Solid Mechanics

An All-Terrain Vehicle is an Off Road vehicle that has the ability to drive on any terrain
without any difficulty. These vehicles are designed to sustain any sort of extreme loads that act
during off road driving. All-Terrain Vehicles are fitted with large tires with grooves for better
traction on slippery terrains, high ground clearance to avoid obstacles, and usually designed with
high torque rather than speed. As compared to commercial vehicles, these vehicles are designed
to be more strong and durable. These vehicles are vastly used in Deserts, Mountains, forests or any
other gravel surfaces.
All-Terrain Vehicles are vastly used in farming in India. Best examples are tractors and
snow mobiles. The main key for such vehicles is the powertrain assembly which is unique and
differs from other vehicles. The power train includes Engine, Manual Gearbox, Limited slip
Differential or in some cases a reduction gearbox coupled with a continuously Variable
Transmission (CVT).CVT is the new technology being implemented in Off Road Racing.
BAJA vehicles are the most famous Off Road Vehicles in India.However,due to the
extreme loads caused by the gravel tracks and the constraints to reduce the weight, one major
component is susceptible to failure—Reduction Gearbox. Gearbox is a costly component and on
failure, the vehicle is immobile The Power Train assembly is so complicated that it occupies a lot
of space in the vehicle hence increasing the overall weight. The aim of this project is to design a
gearbox that is compact, lightweight and has extended life. Major fields covered are Material
selection, Machine Engineering Design of Gears, Shafts, Gearbox Casing, Bearing selection,
Vibrations caused due to the Engine. These analytical designs have been validated through the
help of CAD and FEA software like SolidWorks and ANSYS. Various Analysis like Structural,
Repeated loads and Computational Fluid Dynamics (CFD) have been performed to study the
behavior of components and oil flow during operations. The design is also finalized with the fits
required for assembly and the final product is expected to be efficient, light weight, compact and
long lasting than the other Gearboxes being used today.
Keywords: All-Terrain Vehicle, Reduction Gearbox, Computer Aided Engineering (CAD),
Computational Fluid Dynamics (CFD), Vibrations, Finite Element Analysis (FEA)

Stereo digital image correlation (StereoDIC) measurements of wrinkle shapes and surface deformations along with numerical simulation results of prepreg tapes steered along paths having different radii of curvature without adhesion to a Teflon-coated substrate are presented. Experiments are conducted at temperatures ranging from 25 to 350°C on thermoplastic prepreg tape (APC-2/AS4) having various widths. For the baseline studies at room temperature (25°C), finite element simulation results from the combined experimental-computational study show very good agreement for both the wrinkling mode shape and general trends in the surface strains. Measurements do indicate a sharp change in the longitudinal and transverse strains for the two smallest steering radii, indicating the development of mesoscale damage in the prepreg. Finally, results from the experiments confirm that there is no significant deviation in the deformation response of the prepreg tape when performing steering at temperatures from 25 to 110°C. However, significant differences in the deformation pattern of the thermoplastic tape are observed above the glass transition temperature, T g , of the matrix material.

The increase in  using the composite materials in engineering applications make it necessary to improve the properties of composite materials to be able doing its work . One of the method to improve the mechanical properties for fibrous composite materials is Pre-stressing the fibers during room temperature curing process . This process give the more stiffness for composite and make it more stronger and able to withstand the loading subjected to it . This paper show that the pre-stressing fiber process on the glass fiber/epoxy composite materials will give the more strength and make the composite able to carry larger load . The pre-stressing fibers level will be  ( 0 , 2 , 4, 6, 8, 10 , 15 , and 20MPa) that applied on the E-glass fibers with epoxy resin during the curing process with fiber volume fraction about (30%) , and after solidification complete the samples tested under tensile loading by using INISTRON machine  . The results show the improvement in tensile strengths reach to(70%) at pre-stressing level (20MPa) while the increasing in maximum tensile strain reach to( 46%) . The modeling of composite samples subjected to tensile load by using finite element method with package (ANSYS V.10) . 
The relation between pre-stress levels and maximum deformation  at different external loading between (1kN to 4.5kN) for composite sample were plotted for experimental and finite element results . The differences between experimental and ANSYS results was about (6-12%).

