Two-dimensional equations for electroelastic plates with relatively large shear deformations

A set of two-dimensional, nonlinear equations for electroelastic plates in moderately large thickness-shear deformations is obtained from the variational formulation of the three-dimensional equations of nonlinear electroelasticity by expanding the mechanical displacement vector and the electric potential into power series in the plate thickness coordinate. As an example, the equations are used to study nonlinear thickness-shear vibrations of a quartz plate driven by an electrical voltage. Nonlinear electrical current amplitude-frequency behavior near resonance is obtained. The equations and results are useful in the study and design of piezoelectric crystal resonators and the measurement of nonlinear material constants of electroelastic materials.     

Damage and its growth in laminated composite circular/rectangular plates undergoing large deformations

A nonlinear finite element static analysis is used to predict the damage in rectangular/circular plates due to large deformation simulating low-velocity impact. The impact due to the ball impactor is modelled by an equivalent static load and load distribution is taken according to Hertzian law. Tsai-Wu criterion is used to predict the failure of each ply at Gauss points. The failure mode (fibre breakage or matrix cracking) is identified using maximum stress criterion. The material properties at each failed Gauss point is degraded appropriately and the stress analysis is carried out again. This iterative process is repeated until no further damage condition occurs. Numerical examples for both circular and rectangular composite plates are presented.     

Local and gross deformations in cracked metallic plates

The local deformation near a crack tip were investigated when a cracked specimen was stressed beyond general yielding. Crack opening displacements, and near tip strains were measured by the moire technique. The gross elongation in the cracked-section was also measured. These three quantities were found to be linearly related to each other. Surface strain distributions near a crack are influenced by the local necking at the crack tip, which is controlled by the plate thickness.

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Zusammenfassung Die lokalen Verformungen in der Umgebung der Spitze eines Kerbes wurden für den Fall einer gerissenen, und Beanspruchungen oberhalb der Elastizitätsgrenze ausgesetzten Probe, untersucht.Die COD und die Dehnungen in der Umgebung des Kerbes wurden nach dem Moiré-Verfahren bestimmt. Außerdem wurden die Gesamtdehnungen in der gerissenen Ebene gemessen. Es ergab sich ein linearer Zusammenhang zwischen diesen drei Größen.Die Verteilung der Verformungen an der Oberfläche in der Nähe eines Risses werden durch die lokale Einschnürung and der Rißspitze beinflußt. Letztere hängt ihrerseits von der Dicke des Bleches ab.

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Résumé Les déformations locales au voisinage de l'extrémité d'une entaille ont été étudiées dans le cas d'éprouvettes fissurées sollicitées au delà de la limite élastique. Les COD et les dilatations près de l'entaille on été mesurées par la technique du Moiré. On a également mesuré les dilatations d'ensemble dans la section fissurée, et l'on a trouvé que ces trois quantités sont liées l'une à l'autre par des relations linéaires.Les distributions des déformations en surface près d'une fissure sont influencées par la striction locale à l'extrémité de la fissure, qui est sous la dépendence de l'épaisseur de la tôle.

     

Small deformations on large deformations in BKZ viscoelastic liquids

Small deformations superposed on large deformations in a BKZ viscoelastic fluid are considered. The general constitutive form for small on large is determined. The base motion is taken to be any continuous isochoric motion while the perturbation is assumed to be of small order, in the sense that terms involving the products of the perturbation and its gradients will be neglected.In the case of steady base motions a reduction in the constitutive form is found. For a certain class of perturbations on steady base flows it is shown that the constitutive relation takes on a convolution form.An exact solution is presented to small in-plane disturbances on steady simple shearing of a special type of BKZ fluid between two parallel walls.     

Hyperelastic large deformations of two-phase composites with membrane-type interface

The composite under investigation consists of two phases which are bonded together through a membrane-type interface. The work reported in this paper aims at studying the hyperelastic large deformations of this composite while accounting for interfacial stress effects. A variational formulation for a general traction boundary value problem of the composite is provided, leading to the local bulk and interfacial equilibrium equations and to the traction boundary conditions. Assuming that each bulk phase is incompressible and characterized by an energy density function depending only on the trace of the right Cauchy-Green bulk strain tensor and that the interface is compressible and defined by an energy density function being isotropic with respect to the right Cauchy-Green surface strain tensor, exact solutions are given for the simple axial extension, simple torsion and out-of-plane shear of a fiber-reinforced cylinder, and a closed-form solution is also found for a hollow composite sphere subjected both to an internal pressure and an external pressure. These analytical results are further specified and discussed in the particular case where each bulk phase is described by an incompressible Neo-Hookean law and the interface is specified by a compressible Neo-Hookean law. Apart from their own usefulness, the results obtained in this work can serve as benchmarks for relevant numerical methods.     

Compatibility equations for large deformations

Compatibility equations for large deformations are developed on the sole basis of the div and curl concepts.     

