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An extension of Boley’s continuum mechanics-based successive approximation method is presented for rectangular beams composed of two isotropic linear elastic layers. The solution is cast into the form of tables, in complete analogy to the tables originally presented by Boley and Tolins for single-layer strips. The first column in these tables corresponds to the classical Bernoulli–Euler theory of beams. The further columns represent comparatively fast converging correction terms of an increasing refinement. Our two-layer formulation automatically satisfies the stress continuity conditions at the interface of the two layers. Enforcing displacement continuity at the interface between the layers, we derive results that do satisfy the equilibrium field equations, the stress continuity conditions at the interface and the stress boundary conditions at the upper and lower edges. When converged, the field constitutive relations and the displacement continuity at the interface between the two layers are also satisfied. We present a compact formulation, which allows writing down the results for more than the three successive steps considered by Boley and Tolins. The elasticity solutions presented subsequently can be used as novel analytic benchmarks for comparison with refined structural mechanics beam theories. Interior solutions for beams with a finite axial extent can be obtained by assigning approximate boundary conditions at the lateral ends. Comparisons to finite element computations for a clamped–clamped beam give strong evidence for the correctness of our analytic results.

Automatic panel benders of Salvagnini Maschinenbau GmbH enable high-quality and high-efficient production of sheet metal components. To achieve the steadily increasing requirements on precision, a model-based adaptive concept has been developed controlling the complete production process as a digital twin, from CAD data to the final sheet metal part. First, an overview of the underlying simulation models with different levels of detail is given. The models consider the elastoplastic deformation of the metal sheets as well as the elastic machine components and mechatronic models of the powertrain. Secondly, an overview is given of the adaptive production concept allowing the real-time adaption of the machine to changing process parameters like material properties. Finally, all production steps are controlled by a digital twin based on this model-based adaptive strategy. The paper demonstrates the successful transfer of scientific results to an industrial application.

The present paper is concerned with bending of extensible and shear-deformable rods, where we treat the case of plane deformations. Our emphasis is laid upon infinitesimal deformations superimposed upon finite deformations. In this framework, we focus on deriving generalized reciprocity theorems that must hold with respect to the superimposed infinitesimal deformations, when produced by different types of loadings. Besides imposed distributed and concentrated force and moment loadings, we consider the action of eigenstrains. Moreover, we also take into account the presence of jumps in kinematic entities, particularly concerning jumps of axial rod positions and of cross-sectional rotations. We first derive local and global universal reciprocity relations, which must hold for rods made of any material. The global universal relation then is specialized to the case of hyperelastic rods. From the latter relation, we prove that reciprocity theorems known from the linear theory, such as the theorems of Betti, Maxwell and Land, as well as Maysel’s formula, formally do hold for infinitesimal deformations superimposed upon finite deformations. As a formulation that appears to be novel also with respect to the purely linear theory, we present a combination of Maysel’s formula with the theory of Land, the former dealing with eigenstrains, the latter involving jumps in kinematic entities.

The search for efficient explicit time integration schemes is a relevant topic in the current literature on dynamic mechanical systems. In this paper, we describe a strategy of utilizing the balance relations of mechanics in their integral form, so-called general laws of balance, where the time-evolution of the integrands is approximated by established computational techniques of the discrete-mechanics-type. In a Picard-type iteration, the outcomes are used for repeating the procedure several times, leading to an increased accuracy. The advantages of the present explicit approach are discussed in the context of linear and nonlinear motions of the mathematical pendulum. We utilize the modern symbolic procedures to obtain the time integration formulae and compare the results of our methods with exact solutions and with the results of higher-order implicit methods and also with a recent explicit formulation from the literature. (VLID)4848919 Submitted version

The contour motion of the belt drive, i.e., the motion with the constant trajectory, is addressed. The belt is considered as a closed Cosserat line whose particles have translational and rotational degrees of freedom. The problem is considered in the framework of geometrically nonlinear formulation with no restrictions on the smallness of displacements and rotations. The spatial (Eulerian) coordinate which is the arc coordinate in the actual configuration is introduced. The belt is divided into four segments: two contact segments on the pulleys and two free spans. The friction forces are assumed to obey the Coulomb law. The study is limited to the stationary case with the constant angular velocities of the pulleys and the equations in components are derived for both contact and free spans. In the contact segment two assumptions are employed to eliminate the unknown contact pressure and friction: (1) the full contact, i.e., coincidence between the pulley and the belt and (2) the stick condition, i.e., the belt velocity is related to the pulley angular velocity. A nondimensional coordinate is introduced in the segments to obtain the boundary value problem with fixed boundaries. The boundary coordinates of the contact zones are the integration constants of the derived problem along with the other constants. (VLID)4851961

Nachfolgend werden einige wissenschaftliche Aspekte der Echtzeit-Materialerkennungsstrategie MAC diskutiert, die in der neuen Maschinenfamilie LEAN PANEL BENDER der Firma Salvagnini Maschinenbau realisiert ist und die eine hochprazise und hocheffiziente Herstellung von komplex geformten Blechprodukten sowohl bei Losgrose 1 als auch in der Serie erlaubt.

