Crystal Micro-Mechanics

Size: px
Start display at page:

Download "Crystal Micro-Mechanics"

Transcription

1 Crystal Micro-Mechanics Lectre Classical definition of stress and strain Heng Nam Han Associate Professor School of Materials Science & Engineering College of Engineering Seol National University Seol 5-744, Korea Tel : Fa : hnhan@sn.ac.kr Homepage :

2 Concept of stress The forces acting within a stressed body are either body forces or srface forces. We assme an eqilibrim state nder body forces or srface forces. F lim A df da

3 Concept of stress Decomposition of stress in the direction of coordinate ais A

4 Concept of stress 4 This constrction atomatically implies the need for 8 stress components to define the state of stress in the volme element.

5 Concept of stress 5

6 Concept of stress 6 M i When we know 6 stress components at a position, we can define the stress tensor at the position in the material.

7 Concept of stress 7 Transformation of Coordinates

8 Transformation of Vector 8 The vector S can be easily resolved into components that are parallel to any set of reference aes.

9 Transformation of Vector 9 The two coordinate systems are related throgh a series of angles. We define them in terms of direction cosines, or or

10 Concept of stress Principal stresses X X X X Stress components acting on the faces of a tetrahedron OABC

11 Principal stresses τ τ y z τ yy τ y yz τ τ zz z yz Seclar or Characteristic Eq. Invariants of stress : I, I, I

12 Principal stresses I II III Stress components acting on the faces of a tetrahedron OABC

13 Eample Find the principal components and principal aes of the stress ij

14 Concept of stress 4 Maimm shear stress

15 Concept of stress 5 HW (de 9/) : Derive the Table and eplain that the maimm shear stress directions bisect the principal stress direction by 45 o.

16 6 Deviatoric stress Mean stress : P I ii III II I zz yy m Hydrostatic stress or pressre Deviatoric stress : m m m zz yz z yz yy y z y zz yz z yz yy y z y ij ij m ij ij δ

17 7 Concept of stress Familiar types of stress state Uniaial tension Uniaial compression Pre shear Hydrostatic press P P P Hydrostatic stress has the same components irrespective of the choice of coordinate aes. by 45 o rotation for o ais

18 8 Deviatoric stress Deviatoric stress is a symmetric second rank tensor.

19 9 Deviatoric stress J J / ij jk ki / ij ij

20 Eqilibrim Eqations for Stress Stress distribtion in an infinitesimal element F j, ij

21 Concept of strain : classical definition (inifinitesimal strain) A C B before deform. after deform. : -dimensional strain

22 Concept of strain M M α M L L J J K α K K -dimensional strain

23 Concept of strain M J M α J M L K α K K L -dimensional strain e e -directional normal γ -directional normal Shear α α

24 4 Concept of strain -dimensional strain e e e e e e e e e e ij

25 Concept of strain 5 (a) (b) Rigid body rotation by ω

26 6 Concept of strain Strain tensor Spin tensor ij ij ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ

27 Concept of strain 7 e a ij ki a lj e kl a a ij ki lj kl ω a a ω ij ki lj kl I II III

28 Volmetric Strain 8 Initial length of side :,, Finial length of side : ( ), ( ) ( ) V V V V V V ( )( )( ) V m

29 9 Deviatoric strain Mean strain : V ii III II I zz yy m Deviatoric strain : m m m zz yz z yz yy y z y zz yz z yz yy y z y ij ij m ij ij δ

30 Special Types of Strain Plane strain specimen Pre shear

31 Concept of strain Simple shear and pre shear e e e e Simple shear Pre shear e (/) ( e e ) (/) ( e - e )

32 Strain compatibility Plane strain condition General condition Mathematical Theory of Elasticity I.S. Solkolnikoff, P. 5

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

Chapter 1: Differential Form of Basic Equations

Chapter 1: Differential Form of Basic Equations MEG 74 Energ and Variational Methods in Mechanics I Brendan J. O Toole, Ph.D. Associate Professor of Mechanical Engineering Howard R. Hghes College of Engineering Universit of Nevada Las Vegas TBE B- (7)

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

1 Differential Equations for Solid Mechanics

1 Differential Equations for Solid Mechanics 1 Differential Eqations for Solid Mechanics Simple problems involving homogeneos stress states have been considered so far, wherein the stress is the same throghot the component nder std. An eception to

