# 3D Elasticity Theory

Size: px
Start display at page:

Transcription

1 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. Beams, plates, and shells are examples of such problems. Here the general equations for equilibrium, material law, and kinematic compatibility of a 3D continuum are established. quilibrium Here we are not concerned with externally applied loads or surface forces but rather equilibrium between the forces within a material particle. This includes stresses and potential forces acting on the volume, such as gravity. Figure 1 shows the stresses acting on an infinitesimally small material volume. The figure is hard to read because it includes the changes in stress-values from one end of the cube to the opposite side. For example, the axial stress σ xx changes by dσ xx from one side to the other. The volume forces are not included in the figure but are denoted f x, f y, and f z with index indicating the direction of the force. quilibrium in the x-direction yields dσ xx dy dz + dσ yx dz + dσ zx dy + f x dy dz = 0 (1) Dividing through by ( dy dz) yields: dσ xx + dσ yx dy + dσ zx dz + f x = 0 () Repeating the exercise in all three axis-directions produces the equilibrium equations that all material particles that are in equilibrium must satisfy: σ ij,i + f j = 0 (3) Moment equilibrium of the infinitesimal cube yields the symmetry of the stress tensor: σ ij = σ ji (4) 3D lasticity Theory Updated January 6, 018 Page 1

2 z, w + d y, v τ σ τ yz + dτ yz τ + dτ xz xy xy xx τ yx + dτ yx σ xx + dσ xx dz τ yx τ zy τ yz τ zy + dτ zy + d + d τ xz + dτ xz dy x, u Figure 1: Stresses on a solid cube. Kinematic Compatibility quations are here sought to relate strains with displacements. As shown in Figure 1 the displacements in the axis directions x, y, z are u v w, respectively. xpressions for longitudinal strains are obtained by studying an infinitesimal material cube. First consider the x-direction. The x-direction displacement at x is u. The x-direction displacement at x+ is u+( u/). Defining strain as change in length divided by original length yields = u Repeating the consideration for the other axis directions yields = v y = w z Now to the shear strains, starting with visualized in Figure and defining the change in angle between originally orthogonal lines. The change in angle has two contributions: (5) (6) 3D lasticity Theory Updated January 6, 018 Page

3 = = ε xy + ε yx = v + u y dy dy = v + u y (7) Here it is understood that the engineering shear strain is twice the corresponding coordinate strains. Repeating the consideration for the other two coordinate planes yields y, v y + v = v z + w y γ xz z + w u y dy (8) ε yx dy v ε xy x, u Figure : Shear strains. The kinematic compatibility equations for both axial and shear strains are summarized by ε ij = 1 ( u i, j + u j,i ) (9) By means of Voight notation the kinematic equations can be written in vector notation, with defined as a matrix differential operator: ε = γ zx = y z y 0 0 z y z 0 u v w = u,x v,y w,z u,y + v,x v,z + w,y u,z + w,x = u (10) 3D lasticity Theory Updated January 6, 018 Page 3

4 Because the compatibility equations contain six strain components and only three displacement components, only certain strain patterns are physically possible. For that reason the strains-displacement equations are sometimes combined into compatibility equations that give conditions for valid deformation patterns. This is done below for D elasticity theory. Sometimes the terms compatibility equation and compatibility condition are maintained even when adding material law and equilibrium equations. Such equations are, together with boundary conditions, sufficient to determine the solution to specific problems. Material Law The theory of elasticity is founded on the assumption of a homogeneous isotropic linear elastic material. For a material particle, the relationship between a uniaxial stress and the corresponding uniaxial strain is give by the modulus of elasticity, sometimes called Young s modulus,, formulated in Hooke s law: σ = ε (11) The strain in the transversal direction is defined by Poisson s ratio, ν: ν ε t ε ε t = ν ε ε t = ν σ where ε t is the transversal strain. Strain expressions that account for transversal strains in the orthogonal directions yield the three-dimensional version of Hooke s law: = σ xx ν ν (1) = ν σ xx ν (13) = ν σ xx ν There are only two independent parameters in the general Hooke s law. However, a special material constant named the shear modulus, G, which is related to and ν, defines the relationship between shear stresses and shear strains: τ ij = G γ ij, i j (14) To determine the relationship between G, and ν, consider an infinitesimally small twodimensional material particle subjected to pure shear τ. Mohr s circle for this case is centred at the origin with radius τ. Consequently, the principal stresses are -τ and τ with axes at 45 o. The deformation of the particle is shown in Figure 3. 3D lasticity Theory Updated January 6, 018 Page 4

5 # l! sin " & \$ % ' ( ) l!" l! l!!! l Figure 3: Derivation of the expression G. The quantity Δ can be expressed in two ways. In the pure shear state: Δ = In the rotated state of pure axial stress: quating the two expressions for Δ yields: l γ = l γ (15) Δ = ε ( l) = τ ν ( τ ) ( l) (16) τ = (1+ν) γ (17)!#" # \$ G Hence, together with q. (13) the following equations complete the general Hooke s law: = G, τ yz = G, = G γ zx (18) In Voight notation it reads ε i =C ij -1 σ j or ε=c -1 σ: γ zx 1 ν ν ν 1 ν = 1 ν ν (1 + ν) (1 + ν) (1 + ν) σ xx τ yz (19) 3D lasticity Theory Updated January 6, 018 Page 5

