Finally, the equilibrium equations are used to develop expressions for the speed of stress waves in steel, aluminum, and rubber. Mechanics of solid stress and strain by kaushal patel 2. Opti 222 mechanical design in optical engineering 17 stress strain relationships tensile testing one basic ingredient in the study of the mechanics of deformable bodies is the resistive. This assumption turns out to be an excellent predictor of the response of components which undergo small deformations. A normal strain is perpendicular to the face of an element, and a shear strain is parallel to it. Solid mechanics is one of the important branches of physical science concerned with the deformation and motion of continuous solid media under applied external loadings such as forces, displacements, and accelerations that result in inertial force in the bodies, thermal changes, chemical interactions, electromagnetic forces, and so on. Lecture 5 calculation of principal stresses from principal strains.
Mechanics of solids simple stress and strain part 3. We will proceed without reference to truss members, beams, shafts in torsion, shells, membranes or whatever structural element might come to mind. Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and other external or internal agents. The constant g is called the shear modulus and relates the shear stress and strain in the elastic region. Mar 10, 2015 stress and strain mechanics of solid 1. It can be shown that if the stress strain curve of the material is convex or linear, the rod deforms uniformly and a.
Definition of coordinate system and loadings on beam. Jul 14, 2017 top 15 items every engineering student should have. Strain is defined as deformation of a solid due to stress. The deformation, expressed by strain, arises throughout the material as the particles molecules, atoms, ions of which. Scott school of mathematics, university of east anglia, norwich, nr4 7tj. In 1822 he formalized the stress concept in the context of a general threedimensional theory, showed its properties as consisting of a 3 by 3 symmetric array of numbers that transform as a tensor, derived the equations. For linear, isotropic materials, e and g are related as 3. In other contexts one may be able to reduce the threedimensional problem to a twodimensional one, andor replace the general stress and strain tensors by simpler models like uniaxial tensioncompression, simple shear, etc. This paper provides a brief overview of the basic concepts and equations that will be. Linear regression in a plot of stress vs strain rate viscosity is slope of regression line in a stressstrain rate plot.
Following development of the equations, applications will be presented that involve airy stress functions and tire mechanics. Fhwa nhi06088 2 stress and strain in soils soils and foundations volume i 2 1 december 2006 chapter 2. The solid mechanics as a subject may be defined as a branch of applied. The documentation set for the structural mechanics module consists of the structural mechanics module users guide, the structural mechanics module model library, and this structural mechanics module reference guide. Determination of volumetric strain for a circular bar subjected to an axial pull. Mechanics of materials for dummies cheat sheet dummies. Stressstrain relationship, hookes law, poissons ratio, shear stress, lecture 4 numerical problems on stressstrain relationship, hookes law, poissons ratio, shear stress lecture 5 shear strain, modulus of rigidity, bulk modulus. Geometry of logarithmic strain measures in solid mechanics patrizio ne 1, bernhard eidel 2 and robert j. In this equation, the proportionality between shear stress and shear strain is known as the shear modulus of a material. These definitions are consistent with those of normal stress and shear stress. Find the minimum inner diameter of a hollow circular shaft in torsion 21. Stressstrain diagram for uniaxial loading of ductile and brittle. We consider an arbitrarily shaped body, a continuous solid body, a solid continuum.
One cannot measure the stress without first specifying the datum plane. Mechanics of solids pdf notes mos pdf notes smartzworld. We then develop a set of stress strain equations for a linear, isotropic, homogenous, elastic solid. 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 youngs elastic modulus. Know how to compute strains and stresses of members. The formula is about the critical force for the elastic beam that is supported by its joints. Derivation is available in textbooks on solid mechanics. The french engineer and physicist charlesaugustin coulomb 17361806 was apparently the. The second section shows how to transcribe tensorial relations. Normal stress shear and bearing stress normal strain hookes law thermal effects indeterminate axial structures the topic menu above allows you to move directly to any of.
