- Email: [email protected]
Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics
International Journal of Mechanical Sciences ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics Shuguang Li Institute for Aerospace Technology, Faculty of Engineering, University of Nottingham, Nottingham, NG7 2RD, UK
art ic l e i nf o
a b s t r a c t
Article history: Received 13 February 2013 Received in revised form 30 June 2014 Accepted 14 July 2014
This paper results from a reflection on the problem of finite rotations in plates and shells and its presentation in finite element formulation, a well-attended subject with a wide diversity of treatments, often extremely complicated. For instance, the very topic attracted 3 out of total 15 chapters over 120 out of total 494 pages in a monograph on nonlinear FEA [5]. In another instance, a 71 page journal paper was published [2] specifically on this topic alone. Review papers on this matter can be found in various sources including Appl Mech Rev, (e.g. Ibrahimbegovic, 1997 [8]). No attempt is to be made on reviewing these sophisticated approaches but the present paper will bring answers to questions, if they have ever been asked, such as whether the sophistication in existing approaches is really necessary and whether there is a much simpler and conceptually more direct and accurate approach. A proper re-examination of the existing approaches would reveal a fundamental inconsistency: rigid body rotations have been assumed without reservation to describe the deformations of deformable plates and shells. After reestablishing the consistency based on the very basics of conventional plate and shell theories as a simple reflection, one can conclude surprisingly that the whole issue on ‘finite rotations’ results from a logic fallacy of faulty generalisation. The so-called ‘rotations’ should be displacement gradients instead. They can be considered as ‘rotations’ as conventionally perceived hitherto under the condition of small deformation. Somehow, the concept of ‘rotations’ got generalised regardless the magnitude of the deformation. Instead of calling the problem as ‘finite displacement gradients’ as it should be called, a falsely generalised term of ‘finite rotations’ have been used. Since they were called ‘rotations’, the definition of those of rigid body kinematics has been taken for granted. Finite displacement gradients should not present any additional problem apart from introducing nonlinear nature into the problem, which can be addressed as a geometrically nonlinear problem in a conventional manner. However, along the line of a falsely generalised concept of finite rigid body rotations, complications have been the norm. The complicated accounts on the ‘finite rotation’ problem in the literature, which might have enhanced the understanding of rigid body kinematics, are entirely unnecessary as far as the deformable plates and shells are concerned, involving infinitesimal or finite deformations. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Finite rotations Plates and shells Shell elements Displacement gradients Reissner–Mindlin Faulty generalisation
不识庐山真面目,只缘身在此山中—A Chinese proverb. (One is incapable of appreciating the true beauty of Lusan Mountain simply because he/she is viewing from deep inside it.)
1. Introduction The development of plate/shell finite elements has encountered two major challenges. At an early stage, C1 continuity troubled users for decades from early sixties. Many significant progresses had been made while combating with this difficulty and some even laid cornerstones for the finite element method in general, e.g. patch test for nonconforming elements. Elegant as E-mail address: [email protected]
they were, the solution to the problem had not come through in this way. Rather, the remedy emerged with C0 element using a degenerate formulation. While it came with problems generic to the formulation, typically shear locking associated with its applications to thin plates and shells. Techniques, such as reduced integration among others, had then been developed to keep them at bay. By now, the campaign is more or less won and C0 plate/ shell elements have been adopted unanimously in most, if not all, commercial finite element codes. The second challenge was the rotations when finite deformation is involved. Various treatments have been proposed at their diversity and yet no unanimous conclusion has been drawn. A review of the state-of-the-art seems to suggest that the nightmare be still haunting the minds of users today [4]. The issue does not appear in small deformation problems because small rotations can be treated as vectors approximately while it can be proved
http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004 0020-7403/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Li S. Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics. Int. J. Mech. Sci. (2014), http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004i
2
S. Li / International Journal of Mechanical Sciences ∎ (∎∎∎∎) ∎∎∎–∎∎∎
that infinitesimal rotations form proper vectors strictly [9]. However, a convenient terminology of ‘rotations’ introduced then has misguided the development profoundly into the finite deformation problem of plates and shells. As the rotations become finite, problems emerge. Finite rigid body rotations are wellknown to be incommunicable and non-additive due to their non-vectorial nature. This hits a brick wall straightaway on following three accounts. (i) Whenever a coordinate transformation is required for the three possible rotational degrees of freedom, they cannot be transformed as a vector. (ii) They cannot be accumulated in a conventional way since the incommunicability rules out the parallelogram law of vectors while the accumulation process is crucial in an incremental and iterative process which is by large the most popular way of solving nonlinear problems. (iii) It also affects the uniqueness in the determination of the displacement field in a finite element, as the in-plane displacements at a point off the reference surface are no longer linear functions of the angles of rotation of the normal to the reference surface [15]. It looks as if one is confronting a formidable difficulty and, in fact, it has been treated this way in the literature, with overwhelmingly sophisticated formulations inevitably as a result. The existing treatments for finite rotations are numerous as reviewed in Crisfield [5] and Ibrahimbegovic [8], for example, quaternion representation as in [1], that employs special orthogonal group, SO3 ([12] and [16]), that based on the concept of a local sphere [15], which is in essence the use of Eulerian angles of rotation [3], and so-called co-rotational coordinates, among others. The co-rotational coordinates approach perhaps accounts for significantly more efforts and systematic developments, as reviewed in Ibrahimbegovic [8] and Crisfield [5]. Elegantly formulated as they have been, no victory has been claimed as yet for the battle combatting against finite rotations. The sheer complexity is often sufficient to deter many from treading on the mess. As an example, the issue of finite rotations is dealt with in three consecutive chapters over 120 page of the monograph by Crisfield [5]. In another instance, a 71 page long paper was published [2] specifically on this topic. Given the scenario, a sober question to be asked is, “Are we striving in a similar trap to that of C1 continuity and is there a remedy like the C0 solution to get us out of the mess?” The objective of the paper is to bring a positive answer to this question. In a short note [11], the vectorial nature of finite nodal rotations was argued in the context of finite beam elements. The outcome was that, under Bernoulli plane-section assumption, the three nodal rotations form a pseudo-vector [14], or axial vector as referred to in [7]. As long as no coordinate transformations are involved between systems of mixed nature, i.e. right-hand and left-hand systems, which is prohibited practically anyway (otherwise, negative determinant of Jacobian will be encountered), the three rotations form a vector, just as ordinary as any other vectors. In the case of Timoshenko beams, the rotations do not form a vector exactly but differ by the transverse shear strains. This should not constitute a substantial problem practically since the magnitudes of transverse shear strains, significant as they might be in comparison with other strain components, would be negligible as compared with the magnitude of rotations as in a finite rotation problem. As a result, the rotations could be regarded as a vector as a reasonable approximation under the condition that transverse shear strains are small enough relative to the magnitude of rotations. The similar argument could be made for plates and shells but it is not pursued in the present paper. A fresh re-examination from a different perspective will resolve the whole issue completely. It can be found that plate and shell theories somehow have fallen in a typical logic fallacy of faulty generalisation in the respect of ‘finite rotations’. To describe the deformation and to construct the displacement fields in
deformable plates and shells, what was actually meant by ‘rotations’ are displacement gradients, as will be shown in the next section. The concept of rigid body rotations should not have been resorted to in the first place. It was adopted because some of the displacement gradients are equivalent to rotations under small deformations, but they are not meant to be universally equivalent to rotations. Simply calling the displacement gradients as rotations may not necessarily form a logic fallacy. However, since these displacement gradients have been called rotations, the properties of rigid body rotations have been imposed to them by default ever since they were introduced in small deformation theories of plates and shells and the concept has been kept unchallenged as the plate and shell theories were extended to finite deformations, e.g. Argyris [2]. All the complications associated with rigid body rotations have been inherited as a result unfortunately. A typical logic fallacy of faulty generalisation formed when rotations as legitimate substitutes of displacement gradients under strict premise of small deformations was generalised to replace displacement gradients in finite deformations which have gone beyond the legitimate equivalence between displacement gradients and rotations. Yet, no one seems to be bothered, hitherto. It is clear from the basics of plate and shell theories as will be shown in the next section that it is displacement gradients that are required in order to construct the displacement field within a plate or a shell, not the ‘rotations’. The very basics will be sufficient to wipe off all the myth cast onto the issue of the so-called finite ‘rotation’ problem for plates and shells. A “back to basics” campaign is thus justified.
