Belytschko T. - Introduction (779635), страница 96

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To avoid thesedifficulties, assumed strain methods with 2 or more quadrature points as described inSection 1.5.6 can be used.1.5 Assumed Strain Hourglass StabilizationIn this section, the stabilization procedure for the quadrilateral will be developed bymeans of the assumed strain methodology. The arguments used in constructing theassumed strain field for this procedure are identical to those used with the Hu-Washizuprinciple.

However, the implementation is much simpler because many of the intermediatematrices which are required in the Hu-Washizu approach can be bypassed. Nevertheless,the results obtained by the assumed strain procedure differ very little from the resultsobtained by the corresponding Hu-Washizu elements.The assumed strain approach can also be used in conjunction with quadratureschemes which use more than one point. This avoids the use of stabilization schemes, butdoes require substantially more effort if the constitutive equations are complex.In addition to describing the assumed strain method, the notion of projection ofstrains is examined further in this section. It is shown that the assumed strain fields whicheliminate volumetric locking and excessive stiffness in bending problems correspond toprojections of the higher order terms in the strain field.511.5.1 Variational principle Assumed strain elements herein are based on a simplified formof the Hu-Washizu variational principle as described by Simo and Hughes (1986) in whichthe interpolated stress is assumed to be orthogonal to the difference between the symmetricpart of the velocity gradient and the interpolated rate-of-deformation.

Therefore, thesecond term of (1.4.19) drops out leaving0 = δπ(e) = ∫Ω δe t CedΩ - δdt f exte(1.5.1)In this form, the interpolated stress does not need to be defined since it no longer appearsin the variational principle.The discrete equations then require only the interpolation of the strain, which werelate to the nodal displacements by B which will be defined later.e(x) ≡ B(x)dSubstituting (2) into (1) gives(1.5.2)0 = δdt ∫Ω B t CBdΩd - δdt f ext(1.5.3)eso the arbitrariness of δd leads tof int = f ext(1.5.4)wheref int = K ed(1.5.5)andK e = ∫Ω B t CBdΩ(1.5.6)eThe stiffness matrix of the fully integrated isoparametric element is found by (1.2.28).The application of the assumed strain method to the development of a stabilizationprocedure for an underintegrated element then involves the construction of an appropriateform for the B matrix which avoids locking.1.5.2 Elimination of Volumetric Locking. To eliminate volumetric locking, the strain fieldmust be projected so that the volumetric strain energy always vanishes in the hourglassmode.

For this purpose, we consider a general form of the assumed strain ε ox + q xe1h, x + q ye2h, y   ε ox+ε x  e =  ε oy + q xe2h, x + q ye1h, y  ≡  ε oy+ε y 2ε o + q xe3h, y + q ye3h, x 2ε o +2ε xy xy  xy(1.5.7a)qx = g t ux(1.5.7b)qy = g t uywhere e1 , e2 , and e3 are arbitrary constants, and qx and qy are the magnitudes of thehourglass modes, which vanish except when the element is in the hourglass mode.

In (7a)and subsequent equations, commas denote derivatives with respect to the variables thatfollow. Substituting (1.4.39) and (7b) into (7a), the assumed strain field is put into Bform as follows:52e = BdB=btx +e1 h, x g te2 h, y g te2 h, x g tbTy+e1 h, y g tbTy+e3 h, y g tbTx+e3 h, x g t(1.5.8)For the purpose of illustrating the projections, the symmetric displacement gradient(1.4.27b) is written asDu =uo x,yuox,x + ux,xouy,y + uy,y+ ux,y + u oy,x +=uy,x ε x+qxh, xε oy+qyh, y2ε o + q h, +x yxyoq yh, x(1.5.9)The dilatation of the assumed strain field given by Eq. (2a), which is denoted by ∆,vanishes in the hourglass mode if e1 = −e2.

