Computational Mechanics 18 (1996) 413-428 �9 Springer-Vedag 1996

On enhanced strain methods for small and finite deformations of solids

P. Wriggers, J. Korelc 1

Abstract Numerical simulations of engineering problems require robust elements. For a broad range of applications these elements should perform well in bending dominated situations and also in cases of incompressibility. The element should be insensitive against mesh distortions which frequently occur due to modern mesh generation tools or during finite deformations. Possibly the elements should not lock in the thin limits and thus be applicable to shell problems. Furthermore due to efficiency reasons a good coarse mesh accuracy is required in nonlinear analysis. In this paper we discuss the family of enhanced strain elements in order to depict the positive and negative aspects related to these elements. Throughout this discussion we use numerical examples to underline the theoretical results.

1 Introduction The search for elements which provide a general tool for solving arbitrary problems in solid mechanics has a long history. This can be seen be from the large number of papers which have been published on the subject. The main goal is to find general ele- ment formulations which fulfill the following requirements

1. no locking for incompressible materials, 2. good bending behaviour, 3. no locking in the limit very thin elements, 4. distortion insensitivity, 5. good coarse mesh accuracy, 6. simple implementation of nonlinear constitutive laws, 7. efficiency.

These requirements have different origins. The first two result from the necessity to obtain acceptable answers for the mentioned problems, especially the first point is essential for the analysis of rubberlike materials or of classical]2-elasto-plasticity problems. Thus a wide range of applications needs this sort of elements. The third point becomes more and more important since it enables the user of such elements to simulate shell problems by three-dimensional elements, which is simpler for

Communicated by S. N. Atluri, 26June 1996

P. Wriggers, I- Korelc Institut fiir Mechanik, Technische Hochschule Darmstadt, Hochschulstr. 1 64289 Darmstadt, Germany

1 I. Korelc University of Ljubljana, Dept. of Civil Engineering, lamova 2, Ljubljana, Slovenia

Dedicated to the 10th anniversary of Computational Mechanics

complicated structures. This spares the need for introducing finite rotations as variables in thin shell problems, results in simpler contact detection on upper and lower surfaces, and provides the possibility to apply three-dimensional constitutive equations straightaway. The fourth point is essential since modern mesh generation tools yield, for arbitrary geometries, unstructured meshes which always include distorted elements. Also elements will be highly distorted during nonlinear simula- tions including finite deformations. The fifth point results from the fact that many engineering problems have to he modeled as three-dimensional problems. Due to computer limitations, quite coarse meshes have to be used often to solve these problems. Thus an element which provides a good coarse mesh accuracy is valuable in these situations. Finally point six is associated with the fact that more and more nonlinear computations invol- ving nonlinear constitutive models have to be performed to design engineering structures. Thus an element formulation which allows a straightforward implementation of such consti- tutive equations is desirable. Last, but not the least, the efficiency of the element formulation, mentioned in the last point, is of great importance when finite element meshes with several hund- reds thousands of elements have to be used to solve complex engineering problems.

1.1

Discussion of different element formulations To construct elements which fulfil most of these require- ments, and possibly all of them, different approaches have been followed throughout the last years. Among these are

�9 techniques of underintegration, �9 stabilization methods, �9 hybrid or mixed variational principles for stresses and

displacements, involving the use of complementary energy, �9 mixed Hu-Washizu variational principles, �9 mixed variational principles for rotation fields, �9 mixed variational principles for selected quantities

In the following we discuss these methods in more detail, and try to state which of the afore mentioned requirements are fulfilled by the different methods.

Underintegration techniques and stabilization The most simple but sometimes very efficient approach is provided by the underintegration of the stiffness matrix or of parts of it (then called selective reduced integration). A vast part of literature has been devoted to this topic from which we cite some of the most prominent ones. The development started with a pure application of reduced integration techniques, see e.g. Hughes (1980). In some approaches rank deficiency of underintegrated

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elements occurs and then leads to hourglassing. This can be bypassed by stabilization techniques, see e.g. Belytschko, Ong, Liu and Kennedy (1984) or Flanagan and Belytschko (1981).

By using these techniques, elements can be constructed which fulfil the reguirements numbered 1., 4., 5, 6. and 7. that means they do not lock in case of incompressibility, have a good coarse mesh accuracy, are distortion insensitive, and allow a simple and efficient implementation of arbitrary constitutive equations. Finally they provide the most efficient element for- mulation since the order of integration is reduced and with this the total number of function evaluations is minimal. However one major drawback of most of the formulations is the depen- dence of their performance on stabilization parameters which can effect the accuracy, and in the worst case, lead to a direct dependence of the results on these parameters.

Hybrid or mixed variational principles for stresses and displacements When mixed variational principles are ap- plied to construct finite elements several points of departure are possible. For cases of linear elasticity the mixed formulation has been pioneered by Plan (1964), which lead to a number of different formulations, as discussed in detail by Atluri (1975). In this line of research, Rubinstein, Punch and Atluri (1983), Plan and Sumihara (1984), have provided a very efficient and accurate element which fulfils all mentioned points. For arbi- trary nonlinear material including hyperelastic materials and elastic-plastic materials, the hybrid and mixed finite element method require the construction of a complementary energy density. While the inversion of stress-strain relations is not in general possible, to perform a Lagendre transformation to con- struct the total complementary energy density, a large number of papers (see Atluri (1980) and Seki and Atluri (1995)) exist in the literature in the construction of incremental (or of rate) complementary energy for elastic as well as inelastic materials. Incremental hybrid and mixed finite element methods can then be constructed. Also for a semilinear constitutive equation Atluri (1973) has constructed a nonlinear element based on a mixed form in second Piola-Kirchhoff stresses and displace- ments for the so-called St. Venant constitutive equation which relates the second Piola-Kirchhoff stresses linearly to the Green strains.

Assumed or enhanced strain formulation based on Hu-Washizu principle Lately, in their pioneering work, Simo and Rifai (1990) in the linear case and Simo and Armero (1992) or Simo, Armero and Taylor (1993) in the nonlinear case have developed a family of elements which are based on the Hu- Washizu variational principle. These elements are extensions of the incompatible QM6 element developed by Taylor, Beresford and Wilson (1976). They do not seem to have any rank defi- ciency and perform well in bending situations as well as in the case of incompressibility. Thus these elements are general applicable. On top of that, comparisons to other element formu- lations show a very good coarse mesh accuracy. Furthermore, due to the construction of the element with the enhanced strain modes, the implementation of ineleastic material models is straightforward. Following the work of Simo and Rifai (1990), several other authors have also developed similar element for- mulations for small strain application, see e.g. Andelfinger, Ramm, Roehl (1992). It is now well established that the enhan-

ced strain elements have a superb overall performance in the area of small strains. In total these elements meet for linear applications almost all requirements mentioned above, they even can be formulated such that they are numerically efficient, see e.g. Korelc and Wriggers (1995).

Mixed variational principles for rotational fields Several formulations involving rotations as independent field variables have been proposed so far in the literature. Hughes and Brezzi (1989) have constructed mixed elements for linear elasticity with drilling degrees of freedom based on sound mathematical back- ground. Afluri and Murakawa (1977) have introduced varia- tional principles with rotations for nonlinear elasticity. Atluri and Cazzani (1995) discuss regularized primal and mixed varia- tional principles involving rotations and drilling degrees of freedom for finitely deformed solids and shells. Seki and Atluri (1994 and 1995) give a comprehensine finite elasticity, strain localization and shear band formation. The use of drilling de- grees of freedom for finite deformations has also been intro- duced in Ibrahimbegovic, Taylor and Wilson (1990), Iura and Atluri (1992), Atluri and Cazzani (1995) and Gruttmann, Wagner and Wriggers (1992), where the last paper is mainly devoted to shell analysis.

Mixed variational principles for selected quantities Often special constraint conditions have to be fulfilled throughout numerical simulations. In these cases it is often advantageous to apply special mixed principles to overcome mathematical difficulties associated with the incorporation of constraints. One prominent example is incompressibility which has to be fulfilled within solid mechanics for rubber elasticity or ]2 plasticity but it is also imminent in incompressible flow. These special mixed principles rely on a kinematical split of the deformation. As an example we like to mention the selected three field Hu-Washizu principle which leads to a nonlinear version of the Q1-P0 element, see Simo, Taylor and Pistar (1985) a similar principle is also stated in Atluri and Reissner (1989). The associated ele- ments fulfill points 1., 4., 5., and 7. Due to the kinematical split the formulation of the constitutive relation becomes more com- plicated, especially the incremental form being needed within Newton's method for the iterative solution of the problem.

1.2 Enhanced elements From the above discussion we see that enhanced strain elements meet most of the requirements stated in the beginning of this overview. Thus we will concentrate in the following on these elements which seem to be the preferable choice for an element which can be applied to different types of problems. However this concept has also several disavantages when applied to non- linear problems which will be mentioned here briefly and then be discussed at length together with the advantages of the en- hancement procedure in the next sections.