2. Basics of plate and shell theories The very objective of any plate or shell theory is to reduce the dimension of the problem from 3-D to 2-D by eliminating the dimension in the direction of the thickness of the plate or shell from the 3-D elasticity governing equations. Using the conventional coordinate system x, y and z, with z being in the thickness direction, the displacement field can be expressed as Taylor series with respect to z as follows [10]: uðx; y; zÞ ¼ uðx; y; 0Þ þz vðx; y; zÞ ¼ vðx; y; 0Þ þz
∂uðx; y; 0Þ 1 2 ∂2 uðx; y; 0Þ 1 3 ∂3 uðx; y; 0Þ þ z þ z þ ⋯⋯ ∂z 2 6 ∂z2 ∂z3
∂vðx; y; 0Þ 1 2 ∂2 vðx; y; 0Þ 1 3 ∂3 vðx; y; 0Þ þ z þ z þ ⋯⋯ ∂z 2 6 ∂z2 ∂z3
wðx; y; zÞ ¼ wðx; y; 0Þ þ z
∂wðx; y; 0Þ 1 2 ∂2 wðx; y; 0Þ 1 3 ∂3 wðx; y; 0Þ þ z þ z þ ⋯⋯ ∂z 2 6 ∂z2 ∂z3
ð1Þ
The in-plane coordinates x and y can be curvilinear as appropriate (arc lengths). For the ease of discussion to follow, in the case of curvilinear coordinates for curved surfaces, x and y are assumed to be along the principle directions of the reference surface of the shell. The series in (1) can be finitely truncated, leading to various shell theories, based on the thickness of the plate or shell. For instance, Reissner-Mindlin theory [17] for plates and shells of moderate thickness is based on the following truncations: uðx; y; zÞ ¼ uðx; y; 0Þ þ z vðx; y; zÞ ¼ vðx; y; 0Þ þ z wðx; y; zÞ ¼ wðx; y; 0Þ:
∂uðx; y; 0Þ ∂z
∂vðx; y; 0Þ ∂z ð2Þ
The so-called high order theories truncate at higher orders, e.g. Naghdi [13], while Love–Kirchhoff theory suitable for thin plates and shells is obtained by further introducing the following in addition to that of the Reissner–Mindlin theory (presented in
Please cite this article as: Li S. Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics. Int. J. Mech. Sci. (2014), http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004i
S. Li / International Journal of Mechanical Sciences ∎ (∎∎∎∎) ∎∎∎–∎∎∎
logical sequence here rather than historical one). ∂w ∂u ∂u ∂w þ ¼ 0 hence ¼ γ xz ¼ ∂x ∂z ∂z ∂x ∂w ∂v ∂v ∂w þ ¼ 0 hence ¼ and γ yz ¼ ∂y ∂z ∂z ∂y
ð3Þ
Using (3), the displacement field (2) is expressed as follows: ∂wðx; y; 0Þ ∂x ∂wðx; y; 0Þ vðx; y; zÞ ¼ vðx; y; 0Þ z ∂y wðx; y; zÞ ¼ wðx; y; 0Þ: uðx; y; zÞ ¼ uðx; y; 0Þ z
ð4Þ
The complete formulation of a plate or shell theory proceeds from the discretised (with respect to z) displacement field, e.g. (2) or (4). Green strains can be expressed from the displacements as given above. 8 2 9 2 > ∂u 1 ∂u 2 > > > þ 2 ∂x þ 12 ∂u þ 12 ∂u 9 8 > > ∂x ∂y ∂z > > > > ε x > > > > > > > > 2 > > > > 2 2 > > > > ∂v 1 ∂v 1 ∂v 1 ∂v > > > > þ þ þ εy > > > > ∂y 2 ∂x 2 ∂y 2 ∂z > > > > > > > > > > > = = < < εz > 2 2 2 1 ∂w 1 ∂w 1 ∂w þ þ þ ð5Þ ¼ ∂w ∂z 2 ∂x 2 ∂y 2 ∂z γ yz > > > > > > > > > > > ∂w ∂u ∂u ∂u ∂v ∂v ∂w ∂w > > > > > > > > > > ∂y þ ∂z þ ∂y ∂z þ ∂y ∂z þ ∂y ∂z > > γ xz > > > > > > > > > > > ∂u ∂v ∂v ∂w ∂w > ; > : γ xy > þ ∂w þ ∂u þ ∂z ∂x þ ∂z ∂x > > > ∂u > > ∂z ∂x ∂z ∂x > > > ; : ∂v þ ∂u þ ∂u ∂u þ ∂v ∂v þ ∂w ∂w > ∂x
∂y
∂x ∂y
∂x ∂y
∂x ∂y
By substituting the displacement field into the Green strains, generalised strains (independent of z) can be introduced to describe the deformation in the plate or shell, such as membrane strains and curvatures. By integrating various governing equations (equilibrium and constitutive) in their 3-D forms with respect to z, one eliminates z as an active dimension defining the problem, leading to a set of governing equations in 2-D form which is conventionally called a plate or shell theory. Throughout the process as described above, there is not a single point where rigid body rotations are involved. It is only under small deformation condition, displacement gradients ð∂w=∂xÞ and ð∂w=∂yÞ can be approximated by rotations of the reference surface about the y and x axes and ð∂u=∂zÞ and ð∂v=∂zÞ by rotations of the normal to the reference surface about the y and x axes ([10], Section 2.4). Since the introduction of the terminology of ‘rotations’ in the small deformation theories of plates and shells, these displacement gradients have been referred to as rotations. It is clear now that these displacement gradients were not rigid body rotations in the first place and their equivalence to rigid body rotations is not unconditional, either. Rotations are angles by definition. Mathematically, gradients or derivatives are the tangent of angles. The tangent of an angle is approximately equal to the angle if the angle is small. The so-called small rotations came to light this way which was undoubtedly encouraged by a nice feature of small rotations which behave as vectors [9]. Really, it should have been called small displacement gradients, instead, to be precise. It should be noted that the justification underlying the finite truncations introduced in any form as above, as well as the approximation of Love–Kirchhoff hypothesis, rests on the smallness of the thickness relative to the in-plane dimensions of the plate or shell. It has little to do with the magnitude of deformation or rotations. The magnitude of the deformation only dictates whether the second order terms should be included in the Green strains. The argument to be put forward is that plate and shell theories should not have required rotations as a part of their formulations. Some quantities (displacement gradients) have been called ‘rotations’ because of their approximate equivalence when the
3
displacement gradients are small. Assigning the properties of rigid body rotation to these ‘rotations’ has been a questionable generalisation already even under small deformations. The step of imposing the equivalence for finite deformations between displacement gradients and rigid body rotations has never been justified and cannot be justified. It is the time to return the real identity of these ‘rotations’ back to them and clear the mess behind them. Displacement gradients are much better behaved quantities than rigid body rotations. They are components of a tensor. Issues of rigid body rotations, such as the lack of communicability and additivity, and inapplicability of vectorial coordinate transformation should not have existed in the first place. In most modern commercial FE codes available, either small rotations (for each increment in an incremental analysis) are assumed or complicated approaches, such as the co-rotational method, have been adopted. It seems that they have been misguided by the faulty generalisation from displacement gradients to rigid body rotations. It might be too late to issue a wakeup call but late is still better than never and someone has to blow the whistle which is the purpose of this paper.
3. Formulation of a plate/shell finite element The argument as presented in the previous section applies equally in the formulation of finite element for plates and shells. To demonstrate this, a widely adopted 3-D degenerate 8-noded shell element [18] will be employed for this purpose without losing generality. Following a 3D degenerate formulation, the geometry of the shell element can be defined in terms of its nodal coordinates, nodal thicknesses and unit normal vectors at nodes as follows: 8 9 08 i 9 1 > > = = hi 8 i ! C B Y ¼ ∑ Ni ðξ; ηÞ@ Y i þ ζ e 3 A ð6Þ > > : > ; i¼1 : i> ; 2 Z Z where (X, Y, Z) are the global Cartesian coordinates of an arbitrary point in the element, (ξ, η, ζ) nondimensional coordinates ( 1 r ξ, η, ζ r 1) of the same point, which may not necessarily be orthogonal, (Xi, Yi, Zi) global Cartesian coordinates of i-th node, Ni the shape function for the i-th node, e.g. from the 2D serendipity family, hi the thickness of the shell at the i-th !i node, and e 3 is the unit normal vector to the shell at i-th node which defines the direction of coordinate ζ. For the construction of the displacement field in the element later, two more unit !i !i vectors e 1 and e 2 at the i-th node tangential to the reference surface are required, assumed to be aligned with the principal !i !i !i directions of the reference surface, so that e 1 , e 2 and e 3 form an orthogonal right handed triad. Denoted as 9 9 9 8 8 8 > > > ei11 > ei21 > ei31 > > > > > > > = = = < < < !i !i !i e 2 ¼ ei22 e 3 ¼ ei32 ð7Þ e 1 ¼ ei12 > > > > > > > > > ; ; ; : ei > : ei > : ei > 13
23
and apparently
½T ¼
h
! e1
! e2
i
! e3 ¼
2 ∂x
∂X 6 ∂x 6 ∂Y 4 ∂x ∂Z
33
∂y ∂X ∂y ∂Y ∂y ∂Z
∂z ∂X ∂z ∂Y ∂z ∂Z
3 7 7 5
ð8Þ
with (x, y, z) being the local principle coordinate system on the reference surface of the plate or shell. All quantities as introduced above are illustrated in Fig. 1. From the interpolated coordinates, Jacobian matrix can be obtained in the
Please cite this article as: Li S. Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics. Int. J. Mech. Sci. (2014), http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004i
S. Li / International Journal of Mechanical Sciences ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
ζ
η ξ
Fig. 1. A 3D degenerate shell element.
