1,721,059 research outputs found
Materials
No full textMaterials is an open access journal of related scientific research and technology development. It publishes reviews, regular research papers (articles) and short communications. Its aim is to encourage scientists to publish their experimental and theoretical results in as much detail as possible. Therefore, there is no restriction on the length of the papers. The full experimental details must be provided so that the results can be reproduced.
Materials provides a forum for publishing papers which advance the in-depth understanding of the relationship between the structure, the properties or the functions of all kinds of materials. Chemical syntheses, chemical structures and mechanical, chemical, electronic, magnetic and optical properties and various applications will be considered
International Journal Of Engineering & Applied Sciences
No full textThe journal presents its readers with broad coverage across some branches of engineering and science of the latest development and application of new solution algorithms, artificial intelligent techniques innovative numerical methods and/or solution techniques directed at the utilization of computational methods in solid and nano-scaled mechanics
Science and Engineering of Composite Materials
No full textIn view of the rapid growth of the science and technology of composite materials, there is a need for published documentation on their structure, properties, and the integration of structure-property relations with processing, design and fabrication. Science and Engineering of Composite Materials is a quarterly publication which provides a forum for discussion of all aspects related to the structure and performance under simulated and actual service conditions of composites. The publication covers a variety of subjects, such as macro and micro and nano structure of materials, their mechanics and nanomechanics, the interphase, physical and chemical aging, fatigue, environmental interactions, and process modeling. The interdisciplinary character of the subject as well as the possible development and use of composites for novel and specific applications receives special attention
Mathematical Problems in Engineering
No full textMathematical Problems in Engineering is a peer-reviewed, Open Access journal that publishes results of rigorous engineering research carried out using mathematical tools. Contributions containing formulations or results related to applications are also encouraged. The primary aim of Mathematical Problems in Engineering is rapid publication and dissemination of important mathematical work which has relevance to engineering. All areas of engineering are within the scope of the journal. In particular, aerospace engineering, bioengineering, chemical engineering, computer engineering, electrical engineering, industrial engineering and manufacturing systems, and mechanical engineering are of interest. Mathematical work of interest includes, but is not limited to, ordinary and partial differential equations, stochastic processes, calculus of variations, and nonlinear analysis
Materials Proceedings
No full textMaterials Proceedings is a journal that publishes proceedings abstracts and reports resulting from academic conferences in all areas of material sciences. Published items are approved by the conference committee and original research content is peer reviewed
Composites Part C: Open Access
No full texteditorial channel:
I. Sustainable Composites
II. Multi-functional Composites
III. Composite Structure
Anisotropic Doubly-Curved Shells - Higher-Order Strong and Weak Formulations for Arbitrarily Shaped Shell Structures
No full textThe title, Anisotropic Doubly-Curved Shells, illustrates the themes followed in the present volume. The main aim of this book is to analyze the static and dynamic behavior of doubly-curved shells made of anisotropic materials applying the Differential Quadrature (DQ) and Integral Quadrature (IQ) techniques.
The major structural theories for the analysis of the mechanical behavior of doubly-curved shell structures are presented in depth in the volume. In particular, the strong and weak formulations of the corresponding governing equations are discussed and illustrated. In addition, several numerical applications are developed to support the theoretical models. The book is made up of eight chapters, in which both the structural models and the numerical techniques are examined.
The first chapter includes the mathematical fundamentals of the Differential Quadrature (DQ) method. In particular, the Generalized Differential Quadrature (GDQ) method developed by Shu is discussed. He had the aim of improving the DQ technique presented by Bellman at the beginning of the 1970s. In particular, the weighting coefficient computations, used in the derivative approximation of any order, will be shown. Thus, the differential quadrature method based on the Lagrange polynomials (Polynomial Differential Quadrature or PDQ) and the one based on the Fourier series (Harmonic Differential Quadrature or HDQ) are described. As it is well-known, differential quadrature methods, available in the literature, differ from the choice of the approximating functions (or basis functions) and for the type of grid distributions, which locate discrete points on the domain where the derivative of a function has to be evaluated. For this reason, a general approach to the differential quadrature is proposed. The weighting coefficients for different basis functions and grid distributions are determined. Furthermore, the expressions of the principal approximating polynomials and grid distributions, available in the literature, are shown. Besides the classic orthogonal polynomials, a new class of basis functions, which depend on the radial distance between the discretization points, is presented. They are known as Radial Basis Functions (or RBFs). The general expressions for the derivative evaluation can be utilized in the local form to reduce the computational cost. From this concept the Local Generalized Differential Quadrature (LGDQ) method is derived.