Number of enhanced modes First it is interesting to note that not all types of elements can be enhanced by assumed strains or incompatible modes. In this context we like to men- tion the work of Reddy and Simo (1995) who have shown that enhancement procedures do not work for triangles. Thus only quadrilaterals can be enhanced. For these elements many dif- ferent formulations have been developed in recent years. They

all improve the behavior of low-order elements, see e.g. Simo and Rifai (1990), Andelfinger and Ramm (1993), or Korelc and Wriggers (1995, 1996a). It is generally accepted that four en- hanced modes are sufficient to design a locking free two-dimen- sional element which is objective and always satisfies the patch test. Within three-dimensional analysis it will be derived in this paper that at least 9 modes are necessary to design a locking free element (even in the thin plate limit), however this element does not completely satisfy the objectivity requirement, see section 3. The element which fulfills all necessary requirements and does not lock in the thin limit needs 21 enhanced modes and thus is extremely inefficient.

The number of enhanced modes can also be decreased if the enhanced strain method is combined with the hybrid stress method. However the number of internal parameters is larger and due to this, the efficiency of the element formulation may be lost. Furthermore the implementation of general nonlinear constitutive equations is not straight forward any more, see e.g. Piltner and Taylor (1995).

Here u, a, ~ represent displacements, stresses, and strains respectively, C stands for the elasticity matrix and B is strain- displacement matrix. The integration is performed over the domain B 0. By applying the operator B the strains are calculated from the compatible displacement field u. U ~xT is the total potential energy of the external loading. We consider a strain field e in the following form

companble

where g is an independent incompatible enhanced strain field. Introducing (2) in (1) yields

Bo \

(3)

If the enhanced strain field ~ is chosen such that it is orthogonal to the stress field,

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Problems in nonlinear applications Undesirable modes associated with enhanced element formulations were first obser- ved by Simo and Armero (1992) in the presence of high strains in case of plasticity in their pioneering work regarding the non- linear version of the Q1/E9 element. Later Simo, Armero and Taylor (1993) discussed a version of the enhanced strain element denoted QM1/E12 and concluded that possible loss of rank for is caused by underintegration of the element stiffness matrix. However, as noted first in Wriggers and Reese (1996a), appear- ance of spurious modes (hour-glassing) can be observed, even for very simple deformation states. In the same paper, the pro- blem was analyzed analytically using a single element which already shows all basic features of the phenomenon. A special stabilization technique was suggested in Wriggers and Reese (1996b) to overcome this problem but a general solution still remains unknown. Associated phenomena also occured in the presence of large deformations in several practical applications, see e.g. Neto, Peric, Huang and Owen (1995). One suggestion to overcome this problem was associated with the reduction of the number of enhanced modes. It was observed that a two enhanc- ed mode model does not exhibit hour-glassing, however two modes are not enough to avoid shear as well as volumetric lock- ing, see Crisfield (1995). A new method to overcome these short- comings has been presented in Korelc and Wriggers (1996a) by introducing a new enhancement procedure using consistent gradients which will be discussed in detail in the following.

2 Enhanced strain formulation for linear problems

2.1 Enhanced Assumed Strain (EAS) formulation by Simo and Rifai Let us briefly recall the well known EAS formulation in the case of linear elasticity, first proposed by Simo and Rifai (1990). The variational basis for the formulation is the well known three- field Hu-Washizu functional. For linear elasticity it can be written as

U (u, e a) = ~ (~eTce --aT~: + aTBu)Vo + U EXT. Bo \

(1)

~re~ V0 = 0 (4) B0

then the stresses can be eliminated from (1) and a new simplified functional is obtained

U(u, ~) = ~ l ( B u + a~)TC(Bu + ~e) Vo + u~X'Z (5) B0 :Z

For a typical element continuous displacements and the dis- continuous strain field e e are approximated on the element domain as follows

u = N(~)d e (6)

~e = Ga L (7)

where N (~) is the matrix containing standard isoparametric shape functions and d e is the vector of the element nodal dis- placements. The enhanced strain field ar is defined by the inter- polation matrix G and the vector of internal strain parameters ~. The strain parameters ~e are local for each individual element and can be eliminated at the element level.

The enforcement of full orthogonality on the stress field in (4) at the variational level leads to an over-restrained model (indeed we will get back the original displacement formulation since (4) yields then ~e = 0). Thus orthogonality is only enforced for a constant stress field 6 over the element domain, leading from (4) with the discontinous strains, (7), to

edVo:O y edVo:O S GaVo:O Bo Bo Bo

(8)

Equation (8) ensures the satisfaction of the patch test and convergence of the enhanced strain method for linear elasticity, however no variational justification for this step can be made.

If we simply express the enhanced strain interpolation matrix in reference coordinates of the element (Gr then the orthogona- lity condition will not be fulfilled for the actual frame. Taylor, Beresford and Wilson (1976) have proposed a multiplication of the enhanced strain interpolation matrix expressed in local coordinates, G~, with the factor ]0/] in order to fulfill the ortho-

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gonality condition (8) for both, reference and actual frame. ] is the determinant of the ]acobian of the isoparametric mapping and ]0 is the determinant evaluated at the element center. Addi- tional transformation of the enhanced gradient to the global frame improves the bending characteristics of the element for distorted meshes, although this transformation is not prerequi- site by variational consistency. That leads to the final form for a classical EAS interpolation matrix

G =-~TorG(~) (9)

where T O is transformation matrix which steams from the tensor transformation of the strains when ~ is written in a more convenient vector form ~ = {~x, ~y, ~=, ;;~y, 7~, 7y~}, see also Simo and Rifai (1990).

Still the major question remains, how to choose the inter- polation matrix G(~) in order to obtain an element with best overall characteristics. Basic possibilities, but not all, for two- dimensional elements are

! 0 0 0 ~ 1 G(~)-- r/ 0 0 --~r/ ... (10)

0 ~ r/ 1~-r Q1/E4q Q1/E5q

leading to different elements. For three dimensional elements we can choose

tional modes improve the element behavior in the incompres- sible limit for distorted element shapes. Efficiency of those ele- ments is closely related to the number of enhanced modes thus, increasing the number of modes decreases the efficiency. The elements will be compared in the examples at the end, however none of them is unconditionally free of shear and volumetric locking.

2.2 Behavior of enhanced elements in the incompressible limit Change of volume, as a consequence of the compatible strain field ed --_--_ Bu, is for general distorted element shapes a rational function of the coordinates ~, ~, (. It's Taylor expansion leads to

V d

~_ Co A V C1 ~ _~ C2rl Ay C3 ~ _~_ C 4 ~ ] A V C 5 ~ _}_ C6l~ ~

terms arising for rectangular and paralelogram type shapes

+ ... higher order terms (12)

where the constants C, depend on the nodal displacements d. Change of volume due to the enhanced strain field is then ex- pressed in a similar way as

We e e e

= c ~ + c;~ + c;r + c ; ~ + . . . (13)

G(4) =

" 4 0 0 0 0 0 0 0 0 ~1 0 0 0 0 0 0 o o ~ o o o o o o o o ~ o o o o o o o o o ~ o o 0 0 0 0 0 0 0 ~1 ~

0 0 0 0 0 0

0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 o ~ ~ o o 0 0 0 ~ ~

0 iv 0 0

0 0

0 0

r o o o o

o in ~ o o o o o ~ ~r 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

QI/E9q Ql/E15q Ql/E21q (11)

which also leads to different formulations. Simo and Rifai (1990) took an assumed stress field as a point of departure for construc- ting a four node element by enforcing orthogonality (4) on the element level. This resulted in a element with five modes QI/E5, s e e (10). However, as mentioned, in the 2D case four modes (Q1/E4 element) are already sufficient for full locking free ele- ments. Thus additional modes do not significantly improve the element behavior.

Li, Crook and Lyons (1993) then used that procedure and obtained several 3D EAS elements. Different possibilities in- cluding also the influence of quadratic modes were systema- tically analyzed by Andelfinger and Ramm (1993). They propo- sed elements from 9 up to 30 enhanced modes (designated 3D. EAS-9 to 3D. EAS-30).

Let us now consider the general interpolation (11) for the 3D-case, proposed by Andelfinger and Ramm (1993). The first nine modes are needed to complete the incomplete polynomials of the isoparametric displacement model (Q1/E9 element). The next six modes (Q1/E15 element) improve the bending behavior of the element for distorted element shapes. The last six addi-

where the constants C ~ depend on enhanced parameters ~e" According to Simo, Armero and Taylor (1993) an element will be free of volumetric locking, if the enhanced strain interpolation is constructed such that it leads for the incompressible limit to C, = - C e for all nonzero terms in (12). Thus only constant change of volume remains.

