Belytschko T. - Introduction (779635), страница 44

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These applications are often known as small-strain, large rotationproblems; see Wempner (1969) and Belytschko and Hsieh(1972).The components of a vector v in the corotational system are related to the globalcomponents byvˆ i = R ji vjvˆ = RTvorand v = Rvˆ(4.6.1)where R is an orthogonal transformation matrix defined in Eqs. (3.2.24-25) and the superposed“^” indicates the corotational components.The corotational components of the finite element approximation to the velocity field can bewritten asvˆ i (ξ,t ) = NI (ξ)vˆ iI ( t)(4.6.2)This expression is identical to (4.4.32) except that it pertains to the corotational components.Equation (4.6.2) can be obtained from (4.4.32) by multiplying both sides by RT .The corotational components of the velocity gradient tensor are given byˆ = ∂vˆ i = ∂N I (ξ ) vˆ (t ) = Bˆ vˆLijjI iI∂ˆx j∂ˆx j iIˆ = ˆv ∂N I = ˆv N T = vˆ Bˆ Tor LI ˆI I ,ˆxI I∂x(4.6.3)where∂NBˆ jI = I∂ xˆ j(4.6.4)The corotational rate-of-deformation tensor is then given byˆ = 1 Lˆ + Lˆ = 1  ∂vˆ i + ∂ˆvj Dijji2 ij2  ∂xˆ j ∂xˆ i ()(4.6.5)The corotational formulation is used only for the evaluation of internal nodal forces.

Theexternal nodal forces and the mass matrix are sually evaluated in the global system as before. The4-50T. Belytschko, Lagrangian Meshes, December 16, 1998the semi-discrete equations of motion are treated in terms of global components. We thereforeconcern ourselves only with the evaluation of the internal nodal forces in the corotationalformulation.The expression for ˆf Iint in terms of corotational components is developed as follows.

Westart with the standard expression for the nodal internal forces, Eq. (4.5.5):fiIint =∂N∫ ∂x Ij σ ji dΩorΩ(f intI ) = ∫ NIT, x σdΩ(4.6.6)NI , x = RN I ,ˆx(4.6.7)TΩBy the chain rule and Eq. (4.6.1)∂NI ∂N I ∂xˆ k ∂N I==R∂x j ∂ xˆ k ∂x j ∂ xˆ k jkorSubstituting the transformation for the Cauchy stress into the corotational stress, Box 3.2, and Eq.(4.6.7) into Eq. (4.6.6), we obtain(f intI ) = ∫ NIT, ˆx RT RˆσR T dΩT(4.6.8)Ωand using the orthogonality of R, we have(f intI ) = ∫ N IT,ˆxσˆ RT dΩT[ fiIint ]TorΩ= f Iiint =∂N∫ ∂xˆ jI σˆ jk Rki dΩT(4.6.9)ΩComparing the above to the standard expression for the nodal internal forces, (4.6.5), we can seethat the expressions are similar, but the stress is expressed in the corotational system and therotation matrix R now appears.

In the expression on the right, the indices on f int have beenexchanged so that the expression can be converted to matrix form.If we use the Bˆ matrix defined by Eq. (4.6.4) we can write(f intI ) = ∫ Bˆ IT σˆ RT dΩT∫Tˆ RT dΩfint= B TσΩ(4.6.10)Corresponding relations for the internal nodal forces can be developed in Voigt notation:ˆ T {ˆσ}dΩf Iint = ∫ RT BIΩwhere{Dˆ } = Bˆ ˆvI I(4.6.11)ˆ is obtained from Bˆ by the Voigt rule.and BIIThe rate of the corotational Cauchy stress is objective (frame-invariant), so the constitutiveequation can be expressed directly as a relationship between the rate of the corotational Cauchystress and the corotational rate-of-deformation4-51T. Belytschko, Lagrangian Meshes, December 16, 1998(ˆDσˆ ˆ ˆ= Sσ D D, σˆ , etcDt)(4.6.12)In particular, for hypoelastic material,ˆ ˆ ˆDσ= C: DDtorDσˆ ijDtˆ Dˆ=Cijkl kl(4.6.13)where the elastic response matrix is also expressed in terms of the corotational components.

