Andreas Öchsner

This monograph provides a compact introduction into the classical, i.e. rate-independent, plasticity theory. Starting from the engineering stress-strain diagram, the concept of elastic and elasto-plastic material behavior is introduced, as well as the concept of uniaxial and multiaxial stress states. Continuum mechanical modeling in the elasto-plastic range requires, in regards to the constitutive equation, in addition to the elastic law (e.g. Hooke’s law), a yield condition, a flow rule and a hardening rule. These basic equations are thoroughly introduced and explained for one-dimensional stress states. Considering three-dimensional plasticity, different sets of stress invariants to characterize the stress matrix and the decomposition of the stress matrix in its hydrostatic and deviatoric part are introduced. Furthermore, the concept of the yield condition, flow rule and hardening rule is generalized for multiaxial stress states. Some typical yield conditions are introduced and theirgraphical representation in different stress spaces is discussed in detail. The book concludes with an introduction in the elasto-plastic finite element simulation of mechanical structures. In the context of numerical approximation methods, the so-called predictor-corrector methods are used to integrate the constitutive equations. This is again introduced in detail based on one-dimensional stress states and afterwards generalized to the three-dimensional case. Test your knowledge with questions and answers about the book in the Springer Nature Flashcards app.

This book provides a systematic introduction to composite materials, which are obtained by a layer-wise stacking of one-dimensional bar/beam elements. Each layer may have different mechanical properties but each single layer is considered as isotropic. The major idea is to provide a simplified theory to easier understand the classical two-dimensional laminate theory for composites based on laminae with unidirectional fibers. In addition to the elastic behavior, failure is investigated based on the maximum stress, maximum strain, Tsai-Hill, and the Tsai-Wu criteria. Partial differential equations lay the foundation to mathematically describe the mechanical behavior of any classical structural member known in engineering mechanics, including composite materials. The so-called classical laminate theory provides a simplified stress analysis, and a subsequent failure analysis, without the solution of the system of coupled differential equations for the unknown displacements. Theprocedure provides the solution of a statically indeterminate system based on a generalized stress–strain relationship under consideration of the constitutive relationship and the definition of the so-called stress resultants. This laminate theory is typically provided for two-dimensional plane problems, where the basic structural element is a simple superposition of a classical plane elasticity element with a thin plate element under the consideration of an orthotropic constitutive law. This two-dimensional approach and the underlying advanced continuum mechanical modeling might be very challenging for some students, particularly at universities of applied sciences. Thus, a reduced approach, the so-called simplified classical laminate theory, has been developed. The idea is to use solely isotropic one-dimensional elements, i.e., a superposition of bar and beam elements, to introduce the major calculation steps of the classical laminate theory. Understanding this simplified theory is much easier and the final step it to highlight the differences when moving to the general two-dimensional case.

This book in the advanced structured materials series provides first an introduction to the mircomechanics of fiber-reinforced laminae, which deals with the prediction of the macroscopic mechanical lamina properties based on the mechanical properties of the constituents, i.e., fibers and matrix. Composite materials, especially fiber-reinforced composites, are gaining increasing importance since they can overcome the limits of many structures based on classical metals. Particularly, the combination of a matrix with fibers provides far better properties than the constituents alone. Despite their importance, many engineering degree programs do not treat the mechanical behavior of this class of advanced structured materials in detail, at least on the bachelor’s degree level. Thus, some engineers are not able to thoroughly apply and introduce these modern engineering materials in their design process. The second part of this book provides a systematic and thorough introduction to the classical laminate theory based on the theory for plane elasticity elements and classical (shear-rigid) plate elements. The focus is on unidirectional lamina which can be described based on orthotropic constitutive equations and their composition to layered laminates. In addition to the elastic behavior, failure is investigated based on the maximum stress, maximum strain, Tsai-Hill, and the Tsai-Wu criteria. The introduced classical laminate theory provides a simplified stress analysis, and a subsequent failure analysis, without the solution of the system of coupled differential equations for the unknown displacements in the three coordinate directions. The book concludes with a short introduction to a calculation program, the so-called Composite Laminate Analysis Tool (CLAT), which allows the application of the classical laminate based on a sophisticated Python script.