In the present paper, we consider nonlocal linear elastic Kirchhoff plates, where we restrict to thin, isotropic, and homogeneous plates under the action of static transverse forces, utilizing the differential equation form of the Eringen nonlocal continuum theory. For the case of simply supported plates of polygonal planform, we derive analogies between the solutions of the nonlocal Kirchhoff theory and its local counterpart. First, we extend the Marcus decomposition method for local Kirchhoff plates, where we show that, analogous to the local case considered by Marcus, the moment sum and the nonlocal deflection are both governed by Poisson boundary value problems, which correspond to auxiliary (local) membrane problems. In the present context, it eventually follows that there is a nonvanishing term responsible for a correction of the deflection due to the nonlocal effect, while the moments and shear forces of the local and the nonlocal plate coincide. The nonlocal correction of the deflection turns out to be governed by a membrane-type boundary value problem again. From this fact, it follows immediately that the correction for the deflection due to the nonlocal effect can be derived alternatively from both the local deflection and the local moment sum, which represents a substantial simplification of the nonlocal computations. For the sake of demonstration, we consider examples with closed form solutions. We first consider (as limiting case) infinite plates under the action of single forces, where the singular behavior of Green’s function of the nonlocal Kirchhoff theory can be clarified. Then we discuss the bending of nonlocal plate strips for comparison sake, as well as equilateral triangular plates. To our best knowledge, no results for bending of nonlocal triangular plates have been presented in the literature so far. The present paper has been strongly influenced by a former contribution on analogies between simply supported polygonal Kirchhoff plates rigid in shear and shear-deformable plate solutions. This paper was co-authored together with our teacher, the late Professor Franz Ziegler, to whom we dedicate the present paper. In light of this latter contribution, and extending an idea promoted by Professor Franz Ziegler and his co-workers in the framework of various other fields, we finally show in the Appendix that the deflections of simply supported nonlocal Kirchhoff plates can be considered as deflections of a local “background” Kirchhoff plate under the action of additional fictitious eigenstrain loadings, such as thermal moments.

Boley's method is utilized in order to show that the elementary Bernoulli–Euler beam theory can be enhanced such that exact solutions of the plane-stress theory of linear elasticity are obtained for force loaded rectangular beams. An equivalent enhancement is derived for the elementary Timoshenko theory of beams. The enhancement terms act analogous to thermal loadings; they follow from the force loading of the rectangular beam in an explicit form. The resulting boundary value problem of fourth order can be efficiently solved by means of symbolic computer codes. As an illustrative example, a redundant beam is studied, which is simply supported at one end, and which is clamped at the other end. Outcomes for three alternative clamped end boundary conditions are compared.

The setting of a looped drive belt on two equal pulleys is considered. The belt is modelled as a Cosserat rod, and a geometrically nonlinear model with account for tension and transverse shear is applied. The pulleys are considered as rigid bodies, and the belt–pulley contact is assumed to be frictionless. The problem has two axes of symmetry; therefore, the boundary value problem for the system of ordinary differential equations is formulated and solved for a quarter of the belt. The considered part consists of two segments, which are the free segment without the loading and the contact segment with the full frictionless contact. The introduction of a dimensionless material coordinate at both segments leads to a ninth-order system of ordinary differential equations. The boundary value problem for this system is solved numerically by the shooting method and finite difference method. As a result, the belt shape including the rotation angle, forces, moments, and the contact pressure are determined. The contact pressure increases near the end point of the contact area; however, no concentrated contact force occurs.

