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## Über dieses Buch

The deformation near a material particle of the classical continuum is produced by successive superposition of a rigid-body translation, a pure stretch along principal directions of strain and a rigid-body ro­ tation of those directions. The rotational part of deformation is par­ ticularly important in the non-linear analysis of thin-walled solid structures such as ~eams, thin-walled bars, plates and shells, since in this case finite rotations may appear even if the strains are infinite­ simal. It seems that the research concerning the application of finite ro­ tations is carried out independently in different fields of structural mechanics. Theoretical and numerical methods developed and the results obtained for a particular type of the structure or for a particular ma­ terial behaviour not always are used to analyse similar problems for other types of structures or for another material behaviour. Since the research in this field had been growing rapidly, it was decided to organize an informal international meeting, under the auspi­ ces of the European Mechanics Co~mittee, entitled: Euromech Colloquium 197 "Finite Rotations in Structural Mechanics". The meeting was held on 17 - 20 September 1985 in Jablonna, a small suburbian area of Warsaw.

## Inhaltsverzeichnis

### Intrinsic Shell-Theory Formulation Effective for Large Rotations and an Application

Abstract
Specialization of the general thin-shell theory for the class of large deformations realizable by small strain is considered. This leads in three different ways to the intrinsic nonlinear theory of flexible shells. As an illustration, large displacements and rotations culminating in collapse of finite-length tubes are investigated.
E. L. Axelrad, F. A. Emmerling

### On Geometrically Non-Linear Theory of Elastic Shells Derived from Pseudo-Cosserat Continuum with Constrained Micro-Rotations

Abstract
The set of equations for the geometrically non-linear theory of thin elastic shells is usually expressed in terms of displacements as basic independent variables of the shell deformation. Various general and reduced displacemental forms of bending shell equations are summarized, for example, by MUSHTARI and GALIMOV [1], KOITER [2], PIETRASZKIEWICZ [3, 4], SCHMIDT [5] and BA§AR and KRÄTZIG [6], where further references may be found. When displacement field is determined from the shell equations, strains, rotations and stresses may be obtained by prescribed algebraic or differential procedures.

### Some Mathematical Results Related to Nonlinear Thin Shell Problems

Abstract
There exist many different nonlinear models of thin shell problems depending of the geometry of the shell, of the magnitude of the displacement or of the strains. An account of these models can be found in KOITER [15]. More recently it appears to be meaningful to decompose the motion of a shell into translation, rigid-body rotation and strain. Such decomposition is particularly interesting for shells whose deformation leads to large rotations but small strains. In this way, PIETRASZKIEWICZ [19], PIETRASZKIEWICZ and SZWABOWICZ [20] derived several models according to the magnitude of the rigid-body rotations, with and without restrictions on the magnitude of the strains.

### Postcritical Deformations of Meridional Cross-Section of Elastic Toroidal Shells Subject to External Pressure

Abstract
A thin-walled, rotationally symmetric, elastic toroidal shell subject to external pressure and in-plane bending is considered. Geometrically nonlinear theory of finite displacements and rotations is applied. Both, symmetric as well as nonsymmetric forms of equilibrium of meridional cross-section are taken into account. The method of detecting of critical loadings is proposed and the analysis of post critical forms is performed. The influence of geometrical imperfections (jump-like thickness variation) on the equilibrium forms is discussed.
Jan Bielski

### The Complementary Energy Principle in Finite Elastostatics as a Dual Problem

Abstract
A revival of interest in a formulation of the principle of complementary energy for finitely deformed hyperelastic bodies should be attributed to LEVINSON [44] . Concerning earlier investigations the reader may refer to Refs [26,49,58,61,64]. A critical review of existing formulations is out of scope of the present paper. Here we shall only comment on various approaches to the problem.
W. R. Bielski, J. J. Telega