This book first provides a systematic and thorough introduction to the classical laminate theory for composite materials based on the theory for plane elasticity elements and classical (shear-rigid) plate elements. The focus is on unidirectional lamina which can be described based on orthotropic constitutive equations and their composition to layered laminates. In addition to the elastic behavior, failure is investigated based on the maximum stress, maximum strain, Tsai-Hill, and the Tsai-Wu criteria. The solution of the fundamental equations of the classical laminate theory is connected with extensive matrix operations, and many problems require in addition iteration loops. Thus, a classical hand calculation of related problems is extremely time consuming. In order to facilitate the application of the classical laminate theory, we decided to provide a Python-based computational tool, the so-called Composite Laminate Analysis Tool (CLAT) to easily solve somestandard questions from the context of fiber-reinforced composites. The tool runs in any standard web browser and offers a user-friendly interface with many post-processing options. The functionality comprises stress and strain analysis of lamina and laminates, derivation of off-axis elastic properties of lamina, and the failure analysis based on different criteria.

This book treats the mechanical behavior of one-dimensional sandwich structures, a typicaloncept in the context of lightweight design. Such structures are composed of different constituent (e.g., layers) in order to achieve overall properties, which are better than for a single component alone. This book covers the basic mechanical load cases, i.e., tension/compression, bending, and shear. Based on this knowledge, different failure modes, i.e., plastic yielding, and global and local instabilities are investigated. In addition, an introduction to classic optimization problems, i.e., the formulation of an objective function (e.g., the weight of a structure) and corresponding restrictions, is included. The consideration here is limited to one- or two-dimensional design spaces, i.e., with a maximum of two design variables. For such simple cases, the minimum of the objective function can often be determined using analytical or graphical methods.

This book is the 3rd edition of an introduction to modern computational mechanics based on the finite element method. This third edition is largely extended, adding many new examples to let the reader understand the principles better by performing calculations by hand, as well as numerical example to practice the finite element approach to engineering problems. The new edition comes together with a set of digital flash cards with questions and answers that improve learning success. Featuring over 100 more pages, the new edition will help students succeed in mechanics courses by showing them how to apply the fundamental knowledge they gained in the first years of their engineering education to more advanced topics. In order to deepen readers’ understanding of the equations and theories discussed, each chapter also includes supplementary problems. These problems start with fundamental knowledge questions on the theory presented in the respective chapter, followed by calculation problems. In total, over 80 such calculation problems are provided, along with brief solutions for each. Test your knowledge with questions and answers about the book in the Springer Nature Flashcards app.

This book contains a comprehensive treatment of advanced LaTeX features. The focus is on the development of high quality documents and presentations, by revealing powerful insights into the LaTeX language. The well-established advantages of the typesetting system LaTeX are the preparation and publication of platform-independent high-quality documents and automatic numbering and cross-referencing of illustrations or references. These can be extended beyond the typical applications, by creating highly dynamic electronic documents. This is commonly performed in connection with the portable document format (PDF), as well as other programming tools which allow the development of extremely flexible electronic documents.

Diese Studienhilfe zu numerischen Optimierungsverfahren richtet sich an Studierende des Maschinenbaus im Grundstudium und im Hauptstudium. Optimierungsverfahren gewinnen zunehmend an Bedeutung für den Leichtbau, wo eine Gewichtsreduzierung z. B. im Automobilbau oder in der Luft- und Raumfahrtindustrie zu einem geringeren Kraftstoffverbrauch und einer entsprechenden Senkung der Betriebskosten sowie zu positiven Auswirkungen auf die Umwelt führen kann. Basierend auf dem freien Computeralgebrasystem Maxima stellen die Autoren Verfahren zur numerischen Lösung ingenieurmathematischer Probleme sowie Anwendungen aus traditionellen Lehrveranstaltungen zur Festigkeit von Werkstoffen vor. Die mechanischen Theorien konzentrieren sich auf die typischen eindimensionalen Strukturelemente, d.h. Federn, Stäbe und Euler-Bernoulli-Balken, um die Komplexität des numerischen Rahmens zu reduzieren und den resultierenden Entwurf auf eine geringe Anzahl von Variablen zu beschränken. Die Verwendung eines Computeralgebrasystems und der darin enthaltenen Funktionen, z. B. für Ableitungen oder Gleichungslösungen, ermöglicht eine stärkere Konzentration auf die Methodik der Optimierungsverfahren und nicht auf Standardverfahren. Das Buch enthält auch zahlreiche Beispiele, darunter einige, die mit Hilfe eines grafischen Ansatzes gelöst werden können, um dem Leser ein besseres Verständnis der Computerimplementierung zu vermitteln.