In the present paper, a formulation of Lagrange’s equations, written in the framework of the Lagrange (material) description of Continuum Mechanics, is provided for open systems. An open system is constituted by (the portion of) a continuous material body that is enclosed by a nonmaterial surface. Such a surface, through which a flow of mass takes place, is also denoted as a control surface, and the corresponding volume is called a control volume. In order to apply Lagrange’s equations of analytical mechanics, the motion and the deformation of the body is modeled in the framework of the Ritz approximation technique by means of a finite number of generalized coordinates. However, since mass may not be conserved in an open system due to the flow of mass through the control surface, the original form of Lagrange’s equations must be accomplished by proper flux terms to be considered at the control surface. In order to derive this extended form, a local version of Lagrange’s equations is derived first, using a proper mathematical manipulation of the local relation of balance of linear momentum written in the Lagrange description of Continuum Mechanics. This local form is integrated over the volume that instantaneously is enclosed by the image of the control surface in a properly chosen reference configuration. In the integrated form, the Truesdell–Toupin method of fictitious particles and generalized Reynolds transport theorems are utilized in order to exchange the integrals and the partial derivatives with respect to time and generalized coordinates and velocities. This yields the desired form of Lagrange’s equations for open systems, written in the Lagrange description of Continuum Mechanics. Illustrative examples demonstrate the consistence of this novel form with the Euler (spatial) version of Lagrange’s equations for open systems, which was derived earlier by the present authors.

In the present paper a concept of static and dynamic shape control for slender piezoelastic beams is addressed. The notion shape control means to find a weighting sequence for the piezoelectric control units, such that force-induced lateral deflections are completely annihilated along the beam axis. In our case this spatial control intensity for the piezoelectric transducers is realized with so-called resistive electrodes: by prescribing only the maximum and minimum voltage level, any desirable voltage distribution along the beam axis may be obtained. For the derivation of the shape control method an extended version of the Bernoulli–Euler beam theory is used, which unifies the kinematics of a slender piezoelectric beam and the Telegrapher’s equations for moderately conductive electrodes. The outcome is an actuator equation for the displacement, which is fourth order in space and second order in time, and a sensor equation, which is second order in space and first order in time (diffusion equation for the voltage). The developed shape control method is verified by several numerical examples in the static case as well as in the dynamic frequency regime for a clamped-free beam. We show that the presented method is perfectly suited for static loads, being able to cancel force-induced structural deformations along the beam axis, whereas the deflections in the higher frequency regime can be nearly canceled out. The method is efficient, as long as a non-dimensional constant, involving the resistance, the capacitance per unit length, the length of the electrodes and the excitation frequency, is small.

The drive belt set on two pulleys is considered as a nonlinear elastic rod deforming in plane. The modern equations of the nonlinear theory of rods are used. The static frictionless contact problem for the rod is derived. The nonlinear boundary value problems for the ordinary differential equations are solved by the finite differences method and by the shooting method by means of computer mathematics. The belt shape and the stresses are determined in the nonlinear formulation which delivers the contact reaction and the contact area. The developed method allows performing calculations for any set of geometrical and stiffness parameters. (VLID)4844522

A beam resting on spatially fixed supports may slide relatively to these as soon as external forces are applied. Consequently, the length of the portion of the reference configuration, which is currently located between the supports, depends on the loading and therefore is not known in advance. In the present paper, the problem of a slender beam under a uniformly distributed force is investigated, which is clamped at one side but may slide through another clamping device in axial direction at the opposite side. In combination with a suitable coordinate transformation by which the numerical treatment is simplified, a finite element approach is utilized to determine the equilibrium shape for the maximum critical load that can be imposed on the beam. In the course of this, the influence of the extensibility of the beam axis is studied. A theory based on Reissner’s geometrically exact relations for the plane deformation of beams is adapted such that it allows constitutive relations on stress–strain level to be integrated consistently. In addition to the classical equations of the extensible elastica, a constitutive model derived from the St. Venant–Kirchhoff material of non-linear continuum mechanics is studied. The results obtained in this survey are finally compared to those from the linear beam theory, which turns out to be incapable of describing the problem under consideration in a satisfactory manner.

This article deals with the non-linear modelling of beams that are bent, sheared and stretched by external forces and moments. In the following, we restrict to plane-deformations and static conditions. Our task is to present a continuum mechanics-based interpretation of the celebrated large displacement finite deformation structural mechanics theory, which was presented by Eric Reissner [On one-dimensional finite-strain beam theory: the plane problem, J. Appl. Math. Phys. 23 (1972), pp. 795–804]. The latter formulation was restricted to the notions of structural mechanics and thus did not use the notions of stress and strain, which are fundamental for continuum mechanics. Thus, the common continuum mechanics-based constitutive modelling at the stress–strain level cannot be utilized in connection with Reissner's original theory. Instead, Reissner suggested that constitutive relations between certain generalized strains (bending, shear and axial force strains) and generalized static entities (bending moment...