More information

(MPa) compute (a) The traction vector acting on an internal material plane with normal n ( e1 e

(MPa) compute (a) The traction vector acting on an internal material plane with normal n ( e1 e EN10: Continuum Mechanics Homework : Kinetics Due 1:00 noon Friday February 4th School of Engineering Brown University 1. For the Cauchy stress tensor with components 100 5 50 0 00 (MPa) compute (a) The

More information

Module #4. Fundamentals of strain The strain deviator Mohr s circle for strain READING LIST. DIETER: Ch. 2, Pages 38-46

Module #4. Fundamentals of strain The strain deviator Mohr s circle for strain READING LIST. DIETER: Ch. 2, Pages 38-46 HOMEWORK From Dieter 2-7 Module #4 Fundamentals of strain The strain deviator Mohr s circle for strain READING LIST DIETER: Ch. 2, Pages 38-46 Pages 11-12 in Hosford Ch. 6 in Ne Strain When a solid is

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

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

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

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

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

Properties of the stress tensor

Properties of the stress tensor Appendix C Properties of the stress tensor Some of the basic properties of the stress tensor and traction vector are reviewed in the following. C.1 The traction vector Let us assume that the state of stress

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

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

Lecture #7: Basic Notions of Fracture Mechanics Ductile Fracture

Lecture #7: Basic Notions of Fracture Mechanics Ductile Fracture Lecture #7: Basic Notions of Fracture Mechanics Ductile Fracture by Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling of Materials in Manufacturing

More information

Bone Tissue Mechanics

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

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

σ = F/A. (1.2) σ xy σ yy σ zy , (1.3) σ xz σ yz σ zz The use of the opposite convention should cause no problem because σ ij = σ ji.

σ = F/A. (1.2) σ xy σ yy σ zy , (1.3) σ xz σ yz σ zz The use of the opposite convention should cause no problem because σ ij = σ ji. Cambridge Universit Press 978-0-521-88121-0 - Metal Forming: Mechanics Metallurg, Third Edition Ecerpt 1 Stress Strain An understing of stress strain is essential for analzing metal forming operations.

More information

σ = F/A. (1.2) σ xy σ yy σ zx σ xz σ yz σ, (1.3) The use of the opposite convention should cause no problem because σ ij = σ ji.

σ = F/A. (1.2) σ xy σ yy σ zx σ xz σ yz σ, (1.3) The use of the opposite convention should cause no problem because σ ij = σ ji. Cambridge Universit Press 978-1-107-00452-8 - Metal Forming: Mechanics Metallurg, Fourth Edition Ecerpt 1 Stress Strain An understing of stress strain is essential for the analsis of metal forming operations.

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

Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor

Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor Somnath Bhowmick Materials Science and Engineering, IIT Kanpur April 6, 2018 Tensile test and Hooke s Law Upto certain strain (0.75),

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

Period #5: Strain. A. Context. Structural mechanics deals with the forces placed upon mechanical systems and the resulting deformations of the system.

Period #5: Strain. A. Context. Structural mechanics deals with the forces placed upon mechanical systems and the resulting deformations of the system. Period #5: Strain A. Contet Strctral mechanics deals with the forces placed pon mechanical sstems and the reslting deformations of the sstem. Solid mechanics on the smaller scale relates stresses in the

More information

M E 320 Professor John M. Cimbala Lecture 10

M E 320 Professor John M. Cimbala Lecture 10 M E 320 Professor John M. Cimbala Lecture 10 Today, we will: Finish our example problem rates of motion and deformation of fluid particles Discuss the Reynolds Transport Theorem (RTT) Show how the RTT

More information

Concept Question Comment on the general features of the stress-strain response under this loading condition for both types of materials

Concept Question Comment on the general features of the stress-strain response under this loading condition for both types of materials Module 5 Material failure Learning Objectives review the basic characteristics of the uni-axial stress-strain curves of ductile and brittle materials understand the need to develop failure criteria for

More information

Tensor Transformations and the Maximum Shear Stress. (Draft 1, 1/28/07)

Tensor Transformations and the Maximum Shear Stress. (Draft 1, 1/28/07) Tensor Transformations and the Maximum Shear Stress (Draft 1, 1/28/07) Introduction The order of a tensor is the number of subscripts it has. For each subscript it is multiplied by a direction cosine array

