Belytschko T. - Introduction (779635), страница 43

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4.7. Current configuration of quadrilateral axisymmetric element; the element consists of the volume generatedby rotating the quadrilateral 2π radians about the z-axis.In this case, the isoparametric map relates the cylindrical coordinates [ r, z] to the parentelement coordinates [ξ, η ] :r (ξ,η,t )  rI( t) r =  NI (ξ,η) z(ξ,η,t )  z I ( t) (E4.4.1)4-45T.

Belytschko, Lagrangian Meshes, December 16, 1998where the shape functions NI are given in (E4.2.20. The expression for the rate-of-deformation isbased on standard expressions of the gradient in cylindrical coordinates (the expression areidentical to the expressions for the linear strain):∂ Dr   ∂r Dz   0= 1Dθ 2Drz   r∂ ∂z ∂v r 0 ∂r∂∂v z vr  ∂z   =  ∂zvr0  vz  r∂ ∂v r + ∂v z  ∂z ∂r ∂r (E4.4.2)The conjugate stress is{σ}T = [ σ r ,σ z , σθ ,σ rz ](E4.4.3)The velocity field is given byv r vrI  N1  = NI (ξ, η)   = v z  vzI   00 N2N1 00N2N300 N4N3 0˙dT = [v ,v , v , v , v , v , v , v ]r1 z1 r 2 z2 r3 z3 r4 z 40 ˙dN4 (E4.4.4)(E4.4.5)The submatrices of the B matrix are given from Eq. (E4.4.2) by ∂NI ∂r 0[ B ]I =  N I r ∂NI ∂z0 ∂N I ∂z 0 ∂N I ∂r (E4.4.6)The derivatives in (E4.4.6) now have to be expressed in terms of derivates with respect to theparent element coordinates.

Rather than obtaining these with a matrix product, we just write outthe expressions using (E4.2.7c) with x,y replaced by r,z, which gives∂NI 1  ∂z ∂N I ∂r ∂NI = −∂rJξ  ∂η ∂ξ ∂η ∂η (E4.4.7a)∂NI 1  ∂r ∂N I ∂z ∂NI = −∂zJξ  ∂ξ ∂η ∂ξ ∂ξ (E4.4.7b)where4-46T. Belytschko, Lagrangian Meshes, December 16, 1998∂z∂N= zI I∂η∂η∂z∂N=z I I∂ξ∂ξ(E4.4.8a)∂r∂N=rI I∂η∂η∂r∂N= rI I∂ξ∂ξ(E4.4.8b)The nodal forces are obtained from (4.5.14), which yieldsTTf intI = ∫ BI {σ} dΩ= 2π ∫ BI {σ}Jξ rd∆Ω(E4.4.9)∆where B I is given by (E4.4.6) and we have used dΩ = 2πrJξ d∆ where r is given by Eq.(E4.4.1).

The factor 2π is often omitted from all nodal forces, i.e. the element is taken to be thevolume generated by sweeping the quadrilateral by one radian about the z-axis in Fig. 4.7.Example 4.5.Master-Slave Tieline. A master slave tieline is shown in Figure 4.5.Tielines are frequently used to connect parts of the mesh which use different element sizes, for theyare more convenient than connecting the elements of different sizes by triangles or tetrahedra.Continuity of the motion across the tieline is enforced by constraining the motion of the slavenodes to the linear field of the adjacent edge connecting the master nodes. In the following, theresulting nodal forces and mass matrix are developed by the transformation rules of Section 4.5.5.master nodesslave nodes3421Fig.

4.8. Exploded view of a tieline; when joined together, the velocites of nodes 3 and 5 equal the nodal velocitiesof nodes 1 and 2 and the velocity of node 4 is given in terms of nodes 1 and 2 by a linear constraint.The slave node velocities are given by the kinematic constraint that the velocities along thetwo sides of the tieline must remain compatible, i.e. C 0 . This constraint can be expressed as alinear relation in the nodal velocities, so the relation corresponding to Eq. (4.5.35) can be writtenas4-47T. Belytschko, Lagrangian Meshes, December 16, 1998vˆ   I  M  =   {v M } vˆ S   A IT=  Aso(E4.5.1)where the matrix A is obtained from the linear constraint and the superposed hats indicate thevelocities of the disjoint model before the two sides are tied together. We denote the nodal forcesof the disjoint model at the slave nodes and master nodes by ˆfS and ˆf M , respectively.

Thus, ˆfS isthe matrix of nodal forces assembled from the elements on the slave side of the tieline and ˆf M is thematrix of nodal forces assembled from the elements on the master side of the tieline. The nodalforces for the joined model are then given by Eq. (4.5.36):ˆf  fS {f M } = TT ˆM  = [Iˆf A T  ˆM  fS ](E4.5.2)where T is given by (E4.5.1).