An extension of Mohr’s analogy to bending of shear-deformable beams with eigenstrain-type actuation, such as a piezoelectric actuation, is presented first. Various refined shear-deformable beam theories are included by means of a theory-dependent parameter. The one-dimensional version of Reissner’s sixth-order plate theory is exemplarily addressed. The Bernoulli–Euler theory of beams rigid in shear, as well as the shear-deformable theory of Timoshenko, are included as special cases. Afterwards, the extended Mohr analogy is applied to the bending of smart beams with piezoelectric patch actuators. The following special problem of static shape control is solved: Seek a placement of single patch actuators, such that the displacement and the cross-sectional rotation vanish at some pre-selected locations of the beam, despite the beam is loaded by external forces. Using Mohr’s analogy, it is shown that the auxiliary loading of the adjoint beam must form a self-equilibrated system of loading in order to achieve the latter goal. The high potential of the proposed actuator placement is demonstrated for the case of a cantilever beam with a single force acting at the tip. Placements of single actuators are represented such that the tip displacement and the tip cross-sectional rotation vanish. The outcomes of shear-deformable theories are compared to the Bernoulli–Euler theory and to a Finite Element computation using piezoelectrically coupled elements.

The mechanical response of complex elastic structures that are assembled of substructures is significantly influenced by joints such as bolted joints, spot-welded seams, adhesive-glued joints, and others. In this respect, computational techniques, which are based on the direct finite element method or on classical modal reduction procedures, unfortunately showan inefficient balance between computation time andaccuracy. In the present paper, a novel reduction method for the physical (nodal) joint interface degrees of freedom is presented, which we call joint interface modes. For the computation of the joint interface modes, Newton’s third law (principle of equivalence of forces) across the joint is explicitly accounted for the mode generation. This leads to a dimension of the generalized joint interface degrees of freedom in the reduced system, which is a factor of 2 or more smaller than in conventional reduction methods, which do not consider Newton’s third law. Two different approaches for the computation of the joint interface modes are presented. Numerical studies with bolted joints of different complexities are performed using a simple but representative constitutive joint model. It is demonstrated that the new joint-interface-mode formulation leads to both excellent accuracy and high computational efficiency.

In the present paper, we present a continuum mechanics based derivation of Reissner’s equations for large-displacements and finite-strains of beams, where we restrict ourselves to the case of plane deformations of originally straight Bernoulli–Euler beams. For the latter case of extensible elastica, we succeed in attaching a continuum mechanics meaning to the stress resultants and to all of the generalized strains, which were originally introduced by Reissner at the beam-theory level. Our derivations thus circumvent the problem of needing to determine constitutive relations between stress resultants and generalized strains by physical experiments. Instead, constitutive relations at the stress–strain level can be utilized. Subsequently, this is exemplarily shown for a linear relation between Biot stress and Biot strain, which leads to linear constitutive relations at the beam-theory level, and for a linear relation between the second Piola–Kirchhoff stress and the Green strain, which gives non-linear constitutive relations at the beam theory level. A simple inverse method for analytically constructing solutions of Reissner’s non-linear relations is shortly pointed out in Appendix I.

The present contribution intends to provide a rational framework for manipulating the general differential laws of balance and jump of continuum mechanics. At first, multiplications of the latter laws by proper physical fields are introduced, together with identical expansions. We then point out that the results of these mathematical manipulations, while being of the form of the general differential balance law again, are characterized by a possible lack of integrability with respect to volume production terms in the presence of a singular surface, the production terms being necessary for consistency. Lack of integrability is not to be expected when the general law is specified to the fundamental laws of balance of mass, momentum, total energy and entropy. When it must be supposed that integrability could be missing, however, the integral form of the general law of balance cannot be written down for a region containing a singular surface, and the so-called pillbox procedure fails for deriving the general relation of jump at the singular surface. The specific relations of balance and jump then have to be derived from proper relations that follow from consistent mathematical manipulations of the general differential law of balance and the general law of jump. In the present paper, these general relations are listed explicitly for balance of scalar and vectorial entities, multiplied and expanded by scalar, vector or tensor fields, respectively. Based on this rationale, various important formulations on configurational forces known from the literature on the mechanics in material space are recalled: The specific relation of balance of pseudomomentum, the configurational forces and the driving force at an interface are derived from the above general relations by involving the fundamental laws of balance and jump of linear momentum. The driving traction at an interface is then set into connection with the derived relations of balance and jump of kinetic, internal and free energy, respectively. The latter relations follow from the general relations by involving the fundamental laws of balance of linear momentum, total energy and entropy. As a result, we present relations between the driving traction at an interface and the surface production terms that appear in the various relations of jump, which should allow new interpretations of configurational formulations.

