These lecture notes constitute the core of the course solid mechanics. Pdf mechanics of solids mcq question on simple stress. Linear stress strain relations in nonlinear elasticity. Normal strain occurs when the elongation of an object is in response to a normal stress i. Analysis of three dimensional stresses and strains. Direction of the unit normal vectors nand nat the surface of a rock. Geometry of logarithmic strain measures in solid mechanics the hencky energy is the squared geodesic distance of the deformation. For linear, isotropic materials, e and g are related as 3. An exception to this was the varying stress field in the loaded beam, but there a simplified set of elasticity equations was used.
Like many other more complicated problems, the stress here does not depend on the material properties, but the displacement always does, u l 2. Mechanics of materials 3g stress and strain torsion for a body with radius r being strained to an angle. Fundamentals of solid mechanics krzysztof wilmanski. Jaerisch derived the equations of general vibration of an elastic sphere in the. Among many other ideas he proposed a linear relation between stress. The liquid and gas phases occupy the voids between the solid particles as shown in figure 21a. Once ux is known, the strain and stress fields in the solid can be deduced.
This chapter is concerned with deformation of a structural member under axial. The deformation corresponding to a 3d rigid rotation about an axis through the origin is yrx. Written out in matrix notation, this index equation is. Thus, a rigid body displacement induces no strain, and hence no stress, in the solid. It is the ratio of normal stress to normal strain i. Hookes law, poissons ratio, shear stress lecture 5 shear strain, modulus of rigidity, bulk modulus. Strain displacement relations, equilibrium equations, compatibility conditions and airys stress function. Jones, mechanics of composite materials, mcgrawhill, 1975. Shear stress and strain are related in a similar manner as normal stress and strain, but with a different constant of proportionality. Write down the equation for strain energy stored in a body and explain the terms. Numerous most important contributions were made by the swiss mathematician and mechanician leonhard euler 17071783, who was taught mathematics by jacob brother johann bernoulli 16671748. Fhwa nhi06088 2 stress and strain in soils soils and foundations volume i 2 1 december 2006 chapter 2. Ax fl e a graph of stress against strain will be a straight line with a gradient of e.
Stress, strain, and the basic equations of solid mechanics request. Chapter, a number of differential equations will be derived, relating the stresses and body. Relationship between material properties of isotropic materials. Oct 02, 2019 here you can download the free lecture notes of mechanics of solids pdf notes mos pdf notes materials with multiple file links to download. Stress, strain, and material relations normal stress. Lecture 4 singularities 2011 alex grishin mae 323 lecture 4 plane stress strain and singularities 12 the stress equilibrium equation similarly, repeating the previous three steps in the ydirection yields. The stress strain relationship the stress strain relationship of a material usually can be obtained from tensile or compression test on a specimen of the material. The sign convention is as follows we will plot two points. And, once again, even though we wont go thru the steps, we will simply point. The constant g is called the shear modulus and relates the shear stress and strain in the elastic region. An alternative to using these equations for the principal stresses is to use a graphical method known as mohrs circle.
What is the difference between a linear elastic stressstrain law and a hyperelastic stress strain law. Stress principal stress for the case of plane stress principal directions, principal stress the normal stresses. It is left as an exercise for the reader to show that this process leads to the familiar two dimensional expressions found in the first course in solid mechanics see problem 2. The solid mechanics as a subject may be defined as a branch of applied mechanics that deals with. Strain is a unitless measure of how much an object gets bigger or smaller from an applied load.
Further, in chapter 2, the strain components were related to the displacement components. Stress strain relationship, hookes law, poissons ratio, shear. Write down the equation for strain energy stored due to shear stress and explain the terms. Stress strain relationships tensile testing one basic ingredient in the study of the mechanics of deformable bodies is the resistive properties of materials. We then develop a set of stress strain equations for a linear, isotropic, homogenous, elastic solid. The finite element method fem is a computer technique for solving partial differential equations. We begin our discussion on governing equations with. This enables us to define the displacements, strains and stresses at every point of the body. Shear strain occurs when the deformation of an object.
Thisisthecharacteristic equation forstress,wherethecoe cientsare i 1. Since the stress matrix is symmetric, one can express cauchys law in the form. If direct and shear strains along x and y directions are known, normal strain and the shear strain at angle. Solid mechanics part ii 206 kelly 3 2 1 23 33 12 22 32 11 21 31 3 2 1 n n n t t t.
