F. Ziegler

Eigenstrain Without Stress and Static Shape Control of Structures

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.

Modal analysis of elastic-plastic plate vibrations by integral equations

Abstract A direct boundary element method for the vibration problems of thne elastic-plastic plates is presented. Dynamic fundamental solutions of a suitably shaped finite domain are used in modal form. The series Greens functions are separated into a quasistatic and a dynamic part. Often the series of the quasistatic part can be written in a faster converging form than the equivalent modal series. Analytical integration in the vicinity of the singularity is performed on the closed form fundamental solutions of the infinite domain, and only the non-singular differences from the actual Greens functions are represented in series form. This paper gives a general formulation of this method for Kirchhoff plates on an arbitrary elastic foundation. After integration, the resulting algebraic equations are arranged in a form most convenient for a time-stepping analysis of inelastic response. This rearrangement has to be performed only once, if the time step is kept constant. Constitutive equations are integrated by an implicit backward Euler scheme for plane stress. Applications are shown for impacted circular plates on several different foundations.

Application of the Green's function method to thin elastic polygonal plates

SummaryRecently, the Greens function method has been applied successfully to problems of plane elasticity, using influence functions of some finite basic domain of simple geometrical shape, which contains the given one as a subdomain. The result of this formulation is a pair of integral equations, which have to be defined only along that part of the boundary not coinciding with the border of the basic domain.A rather general formulation for the solution of bending of plates of arbitrary convex planform and loading is presented, where, for the sake of brevity, plates of polygonal shape are considered. The polygonal plate is embedded in a rectangular domain, thereby applying coincidence of boundaries as far as possible. Those boundary conditions in the actual problem, which are not already satisfied, lead to a pair of coupled integral equations for a density function vector with components to be interpreted as line loads and moments distributed in the basic domain along the actual boundary. Thus, the kernel is the corresponding Greens matrix. Hence, having solved the integral equations, deflections and stresses in the actual problem are explicitly known.Solution of the integral equations is generally achieved by a numerical procedure.The method is tested in example problems by considering a trapezoidal plate under various boundary conditions.ZusammenfassungDie Methode der Greenschen Funktion wurde in jüngster Zeit erfolgreich auf Probleme der ebenen Elastizitätstheorie angewendet. Dabei fanden Einflußfunktionen eines endlichen Grundbereiches einfacher geometrischer Form, der den gegebenen Bereich einschließt, Verwendung. Das Resultat dieser Formulierung ist ein Integralgleichungspaar, welches entlang dem Teil des Randes zu erstrecken ist, der nicht mit dem Rand des Grundbereiches bereits zusammenfällt.Eine allgemein gehaltene Formulierung der Biegelösung von Platten mit konvexem Grundriß unter beliebiger Belastung wird angegeben, wobei allerdings der Kürze halber eine Beschränkung auf polygonplatten erfolgt. Die Polygonplatte wird in einen Rechteckbereich eingebettet, wobei soviele Ränder wie möglich zusammenfallen sollen. Jene Randbedingungen des wirklichen Problems, welche dann noch nicht erfüllt sind, führen auf ein gekoppeltes Integralgleichungspaar für den Dichtefunktionsvektor, dessen Komponenten als im Grundbereich entlang der wirklichen Berandung verteilte Linienlasten und Linienmomente gedeutet werden. Damit wird der Kern zur korrespondierenden Greenschen Matrix. Weiters sind, nach Lösung der Integralgleichungen, Biegefläche und Spannungen des wirklichen Problems explizit bekannt. Die Lösung der Integralgleichungen erfolgt im allgemeinen numerisch.Die Methode wird an Beispielen getestet, wobei eine Trapezplatte unter verschiedenen Randbedingungen untersucht wird.

Maysel's formula generalized for piezoelectric vibrations - Application to thin shells of revolution

Either some or each of the layers of a composite shell made of piezoelectric materials behave as distributed actuators so that the shell becomes an intelligent or smart structure. To effectively suppress the vibrations of the shell, an electrical field with a proper control must be applied. The most efficient calculation of thermoelastic deformations is performed by Maysels formula, i.e., within a multiple-field analysis in the isothermal background. By generalizing Maysels formula, it becomes possible to include both the piezoelectric effects and inertia. A version is presented that allows the construction of the best auxiliary problem in the background, particularly preserving all kinds of symmetries present in the actual coupled problem. Applications of the three-dimensional formulation are illustrated for the special case of thin-layered shells of revolution, and, in addition, for a circular cylindrical shell with the piezoelectric influence function being presented. Because solutions of the auxiliary problem are more easily obtained in frequency space, the time convolution is replaced by a Fourier integral, which is eventually subjected to fast Fourier transform.