This book is intended as a study aid for the finite element method. Based on the free computer algebra system Maxima, we offer routines to symbolically or numerically solve problems from the context of two-dimensional problems. For this rather advanced topic, classical ‘hand calculations’ are difficult to perform and the incorporation of a computer algebra system is a convenient approach to handle, for example, larger matrix operations. The mechanical theories focus on the classical two-dimensional structural elements, i.e., plane elements, thin or classical plates, and thick or shear deformable plate elements. The use of a computer algebra system and the incorporated functions, e.g., for matrix operations, allows to focus more on the methodology of the finite element method and not on standard procedures. Furthermore, we offer a graphical user interface (GUI) to facilitate the model definition. Thus, the user may enter the required definitions in a source code manner directly in wxMaxima or use the GUI which is able to execute wxMaxime to perform the calculations.

This monograph provides a compact introduction into the classical, i.e. rate-independent, plasticity theory. Starting from the engineering stress-strain diagram, the concept of elastic and elasto-plastic material behavior is introduced, as well as the concept of uniaxial and multiaxial stress states. Continuum mechanical modeling in the elasto-plastic range requires, in regards to the constitutive equation, in addition to the elastic law (e.g. Hooke’s law), a yield condition, a flow rule and a hardening rule. These basic equations are thoroughly introduced and explained for one-dimensional stress states. Considering three-dimensional plasticity, different sets of stress invariants to characterize the stress matrix and the decomposition of the stress matrix in its hydrostatic and deviatoric part are introduced. Furthermore, the concept of the yield condition, flow rule and hardening rule is generalized for multiaxial stress states. Some typical yield conditions are introduced and theirgraphical representation in different stress spaces is discussed in detail. The book concludes with an introduction in the elasto-plastic finite element simulation of mechanical structures. In the context of numerical approximation methods, the so-called predictor-corrector methods are used to integrate the constitutive equations. This is again introduced in detail based on one-dimensional stress states and afterwards generalized to the three-dimensional case. Test your knowledge with questions and answers about the book in the Springer Nature Flashcards app.

Dieses Lehrbuch stellt die unterschiedlichen Leichtbaukonzepte anhand einfacher eindimensionaler Strukturen in sehr verständlicher Weise dar und ermöglicht einen leichten Einstieg in das Thema. Es werden nachvollziehbare Informationen und Hinweise zur Werkstoffauswahl und geometrischen Gestaltung von Bauteilen gegeben.

This book provides a systematic and thorough introduction to the classical laminate theory based on the theory for plane elasticity elements and classical (shear–rigid) plate elements. It focus on unidirectional lamina which can be described based on orthotropic constitutive equations and their composition to layered laminates. In addition to the elastic behavior, failure is investigated based on the maximum stress, maximum strain, Tsai–Hill and the Tsai–Wu criteria.

This book provides a systematic and thorough overview of the classical bending members based on the theory for thin beams (shear-rigid) according to Euler-Bernoulli, and the theories for thick beams (shear-flexible) according to Timoshenko and Levinson. The understanding of basic, i.e., one-dimensional structural members, is essential in applied mechanics. A systematic and thorough introduction to the theoretical concepts for one-dimensional members keeps the requirements on engineering mathematics quite low, and allows for a simpler transfer to higher-order structural members. The new approach in this textbook is that it treats single-plane bending in the x-y plane as well in the x-z plane equivalently and applies them to the case of unsymmetrical bending. The fundamental understanding of these one-dimensional members allows a simpler understanding of thin and thick plate bending members. Partial differential equations lay the foundation to mathematically describe the mechanical behavior of all classical structural members known in engineering mechanics. Based on the three basic equations of continuum mechanics, i.e., the kinematics relationship, the constitutive law, and the equilibrium equation, these partial differential equations that describe the physical problem can be derived. Nevertheless, the fundamental knowledge from the first years of engineering education, i.e., higher mathematics, physics, materials science, applied mechanics, design, and programming skills, might be required to master this topic.

This book presents important contributions to modern theories concerning the distribution theory applied to convex analysis (convex functions, functions of lower semicontinuity, the subdifferential of a convex function). The authors prove several basic results in distribution theory and present ordinary differential equations and partial differential equations by providing generalized solutions. In addition, the book deals with Sobolev spaces, which presents aspects related to variation problems, such as the Stokes system, the elasticity system and the plate equation. The authors also include approximate formulations of variation problems, such as the Galerkin method or the finite element method. The book is accessible to all scientists, and it is especially useful for those who use mathematics to solve engineering and physics problems. The authors have avoided concepts and results contained in other books in order to keep the book comprehensive. Furthermore, they do not present concrete simplified models and pay maximal attention to scientific rigor.