From (12) we can conclude that for an undistorted element shape, 6 enhanced modes are sufficient for an element that is free of volumetric locking, a fact which was explicitly proved by Freischl~iger and Schweizerhof (1995) (but not, at the same time, for a shear locking free element). Unfortunately, for gene- ral distorted case, this number depends on the order of numeri- cal integration. For an eight-noded element, integrated with a standard eight point integration rule, we need 30 enhaced modes for a full locking-free-element. This element is then identical to the HR-18 element with 18 independent stress modes by Plan and Tong (1986) which is also volumetric locking free. It is obvious that full-locking-free elements formulated with the classical EAS methodology become very inefficient and thus some additional modifications have to be performed.

2.3 Modified enhanced strain formulations The above problem can be solved with modified enhanced strain formulations. Within the classical EAS methodology proposed by Simo and Rifai (1990) the derivatives of basic compatible isoparametric functions remain unchanged and the only design parameter is interpolation of the enhanced strain field ee. Different modifications were proposed in the literature during the recent years with some common points. To obtain a volumetric-locking-free element, regardless of the order of numerical integration and distortion of element, the derivatives of the basic isoparametric shape functions have to be modified. Let us recall some possibilities.

Simo, Armero and Taylor (1993) modified the gradients of the isoparametric shape functions in the same way as the gradient of the enhanced strain field. Thus they obtained the formulation

discussed in the previous section. Introducing (14) in (1) yields a new functional

j" ~ ( B u + ~'~ + ~*)r C ( B u + ~:'~ + a~)dV o + U = uExr Bo

with two orthogonality conditions

or~'"dVo = O, ~6re~dVo = 0 (15) Bo Bo

We approximate the additional enhanced field ~" by

~:~u = ~c _ Bu + a TM (16)

Now the assumed strain fields K and e~" have to satisfy the following orthogonality conditions.

417

GRAD x (N,) = ~ Jo I GRADe (N,)

The resulting element, designated QM/E12, is locking free and has 12 enhanced modes. Unfortunately this modification is not consistent and the resulting element fails to satisfy the patch test, as well as the thin plate bending problem (see examples at the end).

Freischl~iger and Schweizerhof (1995) proposed a method wherein the stiffness matrix is split-up into a constant matrix Koch, which ensures convergence, and a stabilization matrix K~ b. For the evaluation of K~,~b, ] is replaced by/0 in all terms. The derived elements with nine and twelve enhanced modes fulfil the patch test, however they depict either shear or volumetric locking.

A special case of this procedure is the family of under integrated and stabilized elements, introduced by Liu, Ong and Uras (1985). There stabilization is performed by Taylor expan- sion of the basic isoparametric shape functions. No enhanced modes were used, however since the stabilization field is in this case orthogonal to a constant stress field, this formulation can also be viewed as a special case of the enhanced strain method. In the following we will describe a more general for- mulation using Taylor expansion which includes also enhan- ced strain fields. It was proposed by Korelc and Wriggers (1995, 1996b) for the construction of two dimensional element with four enhanced modes and three dimensional elements with 9 modes.

2.3.1 Formulation of enhanced strain elements with Taylor expansion of the shape function derivatives Starting again with the three-field Hu-Washizu functional (1), we consider a strain field ~ in the following form

6T(~ ~ -- B u ) d V 0 = 0, ~ ~r s = 0 (17) B0 B0

By the mean value theorem, e~ is identical with Bu in at least one point in B 0, thus it will be designated in the following as 'local compatible' strain field. The enhanced strain field ~ should be designed such that it leads together with ~ to a stable tangent matrix without rank deficiency. Thus it can be named as 'stabilization' strain field. Observe that the new functional leads to a three-field variational problem, where ~, ~'~ and ae have to be approximated. But in contradiction with the standard Hu-Washizu functional, stresses do not explicitly appear as independent fields.

For a typical element the continuous displacements and discontinuous strain fields e ~, ~'~, ~ are interpolated as follows

gC=GC({)de, g~=GS"({)de, ~=G*({)~te (18)

Then from (15)~, (17)~ and (17)2 the final design conditions are expressed as

~ ( G ~ - B ) d V o = 0 , ~ G ~ d V o = 0 , SGedVo--O (19) BO Bo Bo

Korelc and Wriggers (1996b, 1996c) proposed modified Taylor expansion of the eight isoparametric shape functions

1 N . = - ( I + s ,) i = 1 ..... 8 ~ 8 ~ ~ ~

and the following incompatible modes

N~ = 1 - GN~o = 1 - G N . = 1 - ~2

n,~ = ~ ( ~ + ~) + ~ ( ~ + ~) + ~ ( ~ + ~)

~ = Bu +~su+ge (14)

compatible enhanced

Compared to Simo and Rifai (1990) the strain ~:,u is a new ingredient which represents the incompatible enhanced strain field related to the compatible displacement field u. ae is the standard independent incompatible enhanced strain field as

as a basis for the construction of G c, G % and G e. To satisfy the orthogonality conditions on both the reference and actual frames the following approximate integration formula will be applied for the evaluation of all integrals

1 1 1

Sf(X,Y,Z)dVo..~ ~ ~ ~f(~,fl,~)]od~dtld~ (20) B o - - i - - I - - I

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In order to make the modified integration rule consistent with Taylor expansion Korelc and Wriggers (1996b) derived 'local compatible' strain interpolation matrix G ~, which always fulfills the patch test, of the form

G~= f-V-l y BdV ~ VoV~o

(21)

The Taylor series expansion of a general functionfis expressed up to the second order by

Ta ( f ) =f]o +f,r162 +f,,Ion +f, clor (22)

1 T 2 ( / ) = T ' (S ) + -~ (f,r162 i0~ 2 "q-f.;/ [0~ 2 -~ f,~{ ]0~ 2)

+f.r +f.r162 +f.,cl0n~

where underlined terms have to be neglected in order to fullfil orthogonality condition, leading to ~i and "T 2. The introduction of Taylor expansion allows the formulation of an enhanced strain element such as a B-bar type method where shape func- tion derivatives in the classical B matrix are replaced by appro- priate terms of a modified Taylor expansion and the constant part is replaced by G ~. The rank 3 tangent submatrix for the shape functions i andj is then expressed by

1 I 1

!~0 = ~ I ~ ]oBTD~ d~dlld~ i,j = 1 ..... 11 (23) - - I I 1

The element will be stable if B,, i = 1 ..... 12 in (24) contains all linear and bilinear terms. The order of Taylor expansion for interpolation of normal strains (n) and shear strains (m) are design parameters. In Table 1 two possibilities are listed. The element with 9 enhanced modes (QS/Eg) has, by construction, uncoupled enhanced modes. The second element (QS/E12), with 12 enhanced modes, has superior performances for very thin plates, even if additionaly incompressibility constraints are present, but coupled enhanced modes.

~, =

"T ~ (N,,x) o o o ~m(N,#) 0 0 0 Tm(N,,z)

~m(N,,y) fro(N,, x) 0 Tm(N,,z) 0 Tm(Ni, x)

I _ 0 f,~ (Ni,z) ~m (Ni.y)"

(24)

Taylor expansion enables us to explicitly control the change of volume. Another advantage is that the resulting formulas for the stiffness matrix are simple polynomials which can be sym- bolically integrated leading to very efficient formulation (see Korelc and Wriggers (1996b)). In the first case (QS/E9) we have

Table l. Enhanced strain interpolation matrices

element i= 1,2,...,8 i= 9,10,11 i = 12

also a more efficient static condensation procedure due to an uncoupling of the enhanced modes, resulting in 3 inversions of a 3 x 3 matrix instead of one inversion of a 9 x 9 matrix.

In the two dimensional case of a quadrilateral elements, first order Taylor expansion (25) is sufficient (see Korelc and Wriggers (1996c)) and thus leads to

i = 1 . . . . . 6 (25)

3 Solution of two and three dimensional linear elastic problems with enhanced strain formulations The performance of the proposed elements for highly distorted meshes, in the presence of incompressibility and for the simula- tion of thin shell structures, will be assessed on a set of standard test problems. The following elements are considered for parti- cular tests:

Q1 Isoparametric displacement formulation of the four node plane and tri-linear brick elements. Numerical integration is performed with standard four (2D) or eight (3D) point inte- gration rule.

Q1E4 and QI/E9 Standard EAS elements described by Simo and Rifai, (1990) and Simo and Armero (1992). Q 1/E4 is a 2D element with 4 and Q l/E9 is a 3D element with nine enhanced modes. Derivatives of compatible shape functions remain unchanged for this element. The standard four or eight point integration rule is employed.

QMI/E12 3D enhanced strain element with 12 enhanced modes, described by Simo, Armero and Taylor (1993). The derivatives of compatible shape functions are modified in order to get a volumetric locking free element. Special 9-point quadrature is used, as recommended by the authors.

QP6 2D enhanced element with four enhanced modes based on first order Taylor expansion of shape function derivatives (Korelc and Wriggers 1996c). Exact symbolic integration is implemented.

QS/E9 3D enhanced strain element with nine enhanced modes based on Taylor expansion. Interpolation matrices are ex- pressed with the minimum number of Taylor expansion terms. Exact symbolic integration is implemented.

QS/E12 3D enhanced strain element with 12 enhanced modes based on Taylor expansion (see Table 1). Again the element is integrated exactly.

SHELL Bi-linear shell element described by Simo, Fox and Rifai (1989) with mixed formulation for membrane and bending stresses and 2 x 2 integration rule.

HR-5 Standard 2D five parameter stress mixed element by Pian and Sumihara (1984) and Rnbinstein, Punch and Atluri (1983).

HR-18 18 parameter stress mixed element by Pian and Tong (1986) which is one of the most accurate 3D elements based on stress mixed variational principles.

3.1 Linearized eigenvalue analysis The goal of this test is to assess the performance of the enhanced strain elements in extreme situations such as nearlyincompress-

Q S / E 9 n=l ,m=2 n = l , m = l -- QS/E12 n =2, m = 1 n =2, m = 1 n =2, m = 1

ible materials or severely distorted element shapes. Three different configurations are considered. The first one, shown in Figure la, corresponds to an undistorted element, the second one to a moderate distorted element while the third one, shown in Figure lc, represents a severely distorted element. For the incompressible case a ratio of the Lame constants, A/# = 108, enforces the incompressibility constraint effectively in the sense of a penalty method. The number of eigenvalues 2 going to infi- nity with increasing A/# and the number of eigenvalues 2 less or equal zero are presented in Table 2 for 3D elements and in Table 3 for 2D elements. For a volumetric locking free behavior the element can have only one eigenvalue going to infinity and for a stable behavior the number of zero eigenvalues has to be equal to the number of kinematic degrees of freedom.

We observe that in the 3D case with classical EAS formulation proper behavior in the incompressible limit cannot be achieved even with 21 additional modes, while the modified enhanced strain elements exhibit no locking. Only elements based on Taylor expansion have the proper number of zero eigenvalues also for severely distorted element shapes.

5

a 3 b c

Fig. 1. 2D and 3D element shapes for a single element eigenvalue analysis for a) undistorted, b) moderate distorted and c) severly distorted shape

Table 2. Number of zero and infinite eigenvalues of 3D elements for undistorted, moderate and several distorted element shapes. The ratio of A/# = 108 enforces the incompressibility constraint

element

undist, moderate, dist. severly dist.

2~0o 2~<0 2~0o 2~<0 2~0o 2~<0

Q1 7 6 8 6 8 8 Q1/E9 4 6 5 6 5 8 Q1/E21 1 6 2 6 2 - - QM/E12 1 6 1 6 1 11 QS/E9 1 6 1 6 1 6 QS/E12 i 6 1 6 1 6

Table 3. Number of zero and infinite eigenvalues of 2D elements for undistorted, moderate and several distorted element shapes for 2D elements

element

undist, moderate, dist. severly dist.

2~c~ 2~<0 2--*0o 2~<0 2~0o 2~<0

Q1 3 3 3 3 3 4 Qt/E4 1 3 1 3 1 4 QP6 1 3 1 3 1 3

For two dimensional elements as mentioned before both EAS methodology and modified formulation with Taylor expan- sion lead to a volumetric locking free element with only four enhanced modes. Again the proper number of zero eigenvalues can be obtained only by the Taylor expansion formulation for a severely distorted mesh.

3.2 2D d is tor t ion test

Several tests were proposed in literature in order to demonstrate the sensitivity of elements against mesh distortion. Among them area a two-element distortion test in plane strain and plane stress, a five-element distortion test with parallelogram and trapezoidal meshes etc. In general none of the known elements yields optimal results for all tests at the same time. Herein we will try to asses the overall performance of elements in the distorted regime by analyzing the example presented in Fig. 2 with randomly generated distorted meshes.

The procedure to generate the mesh is presented in Fig. 3a. We take a uniform, undistorted mesh of size 2 as a basis. Around every node of the undistorted mesh we define a bounding rect- angle of the size 2 a, where a represents the distortion para- meter. The node is then moved on the bounding rectangle in the direction determined by a randomly generated angle ~Pr,,,,do,, (0 < q~ra,dom < 2n). For a distortion parameter a = 0.5 some ele- ments of the mesh can degenerate into triangles as depicted in Fig. 3b. Between 50-500 samples were needed for every value a of distortion in order to get an accurate mean value.

The problem, presented in Fig. 2 was analyzed with 10 x 2 randomly distorted elements. Some of the meshes are depicted in Fig. 3.c, and 3.d. Statistically evaluated results for several elements are compared in Fig. 4. The displacement element and

2O

2 20 g=1500 > Vexact =1,0

lo :q

Fig. 2. Element mesh for higher order patch test on element robustness

.i

aIi .... ~ ;,'_~_,~ d~Randorn --aI-,-.-.-.-_L- ) .... ,

I 1 a ' a

I I I

2 2 Critical distortion

b a=0.5

o ~ @ ~ @ ~ @ ] a=0.4

d ~ a=0.8

Fig. 3. Generation of randomly distorted mesh (a), critical distortion (b) and examples for meshes with a = 0.4 (c), a = 0.8 (d).

419

420

1 , 0 c

< r

5 0.9 r

c

r -

E

_~ 0.8

O

--o-- HR-5

QP6 Q1/E4 Disp.

0.7 . . . . : . . . . ~ . . . . ~ . . . . ', . . . . 0 0.2 0.4 0.6 0.8 1.0

Distortion "a"

Fig. 4. Test on overall behaviour of elements in high distorted regime

S r ~ /2

Fig. 5. Problem description of plate bending example

Q1/E4 element were calculated only up to the point of critical distortion a = 0.5, since after this point solutions obtained by this type of elements are not unique any more. The HR-5 ele- ment can be used furthermore due to the analytical integration, but only QP6 element is general usable for arbitrary distorted meshes regardless on the type of integration.

3.3 Plate bending example Behavior of 3D solid enhanced elements in plate bending analy- sis will be assessed on the basis of the classical example depicted in Fig. 5. A square plate with dimensions a x a = 100 x 100, clamped edges and elastic moduli E = 104, is loaded at the centre of the plate with the load P = 16.367. For the plate thickness t = 1 and Poisson ratio v = 0.3 Kirchoff plate theory gives an analy- tical solution for the displacement at the centre of the plate: w = 1. One quadrant of the plate is discretized with 4 elements and only one element in thickness direction. The eight point integration rule was used for all elements.

The centre deflection with respect to different ratios a/t is depicted in Fig. 6. Severe locking can be observed for the dis-

9

O

E E

EL

O

>

1000

800-

600-

400.

200-

0 100

Theo

OS/E9 A / QSN/E9 / t f~" QM/E12 /c(~ \

\ .R-18 \ ~ Q1/E21

300 500 700 900 Ratio a/t

Fig. 6. Vertical displacement at the centre of the clamped plate. Undistorted mesh, v = 0.3

placement element and enhanced elements with nine modes. We can see that the Taylor expansion element QS/E12 with only 12 enhanced modes performs equivalently well as the HR ele- ment with 18 additional internal modes and the EAS element with 21 enhanced modes. This is valid also for the incompres- sible case and with the distorted element shape (see Table 4).

3.4

Pinched cylinder with end diaphragm Figure 7 shows a pinched cylinder with end diaphragms subjected to a unit load at the center. Only one-eighth of the cylinder is modelled with different meshes. The results for the vertical displacement at the center of cylinder are presented in Table 5. The pinched cylinder is one of the most demanding shell tests. The results are normalized against the analytical solution of 1.82488 x 10 -S. Similar behavior as for platebending example can be observed also for the shell example: All the elements show good convergence behaviour and insensitivity to the number of elements in the thickness direction.

4 Numerical equivalence between enhanced strain and other formulations It is interesting to note that for mixed stress-displacements elements based on linear elasticity there exist a very close relation between Hellinger-Reissner principle and the enhanced strain formulation. In the article of Andelfinger and Ramm (1993) the following equivalence table for two dimensional elements based on enhanced strain and HR formulation and integrated with 2 x 2 integration rule can be found where G(~)

element

d v theory disp. Q1/E9 Ql/E21 HR-18 QS/E9 QS/E12

0 0.3 1.00 0.007 0.073 0.888 0.888 0.073 0.874 0 0.499 0.827 0.005 0.050 0.740 0.740 0.050 0.719 2.5 0.3 1.00 0.007 0.066 0.581 0.581 0.066 0.538 2.5 0.499 0.827 0.005 0.042 0.536 0.536 0.042 0.514

Table 4. Centerpoint deflection of clamped plate for undistorted and distorted mesh

z T

e,%•z•F= 1.0 L=600 ,.9, R=300

,~'~'/~-'~.z~-,,. '~-~4 t=3

Rigid ~ ( ' ~ A y Diaphragm ' ~ ' ~ e ~ '

/x Fig. 7. Problem description of the pinched cylinder example

is the enhanced strain interpolation matrix and P (3) the stress interpolation matrix. It is an important fact that two EAS and HR elements which are equivalent for a specific integration rule are no longer equivalent for a higher order integration rule in the case of arbitrary distorted element shapes. For HR ele- ments the number of stress modes is limited by the limitation principle of Fraeijs de Veubeke (1965) while the number of addi- tional strain interpolation modes is not restricted for the EAS formulation. However higher modes require a higher order of numerical integration as well as additional computational effort for the static condensation.

No equivalence exists for materially and geometrically nonlinear problems.

5 Enhanced strain formulation for nonlinear problems The development of nonlinear versions of enhanced elements is based on general mixed variational principles. Several lines can be followed to enhance the strains within the element formulation. This is due to the fact that different strain measures can be applied to formulate the theoretical model for finite deformation problems. Whereas from the point of view of continuum mechanics all different formulation should yield the same results, it makes a difference when the enhancement is applied to different strain measures. This will be discussed in the following where we like to mention some possibilities for the enhancement of the strain field:

�9 Enhance the Green Strain tensor E = 1/2 (FrF -- 1) = 1/2(C -- 1). For this case it was shown in Moita (1994) that this approach leads to problems for cases where finite strains are involved. Thus this enhancement does not seem to work in general.

�9 Enhance the right stretch tensor U = RrF arising from the polar decomposition of the deformation gradient F. This does work for different possibilities, see Wriggers, Crisfield

Table 6. Numerically equivalent pairs of EAS and HR elements

6(~) =

6(r =

EAS HR i!ooo o Ool o o o ~ v ( ~ ) = 1 o o

0 ( ~ 0 o ~ o l o QI/E7 HR-5

~/ 0 P(~)= 0 1 0 0 ~ 0 ~/ 0 ~ 0 0 1 0 0 0 0 ~t/

Q1/E4 HR-8

G(~)= P ( ~ ) = 0 1 0 0 0 0 ~ ~ ~ 0 0

0 0 1 0 0 0 0 0 0 ~ t/ ~t/

disp. HR-12

and Moita (1996) for a discretization using classical polar decomposition. For a special polar decomposition in which the rotation tensor is evaluated at the mid point of the element Crisfield, Moita, Jeleni4 and Lyons (1995) devel- oped an enhanced strain element which can also be viewed as a formulation within a rigidly rotated frame. Another nonlinear formulation for the enhanced elements was based on the displacement gradient H = Grad u, see Simo and Armero (1992).

The first two strain measures can be deduced from the following general formula

1 E ~ = - ( U ~ -- 1)

(g

which yields for a = 1 the strain measure U - 1 and for :r = 2 the Green strain tensor. In the following we will denote by D(u) one of these strain measures.

As shown in Simo and Armero (1992), the Hu-Washizu principle can be used as a point of departure. This principle can be formulated in terms of three independent fields: the displacement u, a strain field A and a stress field T. In general we can associate with the strain field different measures, e. g. the nonsymmetrical displacement gradient H, the symmetrical Green strains E or the left stretch tensor U which is also sym- metric. The stress tensors which are work conjugate to these strain fields are for H the unsymmetric first Piola-Kirchhoff stress tensor P, for E the second Piola-Kirchhoff stress tensor S and finally for U the symmetrical part of the Biot stress T B, respectively. Now let us denote one of the strain fields discussed above by A and the associated stress field by T then the

421

mesh Q1 SHELL HR-18

5 x 5 x 1 0.0437 0.39 0.192 9 x 9 x 1 0.0783 0.76 0.597 17 x 17 x 1 0.159 0.94 0.927 17 x 17 x 2 0.159 - - 0.920 17 x 17 x 3 0.159 - - 0.919 30 x 30 x 1 0.296 - - 0.986

QI/E9 Ql/E21 QS/E9 QS/E12

0.148 0.165 0.154 0.178 0.493 0.544 0.506 0.572 0.858 0.893 0.864 0.922 0.55 0.888 0.859 0.915 0.854 0.887 0.858 0.914 0.968 0.978 0.971 0.986

Table 5. Normalized displacement at the centre of the pinched cylin- der

422

Hu-Washizu principle can be expressed as

H(u, A,T) = ~ [W(A) + T.(D(u) -- A)]dV Bo

-- ~ u . b d V - I u'tdA Bo ~Ba

(26)

W (A) denotes the strain energy function of the considered elastic material. D(u) is introduced as a displacement depending differential operator to define the strain measure, e. g. for the Green strains we have D(u) = 1/2 (Grad u + Gradru + Grad r u Gradu), for A = H the operator D(u) will be equal to Grad u.

The integral containing b is related to the body forces and the surface integral containing i is associated with the pre- scribed traction loads. For simplicity, the last two terms in (27) will be summarized under the expression for the external loads, P~xr" Now the idea due to Simo and Rifai (1990) is, to additively split the strain measure in two parts. In our formulation this enhances the local strain measure D(u) by an independent gradient k, leading to

For an efficient implementation of the element method it is advantageous to transform all quantities in Eq. (29) to the current configuration to obtain sparse matrices within the resi- dual and tangent stiffness calculations, see e.g. Simo and Armero (1992). However for this overview we do not want to go into the details.

To complete the model, the strain energy function, W, has to be specified. There is no restriction in the element formulation with respect to W, however we want to restrict ourselves here to a model problem. Thus from the various strain energy functions which can be found in the literature, e.g. Marsden and Hughes (1983) or Ogden (1984), we apply a compressible Neo-Hooke model and a St. Venant constitutive relation. The latter denotes a linear relationship between Green strains and the second Piola-Kirchhoff stress

S=C~E (31)

and is valid for small strains but large deflections. A similar material equation can also be postulated for the Biot stress

A = D ( u ) + A, (27) T , = ~ f ( U _ I ) (32)

Note that the strain energy W can be written with Eq. (27) as follows W(A) = W(D(u) + A) and thus the first variation of (27) yields

0W ~ D ( u ) . - ~ d V - ~PExT = 0

Bo

bT.~tdV = 0 (28) Bo

The first equation is the standard weak form of equilibrium. The second equation yields the constitutive equation. The third equation is an orthogonality condition which has to be fulfilled by the stress tensor T and the enhanced strain part A. Since we also have the requirement that the virtual work due to the en- hanced strains has to be zero the term ~T.~;tdV in (292) has to vanish. This leads to the final form of the reduced Hu-Washizu principle for enhanced strain methods

OW bD(u) . - ~ d V - bPEx~: = 0

Bo

however according to the different strain measures (32) and (31 ) are not equal.

The strain energy function of the compressible Neo-Hooke model can be written in terms of the right Cauchy-Green strain tensor C as

A W = ~ ( ] F - 1) 2 + ~p[tr(C) --31 - / 2 ln]F, (33)

see Wriggers (1993), where IF = det F denotes the Jacobian of the deformation gradient. A and # are the so called Lame material constants. Using this strain energy function the second Piola-Kirchhoff stresses are derived as

0W S = 2 - ~ = A I F ( b - 1 ) C - I - # ( C 1--1) (34)

This constitutive equation is now transformed to an equation for the first Piola-Kirchhoff stress tensor P using the relation P = FS which yields

0W P = A i r (IF - 1 ) F -T - - p ( F - v - - F) = ~-~ (35)

~5X.~W_ dV OA = 0 B0

together with the orthogonality condition

(29) The strain energy function (33) can also be written in U:

A W=-~(JF-- 1)2 + ~# [tr (U 2) --3] -121nJ F, (36)

~6T.AdV = 0 (30) Bo

The discretization procedure will exhibit that equations (29) and (30) provide a variational basis to include enhanced strain modes into the element formulation.

where It = det U = det F denotes again the Jacobian of the deformation gradient. Using this strain energy function the Biot stresses that appear in (29) 2 are derived as

~ W T, =-b-- f f = A I F ( L - 1)u -I - ~ ( u l - u ) (37)

6 Element design As discussed in the introduction, the standard formulation of an enhanced element (two-dimensional as Q1/E4 or Q1/E5 or three-dimensional as Q1/E9 or QMI/E12) leads to a rank deficiency for stress states in compression. Thus we will analyse here this problem and state a new method developed in Korelc and Wriggers (1996a). Before doing so, we will discuss the hour-glass instability in detail.

Hour-glass eigenmodes appear for constant states of stresses and deformations but also for general stress states. In the simple constant stress state we have additionally no shear deforma- tions, furthermore, the rotational part of the deformation tensor F is zero, so the different strains A and stress measures T are equal in this case or at most different by a constant factor. Thus different types of strain and stress measures have no influence on the element behaviour. As mentioned before the hour-glass phenomenon appears for a rectangular undistorted element geometry, thus a cure can be expected only for a formulation which does not degenerate to the classical Q1E4 element in the undistorted case. It should also be mentioned that a higher order of integration does not help, since in the simple case discussed almost all quantities are constant and thus low order integration rules are sufficient. We like to discuss different possibilities to overcome this problem:

�9 A common procedure to eliminate hour-glassing is the so called 'artificial hour-glass control'. Stabilization of the element behaviour is achieved by adding artificial stabi- lization matrices to the underintegrated element tangent matrix. Stabilization, applied directly to the enhanced modes of the Q1E4 element, improves the element beha- viour, but it is not clear how to design this procedure for arbitrary distorted elements, see Wriggers and Reese (1995).