conventional manner, which will be useful for transformation of displacement gradient matrix with respect to (ξ, η, ζ) to that with respect to the global coordinate system (X, Y, Z). 2
∂X ∂ξ
∂X ∂η
∂X ∂ζ
∂Z ∂ξ
∂Y ∂η ∂Z ∂η
∂Y ∂ζ ∂Z ∂ζ
6 6 ∂Y ½J ¼ 6 ∂ξ 4
3
2
∂N i i X 6 ∂ξ
7 n 6 7 ∂N i i 7¼ ∑ 6 ∂ξ Y 5 i¼16 4 i i ∂N ∂ξ Z
∂N i i ∂η X ∂N i i ∂η Y ∂N i i ∂η Z
i
N i h2 ei31
3
7 7 i N i h2 ei32 7 7 5 i N i h2 ei33
!i e2
i
(
∂X
6 ∂U ½G ¼ 6 4 ∂Y ∂U ∂Z
ð9Þ
φ
i
ψi
)
C A
∂ξ 6 ∂U ½Γ ¼ 6 ∂η 4 ∂U ∂ζ
∂X ∂W ∂Y ∂W ∂Z
∂V ∂ξ ∂V ∂η
∂W ∂ξ ∂W ∂η
∂V ∂ζ
∂ζ
∂ξ ∂Z
2
3
6 n 6 7 7¼ ∑ 6 5 6 i¼14 ∂W
∂X ∂ζ ∂Y ∂ζ ∂Z
∂ξ
∂ξ ∂V ∂η
∂W ∂ξ ∂W ∂η
∂U ∂ζ
∂V ∂ζ
∂W ∂ζ
7 6 ∂U 7 6 ∂η 5 4
∂Ni i ∂ξ U
∂N i i ∂η U
∂N i i ∂ξ V
∂N i i ∂η V
i ∂N i ∂ξ W
i ∂Ni ∂η W
2
∂X
6 ∂ξ 6 ∂Y ¼ ð½J 1 ÞT and ½J 1 ¼ 6 ∂ξ 4 ∂Z ∂ξ
ð10Þ
where U, V, and W are the displacements in the element in the global X, Y and Z directions, respectively. At a node, e.g. the i-th one, there are five active degrees of freedom, Ui, Vi, Wi, φi and ψ i . While translational nodal displacements Ui, Vi and Wi are given in the global coordinate system, the ‘rotational’ ones φi and, ψ i in fact the values of displacement gradients ð∂u=∂zÞ and ð∂v=∂zÞ at the i-th node, are still in the local coordinate system (x, y, z), u and v being the in-plane displacement in the local coordinate system. φi and ψ i contribute to in-plane displacements u and v and their contributions have been transformed to the global !i !i coordinate system in (10) using e 1 and e 2 . With these nodal 0 degrees of freedom, the C continuity of U, V and W across adjacent elements is guaranteed by the fact that the elements on both sides of the border share the common nodal degrees of freedom as defined, regardless the magnitude of the deformation. It is important that a common node in different elements !i !i !i share the same base vectors e 1 , e 2 and e 3 . The displacement field as expressed in (10) is in agreement with that of (2) and the difference is that the displacement field (u, v, w) in (2) is given in the shell's local principle coordinate system (x, y, z) while that (U, V, W) in (10) in the global Cartesian system (X, Y, Z). The use of φi and ψ i (ð∂u=∂zÞ and ð∂v=∂zÞ) in (10) can be justified in the same way as those in (2) before, where the underlying consideration is the smallness of the thickness relative to the inplane dimensions of plate or shell to be analysed while having little to do with the magnitudes of ð∂u=∂zÞ and ð∂v=∂zÞ themselves. No matter how large or small the deformation is, all the information about the deformation is contained in the
∂X ∂η ∂Y ∂η ∂Z
∂X
7 6 ∂ξ 7 ¼ 6 ∂Y 5 4
3 7 7 ¼ ½J T ½Γ 5
i
N i h2 ðei11 φi þ ei21 ψ i Þ
ð11Þ
3
7 7 i N i h2 ðei12 φi þ ei22 ψ i Þ 7 7 5 i N i h2 ðei13 φi þ ei23 ψ i Þ ð12Þ
½J T
1
∂X ∂V ∂Y ∂V ∂Z
where 2 ∂U
The displacements interpolation is based on 3-D brick isoparametric elements in which all the necessary convergence requirements have been met in the sense of isoparametric finite elements. The Reissner–Mindlin hypothesis is introduced following the procedure of degenerate formulation. The displacement field within the element can be expressed as 8 9 08 i 9 > > = = hi h i n B V ¼ ∑ N i ðξ; ηÞ@ V i þ ζ ! e1 > > : > ; i¼1 : i> ; 2 W W
displacement gradient tensor as follows: 2 ∂U ∂V ∂W 3 2 ∂ξ ∂η ∂ζ 3 2 ∂U ∂V
∂X ∂η
∂X ∂ζ
∂Y ∂η ∂Z ∂η
∂Z ∂ζ
31
7 ∂Y 7 ∂ζ 7 5
2 ∂ξ
∂X 6 ∂η ¼6 4 ∂X ∂ζ ∂X
∂ξ ∂Y ∂η ∂Y ∂ζ ∂Y
∂ξ ∂Z ∂η ∂Z ∂ζ ∂Z
3 7 7: 5
ð13Þ
Green strain can always be evaluated from the displacement gradients [G] in the global coordinate system. 9 8 9 > ∂U þ 1∂U 2 þ 1∂U 2 þ 1∂U 2 > 8 > > ∂X 2 ∂X 2 ∂Y 2 ∂Z εX > > > > > > > > > > > > > > > ∂V 1 ∂V 2 1 ∂V 2 1 ∂V 2 > > > > > þ þ þ ε > > > > Y > ∂Y 2 ∂X 2 ∂Y 2 ∂Z > > > > > > > > > > = < εZ = < ∂W 1 ∂W 2 1 ∂W 2 1 ∂W 2 > þ 2 ∂X þ 2 ∂Y þ 2 ∂Z ∂Z ð14Þ ¼ γ > > > > ∂W ∂U ∂U ∂U ∂V ∂V ∂W ∂W YZ > > > > > > > ∂Y þ ∂Z þ ∂Y ∂Z þ ∂Y ∂Z þ ∂Y ∂Z > > > > > > > > > > > γ XZ > > > > ∂U þ ∂W þ ∂U ∂U þ ∂V ∂V þ ∂W ∂W > > > > > > > > ∂Z ∂X ∂Z ∂X ∂Z ∂X ∂Z ∂X > > > : > γ XY ; > ; : ∂V þ ∂U þ ∂U ∂U þ ∂V ∂V þ ∂W ∂W > ∂X
∂Y
∂X ∂Y
∂X ∂Y
∂X ∂Y
When dealing with anisotropic materials, there will be a need to express strains in the local coordinate system (x, y, z). This can be achieved by transforming the displacement gradient matrix (11): 2 ∂u ∂v ∂w 3 ∂x
6 ∂u ½g ¼ 6 4 ∂y ∂u ∂z
∂x ∂v ∂y ∂v ∂z
∂x ∂w 7 ∂y 7 5 ∂w ∂z
¼ ½T½G½TT
ð15Þ
where the transformation matrix [T] has been given in (8). Green strain in the local coordinate system can be obtained from [g] as shown in (5) previously. Given (12), it is conceivable that these strains, either in (5) or (14), can be somehow expressed in terms of nodal displacements, denoted as h iT fδg ¼ ⋯ U i V i W i ϕi ψ i ⋯ ð16Þ Taking those in (5) as an example, to construct the tangential stiffness matrix, as is often required for nonlinear analysis, the
Please cite this article as: Li S. Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics. Int. J. Mech. Sci. (2014), http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004i
S. Li / International Journal of Mechanical Sciences ∎ (∎∎∎∎) ∎∎∎–∎∎∎
strains and nodal displacements relationship can be first expressed in incremental form [18]. dfεg ¼ ½BðfδgÞdfδg
ð17Þ
where [B] is called geometric matrix in finite element method which is a linear function of nodal displacement fδg. Elemental tangential stiffness matrix is then given as Z ½BðfδgÞT ½D½BðfδgÞdV ð18Þ ½K t ¼ Ve
where Ve is the volume of the element. There is nothing special in the above elaboration in this section. The reproduction of these fairly straightforward steps is to make a clear point that rigid body rotations do not play any part in the deformation kinematics of shell element. Another important point to make is the fact that the validity of the displacement field (10) as well as the above procedure does not rely on the magnitude of deformation nor on that of the displacement gradients which had been mistakenly called ‘rotations’, hitherto. The magnitude of deformation affects only the consideration whether the first order terms of {δ} in [B] as in (17) should be kept or not, which introduces geometric nonlinearity into the problem as in conventional 3D solids. If φi and ψ i were rigid body rotations, the formulation would hit fatal difficulties. Eq. (6) would not determine the displacement field uniquely under finite deformation because the final position !i of a point on the normal e 3 would depend on the sequence of φi i and ψ . Secondly, they could not be accumulated in a conventional way in an incremental analysis. Interestingly, no one has seemed to be bothered about the apparent nature of (6) where nothing compromises the apparent uniqueness of displacement field and suitability of accumulating them in an incremental analysis based on the algebraic presentation as it stands. This is perhaps a different way of illustrating the farfetched nature of the concept of ‘rigid body rotations’ in the plate and shell theories, as well as in the finite element formulation for plates and shells. If there is any consideration which may compromise the validity of the theory, it is the thickness of the plate or shell. The only assumption involved in plate and shell theories, as well as finite elements for plates and shells, is that the thickness of the plate or shell that should be small enough in comparison with other dimensions of the plate or shell. This allows the truncation of (1) for the displacement field to be represented in (2), such that the Reissner–Mindlin hypothesis remains valid. With greater thickness beyond the applicability of Reissner–Mindlin hypothesis, more advanced shell theories, such as that in [13], may be resorted to. However, it is not common in FE approaches, especially for large deformation problems. FE users are more inclined to use 3-D elements instead nowadays, given the ever increasing computing power. Incidentally, once 3-D elements are employed, there will be no need to worry about the magnitude of rotations involved in the deformation anymore as all the information about the deformation have been encapsulated in the Green strains. This is certainly another indication of the irrelevance of rigid body rotations to the deformation kinematics of deformable bodies.
