Fracture mechanics of plates and shells applied to fail-safe analysis of fuselage Part II: Computational results

In this paper, the problem of the fracture of a fuselage stiffened by longitudinal longerons and circumferential frames is analyzed by means of the finite element method. Our research is motivated by the fail-safety design concept of fuselage for civil aircraft. In this study, the total energy release rate are evaluated for five types of basic loading, namely, axial extension, pure bending, twisting, transverse shearing, and radial expansion due to internal pressure. The crack is located either at the mid-point or near the end of the fuselage. It extends in two bays with the stiffener at its center. The stiffener which bisects the crack is assumed to be broken at the location of the crack. Computational results indicate that the total energy release rate G_t increases with the increasing crack length. However, when the crack tip approaches the stiffener, the value of G_t decreases as a result of the reinforcement from the stiffener. For a crack near the end of the fuselage, as a result of boundary effect, the value of G_t is larger in comparison with the case of the crack at the mid-point of the fuselage. We also find that the effect of geometrical nonlinearity can reduce the value of G_t for the fuselage under axial tension or pure bending. For the fractured fuselage under pure bending, shell buckling can occur at the concave side of the fuselage prior to crack growth. The maximum tensile stress in the stiffener in front of the crack tip is also investigated.     

A Bending-Gradient model for thick plates. Part I: Theory

This is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In part two (Lebée and Sab, 2011), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner–Mindlin theory and to full 3D Pagano’s exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

Non-linear analysis of functionally graded fiber reinforced composite laminated plates,Part I: Theory and solutions

This paper deals with the large amplitude vibration, non-linear bending and postbuckling of fiber reinforced composite laminated plates resting on an elastic foundation in hygrothermal environments. Two kinds of fiber reinforced laminated plates, namely, uniformly distributed and functionally graded reinforcements, are considered. The material properties of fiber reinforced laminated plates are estimated through a micromechanical model and are assumed to be temperature-dependent and moisture-dependent. The motion equations are based on a higher order shear deformation plate theory that includes plate-foundation interaction and the hygrothermal effect. A two-step perturbation technique is employed to determine the non-linear to linear frequency ratios of plate vibration, the load-deflection and load-bending moment curves of plate bending, and postbuckling equilibrium paths of laminated plates.     

Fracture behavior of brittle plates and cylindrical shells subjected to concentrated impulse loading

The influence of impact velocity and geometry in the fracture patterns produced by a concentrated impulse loading on brittle plates and cylindrical shells has been studied both experimentally and theoretically. The experiments were performed by impacting plates and cylindrical shells made of plaster with a steel ball. The fracture behavior was photographed by a camera with a flash. The crack-initiation time was measured using a memoriscope. The fracture behavior is explained using the theory of flexural motion of a plate and a cylindrical shell. With the addition of impact-fracture criteria to these theories, the fracture patterns of brittle plates and cylindrical shells are predicted and the resemblance is discussed.     

Matrix displacement analysis of plates and shells

Summary The paper presents an aperçu of a theory based on the Matrix Displacement method developed by the author for the analysis of plates and shells of arbitrary form. The philosophy of the method is exclusively founded on the use of a modern large digital computer and a sophisticated matrix interpretative scheme backed up by an allied structural language, both of which were established at the author's Stuttgart Institute. Of primary importance is the concept and use of natural modes of straining to define the deformation of elements.The first part investigates inter alia the statics and dynamics of anisotropic parallelogram plates and shells idealized into an assembly of such elements. Use is made of a so-called natural or oblique system of co-ordinates both with respect to forces, displacements, moments and rotations, all of which obey their own rule of obliquity. A kinematically consistent lumped mass matrix is given which yields a high precision in the estimation of eigenmodes and eigenfrequencies. Comparison with experimental and analytical results (where available) show an excellent agreement. The paper develops also the analysis of anisotropic triangular plates under bending and membrane action which are ideal elements to represent shells and plates of arbitrary configuration. Here too a satisfactory degree of agreement is reached with previous special analytical results. In this first part transverse shear strains are ignored and displacements assumed to be small. A generalization allowing for shear strains and large displacements is to be included in a subsequent publication which will also explore further refinements of the theory.