�9 A further possibility is to change the model equations and enhance the right stretch tensor U instead of the displace- ments gradient H, for a detailed discussion see Crisfield, Moita, Jeleni6 and Lyons (1995). These authors conclude that a variant which is directly related to the co-rotational formulation (using a constant rotation tensor within the element) does not exhibit hour-glassing in compression.

�9 Finally the modes which are used for the enhancement of the elements can be changed. For this purpose one can look for a constructive method to develop orthogonality condi- tions which have to be fulfilled for elements which should not depict the hour-glassing eigenmodes. This line of research has been followed up in Korelc and Wriggers (1996a).

The instability phenomenon appears for the simple neo- Hookean hyperelastic constitutive equation. Thus it can be ex- pected that for more elaborate models such as the Ogden or Mooney-Rivlin material also it will lead to hour-glass eigen- modes.

From the above discussion we can conclude that a change of the basic enhanced strain interpolation A is needed for a solu- tion of the hour-glassing problem. In the following we first will state the standard approach which has been developed in Simo and Armero (1992) and then derive additional orthogonality conditions which have to be fulfilled to obtain a stable enhanced strain element formulation.

6.1 Enhancement of the displacement gradient For the discretization process we subdivide the initial configura- tion B 0 of the body under consideration into n e finite elements t2 e, thus we have

ne

B0 ~ U Sr~e (38) t = l

As in the classical formulation of the Q1 element the displace- ment field u is interpolated by standard isoparametric shape functions N,. Let us introduce the unknown nodal displacement vector d then the interpolation

, = I ( N , v , J (39)

follows. The isoparametric mapping from the reference coor- dinates 4, ~/, defined on the unit square [] = [ - 1,1] • [ - 1,1], to the coordinates X, Y on the actual element domain ~2 e, see Fig. 8, is defined by bi-linear shape functions

1 NI= 5(1 + ~z) (1 + ~th) , l = 1...4 (40)

with this interpolation the displacement gradient can be written as

iUl, 0]{ t = ---- B(~,~/)v

L o Nz, yJ

(41)

This non-standard form has to be used since H is not symmetric. Now the enhanced gradient can be interpolated in a similar

way by H = H (00 where ~ are the unknowns due to the en- hanced modes which are only local parameters on element level. The standard enhanced interpolation using four modes with respect to the reference coordinates yields

LR(~, ~)~ v4(~,,)~4A

The condition

(42)

I 1

~ P,d~ dtl = ~P, ac] = 0 (43) - 1 - 1 []

has to be imposed on the shape functions P, to ensure satis- faction of the orthogonality condition (30).

(-1,-1) 0,-1) �9 X

Reference frame Actual frame

Fig. 8. Isoparametric element formulation

423

426

' \ \ \ i t i " i t - /

IIII IllLII II1 I l l l l l I II

Disp.

DIsp.

Incompressible

Q] 1::4 GG4

Compressible

Q 1 E4 CG4

Fig. 10. Saint-Venant material law for compressible and incom- pressible case. Eigenmodes at the second bifurcation point

A lcI < >< >

10 Up

Fig. 11. System and deformations ofa hyperelastic strip

the non-linear problem was checked by the residual norm in the current iteration. The number of Newton iterations needed to achieve a residual norm less then 10-8 was used to compare the robustness of the different element formulations. The sign oo was used to denote the fact that the above mentioned tolerance was not obtained within 20 iteration steps. Two meshes, with 10 • 10 and 20 x 20 elements were employed to also show the dependence of the convergence on the number of elements. Table 8 depicts that significantly larger load steps can be applied with the Q1/CG4 element in comparison with the Q1/E4 element. It should however be noted that the standard Q1 element is still more robust. The simulations with the refined 20 • 20 element mesh show clearly that a considerably smaller load level can be obtained within one load step. This is true for all formulations.

9 Conclusion In this article an overview has been presented of the enhanced strain methods for small and large strains. It has been shown that different formulations are possible within this method- ology. In the linear case it is essentially a question of efficiency and robustness against severly distorted meshes which will lead to a decision for one or the other types of enhanced for- mulations.

When enhanced elements are applied to nonlinear problems, the situations changes. There always exists the possibility that rank deficiency occurs in compressive deformation states. The consistent gradient formulation is one way to overcome this problem, however this element is less efficient, since the matrix operator G is no longer sparse as in the original method by Simo, Armero (1992). In the three dimensional case, there is so far no analytical expression for the correct orthogonality conditions which are the basis for the element design. Nevertheless, the CG9 element performs well in applications. It should however be noted that also the method based on the co-rotational formu- lation does not exhibit a loss of rank. However it is more compli- cated to apply the resulting element to materials exhibiting large inelastic strains.

References Andelfinger, U.; Ramm, E. 1993: EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. Int. ]. Numerical Methods in Engineering, 36:1311-1337 Andelfinger, U.; Ramm, E.; Roehl, D. 1992: 2D- and 3D-enhanced assumed strain elements and their application in plasticity. In D.R.I. Owen, E. Onate, and E. Hinton, editors, Computational Plasticity, Proceedings of the 4th International Conference, pages 1997-2007. Pineridge Press, Swansea, U. K. Atluri, S. N. 1973: On the hybrid stress finite element model in incremental analysis of large deflection problems. Int. ]. Solids and Structures, 9:1188-1191 Afluri, S. N. 1975: On hybrid finite-element models in solid mechanics. In R. Vishnevetsky, editor, Advances in Computer Methods for Partial Differential Equations, pages 346-356, AICA, Rutgers University, New Brunswick, New Jersey Afluri, S. N. 1980: On some new general and complementary energy theorems for the rate problems in finite strain, classical elastoplasticity. J. Struct. Mech., 8(1): 61-92 Afluri, S. N.; Cazzani, A. 1995: Rotations in computational solid mechanics. Archives of Computational Methods in Engineering, 2 (1): 49-138 Afluri, S. N.; Murakawa, H. 1977: On hybrid finite element models in nonlinear solid mechanics. In P. Bergan, editor, Finite Elements in Nonlinear Mechanics, pages 3-40. Tapir, Trondheim Afluri, S. N.; Reissner, E. 1989: On the formulation of variational theorems involving volume constraints. Computational Mechanics, 5: 337-344

up=6. up=7. ue=8. up=9. %=10.

elem. mesh v A. N N N N N u pmax

disp. 10 x 10 1.708 6 6 6 7 7 12. Q1E4 10 x 10 1.703 7 7 10 oo oo 8. CG4 10 x 10 1.703 6 6 6 7 7 10. disp. 20 x 20 1.695 6 6 7 7 8 10. Q1E4 20 x 20 - - oo oo oo oo oo 5. CG4 20 x 20 1.693 6 7 9 oo oo 8.

Table 8. The number of Newton iterations needed to achieve convergence in one load step for different load levels