Anˆ matrix for anisotropic materials need not beattractive feature of the above relation is that the Cchanged to reflect rotations. Since the coordinate system rotates with the material, material rotationˆ . On the other hand, for an anisotropic material, the C matrix changes as thehas no effect on Cmaterial rotates.Example 4.6. Rods in Two Dimensions. A two-node element is shown in Fig. 4.9. Theelement uses linear displacement and velocity fields.

The corotational coordinate ˆx is chosen tocoincide with the axis of the element at all times as shown. Obtain an expression for thecorotational rate-of-deformation and the internal nodal forces. Then the methodology is extendedto a three-node rod.yyˆxΩ0ˆyθ2Ωˆyˆx211xxFig. 4.9. Two-node rod element showing initial configuration and current configuration and the corotationalcoordinate.The displacement and velocity fields are linear in xˆ and given by4-52T. Belytschko, Lagrangian Meshes, December 16, 1998x2 1− ξ y2  ξ  x   x1 = y   y1ˆvx  vˆ x1 vˆ x 2  1− ξ ˆ  = ˆ vy   v y1 vˆ y2  ξ ˆxξ=l(E4.6.1)where l is the current length of the element. The corotational velocities are related to the globalcomponents by the vector transformation Eq.

(E4.6.1): vˆ xI  v xI  Rxˆx  = R ˆ , R = vyI vyI  RyxˆRxˆy  cos θ=Ryˆy  sinθ− sin θ 1  x21=cosθ  l y21−y21 x 21 (E4.6.2)A state of uniaxial stress is assumed; the only nonzero stress is σˆ x which is the stressalong the axis of the bar element. Since ˆx rotates with the bar element, σˆ x is the axial stress forˆ ,any orientation of the element. Only the axial component of the rate-of-deformation tensor, Dxcontributes to the internal power. It is given the derivative of the velocity field (E4.6.1):ˆ = ∂vˆ x = N  vˆ x1  = 1 [−1 +1]vˆ x1  = Bˆ vˆDxI , ˆx ˆ∂xˆv x 2  lˆvx 2 []ˆ = [ N ] = 1 [−1 +1]BI, ˆxl(E4.6.3)Nodal Internal Forces. The nodal internal forces are obtained from Eq. (4.6.8), which can berewritten as∂N∂N[ fIi ]int = ∫ ∂ˆx I σˆ jk RkiT dΩ= ∫ ∂ˆx I σˆ xx RTˆx idΩ = ∫ ˆBT σˆ xx RˆxTi dΩΩjΩ(E4.6.4)Ωwhere the second expression omits the many zeros which appear in the more general expression;the subscripts on the internal nodal forces have been interchanged.

Substituting (E4.6.2) and(E4.6.3) into the above gives[ fIi ]int1 −1  ˆ= ∫  [ σ][cos θ sin θ ]dΩl +1 x(E4.6.6)If we assume the stress is constant in the element, we can evaluate the integral by multiplying theintegral by the volume of the element, V = Al , which gives f1 x[ fIi ]int =  f 2xf1 y −cos θ= Aσˆ x f2y  cosθ−sin θ sin θ (E4.6.7)The above result shows that the nodal forces are along the axis of the rod and equal and opposite atthe two nodes.4-53T. Belytschko, Lagrangian Meshes, December 16, 1998The stress-strain law in this element is computed in the corotational system. Thus, the rateform of the hypoelastic law isDσˆ xˆ= EDxDt(E4.6.8)where E is a tangent modulus in uniaxial stress.

The rotation terms which appear in the objectiverates are not needed, since the coordinate system is corotational.To evaluate the nodal forces, the current cross-sectional area A must be known. Thechange in area can then be expressed in terms of the transverse strains; the exact formula dependson the shape of the cross-section. For a rectangular cross-section(ˆ +Dˆ˙ =ADAyZ)(E4.6.9a)Computation of internal nodal forces from one-dimensional rod. The internal nodal forces canalso be obtained by computing the corotational components as in Example 2.8.1, Eq.

(E2.2.8) andthen transforming by Eq. (4.5.40). In the corotational system, the nodal forces are given by Eq.(E2.8.8), so we write this equation in the corotational system:ˆ intˆf int =  fx1  = 1−1σˆ Adx∫ l  +1 x ˆf x2 Ω(E4.6.10)Since the we are considering a slender rod with no stiffness normal to its axis, the transversesnodal forces vanish, i.e. ˆf y1 = ˆf y2 = 0.Voigt notation. In Voigt procedures, the element equations are usually developed by starting withthe equations in the local, corotational cooordinates.