This is shown as follows. Consider the nodaldisplacements that correspond to the hourglass mode of deformation.ux = α 3x huy = α 3y h(1.5.10)Evaluating the strain by (2), we obtain the dilatation as∆= ε x + ε y = (e1+e2)(α 3xh, x+α 3yh, y)(1.5.11)So for e1 = −e2, ∆ = 0. Thus, with this projected strain, the dilatation vanishes throughoutthe element in the hourglass mode. Furthermore, it can easily be shown that for the meshesin Figs. 6 and 7 with the nodal displacements given by (1.4.10), the dilatation ∆ vanishesthroughout the element.For linear elastic material with a constitutive matrix given by (1.4.40b), and the nodaldisplacements given in Eq. (10), the strain energy of the assumed strain element withe2=-e1 isU = 12∫Ωee t CedΩ(1.5.12)=µe21 (α 23x Hxx +α 3x α 3y Hxy +α 23y Hyy )+ 12 µe23 (α 23y Hxx +α 3x α 3y Hxy +α 23x Hyy )which is independent of the bulk modulus.

Thus the volumetric energy in this element isalways finite and the element will not be subject to volumetric locking.The portion of the volumetric strain which has been eliminated by this projection isoften called "spurious" or "parasitic" volumetric strain. Whatever the name, it is certainlyundesirable for the treatment of incompressible materials. Since in the nonlinear range,many materials are incompressible, its elimination from the element is crucial.The character of this projection for various values of e1 (when e1=-e2) is shown inFig.

9. The two axes represent the nonconstant terms in ux,x and uy,y , which are denotedby ux,x and uy,y , and the corresponding terms in the assumed strain (compare Eqs. (8) and(9)) ε x and ε y respectively. The square represents an example of a point in (ux,x , uy,y )53space, while the circles represent corresponding points in (ε x, ε y) space. From the formularelating these quantities, namelyε x = e1(ux,x - uy,y )(1.5.13a)ε y = e1(uy,y - ux,x )(1.5.13b)1it can be seen the e1=2 corresponds to a normal projection of the functions ux,x , uy,y ontothe line ε x + ε y = 0, which is the line on which the higher order terms in the assumedstrain field posses no dilatation. Other values of e1 shift the higher order terms of theassumed strain along the same line.uy,y, εyεx = -εyux,x, εxe 1 = 1 (normal projection)2e1 = 1is a point representing ux,x , uy,yare points representing ε x, ε yFigure 9.

Depiction of projection of nonconstant part of displacement gradient ux,x , uy,yonto isochoric assumed strain fields1.5.3 Shear Locking and its Elimination. Shear locking in the four-node quadrilateralmay be explained and eliminated by projection in a similar manner. It should bementioned, and this will become clear from the results, that the effect of "spurious" shear issomewhat different than that of "spurious" strains in volumetric locking. In volumetriclocking, the results completely fail to converge; with spurious shear, the solutionsconverge but rather slowly.

Thus the term "excessive shear stiffness" is probably moreprecise, but the term shear locking is also a useful description.To understand shear locking and its elimination, consider a beam represented by asingle row of elements which is in pure bending as shown in Fig. 10. In pure bending, themoment field is constant and as is well known to structural engineers, the shear mustvanish, since the shear is the derivative of the moment with respect to x: s=m,x.54mmyyxxFigure 10. A beam in pure bending showing that the deformation is primarily into thehourglass modeTo eliminate shear locking, the portion of the shear field which is triggered by anynodal displacements which are not orthogonal to g must be eliminated.

Since only h is notorthogonal to g, this is another way of saying that the shear associated with the hourglassmode must be eliminated. This can be accomplished by letting e3=0 in Eq. (8). In purebending, the nodal displacements in the local coordinate system of the element defined asshown in Fig. 10 are given byux=chuy=0(1.5.14)where c is an arbitrary constant.

