NPTEL :: Mechanical Engineering - Strength of Materials
The following quotes are from the second edition, unless otherwise noted. The discussion of the stress-strain relations rests upon Hooke's Law as an axiom . his modulus of elasticity, was the first to consider shear as an elastic strain. (quote when citing this article) Stress dilatancy relationship of sand under cyclic loading including cyclic mobility and that of liquefied Liquefaction; cyclic mobility; stress dilatancy; sand; bulk modulus; shear modulus; simple shear. Please quote as: Tasevski D., Fernández Ruiz M., Muttoni A., Behaviour of concrete in compression and shear under varying strain rates: from rapid to long- term Acoustic emission measurements clearly confirm the relationship of this . Generalised stress-strain behaviour for a high and a low strain rate.
Remember, strength measures how much stress the material can handle before permanent deformation or fracture occurs, whereas the stiffness measures the resistance to elastic deformation.
A Treatise on the Mathematical Theory of Elasticity - Wikiquote
Simple stress-strain curves illustrating stiff vs non-stiff behavior. Curves A and B correspond to stiffer materials, whereas curve C represents a non-stiff material.
Image source Understanding the Concepts of Stress and Strain Stress is an internal force resulting from an applied load; it acts on the cross-section of a mechanical or structural component. Strain is the change in shape or size of a body that occurs whenever a force is applied.
Yield strength is used in materials that exhibit an elastic behavior. Ultimate strength refers to the maximum stress before failure occurs.
Fracture strength is the value corresponding to the stress at which total failure occurs. Stiffness is how a component resists elastic deformation when a load is applied.
Hardness is resistance to localized surface deformation. Ultimate tensile strength, yield strength, and fracture stress Failure Point. The strength of a material can refer to yield strength, ultimate strength, or fracture strength.
Tensile strength can be calculated from hardness and is convenient because hardness tests—such as Rockwell—are usually simple to do, inexpensive, and nondestructive. Only a small penetration is performed on the specimen.
- Mechanics of Materials: Strain
- A Treatise on the Mathematical Theory of Elasticity
For many metals, tensile strength increases as hardness increases. A reliable online source is www. Stiffness Beam deflection The image below is a finite element analysis FEA of a beam subjected to a loading. FEA analysis of beam deflection. In the analysis of strain I have thought it best to follow Thomson and Tait 's Natural Philosophybeginning with the geometrical or rather algebraical theory of finite homogeneous strain, and passing to the physically most important case of infinitesimal strain.Shearing stress over shear strain is shear modulus - Video in HINDI - EduPoint
The discussion of the stress-strain relations rests upon Hooke's Law as an axiom generally verified in experience, and on Sir W. Thomson ['s] thermodynamical investigation of the existence of the energy-function. The theory of elastic crystals adopted is that which has been elaborated by the researches of F.
The conditions of rupture or rather of safety of materials are as yet so little under stood that it seemed best to give a statement of the various theories that have been advanced without definitely adopting any of them. In most of the problems considered in the text Saint-Venant 's "greatest strain" theory has been provisionally adopted. In connexion with this theory I have endeavoured to give precision to the term " factor of safety ".
Among general theorems I have included an account of the deduction of the theory from Boscovich 's point-atom hypothesis.
This is rendered necessary partly by the controversy that has raged round the number of independent elastic constants, and partly by the fact that there exists no single investigation of the deduction in question which could now be accepted by mathematicians. In spite of the work of Prof.
Pearson it seems not yet to be understood by English mathematicians that the cross-sections of a bent beam do not remain plane. The old-fashioned notion of a bending moment proportional to the curvature resulting from the extensions and contractions of the fibres is still current.
Against the venerable bending moment the modern theory has nothing to say, but it is quite time that it should be generally known that it is not the whole stress, and that the strain does not consist simply of extensions and contractions of the fibres.
In explaining the theory I have followed Clebsch 's mode of treatment, generalising it so as to cover some of the classes of aeolotropic bodies treated by Saint-Venant. The theory leads in every special case to a system of partial differential equations, and the solution of these subject to conditions given at certain bounding surfaces is required.
The general problem is that of solving the general equations with arbitrary conditions at any given boundaries.
In discussing this problem I have made extensive use of the researches of Prof. Betti of Pisa, whose investigations are the most general that have yet been given The case of a solid bounded by an infinite plane and otherwise unlimited is investigated on the lines laid down by Signor Valentino Cerruti, whose analysis is founded on Prof.
Betti's general method, and some of the most important particular cases are worked out synthetically by M. Boussinesq 's method of potentials. In this connexion I have introduced the last-mentioned writer's theory of "local perturbations", a theory which gives the key to Saint-Venant 's "principle of the elastic equivalence of statically equipollent systems of load".