6 Or, inversely σ i =C ij ε j or σ=cε: σ xx τ yz (1 ν) ν ν ν (1 ν) ν ν ν (1 ν) ν = (1+ ν)(1 ν) ν ν In index notation with the original strain and stress tensors, Hooke s law is written γ zx (0) σ ij = λ ε kk δ ij + µ ε ij (1) where δ ij is the unit matrix and λ and µ are the Lame parameters: µ = G λ = ν (1 + ν)(1 ν) In addition to, ν, G, µ, and λ, the bulk modulus, K, is employed in the study of volume change under hydrostatic pressure. Let ε kk =ε 11 +ε +ε 33 denote the dilatation, i.e., the change in volume of an infinitesimally small cube. The pressure, p, is ε kk /3. The bulk modulus relates the pressure to the dilatation: p=-k. ε kk, where K = 3(1 ν) () (3) 3D lasticity Theory Updated January 6, 018 Page 6

### 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]

### 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

### 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

### 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

### 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

### Basic Equations of Elasticity

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

### 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

### 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

### 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

### 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

### 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

### 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

### Exercise: concepts from chapter 5

Reading: Fundamentals of Structural Geology, Ch 5 1) Study the oöids depicted in Figure 1a and 1b. Figure 1a Figure 1b Figure 1. Nearly undeformed (1a) and significantly deformed (1b) oöids with spherulitic

### 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

### 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

### 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

### 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

### 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

### CHAPTER 4 Stress Transformation

CHAPTER 4 Stress Transformation ANALYSIS OF STRESS For this topic, the stresses to be considered are not on the perpendicular and parallel planes only but also on other inclined planes. A P a a b b P z

### CHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES

CHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES * Governing equations in beam and plate bending ** Solution by superposition 1.1 From Beam Bending to Plate Bending 1.2 Governing Equations For Symmetric

### 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

### 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

### 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

### 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)

### 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

### Understand basic stress-strain response of engineering materials.

Module 3 Constitutive quations Learning Objectives Understand basic stress-strain response of engineering materials. Quantify the linear elastic stress-strain response in terms of tensorial quantities

### 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.

### 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

### 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

### 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

### 4. Mathematical models used in engineering structural analysis

4. Mathematical models used in engineering structural analysis In this chapter we pursue a formidable task to present the most important mathematical models in structural mechanics. In order to best situate

### 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

### 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

### 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

### Stress transformation and Mohr s circle for stresses

Stress transformation and Mohr s circle for stresses 1.1 General State of stress Consider a certain body, subjected to external force. The force F is acting on the surface over an area da of the surface.

### 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

### 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

### 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

### Equilibrium of Deformable Body

Equilibrium of Deformable Body Review Static Equilibrium If a body is in static equilibrium under the action applied external forces, the Newton s Second Law provides us six scalar equations of equilibrium

### 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

### 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 =

### 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

### 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:

### FINAL EXAMINATION. (CE130-2 Mechanics of Materials)

UNIVERSITY OF CLIFORNI, ERKELEY FLL SEMESTER 001 FINL EXMINTION (CE130- Mechanics of Materials) Problem 1: (15 points) pinned -bar structure is shown in Figure 1. There is an external force, W = 5000N,

### 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

### 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

### 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

### 3. The linear 3-D elasticity mathematical model

3. The linear 3-D elasticity mathematical model In Chapter we examined some fundamental conditions that should be satisfied in the modeling of all deformable solids and structures. The study of truss structures

### 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

### 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

### Problem " Â F y = 0. ) R A + 2R B + R C = 200 kn ) 2R A + 2R B = 200 kn [using symmetry R A = R C ] ) R A + R B = 100 kn

Problem 0. Three cables are attached as shown. Determine the reactions in the supports. Assume R B as redundant. Also, L AD L CD cos 60 m m. uation of uilibrium: + " Â F y 0 ) R A cos 60 + R B + R C cos

### 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

### 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

### 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

### Samantha Ramirez, MSE. Stress. The intensity of the internal force acting on a specific plane (area) passing through a point. F 2

Samantha Ramirez, MSE Stress The intensity of the internal force acting on a specific plane (area) passing through a point. Δ ΔA Δ z Δ 1 2 ΔA Δ x Δ y ΔA is an infinitesimal size area with a uniform force

### Use Hooke s Law (as it applies in the uniaxial direction),

0.6 STRSS-STRAIN RLATIONSHIP Use the principle of superposition Use Poisson s ratio, v lateral longitudinal Use Hooke s Law (as it applies in the uniaxial direction), x x v y z, y y vx z, z z vx y Copyright

### 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

### ELASTICITY (MDM 10203)

ELASTICITY () Lecture Module 3: Fundamental Stress and Strain University Tun Hussein Onn Malaysia Normal Stress inconstant stress distribution σ= dp da P = da A dimensional Area of σ and A σ A 3 dimensional