This book is focused on the introduction of the finite difference method based on the classical one-dimensional structural members, i.e., rods/bars and beams. It is the goal to provide a first introduction to the manifold aspects of the finite difference method and to enable the reader to get a methodical understanding of important subject areas in structural mechanics. The reader learns to understand the assumptions and derivations of different structural members. Furthermore, she/he learns to critically evaluate possibilities and limitations of the finite difference method. Additional comprehensive mathematical descriptions, which solely result from advanced illustrations for two- or three-dimensional problems, are omitted. Hence, the mathematical description largely remains simple and clear.

This book contains a comprehensive treatment of advanced LaTeX features. The focus is on the development of high quality documents and presentations, by revealing powerful insights into the LaTeX language. The well-established advantages of the typesetting system LaTeX are the preparation and publication of platform-independent high-quality documents and automatic numbering and cross-referencing of illustrations or references. These can be extended beyond the typical applications, by creating highly dynamic electronic documents. This is commonly performed in connection with the portable document format (PDF), as well as other programming tools which allow the development of extremely flexible electronic documents.

This book provides an introduction to the micromechanics of fiber-reinforced laminae, which deals with the prediction of the macroscopic mechanical lamina properties based on the mechanical properties of the constituents, i.e., fibers and matrix. Composite materials, especially fiber-reinforced composites, are gaining increasing importance since they can overcome the limits of many structures based on classical metals. Particularly, the combination of a matrix with fibers provides far better properties than the components alone. Despite their importance, many engineering degree programs do not treat the mechanical behavior of this class of advanced structured materials in detail, at least on the Bachelor degree level. Thus, some engineers are not able to thoroughly apply and introduce these modern engineering materials in their design process. The focus is on unidirectional lamina which can be described based on orthotropic constitutive equations. Three classical approaches to predict the elastic properties, i.e., the mechanics of materials approach, the elasticity solutions with contiguity after Tsai, and the Halpin–Tsai relationships, are presented. The quality of each prediction is benchmarked based on two different sets of experimental values. The book concludes with optimized representations, which were obtained based on the least square approach for the used experimental data sets.

This book is intended as a study aid for the finite element method. Based on the free computer algebra system Maxima, we offer routines to symbolically or numerically solve problems from the context of two-dimensional problems. For this rather advanced topic, classical ‘hand calculations’ are difficult to perform and the incorporation of a computer algebra system is a convenient approach to handle, for example, larger matrix operations. The mechanical theories focus on the classical two-dimensional structural elements, i.e., plane elements, thin or classical plates, and thick or shear deformable plate elements. The use of a computer algebra system and the incorporated functions, e.g., for matrix operations, allows to focus more on the methodology of the finite element method and not on standard procedures. Furthermore, we offer a graphical user interface (GUI) to facilitate the model definition. Thus, the user may enter the required definitions in a source code manner directly in wxMaxima or use the GUI which is able to execute wxMaxime to perform the calculations.

This book uses a novel concept to teach the finite element method, applying it to solid mechanics. This major conceptual shift takes away lengthy theoretical derivations in the face-to-face interactions with students and focuses on the summary of key equations and concepts; and to practice these on well-chosen example problems. For this new, 2nd edition, many examples and design modifications have been added, so that the learning-by-doing features of this book make it easier to understand the concepts and put them into practice. The theoretical derivations are provided as additional reading and students must study and review the derivations in a self-study approach. The book provides the theoretical foundations to solve a comprehensive design project in tensile testing. A classical clip-on extensometer serves as the demonstrator on which to apply the provided concepts. The major goal is to derive the calibration curve based on different approaches, i.e., analytical mechanics and based on the finite element method, and to consider further design questions such as technical drawings, manufacturing, and cost assessment. Working with two concepts, i.e., analytical and computational mechanics strengthens the vertical integration of knowledge and allows the student to compare and understand the different concepts, as well as highlighting the essential need for benchmarking any numerical result.

This book provides a systematic and thorough introduction to the classical laminate theory based on the theory for plane elasticity elements and classical (shear–rigid) plate elements. It focus on unidirectional lamina which can be described based on orthotropic constitutive equations and their composition to layered laminates. In addition to the elastic behavior, failure is investigated based on the maximum stress, maximum strain, Tsai–Hill and the Tsai–Wu criteria.