5
Nodal degrees of freedom φi and ψ i are best to be left as they are as gradients of in-plane displacements with respect to the coordinate in the thickness direction, all expressed in the local coordinate system (x, y, z). If they had to be transformed to a different coordinate systems for whatever reason, they should be transformed as displacement gradients in the form of (13). This would require the complete displacement gradient matrix to be obtained in the local coordinate system first. One ought to be aware that such displacement gradient matrix at a node may be evaluated to different values from different elements involving this node. A certain treatment would be required, as in the case when stresses at nodes are evaluated in finite element analysis, where a typical way is to take average from all elements involving the node concerned. One example where such transformation may become necessary is for nodes along a ridge in a folded or boxed plate or shell. In this case further approximations are inevitable [6], whatever theory is used, unless one is prepared to go back to a 3D model. This should be addressed as a separate issue without distracting the focus of this paper. Given the above discussion, most existing plate and shell theories for finite deformations should give a correct description of the behaviour of the plate or shell after ignoring the complications illogically introduced by mistakenly perceived finite ‘rotations’, provided that the theory is justifiable for the given thickness in relation to the other dimensions of the plate or shell. The elaboration as above in this section based on a 3-D degenerate isoparametric finite shell element can also serve as a roadmap to develop the formulation fully such that numerical results can be obtained to illustrate the suitability of the theoretical framework proposed here after the confusions on the subject of finite deformation of plates and shells have been clarified as in this paper. However, such development will have to be pursued as a future development without distracting the main focus of this paper.
4. Conclusions The message out of the elaboration and discussion in this paper, based purely on the very basics of plate and shell theories, can be summarised in Table 1 below The paper can be concluded with the following remarks. 1) Rigid body rotations are far-fetched into the formulation of plate and shell theories. The real identity of the so-called ‘rotations’ is displacement gradients. Although the equivalence in between can be established under infinitesimal deformations, mistaking displacement gradients by rigid body rotations under finite deformations forms a logic fallacy of faulty generalisation. While it is conceptually wrong and logically defective, it also results in a whole range of complications which are completely unnecessary if their true identity has been incorporated in the development of plate and shell theories for finite deformations, although the complicated and unnecessary approaches have dominated the stage of the finite deformation theories of plates and shells, hitherto.
Table 1 The relationship between displacement gradients and rotations in the formulation of the theory of plates and shells. Formulation of the theory of plates and shells
Required
Actually employed
Relationship between required and employed
Correctness
Consequence
Infinitesimal ‘rotation’ problem Existing finite ‘rotation’ problem Finite displacement gradient problem as proposed in this paper
Displacement gradients Displacement gradients Displacement gradients
Rotations Rotations Displacement gradients
Equivalent Not equivalent Identical
Yes No Yes
Easy to visualise Complicated treatments Direct treatment without undue complications
Please cite this article as: Li S. Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics. Int. J. Mech. Sci. (2014), http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004i
S. Li / International Journal of Mechanical Sciences ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
2) The validity of the theoretical framework as proposed in this paper, including the Reissner–Mindlin hypothesis, the use of displacement gradients to construct displacement field in plates and shells, the finite element presentation and the application to finite deformations are controlled by the thickness of the plate or shell, not the magnitude of ‘rotations’ (displacement gradients). Traditionally, finite ‘rotation’ problems attract two sources of complications. One is ‘finite rotations’ due to their non-vectorial nature which has cast a huge implication on the representation of these rotations and their incorporation in the construction of the displacement field in plates and shells. This has been shown in this paper as a fabricated and unnecessary problem out of a logic fallacy and thus can be ruled out completely. The second source is the finite deformation due to the second order terms in the Green strains which involve displacement gradients but nothing else. This is controlled by the magnitude of deformation alone without the complications associated with the first source. 3) In finite elements for plates and shells, the nodal displacement gradient degrees of freedom are best kept in their local coordinate system as originally introduced in the formulation of plate and shell element. If they have to be transformed into a different coordinate system for whatever reason, they should be transformed as components of the second rank displacement gradient tensor, not any form of vectors.
Appendix The following proverb was quoted in Argyris [2]. 踏破铁鞋无觅处, 得来全不费功夫. It is translated as follows. One has worn his/her metal shoes in searching for, often without joy. However, when it eventually turns up, it is obtained
rather effortless. The proverb emphasises the fact of having obtained the object and the easiness in obtaining the object.
References [1] Abaqus. Abaqus 6.13 Theory Guide; 2013. [2] Argyris J. An excursion into large rotations. Comput Methods Mech Eng 1982;32:85–155. [3] Arnold RN, Maunder L. Gyrodynamics and its engineering applications. London: Academic Press; 1961. [4] Cai YC, Atluri SN. Large Rotation analyses of plate/shell structures based on the primal variational principle and a fully nonlinear theory in the updated lagrangian co-rotational reference frame. Comput Model Eng Sci 2012;83: 249–273. [5] Crisfield MA. Nonlinear finite element analysis of solids and structures, 2. Chichester: Wiley; 1997. [6] Cook RD, Malkus DS, Plesha ME. Concepts and applications of finite element analysis. 3rd ed. London: Wiley; 1989. [7] Hoffmann B. About vectors. Englewood Cliff, New Jersey: Prentice-Hall; 1966. [8] Ibrahimbegovic A. Stress resultant geometrically exact shell theory for finite rotations and its finite element implementation. ASME Appl Mech Rev 1997;50:199–225. [9] Kleppner D, Kolenkow RJ. An introduction to mechanics. London: McGrawHill; 1978. [10] Kraus H. Thin Elastic Shells. New York: Wiley; 1967. [11] Li S. On the vectorial nature of the rotational degrees of freedom in spatial beam elements. Comput Struct 1996;59:1199–200. [12] Libai A, Simmonds JG. The nonlinear theory of elastic shells. 2nd ed. Cambridge: Cambridge University Press; 1998. [13] Naghdi PM. On the theory of thin elastic shells. Q Appl Math 1957;14:369–80. [14] Novozhilov VV. Theory of elasticity. Oxford: Pergamon; 1961. [15] Parisch H. Geometrical nonlinear analysis of shells. Comput Methods Appl Mech Eng 1978;14:159–78. [16] Pietraszkiewicz W. Finite rotations in shells. In: Koiter WT, Mikhailov GK, editors. Theory of shells. Amsterdam: North-Holland Publishing Company; 1980. p. 445–71. [17] Reissner E. Stress strain relations in the theory of thin elastic shells. J Math Phys 1952;31:109–19. [18] Zienkiewicz OC, Taylor RL. The finite element method. 4th ed. . London: McGraw-Hill; 1998.