### Finite Rotations and Complementary Extremum Principles

Abstract
The equations of nonlinear elasticity can be written in terms of the following conjugate quantities (BUFLER [ 1 ] ) : (a) Deformation (or displacement) gradients and 1.Piola-Kirchhoff stresses, (b) Green’s deformations (or strains) and 2.Piola-Kirch-hoff stresses, (c) stretches (or extensions) and Biot stresses. In connection with the complementary energy theorem in nonlinear elasticity frequently the first group of variables is used, see f.i. KOITER [2]. In this case (a), however, the “stress and strain quantities” (deformation gradients) are not objective and the inversion of the necessarily nonlinear constitutive equations generally yields severe difficulties (OGDEN [3]) which do not appear if restriction is made to moderate rotations (STUMPF [4]). In the formulations (b) and (c), on the other hand, objective quantities are involved and a linear material behaviour (Kirchhoff and semilinear material respectively) can be used. Formulation (c) seems to be advantageous for two further reasons: Firstly the stretches (or extensions) and the conjugate stresses (called engineering stresses by F. DE VEUBEKE [5]) are to be considered as “natural quantities” [3] — indeed these ones are frequently used for large deformation problems of plates, membranes and shells, and the assumption of a convex strain energy density with respect to the extensions is physically meaningful — and secondly the rotations are taken into account explicitely. Furthermore the associated most general variational principle does allow independent variations of the displacements, stretches, stresses, reactive forces and rotations the corresponding Euler equations representing the force equilibrium (including the statical boundary conditions and the reactive forces), the constitutive equations, the kinematical field equations, the kinematical boundary conditions and the moment equilibrium, see REISSNER [6] and BUFLER [7] [8].
H. Bufler

### Deformation of the Shell Boundary

Abstract
Within the linear theory of shells, several problems associated with an infinitesimal deformation of the shell boundary were discussed by the author in [1,2]. In this paper the results are extended into the general case of large strains and rotations in thin shells.
K. F. Chernykh

### Comparison of Numerical Results for Nonlinear Finite Element Analysis of Beams and Shells Based on 2-D Elasticity Theory and on Novel Finite Rotation Theories for Thin Structures

Abstract
In recent years a novel approach to the derivation of geometrically nonlinear Kirchhoff-Love type theories for thin elastic shells was initiated by PIETRASZKIEWICZ [10–13]. Based on the polar decomposition theorem he developed an exact theory of finite rotations in thin-walled structures. In [10–13] the rotational part of the shell deformation was described by a finite rotation vector, which in turn was expressed in terms of displacements of the shell middle surface and their gradients. This provides a sound basis for the derivation of constrained kinematic relations for thin elastic shells undergoing small strains ( of 0(η), η«l) accompanied by small (0(η)), moderate (0(η1/2)), large (0(η1/4)) or unrestricted rotations. Based on this approach a complete family of geometrically nonlinear Kirchhoff-Love type shell theories has been systematically derived in the works of PIETRASZKIEWICZ [11,13–18] and SCHMIDT [22–29]. In [22–29] special interest was focussed on the derivation of fully variationally consistent theories and associated energy principles. The aforementioned hierarchical set of shell equations consists of theories for unrestricted rotations, large rotations (with variants for large rotations of the normal only accompanied by moderate or small in-surface rotations) or moderate rotations. A shell theory for unrestricted rotations has been also developed by IURA and HIRASHIMA [4], while a variant for large rotations of the normal accompanied by small in-surface rotations has been also given by NOLTE and STUMPF [9]. STEIN [30]derived geometrically nonlinear theories for beams undergoing moderate or large rotations, respectively, while several other ones obtained by specializing the aforenamed shell theories for the one-dimensional case have been given recently by NOLTE [8].
J. Chróścielewski, R. Schmidt

### Fundamental Equations and Extremum Principles in the Theory of Thin Shells

Abstract
The present paper contributes to the non-linear theory of thin elastic shells in the frame of small strains but finite displacements and rotations (see [2], [3]). In describing the kinematic of deformations as well as in deducing the equilibrium conditions from the conservation laws of continuum mechanics the approximation is consequently applied that strains are small in every material point of the shell. The classification of occuring rotations as e.g. in [4] or restrictions in magnitude are not necessary.
R. De Boer, W. Walther

### Inhomogeneity and Rotation

Abstract
An elastic body B is said to be materially uniform [1, 2] if all its points are made of the same elastic material. Formally, there exists a smooth distribution of local configurations, i.e. a smooth distribution of linear maps of the form
$${\rm{P}}({\rm{X}}):{\rm{V}} \to {{\rm{T}}_{\rm{X}}}{\rm{B}},$$
such that, when referred to those local configurations, the elastic response in terms of the stress tensor
$${\rm{T}} = {\rm{T}}({\rm{F}},{\rm{X}}),$$
is identical for all points X of the body B. In other words, the following condition holds for all pairs of points X, Y of B :
$${\rm{T}}({\rm{F}}{{\rm{P}}^{ - 1}}({\rm{X}}),{\rm{X}}) = {\rm{T}}({\rm{F}}{{\rm{P}}^{ - 1}}({\rm{Y}}),{\rm{Y}}).$$
In the above formulae V is the standard Euclidean translation space, TXB is the tangent space of the body manifold B at the point X, and F denotes an arbitrary deformation gradient. The composition
$${{\rm{P}}_{\rm{X}}}({\rm{Y}}) = {\rm{P}}({\rm{Y}})\,{{\rm{P}}^{ - 1}}({\rm{X}}):{{\rm{T}}_{\rm{X}}}{\rm{B}} \to {{\rm{T}}_{\rm{Y}}}{\rm{B}},$$
called a material isomorphism between the points X and Y, represents a deformation of a neighbourhood of X which will render it, as far as the mechanical response is concerned, indistinguishable from a neighbourhood of Y. Obviously,
$${\rm{T}}({\rm{F}},{\rm{Y}}) = {\rm{T}}({\rm{F}}{{\rm{P}}_{\rm{X}}}({\rm{Y}}),{\rm{X}}).$$
Marcelo Epstein

