = Nd0 ξ3 (E4.1.8)where d is the column matrix of nodal displacements, which is given by[dT = ux1, uy1,u x2 ,uy 2 ,ux 3 ,uy 3](E4.1.9)We will generally not use this form, since it includes many zeroes and write the displacement in aform similar to (E4.4.1).

The velocities are obtained by taking the material time derivatives of thedisplacements, giving v x  ξ1 0 ξ2 = vy   0 ξ1 00 ξ3 0  ˙dξ2 0 ξ3 [˙dT = v ,v , v ,v , v ,vx1 y1 x 2 y2 x 3 y3(E4.1.10)](E4.1.11)The nodal velocities and nodal forces of the element are shown in Fig. 4.3.3yf3f3y3f3 xf221v3 v3yv 3xf2 y2f2 xf1f1y1f1xv1v1yv1xxFig. 4.3. Triangular element showing the nodal force and velocity components.4-33v2 v2yv 2xT. Belytschko, Lagrangian Meshes, December 16, 1998The rate-of-deformation and stress column matrices in Voigt form areσ  xx {σ} = σ yy σ  xy  Dxx {D} =  Dyy 2D  xy (E4.1.12)where the factor of 2 on the shear velocity strain is needed in Voigt notation; see the Appendix B.Only the in-plane stresses are needed in either plane stress or plain strain, since σ zz = 0 in planestress whereas Dzz = 0 in plane strain, so Dzzσ zz makes no contribution to the power in eithercase.

The transverse shear stresses, σ xz and σ yz , and the corresponding components of the rateof-deformation, Dxz and Dyz , vanish in both plane stress and plane strain problems.By the definition of the rate-of-deformation, Equations (3.3.10) and the velocityapproximation, we have∂vx ∂N I=vIx∂x∂x∂v ∂NDyy = y = I v Iy∂y∂y∂v ∂v ∂N∂N2Dxy = x + y = I vIx + I v Iy∂y ∂x∂y∂xDxx =(E4.1.13)In Voigt notation, the B matrix is developed so it relates the rate-of-deformation to the nodalvelocities by {D} = Bd˙ , so using (E4.1.13) and the formulas for the derivatives of the triangularcoordinates (E4.1.5), we haveN I,xBI =  0 NI , y0 NI , y NI , x [ B] = [B1 B2 y231 B3 ] =02A x320x 32y23y310x130x13y31y120x210 x21 y12 (E4.1.14)The internal nodal forces are then given by (4.5.14): f x1  y23f  0 y1  fx2 a  y31  = ∫ BT {σ}dΩ= ∫2A  0 fy 2  ΩΩ fx 3  y12 0 fy 3 0x320x130x 21x32 y23  σ x13   xx σ dAy31  yy  σx21  xy y12 (E4.1.15)where a is the thickness and we have used dΩ = adA ; if we assume that the stresses and thicknessa are constant in the element, we obtain4-34T.

Belytschko, Lagrangian Meshes, December 16, 1998int f x1 f  y1  fx2   fy 2  fx 3   fy 3  y230a  y31= 2 0y12 00x320x130x21x32 y23 σ x13   xx σ y31  yy  σx21  xy y12 (E4.1.16)In the 3-node triangle, the stresses are sometimes not constant within the element; for example,when thermal stresses are included for a linear temperature field, the stresses are linear. In thiscase, or when the thickness a varies in the element, one-point quadrature is usually adequate.

Onepoint quadrature is equivalent to (E4.1.16) with the stresses and thickness evaluated at the centroidof the element.Matrix Form based on Indicial Notation. In the following, the expressions for theelement are developed using a direct translation of the indicial expression to matrix form. Theequations are more compact but not in the form commonly seen in linear finite element analysis.Rate-of-Deformation. The velocity gradient is given by a matrix form of (4.4.7)[ ][L = Lij = [viI ] N I , j=]v x1 v x 2=vy1 vy 21  y23vx1 + y31v x 2 + y12 vx 32 A  y23v y1 + y31vy 2 + y12v y3yvx 3  1  23yvy 3 2 A  31 y12x32 x13  =x21 x32 vx1 + x13v x 2 + x21vx 3 x 32v y1 + x13 vy 2 + x21vy 3 (E4.1.19)The rate-of-deformation is obtained from the above by (3.3.10):D=(1L +LT2)(E4.1.20)As can be seen from (E4.1.19) and (E4.1.20), the rate-of-deformation is constant in the element;the terms xIJ and yIJ are differences in nodal coordinates, not functions of spatial coordinates.Internal Nodal Forces.

The internal forces are given by (4.5.10) using (E4.1.5) for thederivatives of the shape functions:Tfint= [ f Ii ]intf 1x=  f2x f3 xint y23f1 y 1 f2 y  = ∫ N I , j σ ji dΩ = ∫y312AΩAy12f3 y [][ ]x32  σ xx σ xy x13 a dA(E4.1.21)σ xy σ yy x 21 where a is the thickness. If the stresses and thickness are constant within the element, theintegrand is constant and the integral can be evaluated by multiplying the integrand by the volumeaA, giving4-35T. Belytschko, Lagrangian Meshes, December 16, 1998Tfint y23a=  y312 y12y 23σ xx + x32σ xyx32  σ xx σ xy a x13  =y σ +x σσ xy σ yy  2  31 xx 13 xy y12σ xx + x21σ xyx21 y23σ xy + x32σ yy y31σ xy + x13σ yy y12σ xy + x21σ yy (E4.1.22)This expression gives the same result as Eq.