The paper concerns the thermoelastic problems in a thermosensitive elliptical plate subjected to the activity of a heat source which changes its place on the plate surface with time. The solution of conductivity equation and the corresponding initial and boundary conditions is obtained by employing a new integral transform technique. In addition, the intensities of bending moments, resultant force, etc. are formulated involving the Mathieu and modified functions and their derivatives. The analytical solution for the thermal stress components is obtained in terms of resultant forces and resultant moments.

Lorsqu'un corps est soumis à des contraintes externes, celui-ci subit des déformations qui dépendent de l'intensité de ces contraintes. Si ces dernières sont faibles, on observe expérimentalement que les déformations sont proportionnelles aux tensions appliquées. La constante de proportionnalité est une caractéristique du matériau et du type de déformation subi par celui-ci.

Preface

This book collects most of the lecture notes I have developed and I have been using in a course on Continuum Mechanics for the Master of Science Course on Computational Mechanics and the Master Course on Numerical Methods in Engineering taught at the Barcelona School of Civil Engineering, Technical University of Catalonia, UPC BarcelonaTech, in Barcelona, Spain.
When the Master of Science Course on Computational Mechanics began, the Director of the Master asked me to take charge and to develop a course on Continuum Mechanics. The course should have around 40-45 class hours and  would be scheduled to be taught in the first semester of the first year of the Master. I embarked on this path by designing and developing what has been my flagship and signature course on Continuum Mechanics.

This book has been designed as an updated and expanded version of my Continuum Mechanics lecture notes and has been structured in 10 chapters. As an introduction, Chapters 1 and 2 are devoted to Tensor Algebra and Tensor Analysis. Part I of the book includes the next 3 chapters. Chapter 3 is devoted to Nonlinear Kinematics, Chapter 4 deals with Stresses, and Chapter 5 addresses the Fundamental Conservation/Balance Laws. Part II includes the next 4 chapters. Chapter 6 is devoted to Finite Deformation Hyperelasticity, Chapter 7 to Finite Deformation Plasticity, Chapter 8 to Finite Deformation Coupled Thermoplasticity, and Chapter 9 to Finite Deformation Contact Mechanics. The last chapter, Chapter 10, deals with Variational Methods in Solid Mechanics, including coupled thermoplasticity and contact problems.

The common feature of Part I is that all the contents are always valid, regardless of the material, be it solid or fluid. At the end of Part I we should be able to set up in local spatial/material form, the fundamental governing equations and inequalities for a Continuum Mechanics problem. The fundamental governing equations and inequalities arise from the fundamental conservation/balance principles and from the constraints imposed by the second law of thermodynamics, and are valid for any material. At the end of Part I, we should be able to check how many governing equations we have, and how many unknowns are involved in those governing equations and inequalities. We will realize that we have more unknowns than equations. So, to have a well posed problem, we would need to add a number of equations, so that the final number of equations is equal to the number of unknowns. The equations we should add would depend on the material and for some particular material models will be considered in Part II of the book.

Part II of the book is devoted to presenting some nonlinear constitutive models for Nonlinear Solid Mechanics, including Finite Deformation Hyperelasticity, Finite Deformation Plasticity, Finite Deformation Coupled Thermoplasticity, and Finite Deformation Contact Mechanics. Constitutive equations are derived within a thermodynamically consistent framework. Finite deformation elastoplasticity models are based on the multiplicative decomposition of the deformation gradient and the notion of an intermediate configuration. Different formulations based on the intermediate configuration, the current or spatial configuration, and the material configuration, are considered. The last chapter is devoted to Variational Methods in Solid Mechanics, a fundamental topic in Computational Mechanics.

The book may be used as a text book for a Master’s course on Nonlinear Continuum Mechanics for graduate students in Civil, Mechanical or Aerospace Engineering, Applied Mathematics or Applied Physics, with interest in Continuum Mechanics and Computational Mechanics.

I am grateful to my colleagues and friends Jean-Philippe Ponthot at University of Liege, and Mariana Seabra at University of Porto for their careful reading of the manuscript.

I hope you enjoy reading the book as much as I have enjoyed writing it.

Carlos Agelet de Saracibar
Barcelona, Spain, 22 March 2022

W szczegółowy sposób omówione zostały założenia ogólnej nieliniowej teorii sprężystości i wyprowadzenia podstawowych elementów mechaniki ośrodka ciągłego, a także przytoczone zostały dowody fundamentalnych twierdzeń rozważanej teorii w ramach kinematyki, statyki i dynamiki ciał odkształcalnych. Omówiono kwestie formułowania związków konstytutywnych dla ciał izotropowych i anizotropowych zarówno o liniowej, jak i nieliniowej charakterystyce, ze szczególnym uwzględnieniem zagadnienia energetycznego sprzężenia miar naprężenia i odkształcenia oraz zastosowania rozkładu widmowego tensorów sprężystości. Osobno omówiono liniowy wariant teorii z różnymi jego sformułowaniami oraz metodami analitycznymi, m.in. zastosowaniem funkcji naprężeń, potencjałów, metod półodwrotnych oraz analizy zespolonej. Rozważania te poszerzono o opis teorii ogólniejszych – mechanikę ośrodka mikropolarnego Cosseratów oraz mechanikę ośrodka mikromorficznego Mindlina. W ramach teorii liniowej przedstawiono matematyczny opis zjawisk elastooptyki, propagacji płaskich i sferycznych fal sprężystych, a także propagacji sprężystych fal powierzchniowych. Przytoczono i udowodniono twierdzenia wariacyjne teorii liniowej oraz omówiono ich zastosowanie w ramach numerycznych metod rozwiązywania zagadnień tej teorii. Przeanalizowano rozwiązania trzynastu klasycznych zagadnień liniowej teorii sprężystości oraz zagadnień pokrewnych, ze szczególnym uwzględnieniem zagadnień dynamicznych. Całość uzupełniona została rozbudowanym dodatkiem poświęconym rachunkowi tensorowemu i absolutnemu rachunkowi różniczkowemu.

1. Define stress. When an external force acts on a body, it undergoes deformation. At the same time the body resists deformation. The magnitude of the resisting force is numerically equal to the applied force. This internal resisting force per unit area is called stress. Stress = Force/Area 2. Define strain When a body is subjected to an external force, there is some change of dimension in the body. Numerically the strain is equal to the ratio of change in length to the original length of the body.= P/A unit is N/mm^2 Strain = Change in length/Original length e = δL/L 3. State Hooke's law. It states that when a material is loaded, within its elastic limit, the stress is directly proportional to the strain. Stress α Strain σ α e σ = Ee E = σ/e unit is N/mm^2 Where, E-Young's modulus σ-Stress e-Strain 4. Define shear stress and shear strain. The two equal and opposite force act tangentially on any cross sectional plane of the body tending to slide one part of the body over the other part. The stress induced is called shear stress and the corresponding strain is known as shear strain. 5. Define Poisson's ration. When a body is stressed, within its elastic limit, the ratio of lateral strain to the longitudinal strain is constant for a given material. Poisson' ratio (µ or 1/m) = Lateral strain /Longitudinal strain 6. State the relationship between Young's Modulus and Modulus of Rigidity. E = 2G (1+1/m) Where, E-Young's Modulus K-Bulk Modulus 1/m-Poisson's ratio

Course intended for Undergraduate and Post Graduate Level Mechanical Engineering Vibrations course including Undamped free vibrations, Damped free vibrations, Transverse vibrations, Torsional Vibrations of multi degree of freedom and geared systems, Forced Vibrations,Determining equation of Motion of Two and Multi degree freedom systems by Euler- Lagrangian Method and Vibration measurements.

Young's modulus is property that measures the elastic strength of a given material.  There are different methods to find the elasticity of materials like bulk modulus and modulus of rigidity.  Young's modulus can be determined by the application of deforming force along the length of material.  The following article educates you, how the young's experiment has been performed.  The students of Engineering and B.Sc., can be benefited from this article.