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

Lecture 5. Differential Analysis of Fluid Flow Navier-Stockes equation

Lecture 5. Differential Analysis of Fluid Flow Navier-Stockes equation Lectre 5 Differential Analsis of Flid Flo Naier-Stockes eqation Differential analsis of Flid Flo The aim: to rodce differential eqation describing the motion of flid in detail Flid Element Kinematics An

More information

3 2D Elastostatic Problems in Cartesian Coordinates

3 2D Elastostatic Problems in Cartesian Coordinates D lastostatic Problems in Cartesian Coordinates Two dimensional elastostatic problems are discssed in this Chapter, that is, static problems of either plane stress or plane strain. Cartesian coordinates

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

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

Theories of Straight Beams

Theories of Straight Beams EVPM3ed02 2016/6/10 7:20 page 71 #25 This is a part of the revised chapter in the new edition of the tetbook Energy Principles and Variational Methods in pplied Mechanics, which will appear in 2017. These

More information

1. Background. is usually significantly lower than it is in uniaxial tension

1. Background. is usually significantly lower than it is in uniaxial tension NOTES ON QUANTIFYING MODES OF A SECOND- ORDER TENSOR. The mechanical behavior of rocks and rock-like materials (concrete, ceramics, etc.) strongly depends on the loading mode, defined by the values and

More information

Physics of Continuous media

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

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

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

MEG 741 Energy and Variational Methods in Mechanics I

MEG 741 Energy and Variational Methods in Mechanics I MEG 74 Energ and Variational Methods in Mechanics I Brendan J. O Toole, Ph.D. Associate Professor of Mechanical Engineering Howard R. Hghes College of Engineering Universit of Nevada Las Vegas TBE B- (7)

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

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

**********************************************************************

********************************************************************** 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

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

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

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

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

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

Lecture Triaxial Stress and Yield Criteria. When does yielding occurs in multi-axial stress states?

Lecture Triaxial Stress and Yield Criteria. When does yielding occurs in multi-axial stress states? Lecture 5.11 Triaial Stress and Yield Criteria When does ielding occurs in multi-aial stress states? Representing stress as a tensor operational stress sstem Compressive stress sstems Triaial stresses:

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

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

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

Classification of Prostate Cancer Grades and T-Stages based on Tissue Elasticity Using Medical Image Analysis. Supplementary Document

Classification of Prostate Cancer Grades and T-Stages based on Tissue Elasticity Using Medical Image Analysis. Supplementary Document Classification of Prostate Cancer Grades and T-Stages based on Tissue Elasticity Using Medical Image Analysis Supplementary Document Shan Yang, Vladimir Jojic, Jun Lian, Ronald Chen, Hongtu Zhu, Ming C.

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

Lecture Notes 5

Lecture Notes 5 1.5 Lecture Notes 5 Quantities in Different Coordinate Systems How to express quantities in different coordinate systems? x 3 x 3 P Direction Cosines Axis φ 11 φ 3 φ 1 x x x x 3 11 1 13 x 1 3 x 3 31 3

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

MEC-E8001 Finite Element Analysis, Exam (example) 2017

MEC-E8001 Finite Element Analysis, Exam (example) 2017 MEC-E800 Finite Element Analysis Eam (eample) 07. Find the transverse displacement w() of the strctre consisting of one beam element and po forces and. he rotations of the endpos are assmed to be eqal

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

Topics. GG612 Structural Geology Sec3on Steve Martel POST 805 Lecture 4 Mechanics: Stress and Elas3city Theory

Topics. GG612 Structural Geology Sec3on Steve Martel POST 805 Lecture 4 Mechanics: Stress and Elas3city Theory GG612 Structural Geology Sec3on Steve Martel POST 805 smartel@hawaii.edu Lecture 4 Mechanics: Stress and Elas3city Theory 11/6/15 GG611 1 Topics 1. Stress vectors (trac3ons) 2. Stress at a point 3. Cauchy

More information

Mechanics of Solids (APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY) And As per Revised Syllabus of Leading Universities in India AIR WALK PUBLICATIONS

Mechanics of Solids (APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY) And As per Revised Syllabus of Leading Universities in India AIR WALK PUBLICATIONS Advanced Mechanics of Solids (APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY) And As per Revised Syllabus of Leading Universities in India Dr. S. Ramachandran Prof. R. Devaraj Professors School of Mechanical

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

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

Chapter 3: Stress and Equilibrium of Deformable Bodies

Chapter 3: Stress and Equilibrium of Deformable Bodies Ch3-Stress-Equilibrium Page 1 Chapter 3: Stress and Equilibrium of Deformable Bodies When structures / deformable bodies are acted upon by loads, they build up internal forces (stresses) within them to

More information

* Many components have multiaxial loads, and some of those have multiaxial loading in critical locations

* Many components have multiaxial loads, and some of those have multiaxial loading in critical locations Why do Multiaxial Fatigue Calculations? * Fatigue analysis is an increasingly important part of the design and development process * Many components have multiaxial loads, and some of those have multiaxial

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

Failure Mechanism and Evolution Law of Fractured-Rocks based on Moment Tensor Inversion

Failure Mechanism and Evolution Law of Fractured-Rocks based on Moment Tensor Inversion Failure Mechanism and Evolution Law of Fractured-Rocks based on Moment Tensor Inversion Jinfei Chai 1), Yongtao Gao 1), Shunchuan Wu 1), Langqiu Sun 2), Yu Zhou 1) 1) School of Civil and Environmental

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

Lecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity

Lecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity Lecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity by Borja Erice and Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling

More information

Stress Integration for the Drucker-Prager Material Model Without Hardening Using the Incremental Plasticity Theory

Stress Integration for the Drucker-Prager Material Model Without Hardening Using the Incremental Plasticity Theory Journal of the Serbian Society for Computational Mechanics / Vol. / No., 008 / pp. 80-89 UDC: 59.74:004.0 Stress Integration for the Drucker-rager Material Model Without Hardening Using the Incremental

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

TENSOR TRANSFORMATION OF STRESSES

TENSOR TRANSFORMATION OF STRESSES GG303 Lecture 18 9/4/01 1 TENSOR TRANSFORMATION OF STRESSES Transformation of stresses between planes of arbitrar orientation In the 2-D eample of lecture 16, the normal and shear stresses (tractions)

More information

ME 7502 Lecture 2 Effective Properties of Particulate and Unidirectional Composites

ME 7502 Lecture 2 Effective Properties of Particulate and Unidirectional Composites ME 75 Lecture Effective Properties of Particulate and Unidirectional Composites Concepts from Elasticit Theor Statistical Homogeneit, Representative Volume Element, Composite Material Effective Stress-

More information

3.032 Problem Set 2 Solutions Fall 2007 Due: Start of Lecture,

3.032 Problem Set 2 Solutions Fall 2007 Due: Start of Lecture, 3.032 Problem Set 2 Solutions Fall 2007 Due: Start of Lecture, 09.21.07 1. In the beam considered in PS1, steel beams carried the distributed weight of the rooms above. To reduce stress on the beam, it

More information

Stress transformation and Mohr s circle for stresses

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.

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

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

Quasi-Harmonic Theory of Thermal Expansion

Quasi-Harmonic Theory of Thermal Expansion Chapter 5 Quasi-Harmonic Theory of Thermal Expansion 5.1 Introduction The quasi-harmonic approximation is a computationally efficient method for evaluating thermal properties of materials. Planes and Manosa

More information

GG611 Structural Geology Sec1on Steve Martel POST 805

GG611 Structural Geology Sec1on Steve Martel POST 805 GG611 Structural Geology Sec1on Steve Martel POST 805 smartel@hawaii.edu Lecture 5 Mechanics Stress, Strain, and Rheology 11/6/16 GG611 1 Stresses Control How Rock Fractures hkp://hvo.wr.usgs.gov/kilauea/update/images.html

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

GG612 Lecture 3. Outline

GG612 Lecture 3. Outline GG61 Lecture 3 Strain and Stress Should complete infinitesimal strain by adding rota>on. Outline Matrix Opera+ons Strain 1 General concepts Homogeneous strain 3 Matrix representa>ons 4 Squares of line

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

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

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

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

HÅLLFASTHETSLÄRA, LTH Examination in computational materials modeling

HÅLLFASTHETSLÄRA, LTH Examination in computational materials modeling HÅLLFASTHETSLÄRA, LTH Examination in computational materials modeling TID: 2016-28-10, kl 14.00-19.00 Maximalt 60 poäng kan erhållas på denna tenta. För godkänt krävs 30 poäng. Tillåtet hjälpmedel: räknare

More information

Continuum Mechanics Fundamentals

Continuum Mechanics Fundamentals Continuum Mechanics Fundamentals James R. Rice, notes for ES 220, 12 November 2009; corrections 9 December 2009 Conserved Quantities Let a conseved quantity have amount F per unit volume. Examples are

More information

s ij = 2ηD ij σ 11 +σ 22 +σ 33 σ 22 +σ 33 +σ 11

s ij = 2ηD ij σ 11 +σ 22 +σ 33 σ 22 +σ 33 +σ 11 Plasticity http://imechanicaorg/node/17162 Z Suo NONLINEAR VISCOSITY Linear isotropic incompressible viscous fluid A fluid deforms in a homogeneous state with stress σ ij and rate of deformation The mean

More information

Law of behavior very-rubber band: almost incompressible material

Law of behavior very-rubber band: almost incompressible material Titre : Loi de comportement hyperélastique : matériau pres[...] Date : 25/09/2013 Page : 1/9 Law of behavior very-rubber band: almost incompressible material Summary: One describes here the formulation

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

Chapter 9: Differential Analysis of Fluid Flow

Chapter 9: Differential Analysis of Fluid Flow of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known

More information

Module #3. Transformation of stresses in 3-D READING LIST. DIETER: Ch. 2, pp Ch. 3 in Roesler Ch. 2 in McClintock and Argon Ch.

Module #3. Transformation of stresses in 3-D READING LIST. DIETER: Ch. 2, pp Ch. 3 in Roesler Ch. 2 in McClintock and Argon Ch. HOMEWORK From Dieter -3, -4, 3-7 Module #3 Transformation of stresses in 3-D READING LIST DIETER: Ch., pp. 7-36 Ch. 3 in Roesler Ch. in McClintock and Argon Ch. 7 in Edelglass The Stress Tensor z z x O

More information

Plasticity R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur

Plasticity R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur Plasticity R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur-613 401 Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 9 Table of Contents 1. Plasticity:... 3 1.1 Plastic Deformation,

More information

LECTURE 4 Stresses in Elastic Solid. 1 Concept of Stress

LECTURE 4 Stresses in Elastic Solid. 1 Concept of Stress V. DEMENKO MECHANICS OF MATERIALS 2015 1 LECTURE 4 Stresses in Elastic Solid 1 Concept of Stress In order to characterize the law of internal forces distribution over the section a measure of their intensity

More information

L8. Basic concepts of stress and equilibrium

L8. Basic concepts of stress and equilibrium L8. Basic concepts of stress and equilibrium Duggafrågor 1) Show that the stress (considered as a second order tensor) can be represented in terms of the eigenbases m i n i n i. Make the geometrical representation

More information

COMP003 Test of behaviors specific to the concretes. Simulation in a Summarized

COMP003 Test of behaviors specific to the concretes. Simulation in a Summarized Titre : COMP003 - Test de comportements spécifiques aux bé[...] Date : 30/01/2013 Page : 1/10 COMP003 Test of behaviors specific to the concretes. Simulation in a Summarized material point: This test implements

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

CH.4. STRESS. Continuum Mechanics Course (MMC)

CH.4. STRESS. Continuum Mechanics Course (MMC) CH.4. STRESS Continuum Mechanics Course (MMC) Overview Forces Acting on a Continuum Body Cauchy s Postulates Stress Tensor Stress Tensor Components Scientific Notation Engineering Notation Sign Criterion

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

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS Fifth SI Edition CHTER 1 MECHNICS OF MTERILS Ferdinand. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Introduction Concept of Stress Lecture Notes: J. Walt Oler Teas Tech University Contents

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

Formal Methods for Deriving Element Equations

Formal Methods for Deriving Element Equations Formal Methods for Deriving Element Eqations And the importance of Shape Fnctions Formal Methods In previos lectres we obtained a bar element s stiffness eqations sing the Direct Method to obtain eact

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

1. Shear force and bending moment diagrams

1. Shear force and bending moment diagrams . Shear force and bending moment diagrams Internal Forces in solids Sign conventions Shear forces are given a special symbol on V y and V z The couple moment along the ais of the member is given M T Torque

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