Formulation of a semi-analytical approach based on Gurtin variational principle for dynamic response of general thin plates

4 semi-analytical approach for the dynamic response of general thin plates which employes finite element discretization in space domain and a series of representation in time domain is developed on the basis of Gurtin variational principles. The formulation of time series is also investigated so that the dynamic response of plates with arbitrary shape and boundary constraints can be achieved with adequate accuracy.Project supported by the National Natural Science Foundation of China     

Elastic nonlinear stability analysis of thin rectangular plates through a semi-analytical approach

A semi-analytical approach to the elastic nonlinear stability analysis of rectangular plates is developed. Arbitrary boundary conditions and general out-of-plane and in-plane loads are considered. The geometrically nonlinear formulation for the elastic rectangular plate is derived using the thin plate theory with the nonlinear von Kármán strains and the variational multi-term extended Kantorovich method. Emphasis is placed on the effect of destabilizing loads and on the derivation of the solution methodologies required for tracking a highly nonlinear equilibrium path, namely: parameter continuation and arc-length continuation procedures. These procedures, which are commonly used for the solution of discretized structural systems governed by nonlinear algebraic equations, are augmented and generalized for the direct application to the PDE. The boundary value problem that results from the arc-length continuation scheme and consists of coupled differential, integral, and algebraic equations is re-formulated in a form that allows the use of standard numerical BVP solvers. The performance of the continuation procedures and the convergence of the multi-term extended Kantorovich method are examined through the solution of the two-dimensional Bratu–Gelfand benchmark problem. The applicability of the proposed approach to the tracking of the nonlinear equilibrium path in the post-buckling range is demonstrated through numerical examples of rectangular plates with various boundary conditions.     

Nonlinear primary resonance analysis for a coupled thermo-piezoelectric-mechanical model of piezoelectric rectangular thin plates

A model of piezoelectric rectangular thin plates with the consideration of the coupled thermo-piezoelectric-mechanical effect is established. Based on the von Karman large deflection theory, the nonlinear vibration governing equation is obtained by using Hamilton's principle and the Rayleigh-Ritz method. The harmonic balance method(HBM) is used to analyze the first-order approximate response and obtain the frequency response function. The system shows non-linear phenomena such as hardening nonlinearity, multiple coexistence solutions, and jumps. The effects of the temperature difference,the damping coefficient, the plate thickness, the excited charge, and the mode on the primary resonance response are theoretically analyzed. With the increase in the temperature difference, the corresponding frequency jumping increases, while the resonant amplitude decreases gradually. Finally, numerical verifications are carried out by the Runge-Kutta method, and the results agree very well with the theoretical results.     

THE MATHEMATICAL MODELS AND GENERALIZED VARIATIONAL PRINCIPLES OF NONLINEAR ANALYSIS FOR PERFORATED THIN PLATES

ＴＨＥＭＡＴＨＥＭＡＴＩＣＡＬＭＯＤＥＬＳＡＮＤＧＥＮＥＲＡＬＩＺＥＤＶＡＲＩＡＴＩＯＮＡＬＰＲＩＮＣＩＰＬＥＳＯＦＮＯＮＬＩＮＥＡＲＡＮＡＬＹＳＩＳＦＯＲＰＥＲＦＯＲＡＴＥＤＴＨＩＮＰＬＡＴＥＳＣｈｅｎｇＣｈａｎｇｉｕｎ（程昌钧）；ＹａｎｇＸｉａｏ...     

Vibration analysis of plates by real-time stroboscopic holography

The application of time-averaged holography to the analysis of vibrating surfaces is now a widespread and very powerful experimental technique. However, the realtime method of exactly superimposing the reconstructed static image on the vibrating object and observing the resulting fringe patterns under stroboscopic illumination is becoming more common. In fact, by introducing an initial family of interference fringes by the rotation of the object (or image) before vibration begins, the advantages of both methods are combined.     

A rectangular element of thin plates based upon the generalized variational principles

Based on generalized variational principles, an element called M R-12 was constructed for the static and dynamic analysis of thin plates with orthogonal anisotropy. Numerical results showed that this incompatible element converges very rapidly and has good accuracy. It was demonstrated that generalized variational principles are useful and effective in founding incompatible element. Moreover, element MR-12 is easy for implementation since it does not differ very much from the common rectangular element R-12 of thin plate.     

A variational principle of perturbed motion on viscoelastic thin plates with applications

In this paper, in the light of the Boltzmann superposition principle in linear viscoelasticity, a mathematical model of perturbed motion on viscoelastic thin plates is established. The corresponding variational principle is obtained in a convolution bilinear form. For application the problems of free vibration, forced vibration and stability of a viscoelastic simply-supported rectangular thin plate are considered. The results show that numerical solutions agree well with analytical solutions. The project was supported by the National Natural Science Foundation of China (No. 19772027) and the Shanghai Municipal Development Foundation of Science and Technology (No. 98JC14032).     

A new approach to an analysis of large deflections of thin elastic plates

For thin plates undergoing large deflections, a modified energy expression has been suggested and a new set of differential equations has been obtained in a decoupled form. Accuracy of the equations has been tested for a circular and a square plate with immovable as well as movable edge conditions under a uniform static load. Results obtained are in excellent agreement with other known results. These new equations are more advantageous than Berger's decoupled equations which fail to give meaningful results for movable edge conditions.     