إنَّ مؤلف هذا الكتاب وإيماناً منه بالدور العظيم والمقَّدر الذي يقوم به الأستاذ الجامعي في إثراء حركة التأليف والتعريب والترجمة للمراجع والكتب الهندسية القِّيمة يأمل أن يفي هذا الكتاب بمتطلبات برامج البكالوريوس ، الدبلوم العالي والدبلوم العام لطلاب الهندسة الميكانيكية ، الهندسة المدنية وهندسة الإنتاج أو التصنيع لما له من أهمية كبيرة في تغطية جزء من مقررات عمليات التصنيع وعلم المواد وخواص المواد الهندسية وميكانيكا ومتانة المواد .
يتفق هذا الكتاب لغوياً مع القاموس الهندسي الموّحد السوداني ويُعدُّ مرجعاً في مجاله حيث يمكن أن يستفيد منه الطالب والمهندس والباحث . معظم مادة هذا الكتاب مقتبسة من محاضرات ومذكرات مؤلفة في تدريسه لهذا المقرر لفترة تزيد قليلاً عن عشرون عاماً .
يهدف هذا الكتاب لتعريف طالب الهندسة ببعض عمليات التشكيل والتشغيل اللدن للمعادن ذات الطابع الكلاسيكي (i.e. التقليدي) أو الحديث .
يشتمل هذا الكتاب على ستة فصول. حيث يستعرض الفصل الأول أهمية المعادن في الصناعة ، الخواص الفيزيائية والميكانيكية للمعادن ، بعض الاختبارات الهامة التي تُجرى على المواد الهندسية مثل اختبار الشد ، اختبار مقاومة الصدمات واختبار الصلادة بالإضافة إلى طرق تحضير وتجهيز المعادن لعمليات التشكيل أو التشغيل اللدن . أما الفصل الثاني فيؤكد على أهمية عملية توليد الرائش في المعادن ذات اللدونة العالية أو المنخفضة وذلك من حيث أنواع الرائش ، الظروف المؤاتية لحدوثه ، تأثير الحرارة على الحد القاطع لأدوات وعدد القطع بالإضافة لبعض عمليات التشكيل بدون توليد رائش .
يناقش الفصل الثالث بشيء من التفصيل اختبارات الشد والانضغاط من وجهة نظر شكل مقطع العيِّنة ، كيفية ربطها أو تثبيتها على ماكينة الاختبار ، تأثير معدَّل التحميل على العيِّنة ، تفاوت الاستطالة مع الطول الاسمي ومساحة المقطع العرضي ، الانفعال الزائد والتحميل التكراري . أما الفصل الرابع فيناقش أساليب طاقة الانفعال بتعريف طاقة الانفعال الداخلية المخزونة خلال جسم مسلط عليه حمل شد أو عزم إلتواء أو عزم إنحناء . يتم خلال هذا الفصل إيجاد تعريف مناسب لنظرية كاستيقليانو وإشتقاق صيغتها النهائية لإستخدامها في تحديد الإنحراف والميل ، ردود الأفعال والعزوم تحت نقطة تسليط الأحمال المختلفة . 
يستعرض الفصل الخامس من هذا الكتاب الإنهيار اللدن في العارضات والهياكل البسيطة بمناقشة أساليب إيجاد الحمل الحدي قبل إنهيار المنشأة بطريقة مخطط عزم الإنحناء وبطريقة الشغل الإفتراضي . أما الفصل السادس فيتناول بشيء من التفصيل نظرية اللدونة للإنحناء من وجهة نظر الإنحناء فيما بعد إجهاد الخضوع ، الإفتراضات المأخوذة في نظرية اللدونة ، وعزم المقاومة عند المفصلة اللدنة .
خلال كل فصل من فصول هذا الكتاب هنالك العديد من الأمثلة والمسائل المحلولة التي تبسِّط للقارئ وطالب الهندسة في المقام الأول فهم وإستيعاب هذه المادة وبالتالي هضمها بسهولة ويسر وتقريب المفاهيم النظرية للمادة إلى مفاهيم أقرب للواقع العملي . 
  يأمل الكاتب أن يحقِّق هذا الكتاب الهدف الذي كُتِبَ من أجله ألا وهو تبسيط الفهم والاستيعاب لهذه المادة وهضمها بسهولة حتى تصبح معيناً للطالب والمهندس والباحث وكل قارئ لهذا الكتاب .
في الختام أسأل الله التوفيق والسداد .
اللهـم لا سـهل إلاَّ مـا جعلـته سهـلاً
وأنت تجـعل الحـزن إذا شـئت سهـلاً
والله الموفق
                                                                المؤلف
                                                              أسامة محمد المرضي سليمان خيال
Osama Mohammed Elmardi Suleiman Khayal
                                                                أكتوبر 2016 م