If the strain energy is computed using Eq. (8) for arbitrarye3 , we find that the shear strain energyUshear = 12µe23 c2 Hyy = 0(1.5.15)so it vanishes as expected when e3 =0; parasitic shear in bending is thus eliminated. Thiscorresponds to the projection illustrated in Fig. 11. The shear field emanating from theodisplacement field can be written in terms of the 3 parameters ε xy , qx , and qy .

The secondand third parameters are associated with the parts of the shear which are triggered only bythe hourglass mode of deformation. The assumed shear strain field, ε xy, is the projectionof the strain field emanating from the displacement field onto the line of constant shearstrain fields, as shown in Fig.

11.552εoxy2εoxy + qx h,y + qy h,xqxqyFigure 11. Projection of higher order shear terms in assumed strain elementsTable 2 lists the arbitrary constants for Eq. (8) for the assumed strain elementsconsidered in this paper. Note that the fully integrated QUAD4 element can be obtained bystabilization with one point quadrature for linear materials. It can be shown that ASMDstabilization is identical to the mean dilatation approach of Nagtegaal et al. (1974) for linearmaterials.

ASQBI and ASOI are identical to the mixed method QBI and OI stabilization ofBelytschko and Bachrach (1986) for rectangular elements.Table 2. Constants to define the assumed strain fieldElementQUAD4ASMDe11ASQBIASOIADS111212e20− 12−ν−1− 12e3110001.5.4 Stiffness Matrices for Assumed Strain Elements. The stiffness matrix for all of theassumed strain elements can be obtained by (16). If we take advantage of (1.4.16), thenK e = K 1ewhere K 1eptpt+ K stabe(1.5.16)is the stiffness obtained by one-point quadrature with the quadrature pointξ=η=0, and K stabis the rank 2 stabilization stiffness, which is given bye (c H +c H )γ γ tc3 Hxy γ γ tK stab= 2µ  1 cxxH 2 γ γyytte(c1 Hyy +c2 Hxx )γ γ 3 xy(1.5.17)where the constants c1 , c 2 , and c3 are given by Table 3. Constants are given not only forthe elements listed in Table 2, but also for the ASSRI stabilization which behaves like the56SRI element of Hughes (1987) with elastic material.

SRI stabilization cannot be derived bythe assumed strain approach. It is obvious that the plane strain QUAD4 element will lockfor nearly incompressible materials since c1 and c3 get very large. The projection toeliminate excessive shear stiffness corresponds to c2=0. When both projections are made,then c1=-c3.Table 3. Constants for assumed strain stabilizationElementQUAD4 (plane strain)QUAD4 (plane stress)ASSRIASMDASQBIASOIADSc11-ν1-2ν11-ν1121+ν212c212121212000c312(1-2ν)1+ν2(1-ν)120-ν(1+ν)-2-1 21.5.5 Nonlinear Hourglass Control.

The nonlinear counterpart of the Simo-Hughes(1986) principle has been given by Fish and Belytschko (1988) as the following weakform:∫∫0 = δΠ = Ω eδe° t s (e° ,s ,...)dΩ + δ Ω ett (Dv − e° )dΩ - δv t f ext(1.5.18)where e° is the interpolated velocity strain (rate-of-deformation), s the Cauchy stress whichis computed from the velocity strain and other state variables by the constitutive equation, tthe interpolated Cauchy stress, and Dv is the symmetric part of the velocity gradient; thelatter would be equivalent to the rate-of-deformation in a standard displacement method, butin mixed methods, the velocity gradient is projected on a smaller space to avoid locking.Note that s was the symbol for the interpolated stress in Section 1.4, but has a newmeaning here. The superposed circle on the symbol for the rate of deformation, e° does notindicate a time derivative.The velocity and strain-rate (rate-of-deformation) are interpolated byneNv = ∑N I(ξ,η)v I ≡ Nv(1.5.19)I=1neNe = ∑B I(ξ,η)v I ≡ Bv°I=1(1.5.20)where neN is the number of nodes per element.

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