Reaction force b equilibrium condition c stress and strain d material properties. Geometry of logarithmic strain measures in solid mechanics. Stress, strain, and the basic equations of solid mechanics request. Introduction a class of nonlinear elasticity models allowing a nonlinear strainstress response due to limiting small strain was introduced by rajagopal 1 and developed in later works by the author. Relationship between material properties of isotropic materials. In our derivations that follow, we limit our attention to two dimensions.
Tensile load, there will be a decrease in crosssectional area and an increase in length of the body. When a body is subjected to two equal and opposite axial pulls p also called tensile load, then the stress induced at any section of the body is known as tensile stress. This linear, elastic relationship between stress and strain is known as hookes law. Determination of volumetric strain for an object subjected to triaxial loading. Ax fl e a graph of stress against strain will be a straight line with a gradient of e. Chapter, a number of differential equations will be derived, relating the stresses and body forces equations of motion, the strains and displacements strain. Pdf basic concepts and equations of solid mechanics. 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. Stress, strain, and the basic equations of solid mechanics. The liquid and gas phases occupy the voids between the solid particles as shown in figure 21a. An exception to this was the varying stress field in the loaded beam, but there a simplified set of elasticity equations was used. The above equation is a simple linear model for the 1d analysis of materials operating in the elastic region of behavior. Kelvinvoigt viscoelastic solid, limiting small strain, crack, variational problem, generalized solution 1.
Mathematics and mechanics of solids on the states of stress. If we require a 3d analysis of materials, we must use a more. All books are available in pdf and html versions from the comsol help desk. Shear strain change in angle between two line segments originally perpendicular. Normal stress shear and bearing stress normal strain hookes law thermal effects indeterminate axial structures the topic menu above allows you to move directly to any of the four sections for each topic. Ferc mechanics of materials 1 stress strain curve for mild steel. This value can vary greatly from 1 kpa for jello to 100 gpa for steel. Stressstrain relationship, hookes law, poissons ratio, shear stress, shear strain, modulus of rigidity. As part of this work, cauchy also introduced the equations which. Plastic no strain until some critical stress value has been reached. Stress transformations using equation method youtube. Top 15 items every engineering student should have. Hookes law in shear looks very similar to the equation we saw for normal stress and strain. Strain, in physical sciences and engineering, number that describes relative deformation or change in shape and size of elastic, plastic, and fluid materials under applied forces.
Find the minimum diameter of a solid circular shaft in torsion 20. Mechanics of materials analysis is based on several basic concepts such as. For isotropic solids, principal strain axes coincide with the principal stress axes definition of principal strain axes. The concept of homogeneity in mechanics means independence of the solution on the spatial coordinates system, the rod axis in the present case. Formulas in solid mechanics division of solid mechanics. Mechanics of materials describes how solid materials will deform change shape and how they will fail break when subjected to applied forces. The stress at point p can be defined using the same equation as was used for t. Dec 28, 2018 determination of volumetric strain for an object subjected to triaxial loading. Formulas in solid mechanics tore dahlberg solid mechanicsikp, linkoping university linkoping, sweden this collection of formulas is intended for use by foreign students in the course tmhl61, damage mechanics and life analysis, as a complement to the textbook dahlberg and. Shear stress and strain are related in a similar manner as normal stress and strain, but with a different constant of proportionality.
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. Stress therefore can be interpreted as internal tractions that act on a defined internal datum plane. These equation expresses the relationship between stress and strain hooks law for. Definition of stress, stress tensor, normal and shear stresses in axially loaded members. Determination of the stress distribution within a member. Normal strain elongation or contraction of a line segment.
Three mutually perpendicular directions in the body which remain mutually perpendicular during deformation. Shearing stress due to torsion in a hollow circular shaft 19. Stresses in thin cylinder and thin spherical shells under internal pressure. Fundamentals of solid mechanics krzysztof wilmanski. The question is whether the resulting strain eld is homogeneous or not. Students and professional engineers in the mechanical sciences know that mechanics of materials deals extensively with stress on objects from determining stress at a particular point to finding stresses in columns. We first con struct a set of strain measures in terms of the x,y and z components of displace ment at a point. United kingdom 1 introduction in a solid material e. For an isotropic material that obeys hookes law, a normal stress will cause a normal strain. It we just examine the earlier tension curve one can notice that the extension of the.
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