### On a General Theory of Large Rotations and Small Strain with Application to Three-Dimensional Beam Structures

Abstract
For economical calculations of slender three-dimensional (3D) beam structures it becomes increasingly necessary to apply a more accurate method than the so-called 2nd order theory in structural engineering. To this kind of structures belong f.e. tower cranes, dredgers, masts and antennas. Because of the application of high-tensile steel (with yield stresses 800 — 1000 N/mm2) very slender constructions may undergo large displacements and rotations without coming close to the yield limit at any point. Therefore the strain will also be limited by the order of 10-3 .
Th. Hinkelmann, G. Lumpe, H. Rothert

### Finite Displacement Theory of Naturally Curved and Twisted Beams with Finite Rotations

Abstract
A finite rotation is not a quantity in a vector space. Various approaches, therefore, exist for evaluating the finite rotation. Euler angles [1–4], finite rotation tensors [5] and finite rotation vectors [6–8] have been employed as a measure for finite rotations. The exact finite displacement functions of beams have been derived in terms of the three translation and three rotation parameters. It is widely accepted, however, that four parameters are necessary and sufficient for formulating a beam theory under the Bernoulli-Euler hypotheses. When the number of parameters decreases from six to four, some approximations are introduced in the most of existing literature. Consequently, the most of existing equations are available only for the analyis of geometrically nonlinear behavior of beams with moderate rotations.
M. Iura

### Higher-Order Moderate Rotation Theories for Elastic Anisotropic Plates

Abstract
A great deal of interest in the substantiation of refined theories of elastic an-isotropic plates and shells has been manifested in the specialized literature in the last two decades. This interest is largely due to the need for more adequate methods of analysis of structural elements exposed to severe and complex operational conditions in various branches of the advanced technology. In addition, the increased use of new exotic composite materials has provided a new impetus for such refined theories. As it was conclusively shown, the classical methods of analysis based on the Kirchhoff-Love assumptions are inadequate in many important instances. This is especially true whenever the material of the structure exhibits high degrees of anisotropy in its physical and mechanical properties. Such features are typical for the composite and refractory type materials used with increased frequency in the aerospace, naval, nuclear industries, etc. In such cases, refined models allowing a more adequate description of the structural response are needed. They should include transverse shear and transverse normal deformations and should account for the higher-order effects.
Liviu Librescu, Rüdiger Schmidt

### Finite Strains and Rotations in Shells

Abstract
In this article we present an exact and systematic derivation of a theory of finite strain deformation of shells from the principles of classical continuum mechanics. We assume that the three-dimensional deformation of the shell satisfies the following kinematical constraints; I) material fibres initially normal to the reference surface of the shell remain straight during the deformation, II) deformation is isochoric (volume preserving). These are the only assumptions made in our developments out which the first one is of simplifying nature while the other one reflects merely the real property of many materials. We show that the resulting theory is characterized by the following features; a) a rational incorporation of transverse shear deformation and exact incorporation of transverse normal deformation, b) the constitutive equations for the stress resultants and stress couple as nonlinear functions of appropriate strain measures and their surface derivatives, c) a sufficient geometric structure to account for a non-uniform change in the shell thickness at the boundary.
J. Makowski, H. Stumpf

### Theory of Thin-Walled Elastic Beams with Finite Displacements

Abstract
The well known linear theory of thin-walled elastic beams of open cross-section (Vlasov, 1940) is a useful tool which can be used to treat a wide range of problems involving torsional-flexural interaction in thin-walled beams. However, several problems of practical importance (such as the postbuckling behaviour of thin-walled beams) involve the effects of finite displacements and are therefore outside the range of a purely linear theory. It follows that there is a need for a nonlinear theory of thin-walled beams capable of accounting for the effects of finite displacements. In recent years, several attempts have been made to develop such a nonlinear theory (see e.g. [1], [2], [3], [4], [9], [10]). It will be recalled that the linear theory of thin-walled beams of open cross-section is based on the following fundamental geometrical assumptions (Vlasov’s constraints):
1)
The shearing strains in the middle surface of the beam are negligible.