Dynamic analysis of polygonal Mindlin plates on two-parameter foundations using classical plate theory and an advanced BEM

Forced vibrations of moderately thick plates on two-parameter, Pasternak-type foundations are considered. Influence of plate shear and rotatory inertia are taken into account according to Mindlin. Excitations are of the force as well as of the support motion type. Formulation is in the frequency domain. An analogy to thin plates without foundations is given. This analogy to classical plate theory is complete in the case of polygonal plan-forms and hinged support conditions. In that case the higher order Mindlin-problem is reduced to two (second order) Helmholtz-Klein- Gordon boundary value problems. An advanced BEM using Greens functions of rectangular domains is applied to the latter, thereby satisfying boundary conditions exactly as far as possible. This problem oriented strategy provides the frequency response functions for the deflection of the undamped Mindlin plate with high numerical accuracy. Structural damping is built in subsequently, and Fast Fourier Transform is applied for calculation of the transient response.

Wave propagation in periodic and disordered layered composite elastic materials

Abstract A powerful complex transfer matrix approach to wave propagation perpendicular to the layering of a composite of periodic and disordered structure is worked out showing propagating and stopping bands of time-harmonic waves and the singular cases of standing waves. A state ratio of left- and right-going plane waves is defined and interpreted geometrically in the complex plane in terms of fixed points and flow lines. For numerical considerations and extension of the approach to higher dimensional problems a continued fraction expansion of the state ratio mapping is presented. Impurity modes of wave propagation in composites with widely spaced impurity cells of different elastic materials are discussed. Stopping bands in the frequency spectrum of global waves in fully disordered composites are found to exist in the range of frequencies corresponding to common gaps in the spectrum of cnstituent regular periodic composites which are constructed from the cells of the disordered system. For those frequencies, waves propagate only a (short) finite distance and are therefore strongly localized modes in a composite of fairly large extent.

Transient elastic waves in a wedge-shaped layer

SummaryThe theory of generalized ray is applied to analyzing transient elastic waves in a layered half-space with non-parallel interface. The propagation, reflection and refraction of longitudinal (P-) and transverse (SV-) waves which are generated by a line source in the surface layer of a two layer model are considered, each of the two homogeneous and isotropic layers having different density and inverse of wave speeds. Generalized ray integrals for multi-reflected rays in the top layer are formulated by using two rotated coordinate systems, one for each interface, and are expressed in terms of local wave slowness along each interface. Through a series of transformations of the local slowness, all ray integrals are expressible in a common slowness variable. Special attention is given to wave mode changes during reflection. The arrival time of each ray is then determined from the stationary value of the phase function with common slowness of the ray integral. Arrivals of head waves corresponding to rays refracted at a fast bottom are calculated from proper branch points of the Cagniard-mapping.

Thermally induced vibrations of composite beams with interlayer slip

Thermally excited vibrations of composite viscoelastic two-layer beams with interfacial slip are analyzed. Geometrically linearized conditions are considered, and the Bernoulli-Euler hypothesis is applied to each layer. At the interface a linear viscoelastic slip law is assigned. The resulting sixth-order initial boundary value problem of the deflection is solved in the time domain by separating the dynamic response in a quasistatic and a complementary dynamic portions. The quasistatic solution is determined in closed form, and the remaining complementary dynamic part is approximated by a truncated modal series that exhibits accelerated convergence. Numerical results are obtained for single-span composite beams with interlayer slip by means of a time-stepping procedure based on the linear interpolation of the driving terms within the time intervals.Thermally excited vibrations of composite viscoelastic two-layer beams with interfacial slip are analyzed. Geometrically linearized conditions are considered, and the Bernoulli-Euler hypothesis is applied to each layer. At the interface a linear viscoelastic slip law is assigned. The resulting sixth-order initial boundary value problem of the deflection is solved in the time domain by separating the dynamic response in a quasistatic and a complementary dynamic portions. The quasistatic solution is determined in closed form, and the remaining complementary dynamic part is approximated by a truncated modal series that exhibits accelerated convergence. Numerical results are obtained for single-span composite beams with interlayer slip by means of a time-stepping procedure based on the linear interpolation of the driving terms within the time intervals.

Forced vibrations of an elasto-plastic and deteriorating beam

SummaryA solution method for elastoplastic vibrating beams including damage accumulation is shown, where inelastic behavior of the structure is represented by an additional loading due to sources of selfstresses acting upon the linear elastic structure of time-invariant stiffness. Response due to this additional loading is evaluated using proper Greens functions. Thus, integral relations are set up, similar to Maysels formula. Theory is applied to a two span sandwich beam with elastoplastic degrading flanges and elastic core material.

Nonlinear random vibrations of thermally buckled skew plates

Abstract Random vibrations in the postbuckling range have chaotic properties superposed. For hard and simply supported polygonal plates a multi-modal projection by the Galerkin-procedure renders as a result of a proper non-dimensional formulation a set of nonlinearly coupled ordinary differential equations. Exact unifying solutions of the stationary F-P-K equation are constructed for that class of problems where the nonlinear restoring forces are derived from a potential function. Assuming an effective white noise excitation, the probability of first occurrence of dynamic snap-through is determined for a single mode approximation. Using a two-mode expansion the probability distribution of the asymmetric snap-buckling is also evaluated.