The present paper is concerned with an elastic body in a static intermediate configuration, the body being loaded by an additional distribution of transient eigenstrain. For the sake of convenience, we use the notion of actuation stress in the present paper in order to describe the action of eigenstrain. It is assumed that the incremental displacements produced by the additional actuation stress can be prescribed within the framework of the theory of infinitesimally small dynamic deformations superimposed upon a static strain. The intermediate configuration thus may represent a possibly large deformation from an undistorted state of the body. It is the scope of the present contribution to derive distributions of actuation stress, which do not produce any incremental displacements from the intermediate configuration. Solutions of this problem are not unique in general. In the present paper, we particularly seek for a displacement-free distribution of actuation stress that acts within a sub-region of the body only. We show that any statically admissible stress, which is produced in the sub-region by a possibly transient eigenstrain, and which satisfies appropriate dynamic boundary conditions, may be used as actuation stress in order that the incremental displacements vanish throughout the whole body. The body then remains in the static intermediate configuration for all times. Appropriate boundary-value problems are suggested for computing the statically admissible stress in the sub-domain. Other than elastic constitutive equations might be used for selecting a suitable boundary-value problem for the actuation stress. It is furthermore shown that the incremental first Piola-Kirchhoff stress of the elastic body is equal to the actuation stress inside the sub-region under consideration, and that it vanishes outside. As an example, we present successful Finite Element computations for the vibrations of a rectangular domain in a state of plane strain, where the rectangle is fixed at two adjacent edges, the sub-region also being taken as rectangular.

Dieser Beitrag beschreibt computergestutzte und messtechnische Untersuchungen, welche als eine wissenschaftliche Grundlage fur die Analyse und Entwicklung einer neuen Generation von Abkantautomaten zur Produktion von kompliziert geformten Profilen aus ebenen Blechteilen dienen. Es erfolgt einerseits eine Modellbildung unter Verwendung der Mehrkorperdynamik, um das Verhalten von starren Maschinenteilen zu modellieren, andererseits werden elastische Komponenten uber eine Diskretisierung mittels nichtlinearer Finite Elemente abgebildet. Fur die Beschreibung der Blechumformung werden Formulierungen der Plastizitatstheorie und der Kontaktmechanik herangezogen. Besonderer Wert wird auf die Verbindung dieser fortschrittlichen computergestutzten Methoden mit entsprechend konzipierten Messungen an gebauten Prototypen, sowie mit dem Expertenwissen gelegt. Methoden der Schwingungslehre und Festigkeitslehre werden interaktiv in den Untersuchungen und als Grundlage fur den Konstruktionsprozess angewendet. Im Rahmen von umfangreichen Messerien wurde eine ausgezeichnete Ubereinstimmung zwischen den Rechenergebnissen und den Versuchsergebnissen erzielt und daraus Verbesserungsvorschlage hinsichtlich des Entwurfs und des Betriebs abgeleitet. Das Studium von Detailproblemen im Rahmen von Modellverfeinerungen resultiert in neuen Erkenntnissen fur den Umformprozess.

Die computergestutzte Simulation des mechanischen Verhaltens von Maschinenteilen nimmt in der industriellen Entwicklung einen hohen Stellenwert ein. Mit der derzeit zur Verfugung stehenden Software ist jedoch die Simulation einer gesamten Maschine oft nur unter groben Vernachlassigungen moglich. Der vorliegende Beitrag beschaftigt sich mit der systematischen Gesamtsimulation einer Maschine, bei welcher einzelne Teile grose Bewegungen durchfuhren. In Erweiterung der vorliegenden Literatur werden grose Starrkorperbewegungen und elastische Verformungen mit hydraulischen Stellgliedern und einem entsprechenden Regler kombiniert.

Im vorliegenden Beitrag wird ein aussagekraftiges Stosdampfermodell vorgestellt, welches erlaubt, fahrdynamische und komfortrelevante Untersuchungen am Gesamtfahrzeug mit Hilfe der Mehrkorper-Software ADAMS wirklichkeitsnahe durchzufuhren. Das neue Modell ist so einfach wie moglich gehalten, basiert aber auf hydrodynamischen Betrachtungen; im Gegensatz zu herkommlichen Beschreibungen wird bei der neuen Formulierung aufgrund einer robusten und zutreffenden physikalischen Modellierung ein weiter Amplituden- und Frequenzbereich abgedeckt. Zur Parameteridentifikation und Verifikation des Modells wurden umfangreiche Messungen an fahrzeugtypischen Dampfern durchgefuhrt.

