Basic Equations of Elasticity

Size: px
Start display at page:

Download "Basic Equations of Elasticity"

Transcription

1 A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ xz, where the first three are the normal components and the latter six are the components of shear stress. The equations of internal equilibrium in terms of the nine components of stress can be derived by considering the equilibrium of moments and forces acting on the elemental volume shown in Fig. A.1. The equilibrium of moments about the x, y, an axes, assuming that there are no bo moments, leads to the relations σ yx = σ xy, σ zy = σ yz, σ xz = σ zx A.1) Equations A.1) show that the state of stress at any point can be defined completely by the six components σ xx,σ yy,σ zz,σ xy,σ yz,andσ zx. A.2 STRAIN DISPLACEMENT RELATIONS The deformed shape of an elastic bo under any given system of loads can be described completely by the three components of displacement u, v, and w parallel to the directions x, y, an, respectively. In general, each of these components u, v, andw is a function of the coordinates x, y, an. The strains and rotations induced in the bo can be expressed in terms of the displacements u, v, andw. We shall assume the deformations to be small in this work. To derive the expressions for the normal strain components ε xx and ε yy and the shear strain component ε xy, consider a small rectangular element OACB whose sides of lengths and ) lie parallel to the coordinate axes before deformation. When the bo undergoes deformation under the action of external load and temperature distribution, the element OACB also deforms to the shape O A C B, as shown in Fig. A.2. We can observe that the element OACB has two basic types of deformation, one of change in length and the other of angular distortion. Since the normal strain is defined as change in length divided by original length, the strain components ε xx and ε yy can be found as change in length of the fiber OA which lies in the x direction before deformation ε xx = original length of the fiber = { + [u + u/)] u} = u A.2) 700 Vibration of Continuous Systems. Singiresu S. Rao 2007 John Wiley & Sons, Inc. ISBN:

2 A.2 Strain Displacement Relations 701 z s zz + s zz s yy s yz s yx s zx + s zx 0 s xy s xz + s xz s xy + s xy s zy + s zy s xz s zx s xx s yz + s yz s yx + s yx s yy + s yy y s xx + s xx s zy s zz x Figure A.1 Stresses on an element of size. u + u B C B v + v C q 2 A y O v u O A q 1 u + u v + v x Figure A.2 Deformation of an element.

3 702 Basic Equations of Elasticity change in length of the fiber OB which lies in the y direction before deformation ε yy = original length of the fiber OB = { + [v + v/)] v} = v A.3) The shear strain is defined as the decrease in the right angle between fibers OA and OB, which were at right angles to each other before deformation. Thus, the expression for the shear strain ε xy can be obtained as [v + v/)] v ε xy = θ 1 + θ 2 + [u + u/)] u + [u + u/)] u + [v + v/)] v A.4) If the displacements are assumed to be small, ε xy canbeexpressedas ε xy = u + v A.5) The expressions for the remaining normal strain component ε zz and shear strain components ε yz and ε zx can be derived in a similar manner as ε zz = w A.6) ε yz = w + v A.7) ε zx = u + w A.8) A.3 ROTATIONS Consider the rotation of a rectangular element of sides and as a rigid bo by a small angle, as shown in Fig. A.3. Noting that A D and C E denote the displacements of A and C along the y and x axes, the rotation angle α canbeexpressedas α = v = u A.9) Of course, the strain in the element will be zero during rigid-bo movement. If both rigid-bo displacements and deformation or strain occur, the quantity ω z = 1 v 2 u ) A.10) can be seen to represent the average of angular displacement of and the angular displacement of, and is called rotation about the z axis. Thus, the rotations of an elemental bo about the x, y, an axes can be expressed as ω x = 1 w 2 v ) A.11) ω y = 1 u 2 w ) A.12) ω z = 1 v 2 u ) A.13)

