Nonlinear bending of FGM plates subjected to combined loading and resting on elastic foundations

A nonlinear bending analysis is presented for a simply supported, functionally graded plate resting on an elastic foundation of Pasternak-type. The plate is exposed to elevated temperature and is subjected to a transverse uniform or sinusoidal load combined with initial compressive edge loads. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The formulations are based on a higher-order shear deformation plate theory and general von Kármán-type equation that includes the plate-foundation interaction and thermal effects. A two step perturbation technique is employed to determine the load–deflection and load–bending moment curves. The numerical illustrations concern nonlinear bending response of functional graded plates with two constituent materials resting on Pasternak elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The results reveal that the characteristics of nonlinear bending are significantly influenced by foundation stiffness, temperature rise, transverse shear deformation, the character of in-plane boundary conditions and the amount of initial compressive load. In contrast, the effect of volume fraction index N becomes weaker when the plate is supported by an elastic foundation.     

Buckling analysis of functionally graded plates subjected to uniaxial loading

Elastic bifurcational buckling of functionally graded plates under in-plane compressive loading is studied. It is supposed that the gradients of material properties throughout the structure are produced by a spatial distribution of the local reinforcement volume fraction v_f = v_f(x, y, z). To analyze the problem, a method based on a combination of micromechanical and structural approaches is employed. This establishes the effective constitutive behavior at every point of a nonhomogeneous composite plate and provides a buckling criterion. The derived criterion enables one to calculate the critical buckling load R_x^cr for a given distribution v(x, y, z).

Furthermore, with the aim to improve the buckling resistance of the functionally graded plate, the functional R_x^cr(v_f) is maximized. This yields an optimal spatial distribution v_f(x, y, z) of the reinforcement phase.

Results are presented for both short- and long-fiber SiC/Al plates in which the fibers are nonuniformly distributed in the x-, y-, or z-directions. The effects of length-to-width ratio of the plate, and of different types of boundary conditions are studied. Buckling load improvements of up to 100%, as compared to the corresponding uniformly reinforced structure, are shown.     

Shape factors of self-compacting concrete specimens subjected to uniaxial loading

One of the problems encountered when comparing the mechanical properties of self-compacting concrete (SCC) is the use of different specimen sizes all over the world. For vibrated concrete (VC), conversion factors are defined to convert the obtained compressive strength on one specimen type to another. In order to investigate the applicability of these factors for SCC, a total of 2 VC and 10 SCC mixtures were selected varying in cement type, cement content, water-to-cement ratio and water-to-powder ratio. Beside cubes with sides of 100 mm, 150 mm and 200 mm, cylinders with a diameter of 100 mm and diameter 150 mm were cast and cores with a diameter of 100 mm, 80 mm and 50 mm were drilled. A significant difference of about 10% in the shape factors between SCC and VC has been found. Mix design parameters, such as the fraction of powder, the cement-to-powder ratio and the water-to-cement ratio, seem to have little influence on the obtained shape factors.     

Three-dimensional stress state around quarter-elliptical corner cracks in elastic plates subjected to uniform tension loading

Detailed full-field three-dimensional (3D) finite element analyses have been conducted to study the out-of-plane stress constraint factor T_z around a quarter-elliptical corner crack embedded in an isotropic elastic plate subjected to uniform tension loading. The distributions of T_z are studied in the forward section (0° ? θ ? 90°) of the corner cracks with aspect ratios a/c of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. In the normal plane of the crack front line, T_z drops radially from Poisson’s ratio at the crack tip to zero beyond certain radial distances. Strong 3D zones (T_z > 0) exist within a radial distance r/a of about 4.6-0.7 for a/c = 0.2-1.0 along the crack front, despite the stress-free boundary conditions far away. At the same radial distance along the crack front in the 3D zones, T_z increases from zero on one free surface to a peak value in the interior, and then decreases to zero on another free surface. The distributions of T_z near the corner points are also discussed. Empirical formulae describing the 3D distributions of T_z are obtained by fitting the numerical results, which prevail with a sufficient accuracy in the valid range of 0.2 ? a/c ? 1.0 and 0° ? θ ? 90° except very near the free surfaces where T_z is extremely low. Combined with the K-T solution, the transition of approximate plane-stress state near the surfaces to plane-strain state in the interior can be characterized more accurately.     