The present paper is concerned with the motion of an elastically supported beam that carries an elastic beam moving at constant speed. This problem provides a limiting case to the assumptions usually considered in the study of trains moving on rail tracks. In the literature, the train is commonly treated as a moving line-load with space-wise constant intensity, or as a system of moving rigid bodies supported by single springs and dampers. In extension, we study an elastically supported infinite beam, which is mounted by an elastic beam moving at a constant speed. Both beams are considered to have distributed stiffness and mass. The moving beam represents the train, while the elastically supported infinite beam models the railway track. The two beams are connected by an interface modeled as an additional continuous elastic foundation. Here, we follow a strategy by Stephen P. Timoshenko, who showed that a beam on discrete elastic supports could be modeled as a beam on a continuous elastic Winkler (one-parameter) foundation without suffering a substantial loss in accuracy. The celebrated Timoshenko theory of shear deformable beams with rotatory inertia is used to formulate the equations of motion of the two beams under consideration. The resulting system of ordinary differential equations and boundary conditions is solved by means of the powerful methods of symbolic computation. We present a nondimensional study on the influence of the train stiffness and the interface stiffness upon the pressure distribution between train and railway track. Considerable pressure concentrations are found to take place at the ends of the moving train.

Abstract: In this paper, flexural vibrations of smart circular plates are considered. Distributed actuators and sensors are realized by means of spatially shaped piezoelastic layers. We use piezoelectric actuating layers shaped in order to annihilate deflections due to known external transverse forces. Such spatial shape functions correspond to the distribution of the static bending moment in the form of the so-called Marcus moment of the plate due to the external forces. When only the spatial distribution of the external forces is known, but their time evolution may be arbitrary, an automatic control system must be used in order to minimize the plate vibrations. To utilize the concept of collocated sensing, a shaped piezoelectric sensor is required that measures the so-called natural output. It is shown that the above shape function of the actuator can be used as the shape function of the sensor in order to achieve this goal. Hence, the shaped piezoelectric layer can be used as a self-sensing actuator without violating the requirements of collocated control. We develop the corresponding transfer function for the case of a clamped circular plate with a space-wise constant transverse force. This transfer function is used for the design of a self-sensing PD controller. It is proven that the energy of the closed-loop system becomes a positive definite function, its time derivative being negative semi-definite, such that the PD-controlled plate is stable. In a numerical study, output and input signals of the closed loop are discussed. This study successfully demonstrates the ability of the proposed method.

When a singular surface is present in the region under consideration, the local forms of the equations of balance must be accompanied by jump relations in order to connect the different values of the discontinuous entities at the two sides of the singular surface. The classical general form of these jump relations is provided by the Kotchine relation. In the present contribution, we exemplary deal with balance of mass, momentum and kinetic energy, as well as with balance of total energy, denoted as the first law of thermodynamics, and with balance of internal energy. We first derive an extension of the Kotchine relation with respect to surface growth terms. By a surface growth term we mean an entity which equivalently describes the balance of some quantity associated with the material that is instantaneously located at or in the vicinity of the singular surface. In a more detailed modelling, a singular surface is often described by a thin shell-type region or layer of transition. Surface growth terms may represent a non-vanishing rate of change of the respective quantity, or they may characterise some sources of this quantity. As a main result of the present paper, we show that surface growth terms are needed in order to ensure consistency between the jump relations for balance of mass, momentum and kinetic energy, and between the jump relations for balance of total energy and balance of internal energy. Even when surface growth terms for mass, balance of momentum and total energy are absent, one generally must take into account surface growth terms for balance of kinetic energy and balance of internal energy. The presented results refer to both, a singular surface, as well as to a thin region of transition. The jump relations in the latter case are referred to an equivalent singular surface, replacing the region of transition. The uni-axial flow of a fluid in a diffusor is used to demonstrate that the present results may be used even in cases in which the region of transition is not thin.

A numerical strategy for vibrations of elasto-plastic beams with rigid-body degrees-of-freedom is presented. Beams vibrating in the small-strain regime are considered. Special emphasis is laid upon the development of plastic zones. An elasto-plastic beam performing plane rotatory motions about a fixed hinged end is used as example problem. Emphasis is laid upon the coupling between the vibrations and the rigid body rotation of the pendulum. Plastic strains are treated as eigenstrains acting in the elastic background structure. The formulation leads to a non-linear system of differential algebraic equations which is solved by means of the Runge–Kutta midpoint rule. A low dimension of this system is obtained by splitting the flexural vibrations into a quasi-static and a dynamic part. Plastic strains are computed by means of an iterative procedure tailored for the Runge–Kutta midpoint rule. The numerical results demonstrate the decay of the vibration amplitude due to plasticity and the development of plastic zones. The pendulum approaches a state of plastic shake-down after sufficient time.