4 A.4 Stress Strain Relations 703 y B C C E A B a a O D A x Figure A.3 Rotation of an element. A.4 STRESS STRAIN RELATIONS The stress strain relations, also known as the constitutive relations, of an anisotropic elastic material are given by the generalized Hooke s law, based on the experimental observation that strains are linearly related to the applied load within the elastic limit. The six components of stress at any point are related to the six components of strain linearly as σ xx σ yy σ zz σ yz σ zx σ xy = C 11 C 12 C 13 C 16 C 21 C 22 C 23 C 26 C 31 C 32 C 33 C 36 C 61 C 62 C 63 C 66 ε xx ε yy ε zz ε yz ε zx ε xy A.14) where the C ij denote one form of elastic constants of the particular material. Equation A.14) has 36 elastic constants. However, for real materials, the condition for the elastic energy to be a single-valued function of the strain requires the constants C ij to be symmetric; that is, C ij = C ji. Thus, there are only 21 different elastic constants in Eq. A.14) for an anisotropic material. For an isotropic material, the elastic constants are invariant, that is, independent of the orientation of the x, y, an axes. This reduces to two the number of independent elastic constants in Eq. A.14). The two independent elastic constants, called Lamé s elastic constants, are commonly denoted as λ and µ. The Lamè constants are related

5 704 Basic Equations of Elasticity to C ij as follows: C 11 = C 22 = C 33 = λ + 2µ C 12 = C 21 = C 31 = C 13 = C 32 = C 23 = λ C 44 = C 55 = C 66 = µ all other C ij = 0 Equation A.14) can be rewritten for an elastic isotropic material as A.15) σ xx = λ + 2µε xx σ yy = λ + 2µε yy σ zz = λ + 2µε zz σ yz = µε yz A.16) σ zx = µε zx σ xy = µε xy where = ε xx + ε yy + ε zz A.17) denotes the dilatation of the bo and denotes the change in the volume per unit volume of the material. Lamé s constants λ and µ are related to Young s modulus E, shear modulus G, bulk modulus K, and Poisson s ratio ν as follows: µ3λ + 2µ) E = λ + µ A.18) G = µ A.19) K = λ µ A.20) or λ = µ = ν = λ 2λ + µ) νe 1 + ν)1 2ν) A.21) A.22) E 21 + ν) = G A.23) A.5 EQUATIONS OF MOTION IN TERMS OF STRESSES Due to the applied loads which may be namic), stresses will develop inside an elastic bo. If we consider an element of material inside the bo, it must be in namic equilibrium due to the internal stresses developed. This leads to the equations of motion of a typical element of the bo. The sum of all forces acting on the element shown in Fig. A.1 in the x direction is given by Fx = σ xx + σ ) xx σ xx + + σ zx + σ zx ) σ zx σ xy + σ ) xy σ xy = σ xx + σ xy + σ zx A.24)

6 A.6 Equations of Motion in Terms of Displacements 705 According to Newton s second law of motion, the net force acting in the x direction must be equal to mass times acceleration in the x direction: Fx = ρ 2 u A.25) where ρ is the density, u is the displacement, and 2 u/ is the acceleration parallel to the x axis. Equations A.24) and A.25) lead to the equation of motion in the x direction. A similar procedure can be used for the y and z directions. The final equations of motion can be expressed as σ xx σ xy σ zx + σ xy + σ yy + σ yz + σ zx + σ yz + σ zz = ρ 2 u = ρ 2 v = ρ 2 w A.26) A.27) A.28) where u, v, andw denote the components of displacement parallel to the x, y, an axes, respectively. Note that the equations of motion are independent of the stress strain relations or the type of material. A.6 EQUATIONS OF MOTION IN TERMS OF DISPLACEMENTS Using Eqs. A.16), the equation of motion, Eq. A.26), can be expressed as λ + 2µε xx) + µ ε xy) + µε xz) = ρ 2 u A.29) Using the strain displacement relations given by Eqs. A.2), A.4), and A.8), Eq. A.29) can be written as λ + 2 µ u ) + which can be rewritten as [ v µ + u )] + [ w µ + u )] = ρ 2 u A.30) λ + µ) + µ 2 u = ρ 2 u A.31) where is the dilatation and 2 is the Laplacian operator: 2 = A.32)

7 706 Basic Equations of Elasticity Using a similar procedure, the other two equations of motion, Eqs. A.27) and A.28), can be expressed as λ + µ) + µ 2 v = ρ 2 v λ + µ) + µ 2 w = ρ 2 w A.33) A.34) The equations of motion, Eqs. A.31), A.33), and A.34), govern the propagation of waves as well as the vibratory motion in elastic bodies.

Macroscopic theory Rock as 'elastic continuum'

Macroscopic theory Rock as 'elastic continuum' Elasticity and Seismic Waves Macroscopic theory Rock as 'elastic continuum' Elastic body is deformed in response to stress Two types of deformation: Change in volume and shape Equations of motion Wave

More information

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16. CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo

More information

3D Elasticity Theory

3D Elasticity Theory 3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.