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Übersicht Es wird ein Überblick über eine neue Berechnungsmethode von willkürlich geformten Platten und Schalen gegeben, die auf der vom Autor entwickelten Matrizenverschiebungstheorie beruht. Die grundlegende Voraussetzung der Theorie ist die Benutzung eines modernen, großen Digitalautomaten und eines ausgeklügelten Matrizen-Interpretativ-Systems, gekoppelt mit einer Tragwerksaufbausprache, die im Stuttgarter Institut des Authors aufgestellt wurden. Von besonderer Bedeutung ist der Gedanke und die Benutzung natürlicher Verzerrungszustände, um die Deformation der Elemente zu beschreiben.Im ersten Teil wird unter anderem die Statik und Dynamik von anisotropen Parallelogrammplatten und-schalen untersucht, die in einem von Parallelogrammelementen aufgebauten System idealisiert werden können. Es werden die sogenannten natürlichen oder schiefen Koordinaten verwendet mit Bezug auf Kräfte, Verschiebungen, Momente und Verdrehungen, die alle einem besonderen Gesetz der Schiefe gehorchen. Eine kine-matisch konsequente konzentrierte Massenmatrix wird entwickelt, welche eine hohe Genauigkeit in der Berechnung von Eigenschwingungsformen und Eigenfrequenzen ergibt. Vergleiche mit experimentellen und analytischen Ergebnissen (wo vorhanden) zeigen ausgezeichnete Übereinstimmung. Ferner wird die Matrizenberechnung von anisotropen Dreiecksplatten allgemeiner Form entwickelt, die sowohl einem Biege- wie einem Membran-Zustand ausgesetzt sind, und ideale Elemente für den Aufbau von Schalen willkürlicher Formen ergeben. Eine zufriedenstellende Übereinstimmung mit vorhandenen Ergebnissen zeigt wieder die Genauigkeit der Methode. In diesem ersten Teil der Arbeit werden die transversalen Schubverzerrungen vernachlässigt und kleine Verschiebungen vorausgesetzt. Eine Verallgemeinerung, welche diese transversalen Schubverzerrungen, sowie große Verschiebungen einschließt, wird in einer zukünftigen Veröffentlichung erfaßt werden, die auch weitere Verfeinerungen der Methode enthalten wird.

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The author wishes to thank his collaborators K. E. Buck and Kariappa for theier imaginative support, especially in programming the examples of this paper.     

Asymptotic analysis of laminated plates and shallow shells

It was noted long ago [1] that the material strength theory develops both by improving computational methods and by widening the physical foundations. In the present paper, we develop a computational technique based on asymptotic methods, first of all, on the homogenization method [2, 3]. A modification of the homogenization method for plates periodic in the horizontal projection was proposed in [4], where the bending of a homogeneous plate with periodically repeating inhomogeneities on its surface was studied. A more detailed asymptotic analysis of elastic plates periodic in the horizontal projection can be found, e.g., in [5, 6]. In [6], three asymptotic approximations were considered, local problems on the periodicity cell were obtained for them, and the solvability of these problems was proved. In [7], it was shown that the techniques developed for plates periodic in the horizontal projection can also be used for laminated plates. In [7], this was illustrated by an example of asymptotic analysis of an isotropic plate symmetric with respect to the midplane.     

Limit analysis of multi-layered plates. Part I: The homogenized Love-Kirchhoff model

The purpose of this paper is to determine , the overall homogenized Love-Kirchhoff strength domain of a rigid perfectly plastic multi-layered plate, and to study the relationship between the 3D and the homogenized Love-Kirchhoff plate limit analysis problems. In the Love-Kirchhoff model, the generalized stresses are the in-plane (membrane) and the out-of-plane (flexural) stress field resultants. The homogenization method proposed by Bourgeois [1997. Modélisation numérique des panneaux structuraux légers. Ph.D. Thesis, University Aix-Marseille] and Sab [2003. Yield design of thin periodic plates by a homogenization technique and an application to masonry wall. C. R. Méc. 331, 641-646] for in-plane periodic rigid perfectly plastic plates is justified using the asymptotic expansion method. For laminated plates, an explicit parametric representation of the yield surface is given thanks to the π-function (the plastic dissipation power density function) that describes the local strength domain at each point of the plate. This representation also provides a localization method for the determination of the 3D stress components corresponding to every generalized stress belonging to . For a laminated plate described with a yield function of the form , where σ^u is a positive even function of the out-of-plane coordinate x₃ and is a convex function of the local stress σ, two effective constants and a normalization procedure are introduced. A symmetric sandwich plate consisting of two Von-Mises materials ( in the skins and in the core) is studied. It is found that, for small enough contrast ratios (), the normalized strength domain is close to the one corresponding to a homogeneous Von-Mises plate [Ilyushin, A.-A., 1956. Plasticité. Eyrolles, Paris].     

Theory of limit equilibrium of bimetallic shells of revolution and circular plates

Bimetallic shells and plates are widely used in technology (see [1, 2]). An investigation into the flexure and stability of thin shells and various types of loading within the limits of elasticity has been carried out in [3]. An investigation into the load-carrying capacity of cylindrical bimetallic shells made of materials which equally resist tension and compression was carried out in [4]. In many cases the materials of the base and plating layers of bimetallic constructions possess substantially different plastic resistance under tension and compression [5]. The given paper is devoted to the investigation of the load-carrying capacity of bimetallic axisymmetric shells which are made of materials that have different resistances to tension and compression; it is also devoted to the assessment of their economy in comparison with homogeneous shells.     

Finite element analysis of post-buckling dynamics in plates—Part I: An asymptotic approach

Various static and dynamic aspects of post-buckled thin plates, including the transition of buckled patterns, post-buckling dynamics, secondary bifurcation, and dynamic snapping (mode jumping phenomenon), are investigated systematically using asymptotical and non-stationary finite element methods. In part I, the secondary dynamic instability and the local post-secondary buckling behavior of thin rectangular plates under generalized (mechanical and thermal) loading is investigated using an asymptotic numerical method which combines Koiter’s nonlinear instability theory with the finite element technique. A dynamic multi-mode reduction method—similar to its static single-mode counterpart: Liapunov–Schmidt reduction—is developed in this perturbation approach. Post-secondary buckling equilibrium branches are obtained by solving the reduced low-dimensional parametric equations and their stability properties are determined directly by checking the eigenvalues of the resulting Jacobian matrix. Typical post-secondary buckling forms—transcritical, supercritical and subcritical bifurcations are observed according to different combinations of boundary conditions and load types. Geometric imperfection analysis shows that not only the secondary bifurcation load but also changes in the fundamental post-secondary buckling behavior are affected. The post-buckling dynamics and the global analysis of mode jumping of the plates are addressed in part II.     

General non-linear finite element analysis of thick plates and shells

A non-linear finite element analysis is presented, for the elasto-plastic behavior of thick shells and plates including the effect of large rotations. The shell constitutive equations developed previously by the authors [Voyiadjis, G.Z., Woelke, P., 2004. A refined theory for thick spherical shells. Int. J. Solids Struct. 41, 3747–3769] are adopted here as a base for the formulation. A simple C⁰ quadrilateral, doubly curved shell element developed in the authors’ previous paper [Woelke, P., Voyiadjis, G.Z., submitted for publication. Shell element based on the refined theory for thick spherical shells] is extended here to account for geometric and material non-linearities. The small strain geometric non-linearities are taken into account by means of the updated Lagrangian method. In the treatment of material non-linearities the authors adopt: (i) a non-layered approach and a plastic node method [Ueda, Y., Yao, T., 1982. The plastic node method of plastic analysis. Comput. Methods Appl. Mech. Eng. 34, 1089–1104], (ii) an Iliushin’s yield function expressed in terms of stress resultants and stress couples [Iliushin, A.A., 1956. Plastichnost’. Gostekhizdat, Moscow], modified to investigate the development of plastic deformations across the thickness, as well as the influence of the transverse shear forces on plastic behaviour of plates and shells, (iii) isotropic and kinematic hardening rules with the latter derived on the basis of the Armstrong and Frederick evolution equation of backstress [Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of the multiaxial Bauschinger effect. (CEGB Report RD/B/N/731). Berkeley Laboratories. R&D Department, California.], and reproducing the Bauschinger effect. By means of a quasi-conforming technique, shear and membrane locking are prevented and the tangent stiffness matrix is given explicitly, i.e., no numerical integration is employed. This makes the current formulation not only mathematically consistent and accurate for a variety of applications, but also computationally extremely efficient and attractive.     

Breakage mechanics—Part I: Theory

Different measures have been suggested for quantifying the amount of fragmentation in randomly compacted crushable aggregates. A most effective and popular measure is to adopt variants of Hardin's [1985. Crushing of soil particles. J. Geotech. Eng. ASCE 111(10), 1177-1192] definition of relative breakage ‘B_r’. In this paper we further develop the concept of breakage to formulate a new continuum mechanics theory for crushable granular materials based on statistical and thermomechanical principles. Analogous to the damage internal variable ‘D’ which is used in continuum damage mechanics (CDM), here the breakage internal variable ‘B’ is adopted. This internal variable represents a particular form of the relative breakage ‘B_r’ and measures the relative distance of the current grain size distribution from the initial and ultimate distributions. Similar to ‘D’, ‘B’ varies from zero to one and describes processes of micro-fractures and the growth of surface area. However, unlike damage that is most suitable to tensioned solid-like materials, the breakage is aimed towards compressed granular matter. While damage effectively represents the opening of micro-cavities and cracks, breakage represents comminution of particles. We term the new theory continuum breakage mechanics (CBM), reflecting the analogy with CDM. A focus is given to developing fundamental concepts and postulates, and identifying the physical meaning of the various variables. In this part of the paper we limit the study to describe an ideal dissipative process that includes breakage without plasticity. Plastic strains are essential, however, in representing aspects that relate to frictional dissipation, and this is covered in Part II of this paper together with model examples.     

Variable kinematic models applied to free-vibration analysis of functionally graded material shells

Closed-form solutions of free-vibration problems of simply supported multilayered shells made of Functionally Graded Material have been examined in the present paper. A variable kinematic shell model, which is based on Carrera’s Unified Formulation is extended, in this work, to dynamic shell cases. Classical shell theories are compared to refined ones as well as to layer-wise kinematics and mixed assumptions based on the Reissner mixed variational theorem. A comparison with the few results available in the open literature is presented and conclusions are drawn regarding the accuracy of classical and advanced shell modeling to evaluate lower and higher vibration modes as well as the behavior of these modes in the shell thickness direction.     

各向异性弹塑性中厚度板壳的有限元分析

本文在文献［２，３］的基础上，提出了一个解各向异性弹塑性中厚度板壳问题的有限元方法。考虑材料各向异性的特点，采用了Ｈｉｌｌ推广的Ｈｕｂｅｒ－Ｍｉｓｅｓ屈服准则；借用Ｏｗｅｎ的剪切修正系数，正确计及了叠层复合材料壳体的横向剪切效应；为了避免“自锁”现象，文中采用了９节点的Ｈｅｔｅｒｏｓｉｓ二次壳单元；特别是本文利用插值外推的思想，提出了一个带预测的弧长增量控制法，显著提高了确定变形路径的计算效率。几个数值算例表明本文给出的有限元方法对于各向异性中厚度板壳的弹塑性分析有较好的精度，尤其是对具有复杂变形路径的结构计算，收敛速度提高更快。     

Dynamics of elasto-viscoplastic plates and shells

Summary In this paper, the problem of the elasto-viscoplastic dynamic behaviour of geometrically non-linear plates and shells is studied under the assumption of small strains and moderate rotations. The Chaboche and Bodner-Partom models were chosen among several types of constitutive laws. To avoid the calculation of the stiffness matrix, an effective procedure using the central difference method of solving the equations of motion was applied. The trapezoidal method was used to integrate the constitutive viscoplastic laws. A nine-node isoparametric shell element was utilised for the finite element algorithm.     

A new higher-order theory to laminated plates and shells

In this paper a new higher-order theory to laminated plates and shells is presented and then Symmetric and antisymmetric cross-ply laminated plates, cylindrie bending and bending of spherical shells are also studied. In order to examine the accuracy of the theory, several particular examples have been calculated. The numerical results are in good agreement with the exact solution, which shows the theory is possessed of higher accuracy and is easy to solve a problem with few unknowns.     

A new approach to the analysis of shells,plates and membranes with finite deflections

This paper presents a new perturbation method of analysis applicable to a class of geometrically non-linear problems of shells, plates, and membranes with translationally restrained edges. The perturbation parameter is a linear function of Poisson's ratio. The convergence of successive perturbations (i.e., approximations) is independent of the magnitudes of deflections. The method also offers a rational explanation of the efficacy of Berger's approximate equations, thus placing Berger's method on a firmer foundation while at the same time weakening his hypothesis of vanishing second membrane strain invariant in the strain energy integral. Several solutions and results are obtained for the purposes of illustration and discussion. Whenever possible, calculated values are compared with results obtained by other means.