Numerical investigation of concrete subjected to high rates of uniaxial tensile loading

The present article is concerned with the response of structural concrete prisms to high rates of uniaxial tensile loading. The numerical investigation carried out is based on a finite-element (FE) program capable of carrying out three-dimensional (3D) nonlinear static and dynamic analyses. This program is known to yield realistic predictions to the response of a wide range of plain- and reinforced-concrete structural forms subjected to arbitrary static and earthquake actions. Furthermore, its application has recently been successfully extended in predicting the response of plain-concrete prism elements under high rates of uniaxial compressive loading. The main feature of the FE program is that it incorporates a 3D material model which is characterized by both its simplicity and its attention to the actual physical behaviour of concrete in a structure. Its analytical formulation is based on the assumption that the material properties of concrete are independent of the applied loading rate (strain rate) thus attributing the effect of the applied loading rate on the prism's response to inertia. The validation of this assumption is based on a comparative study between numerical and experimental data which reveals good agreement. This constitutes a major departure from current thinking as regards material modelling of concrete under high-rate loading. In addition, the available data (numerical and experimental) show that the response of the concrete prism elements depends on a number of parameters linked to geometry and material properties of the structural forms under investigation as well as the testing method adopted. This dependence explains, to a significant extent, the scatter that characterizes the available experimental data, and it also suggests that both experimental and numerical results describe structural rather than material behaviour thus raising questions regarding the validity of the use of such data in the constitutive modelling of concrete-material behaviour under high-rate loading conditions.     

Effect of loading rate on the strength of concrete subjected to uniaxial tension

In the context of an international co-operation project between the University of Delft (The Netherlands) and the LCPC, an experimental study was made of rate effects in the behaviour of concrete under tensile stress. Very high speed tests ( between 1 and 80 GPa s⁻¹) were carried out in Delft on a Hopkinson bar, and quasi-static tests ( between 5×10⁻⁵ and 5×10⁻³ GPa s⁻¹) were carried out by the LCPC on a hydraulic press. This investigation had two objectives. 1. To verify on a mini-concrete (diameter of the largest particles 10 mm) a result obtained with a micro-concrete (diameter of the largest particles 2 mm) in the course of a previous study. Rate effects are produced by the presence of pore water in the material. 2. To investigate the influence of the water/cement ratio (i.e., the compressive strength of the concrete) on these rate effects. The three main conclusions that can be drawn from this study are (i) it is indeed the presence of pore water in the concrete which is at the origin of rate effects where this materials is concerned, (ii) the effect of speed on the tensile strengthf ₁ increases with the water/cement ratio, and (iii) in absolute value, the increase in strength (f _tdyn-f _tstat ) seems to be independent of the water/cement ratio.

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Resume Dans le cadre d'une coopération internationale entre l'Université de Delft (Pays-Bas), et le Laboratoire Central des Ponts et Chaussées (LCPC), une étude expérimentale a été menée sur les effects de vitesses dans le comportement en traction du béton. Les essais très rapides ( compris entre 1 et 80 GPa s⁻¹) ont été réalisés à Delft sur une barre d'Hopkinson, les essais quasi-statiques ( compris entre 5×10⁻³ et 5×10⁻⁵ GPa s⁻¹) étant réalisés au LCPC sur une presse hydraulique. Deux objectifs sont visés dans cette étude: (i) vérifier sur un mini-béton (diamètre du plus gros grain égal à 10 mm) un résultant obtenu sur micro-béton (plus gros grain égal à 2 mm) lors d'une étude précédente: c'est la présence d'eau libre au sein du matériau qui induit les effects de vitesse; et (ii) étudier l'influence du rapport eau/ciment (c'est-à-dire de la résistance en compression du béton) sur ces effets de vitesse. Les conclusions principales que l'on peut tirer de cette étude sont les trois suivantes: c'est bien la présence d'eau libre dans les bétons qui est l'origine des effects de vitesse relatifs à ce matériau; l'effet de vitesse sur la résistance en traction f₁ augmente avec le rapport eau/ciment; et en valeur absolue, l'augmentation de la résistance (f _tdyn-f _tstat ) semble indépendante de ce rapport eau/ciment.