Belytschko, T.; Ong, J. S.-I.; Liu, W. K.; Kennedy, J. M. 1984: Hourglass control in linear and nonlinear problems. Computer Methods in Applied Mechanics and Engineering, 43:251-276 Crisfield, M. A. 1995: Incompatible modes, enhanced strain and sub- stitute strains for continuum elements. In N.-E. Wiberg, editor, Ad- vances in finite element technology, A Series of Handbooks on Theory and Engineering Applications of Computational Methods, pages 47-61. CIMNE, Barcelona Crisfield, M. A.; Mofia, G. F.; Jelenic, G.; Lyons, L P. R. 1995: Enhanced lower-order element formulations for large strains. In Computational Plasticity, Fundamentals and Applications, Proceedings of the 4th International Conference, pages 293-320. Pineridge Press, Swansea, U.K. Felippa, C. A.; Haugen, B. 1995: From the individual element test to finite element templates: Evolution of the patch test. Int. J. Numerical Methods in Engineering, 38:199-229 Flanagan, D. P.; Belytschko, T. 1981: A uniform strain hexahedron and quadrilateral with orthogonal hourglass control Int. J. Numerical Methods in Engineering, 17; 679-706 Freischlager, C.; Schweizerhof, K. 1995: On a systematic development of trilinear 3D solid elements based on Simo's enhanced strain formu- lation, submitted to Computers & Structures de Veubeke, B. F. 1965: Displacement and equilibrium models in the finite element method. In O. C. Zienkiewicz and G. C. Holister, editors, Stress Analysis. Wiley, London Gruttmann, F.; Wanger, W.; Wriggers, P. 1992: A nonlinear quadri- lateral shell element with drilling degrees of freedom. Ing. Archiv, 62:474-486 Hughes, T. R. J. 1980: Generalization of selective integration procedures to anisotropic and nonlinear media. Int. J. Numerical Methods in Engineering, 15:1413-1418 Hughes, T. R. J. 1987: The Finite Element Method: Linear Static and Dynamic Finite Elemenet Analysis. Prentice-Hall, Englewood Cliffs, New Jersey Hughes, T. R. J.; Brezzi, F. 1989: On drilling degrees of freedom. Computer Methods in Applied Mechanics and Engineering, 72: 105-121 Hueck, U.; Reddy, B. D.; Wriggers, P. 1993: On the stabilization of the rectangular four-node quadrilateral element. Technical report, Centre for Research in Computational and Applied Mechanics, University of Cape Town, South Africa, 1993 Hueck, U.; Wriggers, P. 1995: A formulation for the 4-node quad- rilateral element. Int. J. Numerical Methods in Engineering, 38: 3007-3037 Ibrahimbegovic, A.; Taylor, R. L.; Wilson, E. L. 1990: A robust quadrilateral membrane element with drilling degrees of freedom. Int. J. Numerical Methods in Engineering, 30:445-457 Iura, M.; Afluri, S. N. 1992: Formulation of a membrane finite element with drilling degrees of freedom. Computational Mechanics, 9:417-428 Korelc, J. 1995: SMS-Symbolic Mechanics System. Technical Report 1/95, Institut for Mechanics, Technical University of Darmstadt, Germany Korelc, J.; Wriggers, P. 1995: Efficient enhanced strain element for- mulation for 2D and 3D problems. In N.-E. Wiberg, editor, Advances in Finite Element Technology, A Series of Handbooks on Theory and Engineering Applications of Computational Methods, pages 22-46. CIMNE, Barcelona Korelc, J4 Wriggers, P. 1996a: Consistent gradient formulation for a stable enhanced strain method for large deformations. Engineering Computations, 13 (1): 103-123 Korelc, J.; Wriggers, P. 1996b: An efficient 3D enhanced strain element with Taylor expansion of the shape functions. To appear in Computa- tional Mechanics Korelc, J.; Wriggers, P. 1996c: Improved enahnced strain four node element with Taylor expansion of the shape functions. To appear in Int. J. Numerical Methods in Engineering Liu, W. K.; Belytschko, T.; Chen, ].-S. 1988: Nonlinear version of flexurally superconvergent elements. Computer Methods in Applied Mechanics and Engineering, 71:241-258 Li, X.; Crook, A. ]. L.; Lyons, L. P. R. 1993: Mixed strain elements for non-linear analysis. Engineering Computations, 10:223-242

Liu, W. K.; Flu, Y.-K.; Belytschko, T. 1994: Multiple quadrature under- integrated finite elements. Int. J. Numerical Methods in Engineering, 37:3263-3289 Liu, W. K.; Ong, J. S. J.; Uras, R. A. 1985: Finite element stabilization matrices - a unified approach. Computer Methods in Applied Mechan- ics and Engineering, 53:13-46 Marsden, J. E4 Hughes, T. R. J. 1983: Mathematical Foundation of Elasticity. Prentice-Hall, Englewood Cliffs, New Jersey Miche, C. 1994: Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int. J. Numerical Methods in Engineering, 37:1981-2004 Moita, G. F. 1994: Non-linear Finite Element Analysis of Continua with Emphasis on Hyperelasticity. Ph. D. Thesis, Imperial College, London. Neto, E. A. S.; Perle, D.; Huang, G. C.; Owen, D. R. J. (1995): Remarks on the stability of enhanced strain elements in finite elasticity and elastoplasticity. In Computational Plasticity, Fundamentals and Appli- cations, Proceedings of the 4th International Conference, pages 361-372. Pineridge Press, Swansea, U. K. Ogden, R. W.: Non-linear Elastic Deformations. Ellis Horwood Limited, Chichester, U. K. Plan, T. H. H. 1964: Derivation of element stiffness matrices by assumed stress distribution. AIAA Journal, 2: 1333-1336 Pian, T. H. H.; Sumihara, K. 1984: Rational approach for assumed stress finite elements. Int. J. Numerical Methods in Engineering, 20: 1685-1695 Plan, T. H. H.; Tong, P. 1986: Relations between incompatible displace- ment model and hybrid stress model. Int. J. Numerical Methods in Engineering, 22:173-181 Piltner, R.; Taylor, R. L. 1995: A quadrilateral mixed finite element with two enhanced strain modes. Int. J. Numerical Methods in Engineering, 38:1783-1808 Punch, E. F.; Afluri, S. N. 1984: Development and testing of stable invariant isoparametric curvilinear 2 and 3-d hybrid-stress elements. Computer Methods in Applied Mechanics and Engineering, 47: 331-356 Reddy, B. D.; Simo, J. C. 1995: Stability and convergence of a class of enhanced strain methods. SIAM J. Numer. Anal., 32 (6): 1705-1728, 1995 Reese, S.; Wriggers, P. 1995: A finite element method for stability problems in finite elasticity. Int. J. Numerical Methods in Engineering, 38:1171-1200 Rubinstein, R.; Punch, E. F.; Afluri, S. N. 1983: An analysis of, and remedies for, kinematic modes in hybrid-stress finite elements: Selection of stable, invariant stress fields. Computer Methods in Applied Mechanics and Engineering, 38:63-92 Seki, W.; Atluri, S. N. 1994: "Analysis of Strain Localization in Strain-Softening layperelastic meterials, using assumed stress hybrid elements" Computational Mechanics, 14, pp 549-585 Seki, W.; Atluri, S. N. 1995: On newly developed assumed stress finite element formulations for geometrically and materially nonlinear problems. Finite Elements in Analysis and Design, 21: 75-110 Simo, J. C.; Armero, F.; Taylor, R. L. 1993: Improved version of assumed enhanced strain tri-linear element for 3D finite deformation problems. Computer Methods in Applied Mechanics and Engineering, 110: 359-386 Simo, J. C.; Rifai, M. S. 1990: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numerical Methods in Engineering, 29:1595-1638 Simo, J. C.; Armero, F. 1992: Geometrically nonlinear enhanced mixed methods and the method of nicompatible modes. Int. J. Numerical Methods in Engineering, 33:1413-1449 Simo, J. C.; Fox, D. D.; Rifai, M. S. 1989: On a stress resultant geo- metrically exact shell model, part II: The linear theory; computational aspects. Computer Methods in Applied Mechanics and Engineering, 73:53-92 Simo, J. C.; Hughes, T. J. R. 1986: On the variational foundations of assumed strain methods. Journal of Applied Mechanics, 53:51-54 Simo, J. C.; Taylor, R. L.; Pister, K. S. 1985: Variational and projection methods for the volume constraint in finite deformation plasticity. Computer Methods in Applied Mechanics and Engineering, 51:177 -208

427

426

Incompressible

I l l l l l l ' l l IIIIll,ll Disp. Q1 E4 CG4

Compressible

Disp. Q1 E4 CG4

Fig. 10. Saint-Venant material law for compressible and incom- pressible case. Eigenmodes at the second bifurcation point

< > < �9 10 Up

Fig. 11. System and deformations ofa hyperelastic strip

the non-linear problem was checked by the residual norm in the current iteration. The number of Newton iterations needed to achieve a residual norm less then 10 -8 was used to compare the robustness of the different element formulations. The sign oo was used to denote the fact that the above mentioned tolerance was not obtained within 20 iteration steps. Two meshes, with 10 x 10 and 20 • 20 elements were employed to also show the dependence of the convergence on the number of elements. Table 8 depicts that significantly larger load steps can be applied with the Q1/CG4 element in comparison with the QI/E4 element. It should however be noted that the standard Q1 element is still more robust. The simulations with the refined 20 x 20 element mesh show clearly that a considerably smaller load level can be obtained within one load step. This is true for all formulations.

9 Conclusion In this article an overview has been presented of the enhanced strain methods for small and large strains. It has been shown that different formulations are possible within this method- ology. In the linear case it is essentially a question of efficiency and robustness against severly distorted meshes which will lead to a decision for one or the other types of enhanced for- mulations.

When enhanced elements are applied to nonlinear problems, the situations changes. There always exists the possibility that rank deficiency occurs in compressive deformation states. The consistent gradient formulation is one way to overcome this problem, however this element is less efficient, since the matrix operator G is no longer sparse as in the original method by Simo, Armero (1992). In the three dimensional case, there is so far no analytical expression for the correct orthogonality conditions which are the basis for the element design. Nevertheless, the CG9 element performs well in applications. It should however be noted that also the method based on the co-rotational formu- lation does not exhibit a loss of rank. However it is more compli- cated to apply the resulting element to materials exhibiting large inelastic strains.

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up=6. up=7. up=8. %=9. %=10.

elem. mesh v A. N N N N N u pmax

disp. 10 • 10 1.708 6 6 6 7 7 12. Q1E4 10 • 10 1.703 7 7 10 ~ cxD 8. CG4 10 x 10 1.703 6 6 6 7 7 10. disp. 20 • 20 1.695 6 6 7 7 8 10. Q1E4 20 • 20 - - ~ ~ oo ~x) cz) 5. CG4 20 • 20 1.693 6 7 9 co c~ 8.