2)
Any cross-section of the beam is not deformable in its own plane.

In the following, a consistent nonlinear theory of thin-walled elastic beams of open cross-section is presented. The theory is based on Vlasov ’s constraints and is valid for large displacements and rotations, but the strains are assumed to be small throughout the beam. A detailed derivation of the theory is given in [7].
H. Møllmann

### One-Dimensional Finite Rotation Shell Problems in Displacement Formulation

Abstract
The present report deals with certain consequences for the displacement formulation of nonlinear first approximation shell theories if shell problems are concerned in which the partial differential equations reduce to ordinary ones. In these cases a necessary geometrical constraint is that the reference middle surface admit one-dimensional strain fields. Accordingly the shell strains, depending on differences of the metric and curvature tensors in the deformed and undeformed configuration are functions of one independent variable only. It has been shown by Simmonds [1] that then during the deformation process the shell middle surface must be a general helicoid, which additionally implies special boundary conditions, material properties and type of loadings. There are several reasons for the analysis of one-dimensional reduced shell problems. By that general nonlinear shell equations remain rather complicated and require approximate solutions by using large computer codes such that the associated one-dimensional equations may considerably reduce the cost and/or time of the nonlinear solution. Besides, the derivation of general and simplified one-dimensional shell theories gains a good insight into similar investigations of the general theory of shells.
L.-P. Nolte

### On the Derivation and Efficient Computation of Large Rotation Shell Models

Abstract
Within the past few years remarkable progress has been achieved on the nonlinear analysis of so termed flexible shells, mainly influenced by the fundamental results of Reissner, John and Koiter [1–4]. A novel approach to the displacement formulation of general and constrained geometrically nonlinear shell equations has been given in recent years by Pietraszkiewicz [5–7]. In particular, based on polar decomposition of shell strains and rotations it was suggested to classify small strain shell models according to the magnitude of the rotation angle of the material elements. It is known from the literature that most of the engineering shell problems may be tackled accurately enough with the help of moderate rotation shell theories [5–12]. Numerical applications of corresponding shell equations may be found e.g. in [12–14], There are, however, various one- and two-dimensional shell problems to which large rotation shell models with nonlinear membran and bending strains should be applied. In these situations the rotational part of deformation dominates. Several shell theories with nonlinear change of curvature expressions have been already given (see [15–17] and literature cited therein). It turned out, however, that most of the corresponding field equations are very complicated and virtually useless for numerical applications. On the other hand we have shown recently [16] that various approaches used in engineering practice may lead to inconsistent shell equations.
L.-P. Nolte

### Rotations as Primary Unknowns in the Nonlinear Theory of Shells and Corresponding Finite Element Models

Abstract
A consistent geometrically nonlinear theory of shells is derived based on the formulation of a generalized variational principle. In addition to displacements and stresses, the components of a finite rotation vector are introduced as primary unknowns. This is accomplished by applying the polar decomposition of the deformation gradient. The basic equations are described in an incremental Lagrangian frame and are given in clear operator form to emphasize the additional equations of the angular momentum balance and the rigid body kinematics as a special feature of the present theory. Mixed variant tensorial components render advantages in the reduction to the shell equations in which shear effects are included. The two-dimensional principle serves then as the adequate basis for the formulation of different mixed-type finite element models. Numerical results are presented.
L. Recke, W. Wunderlich

### Polar Decomposition and Finite Rotation Vector in First — Order Finite Elastic Strain Shell Theory

Abstract
While large strain membrane theories are well established in literature since more than three decades, there exist considerably less papers which deal with large elastic strain shell theory incorporating also the bending effects into the nonlinear analysis. Recently, however, this topic has gained considerable interest. Important contributions have been given by CHERNYKH [1,2] , LIBAI and SIMMONDS [5,6] , and SIMMONDS [20] , where also additional references on related works may be found. The aforementioned authors agree that such a theory should be based on a refined Kirchhoff-Love type model which admits at least changes in shell thickness. Due to bending this thickness change is in general asymmetric about the undeformed midsurface so that its deformed configuration is no longer the geometrical midsurface of the deformed shell. This requires a representation of the position vector of the deformed shell space which incorporates at least quadratic terms with respect to the thickness coordinate.
R. Schmidt