The Lagrange equations are extended with respect to a non-material volume which instantaneously coincides with some material volume of a continuous body. The surface of the non-material volume is allowed to move at a velocity which is different from the velocity of the material surface. The non-material volume thus represents an arbitrarily moving control volume in the terminology of fluid mechanics. The extension of the Lagrange equations to a control volume is derived by using the method of fictitious particles. Within a continuum mechanics based framework, it is assumed that, the instantaneous positions of both, the original particles included in the material volume, and the fictitious particles included in the control volume, are given as function of their positions in the respective reference configurations, of a set of time-dependent generalized coordinates, and of time. The corresonding spatial formulations are also assumed to be available. Imagining that the fictitious particles do transport the density of kinetic energy of the original particles, the partial derivatives of the total kinetic energy included in the material volume with respect to generalized coordinates and velocities are related to the respective partial derivatives of the total kinetic energy contained in the control volume. Hence follow the Lagrange equations for a control volume by substituting the above relations into the classical formulations for a material volume. In the present paper, holonomic problems are considered. The correction terms in the newly derived version of the Lagrange equations contain the flux of kinetic energy appearing to be transported through the surface of the control volume. This flux comes into the play in the form of properly formulated partial derivatives. Our version of the Lagrange equations is tested using the rocket equation and a folded falling string as illustrative examples.

The present paper is devoted to the coupling between electrical and mechanical fields in piezoelastic structures. In the present contribution, an electromechanically coupled technical theory for flexural and extensional deformations of piezoelastic composite beams is developed. Such a technical theory should be of special interest for control applications, where a lower-order but sufficiently accurate modelling is required. In a first step, an equivalent single-layer theory of the Timoshenko-type for composite beams is utilized. The influence of shear, rotatory inertia as well as the influence of the electric field is taken into account in this technical beam theory. The electric field is unspecified so far in this formulation, but is coupled to the deformation by means of the charge equation of electrostatics. In order to incorporate this coupling, the electric potential is approximated by a power series in the thickness direction of the beam. Terms up to an order of two are considered in the approximation. The formulation then is adapted to the electric boundary conditions at the upper and lower sides of the electroded piezoelectric layers, namely that the electrodes have to be equipotential areas. Putting this distribution into an electrical variational principle, a weak one-dimensional formulation of the charge equation of electrostatics is obtained for the axial distribution of the electric potential. Prescribing the electric potential at the electrodes, and specifying the electrical boundary conditions at the vertical ends of the layer, this weak form completes the proposed electromechanically coupled technical theory for composite piezoelastic beams. In order to demonstrate the influence of the coupling between deformation and electric field, the quasi-static behavior and free flexural vibrations of a symmetrically laminated 3-layer beam are studied in detail. Results are compared to results of coupled finite element computations as well as to results obtained by a simplified theory, previously developed by the authors.

A literature review is presented for shape control of structures, where special emphasis is laid upon smart structures with piezoelectric control actuation. In shape control one intends to specify the spatial distribution, or the shape, of an actuating control agency, such that the displacement field of a structure distorted from its original shape eventually vanishes, or such that the structure follows some desired field of trajectories. The disturbances that distort the shape of structures may be transient, or they may be slowly varying in time. While the problem of (quasi-) static shape control refers to the latter situation, one speaks about dynamic shape control in the former case. In the present review, some structural aspects are emphasized that contribute to the complexity of shape control. Particularly, we discuss the necessity of a distributed actuation, and we refer to the inverse character of the shape control problem. We shortly mention the requirements posed by the latter two aspects upon methods of automatic control for smart structures. Since an analogy exists between piezoelectric actuation and thermal actuation, some references are cited concerning the latter topic. In order to stimulate interdisciplinary work, a short account is further presented with respect to the similar strategies of shape control by shape memory alloys and shape control by distributed pre-stress.

The present contribution explores two fundamental aspects of eigenstrain analysis in three-dimensional bodies. At first, distributions of eigenstrain are derived that do not cause stresses, so-called stress-free or impotent eigenstrains. We consider bodies of finite extent with geometric surface constraints, such as imposed by immovable supports or rigidly clamped boundaries. Within the setting of anisotropic linear elastic bodies, it is verified that a field of eigenstrains that is equal to the field of strains produced by external forces is a stress-free one and that the deformations caused by these eigenstrains and the deformations caused by the forces are equal. Hence, the stress-free eigenstrain load represents an exact solution for the static shape control problem of bodies acted upon by forces. Additionally, nonuniqueness of this shape control problem is demonstrated, and three-dimensional eigenstrains responsible for that nonuniqueness are identified. This is performed by showing that incompatible distributions of eigenstrain and the strains generated by these fields, when applied as a compatible distribution of eigenstrain, result in identical deformations and stresses. Deformation-free fields then result by applying the difference between those fields of eigenstrain.