Please cite this article as: Li S. Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics. Int. J. Mech. Sci. (2014), http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004i
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics Shuguang Li Institute for Aerospace Technology, Faculty of Engineering, University of Nottingham, Nottingham, NG7 2RD, UK
art ic l e i nf o
a b s t r a c t
Article history: Received 13 February 2013 Received in revised form 30 June 2014 Accepted 14 July 2014
This paper results from a reflection on the problem of finite rotations in plates and shells and its presentation in finite element formulation, a well-attended subject with a wide diversity of treatments, often extremely complicated. For instance, the very topic attracted 3 out of total 15 chapters over 120 out of total 494 pages in a monograph on nonlinear FEA [5]. In another instance, a 71 page journal paper was published [2] specifically on this topic alone. Review papers on this matter can be found in various sources including Appl Mech Rev, (e.g. Ibrahimbegovic, 1997 [8]). No attempt is to be made on reviewing these sophisticated approaches but the present paper will bring answers to questions, if they have ever been asked, such as whether the sophistication in existing approaches is really necessary and whether there is a much simpler and conceptually more direct and accurate approach. A proper re-examination of the existing approaches would reveal a fundamental inconsistency: rigid body rotations have been assumed without reservation to describe the deformations of deformable plates and shells. After reestablishing the consistency based on the very basics of conventional plate and shell theories as a simple reflection, one can conclude surprisingly that the whole issue on ‘finite rotations’ results from a logic fallacy of faulty generalisation. The so-called ‘rotations’ should be displacement gradients instead. They can be considered as ‘rotations’ as conventionally perceived hitherto under the condition of small deformation. Somehow, the concept of ‘rotations’ got generalised regardless the magnitude of the deformation. Instead of calling the problem as ‘finite displacement gradients’ as it should be called, a falsely generalised term of ‘finite rotations’ have been used. Since they were called ‘rotations’, the definition of those of rigid body kinematics has been taken for granted. Finite displacement gradients should not present any additional problem apart from introducing nonlinear nature into the problem, which can be addressed as a geometrically nonlinear problem in a conventional manner. However, along the line of a falsely generalised concept of finite rigid body rotations, complications have been the norm. The complicated accounts on the ‘finite rotation’ problem in the literature, which might have enhanced the understanding of rigid body kinematics, are entirely unnecessary as far as the deformable plates and shells are concerned, involving infinitesimal or finite deformations. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Finite rotations Plates and shells Shell elements Displacement gradients Reissner–Mindlin Faulty generalisation
不识庐山真面目,只缘身在此山中—A Chinese proverb. (One is incapable of appreciating the true beauty of Lusan Mountain simply because he/she is viewing from deep inside it.)
1. Introduction The development of plate/shell finite elements has encountered two major challenges. At an early stage, C1 continuity troubled users for decades from early sixties. Many significant progresses had been made while combating with this difficulty and some even laid cornerstones for the finite element method in general, e.g. patch test for nonconforming elements. Elegant as E-mail address: [email protected]
they were, the solution to the problem had not come through in this way. Rather, the remedy emerged with C0 element using a degenerate formulation. While it came with problems generic to the formulation, typically shear locking associated with its applications to thin plates and shells. Techniques, such as reduced integration among others, had then been developed to keep them at bay. By now, the campaign is more or less won and C0 plate/ shell elements have been adopted unanimously in most, if not all, commercial finite element codes. The second challenge was the rotations when finite deformation is involved. Various treatments have been proposed at their diversity and yet no unanimous conclusion has been drawn. A review of the state-of-the-art seems to suggest that the nightmare be still haunting the minds of users today [4]. The issue does not appear in small deformation problems because small rotations can be treated as vectors approximately while it can be proved
http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004 0020-7403/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Li S. Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics. Int. J. Mech. Sci. (2014), http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004i
2
S. Li / International Journal of Mechanical Sciences ∎ (∎∎∎∎) ∎∎∎–∎∎∎
that infinitesimal rotations form proper vectors strictly [9]. However, a convenient terminology of ‘rotations’ introduced then has misguided the development profoundly into the finite deformation problem of plates and shells. As the rotations become finite, problems emerge. Finite rigid body rotations are wellknown to be incommunicable and non-additive due to their non-vectorial nature. This hits a brick wall straightaway on following three accounts. (i) Whenever a coordinate transformation is required for the three possible rotational degrees of freedom, they cannot be transformed as a vector. (ii) They cannot be accumulated in a conventional way since the incommunicability rules out the parallelogram law of vectors while the accumulation process is crucial in an incremental and iterative process which is by large the most popular way of solving nonlinear problems. (iii) It also affects the uniqueness in the determination of the displacement field in a finite element, as the in-plane displacements at a point off the reference surface are no longer linear functions of the angles of rotation of the normal to the reference surface [15]. It looks as if one is confronting a formidable difficulty and, in fact, it has been treated this way in the literature, with overwhelmingly sophisticated formulations inevitably as a result. The existing treatments for finite rotations are numerous as reviewed in Crisfield [5] and Ibrahimbegovic [8], for example, quaternion representation as in [1], that employs special orthogonal group, SO3 ([12] and [16]), that based on the concept of a local sphere [15], which is in essence the use of Eulerian angles of rotation [3], and so-called co-rotational coordinates, among others. The co-rotational coordinates approach perhaps accounts for significantly more efforts and systematic developments, as reviewed in Ibrahimbegovic [8] and Crisfield [5]. Elegantly formulated as they have been, no victory has been claimed as yet for the battle combatting against finite rotations. The sheer complexity is often sufficient to deter many from treading on the mess. As an example, the issue of finite rotations is dealt with in three consecutive chapters over 120 page of the monograph by Crisfield [5]. In another instance, a 71 page long paper was published [2] specifically on this topic. Given the scenario, a sober question to be asked is, “Are we striving in a similar trap to that of C1 continuity and is there a remedy like the C0 solution to get us out of the mess?” The objective of the paper is to bring a positive answer to this question. In a short note [11], the vectorial nature of finite nodal rotations was argued in the context of finite beam elements. The outcome was that, under Bernoulli plane-section assumption, the three nodal rotations form a pseudo-vector [14], or axial vector as referred to in [7]. As long as no coordinate transformations are involved between systems of mixed nature, i.e. right-hand and left-hand systems, which is prohibited practically anyway (otherwise, negative determinant of Jacobian will be encountered), the three rotations form a vector, just as ordinary as any other vectors. In the case of Timoshenko beams, the rotations do not form a vector exactly but differ by the transverse shear strains. This should not constitute a substantial problem practically since the magnitudes of transverse shear strains, significant as they might be in comparison with other strain components, would be negligible as compared with the magnitude of rotations as in a finite rotation problem. As a result, the rotations could be regarded as a vector as a reasonable approximation under the condition that transverse shear strains are small enough relative to the magnitude of rotations. The similar argument could be made for plates and shells but it is not pursued in the present paper. A fresh re-examination from a different perspective will resolve the whole issue completely. It can be found that plate and shell theories somehow have fallen in a typical logic fallacy of faulty generalisation in the respect of ‘finite rotations’. To describe the deformation and to construct the displacement fields in
deformable plates and shells, what was actually meant by ‘rotations’ are displacement gradients, as will be shown in the next section. The concept of rigid body rotations should not have been resorted to in the first place. It was adopted because some of the displacement gradients are equivalent to rotations under small deformations, but they are not meant to be universally equivalent to rotations. Simply calling the displacement gradients as rotations may not necessarily form a logic fallacy. However, since these displacement gradients have been called rotations, the properties of rigid body rotations have been imposed to them by default ever since they were introduced in small deformation theories of plates and shells and the concept has been kept unchallenged as the plate and shell theories were extended to finite deformations, e.g. Argyris [2]. All the complications associated with rigid body rotations have been inherited as a result unfortunately. A typical logic fallacy of faulty generalisation formed when rotations as legitimate substitutes of displacement gradients under strict premise of small deformations was generalised to replace displacement gradients in finite deformations which have gone beyond the legitimate equivalence between displacement gradients and rotations. Yet, no one seems to be bothered, hitherto. It is clear from the basics of plate and shell theories as will be shown in the next section that it is displacement gradients that are required in order to construct the displacement field within a plate or a shell, not the ‘rotations’. The very basics will be sufficient to wipe off all the myth cast onto the issue of the so-called finite ‘rotation’ problem for plates and shells. A “back to basics” campaign is thus justified.