### Nonlinear Models of Deformed Thin Bodies with Separation of the Finite Rotation Field

Abstract
In scientific literature there are many works concerning the formulation of the two-dimensional shell and one-dimensional rod deformation models on the basis of directed material surface and curve conception [1–4]. Such formalism is mathematically strong. However this is not satisfactory from the physical point of view because it breaks the natural connections with spatial formulation. As a result the problem of the construction of reological equations and of the reconstruction of the displacement, strain and stress spatial fields has been outside the scope of such formal approach.
L. I. Shkutin

### Ultimate Load Analysis of thin Walled Steel Structures with Elastoplastic Deformation Properties Using Fem — Theoretical, Algorithmic and Numerical Investigations —

Abstract
For the prediction of static ultimate loads of thin walled stiffened structures it is necessary to take into account geometrical and material non-linearities. An adequate method for the calculation of different kinds of structures, e.g. bridges, vessels, tall buildings and masts, is the finite element method (FEM). The geometrical non-linearity is treated by using an updated Lagrangian incremental formulation [1] assuming small strains but moderately large displacements and rotations in the increments [2].
E. Stein, K. H. Lambertz, L. Plank

### Compatibility of Rotations with the Change-of-Metric Measures in a Deformation of a Material Surface

Abstract
Considering a deformation of a thin shell, modelled by a flexible material surface, one is confronted with diversity of shapes that can be attained by stretching and bending. Yet, not all conceivable shapes are accessible in reality. There is a reasonable limit to this richness, be it due to the fact that any material can withstand stretching only up to a certain point and then falls. Once the stretches are restricted, some deeper reaching consequences of this may be detected. For instance, a flat plate cannot be deformed into a strongly curved cap without generating large stretches. Apparently, the nature of this restraint involves something more than just mere restriction on the elongation of the material fibres of the shell. Curvature must be liable to some restrictions of a similar type and, consequently, so must be the rotations.
M. L. Szwabowicz

### Finite Rotations, Variational Principles and Buckling in Shell Theory

Summary
A non linear principle including stresses and finite rotations using a direct surface polar decomposition, is presented with some variants, following the three-dimensional approach of FRAEIJS DE VEUBEKE.
A mixed stability criterion which has to be applied for mixed functionals is also derived. Its applications holds for the principle variants.
R. Valid

### Numerical Analysis of Thin-Walled Structure Finite Displacements

Abstract
The behavior of the structure thin-walled elements upon finite displacements is described in terms of the boundary value problem for non-linear differential equations. The dependence of the mechanical system state vector $$\mathop {\rm{X}}\limits^ \to$$ on external effect vector $$\mathop {\rm{Q}}\limits^ \to$$ may be very complicated. Thus Fig.I shows the relation between deflection Wo in the vertex of the gently sloping spherical shell and load parameter q [1]. To obtain the solution of such problems numerical methods are usually used.
N. V. Valishvili, A. K. Tvalchrelidze

### Elastic-Plastic Structures under Variable Loads at Small Strains and Moderate Rotations

Abstract
The long-time behaviour of elastic-plastic structures under variable loads is investigated. In particular, shakedown-conditions are derived for shell-like structures undergoing moderate rotations at small strains.
D. Weichert

### Finite Rotations in the Approximation of Shells

Abstract
In the spirit of this colloquium, our attention is focused upon finite rotations in structures and, specifically, their role in the approximation of thin shells. Here, the theory of shells is recast; the motion is decomposed into strains and rotations with no restrictions on their magnitudes. With a view toward the further approximation via finite elements, the general theory is couched in alternative forms: A potential admits variations of displacements whereas a complementary functional admits variations of displacements, stresses and strains. To admit very simple approximations, transverse-shear deformations are included. Interelement continuity is then preserved even when kinks occur in the surface.
Gerald Wempner

### Finite Rotations of Linear Elastic Bodies

Abstract
The purpose of the paper is to derive from the nonlinear elastodynamics the governing relations for problems in which small strains and finite rotations and/or finite displacements are involved. Such problems arise, for example, when a motion of a linear elastic body is restricted by a system of obstacles. The main result is a proof of the two following facts:
1∘
The dynamics of a linear elastic body subject to finite motion can be described by the equations of the rigid body dynamics coupled with the equations of the linear elastodynamics,

2∘
The aforementioned coupling is due to the interaction between the body and the obstacles.

Cz. Woźniak

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