     

Temporal homogenization of viscoelastic and viscoplastic solids subjected to locally periodic loading

 As a direct extension of the asymptotic spatial homogenization method we develop a temporal homogenization scheme for a class of homogeneous solids with an intrinsic time scale significantly longer than a period of prescribed loading. Two rate-dependent material models, the Maxwell viscoelastic model and the power-law viscoplastic model, are studied as an illustrative examples. Double scale asymptotic analysis in time domain is utilized to obtain a sequence of initial-boundary value problems with various orders of temporal scaling parameter. It is shown that various order initial-boundary value problems can be further decomposed into: (i) the global initial-boundary value problem with smooth loading for the entire loading history, and (ii) the local initial-boundary value problem with the remaining (oscillatory) portion of loading for a single load period. Large time increments can be used for integrating the global problem due to smooth loading, whereas the integration of the local initial-boundary value problem requires a significantly smaller time step, but only locally in a single load period. The present temporal homogenization approach has been found to be in good agreement with a closed-form analytical solution for one-dimensional case and with a numerical solution in multidimensional case obtained by using a sufficiently small time step required to resolve the load oscillations. Received: 22 November 2001 / Accepted: 21 May 2002     

The multiaxial weld root fatigue of butt welded joints subjected to uniaxial loading

In this study, the fatigue strength of inclined butt welds subjected to a proportional multiaxial stress state generated by uniaxial loading is studied in nominal and local stress concepts. The local methodologies studied included principal stress hypothesis, von Mises stress hypothesis and modified Wöhler curve method. Nominal methodologies included modified Gough–Pollard interaction equation, the design equation in Eurocode3 and the interaction equation in DNV standard. Results are evaluated along with data published in relevant literature. It is observed that both local and nominal stress assessment methods are able to estimate multiaxial fatigue strength. No obvious difference in fatigue strength is observed in the nominal stress concept, but the notch stress concept is able to capture a decrease in fatigue strength in shear‐dominated joints. It is concluded that modified Wöhler curve method is a suitable tool for the evaluation of fatigue strength in joints dominated by both normal and shear stresses.     

Flexoelectric effects: Charge separation in insulating solids subjected to elastic strain gradients

After a brief historical introduction this article will present a summary of experimental work carried through at Penn State to explore the flexoelectric coefficients μ _ijkl in ferroelectric, incipient ferroelectric and relaxor ferroelectric perovskites. The initial objective was to understand the magnitude of flexoelectricity in these systems to see whether it would be possible to develop a piezoelectric composite containing no piezoelectric element, which nonetheless could have practically useful properties. Recent discussions of the thermodynamic converse effect, ie. the generation of elastic strain by an electric field gradient, now suggest that such composites might be designed to have unique properties such as a direct but no converse effect, or vice-versa, and materials with this character could have important practical application. Present data already suggest that the direct effect may make an important contribution to the properties of epitaxial thin films where mismatch can give rise to very steep elastic strain gradients. Clearly, more work is needed to fully quantify the flexoelectric behavior. It will be important to measure single crystals in the ceramic systems which have been studied and to characterize the converse effect as a check of the measured values.     

Nonlinear behavior of bumper foams under uniaxial compressive cyclic loading

Theoretical and experimental studies on the nonlinear behaviors of bumper foams under cyclic loading are carried out in this paper. To study the compressible materials, the incompressible viscoelastic model proposed by Rajagopal and Srinivasa in 2000 is modified and expressed as a function of the principal stretches. The modified model is used to describe bumper foams for the first time. Besides, in order to better predict the nonlinear process of bumper foams under cyclic loading, a new compressible viscoplastic model is proposed, which is expressed separately as the invariants of stretches and the principal stretches. Then the compressible viscoelastic model and the compressible viscoplastic model are used to describe the response of bumper foams under cyclic loading with constant and variable amplitudes, respectively. The experimental results demonstrate that the compressible viscoelastic model and the compressible viscoplastic model are both suitable to describe the response of bumper foams under cyclic loading, the new proposed compressible viscoplastic model is more suitable to describe the deformation at the end of each cycle.     