2. Basics of plate and shell theories The very objective of any plate or shell theory is to reduce the dimension of the problem from 3-D to 2-D by eliminating the dimension in the direction of the thickness of the plate or shell from the 3-D elasticity governing equations. Using the conventional coordinate system x, y and z, with z being in the thickness direction, the displacement field can be expressed as Taylor series with respect to z as follows [10]: uðx; y; zÞ ¼ uðx; y; 0Þ þz vðx; y; zÞ ¼ vðx; y; 0Þ þz
∂uðx; y; 0Þ 1 2 ∂2 uðx; y; 0Þ 1 3 ∂3 uðx; y; 0Þ þ z þ z þ ⋯⋯ ∂z 2 6 ∂z2 ∂z3
∂vðx; y; 0Þ 1 2 ∂2 vðx; y; 0Þ 1 3 ∂3 vðx; y; 0Þ þ z þ z þ ⋯⋯ ∂z 2 6 ∂z2 ∂z3
wðx; y; zÞ ¼ wðx; y; 0Þ þ z
∂wðx; y; 0Þ 1 2 ∂2 wðx; y; 0Þ 1 3 ∂3 wðx; y; 0Þ þ z þ z þ ⋯⋯ ∂z 2 6 ∂z2 ∂z3
ð1Þ
The in-plane coordinates x and y can be curvilinear as appropriate (arc lengths). For the ease of discussion to follow, in the case of curvilinear coordinates for curved surfaces, x and y are assumed to be along the principle directions of the reference surface of the shell. The series in (1) can be finitely truncated, leading to various shell theories, based on the thickness of the plate or shell. For instance, Reissner-Mindlin theory [17] for plates and shells of moderate thickness is based on the following truncations: uðx; y; zÞ ¼ uðx; y; 0Þ þ z vðx; y; zÞ ¼ vðx; y; 0Þ þ z wðx; y; zÞ ¼ wðx; y; 0Þ:
∂uðx; y; 0Þ ∂z
∂vðx; y; 0Þ ∂z ð2Þ
The so-called high order theories truncate at higher orders, e.g. Naghdi [13], while Love–Kirchhoff theory suitable for thin plates and shells is obtained by further introducing the following in addition to that of the Reissner–Mindlin theory (presented in
Please cite this article as: Li S. Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics. Int. J. Mech. Sci. (2014), http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004i
S. Li / International Journal of Mechanical Sciences ∎ (∎∎∎∎) ∎∎∎–∎∎∎
logical sequence here rather than historical one). ∂w ∂u ∂u ∂w þ ¼ 0 hence ¼ γ xz ¼ ∂x ∂z ∂z ∂x ∂w ∂v ∂v ∂w þ ¼ 0 hence ¼ and γ yz ¼ ∂y ∂z ∂z ∂y
ð3Þ
Using (3), the displacement field (2) is expressed as follows: ∂wðx; y; 0Þ ∂x ∂wðx; y; 0Þ vðx; y; zÞ ¼ vðx; y; 0Þ z ∂y wðx; y; zÞ ¼ wðx; y; 0Þ: uðx; y; zÞ ¼ uðx; y; 0Þ z
ð4Þ
The complete formulation of a plate or shell theory proceeds from the discretised (with respect to z) displacement field, e.g. (2) or (4). Green strains can be expressed from the displacements as given above. 8 2 9 2 > ∂u 1 ∂u 2 > > > þ 2 ∂x þ 12 ∂u þ 12 ∂u 9 8 > > ∂x ∂y ∂z > > > > ε x > > > > > > > > 2 > > > > 2 2 > > > > ∂v 1 ∂v 1 ∂v 1 ∂v > > > > þ þ þ εy > > > > ∂y 2 ∂x 2 ∂y 2 ∂z > > > > > > > > > > > = = < < εz > 2 2 2 1 ∂w 1 ∂w 1 ∂w þ þ þ ð5Þ ¼ ∂w ∂z 2 ∂x 2 ∂y 2 ∂z γ yz > > > > > > > > > > > ∂w ∂u ∂u ∂u ∂v ∂v ∂w ∂w > > > > > > > > > > ∂y þ ∂z þ ∂y ∂z þ ∂y ∂z þ ∂y ∂z > > γ xz > > > > > > > > > > > ∂u ∂v ∂v ∂w ∂w > ; > : γ xy > þ ∂w þ ∂u þ ∂z ∂x þ ∂z ∂x > > > ∂u > > ∂z ∂x ∂z ∂x > > > ; : ∂v þ ∂u þ ∂u ∂u þ ∂v ∂v þ ∂w ∂w > ∂x
∂y
∂x ∂y
∂x ∂y
∂x ∂y
By substituting the displacement field into the Green strains, generalised strains (independent of z) can be introduced to describe the deformation in the plate or shell, such as membrane strains and curvatures. By integrating various governing equations (equilibrium and constitutive) in their 3-D forms with respect to z, one eliminates z as an active dimension defining the problem, leading to a set of governing equations in 2-D form which is conventionally called a plate or shell theory. Throughout the process as described above, there is not a single point where rigid body rotations are involved. It is only under small deformation condition, displacement gradients ð∂w=∂xÞ and ð∂w=∂yÞ can be approximated by rotations of the reference surface about the y and x axes and ð∂u=∂zÞ and ð∂v=∂zÞ by rotations of the normal to the reference surface about the y and x axes ([10], Section 2.4). Since the introduction of the terminology of ‘rotations’ in the small deformation theories of plates and shells, these displacement gradients have been referred to as rotations. It is clear now that these displacement gradients were not rigid body rotations in the first place and their equivalence to rigid body rotations is not unconditional, either. Rotations are angles by definition. Mathematically, gradients or derivatives are the tangent of angles. The tangent of an angle is approximately equal to the angle if the angle is small. The so-called small rotations came to light this way which was undoubtedly encouraged by a nice feature of small rotations which behave as vectors [9]. Really, it should have been called small displacement gradients, instead, to be precise. It should be noted that the justification underlying the finite truncations introduced in any form as above, as well as the approximation of Love–Kirchhoff hypothesis, rests on the smallness of the thickness relative to the in-plane dimensions of the plate or shell. It has little to do with the magnitude of deformation or rotations. The magnitude of the deformation only dictates whether the second order terms should be included in the Green strains. The argument to be put forward is that plate and shell theories should not have required rotations as a part of their formulations. Some quantities (displacement gradients) have been called ‘rotations’ because of their approximate equivalence when the
3
displacement gradients are small. Assigning the properties of rigid body rotation to these ‘rotations’ has been a questionable generalisation already even under small deformations. The step of imposing the equivalence for finite deformations between displacement gradients and rigid body rotations has never been justified and cannot be justified. It is the time to return the real identity of these ‘rotations’ back to them and clear the mess behind them. Displacement gradients are much better behaved quantities than rigid body rotations. They are components of a tensor. Issues of rigid body rotations, such as the lack of communicability and additivity, and inapplicability of vectorial coordinate transformation should not have existed in the first place. In most modern commercial FE codes available, either small rotations (for each increment in an incremental analysis) are assumed or complicated approaches, such as the co-rotational method, have been adopted. It seems that they have been misguided by the faulty generalisation from displacement gradients to rigid body rotations. It might be too late to issue a wakeup call but late is still better than never and someone has to blow the whistle which is the purpose of this paper.