Moreover, a procedure for numerical integration, based on the weighting coefficients utilized for the differential quadrature, is illustrated. This is done because several numerical applications require the evaluation of integrals in their formulation. Thus, the Generalized Integral Quadrature (GIQ) technique, introduced by Shu following the same concept of the GDQ method, is shown. This method can be used employing several basis functions, without any restriction on the point distributions for the given definition domain. To better underline these concepts some classical numerical integration schemes are reported, such as the trapezoidal rule or the Simpson method. The general approach to the integral and differential quadrature can be extended directly to the multidimensional case. For this reason, a simple and compact mathematical formulation, for the computation of the weighting coefficients and the numerical evaluation of derivatives and integrals for a two-dimensional space, is presented. An alternative approach based on Taylor series is also illustrated to approximate integrals. This technique is named as Generalized Taylor-based Integral Quadrature (GTIQ) method. Finally, the accuracy of the GDQ and GIQ methods is demonstrated through few numerical applications.
The second chapter of the book, instead, introduces the bases of the Differential Geometry, a fundamental tool for the analysis of the doubly-curved structures at issue. In fact, it allows to obtain those geometric parameters that are involved in the writing of shell governing equations. Then, the chapter continues analyzing the main shell structures that are commonly employed in engineering. For each considered doubly-curved, singly-curved or degenerate shell structure, the main geometric features are presented.
A peculiar procedure is also introduced to deal with distorted domains. For this purpose, a proper coordinate transformation is presented to define arbitrarily shaped curved surfaces. This technique is known as mapping procedure. In other words, this approach aims to convert a regular domain, described by the principal coordinates, into a distorted element. Therefore, the original structural problem is transferred in the computational domain, known as parent space, which is described by the so-called natural coordinates. In this book, a mapping procedure based on the use of Non-Uniform Rational Basis Splines (NURBS) is presented. Such techniques is known as Isogeometric Mapping.
In the third chapter, the 3D Elasticity Equations in Principal Curvilinear Coordinates are presented for shell structures. They represent the basis for the development of several engineering theories, which allow to analyze the mechanical behavior of various doubly-curved shell structures. In this chapter, the kinematic equations are obtained for a generic three-dimensional solid described in a principal curvilinear coordinate reference system. Then, the strain components are related to the corresponding stress components by means of the generalized Hooke laws for several kinds of materials. The indefinite equilibrium equations, as well as the natural boundary conditions, are eventually worked out by the Hamilton principle. The kinematic, constitutive and dynamic equilibrium equations are modified through simple geometric considerations in order to obtain the whole set of three-dimensional equations which governs the mechanical behavior of different shell structures. By substituting the kinematic and the constitutive relations into the equilibrium equations, the behavior of a generic shell structure can be described through the indefinite equilibrium equations expressed in terms of generalized displacement components, which represent the degrees of freedom of the problem at issue. The relations that are deduced through all these substitutions are known as fundamental equations and include the three aspects of the elastic problem, that are kinematic, constitutive and dynamic equilibrium, of thick and moderately thick composite shells, within a unique fundamental system.
These 3D Elasticity equations contemplate a three-dimensional structural model without taking into account any hypothesis about the displacement field. In the third chapter, the three-dimensional problem is reduced to a two-dimensional problem defined on the shell middle surface, according to the principles of the Equivalent Single Layer (ESL) approach, by means of a proper kinematic model. According to the choice of the kinematic model in hand, several Higher-order Shear Deformation Theories (HSDTs) are obtained. This approach, as it will be examined in depth in the following, represents a peculiar technique that allows to consider and study several higher-order kinematic models in a unified manner. In fact, the expansion order of the three dimensional displacements is taken as a free parameter and it can include both several thickness functions and the so-called Zig-Zag effect (known also as Murakami’s function). Inserting this general displacement field into the three-dimensional kinematic equations, these relations can be expressed in terms of generalized strain characteristics, which are defined on the shell reference surface.