均匀热载荷作用下功能梯度圆板的非线性振动

基于经典板理论,研究了热载荷作用下功能梯度圆板的大幅振动问题.在经典板理论下利用物理中面概念,导出了功能梯度圆板的非线性运动方程.利用Ritz?Kantorovich方法消去时间变量,将非线性运动方程转换成了一组关于空间变量的非线性常微分方程.采用打靶法数值求解所得方程,并利用数值结果研究了热载荷作用下功能梯度圆板静态...     

Stress state of solids containing thin elastic crooked inclusions

The mathematical models, integral equations and the corresponding numerical schemes, required for the study of the stress state of plates containing thin crooked defects are considered and explicitly written. The results obtained using the proposed line model for a circular arc inclusion are in good agreement with those obtained by the direct approach. Numerical examples are also presented for S-like and kinked inclusions.     

Study of the stress state of ribbed shells subjected to local loading

The methods of photoelasticity and extensometry are used to study the stress state of cylindrical shells provided with stringers and rings. The results of the study are compared with theoretical data. It is found that the experimental and theoretical results agree satisfactorily.Translated from Problemy Prochnosti, No. 2, pp. 71–74, February, 1990.     

Boundary-integral equation analysis of twisted internally cracked axisymmetric bimaterial elastic solids

 Green's function is obtained for the infinite bimaterial elastic solid, containing an internal circular interface crack, loaded by a unit tangential co-axial circular source. An axisymmetric direct boundary integral equation (BIE) is used for the analysis of a finite bimaterial axisymmetric body containing an internal circular interface crack and a finite homogeneous cracked cylinder, both under torsional loading. Using the proposed technique, no discretization of the crack surface is necessary. Numerical results for both examples as obtained by the proposed method are presented and discussed. Received: 29 October 2001 / Accepted: 29 May 2002     

Computational comparisons of homogeneous and statistical descriptions of AerMet100 steel subjected to high strain rate loading

In order to simulate the effect of material microstructure a statistically compensated Johnson-Cook (JC) fracture model has been implemented into the Eulerian shock physics code, CTH. This model uses a Weibull function to produce a distribution of initial failure strains within the JC fracture model. A parametric analysis where the Weibull modulus was systematically varied was conducted on two sets of experimental fragmentation data. The first experiment consisted of an explosively loaded cylinder of AerMet100. The second was an expanding tube experiment which used a plastic cylinder to load the AerMet100 and provided a problem at a lower strain rate. In both sets of experiments, the fragments were soft captured for later examination. While CTH does not explicitly track fragments, a post processor written at the Naval Surface Warfare Center Dahlgren Division was used to calculate the mass of each of the fragments in the expanding debris cloud. The results were analyzed and compared back to baseline homogeneous calculations. The use of a statistically compensated JC fracture model substantially improved the fragment mass distribution for the explosively loaded cylinder. However, the lower strain rate expanding tube showed only minimal improvement. A probable reason for this limitation and future analysis are discussed.     

Thermodynamic properties of the generalized Murnaghan equation of state of solids

In the Murnaghan approximation, an isothermal relation between pressure (P) and volume (V) for solids is derived from the assumption that the isothermal bulk modulus (B) is a linear function ofP. This paper presents a thermodynamic analysis of a generalized form of the equation, based on treating all its various parameters [viz., V, B, and (B/P) _T atP=0] as functions of temperature. Extending a previous study, the effect ofT upon (B/P) _T is accounted for by using a dimensionless parameter introduced by us, which is used in formulating general expressions for the volume dependence of various thermophysical quantities,viz., the thermal expansion coefficient (), the productB, and the Grüneisen and Anderson-Grüneisen parameters. Some combinations of these parameters are identified, which show a simple dependence uponP. The new expressions are used in analyzing current approximations and the behavior of the solid in the low-compression range. In particular, an expression for at highP is reported which generalizes the Anderson equation and previous results by us.     