The global components of the nodal forcescan then be obtained by the transformation equations, (4.5.40). We first define T by relating thelocal degrees-of-freedom (which are conjugate to ˆf int ) to the four degrees-of-freedom of theelement: vˆ x1  cosθ = ˆvx 2   0v x1 sin θ00  v y1 cos θ  so T = 0cos θ sinθ  vx 2  0 vy 2 sinθ00cosθ0 sin θ(E4.6.11)which defines the T matrix.

Using Eq. (4.5.36), f = TTˆf , and assuming the stress is constant inthe element then givesintf int fx1  f y1 =  fx2  fy2 = TT f int0 cos θ −cos θ sin θ0 −sin θ  −1ˆˆ=Aσ   = A σ x cos θ  x  1  0 cos θ  0 sin θ sin θ4-54(E4.6.12)T. Belytschko, Lagrangian Meshes, December 16, 1998which is identical to (E4.6.7).Three-Node Element. We consider the three-node curved rod element shown in Fig. 4.10. Theconfigurations, displacement, and velocity are given by quadratic fields.

The expression for thenodal internal forces will be developed by the corotational approach.y, Yyexey232Ω0131X (ξ )Ωx (ξ , t)x, Xxξ =-1ξ =+1213parent elementFig. 4.10. Initial, current, and parent elements for a three-node rod; the corotational base vector eˆ x is tangent to thecurrent configuration.The initial and current configurations are given byX(ξ, t ) = XI (t ) NI (ξ)x (ξ,t ) = x I( t) N I (ξ )(E4.6.13)where[ N I ] =  2 ξ(ξ − 1)11− ξ21ξ (ξ + 1)2(E4.6.14)The displacement and velocity are given byu(ξ, t) = uI (t ) NI (ξ )v(ξ,t ) = v I ( t) NI (ξ)(E4.6.15)The corotational system is defined at each point of the rod (in practice it is needed only at thequadrature points).

Let ˆe x be tangent to the rod, so4-55T. Belytschko, Lagrangian Meshes, December 16, 1998eˆ x =x,ξx,ξwhere x,ξ = x I N I ,ξ (ξ)(E4.6.16)The normal to the element is given byˆe y = e z × ˆe x where e z = [ 0, 0, 1](E4.6.17)The rate of deformation is given byˆ = ∂vˆ x = ∂ˆvx ∂ξ = 1 ∂ˆvxDx∂xˆ∂ξ ∂ˆx x, ξ ∂ξmust be explained-may be wrong(E4.6.18)From Eq. (E4.6.15) and Eq. (E4.6.18)vˆ x = NI (ξ)( Rxxˆ vxI + Ryxˆ vyI )(E4.6.19)the rate-of-deformation is given byˆ = 1 N (ξ ) vxI Dxx,ξ I ,ξ  vyI (E4.6.20)The above shows the Bˆ I matrix to be1Bˆ I =Nx,ξ I ,ξ(E4.6.21)The nodal internal forces are then given by(f ) = [ fint TIxIf yI]int1[= ∫ ABˆ Iσˆ x x,ξ Rxxˆ−1]Ryxˆ dξ(E4.6.22)An interesting feature of the above development is that it avoids curvilinear tensors completely.However, the rate-of-deformation as computed here is correct; Exercize ?? shows how Eq.(E4.6.20) reproduces the correct result for a curved bar.Example 4.7.

Triangular Element. Develop the expression for the velocity strain and thenodal internal forces for a three-node triangle using the corotational approach.4-56T. Belytschko, Lagrangian Meshes, December 16, 1998yy32xyy2x3θθ0yx11xcurrent configuration Ωinitial configuration Ω e0Fig. 4.11 Triangular three-node element treated by corotational coordinate system.The element in its initial and current configurations is shown in Fig. 4.11. The corotationalsystem is initially at an angle of θ 0 with the global coordinate system; in the following, θ 0 is oftenchosen to vanish, but for an anistropic material it may be desirable to orient the initial xˆ -axis in adirection of anisotropy, for example, in a composite material it may be useful to orient xˆ in a fiberdirection.

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