As can be seen from the above, the master nodal forces are thesum of the master nodal forces for the disjoint model and the transformed slave node forces.These formulas apply to both the external and internal nodal forces.The consistent mass matrix is given by Eq. (4.5.39): M M 0  I M = TT MT= I AT = M M + AT MsAM s A 0[](E4.5.3)We illustrate these transformations in more detail for the 5 nodes which are numbered in Fig.

4.8.The elements are 4-node quadrilaterals, so the velocity along any edge is linear. Slave nodes 3 and5 are coincident with master nodes 1 and 2, and slave node 4 is at a distance ξl from node 1,where l = x 2 −x1 . Therefore,v 3 = v1, v 5 = v2 , v 4 =ξv 2 + (1− ξ )v1(E4.5.4)and Eq. (E4.5.1) can be written as0 v1   Iv   0I v  2 0  1  v3  =  I v  (1− ξ )I ξI  v2  4  v5   0I 0 I 0I0T = I 1−ξ )I ξI ( 0I The nodal forces are then given by4-48(E4.5.5)T. Belytschko, Lagrangian Meshes, December 16, 1998ˆf1 ˆ f f1   I 0 I (1 − ξ)I 0 ˆ 2 = f f2  0 I 0ξII ˆ 3 f4 ˆf (E4.5.6)5The force for master node 1 isf1 = ˆf1 + ˆf3 + (1 − ξ)ˆf5(E4.5.7)Both components of the nodal force transform identically; the transformation applies to bothinternal and external nodal forces.

The mass matrix is transformed by Eq. (4.5.39) using T asgiven in Eq. (E4.5.1).If the two lines are only tied in the normal direction, a local coordinate system needs to beset up at the nodes to write the linear constraint. The normal components of the nodal forces arethen related by a relation similar to Eq. (4.5.7), whereas the tangential components remainindependent.4.6 COROTATIONAL FORMULATIONSIn structural elements such as bars, beams and shells, it is awkward to deal with fixedcoordinate systems. Consider for example a rotating rod such as shown in Fig.

3.6. Initially, theonly nonzero stress is σ x , whereas σ y vanishes. Subsequently, as the rod rotates it is awkwardto express the state of uniaxial stress in a simple way in terms of the global components of thestress tensor.A natural approach to overcoming this difficulty is to embed a coordinate system in the barand rotate the embedded system with the rod. Such coordinate systems are known as corotationalcoordinates.

For example, consider a coordinate system, xˆ = [xˆ , ˆy] for a rod so that ˆx alwaysconnects nodes 1 and 2, as shown in Fig. 4.9. A uniaxial state of stress can then always bedescribed by the condition that σˆ y = σˆ xy = 0 and that σˆ x is nonzero.

Similarly the rate-ofˆ .deformation of the rod is described by the component DxThere are two approaches to corotational finite element formulations:1. a coordinate system is embedded at each quadrature point and rotated with materialin some sense.2. a coordinate system is embedded in an element and rotated with the element.The first procedure is valid for arbitrarily large strains and large rotations. A majorconsideration in corotational formulations lies in defining the rotation of the material. The polardecomposition theorem can be used to define a rotation which is independent of the coordinatesystem. However, when particular directions of the material have a large stiffness which must berepresented accurately, the rotation provided by a polar decomposition does not necessarily providethe best rotation for a Cartesian coordinate system; this is illustrated in Chapter 5.A remarkable aspect of corotational theories is that although the corotational coordinate isdefined only at discrete points and is Cartesian at these points, the resulting finite element4-49T.

Belytschko, Lagrangian Meshes, December 16, 1998formulation accurately reproduces the behavior of shells and other complex structures. Thus, byusing a corotational formulation in conjunction with a “degenerated continuum” approximation, thecomplexities of curvilinear coordinate formulations of shells can be avoided. This is furtherdiscussed in Chapter 9, since this is particularly attractive for the nonlinear analysis of shells.For some elements, such as a rod or the constant strain triangle, the rigid body rotation isthe same throughout the element. It is then sufficient to embed a single coordinate system in theelement.

For higher order elements, if the strains are small, the coordinate system can beembedded so that it does not rotate exactly with the material as described later. For example, thecorotational coordinate system can be defined to be coincident to one side of the element. If therotations relative to the embedded coordinate system are of order θ , then the error in the strains isof order θ 2 . Therefore, as long as θ 2 is small compared to the strains, a single embeddedcoordinate system is adequate.

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