Nonlinear dynamic analysis of tapered sandwich plates with multi-layered faces subjected to air blast loading

The nonlinear dynamic behavior of simply supported tapered sandwich plates subjected to air blast loading is investigated theoretically and numerically. The plate is supposed to have both tapered core and tapered laminated face sheets and be subjected to uniform air blast load. The theory is based on a sandwich plate theory, which includes von Kármán large deformation effects, in-plane stiffnesses, inertias and shear deformations. The sandwich plate theory for plates with constant thickness which have one-layered face sheets found in the literature is developed to analyze the tapered sandwich plates with multi-layered face sheets. The equations of motion are derived by the use of the virtual work principle. Approximate solution functions are assumed for the space domain and substituted into the equations. The Galerkin method is used to obtain the nonlinear differential equations in the time domain. The finite difference method is applied to solve the system of coupled nonlinear equations. The tapered sandwich plate subjected to air blast load is also modelled by using the finite element method. The displacement–time and strain–time histories are obtained. The theoretical results are compared with finite element results and are found to be in an agreement.     

Failure probability analysis of an elastic orthotropic brittle cylinder subjected to axisymmetric thermal and pressure loading

Procedures for the calculation of the principal stresses in a long hollow circular cylinder of a brittle, orthotropically anisotropic material, subjected to a uniform internal pressure and axisymmetric radial temperature gradient are developed; the Runge-Kutta method is used in the solution, after an initial iterative stage. An equation for the failure probability of the cylinder, derived from the Weibull distribution, is presented, in which allowance is made for the anisotropy in mechanical strength.In order to illustrate their application, the equations are used to determine the stresses in and failure probability of an isotropic cylinder and a particular anisotropic cylinder, subjected to an internal pressure or a radial temperature gradient. The results are critically discussed.

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Résumé On développe des procédures pour le calcul des contraintes principales dans un cylindre circulaire long et creux, constitué d'un matériau fragile et orthotrope et anisotrope sujet à une pression uniforme interne et à un gradient de température radiale axisymétrique. La méthode Runge-Kutta est utilisée dans la solution, après une étape itérative initiale. Une équation pour la probabilité de rupture du cylindre dérivée de la distribution de Weibull, est présentée; dans cette équation, on tient compte de l'anisotropie des propriétés mécaniques. Afin d'illustrer leur application, les équations ont été utilisées pour déterminer les contraintes et la probabilité de fissure pour un cylindre isotrope et un cylindre particulièrement anisotropique, tous deux soumis à une pression interne et à un gradient radial de la température. Les résultats sont discutés sous forme critique.

     

Stress-strain state of an elastic plate subjected to a surface load moving at constant speed

Altai Research Institute of Engineering Technology, Barnaul. Institute of Strength Problems, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Problemy Prochnosti, No. 4, pp. 3–8, April, 1988.     

Nonlinear transient analysis of reinforced concrete slabs subjected to blast loading and retrofitted with CFRP composites

Computational models using the finite element method for nonlinear transient analysis of reinforced concrete (RC) two-way slabs subjected to blast loading are presented. Both as-built and retrofitted slabs with carbon fiber reinforced polymer (CFRP) composite strips are analyzed. The models are used to investigate different parameters including (a) loading duration, and (b) effect of CFRP retrofit on damage accumulation. In this study, damage is globally quantified by the amount of reduction of the first two vibrational frequencies of the slabs. Local representation of damage in terms of reinforcing steel strains is also discussed. The computational models for both the as-built and the retrofitted slabs are verified using experimental results. In these experiments, a slowly increasing uniform pressure is applied to the bottom surface of large-scale RC slab specimens using high-pressure water bag. Experimental results showed that an increase up to 200% in the load carrying capacity is achieved when using the CFRP composite retrofit system. Transient nonlinear analysis results proved the efficiency of the CFRP composite retrofit in improving the slab behavior under blast loading for different loading durations, i.e. for small, medium, and large charge weights at the same applied maximum pressure. In particular, less than 50% reduction of the fundamental frequency due to concrete damage is obtained for the retrofitted slab compared to more than 85% reduction for the as-built slab. Moreover, the maximum displacement is reduced by 40–70% with the CFRP retrofit compared to the as-built slab. As for reinforcing steel strains, the application of CFRP retrofit significantly limited the spread of yielding in time and space. The improved slab behavior with CFRP is best when retrofitting is applied to both sides of the slab.