As far as the constitutive equations are concerned, particular attention is given to anisotrpic materials due to the increasing development in several structural engineering areas. The scientific interest in these materials, that have the high makings of application, suggested the static and dynamic analysis of composite shell structures. A new class of composite materials, recently introduced in literature, is also taken into account. As it is well-known, laminated composite materials are affected by inevitable problems of delamination due to the presence of interfaces where different materials are in contact. On the contrary, Functionally Graded Materials (FGMs) are characterized by a continuous variation of the mechanical properties, such as the elastic modulus, material density and Poisson ratio, along a particular direction. This feature is obtained by varying gradually, along a preferential direction, the volume fraction of the constituent materials with appropriate industrial processes. Therefore, FGMs are non-homogeneous materials, typically composed of metal and ceramic. Their overall mechanical properties are obtained by means of different approaches. In particular, in the present work the classic theory of mixtures is compared to the Mori-Tanaka scheme for the computation of these mechanical properties. Other classes of advanced and innovative constituents, such as Carbon Nanotubes (CNTs) reinforced media and Variable Angle Tow (VAT) composites are illustrated.
The relations which express the dynamic equilibrium of shell structures are obtained once again through the Hamilton variational principle. In order to transform the initial three-dimensional elastic problem in a two-dimensional one, the internal actions in terms of stress resultants for each order of kinematic expansion are introduced. It should be noticed that the Hamilton principle allows to consider also the effect of non-conservative forces. Thus, it is possible to introduce the structural damping into the dynamic equilibrium equations. The procedure for evaluating the damping coefficients is performed according to the Rayleigh proportional damping technique. In addition, the current approach allows to take into account different kinds of external loads. The surface forces applied on the external shell surfaces and the volume forces which are applied in each point of the three-dimensional solid, as well as the seismic excitation and the effect of the non-linear elastic foundation, are converted in statically equivalent loads acting on the middle surface of the shell. Finally, the case of rotating shells is also included in the treatise, by introducing a general approach for arbitrary rotations.
Generally speaking, two formulations of the same system of governing equations can be developed, which are respectively the strong and weak (or variational) formulations. Once the governing equations that rule a generic structural problem are obtained, together with the corresponding boundary conditions, a differential system is written. In particular, the Strong Formulation (SF) of the governing equations is presented in the fourth chapter.
The differentiability requirement, instead, is reduced through a weighted integral statement if the corresponding Weak Formulation (WF) of the governing equations is developed. Thus, an equivalent integral formulation is presented in the fifth chapter, starting directly from the previous one. In particular, the formulation in hand is obtained by introducing a higher-order Lagrangian approximation of the degrees of freedom of the problem, which consists in the nodal displacements defined on the shell middle surface. The theoretical framework is based on a higher-order kinematic expansion of the displacement field, as shown in the fourth chapter.
On the other hand, simpler structural models are considered and illustrated in the sixth chapter. In particular, the strong and weak forms are presented for one-dimensional and two-dimensional problems. As far as one-dimensional models are concerned, a unified approach is developed to deal with several kinds of mechanical problems. The cases of the rod and the Euler-Bernoulli beam are illustrated in depth. On the other hand, the membrane, the Kirchhoff-Love plate, the plane elasticity and the Reissner-Mindlin plate are the two-dimensional problems considered in the chapter. All these structural models are described through the same analytical formulation, highlighting the fact that the various matrices and operators assume different meanings according to the structure under consideration.
The book continues with the seventh chapter, wherein the obtained numerical results are illustrated for several structural components. The dynamic analysis (free vibrations) and static analysis of composite shell structures are presented. The effect of the variation of the mechanical properties on the vibration frequencies and the stress field, for several structural theories shown in the previous chapters, are illustrated. Besides different geometries and different lamination schemes used in the numerical analyses, the convergence and stability trends of this technique are presented. Finally, the DQ and IQ numerical solutions are compared to results from the literature and the same obtained from structural code programs.
In this chapter, a posteriori recovery procedure is introduced to compute the shear and normal stresses from the two-dimensional solution through a numerical integration along the thickness via DQ method of the three-dimensional elasticity equations shown in the previous chapters. A lot of examples show the results of this recovery procedure of the stress state. All these results are useful in the structural design to avoid delamination problems in composite materials.