Table 8. The number of Newton iterations needed to achieve convergence in one load step for different load levels

Belytschko, T.; Ong, J. S.-J.; Liu, W. K.; Kennedy, J. M. 1984: Hourglass control in linear and nonlinear problems. Computer Methods in Applied Mechanics and Engineering, 43:251-276 Crisfield, M. A. 1995: Incompatible modes, enhanced strain and sub- stitute strains for continuum elements. In N.-E. Wiberg, editor, Ad- vances in finite element technology, A Series of Handbooks on Theory and Engineering Applications of Computational Methods, pages 47-61. CIMNE, Barcelona Crisfield, M. A.; Motia, G. F.; Jelenic, G.; Lyons, L. P. R. 1995: Enhanced lower-order element formulations for large strains. In Computational Plasticity, Fundamentals and Applications, Proceedings of the 4th International Conference, pages 293-320. Pinefidge Press, Swansea, U.K. Felippa, C. A.; Haugen, B. 1995: From the individual element test to finite element templates: Evolution of the patch test. Int. J. Numerical Methods in Engineering, 38:199-229 Flanagan, D. P.; Belytschko, T. 1981: A uniform strain hexahedron and quadrilateral with orthogonal hourglass control Int. J. Numerical Methods in Engineering, 17; 679-706 Freischlager, C.; Schweizerhof, K. 1995: On a systematic development of trilinear 3D solid elements based on Simo's enhanced strain formu- lation, submitted to Computers & Structures de Veubeke, B. F. 1965: Displacement and equilibrium models in the finite element method. In O. C. Zienkiewicz and G. C. Holister, editors, Stress Analysis. Wiley, London Gruttmann, F.; Wanger, W.; Wriggers, P. 1992: A nonlinear quadri- lateral shell element with drilling degrees of freedom. Ing. Archiv, 62:474-486 Hughes, T. R. J. 1980: Generalization of selective integration procedures to anisotropic and nonlinear media. Int. J. Numerical Methods in Engineering, 15:1413-1418 Hughes, T. R. J. 1987: The Finite Element Method: Linear Static and Dynamic Finite Elemenet Analysis. Prentice-Hall, Englewood Cliffs, New Jersey Hughes, T. R. J.; Brezzi, F. 1989: On drilling degrees of freedom. Computer Methods in Applied Mechanics and Engineering, 72: 105-121 Hueck, U.; Reddy, B. D.; Wriggers, P. I993: On the stabilization of the rectangular four-node quadrilateral element. Technical report, Centre for Research in Computational and Applied Mechanics, University of Cape Town, South Africa, 1993 Hueck, U.; Wriggers, P. 1995: A formulation for the 4-node quad- rilateral element. Int. J. Numerical Methods in Engineering, 38: 3007-3037 Ibrahimbegovic, A.; Taylor, R. L; Wilson, E. L 1990: A robust quadrilateral membrane element with drilling degrees of freedom. Int. J. Numerical Methods in Engineering, 30:445-457 Iura, M.; Afluri, S. N. 1992: Formulation of a membrane finite element with drilling degrees of freedom. Computational Mechanics, 9:417-428 Korelc, J. 1995: SMS-Symbolic Mechanics System. Technical Report 1/95, Institut for Mechanics, Technical University of Darmstadt, Germany Korelc, J4 Wriggers, P. 1995: Efficient enhanced strain element for- mulation for 2D and 3D problems. In N.-E. Wiberg, editor, Advances in Finite Element Technology, A Series of Handbooks on Theory and Engineering Applications of Computational Methods, pages 22-46. CIMNE, Barcelona Korelc, J.; Wriggers, P. 1996a: Consistent gradient formulation for a stable enhanced strain method for large deformations. Engineering Computations, 13(I): 103-123 Korelc, J.; Wriggers, P. 1996b: An efficient 3D enhanced strain element with Taylor expansion of the shape functions. To appear in Computa- tional Mechanics Korelc, J.; Wriggers, P. 1996c: Improved enahnced strain four node element with Taylor expansion of the shape functions. To appear in Int. J. Numerical Methods in Engineering Liu, W. K.; Belytschko, T.; (;hen, J.-S. 1988: Nonlinear version of flexurally superconvergent elements. Computer Methods in Applied Mechanics and Engineering, 71:241-258 Li, X.; Crook, A. J. L; Lyons, L P. R. 1993: Mixed strain elements for non-linear analysis. Engineering Computations, 10:223-242

Liu, W. K.; Hu, Y.-K.; Belytschko, T. 1994: Multiple quadrature under- integrated finite elements. Int. J. Numerical Methods in Engineering, 37:3263-3289 Liu, W. K.; Ong, J. S. J.; Uras, R. A. 1985: Finite element stabilization matrices - a unified approach. Computer Methods in Applied Mechan- ics and Engineering, 53:13-46 Marsden, J. E.; Hughes, T. R. J. 1983: Mathematical Foundation of Elasticity. Prentice-Hall, Englewood Cliffs, New Jersey Miche, C. 1994: Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int. J. Numerical Methods in Engineering, 37:1981-2004 Moita, G. F. 1994: Non-linear Finite Element Analysis of Continua with Emphasis on Hyperelasticity. Ph.D. Thesis, Imperial College, London. Neto, E. A. S.; Perk, D.; Hnang, G. C.; Owen, D. R. J. (1995): Remarks on the stability of enhanced strain elements in finite elasticity and elastoplasticity. In Computational Plasticity, Fundamentals and Appli- cations, Proceedings of the 4th International Conference, pages 361-372. Pineridge Press, Swansea, U. K. Ogden, R. W.: Non-linear Elastic Deformations. Ellis Horwood Limited, Chichester, U. K. Plan, T. H. H. 1964: Derivation of element stiffness matrices by assumed stress distribution. AIAA Journal, 2:1333-1336 Plan, T. H. H.; Sumihara, K. 1984: Rational approach for assumed stress finite elements. Int. J. Numerical Methods in Engineering, 20: 1685-1695 Plan, T. H. H.; Tong, P. 1986: Relations between incompatible displace- ment model and hybrid stress model. Int. J. Numerical Methods in Engineering, 22: 173-181 Piltner, R.; Taylor, R. L. 1995: A quadrilateral mixed finite element with two enhanced strain modes. Int. J. Numerical Methods in Engineering, 38:1783-1808 Punch, E. F.; Afluri, S. N. 1984: Development and testing of stable invariant isoparametric curvilinear 2 and 3-d hybrid-stress elements. Computer Methods in Applied Mechanics and Engineering, 47: 331-356 Reddy, B. D.; Simo, J. C. 1995: Stability and convergence of a class of enhanced strain methods. SIAM J. Numer. Anal., 32 (6): 1705-1728, 1995 Reese, S.; Wriggers, P. 1995: A finite element method for stability problems in finite elasticity. Int. J. Numerical Methods in Engineering, 38:1171-1200 Rubinstein, R.; Punch, E. F.; Atluri, S. N. 1983: An analysis of, and remedies for, kinematic modes in hybrid-stress finite elements: Selection of stable, invariant stress fields. Computer Methods in Applied Mechanics and Engineering, 38:63-92 Seki, W.; Atluri, S. N. 1994: "Analysis of Strain Localization in Strain-Softening layperelastic meterials, using assumed stress hybrid elements" Computational Mechanics, 14, pp 549-585 Seki, W.; Atluri, S. hi. 1995: On newly developed assumed stress finite element formulations for geometrically and materially nonlinear problems. Finite Elements in Analysis and Design, 21: 75-110 Simo, J. C.; Armero, F.; Taylor, R. L 1993: Improved version of assumed enhanced strain tri-linear element for 3D finite deformation problems. Computer Methods in Applied Mechanics and Engineering, 110: 359-386 Simo, J. C.; Rifai, M. S. 1990: A class of mixed assumed strain methods and the method of incompatible modes. Int. I. Numerical Methods in Engineering, 29:1595-1638 Simo, I. C.; Armero, F. 1992: Geometrically nonlinear enhanced mixed methods and the method of nicompatible modes. Int. J. Numerical Methods in Engineering, 33:1413-1449 Simo, J. C.; Fox, D. D.; Rifai, M. S. 1989: On a stress resultant geo- metrically exact shell model, part II: The linear theory; computational aspects. Computer Methods in Applied Mechanics and Engineering, 73:53-92 Simo, J. C.; Hughes, T. J. R. 1986: On the variational foundations of assumed strain methods~ Journal of Applied Mechanics, 53:51-54 Simo, J. C.; Taylor, R. L.; Pister, K. S. 1985: Variational and projection methods for the volume constraint in finite deformation plasticity. Computer Methods in Applied Mechanics and Engineering, 51: 177 - 208

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