## MECHANICS OF MATERIALS

## MASSACHUSETTS INSTITUTE OF TECHNOLOGY

## STUDY MATERIAL

## FREE DOWNLOAD PDF

**CONTENTS**

**Introduction to elasticity****Atomistics****Basic of Elasticity****Introduction to composite materials****Stress-Strain curves****Trusses****Pressure vessels****Shear and torsion****The kinematic Equations****The equilibrium Equations****Transformation of Stresses and strains****Constitutive Equations Statics of Bendings: Shear and Bending Moment Diagrams****Stresses in Beams****Beam Displacements****Closed-Form Solutions****Experimental Strain Analysis****Finite Element Analysis****Engineering viscoelasticity****Yield and plastic flow****The dislocation basic of yield and creep****Statics of Fracture****Introduction to Fracture Mechanics****Fatigue****Matrix and Index Notation**

**INTRODUCTION TO THIS BOOK :**

**INTRODUCTION TO ELASTICITY**

**Introduction**

**This module outlines the basic mechanics of elastic response a physical phenomenon that materials often (but do not always) exhibit. An elastic material is one that deforms immediately upon loading, maintains a constant deformation as long as the load is held constant, and returns immediately to its original undeformed shape when the load is removed. This module will also introduce two essential concepts in Mechanics of Materials: stress and strain. Tensile strength and tensile stress**

**Perhaps the most natural test of a material's mechanical properties is the tension test, in which a strip or cylinder of the material, having length L and cross-sectional area A, is anchored at one end and subjected to an axial load P { a load acting along the specimen's long axis { at the other. (See Fig. 1). As the load is increased gradually, the axial deflection of the loaded end will increase also. Eventually the test specimen breaks or does something else catastrophic, often fracturing suddenly into two or more pieces. (Materials can fail mechanically in many di erent ways; for instance, recall how blackboard chalk, a piece of fresh wood, and Silly Putty break.) As engineers, we naturally want to understand such matters as how is related to P, and what ultimate fracture load we might expect in a specimen of di erent size than the original one. As materials technologists, we wish to understand how these relationships are influenced by the constitution and microstructure of the material.**

**STRESS-STRAIN CURVES**

**Introduction**

**Stress-strain curves are an extremely important graphical measure of a material’s mechanical properties, and all students of Mechanics of Materials will encounter them often. However, they are not without some subtlety, especially in the case of ductile materials that can undergo substantial geometrical change during testing. This module will provide an introductory discussion of several points needed to interpret these curves, and in doing so will also provide a preliminary overview of several aspects of a material’s mechanical properties. However, this module will not attempt to survey the broad range of stress-strain curves exhibited by modern engineering**

**materials (the atlas by Boyer cited in the References section can be consulted for this). Several of the topics mentioned here — especially yield and fracture — will appear with more detail in later modules.**

**TRUSSES**

**Introduction**

**A truss is an assemblage of long, slender structural elements that are connected at their ends. Trusses find substantial use in modern construction, for instance as towers (see Fig. 1), bridges, scaffolding, etc. In addition to their practical importance as useful structures, truss elements have a dimensional simplicity that will help us extend further the concepts of mechanics introduced in the modules dealing with uniaxial response. This module will also use trusses to introduce important concepts in statics and numerical analysis that will be extended in later modules to more general problems.**

**PRESSURE VESSELS**

**Introduction**

**A good deal of the Mechanics of Materials can be introduced entirely within the confines of uniaxially stressed structural elements, and this was the goal of the previous modules. But of course the real world is three-dimensional, and we need to extend these concepts accordingly. We now take the next step, and consider those structures in which the loading is still simple, but where the stresses and strains now require a second dimension for their description. Both for their value in demonstrating two-dimensional effects and also for their practical use in mechanical design, we turn to a slightly more complicated structural type: the thin-walled pressure vessel. Structures such as pipes or bottles capable of holding internal pressure have been very important in the history of science and technology. Although the ancient Romans had developed municipal engineering to a high order in many ways, the very need for their impressive system of large aqueducts for carrying water was due to their not yet having pipes that could maintain internal pressure. Water can flow uphill when driven by the hydraulic pressure of the reservoir at a higher elevation, but without a pressure-containing pipe an aqueduct must be constructed so the water can run downhill all the way from the reservoir to the destination.**

**SHEAR AND TORSION**

**Introduction**

**Torsionally loaded shafts are among the most commonly used structures in engineering. For instance, the drive shaft of a standard rear-wheel drive automobile, depicted in Fig. 1, serves primarily to transmit torsion. These shafts are almost always hollow and circular in cross section, transmitting power from the transmission to the differential joint at which the rotation is diverted to the drive wheels. As in the case of pressure vessels, it is important to be aware of design methods for such structures purely for their inherent usefulness. However, we study them here also because they illustrate the role of shearing stresses and strains.**

**FINITE ELEMENT ANALYSIS**

**Introduction**

**Finite element analysis (FEA) has become commonplace in recent years, and is now the basis of a multibillion dollar per year industry. Numerical solutions to even very complicated stress problems can now be obtained routinely using FEA, and the method is so important that even introductory treatments of Mechanics of Materials { such as these modules { should outline its principal features. In spite of the great power of FEA, the disadvantages of computer solutions must be kept in mind when using this and similar methods: they do not necessarily reveal how the stresses are influenced by important problem variables such as materials properties and geometrical features, and errors in input data can produce wildly incorrect results that may be overlooked by the analyst. Perhaps the most important function of theoretical modeling is that of sharpening the designer's intuition; users of finite element codes should plan their strategy toward this end, supplementing the computer simulation with as much closed-form and experimental analysis as possible.**

**Finite element codes are less complicated than many of the word processing and spreadsheet packages found on modern microcomputers. Nevertheless, they are complex enough that most users do not find it effective to program their own code. A number of prewritten commercial codes are available, representing a broad price range and compatible with machines from microcomputers to super computers. However, users with specialized needs should not necessarily shy away from code development, and may find the code sources available in such texts as that by Zienkiewicz to be a useful starting point. Most finite element software is written in Fortran, but some newer codes such as felt are in C or other more modern programming languages.**

**YIELD AND PLASTIC ANALYSIS**

**Introduction**

**In our overview of the tensile stress-strain curve in Module 4, we described yield as a permanent molecular rearrangement that begins at a sufficiently high stress, denoted σY in Fig. 1. The yielding process is very material-dependent, being related directly to molecular mobility. It is often possible to control the yielding process by optimizing the materials processing in a way that influences mobility. General purpose polystyrene, for instance, is a weak and brittle plastic often credited with giving plastics a reputation for shoddiness that plagued the industry for years. This occurs because polystyrene at room temperature has so little molecular mobility that it experiences brittle fracture at stresses less than those needed to induce yield with its associated ductile flow. But when that same material is blended with rubber particles of suitable size and composition, it becomes so tough that it is used for batting helmets and ultra-durable children’s toys. This magic is done by control of the yielding process. Yield control to balance strength against toughness is one of the most important aspects of materials engineering for structural applications, and all engineers should be aware of the possibilities.**

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