Connaître la déformation ainsi que les différentes propriétés intrinsèques aux matériaux est essentiel et indispensable à leur utilisation. Pour cela, on étudie ici la déformée d’une poutre reposant
sur deux appuis simples. Dans cette expérience, on va appliquer une charge croissante concentrée en milieu de la poutre en variant les masses et enregistrer par la suite la valeur de la flèche.

This article presents a technical review on the effects of thermal loads evolved at the wheel–rail–brake contact interfaces. These dynamic contact interfaces develop heat transfer conditions of widely varied thermal level. Their modeling to identify the sources for a variety of defect formation, observable on wheel tread or rail surface, is very important. The railway system, in general, has to bear axle load, friction load, and thermal load arising from their contact conditions in addition to traction and dynamic loads. The defects arising from the interaction of thermal load and other loadings may be identified as hot spots, shelling, spalling, rolling contact fatigue (RCF), and corrugation. The mechanisms for the formation of such defects are pivoted over the existing thermal environment of dynamic interacting surfaces. This review summarizes the works of early investigations and recent advances in modeling the heat transfer conditions required to estimate the temperature distribution at the contact zone. The heat partitioning method for both drag and stop braking conditions, in the presence of rail chill effect, is emphasized. Thermal gradient, introduced by localized temperature rise in the contact zone, in the presence of variable friction coefficient, promotes the RCF process. These alter the residual stresses in the contact region to cause a structural shakedown, aggravate plastic  flow and activates ratchetting phenomenon in rails. The evolution of the thermomechanical surface and subsurface fatigue cracks are also discussed for the completeness of this article. The effect of all such defect formation, emerging from thermal loading condition, and their countermeasures for defect mitigation are presented in this review. This abridged technical documentation envisions attracting more research in the area to improve wheel–rail set design and performance standards to extend enhanced safety and comfort to rail transport operation. It is opined that the thermomechanical loading, their effects on promoting defect formation and propagation should be studied in combination instead of the current practice of treating them separately.

This Book is intended to give an illuminating presentation on simulating structural realistic problems using ABAQUS. The problem definition and configuration for the first two examples of this book are taken from ABAQUS manual to give the reader the opportunity to compare results, and also illustrate the ideal modeling algorithm according to manual, then the same modeling concept and procedures are used through the book for a group of comprehensive structural problems, beside the detailed explanation for the new needed tools according to each problem, accompanied by ABAQUS manual explanation if needed.
To give the desired complete understanding for structural simulation and ABAQUS tools, the book is started with the most simplified examples, and then the level of complexity is increased gradually. This book will be published in the form of parts, each part focus on a specific topics, that are explained through  a detailed group of problems.

This paper provides a brief overview of the basic concepts and equations that will be used by other papers in the special issue on Damage and Fracture in Geomaterials. The first section introduces the tensorial notation, defines the basic tensor operations and presents the formalism for the volumetric-deviatoric decomposition of stress and strain. The second section shows how to transcribe tensorial relations in the engineering notation. The basic equations describing a linear elastic deformable body are summarized in the third section, and their discretization by the finite element method is described in the fourth section, with an extension to nonlinear constitutive relations in the fifth section. The last section outlines a thermodynamic framework based on two potentials—the free energy and the dissipation potential. RÉSUMÉ. Cet article présente une introduction sur les idées de base et les équations qui sont utilisées dans cet ouvrage sur l'endommagement et la rupture des géomatériaux. La première partie présente les notations tensorielles, les opérations tensorielles usuelles et le concept de décomposition des contraintes et des déformations en une partie volumique et une partie dé-viatorique. La deuxième partie montre comment transformer ces relations tensorielles en des notations usuelles pour l'ingénieur. Les équations de bases décrivant la déformation d'un milieu élastique et leurs discrétisations par la méthode des éléments finis sont respectivement introduites dans les troisième et quatrième parties avec une extension pour les comportements non linéaires dans la cinquième partie. La dernière partie est dédiée au cadre thermodynamique basé sur deux potentiels, l'énergie libre et le potentiel dissipatif.