3. Formulation of a plate/shell finite element The argument as presented in the previous section applies equally in the formulation of finite element for plates and shells. To demonstrate this, a widely adopted 3-D degenerate 8-noded shell element [18] will be employed for this purpose without losing generality. Following a 3D degenerate formulation, the geometry of the shell element can be defined in terms of its nodal coordinates, nodal thicknesses and unit normal vectors at nodes as follows: 8 9 08 i 9 1 > >
23
and apparently
½T ¼
h
! e1
! e2
i
! e3 ¼
2 ∂x
∂X 6 ∂x 6 ∂Y 4 ∂x ∂Z
33
∂y ∂X ∂y ∂Y ∂y ∂Z
∂z ∂X ∂z ∂Y ∂z ∂Z
3 7 7 5
ð8Þ
with (x, y, z) being the local principle coordinate system on the reference surface of the plate or shell. All quantities as introduced above are illustrated in Fig. 1. From the interpolated coordinates, Jacobian matrix can be obtained in the
Please cite this article as: Li S. Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics. Int. J. Mech. Sci. (2014), http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004i
S. Li / International Journal of Mechanical Sciences ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
ζ
η ξ
Fig. 1. A 3D degenerate shell element.
conventional manner, which will be useful for transformation of displacement gradient matrix with respect to (ξ, η, ζ) to that with respect to the global coordinate system (X, Y, Z). 2
∂X ∂ξ
∂X ∂η
∂X ∂ζ
∂Z ∂ξ
∂Y ∂η ∂Z ∂η
∂Y ∂ζ ∂Z ∂ζ
6 6 ∂Y ½J ¼ 6 ∂ξ 4
3
2
∂N i i X 6 ∂ξ
7 n 6 7 ∂N i i 7¼ ∑ 6 ∂ξ Y 5 i¼16 4 i i ∂N ∂ξ Z
∂N i i ∂η X ∂N i i ∂η Y ∂N i i ∂η Z
i
N i h2 ei31
3
7 7 i N i h2 ei32 7 7 5 i N i h2 ei33
!i e2
i
(
∂X
6 ∂U ½G ¼ 6 4 ∂Y ∂U ∂Z
ð9Þ
φ
i
ψi
)
C A
∂ξ 6 ∂U ½Γ ¼ 6 ∂η 4 ∂U ∂ζ
∂X ∂W ∂Y ∂W ∂Z
∂V ∂ξ ∂V ∂η
∂W ∂ξ ∂W ∂η
∂V ∂ζ
∂ζ
∂ξ ∂Z
2
3
6 n 6 7 7¼ ∑ 6 5 6 i¼14 ∂W
∂X ∂ζ ∂Y ∂ζ ∂Z
∂ξ
∂ξ ∂V ∂η
∂W ∂ξ ∂W ∂η
∂U ∂ζ
∂V ∂ζ
∂W ∂ζ
7 6 ∂U 7 6 ∂η 5 4
∂Ni i ∂ξ U
∂N i i ∂η U
∂N i i ∂ξ V
∂N i i ∂η V
i ∂N i ∂ξ W
i ∂Ni ∂η W
2
∂X
6 ∂ξ 6 ∂Y ¼ ð½J 1 ÞT and ½J 1 ¼ 6 ∂ξ 4 ∂Z ∂ξ
ð10Þ
where U, V, and W are the displacements in the element in the global X, Y and Z directions, respectively. At a node, e.g. the i-th one, there are five active degrees of freedom, Ui, Vi, Wi, φi and ψ i . While translational nodal displacements Ui, Vi and Wi are given in the global coordinate system, the ‘rotational’ ones φi and, ψ i in fact the values of displacement gradients ð∂u=∂zÞ and ð∂v=∂zÞ at the i-th node, are still in the local coordinate system (x, y, z), u and v being the in-plane displacement in the local coordinate system. φi and ψ i contribute to in-plane displacements u and v and their contributions have been transformed to the global !i !i coordinate system in (10) using e 1 and e 2 . With these nodal 0 degrees of freedom, the C continuity of U, V and W across adjacent elements is guaranteed by the fact that the elements on both sides of the border share the common nodal degrees of freedom as defined, regardless the magnitude of the deformation. It is important that a common node in different elements !i !i !i share the same base vectors e 1 , e 2 and e 3 . The displacement field as expressed in (10) is in agreement with that of (2) and the difference is that the displacement field (u, v, w) in (2) is given in the shell's local principle coordinate system (x, y, z) while that (U, V, W) in (10) in the global Cartesian system (X, Y, Z). The use of φi and ψ i (ð∂u=∂zÞ and ð∂v=∂zÞ) in (10) can be justified in the same way as those in (2) before, where the underlying consideration is the smallness of the thickness relative to the inplane dimensions of plate or shell to be analysed while having little to do with the magnitudes of ð∂u=∂zÞ and ð∂v=∂zÞ themselves. No matter how large or small the deformation is, all the information about the deformation is contained in the
∂X ∂η ∂Y ∂η ∂Z
∂X
7 6 ∂ξ 7 ¼ 6 ∂Y 5 4
3 7 7 ¼ ½J T ½Γ 5
i
N i h2 ðei11 φi þ ei21 ψ i Þ
ð11Þ
3
7 7 i N i h2 ðei12 φi þ ei22 ψ i Þ 7 7 5 i N i h2 ðei13 φi þ ei23 ψ i Þ ð12Þ
½J T
1
∂X ∂V ∂Y ∂V ∂Z
where 2 ∂U
The displacements interpolation is based on 3-D brick isoparametric elements in which all the necessary convergence requirements have been met in the sense of isoparametric finite elements. The Reissner–Mindlin hypothesis is introduced following the procedure of degenerate formulation. The displacement field within the element can be expressed as 8 9 08 i 9 > > = = hi h i n B V ¼ ∑ N i ðξ; ηÞ@ V i þ ζ ! e1 > > : > ; i¼1 : i> ; 2 W W
displacement gradient tensor as follows: 2 ∂U ∂V ∂W 3 2 ∂ξ ∂η ∂ζ 3 2 ∂U ∂V
∂X ∂η
∂X ∂ζ
∂Y ∂η ∂Z ∂η
∂Z ∂ζ
31
7 ∂Y 7 ∂ζ 7 5
2 ∂ξ
∂X 6 ∂η ¼6 4 ∂X ∂ζ ∂X
∂ξ ∂Y ∂η ∂Y ∂ζ ∂Y
∂ξ ∂Z ∂η ∂Z ∂ζ ∂Z
3 7 7: 5
ð13Þ
Green strain can always be evaluated from the displacement gradients [G] in the global coordinate system. 9 8 9 > ∂U þ 1∂U 2 þ 1∂U 2 þ 1∂U 2 > 8 > > ∂X 2 ∂X 2 ∂Y 2 ∂Z εX > > > > > > > > > > > > > > > ∂V 1 ∂V 2 1 ∂V 2 1 ∂V 2 > > > > > þ þ þ ε > > > > Y > ∂Y 2 ∂X 2 ∂Y 2 ∂Z > > > > > > > > > > = < εZ = < ∂W 1 ∂W 2 1 ∂W 2 1 ∂W 2 > þ 2 ∂X þ 2 ∂Y þ 2 ∂Z ∂Z ð14Þ ¼ γ > > > > ∂W ∂U ∂U ∂U ∂V ∂V ∂W ∂W YZ > > > > > > > ∂Y þ ∂Z þ ∂Y ∂Z þ ∂Y ∂Z þ ∂Y ∂Z > > > > > > > > > > > γ XZ > > > > ∂U þ ∂W þ ∂U ∂U þ ∂V ∂V þ ∂W ∂W > > > > > > > > ∂Z ∂X ∂Z ∂X ∂Z ∂X ∂Z ∂X > > > : > γ XY ; > ; : ∂V þ ∂U þ ∂U ∂U þ ∂V ∂V þ ∂W ∂W > ∂X
∂Y
∂X ∂Y
∂X ∂Y
∂X ∂Y
When dealing with anisotropic materials, there will be a need to express strains in the local coordinate system (x, y, z). This can be achieved by transforming the displacement gradient matrix (11): 2 ∂u ∂v ∂w 3 ∂x
6 ∂u ½g ¼ 6 4 ∂y ∂u ∂z
∂x ∂v ∂y ∂v ∂z
∂x ∂w 7 ∂y 7 5 ∂w ∂z
¼ ½T½G½TT
ð15Þ
where the transformation matrix [T] has been given in (8). Green strain in the local coordinate system can be obtained from [g] as shown in (5) previously. Given (12), it is conceivable that these strains, either in (5) or (14), can be somehow expressed in terms of nodal displacements, denoted as h iT fδg ¼ ⋯ U i V i W i ϕi ψ i ⋯ ð16Þ Taking those in (5) as an example, to construct the tangential stiffness matrix, as is often required for nonlinear analysis, the
Please cite this article as: Li S. Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics. Int. J. Mech. Sci. (2014), http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004i
S. Li / International Journal of Mechanical Sciences ∎ (∎∎∎∎) ∎∎∎–∎∎∎
strains and nodal displacements relationship can be first expressed in incremental form [18]. dfεg ¼ ½BðfδgÞdfδg
ð17Þ
where [B] is called geometric matrix in finite element method which is a linear function of nodal displacement fδg. Elemental tangential stiffness matrix is then given as Z ½BðfδgÞT ½D½BðfδgÞdV ð18Þ ½K t ¼ Ve
where Ve is the volume of the element. There is nothing special in the above elaboration in this section. The reproduction of these fairly straightforward steps is to make a clear point that rigid body rotations do not play any part in the deformation kinematics of shell element. Another important point to make is the fact that the validity of the displacement field (10) as well as the above procedure does not rely on the magnitude of deformation nor on that of the displacement gradients which had been mistakenly called ‘rotations’, hitherto. The magnitude of deformation affects only the consideration whether the first order terms of {δ} in [B] as in (17) should be kept or not, which introduces geometric nonlinearity into the problem as in conventional 3D solids. If φi and ψ i were rigid body rotations, the formulation would hit fatal difficulties. Eq. (6) would not determine the displacement field uniquely under finite deformation because the final position !i of a point on the normal e 3 would depend on the sequence of φi i and ψ . Secondly, they could not be accumulated in a conventional way in an incremental analysis. Interestingly, no one has seemed to be bothered about the apparent nature of (6) where nothing compromises the apparent uniqueness of displacement field and suitability of accumulating them in an incremental analysis based on the algebraic presentation as it stands. This is perhaps a different way of illustrating the farfetched nature of the concept of ‘rigid body rotations’ in the plate and shell theories, as well as in the finite element formulation for plates and shells. If there is any consideration which may compromise the validity of the theory, it is the thickness of the plate or shell. The only assumption involved in plate and shell theories, as well as finite elements for plates and shells, is that the thickness of the plate or shell that should be small enough in comparison with other dimensions of the plate or shell. This allows the truncation of (1) for the displacement field to be represented in (2), such that the Reissner–Mindlin hypothesis remains valid. With greater thickness beyond the applicability of Reissner–Mindlin hypothesis, more advanced shell theories, such as that in [13], may be resorted to. However, it is not common in FE approaches, especially for large deformation problems. FE users are more inclined to use 3-D elements instead nowadays, given the ever increasing computing power. Incidentally, once 3-D elements are employed, there will be no need to worry about the magnitude of rotations involved in the deformation anymore as all the information about the deformation have been encapsulated in the Green strains. This is certainly another indication of the irrelevance of rigid body rotations to the deformation kinematics of deformable bodies.