(E4.1.16). It is easy to show that the sums of each ofthe components of the nodal forces vanish, i.e. the element is in equilibrium. Comparing(E4.1.21) with (E4.1.16), we see that the matrix form of the indicial expression involves fewermultiplications.

In evaluating the Voigt form (E4.1.16) involves many multiplications with zero,which slows computations, particularly in the three-dimensional counterparts of these equations.However, the matrix indicial form is difficult to extend to the computation of stiffness matrices, soas will be seen in Chapter 6, the Voigt form is indispensible when stiffness matrices are needed.Mass Matrix.

The mass matrix is evaluated in the undeformed configuration by (4.4.52).mass matrix is given by˜ =MIJ∫ ρ0 NI N J dΩ0 = ∫ a0 ρ0ξI ξ J Jξ d∆0Ω0The(E4.1.23)∆where we have used dΩ 0 = a0 Jξ0 d∆ ; the quadrature in the far right expression is over the parentelement domain. Putting this in matrix form gives ξ1  ˜M = a0 ρ0 ξ2 [ξ1∆ ξ3 ∫ξ2ξ3 ]Jξ0 d∆(E4.1.24)where the element Jacobian determinant for the initial configuration of the triangular element isgiven by Jξ0 = 2A0 , where A0 is the initial area. Using the quadrature rule for triangularcoordinates, the consistent mass matrix is:2 1ρAa˜ = 0 0 0 1 2M12 1 1112(E4.1.25)˜ and then usingThe mass matrix can be expanded to full size by using Eq. (4.4.46), MiIjJ = δ ijMIJthe rule of Eq. (1.4.26), which gives20ρ A a 1M= 0 0 012  01 0020101102010010201101020010102 (E4.1.26)4-36T.

Belytschko, Lagrangian Meshes, December 16, 1998The diagonal or lumped mass matrix can be obtained by the row-sum technique, giving 1 0 0ρ0 A0a0 ˜M=0 1 0 3  0 0 1(E4.1.27)This matrix could also be obtained by simply assigning one third of the mass of the element to eachof the nodes.External Nodal Forces. To evaluate the external forces, an interpolation of these forces is needed.Let the body forces be approximated by linear interpolants expressed in terms of the triangularcoordinates asξ bx3  1 ξ by3  2  ξ3 bx   bx1 bx 2 =by  by1 by2(E4.1.28)Interpretation of Equation (4.4.13) in matrix form then givesTfext fx1= fy1fx 2fy 2f x3 ext bx1=fy3 by1bx 2by2ξ bx 3  1 ξ ξ ξby3  ∫  2 [ 1 2Ω  ξ 3ξ3 ] ρadA(E4.1.29)Using the integration rule for triangular coordinates with the thickness and density consideredconstant then givesTfextρAa bx1 bx 2=12 by1 by 22 1bx3  1 2by3  1 1112 (E4.1.30)To illustrate the formula for the computation of the external forces due to a prescribed traction,consider component i of the traction to be prescribed between nodes 1 and 2.

If we approximatethe traction by a linear interpolation, thenti = t i1ξ1 +t i2ξ 2(E4.1.31)The external nodal forces are given by Eq. (4.4.13). We develop a row of the matrix:[ fi1fi 2fi3 ]ext1=∫ tiN I dΓ = ∫ (t i1ξ1 + ti2 ξ2 )[ξ1Γ12ξ2ξ 3]al12dξ1(E4.1.32)0where we have used ds = l12dξ1 ; l12 is the current length of the side connecting nodes 1 and 2.Along this side, ξ 2 = 1 − ξ1 , ξ3 = 0 and evaluation of the integral in (E4.1.32) gives4-37T. Belytschko, Lagrangian Meshes, December 16, 1998[ffi 2i1fi3]ext=al12[2ti1 +ti26ti1 +2t i20](E4.1.33)The nodal forces are nonzero only on the nodes of the side to which the traction is applied. Thisequation holds for an arbitrary local coordinate system.

For an applied pressure, the above wouldbe evaluated with a local coordinate system with one coordinate along the element edge.Example 4.2.Quadrilateral Element and other Isoparametric 2D Elements.Develop the expressions for the deformation gradient, the rate-of-deformation, the nodal forces andthe mass matrix for two-dimensional isoparametric elements.

Detailed expressions are given forthe 4-node quadrilateral. Expressions for the nodal internal forces are given in matrix form.y34ηΩ0e1x ξ , η, t2ξX ξ , η341(-1, -1)2(1, -1)parentelementΩ0e1(1, 1)3xxX, tY(-1, 1)42XFig. 4.4. Quadrilateral element in current and initial configurations and the parent domain.Shape Functions and Nodal Variables. The element shape functions are expressed in terms of theelement coordinates (ξ, η) . At any time t, the spatial coordinates can be expressed in terms of theshape functions and nodal coordinates by x( ξ,t)x I ( t )ξ  = NI ( ξ) , ξ =   yI (t )  y( ξ,t ) η (E4.2.1)For the quadrilateral, the isoparametric shape functions are4-38T.

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