All the results presented in the book are obtained through the DiQuMASPAB software, acronym of “Differential Quadrature for Mechanics of Anisotropic Shells, Plates, Arches and Beams”. This code is generated using MATLAB environment which implement the considered theoretical formulations and aims to analyze the static and dynamic analyses of various shell structures.
The need of studying arbitrarily shaped domains or characterized by mechanical and geometrical discontinuities leads to the development of new numerical approach that divide the structure in finite elements. Then, the strong form or the weak form of the fundamental equations are solved inside each element. The fundamental aspects of this technique, which the authors defined respectively Strong Formulation Finite Element Method (SFEM) and Weak Formulation Finite Element Method (WFEM), are presented in the eighth chapter. The starting points of these approaches are the differential and integral quadrature methods, introduced in the first chapter. The mapping technique, used also in the finite element methods, transforms an arbitrarily shaped element into a regular element (computational element). In other words, the SFEM and WFEM denote two numerical procedures that divide the physical domain in finite elements and use the differential or integral quadrature methods to solve the strong or weak forms of the equations inside each element, mapped into the computational space. Several numerical applications regarding the static and dynamic behavior of arches, beams, plates, membranes, plane stress and strain states are reported to study the convergence and stability characteristics and reliability of these numerical techniques.
Finally, a recap of the main operations that can be performed between matrices and vectors is presented in the appendix, in which some concepts, operations and results of matrix algebra employed in the previous chapters are discussed.
This book is intended to be a reference for experts in structural, applied and computational mechanics. It can be also used as a text book, or a reference book, for a graduate or PhD courses on plates and shells, anisotropic and composite materials, vibration of continuous systems and stress recovery of the previous structures. Finally, the present book has also the same audience of the book by Professor Harry Kraus (1967). Thus, using his words: “The” present “book is aimed primarily at graduate students at the intermediate level in engineering mechanics, aerospace engineering, mechanical engineering and civil engineering, whose field of specialization is solid mechanics. Stress analysts in industry will find the” present “book a useful introduction that will equip them to read further in the literature of solutions to technically important shell problems, while research specialists will find it useful as an introduction to current theoretical work. This volume is not intended to be an exhaustive treatise on the theory of thin” and thick “elastic shells but, rather, a broad introduction from which each reader can follow his own interests further”. In addition, it is opinion of the authors that the present volume represents the continuation and the generalization of the work begun by Kraus in 1967
Modelling of Damaged Laminated and Sandwich Shell Structures by means of Higher-order Shear Deformation Theories
No full textThe main aim of the current research is the development of a mathematical formulation for the modelling of damage in laminated and sandwich composite shells. For this purpose, the damage of some areas of the structures can be seen as concentrated decays of the mechanical properties of the elastic constituents. In general, several kinds of damage can affect the mechanical behavior of a generic laminated structure, such as microcracking, debonding, fiber ruptures, and transverse matrix cracking, as specified in [1].
Without investigating the causes of the damage, the current approach suggests to introduce peculiar functions that multiply directly the mechanical properties of the elastic media, expressed in terms of engineering constants. To this aim, the Gaussian function and an ellipse shaped law are used to model a quick variation of the mechanical properties within the whole structural domain. By setting properly the parameters that characterize these distributions, it is possible to control the intensity of the deterioration and the width of the damaged areas, as well as the point of applications.
The present approach is employed to characterize the damage in some doubly-curved shells characterized by different radii of curvature. The difficulties related to the description of these curved surfaces are overcome by means of an analytical formulation based on differential geometry [2]. As far as the mechanical properties are concerned, several constituents are considered and combined.
The theoretical framework is based on a formulation that allows to develop easily different kinematic models and expansions in a unified manner. Thus, several Higher-order Shear Deformation Theories, which can include also the zig-zag effect, are employed. In fact, it has been proven that peculiar mechanical configurations require an enriched structural model, since lower-order theories could be inadequate to capture the effective mechanical behavior.
Finally, a numerical technique able to solve the strong form of the governing equations is used. For this purpose, the partial derivatives that appear in the fundamental system are directly approximated through the Generalized Differential Quadrature method due to its accuracy [3].