More information

1 Stress and Strain. Introduction

1 Stress and Strain. Introduction 1 Stress and Strain Introduction This book is concerned with the mechanical behavior of materials. The term mechanical behavior refers to the response of materials to forces. Under load, a material may

More information

Chapter 2 Governing Equations

Chapter 2 Governing Equations Chapter Governing Equations Abstract In this chapter fundamental governing equations for propagation of a harmonic disturbance on the surface of an elastic half-space is presented. The elastic media is

More information

Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Standard Solids and Fracture Fluids: Mechanical, Chemical Effects Effective Stress Dilatancy Hardening and Stability Mead, 1925

More information

Finite Element Method in Geotechnical Engineering

Finite Element Method in Geotechnical Engineering Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps

More information

PEAT SEISMOLOGY Lecture 2: Continuum mechanics

PEAT SEISMOLOGY Lecture 2: Continuum mechanics PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a

More information

Mechanics PhD Preliminary Spring 2017

Mechanics PhD Preliminary Spring 2017 Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n

More information

Mechanical Properties of Materials

Mechanical Properties of Materials Mechanical Properties of Materials Strains Material Model Stresses Learning objectives Understand the qualitative and quantitative description of mechanical properties of materials. Learn the logic of

More information

3.2 Hooke s law anisotropic elasticity Robert Hooke ( ) Most general relationship

3.2 Hooke s law anisotropic elasticity Robert Hooke ( ) Most general relationship 3.2 Hooke s law anisotropic elasticity Robert Hooke (1635-1703) Most general relationship σ = C ε + C ε + C ε + C γ + C γ + C γ 11 12 yy 13 zz 14 xy 15 xz 16 yz σ = C ε + C ε + C ε + C γ + C γ + C γ yy

More information

Lecture 8. Stress Strain in Multi-dimension

Lecture 8. Stress Strain in Multi-dimension Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]

More information

Strain Transformation equations

Strain Transformation equations Strain Transformation equations R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur-613 401 Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 8 Table of Contents 1. Stress transformation

More information

Example 3.7 Consider the undeformed configuration of a solid as shown in Figure 3.60.

Example 3.7 Consider the undeformed configuration of a solid as shown in Figure 3.60. 162 3. The linear 3-D elasticity mathematical model The 3-D elasticity model is of great importance, since it is our highest order hierarchical model assuming linear elastic behavior. Therefore, it provides

More information

Continuum Mechanics. Continuum Mechanics and Constitutive Equations

Continuum Mechanics. Continuum Mechanics and Constitutive Equations Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform

More information

ELASTICITY (MDM 10203)

ELASTICITY (MDM 10203) LASTICITY (MDM 10203) Lecture Module 5: 3D Constitutive Relations Dr. Waluyo Adi Siswanto University Tun Hussein Onn Malaysia Generalised Hooke's Law In one dimensional system: = (basic Hooke's law) Considering

More information

Unit IV State of stress in Three Dimensions

Unit IV State of stress in Three Dimensions Unit IV State of stress in Three Dimensions State of stress in Three Dimensions References Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004. Rattan.S.S., "Strength

More information

12. Stresses and Strains

12. Stresses and Strains 12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)

More information

Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth

Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth References: Turcotte and Schubert, Geodynamics, Sections 2.1,-2.4, 2.7, 3.1-3.8, 6.1, 6.2, 6.8, 7.1-7.4. Jaeger and Cook, Fundamentals of

More information

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Thermodynamics Derivation Hooke s Law: Anisotropic Elasticity

More information

Introduction to Seismology Spring 2008

Introduction to Seismology Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Stress and Strain

More information

Continuum mechanism: Stress and strain

Continuum mechanism: Stress and strain Continuum mechanics deals with the relation between forces (stress, σ) and deformation (strain, ε), or deformation rate (strain rate, ε). Solid materials, rigid, usually deform elastically, that is the

More information

Introduction, Basic Mechanics 2

Introduction, Basic Mechanics 2 Computational Biomechanics 18 Lecture : Introduction, Basic Mechanics Ulli Simon, Lucas Engelhardt, Martin Pietsch Scientific Computing Centre Ulm, UZWR Ulm University Contents Mechanical Basics Moment

More information

A short review of continuum mechanics

A short review of continuum mechanics A short review of continuum mechanics Professor Anette M. Karlsson, Department of Mechanical ngineering, UD September, 006 This is a short and arbitrary review of continuum mechanics. Most of this material

More information

[5] Stress and Strain

[5] Stress and Strain [5] Stress and Strain Page 1 of 34 [5] Stress and Strain [5.1] Internal Stress of Solids [5.2] Design of Simple Connections (will not be covered in class) [5.3] Deformation and Strain [5.4] Hooke s Law

More information

Fundamentals of Linear Elasticity

Fundamentals of Linear Elasticity Fundamentals of Linear Elasticity Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy

More information

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is

More information

Chapter 1 Fluid Characteristics

Chapter 1 Fluid Characteristics Chapter 1 Fluid Characteristics 1.1 Introduction 1.1.1 Phases Solid increasing increasing spacing and intermolecular liquid latitude of cohesive Fluid gas (vapor) molecular force plasma motion 1.1.2 Fluidity

More information

Chapter 2 CONTINUUM MECHANICS PROBLEMS

Chapter 2 CONTINUUM MECHANICS PROBLEMS Chapter 2 CONTINUUM MECHANICS PROBLEMS The concept of treating solids and fluids as though they are continuous media, rather thancomposedofdiscretemolecules, is one that is widely used in most branches

More information

Variational principles in mechanics

Variational principles in mechanics CHAPTER Variational principles in mechanics.1 Linear Elasticity n D Figure.1: A domain and its boundary = D [. Consider a domain Ω R 3 with its boundary = D [ of normal n (see Figure.1). The problem of

More information

COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS. Abstract. Introduction

COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS. Abstract. Introduction COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS Hsiang-Chuan Tsai, National Taiwan University of Science and Technology, Taipei, Taiwan James M. Kelly, University of California,

More information

UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich

UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich UNIVERSITY OF SASKATCHEWAN ME 313.3 MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS For Marker s Use Only LAST NAME (printed): FIRST

More information

TRESS - STRAIN RELATIONS

TRESS - STRAIN RELATIONS TRESS - STRAIN RELATIONS Stress Strain Relations: Hook's law, states that within the elastic limits the stress is proportional to t is impossible to describe the entire stress strain curve with simple

More information

Stress, Strain, Mohr s Circle

Stress, Strain, Mohr s Circle Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected

More information

AE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.

AE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept. AE/ME 339 Computational Fluid Dynamics (CFD) 9//005 Topic7_NS_ F0 1 Momentum equation 9//005 Topic7_NS_ F0 1 Consider the moving fluid element model shown in Figure.b Basis is Newton s nd Law which says

More information

Elements of Rock Mechanics

Elements of Rock Mechanics Elements of Rock Mechanics Stress and strain Creep Constitutive equation Hooke's law Empirical relations Effects of porosity and fluids Anelasticity and viscoelasticity Reading: Shearer, 3 Stress Consider

More information

Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body

More information

STRAIN. Normal Strain: The elongation or contractions of a line segment per unit length is referred to as normal strain denoted by Greek symbol.

STRAIN. Normal Strain: The elongation or contractions of a line segment per unit length is referred to as normal strain denoted by Greek symbol. STRAIN In engineering the deformation of a body is specified using the concept of normal strain and shear strain whenever a force is applied to a body, it will tend to change the body s shape and size.

More information

16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations

16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations 6.2 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive quations Constitutive quations For elastic materials: If the relation is linear: Û σ ij = σ ij (ɛ) = ρ () ɛ ij σ ij =

More information

16.20 HANDOUT #2 Fall, 2002 Review of General Elasticity

16.20 HANDOUT #2 Fall, 2002 Review of General Elasticity 6.20 HANDOUT #2 Fall, 2002 Review of General Elasticity NOTATION REVIEW (e.g., for strain) Engineering Contracted Engineering Tensor Tensor ε x = ε = ε xx = ε ε y = ε 2 = ε yy = ε 22 ε z = ε 3 = ε zz =

More information

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»

More information

Surface force on a volume element.

Surface force on a volume element. STRESS and STRAIN Reading: Section. of Stein and Wysession. In this section, we will see how Newton s second law and Generalized Hooke s law can be used to characterize the response of continuous medium

More information

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium

More information

MATERIAL PROPERTIES. Material Properties Must Be Evaluated By Laboratory or Field Tests 1.1 INTRODUCTION 1.2 ANISOTROPIC MATERIALS

MATERIAL PROPERTIES. Material Properties Must Be Evaluated By Laboratory or Field Tests 1.1 INTRODUCTION 1.2 ANISOTROPIC MATERIALS . MARIAL PROPRIS Material Properties Must Be valuated By Laboratory or Field ests. INRODUCION he fundamental equations of structural mechanics can be placed in three categories[]. First, the stress-strain

More information

20. Rheology & Linear Elasticity

20. Rheology & Linear Elasticity I Main Topics A Rheology: Macroscopic deformation behavior B Linear elasticity for homogeneous isotropic materials 10/29/18 GG303 1 Viscous (fluid) Behavior http://manoa.hawaii.edu/graduate/content/slide-lava

More information

EE C247B ME C218 Introduction to MEMS Design Spring 2017

EE C247B ME C218 Introduction to MEMS Design Spring 2017 247B/M 28: Introduction to MMS Design Lecture 0m2: Mechanics of Materials CTN 2/6/7 Outline C247B M C28 Introduction to MMS Design Spring 207 Prof. Clark T.- Reading: Senturia, Chpt. 8 Lecture Topics:

More information

Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship

Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction

More information

ME 243. Lecture 10: Combined stresses

ME 243. Lecture 10: Combined stresses ME 243 Mechanics of Solids Lecture 10: Combined stresses Ahmad Shahedi Shakil Lecturer, Dept. of Mechanical Engg, BUET E-mail: sshakil@me.buet.ac.bd, shakil6791@gmail.com Website: teacher.buet.ac.bd/sshakil

More information

Linearized theory of elasticity

Linearized theory of elasticity Linearized theory of elasticity Arie Verhoeven averhoev@win.tue.nl CASA Seminar, May 24, 2006 Seminar: Continuum mechanics 1 Stress and stress principles Bart Nowak March 8 2 Strain and deformation Mark

More information

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS CHATR Stress MCHANICS OF MATRIALS and Strain Axial Loading Stress & Strain: Axial Loading Suitability of a structure or machine may depend on the deformations in the structure as well as the stresses induced

More information

Lecture 8: Tissue Mechanics

Lecture 8: Tissue Mechanics Computational Biology Group (CoBi), D-BSSE, ETHZ Lecture 8: Tissue Mechanics Prof Dagmar Iber, PhD DPhil MSc Computational Biology 2015/16 7. Mai 2016 2 / 57 Contents 1 Introduction to Elastic Materials

More information

ABHELSINKI UNIVERSITY OF TECHNOLOGY

ABHELSINKI UNIVERSITY OF TECHNOLOGY ABHELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D HELSINKI A posteriori error analysis for the Morley plate element Jarkko Niiranen Department of Structural

More information

Measurement of deformation. Measurement of elastic force. Constitutive law. Finite element method

Measurement of deformation. Measurement of elastic force. Constitutive law. Finite element method Deformable Bodies Deformation x p(x) Given a rest shape x and its deformed configuration p(x), how large is the internal restoring force f(p)? To answer this question, we need a way to measure deformation

More information

CONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS

CONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS Chapter 9 CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS Figure 9.1: Hooke memorial window, St. Helen s, Bishopsgate, City of London 211 212 CHAPTR 9. CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS 9.1 Mechanical

More information

Unit 13 Review of Simple Beam Theory

Unit 13 Review of Simple Beam Theory MIT - 16.0 Fall, 00 Unit 13 Review of Simple Beam Theory Readings: Review Unified Engineering notes on Beam Theory BMP 3.8, 3.9, 3.10 T & G 10-15 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics

More information

EE C245 ME C218 Introduction to MEMS Design Fall 2007

EE C245 ME C218 Introduction to MEMS Design Fall 2007 EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 13: Material

More information

Lecture 7. Properties of Materials

Lecture 7. Properties of Materials MIT 3.00 Fall 2002 c W.C Carter 55 Lecture 7 Properties of Materials Last Time Types of Systems and Types of Processes Division of Total Energy into Kinetic, Potential, and Internal Types of Work: Polarization

More information

In this section, mathematical description of the motion of fluid elements moving in a flow field is

In this section, mathematical description of the motion of fluid elements moving in a flow field is Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small

More information

Elements of Continuum Elasticity. David M. Parks Mechanics and Materials II February 25, 2004

Elements of Continuum Elasticity. David M. Parks Mechanics and Materials II February 25, 2004 Elements of Continuum Elasticity David M. Parks Mechanics and Materials II 2.002 February 25, 2004 Solid Mechanics in 3 Dimensions: stress/equilibrium, strain/displacement, and intro to linear elastic

More information

Chapter 5. The Differential Forms of the Fundamental Laws

Chapter 5. The Differential Forms of the Fundamental Laws Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations

More information

3.22 Mechanical Properties of Materials Spring 2008

3.22 Mechanical Properties of Materials Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 3.22 Mechanical Properties of Materials Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Quiz #1 Example

More information

Stress/Strain. Outline. Lecture 1. Stress. Strain. Plane Stress and Plane Strain. Materials. ME EN 372 Andrew Ning

Stress/Strain. Outline. Lecture 1. Stress. Strain. Plane Stress and Plane Strain. Materials. ME EN 372 Andrew Ning Stress/Strain Lecture 1 ME EN 372 Andrew Ning aning@byu.edu Outline Stress Strain Plane Stress and Plane Strain Materials otes and News [I had leftover time and so was also able to go through Section 3.1

More information

VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA

VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA FAKULTA METALURGIE A MATERIÁLOVÉHO INŽENÝRSTVÍ APPLIED MECHANICS Study Support Leo Václavek Ostrava 2015 Title:Applied Mechanics Code: Author: doc. Ing.

More information

Module III - Macro-mechanics of Lamina. Lecture 23. Macro-Mechanics of Lamina

Module III - Macro-mechanics of Lamina. Lecture 23. Macro-Mechanics of Lamina Module III - Macro-mechanics of Lamina Lecture 23 Macro-Mechanics of Lamina For better understanding of the macromechanics of lamina, the knowledge of the material properties in essential. Therefore, the

More information

Chapter 6: Momentum Analysis

Chapter 6: Momentum Analysis 6-1 Introduction 6-2Newton s Law and Conservation of Momentum 6-3 Choosing a Control Volume 6-4 Forces Acting on a Control Volume 6-5Linear Momentum Equation 6-6 Angular Momentum 6-7 The Second Law of

More information

Applications of Eigenvalues & Eigenvectors

Applications of Eigenvalues & Eigenvectors Applications of Eigenvalues & Eigenvectors Louie L. Yaw Walla Walla University Engineering Department For Linear Algebra Class November 17, 214 Outline 1 The eigenvalue/eigenvector problem 2 Principal

More information

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS Third E CHAPTER 2 Stress MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University and Strain Axial Loading Contents Stress & Strain:

More information

Consider an elastic spring as shown in the Fig.2.4. When the spring is slowly

Consider an elastic spring as shown in the Fig.2.4. When the spring is slowly .3 Strain Energy Consider an elastic spring as shown in the Fig..4. When the spring is slowly pulled, it deflects by a small amount u 1. When the load is removed from the spring, it goes back to the original

More information

SEISMOLOGY I. Laurea Magistralis in Physics of the Earth and of the Environment. Elasticity. Fabio ROMANELLI

SEISMOLOGY I. Laurea Magistralis in Physics of the Earth and of the Environment. Elasticity. Fabio ROMANELLI SEISMOLOGY I Laurea Magistralis in Physics of the Earth and of the Environment Elasticity Fabio ROMANELLI Dept. Earth Sciences Università degli studi di Trieste romanel@dst.units.it 1 Elasticity and Seismic

More information

Analytical Mechanics: Elastic Deformation

Analytical Mechanics: Elastic Deformation Analytical Mechanics: Elastic Deformation Shinichi Hirai Dept. Robotics, Ritsumeikan Univ. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 1 / 60 Agenda Agenda

More information

MEC-E8001 FINITE ELEMENT ANALYSIS

MEC-E8001 FINITE ELEMENT ANALYSIS MEC-E800 FINIE EEMEN ANAYSIS 07 - WHY FINIE EEMENS AND IS HEORY? Design of machines and structures: Solution to stress or displacement by analytical method is often impossible due to complex geometry,

More information

2. Mechanics of Materials: Strain. 3. Hookes's Law

2. Mechanics of Materials: Strain. 3. Hookes's Law Mechanics of Materials Course: WB3413, Dredging Processes 1 Fundamental Theory Required for Sand, Clay and Rock Cutting 1. Mechanics of Materials: Stress 1. Introduction 2. Plane Stress and Coordinate

More information

Smart Materials, Adaptive Structures, and Intelligent Mechanical Systems

Smart Materials, Adaptive Structures, and Intelligent Mechanical Systems Smart Materials, Adaptive Structures, and Intelligent Mechanical Systems Bishakh Bhattacharya & Nachiketa Tiwari Indian Institute of Technology Kanpur Lecture 19 Analysis of an Orthotropic Ply References

More information

Wave and Elasticity Equations

Wave and Elasticity Equations 1 Wave and lasticity quations Now let us consider the vibrating string problem which is modeled by the one-dimensional wave equation. Suppose that a taut string is suspended by its extremes at the points

More information

Chapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature

Chapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte

More information

Composites Design and Analysis. Stress Strain Relationship

Composites Design and Analysis. Stress Strain Relationship Composites Design and Analysis Stress Strain Relationship Composite design and analysis Laminate Theory Manufacturing Methods Materials Composite Materials Design / Analysis Engineer Design Guidelines

More information

Chapter 1 Fracture Physics

Chapter 1 Fracture Physics Chapter 1 Fracture Physics In this chapter we give a brief account of the theory of fracture, restricting ourselves to brittle fracture. The structure of this chapter is as follows. The first part describes

More information

Fig. 1. Circular fiber and interphase between the fiber and the matrix.

Fig. 1. Circular fiber and interphase between the fiber and the matrix. Finite element unit cell model based on ABAQUS for fiber reinforced composites Tian Tang Composites Manufacturing & Simulation Center, Purdue University West Lafayette, IN 47906 1. Problem Statement In

More information

Chapter 3. Load and Stress Analysis

Chapter 3. Load and Stress Analysis Chapter 3 Load and Stress Analysis 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2 Sign Conventions for Bending and Shear Fig. 3 3

More information

NONLINEAR WAVE EQUATIONS ARISING IN MODELING OF SOME STRAIN-HARDENING STRUCTURES

NONLINEAR WAVE EQUATIONS ARISING IN MODELING OF SOME STRAIN-HARDENING STRUCTURES NONLINEAR WAE EQUATIONS ARISING IN MODELING OF SOME STRAIN-HARDENING STRUCTURES DONGMING WEI Department of Mathematics, University of New Orleans, 2 Lakeshore Dr., New Orleans, LA 7148,USA E-mail: dwei@uno.edu

More information

Tensor Visualization. CSC 7443: Scientific Information Visualization

Tensor Visualization. CSC 7443: Scientific Information Visualization Tensor Visualization Tensor data A tensor is a multivariate quantity Scalar is a tensor of rank zero s = s(x,y,z) Vector is a tensor of rank one v = (v x,v y,v z ) For a symmetric tensor of rank 2, its

More information

Solid State Theory Physics 545

Solid State Theory Physics 545 olid tate Theory hysics 545 Mechanical properties of materials. Basics. tress and strain. Basic definitions. Normal and hear stresses. Elastic constants. tress tensor. Young modulus. rystal symmetry and

More information

Lecture notes Models of Mechanics

Lecture notes Models of Mechanics Lecture notes Models of Mechanics Anders Klarbring Division of Mechanics, Linköping University, Sweden Lecture 7: Small deformation theories Klarbring (Mechanics, LiU) Lecture notes Linköping 2012 1 /

More information

Useful Formulae ( )

Useful Formulae ( ) Appendix A Useful Formulae (985-989-993-) 34 Jeremić et al. A.. CHAPTER SUMMARY AND HIGHLIGHTS page: 35 of 536 A. Chapter Summary and Highlights A. Stress and Strain This section reviews small deformation

More information

202 Index. failure, 26 field equation, 122 force, 1

202 Index. failure, 26 field equation, 122 force, 1 Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic

More information

HIGHER-ORDER THEORIES

HIGHER-ORDER THEORIES HIGHER-ORDER THEORIES THIRD-ORDER SHEAR DEFORMATION PLATE THEORY LAYERWISE LAMINATE THEORY J.N. Reddy 1 Third-Order Shear Deformation Plate Theory Assumed Displacement Field µ u(x y z t) u 0 (x y t) +