Stresses in a nonlinearly hardening pipe after expansion to large deformations,inelastic unloading,and removal of its inner layers

The problem of the numerical determination of residual stresses in an elastoplastic nonlinearly hardening pipe of V95ochT2 material in plane strain is examined.It is shown that due to secondary plastic deformations that develop during unloading, the greatest residual circumferential stresses reside at a certain distance from the inner surface of a thick-wall pipe.Removal of the internal layer of the pipe gives rise to a small change in the circumferential compressive stresses on its inner surface. In this case, the zone where the maximum residual circumferential compressive stresses reside is shifted deeper into the pipe with a simultaneous decrease in the distance from this zone to the new inner surface of the piper, which is formed after removal of a layer of metal.Translated from Problemy Prochnosti, No. 11, pp. 111–114, November, 1989.     

On large strain deformations of shells

Summary A nonlinear shell theory is derived for large strain—large beding deformations of shells composed of highly nonlinear materials. Expressions for the stress resultants and stress couples are presented. The equilibrium equations are obtained as weak solution of the stationary principle of total potential energy. A modified Kirchhoff hypothesis is used which accounts for thickness changes as well as for a shift in the location of the original midsurface of the shell. As example the eversion of a spherical shell is calculated numerically.With 4 Figures     

Inverse transient thermoelastic deformations in thin circular plates

This paper deals with the determination of unknown heating temperatures and temperature distributions on the upper surface of a thin circular plate, defined as 0 ≤ r ≤a, —b/2 ≤z ≤b/2. The expressions of unknown heating temperatures and temperature distributions are obtained in series form, involving Bessel’s functions with the help of the integral transform technique. Thermoelastic deformations are discussed with the help of temperature and are illustrated numerically.     

Reconstructive elasticity imaging for large deformations

A method is presented to reconstruct the elastic modulus of soft tissue based on ultrasonic displacement and strain images for comparatively large deformations. If the average deformation is too large to be described with a linear elastic model, nonlinear displacement-strain relations must be used and the mechanical equilibrium equations must include high order spatial derivatives of the displacement. Numerical methods were developed to reduce error propagation in reconstruction algorithms, including these higher order derivatives. Problems arising with the methods, as well as results using ultrasound measurements on gel-based, tissue equivalent phantoms, are given. Comparison to reconstructions using a linear elastic model shows that equivalent image quality can be produced with algorithms appropriate for finite amplitude deformations.     

Stress concentration in plates under tension with local thinnings in elastoplastic deformations and creep

The results are given of a calculation-experimental investigation of the stress concentration in the zone of two-sided and one-sided craters in the form of spherical segments. The elastoplastic problem was solved by the method of variable parameters of elasticity with use of the finite element method. The experiments were made on plates of AL2 alloy with the use of short-base strain gauges. The relationship was obtained of the stress concentration factors to the parameters of thinning and loading at characteristic points. The picture is given of the successive transition of the elements into the plastic state. The simulation of creep based on the method of isochronous curves made it possible to obtain curves of the change in stress concentration factors with time for the characteristic points applicable to the steels used in turbine production from the data of experimental investigation of aluminum elastoplastic models.Khar'kov and Perm'. Translated from Problemy Prochnosti, No. 12, pp. 19–23, December, 1989.     

Dynamic contact of bodies experiencing large deformations

Summary In the paper the statement ad basic features of the large displacement contact problem will be considered. Using the notion and formalism of a material singular surface in continua important physico-chemical phenomena of the contact interface may be described. Our attention is focused on details which follow from the geometrical nonlinearity and from the role of the interfacial layer.Dedicated to Prof. Dr. Dr. h. c. Franz Ziegler on the occasion of his 60th birthday     

Modeling of nonlinear viscoelasticity at large deformations

A constitutive model of finite strain viscoelasticity, based on the multiplicative decomposition of the deformation gradient tensor into elastic and inelastic parts, is presented. The nonlinear response of rubbers, manifested by the rate effect, cycling loading and stress relaxation tests was captured through the introduction of two internal variables, namely the constitutive spin and the back stress tensor. These parameters, widely used in plasticity, are applied in this work to model the nonlinear viscoelastic behaviour of rubbers. The experimental results, obtained elsewhere, related with shear deformation in monotonic and cyclic loading, as well as stress-relaxation, were simulated with a good accuracy.     

Experimental and theoretical study on large ductile transverse deformations of rectangular plates subjected to shock load due to gas mixture detonation

The main purpose of this study is to show that metal plate forming by direct application of gas mixture detonation loads can be considered as an alternative high‐velocity forming method for structures instead of a conventional one. Therefore, in this investigation, a series of experimental tests have been conducted on aluminium alloy and mild steel plates with different thicknesses to examine large ductile transverse deformations of rectangular plates with clamped edge conditions subjected to gas mixture detonation loading. The main aim of the experimental section is to investigate the effects of predetonation pressures of acetylene (C₂H₂) and oxygen (O₂) gasses and different mixture ratios on the dynamic response of specimens. The permanent deflections of plates have widely varied from 21.66 up to 56.31 mm. In theoretical analysis, according to an upper bound solution and energy method, theoretical models have been presented by assuming a zero‐order Bessel function of the first kind in the x and y directions for a transverse displacement profile to predict permanent deflections. To account for material strain rate sensitivity, a Cowper–Symonds model has been used, whereas the material coefficients of this equation are constant values or functions of plate thickness. A comparison of the present models with Jones' theoretical model shows that a good agreement with experimental results can be obtained when constant values are used for material coefficients in the Cowper–Symonds equation.