It is hoped that he will not then fail to understand the subject for lack of examples, nor waste his time in mere problem grinding.
Historical Introduction[ edit ] The Mathematical Theory of Elasticity is occupied with an attempt to reduce to calculation the state of strainor relative displacement, within a solid body which is subject to the action of an equilibrating system of forces, or is in a state of slight internal relative motion, and with endeavours to obtain results which shall be practically important in applications to architecture, engineering, and all other useful arts in which the material of construction is solid.
Alike in the experimental knowledge obtained, and in the analytical methods and results, nothing that has once been discovered ever loses its value or has to be discarded; but the physical principles come to be reduced to fewer and more general ones, so that the theory is brought more into accord with that of other branches of physics, the same general dynamical principles being ultimately requisite and sufficient to serve as a basis for them all.
The first mathematician to consider the nature of the resistance of solids to rupture was Galileo. Although he treated solids as inelastic, not being in possession of any law connecting the displacements produced with the forces producing them, or of any physical hypothesis capable of yielding such a law, yet his enquiries gave the direction which was subsequently followed by many investigators.
This problem, and, in particular, the determination of this axis is known as Galileo's problem. Hooke's Law provided the necessary experimental foundation for the theory. When the general equations had been obtained, all questions of the small strain of elastic bodies were reduced to a matter of mathematical calculation. Hooke and Mariotte occupied themselves with the experimental discovery of what we now term stress-strain relations. Hooke gave in the famous law of proportionality of stress and strain which bears his name, in the words "Ut tensio sic vis; that is, the Power of any spring is in the same proportion with the Tension thereof.
This law forms the basis of the mathematical theory of Elasticity. Hooke does not appear to have made any application of [his law] to the consideration of Galileo's problem. This application was made by Mariottewho in enunciated the same law independently.
He remarked that the resistance of a beam to flexure arises from the extension and contraction of its parts, some of its longitudinal filaments being extended, and others contracted. He assumed that half are extended, and half contracted. His theory led him to assign the position of the axis, required in the solution of Galileo's problem, at one-half the height of the section above the base. In the interval between the discovery of Hooke's law and that of the general differential equations of Elasticity by Navierthe attention of those mathematicians who occupied themselves with our science was chiefly directed to the solution and extension of Galileo's problem, and the related theories of the vibrations of bars and plates, and the stability of columns.
The first investigation of any importance is that of the elastic line or elastica by James Bernoulli inin which the resistance of a bent rod is assumed to arise from the extension and contraction of its longitudinal filaments, and the equation of the curve assumed by the axis is formed. This equation practically involves the result that the resistance to bending is a couple proportional to the curvature of the rod when bent, a result which was assumed by Euler in his later treatment of the problems of the elastica, and of the vibrations of thin rods.
As soon as the notion of a flexural couple proportional to the curvature was established it could be noted that the work done in bending a rod is proportional to the square of the curvature. When a force acts parallel to the surface of an object, it exerts a shear stress.
Let's consider a rod under uniaxial tension. The rod elongates under this tension to a new length, and the normal strain is a ratio of this small deformation to the rod's original length.
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Strain is a unitless measure of how much an object gets bigger or smaller from an applied load. Shear strain occurs when the deformation of an object is response to a shear stress i. Mechanical Behavior of Materials Clearly, stress and strain are related. Stress and strain are related by a constitutive law, and we can determine their relationship experimentally by measuring how much stress is required to stretch a material.
This measurement can be done using a tensile test.
In the simplest case, the more you pull on an object, the more it deforms, and for small values of strain this relationship is linear. This linear, elastic relationship between stress and strain is known as Hooke's Law. If you plot stress versus strain, for small strains this graph will be linear, and the slope of the line will be a property of the material known as Young's Elastic Modulus.
This value can vary greatly from 1 kPa for Jello to GPa for steel. In this course, we will focus only on materials that are linear elastic i. From Hooke's law and our definitions of stress and strain, we can easily get a simple relationship for the deformation of a material.
Intuitively, this exam makes a bit of sense: If the structure changes shape, or material, or is loaded differently at various points, then we can split up these multiple loadings using the principle of superposition. Generalized Hooke's Law In the last lesson, we began to learn about how stress and strain are related — through Hooke's law. But, up until this point we've only considered a very simplified version of Hooke's law: In this lesson, we're going to consider the generalized Hooke's law for homogenousisotropicand elastic materials being exposed to forces on more than one axis.
First things first, even just pulling or pushing on most materials in one direction actually causes deformation in all three orthogonal directions. Let's go back to that first illustration of strain. This time, we will account for the fact that pulling on an object axially causes it to compress laterally in the transverse directions: This property of a material is known as Poisson's ratio, and it is denoted by the Greek letter nu, and is defined as: