Complex Representation in Two-Dimensional Theory of Elasticity

Size: px
Start display at page:

Download "Complex Representation in Two-Dimensional Theory of Elasticity"

Transcription

1 Complex Representation in Two-Dimensional Theory of Elasticity E. Vondenhoff 7-june-2006

2 Literature: Muskhelishvili : Some Basic Problems of the Mathematical Theory of Elasticity, Chapter 5 Prof. ir. C. de Pater: Collegediktaat Elasticiteitstheory, Technische Hogeschool Twente (afdeling der werktuigbouwkunde)

3 This presentation consists of the following parts: 1. Basic equations in theory of elasticity 2. The Airy stress function in 2D problems 3. Cauchy Riemann equations 4. Determination of the displacements from the stress function 5. Representation of the Airy stress function by two complex analytic functions 6. Complex representation of displacements and stresses 7. Example 8. Results

4 1. Basic equations in theory of elasticity Solutions of problems in elasticity have to satisfy Conservation laws Compatibility conditions Constitutive equations Variables: u: displacement vector E: deformation tensor T : stress tensor

5 Conservation of momentum: ( j T ij,j ) + F i = 0 Relation between deformation and displacement: ε ij = 1 2 (u i,j + u j,i ) or E = 1 2 ( u + ut )

6 Constitutive equations (Hooke s law) T ij = λδ ij ( k ε kk ) + 2µε ij or T = λ(tre)i + 2µE where λ and µ are the Lamé parameters.

7 Combining Balance laws and constitutive equations and assuming that there are no body forces we get ( ) µ u i + (λ + µ) u k,ki = 0. These are called the equations of Navier-Cauchy. Differentiating with respect to x i and summing over i we get (tre) = 0 k

8 2. The Airy stress function in 2D problems We assume that there exists a function U = U(x, y) which is called the Airy stress function such that T 11 = 2 U y 2 T 22 = 2 U x 2 T 12 = 2 U x y The balance equations are automatically satisfied: T i1 x + T i2 y = 0

9 From (trt ) = 0 we get U = 0 The function U is called a biharmonic function.

10 3. Cauchy-Riemann equations Let ϕ : C C be a complex function ϕ(z) = p(x, y) + iq(x, y). ϕ is differentiable at z C then there exists a C C such that for small h C ϕ(z + h) = ϕ(z) + Ch + O( h 2 ). Write C = D + ie and h = ξ + iη

11 We get ϕ(z + h) = p(x + ξ, y + η) + iq(x + ξ, y + η) = p(x, y) + iq(x, y) + ξ p x + η p q + iξ y x ϕ(z) + Ch = ϕ(z) + Dξ Eη + i(eξ + Dη). Differentiable functions satisfy the Cauchy-Riemann equations p y p x = D = q y = E = q x + iη q y + O( h 2 ),

12 4. Determination of displacements from the stress function Let u and v denote the components of the displacement vector u in the twodimensional case. From the constitutive equations we get λtre + 2µ u x = 2 U y 2 λtre + 2µ v y = 2 U x 2 ( v µ x + u ) = 2 U y x y

13 We can easily derive 2µ u x = 2 U y 2 λ 2(λ + µ) U because 2µ v y = 2 U x 2 λ 2(λ + µ) U U = trt = 2(λ + µ)tre.

14 Introduce the function P := U. This P is harmonic: P = U = 0. We get 2µ u x = 2 U x 2 + λ + 2µ 2(λ + µ) P 2µ v y = 2 U y 2 + λ + 2µ 2(λ + µ) P

15 Define Q as the harmonic conjugate of P : P x = Q y, P y = Q x Define f : G C C by f(x + iy) = P (x, y) + iq(x, y) f : G C is analytic. This means that for all a G there exist A k s such that in a neighborhood of a f(z) = A k (z a) k. k=0

16 Define ϕ by and define p and q ϕ(z) := 1 4 z 0 f(ζ)dζ. p(z) := Re ϕ(z), q(z) := Im ϕ(z). We have ϕ (z) = p x + i q x = 1 (P + iq). 4 By the Cauchy-Riemann equations we get P = 4 p x = 4 q y Q = 4 p y = 4 q x

17 We can write After integration we get 2µ u x = U 2(λ + 2µ) p 2 + x2 λ + µ x 2µ v y = U 2(λ + 2µ) q 2 + y2 λ + µ y. 2µu = U x 2µv = U y 2(λ + 2µ) + λ + µ p + f 1(y) 2(λ + 2µ) + λ + µ q + f 2(x).

18 From the Cauchy Riemann equations and ( v µ x + u ) = 2 U y x y we get The general form must be f 1(y) + f 2(x) = 0. f 1 (y) = 2µ( εy + α) f 2 (x) = 2µ(εx + β). These terms f 1 and f 2, which only give rigid body displacement, are often omitted.

19 After omitting f 1 and f 2 2µu = U x 2µv = U y 2(λ + 2µ) + λ + µ p + 2(λ + 2µ) λ + µ q.

20 5. Representation of the biharmonic function by two complex analytic functions Theorem: (U px qy) = 0 where ϕ(z) = p(z) + iq(z) as introduced earlier. Write for a harmonic function p 1. U = px + qy + p 1 Let q 1 be the harmonic conjugate of p 1 and define the analytic function χ = χ(z) by χ(z) = p 1 (z) + iq 1 (z).

21 One may write or U = 1 2 U = Re( zϕ(z) + χ(z)) ( ) zϕ(z) + zϕ(z) + χ(z) + χ(z). Every U of this form is biharmonic if ϕ and χ are analytic.

22 The method of deduction by Goursat. Consider the biharmonic equation U = 0. Introduce new variables ξ = x + iy = z and η = x iy = z. Then the preceding equation takes the form whence it follows directly that 4 U ξ 2 η 2 = 0, U = ϕ 1 (z) + ϕ 2 (z) + zχ 1 (z) + zχ 2 (z), where ϕ 1, ϕ 2, χ 1 and χ 2 are "arbitrary" functions. If U is real then it is easily seen that one must put ϕ 2 (z) = ϕ 1 (z) and χ 2 (z) = χ 1 (z).

23 It is easily found that and 2 U x = ϕ(z) + zϕ (z) + ϕ(z) + zϕ (z) + χ (z) + χ (z) 2 U ( ) y = i ϕ(z) + zϕ (z) + ϕ(z) zϕ (z) + χ (z) χ (z). It turns out to be convenient to work with where We have U x + i U y = ϕ(z) + zϕ (z) + ψ(z) ψ = χ. U = 4 Re ϕ (z).

24 6. Complex representation of displacements and stresses We already found the following relation between the displacement vector and U: ( ) U 2µ(u + iv) = x + i U 2(λ + 2µ) + y λ + µ ϕ(z). A more convenient formula is 2µ(u + iv) = κϕ(z) zϕ (z) ψ(z), where κ = λ + 3µ λ + µ.

25 In order to find a representation of the stress components by means of the functions ϕ and ψ we will find an expression for the forces acting on an arc AB lying in the plane Oxy. t : tangential vector n: normal vector.

26 The force (T xn ds, T yn ds) acting on an element ds of the arc AB, will be understood to be the force exerted on the side of the positive normal: T xn = T xx (n, e x ) + T xy (n, e y ) = 2 U y (n, e x) 2 U 2 x y (n, e y), T yn = T xy (n, e x ) + T yy (n, e y ) = 2 U x y (n, e x) + 2 U x (n, e y). 2

27 We have (n, e x ) = (t, e y ) = dy ds, (n, e y ) = (t, e x ) = dx ds. Introducing these values into the preceding formulae, one finds ( ) U T xn = d ds T yn = d ds y ( ) U. x

28 We get T xn + it yn = d ( ) U ds y i U x = i d ( ) U ds x + i U y = i d ( ) ϕ(z) + zϕ ds (z) + ψ(z)

29 Let ds have the direction of Oy, then T xn = T xx and T yn = T xy and T xx + it xy = ϕ (z) + ϕ (z) zϕ (z) ψ (z). Let ds have the direction of Ox, then T xn = T xy and T yn = T yy and T yy it xy = ϕ (z) + ϕ (z) + zϕ (z) + ψ (z).

30 7. Example Use the Airy stress function to calculate displacements.

31 Use a stress function of the form U(x, y) = αxy + βxy 3, such that T 11 = 6βxy, T 22 = 0 and T 12 = α + 3βy 2. The top side and the bottom side are stress free and C C T xy dy = P.

32 We get the stress function the stress tensor T = U(x, y) = 3P 4C xy P 4C 3xy3, ( P 4C 3 xy 3P 4C + 3P 3P 4C + 3P 4C 3 y 2 4C 3 y 2 0 and the deformations can be calculated from ( ) ( ) 1 ( ) ε11 λ + µ λ T11 = ε 22 λ λ + µ T 22 ) and ε 12 = T 12 2µ.

33 After introducing some other boundary conditions, the displacement can be calculated.

34 8. Results Introduction of the Airy stress function Determination of displacements from the stress function Representation of the stress function in terms of two complex analytic functions Complex representation of displacements and stresses

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

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

Complex Variables. Chapter 2. Analytic Functions Section Harmonic Functions Proofs of Theorems. March 19, 2017

Complex Variables. Chapter 2. Analytic Functions Section Harmonic Functions Proofs of Theorems. March 19, 2017 Complex Variables Chapter 2. Analytic Functions Section 2.26. Harmonic Functions Proofs of Theorems March 19, 2017 () Complex Variables March 19, 2017 1 / 5 Table of contents 1 Theorem 2.26.1. 2 Theorem

More information

Math 4263 Homework Set 1

Math 4263 Homework Set 1 Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that

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

Mechanics of materials Lecture 4 Strain and deformation

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

More information

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

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

An analytical method for the inverse Cauchy problem of Lame equation in a rectangle

An analytical method for the inverse Cauchy problem of Lame equation in a rectangle Journal of Physics: Conference Series PAPER OPEN ACCESS An analytical method for the inverse Cauchy problem of Lame equation in a rectangle To cite this article: Yu Grigor ev 218 J. Phys.: Conf. Ser. 991

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

Basic Equations of Elasticity

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

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL Y G y G Y 87 y Y 8 Y - $ X ; ; y y q 8 y $8 $ $ $ G 8 q < 8 6 4 y 8 7 4 8 8 < < y 6 $ q - - y G y G - Y y y 8 y y y Y Y 7-7- G - y y y ) y - y y y y - - y - y 87 7-7- G G < G y G y y 6 X y G y y y 87 G

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

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

MATH 452. SAMPLE 3 SOLUTIONS May 3, (10 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic.

MATH 452. SAMPLE 3 SOLUTIONS May 3, (10 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic. MATH 45 SAMPLE 3 SOLUTIONS May 3, 06. (0 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic. Because f is holomorphic, u and v satisfy the Cauchy-Riemann equations:

More information

Two Posts to Fill On School Board

Two Posts to Fill On School Board Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83

More information

30 Bulletin of TICMI. x i

30 Bulletin of TICMI. x i 30 Bulletin of TICMI ON CONSTUCTION OF APPOXIMATE SOLUTIONS OF EQUATIONS OF THE NON-LINEA AND NON-SHALLOW CYLINDICAL SHELLS Gulua B. I. Veua Institute of Applied Mathematics of Iv. Javahishvili Tbilisi

More information

Lecture Kollbruner Section 5.2 Characteristics of Thin Walled Sections and.. Kollbruner Section 5.3 Bending without Twist

Lecture Kollbruner Section 5.2 Characteristics of Thin Walled Sections and.. Kollbruner Section 5.3 Bending without Twist Lecture 3-003 Kollbruner Section 5. Characteristics of Thin Walled Sections and.. Kollbruner Section 5.3 Bending without Twist thin walled => (cross section shape arbitrary and thickness can vary) axial

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

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.

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,

More information

Leplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math EECE

Leplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math EECE Leplace s Analyzing the Analyticity of Analytic Analysis Engineering Math EECE 3640 1 The Laplace equations are built on the Cauchy- Riemann equations. They are used in many branches of physics such as

More information

Leplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math 16.

Leplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math 16. Leplace s Analyzing the Analyticity of Analytic Analysis Engineering Math 16.364 1 The Laplace equations are built on the Cauchy- Riemann equations. They are used in many branches of physics such as heat

More information

Mathematical Tripos Part IA Lent Term Example Sheet 1. Calculate its tangent vector dr/du at each point and hence find its total length.

Mathematical Tripos Part IA Lent Term Example Sheet 1. Calculate its tangent vector dr/du at each point and hence find its total length. Mathematical Tripos Part IA Lent Term 205 ector Calculus Prof B C Allanach Example Sheet Sketch the curve in the plane given parametrically by r(u) = ( x(u), y(u) ) = ( a cos 3 u, a sin 3 u ) with 0 u

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

Journal of Solid Mechanics and Materials Engineering

Journal of Solid Mechanics and Materials Engineering and Materials Engineering Analysis of In-Plane Problems for an Isotropic Elastic Medium with Many Circular Holes or Rigid Inclusions Mutsumi MIYAGAWA, Jyo SHIMURA, Takuo SUZUKI and Takanobu TAMIYA Dept.of

More information

(z 0 ) = lim. = lim. = f. Similarly along a vertical line, we fix x = x 0 and vary y. Setting z = x 0 + iy, we get. = lim. = i f

(z 0 ) = lim. = lim. = f. Similarly along a vertical line, we fix x = x 0 and vary y. Setting z = x 0 + iy, we get. = lim. = i f . Holomorphic Harmonic Functions Basic notation. Considering C as R, with coordinates x y, z = x + iy denotes the stard complex coordinate, in the usual way. Definition.1. Let f : U C be a complex valued

More information

Fundamentals of Fluid Dynamics: Elementary Viscous Flow

Fundamentals of Fluid Dynamics: Elementary Viscous Flow Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research

More information

Elastic behaviour of an edge dislocation near a sharp crack emanating from a semi-elliptical blunt crack

Elastic behaviour of an edge dislocation near a sharp crack emanating from a semi-elliptical blunt crack Chin. Phys. B Vol. 19, No. 1 010 01610 Elastic behaviour of an edge dislocation near a sharp crack emanating from a semi-elliptical blunt crack Fang Qi-Hong 方棋洪, Song Hao-Peng 宋豪鹏, and Liu You-Wen 刘又文

More information

Game Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost

Game Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit

More information

1. DO NOT LIFT THIS COVER PAGE UNTIL INSTRUCTED TO DO SO. Write your student number and name at the top of this page. This test has SIX pages.

1. DO NOT LIFT THIS COVER PAGE UNTIL INSTRUCTED TO DO SO. Write your student number and name at the top of this page. This test has SIX pages. Student Number Name (Printed in INK Mathematics 54 July th, 007 SIMON FRASER UNIVERSITY Department of Mathematics Faculty of Science Midterm Instructor: S. Pimentel 1. DO NOT LIFT THIS COVER PAGE UNTIL

More information

f (n) (z 0 ) Theorem [Morera s Theorem] Suppose f is continuous on a domain U, and satisfies that for any closed curve γ in U, γ

f (n) (z 0 ) Theorem [Morera s Theorem] Suppose f is continuous on a domain U, and satisfies that for any closed curve γ in U, γ Remarks. 1. So far we have seen that holomorphic is equivalent to analytic. Thus, if f is complex differentiable in an open set, then it is infinitely many times complex differentiable in that set. This

More information

Candidates must show on each answer book the type of calculator used. Log Tables, Statistical Tables and Graph Paper are available on request.

Candidates must show on each answer book the type of calculator used. Log Tables, Statistical Tables and Graph Paper are available on request. UNIVERSITY OF EAST ANGLIA School of Mathematics Spring Semester Examination 2004 FLUID DYNAMICS Time allowed: 3 hours Attempt Question 1 and FOUR other questions. Candidates must show on each answer book

More information

OWELL WEEKLY JOURNAL

OWELL WEEKLY JOURNAL Y \»< - } Y Y Y & #»»» q ] q»»»>) & - - - } ) x ( - { Y» & ( x - (» & )< - Y X - & Q Q» 3 - x Q Y 6 \Y > Y Y X 3 3-9 33 x - - / - -»- --

More information

BACKGROUNDS. Two Models of Deformable Body. Distinct Element Method (DEM)

BACKGROUNDS. Two Models of Deformable Body. Distinct Element Method (DEM) BACKGROUNDS Two Models of Deformable Body continuum rigid-body spring deformation expressed in terms of field variables assembly of rigid-bodies connected by spring Distinct Element Method (DEM) simple

More information

2.4 CONTINUUM MECHANICS (SOLIDS)

2.4 CONTINUUM MECHANICS (SOLIDS) 43.4 CONTINUUM MCHANICS (SOLIDS) In this introduction to continuum mechanics we consider the basic equations describing the physical effects created by external forces acting upon solids and fluids. In

More information

County Council Named for Kent

County Council Named for Kent \ Y Y 8 9 69 6» > 69 ««] 6 : 8 «V z 9 8 x 9 8 8 8?? 9 V q» :: q;; 8 x () «; 8 x ( z x 9 7 ; x >«\ 8 8 ; 7 z x [ q z «z : > ; ; ; ( 76 x ; x z «7 8 z ; 89 9 z > q _ x 9 : ; 6? ; ( 9 [ ) 89 _ ;»» «; x V

More information

Chapter 3 Stress, Strain, Virtual Power and Conservation Principles

Chapter 3 Stress, Strain, Virtual Power and Conservation Principles Chapter 3 Stress, Strain, irtual Power and Conservation Principles 1 Introduction Stress and strain are key concepts in the analytical characterization of the mechanical state of a solid body. While stress

More information

Aircraft Structures Kirchhoff-Love Plates

Aircraft Structures Kirchhoff-Love Plates University of Liège erospace & Mechanical Engineering ircraft Structures Kirchhoff-Love Plates Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 http://www.ltas-cm3.ulg.ac.be/ Chemin

More information

Dynamics of Ice Sheets and Glaciers

Dynamics of Ice Sheets and Glaciers Dynamics of Ice Sheets and Glaciers Ralf Greve Institute of Low Temperature Science Hokkaido University Lecture Notes Sapporo 2004/2005 Literature Ice dynamics Paterson, W. S. B. 1994. The Physics of

More information

Integral equations for crack systems in a slightly heterogeneous elastic medium

Integral equations for crack systems in a slightly heterogeneous elastic medium Boundary Elements and Other Mesh Reduction Methods XXXII 65 Integral equations for crack systems in a slightly heterogeneous elastic medium A. N. Galybin & S. M. Aizikovich Wessex Institute of Technology,

More information

MA 412 Complex Analysis Final Exam

MA 412 Complex Analysis Final Exam MA 4 Complex Analysis Final Exam Summer II Session, August 9, 00.. Find all the values of ( 8i) /3. Sketch the solutions. Answer: We start by writing 8i in polar form and then we ll compute the cubic root:

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

CHAPTER 3. Analytic Functions. Dr. Pulak Sahoo

CHAPTER 3. Analytic Functions. Dr. Pulak Sahoo CHAPTER 3 Analytic Functions BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India E-mail : sahoopulak1@gmail.com 1 Module-4: Harmonic Functions 1 Introduction

More information

Symmetries and invariant solutions of mathematical models of plastic ow during linear friction welding

Symmetries and invariant solutions of mathematical models of plastic ow during linear friction welding Symmetries and invariant solutions of mathematical models of plastic ow during linear friction welding Artur Araslanov Ufa State Aviation Technical University, Ufa, Russia. Laboratory of Group Analysis

More information

On Cauchy s theorem and Green s theorem

On Cauchy s theorem and Green s theorem MA 525 On Cauchy s theorem and Green s theorem 1. Introduction No doubt the most important result in this course is Cauchy s theorem. There are many ways to formulate it, but the most simple, direct and

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

Math 185 Homework Exercises II

Math 185 Homework Exercises II Math 185 Homework Exercises II Instructor: Andrés E. Caicedo Due: July 10, 2002 1. Verify that if f H(Ω) C 2 (Ω) is never zero, then ln f is harmonic in Ω. 2. Let f = u+iv H(Ω) C 2 (Ω). Let p 2 be an integer.

More information

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

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

More information

Hooke s law and its consequences 1

Hooke s law and its consequences 1 AOE 354 Hooke s law and its consequences Historically, the notion of elasticity was first announced in 676 by Robert Hooke (635 73) in the form of an anagram, ceiinosssttuv. He explained it in 678 as Ut

More information

Math 417 Midterm Exam Solutions Friday, July 9, 2010

Math 417 Midterm Exam Solutions Friday, July 9, 2010 Math 417 Midterm Exam Solutions Friday, July 9, 010 Solve any 4 of Problems 1 6 and 1 of Problems 7 8. Write your solutions in the booklet provided. If you attempt more than 5 problems, you must clearly

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

Introduction to fracture mechanics

Introduction to fracture mechanics Introduction to fracture mechanics Prof. Dr. Eleni Chatzi Dr. Giuseppe Abbiati, Dr. Konstantinos Agathos Lecture 6-9 November, 2017 Institute of Structural Engineering, ETH Zu rich November 9, 2017 Institute

More information

Exercise: concepts from chapter 5

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

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

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

WESTERGAARD COMPLEX STRESS FUNCTIONS (16)

WESTERGAARD COMPLEX STRESS FUNCTIONS (16) GG711c 1/1/03 1 WESTERGAARD COMPLEX STRESS FUNCTIONS (16) I Main topics A B Historical perspective Expression of the biharmonic function by harmonic functions C Boundary conditions D Use of symmetry and

More information

Basic Theorems in Dynamic Elasticity

Basic Theorems in Dynamic Elasticity Basic Theorems in Dynamic Elasticity 1. Stress-Strain relationships 2. Equation of motion 3. Uniqueness and reciprocity theorems 4. Elastodynamic Green s function 5. Representation theorems Víctor M. CRUZ-ATIENZA

More information

17th European Conference on Fracture 2-5 September,2008, Brno, Czech Republic. Thermal Fracture of a FGM/Homogeneous Bimaterial with Defects

17th European Conference on Fracture 2-5 September,2008, Brno, Czech Republic. Thermal Fracture of a FGM/Homogeneous Bimaterial with Defects -5 September,8, Brno, Czech Republic Thermal Fracture of a FGM/Homogeneous Bimaterial with Defects Vera Petrova, a, Siegfried Schmauder,b Voronezh State University, University Sq., Voronezh 3946, Russia

More information

KINEMATICS OF CONTINUA

KINEMATICS OF CONTINUA KINEMATICS OF CONTINUA Introduction Deformation of a continuum Configurations of a continuum Deformation mapping Descriptions of motion Material time derivative Velocity and acceleration Transformation

More information

Solution of some problems of linear plane theory of elasticity with mixed boundary conditions by the method of boundary integrals

Solution of some problems of linear plane theory of elasticity with mixed boundary conditions by the method of boundary integrals Solution of some problems of linear plane theory of elasticity with mixed boundary conditions by the method of boundary integrals By Aisha Sleman Melod Gjam A Thesis Submitted to Faculty of Science In

More information

Draft:S Ghorai. 1 Introduction. 2 Invariance. Constitutive equations. The conservation laws introduced before are. Equations of mass conservation

Draft:S Ghorai. 1 Introduction. 2 Invariance. Constitutive equations. The conservation laws introduced before are. Equations of mass conservation 1 Constitutive equations 1 Introduction The conservation laws introduced before are Equations of mass conservation Conservation of linear momentum Conservation of angular momentum Dρ Dt + ρdiv v = 0 ρ

More information

Department of mathematics MA201 Mathematics III

Department of mathematics MA201 Mathematics III Department of mathematics MA201 Mathematics III Academic Year 2015-2016 Model Solutions: Quiz-II (Set - B) 1. Obtain the bilinear transformation which maps the points z 0, 1, onto the points w i, 1, i

More information

Chapter 2: Basic Governing Equations

Chapter 2: Basic Governing Equations -1 Reynolds Transport Theorem (RTT) - Continuity Equation -3 The Linear Momentum Equation -4 The First Law of Thermodynamics -5 General Equation in Conservative Form -6 General Equation in Non-Conservative

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

Inverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros

Inverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros Inverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros Computational Design Forward design: direct manipulation of design parameters Level of abstraction Exploration

More information

The Finite Element Method

The Finite Element Method The Finite Element Method 3D Problems Heat Transfer and Elasticity Read: Chapter 14 CONTENTS Finite element models of 3-D Heat Transfer Finite element model of 3-D Elasticity Typical 3-D Finite Elements

More information

PROPERTIES OF SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS FOR LAVRENT EV-BITSADZE EQUATIONS AT CORNER POINTS

PROPERTIES OF SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS FOR LAVRENT EV-BITSADZE EQUATIONS AT CORNER POINTS Electronic Journal of Differential Equations, Vol. 24 24, No. 93, pp. 9. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PROPERTIES OF SOLUTIONS TO

More information

u xx + u yy = 0. (5.1)

u xx + u yy = 0. (5.1) Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function

More information

Other state variables include the temperature, θ, and the entropy, S, which are defined below.

Other state variables include the temperature, θ, and the entropy, S, which are defined below. Chapter 3 Thermodynamics In order to complete the formulation we need to express the stress tensor T and the heat-flux vector q in terms of other variables. These expressions are known as constitutive

More information

Singular integro-differential equations for a new model of fracture w. curvature-dependent surface tension

Singular integro-differential equations for a new model of fracture w. curvature-dependent surface tension Singular integro-differential equations for a new model of fracture with a curvature-dependent surface tension Department of Mathematics Kansas State University January 15, 2015 Work supported by Simons

More information

Chapter 5 Linear Elasticity

Chapter 5 Linear Elasticity Chapter 5 Linear Elasticity 1 Introduction The simplest mechanical test consists of placing a standardized specimen with its ends in the grips of a tensile testing machine and then applying load under

More information

The Generalized Interpolation Material Point Method

The Generalized Interpolation Material Point Method Compaction of a foam microstructure The Generalized Interpolation Material Point Method Tungsten Particle Impacting sandstone The Material Point Method (MPM) 1. Lagrangian material points carry all state

More information

Math 185 Fall 2015, Sample Final Exam Solutions

Math 185 Fall 2015, Sample Final Exam Solutions Math 185 Fall 2015, Sample Final Exam Solutions Nikhil Srivastava December 12, 2015 1. True or false: (a) If f is analytic in the annulus A = {z : 1 < z < 2} then there exist functions g and h such that

More information

MA3D1 Fluid Dynamics Support Class 5 - Shear Flows and Blunt Bodies

MA3D1 Fluid Dynamics Support Class 5 - Shear Flows and Blunt Bodies MA3D1 Fluid Dynamics Support Class 5 - Shear Flows and Blunt Bodies 13th February 2015 Jorge Lindley email: J.V.M.Lindley@warwick.ac.uk 1 2D Flows - Shear flows Example 1. Flow over an inclined plane A

More information

5 n N := {1, 2,...} N 0 := {0} N R ++ := (0, ) R + := [0, ) a, b R a b := max{a, b} f g (f g)(x) := f(x) g(x) (Z, Z ) bz Z Z R f := sup z Z f(z) κ: Z R ++ κ f : Z R f(z) f κ := sup z Z κ(z). f κ < f κ

More information

Continuum Mechanics and the Finite Element Method

Continuum Mechanics and the Finite Element Method Continuum Mechanics and the Finite Element Method 1 Assignment 2 Due on March 2 nd @ midnight 2 Suppose you want to simulate this The familiar mass-spring system l 0 l y i X y i x Spring length before/after

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

NONLINEAR CONTINUUM FORMULATIONS CONTENTS

NONLINEAR CONTINUUM FORMULATIONS CONTENTS NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell

More information

Some analysis problems 1. x x 2 +yn2, y > 0. g(y) := lim

Some analysis problems 1. x x 2 +yn2, y > 0. g(y) := lim Some analysis problems. Let f be a continuous function on R and let for n =,2,..., F n (x) = x (x t) n f(t)dt. Prove that F n is n times differentiable, and prove a simple formula for its n-th derivative.

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

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t) IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common

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

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

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

examples of equations: what and why intrinsic view, physical origin, probability, geometry

examples of equations: what and why intrinsic view, physical origin, probability, geometry Lecture 1 Introduction examples of equations: what and why intrinsic view, physical origin, probability, geometry Intrinsic/abstract F ( x, Du, D u, D 3 u, = 0 Recall algebraic equations such as linear

More information

INDEX 363. Cartesian coordinates 19,20,42, 67, 83 Cartesian tensors 84, 87, 226

INDEX 363. Cartesian coordinates 19,20,42, 67, 83 Cartesian tensors 84, 87, 226 INDEX 363 A Absolute differentiation 120 Absolute scalar field 43 Absolute tensor 45,46,47,48 Acceleration 121, 190, 192 Action integral 198 Addition of systems 6, 51 Addition of tensors 6, 51 Adherence

More information

Mathematical Background

Mathematical Background CHAPTER ONE Mathematical Background This book assumes a background in the fundamentals of solid mechanics and the mechanical behavior of materials, including elasticity, plasticity, and friction. A previous

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

Existence and uniqueness of the weak solution for a contact problem

Existence and uniqueness of the weak solution for a contact problem Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (216), 186 199 Research Article Existence and uniqueness of the weak solution for a contact problem Amar Megrous a, Ammar Derbazi b, Mohamed

More information

Chapter 9: Differential Analysis

Chapter 9: Differential Analysis 9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control

More information

EFFECT OF ELLIPTIC OR CIRCULAR HOLES ON THE STRESS DISTRIBUTION IN PLATES

EFFECT OF ELLIPTIC OR CIRCULAR HOLES ON THE STRESS DISTRIBUTION IN PLATES EFFECT OF ELLIPTIC OR CIRCULAR HOLES ON THE STRESS DISTRIBUTION IN PLATES OF WOOD OR PLYWOOD CONSIDERED AS ORTHOTROPIC MATERIALS Information Revied and Reaffirmed March 1956 No. 1510 EFFECT OF ELLIPTIC

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 MASSACHUSETTS NSTTUTE OF TECHNOLOGY DEPARTMENT OF MATERALS SCENCE AND ENGNEERNG CAMBRDGE, MASSACHUSETTS 39 3. MECHANCAL PROPERTES OF MATERALS PROBLEM SET SOLUTONS Reading Ashby, M.F., 98, Tensors: Notes

More information

Elasticity in two dimensions 1

Elasticity in two dimensions 1 Elasticity in two dimensions 1 Elasticity in two dimensions Chapters 3 and 4 of Mechanics of the Cell, as well as its Appendix D, contain selected results for the elastic behavior of materials in two and

More information

Chapter 9. Analytic Continuation. 9.1 Analytic Continuation. For every complex problem, there is a solution that is simple, neat, and wrong.

Chapter 9. Analytic Continuation. 9.1 Analytic Continuation. For every complex problem, there is a solution that is simple, neat, and wrong. Chapter 9 Analytic Continuation For every complex problem, there is a solution that is simple, neat, and wrong. - H. L. Mencken 9.1 Analytic Continuation Suppose there is a function, f 1 (z) that is analytic

More information

c. Better work with components of slowness vector s (or wave + k z 2 = k 2 = (ωs) 2 = ω 2 /c 2. k=(kx,k z )

c. Better work with components of slowness vector s (or wave + k z 2 = k 2 = (ωs) 2 = ω 2 /c 2. k=(kx,k z ) .50 Introduction to seismology /3/05 sophie michelet Today s class: ) Eikonal equation (basis of ray theory) ) Boundary conditions (Stein s book.3.0) 3) Snell s law Some remarks on what we discussed last

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

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

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

Lecture Note 1. Introduction to Elasticity Equations

Lecture Note 1. Introduction to Elasticity Equations ME34B Elasticity of Microscopic tructures tanford University Winter 24 Lecture Note. Introduction to Elasticity Equations Chris Weinberger and Wei Cai c All rights reserved January 2, 24 Contents Index

More information