A numerical algorithm for studying the development of plastic and damaged zones in a vibrating structural element with a large, guided rigid-body motion is presented. Beam-type elements vibrating in the small-strain regime are considered. A machine element performing rotatory motions, similar to an element of a slider-crank mechanism, is treated as a benchmark problem. Microstructural changes in the deforming material are described by the mesolevel variables of plastic strain and damage, which are consistently included into a macroscopic analysis of the overall beam motion. The method is based on an eigenstrain formulation, considering plastic strain and damage to contribute to an eigenstrain loading of a linear elastic background structure. Rigid-body coordinates are incorporated into this beam-type structural formulation, and an implicit numerical scheme is presented for iterative computation of the eigenstrains from the mesolevel constitutive behavior. Owing to the eigenstrain formulation, any of the existing constitutive models with internal variables could in principle be implemented. Linear elastic/perfectly plastic behavior is exemplarily treated in a numerical study, where plastic strain is connected to the Kachanov damage parameter by a simple damage law. Inelastic effects like plastic shakedown and damage-induced low-cycle rupture are shown to occur in the examplary problems.

Die dreidimensionalen Beziehungen eines piezothermoelastischen Korpers werden auf die Gleichungen fur die Verschiebungen, mechanischen Spannungen und fur den dielektrischen Verschiebungsvektor einer Scheibe vereinfacht. Ein Korper mit den Materialeigenschaften der Kristallklasse mm2 wird untersucht. Zur Darstellung der gekoppelten elastischen und elektrischen Felder in der Scheibe werden Potential-funktionen eingefuhrt. Es gelingt die Herleitung einfacher, entkoppelter Beziehungen fur die Komponenten eines generalisierten Galerkin-Vektors und eines verallgemeinerten thermoelastischen Verschiebungspotentials. Als Anwendungsbeispiel wird eine streifenformige Scheibe aus drei Schichten mit entsprechenden Rand- und Ubergangsbedingungen betrachtet. Im untersuchten Fall degenerieren die partiellen Differentialgleichungen zu gewohnlichen und konnen streng integriert werden. Die Verschiebungen und die elektrische Potentialverteilung in den Schichten werden graphisch dargestellt. Die Ergebnisse konnen zur Kontrolle der Genauigkeit numerischer Berechnungen herangezogen werden.

Free and forced vibrations of moderately thick, transversely isotropic plates loaded by lateral forces and hydrostatic (isotropic) in-plane forces are analyzed in the frequency domain. Influences of shear, rotatory inertia, transverse normal stress and of a two-parameter Pasternak foundation are taken into account. First-order shear-deformation theories of the Reissner–Mindlin type are considered. These theories are written in a unifying manner using tracers to account for the various influencing parameters. In the case of a general polygonal shape of the plate and hard-hinged support conditions, the Reissner-Mindlin deflections are shown to coincide with the results of the classical Kirchhoff theory of thin plates. The background Kirchhoff plate, which has effective (frequency-dependent) stiffness and mass, is loaded by effective lateral and in-plane forces and by imposed fictitious “thermal” curvatures. These deflections are further split into deflections of linear elastic prestressed membranes with effective stiffness, mass and load. This analogy for the deflections is confirmed by utilizing D'Alembert's dynamic principle in the formulation of Lagrange, which yields an integral equation. Furthermore, the analogy is extended in order to include shear forces and bending moments. It is shown that in the static case, with no in-plane prestress taken into account, the stress resultants for certain groups of Reissner-type shear-deformable plates are identical with those resulting from the Kirchhoff theory of the background. Finally, results taken from the literature for simply supported rectangular and polygonal Mindlin plates are yielded and verified by analogy in a quick and simple manner.

This paper is concerned with flexural vibrations of composite plates, where piezoelastic layers are used to generate distributed actuation or to perform distributed sensing of strains in the plate. Special emphasis is given to the coupling between mechanical, electrical and thermal fields due to the direct piezoelectric effect and the pyroelectric effect. Moderately thick plates are considered, where the influence of shear and rotatory inertia is taken into account according to the kinematic approximations introduced by Mindlin. An equivalent single-layer theory is thus derived for the composite plates. It is shown that coupling can be taken into account by means of effective stiffness parameters and an effective thermal loading. Polygonal plates with simply supported edges are treated in some detail, where quasi-static thermal bending as well as free, forced and actuated vibrations are studied.