A variational principle for the dynamic problem of linear coupled thermoelasticity

Summary A functional is constructed by whose stationarity the mixed problem for linear and dynamic thermoelasticity is obtained. In contrast with previous results[2], [4], [8], neither constrained variations nor previous transformations upon the equations of the problem are used. The unrestricted variational formulation is made possible by means of a convolutive bilinear form.

---

Sommario Si costruisce un funzionale dalla cui stazionarietà si ricava il problema misto della termoelasticità lineare accoppiata. A differenza di precedenti risultati[2], [4], [8] non si ricorre né a variazioni bloccate né a preliminari trasformazioni del problema. La formulazione variazionale non ristretta è resa possibile dall'uso di una forma bilineare convolutiva.

---

---

Work done in the sphere of activity of the Group for Mathematical Research of the (Italian) C.N.R.     

Accurate variational approach for free vibration of simply supported anisotropic rectangular plates

The Ritz method is one of the most elegant and useful approximate methods for analyzing free vibration of laminated composite plates. It is simple to use and also straightforward to implement. However, the Ritz method has its own difficulty in determining the natural frequencies of simply supported laminated anisotropic plates. This is caused by the fact that the natural boundary conditions in the vibration of anisotropic plates can never be exactly satisfied by a solution in a variables separable form. As a result, the calculated natural frequencies would be expected to converge to solutions a little higher than true ones. To overcome this difficulty, this paper presents a simple variational formulation with Ritz procedure in which all the natural boundary conditions are implemented in an averaging manner. It is revealed that the proposed method can produce lower upper bound solutions compared with the conventional Ritz method where the geometric boundary conditions can only be satisfied by the assumed deflection functions.     

A mixed variational formulation for large deformation analysis of plates

A mixed variational formulation for large deformation analysis of plates is introduced. In this formulation the equilibrium and compatibility equations are satisfied identically by means of stress functions and displacement components, respectively, and the constitutive equations are satisfied in a least square sense. An example is solved and the results are compared with those available in the literature. Further, the functional is particularized for buckling analysis of plates and a simple example is solved to illustrate the theory.The results presented here were obtained in the course of research sponsored by the Natural Sciences and Engineering Research Council of Canada Grant No. A-1628.     

Non linear analysis of plates by means of finite element method

Summary The report presents an approach to the numerical solution of bi-harmonic problems in the non-linear field. Within the sphere of more recent methods based on finite elements alone, there are a large number of works that propose solutions to this problem. Here, with reference to an approach produced by Mindlin, assumptions are advanced that will make it possible to analyse thin plates in the presence of deformatory effects due to shear. The problem is also reformulated in terms of three-dimensional elasticity, with appropriate conditions on the components of the stress tensor and of the deformation. This method makes it possible to deal with the punctual quantities inside the solid in question, instead of the intermediate components on the cross-section, which are typical of classic approaches to the matter. Moreover, by means of appropriate numerical integration techniques, it reduces the resolvent system of equations to the usual dimensions for the analysis of flat plates. This procedure is particularly advantageous in the case of non-linear phenomena, which are present as laws constituting the stress-strain relationship. This report produces examples that take as assumptions of non-linearity the not completely elastic behaviour of the material, using VON MISES' limit function within the framework of the plasticity theory.

---

Sommario Viene presentata una formulazione non lineare di elementi finiti di piastra. Le ipotesi adottate consentono di tenere in conto dell'effetto del taglio, seguendo un'impostazione dovuta a Mindlin. La non linearità considerata è quella del materiale per cui è stata adottata, nell'ambito della plasticità classica, come funzione limite quella di Von Mises. Il controllo di condizione di plasticità viene effettuato in un numero finito di punti all'interno di ogni elemento e attraverso lo spessore. Vengono infine presentati e confrontati con soluzioni classiche alcuni esempi risolti con un programma sviluppato all'uopo.

     

Dynamic deformation of a thermoviscoelastic rod of triangular cross section in a coupled formulation

For the coupled model of a thermoviscoelastic rod of equilateral triangular cross section, two exact solutions are obtained for the cases where a normal displacement and a shear stress or a tangential displacement and a normal stress are specified on the lateral surface of the rod. A dimensionless parameter R₀ is introduced to judge the appropriateness of taking into account the coupling in the formulation of the problem. Formulas are given for the velocities and lengths of the temperature, shear, and longitudinal waves, which can be used in experiments to determine the physical properties of thermoviscoelastic materials. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 128–143, July–August, 2007.     

Plastic analysis of thin plates with anisotropic hardening

In this paper we discuss the adoption of the anisotropic hardening model due to the existence of Bauschinger effect when thin plate is applied by repeated loading. The loading condition of thin plates for linear kinematic hardening has been deduced in terms of generalized forces and generalized plastic deformation. And it can be extended to nonlinear kinematic hardening and mixed hardening. Finally as an example the numerical results are obtained for a circular plate.     

Basic relations of the theory of thermoviscoelastic plates with distributed sensors

The basic relations of the thermomechanics of thin-walled viscoelastic plates with distributed piezoelectric sensors under monogarmonic mechanical loading are presented. To describe the thermomechanical behavior of materials, the concept of complex characteristics is used. Numerical and variational methods are used to study the forced resonant vibrations of viscoelastic plates with distributed piezoelectric sensors. The effect of dissipative heating on the readings of the sensors of a circular viscoelastic plate undergoing axisymmetric resonant bending vibrations is analyzed as an example