### 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

### EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading

MA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading MA 3702 Mechanics & Materials Science Zhe Cheng (2018) 2 Stress & Strain - Axial Loading Statics

### Principal Stresses, Yielding Criteria, wall structures

Principal Stresses, Yielding Criteria, St i thi Stresses in thin wall structures Introduction The most general state of stress at a point may be represented by 6 components, x, y, z τ xy, τ yz, τ zx normal

### 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

### Symmetric Bending of Beams

Symmetric Bending of Beams beam is any long structural member on which loads act perpendicular to the longitudinal axis. Learning objectives Understand the theory, its limitations and its applications

### The Kinematic Equations

The Kinematic Equations David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 0139 September 19, 000 Introduction The kinematic or strain-displacement

### 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

### Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA UNESCO EOLSS

MECHANICS OF MATERIALS Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA Keywords: Solid mechanics, stress, strain, yield strength Contents 1. Introduction 2. Stress

### Physics of Continuous media

Physics of Continuous media Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2012 October 26, 2012 Deformations of continuous media If a body is deformed, we say that the point which originally had

### **********************************************************************

Department of Civil and Environmental Engineering School of Mining and Petroleum Engineering 3-33 Markin/CNRL Natural Resources Engineering Facility www.engineering.ualberta.ca/civil Tel: 780.492.4235

### 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

### By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ.

Mohr s Circle By drawing Mohr s circle, the stress transformation in -D can be done graphically. σ = σ x + σ y τ = σ x σ y + σ x σ y cos θ + τ xy sin θ, 1 sin θ + τ xy cos θ. Note that the angle of rotation,

### 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

### Chapter: 5 Subdomain boundary nodes

Chapter: 5 MEK4560 The Finite Element Method in Solid Mechanics II (February 11, 2008) (E-post:torgeiru@math.uio.no) Page 1 of 19 5 Thick plates 3 5.1 ssumptions..................................... 3

### 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) +

### 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

### HIGHER-ORDER THEORIES

HIGHER-ORDER THEORIES Third-order Shear Deformation Plate Theory Displacement and strain fields Equations of motion Navier s solution for bending Layerwise Laminate Theory Interlaminar stress and strain

### 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:

### Bone Tissue Mechanics

Bone Tissue Mechanics João Folgado Paulo R. Fernandes Instituto Superior Técnico, 2016 PART 1 and 2 Introduction The objective of this course is to study basic concepts on hard tissue mechanics. Hard tissue

### 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

### 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

### Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

51 Module 4: Lecture 2 on Stress-strain relationship and Shear strength of soils Contents Stress state, Mohr s circle analysis and Pole, Principal stressspace, Stress pathsin p-q space; Mohr-coulomb failure

### 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,

### ANALYSIS OF STRAINS CONCEPT OF STRAIN

ANALYSIS OF STRAINS CONCEPT OF STRAIN Concept of strain : if a bar is subjected to a direct load, and hence a stress the bar will change in length. If the bar has an original length L and changes by an

### 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

### 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

### 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

### 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

### Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms

Continuum mechanics office Math 0.107 ales.janka@unifr.ch http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Université de Fribourg 1. Constitutive equation: definition and basic axioms Constitutive

### Advanced Structural Analysis EGF Section Properties and Bending

Advanced Structural Analysis EGF316 3. Section Properties and Bending 3.1 Loads in beams When we analyse beams, we need to consider various types of loads acting on them, for example, axial forces, shear

### 3D Stress Tensors. 3D Stress Tensors, Eigenvalues and Rotations

3D Stress Tensors 3D Stress Tensors, Eigenvalues and Rotations Recall that we can think of an n x n matrix Mij as a transformation matrix that transforms a vector xi to give a new vector yj (first index

### Geology 229 Engineering Geology. Lecture 5. Engineering Properties of Rocks (West, Ch. 6)

Geology 229 Engineering Geology Lecture 5 Engineering Properties of Rocks (West, Ch. 6) Common mechanic properties: Density; Elastic properties: - elastic modulii Outline of this Lecture 1. Uniaxial rock

### MECH 401 Mechanical Design Applications

MECH 401 Mechanical Design Applications Dr. M. O Malley Master Notes Spring 008 Dr. D. M. McStravick Rice University Updates HW 1 due Thursday (1-17-08) Last time Introduction Units Reliability engineering

### 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

### Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

50 Module 4: Lecture 1 on Stress-strain relationship and Shear strength of soils Contents Stress state, Mohr s circle analysis and Pole, Principal stressspace, Stress pathsin p-q space; Mohr-Coulomb failure

### THE GENERAL ELASTICITY PROBLEM IN SOLIDS

Chapter 10 TH GNRAL LASTICITY PROBLM IN SOLIDS In Chapters 3-5 and 8-9, we have developed equilibrium, kinematic and constitutive equations for a general three-dimensional elastic deformable solid bod.

### 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

### Mechanics of materials Lecture 4 Strain and deformation

Mechanics of materials Lecture 4 Strain and deformation Reijo Kouhia Tampere University of Technology Department of Mechanical Engineering and Industrial Design Wednesday 3 rd February, 206 of a continuum

### 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

### [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