We of course must consider the deformations even of a determinate structure if we wish to estimate the dis. Stressappliedatanangletothe bersinaonedimensionalply. For this reason, the reader with background in the analysis of stress and strain and the equations for elastic and plastic deformation can proceed to chap. Here the question of varying stress and strain fields in materials is. Therefore, the strain equations of equilibrium can be converted to displacement equations of equilib rium. Shear strain is the complement of the angle between two initially perpendicular line segments. Formulas in solid mechanics division of solid mechanics.
One application is to predict the deformation and stress fields within solid bodies subjected to external forces. The normal and shear stress acting on the right face of the plane make up one point, and the. While in the mechanics of materials course, one was introduced to the various components of the stress and strain, namely the normal and shear, in. Mechanics of solid stress and strain by kaushal patel 2. Stress, strain, and the basic equations of solid mechanics. Show that the lagrange strain associated with this deformation is zero. Forces acting on a a body, b crosssection of the body. This involves creating a graph with sigma as your abscissa and tau as your ordinate, and plotting the the given stress state. Through poissons ratio, we now have an equation that relates strain in the y or z direction to strain in the z direction. All problems in pdf format applied mechanics of solids. The vfm is based on the fundamental equations of solid mechanics. Solid mechanics low cycle fatigue lcf anders ekberg 7 8 stress and strain concentrations stress concentration the stress concentration factor ahead of a notch is defined as k. Hookes law states that the stress and strain are directly proportional to each other. A positive value corresponds to a tensile strain, while negative is compressive.
Chapter an overview of stressstrain analysis for elasticity. The problem of solid mechanics is reduced, as follows from the foregoing derivation, to a set of 15 equations, i. Lecture 6 numerical problems on shear strain, modulus of rigidity lecture 7 stress strain diagram for uniaxial loading of ductile and brittle materials. Geometry of logarithmic strain measures in solid mechanics. Morozov, in advanced mechanics of composite materials third edition, 20 2. True stress and strain are calculated using the instantaneous deformed at a particular load values of the crosssectional area, a, and the length of the rectangle, l, f t 2. Recall that the normal stesses equal the principal. Chapter 6, is expanded, presenting more coverage on electrical strain gages and providing tables of equations for commonly used strain gage rosettes. Introduction to finite element analysis in solid mechanics. Pdf this paper provides a brief overview of the basic concepts and equations. Hookes law modulus of elasticity e aka youngs modulus.
Hence, stress equations of equilibrium can be converted to strain equations of equilibrium. Stress mohrs circle for plane stress mohrs circle introduced by otto mohr in 1882, mohrs circle illustrates principal stresses and stress transformations via a graphical format, the two principal stresses are shown in red, and the maximum shear stress is shown in orange. In the case of the socalled brittle materials, there is no yield zone. Stress strain relations for linearly elastic solids, generalized hookes law. The treatment in this chapter is based on a homogeneous and isotropic continuum. We then develop a set of stressstrain equations for a linear, isotropic, homogenous, elastic solid. In structural mechanics, the governing differential equation will most often be the stress equilibrium equation. Direction of the unit normal vectors nand nat the surface of a rock mass. Mar 10, 2015 stress and strain mechanics of solid 1.
Normal strain is the change in length in a given direction divided by the initial length in that direction. We of course must consider the deformations even of a. Every invertible stress relation t fb for an isotropic elastic material is linear, trivially, in an appropriately. These properties relate the stresses to the strains and can only be determined by experiment. The stress component parallel to the surface are called shear stress component, is indicated by the a b figure 1. In spite of our best efforts, some errors doubtless remain. E106 stress and strain tensor summary page 7 for yielding to occur.
In this problem you will calculate the formula that can be used. We can in turn relate this back to stress through hookes law. In this problem you will calculate the formula that can. Sol mech course text feb10 solid mechanics at harvard. What is the external diameter of the shaft which is subjected to a maximum shear stress of 90 na solid or a hollow shaft subject to a twisting moment t, the torsional shearing stress t is second moment of area. The slope of the straightline portion of the stress strain diagram is called the modulus of elasticity or youngs modulus. Shear stress in direction j on surface with normal direction i.
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