More information

2 Introduction to mechanics

2 Introduction to mechanics 21 Motivation Thermodynamic bodies are being characterized by two competing opposite phenomena, energy and entropy which some researchers in thermodynamics would classify as cause and chance or determinancy

More information

Mechanical Behavior of Composite Tapered Lamina

Mechanical Behavior of Composite Tapered Lamina International Journal of Engineering Research and Development e-issn: 2278-067X, p-issn: 2278-800X, www.ijerd.com Volume 10, Issue 8 (August 2014), PP.19-27 Mechanical Behavior of Composite Tapered Lamina

More information

Exercise: concepts from chapter 8

Exercise: concepts from chapter 8 Reading: Fundamentals of Structural Geology, Ch 8 1) The following exercises explore elementary concepts associated with a linear elastic material that is isotropic and homogeneous with respect to elastic

More information

Numerical Modelling in Geosciences. Lecture 6 Deformation

Numerical Modelling in Geosciences. Lecture 6 Deformation Numerical Modelling in Geosciences Lecture 6 Deformation Tensor Second-rank tensor stress ), strain ), strain rate ) Invariants quantities independent of the coordinate system): - First invariant trace:!!

More information

CLASS NOTES Version 5.0

CLASS NOTES Version 5.0 MAE 477/577 - EXPERIMENTAL TECHNIQUES IN SOLID MECHANICS - CLASS NOTES Version 5.0 by John A. Gilbert, Ph.D. Professor of Mechanical Engineering University of Alabama in Huntsville FALL 2013 Table of Contents

More information

Basic Concepts of Strain and Tilt. Evelyn Roeloffs, USGS June 2008

Basic Concepts of Strain and Tilt. Evelyn Roeloffs, USGS June 2008 Basic Concepts of Strain and Tilt Evelyn Roeloffs, USGS June 2008 1 Coordinates Right-handed coordinate system, with positions along the three axes specified by x,y,z. x,y will usually be horizontal, and

More information

GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS

GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS I Main Topics A Why deal with tensors? B Order of scalars, vectors, and tensors C Linear transformation of scalars and vectors (and tensors) II Why

More information

Crystal Relaxation, Elasticity, and Lattice Dynamics

Crystal Relaxation, Elasticity, and Lattice Dynamics http://exciting-code.org Crystal Relaxation, Elasticity, and Lattice Dynamics Pasquale Pavone Humboldt-Universität zu Berlin http://exciting-code.org PART I: Structure Optimization Pasquale Pavone Humboldt-Universität

More information

SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by

SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e

More information

3D and Planar Constitutive Relations

3D and Planar Constitutive Relations 3D and Planar Constitutive Relations A School on Mechanics of Fibre Reinforced Polymer Composites Knowledge Incubation for TEQIP Indian Institute of Technology Kanpur PM Mohite Department of Aerospace

More information

Rotational Motion. Chapter 4. P. J. Grandinetti. Sep. 1, Chem P. J. Grandinetti (Chem. 4300) Rotational Motion Sep.

Rotational Motion. Chapter 4. P. J. Grandinetti. Sep. 1, Chem P. J. Grandinetti (Chem. 4300) Rotational Motion Sep. Rotational Motion Chapter 4 P. J. Grandinetti Chem. 4300 Sep. 1, 2017 P. J. Grandinetti (Chem. 4300) Rotational Motion Sep. 1, 2017 1 / 76 Angular Momentum The angular momentum of a particle with respect

More information

Constitutive Equations

Constitutive Equations Constitutive quations David Roylance Department of Materials Science and ngineering Massachusetts Institute of Technology Cambridge, MA 0239 October 4, 2000 Introduction The modules on kinematics (Module

More information

INTRODUCTION TO STRAIN

INTRODUCTION TO STRAIN SIMPLE STRAIN INTRODUCTION TO STRAIN In general terms, Strain is a geometric quantity that measures the deformation of a body. There are two types of strain: normal strain: characterizes dimensional changes,

More information

MAE4700/5700 Finite Element Analysis for Mechanical and Aerospace Design

MAE4700/5700 Finite Element Analysis for Mechanical and Aerospace Design MAE4700/5700 Finite Element Analsis for Mechanical and Aerospace Design Cornell Universit, Fall 2009 Nicholas Zabaras Materials Process Design and Control Laborator Sible School of Mechanical and Aerospace

More information