References
[1] Tornabene, F., Fantuzzi, N., Bacciocchi, M., “Linear Static Behavior of Damaged Laminated Composite Plates and Shells”, Materials, 10, 811, 1-52 (2017).
[2] Tornabene, F., Fantuzzi, N., Bacciocchi, M., and E. Viola, Laminated Composite Doubly-Curved Shell Structures. Differential Geometry. Higher-order Structural Theories, Esculapio, Bologna (2016).
[3] Tornabene, F., Fantuzzi, N., Ubertini, F., Viola, E., “Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey”, Applied Mechanics Reviews, 67, 020801-1-55
Strong and Weak Formulations for the Analysis of Arbitrarily Shaped Laminated Composite Structures
No full textA numerical approach is developed to deal with arbitrarily shaped structures. Two different methodologies are used to this aim, which are based on the Differential Quadrature and Integral Quadrature methods, respectively. These numerical methods are able to approximate both derivatives and integrals [1]. Therefore, the strong and weak formulations of the governing equations can be solved. As shown in the paper [2], these approaches are accurate, reliable and stable, when employed to obtain the mechanical response of various kinds of structures, such as plates, shells and membranes. In particular, their effectiveness is proven by means of the comparison with the analytical solutions available in the literature, both for isotropic and composite structures.
With respect to other approaches such as the Finite Element Method (FEM), the proposed methodologies are able to get the solution with few degrees of freedom. In addition, the convergence behavior is faster than the FEM.
A domain decomposition based on Isogeometric analysis is developed to analyze the mechanical behavior of arbitrarily shaped structures. The so-called blending functions are used to deal with discontinuities and distortions by means of a reduced number of elements [3, 4]. Thus, a nonlinear mapping is achieved by employing NURBS curves. According to the numerical method used in the computation, the strong and weak formulations are solved within each element. The effect of distorted meshes on the solution is investigated, as well. The numerical methods at issue are named Strong Formulation Finite Element Method (SFEM) and Weak Formulation Finite Element Method (WFEM).
References
[1] Tornabene, F., Fantuzzi, N., Ubertini, F., Viola, E., "Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey", Applied Mechanics Reviews, 67, 02081-1-55 (2015).
[2] Tornabene, F., Fantuzzi, Bacciocchi, M., "Strong and weak formulations based on differential and integral quadrature methods for the free vibration analysis of composite plates and shells: Convergence and accuracy", Engineering Analysis with Boundary Elements. In press. DOI: 10.1016/j.enganabound.2017.08.020.
[3] Fantuzzi, N., Tornabene, F., "Strong Formulation Isogeometric Analysis (SFIGA) for Laminated Composite Arbitrarily Shaped Plates", Composites Part B - Engineering, 96, 173-203 (2016).
[4] Tornabene, F., Fantuzzi, Bacciocchi, M., "The GDQ Method for the Free Vibration Analysis of Arbitrarily Shaped Laminated Composite Shells Using a NURBS-Based Isogeometric Approach", Composite Structures, 154, 190-218 (2016)
Peculiar Convergence and Accuracy for Laminated Moderately Thick Plates of Arbitrary Shape in Free Vibrations
No full textAs it is well known, engineering theories for plates and shells simplify the three-dimensional (3D) elasticity problem by introducing kinematic hypothesis which lead to simpler mathematical problems. Therefore, such simplified theories have limitations, which are strictly related to the initial hypotheses. The present work is based on the so-called Reissner-Mindlin theory or First-order Shear Deformation, which is used to study “moderately thick” plates. The term “moderately thick” refers to the fact that the plate is not “thin” as in the Classical Laminated Plate Theory (CLPT) or Kirchhoff-Love Theory and not “thick” as in the classical 3D theory of elasticity. Once the physical problem is mathematically well-posed, it is generally solved via numerical methods due to the complexity of finding analytical or semi-analytical solutions.
The present work aims to show a peculiar behavior in the solution of such problems by comparing the results obtained using strong and weak form finite element methods when the plates are in free vibrations. In particular, the authors compare the results obtained with two- and three-dimensional theories as a function of the plate thickness
- …