After completing the course, you should be able to:  Analyze forces and find out the resultant forces in two and three dimension  Differentiate between various type of supports and draw free-body-diagram  Compute the reaction force, internal forces and bending moment at a specific point on a simple structure (beam, frame, truss)

The existing motorcycle helmets are thermally uncomfortable as there is no provision for air flow inside the helmet. A new design of helmet, with grooves in the liner foam and slot in the outer shell and liner foam to improve the ventilation, is proposed. Computational fluid dynamics studies show considerable improvement in air velocities inside the helmet in the presence of grooves and slot. Impact dynamics and biomechanics studies on various motorcycle helmets with deformable and rigid head show that the proposed design meets the requirements of the standards in the drop test. The optimum size of the groove and number of grooves for a motorcycle helmet are decided on the basis of the above studies.

Structural deterioration and collapse during major fire is evident and recorded in many incidents. Such events lead to severe injuries and deaths in addition to major economic loss of property. Research efforts to accurately predict the effect of fire on the structural safety and performance and to identify ways to mitigate such effect has been increasing recently with the increase in the capital investment in buildings and infrastructure sectors. In this thesis the main objective is reaching a reliable algorithm to simulate a reinforced concrete beams under both mechanical and fire loads. The proposed algorithm is performed based on connecting thermal analysis and structural analysis using a sequential coupling technique. These analyses involves non-linearity of both thermal and structural properties such as conductivity, specific heat, stress-strain relationship, thermal expansion to capture the thermal and mechanical response during both heating and cooling phases. The proposed algorithm is achieved through detailed numerical finite element models utilizing ANSYS program. The numerical results of the proposed methodology is verified against that of published experimental work. In that work, two beams were tested. The first one failed during fire where the fire resistance was evaluated in minutes. The second one didn’t fail during fire as it exposed to different fire scenario, so it was cooled and reloaded to evaluate the residual capacity. The results of the numerical model showed good correlation with the experimental results in both thermal and mechanical responses indicating the efficiency of the numerical model. This algorithm of creating the coupled numerical model, is then used to execute parametric study, for establishing the effect of some parameters on beam thermal and structural responses to fire loads.

Many efforts have been made to understand the effects of hydrogen on steels, resulting in an abundance of theoretical models and papers. However, a fully developed and practically applicable predictive physical model still does not exist industrially for predicting and preventing hydrogen damage. In practice, it is observed that different types of damages to industrial boiler components have been associated with the presence and localization of hydrogen in metals. In this paper, a damaged boiler tube made of grade 20 – St.20 (or 20G, equivalent to AISI 1020) was investigated. The experimental research was conducted in two distinctive phases: failure analysis of the boiler evaporator tube sample and subsequent postmortem analysis of the viable hydrogen embrittlement mechanisms (HE) in St.20 steel. Numerous tested samples were cut out from the boiler tubes of fossil fuel power plant, damaged due to high temperature hydrogen attack (HTHA) during service, as a result of the development of hydrogen-induced corrosion process. Samples were prepared for the chemical composition analysis, tube wall thickness measurement, tensile testing, hardness measurement, impact strength testing (on in-strumented Charpy machine), analysis of the chemical composition of corrosion products – deposit and the microstructural characterization by optical and scanning electron microscopy – SEM/EDX. The HTHA damage mechanism is a primary cause of boiler tube fracture. Based on the multi-scale special model, applied in subsequent postmortem investigations, the results indicate a simultaneous action of the hydrogen-enhanced decohesion (HEDE) and hydrogen-enhanced localized plasticity (HELP) mechanisms of HE, depending on the local concentration of hydrogen in investigated steel. The model is based on the correlation of mechanical properties to the SEM fractography analysis of fracture surfaces.