5
Nodal degrees of freedom φi and ψ i are best to be left as they are as gradients of in-plane displacements with respect to the coordinate in the thickness direction, all expressed in the local coordinate system (x, y, z). If they had to be transformed to a different coordinate systems for whatever reason, they should be transformed as displacement gradients in the form of (13). This would require the complete displacement gradient matrix to be obtained in the local coordinate system first. One ought to be aware that such displacement gradient matrix at a node may be evaluated to different values from different elements involving this node. A certain treatment would be required, as in the case when stresses at nodes are evaluated in finite element analysis, where a typical way is to take average from all elements involving the node concerned. One example where such transformation may become necessary is for nodes along a ridge in a folded or boxed plate or shell. In this case further approximations are inevitable [6], whatever theory is used, unless one is prepared to go back to a 3D model. This should be addressed as a separate issue without distracting the focus of this paper. Given the above discussion, most existing plate and shell theories for finite deformations should give a correct description of the behaviour of the plate or shell after ignoring the complications illogically introduced by mistakenly perceived finite ‘rotations’, provided that the theory is justifiable for the given thickness in relation to the other dimensions of the plate or shell. The elaboration as above in this section based on a 3-D degenerate isoparametric finite shell element can also serve as a roadmap to develop the formulation fully such that numerical results can be obtained to illustrate the suitability of the theoretical framework proposed here after the confusions on the subject of finite deformation of plates and shells have been clarified as in this paper. However, such development will have to be pursued as a future development without distracting the main focus of this paper.
4. Conclusions The message out of the elaboration and discussion in this paper, based purely on the very basics of plate and shell theories, can be summarised in Table 1 below The paper can be concluded with the following remarks. 1) Rigid body rotations are far-fetched into the formulation of plate and shell theories. The real identity of the so-called ‘rotations’ is displacement gradients. Although the equivalence in between can be established under infinitesimal deformations, mistaking displacement gradients by rigid body rotations under finite deformations forms a logic fallacy of faulty generalisation. While it is conceptually wrong and logically defective, it also results in a whole range of complications which are completely unnecessary if their true identity has been incorporated in the development of plate and shell theories for finite deformations, although the complicated and unnecessary approaches have dominated the stage of the finite deformation theories of plates and shells, hitherto.
Table 1 The relationship between displacement gradients and rotations in the formulation of the theory of plates and shells. Formulation of the theory of plates and shells
Required
Actually employed
Relationship between required and employed
Correctness
Consequence
Infinitesimal ‘rotation’ problem Existing finite ‘rotation’ problem Finite displacement gradient problem as proposed in this paper
Displacement gradients Displacement gradients Displacement gradients
Rotations Rotations Displacement gradients
Equivalent Not equivalent Identical
Yes No Yes
Easy to visualise Complicated treatments Direct treatment without undue complications
Please cite this article as: Li S. Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics. Int. J. Mech. Sci. (2014), http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004i
S. Li / International Journal of Mechanical Sciences ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
2) The validity of the theoretical framework as proposed in this paper, including the Reissner–Mindlin hypothesis, the use of displacement gradients to construct displacement field in plates and shells, the finite element presentation and the application to finite deformations are controlled by the thickness of the plate or shell, not the magnitude of ‘rotations’ (displacement gradients). Traditionally, finite ‘rotation’ problems attract two sources of complications. One is ‘finite rotations’ due to their non-vectorial nature which has cast a huge implication on the representation of these rotations and their incorporation in the construction of the displacement field in plates and shells. This has been shown in this paper as a fabricated and unnecessary problem out of a logic fallacy and thus can be ruled out completely. The second source is the finite deformation due to the second order terms in the Green strains which involve displacement gradients but nothing else. This is controlled by the magnitude of deformation alone without the complications associated with the first source. 3) In finite elements for plates and shells, the nodal displacement gradient degrees of freedom are best kept in their local coordinate system as originally introduced in the formulation of plate and shell element. If they have to be transformed into a different coordinate system for whatever reason, they should be transformed as components of the second rank displacement gradient tensor, not any form of vectors.
Appendix The following proverb was quoted in Argyris [2]. 踏破铁鞋无觅处, 得来全不费功夫. It is translated as follows. One has worn his/her metal shoes in searching for, often without joy. However, when it eventually turns up, it is obtained
rather effortless. The proverb emphasises the fact of having obtained the object and the easiness in obtaining the object.
References [1] Abaqus. Abaqus 6.13 Theory Guide; 2013. [2] Argyris J. An excursion into large rotations. Comput Methods Mech Eng 1982;32:85–155. [3] Arnold RN, Maunder L. Gyrodynamics and its engineering applications. London: Academic Press; 1961. [4] Cai YC, Atluri SN. Large Rotation analyses of plate/shell structures based on the primal variational principle and a fully nonlinear theory in the updated lagrangian co-rotational reference frame. Comput Model Eng Sci 2012;83: 249–273. [5] Crisfield MA. Nonlinear finite element analysis of solids and structures, 2. Chichester: Wiley; 1997. [6] Cook RD, Malkus DS, Plesha ME. Concepts and applications of finite element analysis. 3rd ed. London: Wiley; 1989. [7] Hoffmann B. About vectors. Englewood Cliff, New Jersey: Prentice-Hall; 1966. [8] Ibrahimbegovic A. Stress resultant geometrically exact shell theory for finite rotations and its finite element implementation. ASME Appl Mech Rev 1997;50:199–225. [9] Kleppner D, Kolenkow RJ. An introduction to mechanics. London: McGrawHill; 1978. [10] Kraus H. Thin Elastic Shells. New York: Wiley; 1967. [11] Li S. On the vectorial nature of the rotational degrees of freedom in spatial beam elements. Comput Struct 1996;59:1199–200. [12] Libai A, Simmonds JG. The nonlinear theory of elastic shells. 2nd ed. Cambridge: Cambridge University Press; 1998. [13] Naghdi PM. On the theory of thin elastic shells. Q Appl Math 1957;14:369–80. [14] Novozhilov VV. Theory of elasticity. Oxford: Pergamon; 1961. [15] Parisch H. Geometrical nonlinear analysis of shells. Comput Methods Appl Mech Eng 1978;14:159–78. [16] Pietraszkiewicz W. Finite rotations in shells. In: Koiter WT, Mikhailov GK, editors. Theory of shells. Amsterdam: North-Holland Publishing Company; 1980. p. 445–71. [17] Reissner E. Stress strain relations in the theory of thin elastic shells. J Math Phys 1952;31:109–19. [18] Zienkiewicz OC, Taylor RL. The finite element method. 4th ed. . London: McGraw-Hill; 1998.
Please cite this article as: Li S. Reflection on ‘finite rotation problem’ in plate and shell theories and in finite element formulation – Back to basics. Int. J. Mech. Sci. (2014), http://dx.doi.org/10.1016/j.ijmecsci.2014.07.004i