Introduction to Condensed Matter Physics

Size: px
Start display at page:

Download "Introduction to Condensed Matter Physics"

Transcription

1 Introduction to Condensed Matter Physics Elasticity M.P. Vaughan

2 Overview

3 Overview of elasticity Classical description of elasticity Speed of sound Strain Stress Young s modulus Shear modulus Poisson ratio The harmonic potential Continuum approximation Symmetry properties

4 Overview of elasticity Symmetry properties The strain tensor Elastic constants Crystal systems Cubic system Waves in elastic media Strain in elastic media

5 Classical description of elasticity

6 Elastic properties - the speed of sound We have already seen that, for small q, the group velocity of acoustic phonons is fairly constant. This corresponds to the speed of sound in the material. In terms of sound waves, longitudinal modes longitudinal waves transverse modes shear waves The speed of sound is related to the elastic properties of the medium.

7 Normal strain Normal strain is defined as the extension of a material divided by its original length (i.e. the fractional change in length). Thus, if a rod of length L is stretched to L + L, then the normal strain is L L In the ith Cartesian direction, this may be written i u x i i..

8 Shear strain

9 Shear strain With reference to the previous image, the shear strain (in 2D) is given as the change in angle at corner A. That is, xy u y x The strain tensor may then be written as u y x. xx yx zx xy yy zy xz yz zz.

10 Normal stress The normal stress s is the force F per unit area applied normally to the crosssectional area A. The normal stress s is defined by s F A.

11 Shear stress The shear stress t is the force F per unit area applied coplanar to the cross-sectional area A. The shear stress t is defined in the same way as normal stress via t F A.

12 Cauchy stress tensor Note s is used to denote normal stress as well as shear stress in the above.

13 Cauchy stress tensor The components of the Cauchy stress tensor are s s s s xx yx zx s s s xy yy zy s s s xz yz zz. The stress vectors T shown in the previous image are obtained from s via T = sn, where n is a normal vector.

14 Young s modulus Young s modulus is defined as the ratio of the tensile stress applied to a material and the tensile strain that it induces. Thus, if a tensile force F is applied along a rod of length L and cross-sectional area A, then Young s modulus Y is given by Y s FL AL, where L/L is the fractional change in length.

15 Shear modulus The shear modulus G is defined as the ratio of shear stress to shear strain. Hence s xy G xy.

16 Poisson ratio The Poisson ration is defined as the negative ratio of transverse to the axial strain.

17 Poisson ratio In the image transversestrain L' L, axial strain L L, so L' n L L L L'. L

18 Bulk modulus The bulk modulus is be described as a materials resistance to compression. In terms of the pressure P and volume V, the bulk modulus B is defined as B V dp dv.

19 The speed of sound Longitudinal waves v L B 4 3 C 1/ 2 Y1 n 1 n 1 2n 1/ 2, where is the density of the material and C is a material constant. Shear waves v S C 1/ 2.

20 The Harmonic Potential

21 The harmonic potential Previously, we found that the harmonic potential could be written in the form U 1 2 n' n u T n D nn' u n', where the potential energy terms are incorporated into the entity D nn.

22 The continuum approximation It can be shown that U may also be rendered in the form U 1 4 n' n T u u D u u. n' n nn' n' n We now consider the continuum limit, in which the material is taken to approximate a classical elastic medium. In line with this assumption, we take u R' ur R' Rur. rr

23 The continuum approximation This yields U 1 2 u n, nst xs x R t R u R E, where E stn is a fourth rank tensor given by stn E stn 1 2 R R s D n R R. t

24 Symmetry properties of E stn Note that due to the commutativity of scalar multiplication R s D Since we also have n D RR R D R R. n t E stn is unaffected by interchange of s and t, or and n. t n R D R, n s

25 Symmetry properties of E stn Due to these symmetries, for a given pair st or n there are six possible values of E stn for the other pair xx, yy, zz, yz, zx, xy. This gives 6 6 = 36 independent numbers required to specify E stn.

26 Rotational properties of E stn The potential energy is unaffected by a rigid rotation. Generally, such a rotation by an infinitesimal angle dw would yield u R dwn R, where n is the unit vector normal to the rotation.

27 The strain tensor It can be shown that this implies that U can only depend on the derivatives in the symmetrical combination s 1 2 u x s u x s. This quantity is known as the strain tensor.

28 The elastic constants Transforming to an integration, the potential energy may then be written U 1 3 s cstn tn d 2 R, nst where the elastic stiffness constants c stn are given by c s tn R s D t R n R D st R n r, 1 Rs DnRt R Dsn Rt, 8 and is the volume of the primitive cell. R

29 The elastic constants Incorporating the symmetry relations of c stn, we find that the number of independent elastic constants is, in general, reduced to 21. This number may be further reduced by the symmetries of the particular crystal system.

30 Crystal systems Lattice system Independent elastic constants Triclinic 21 Monoclinic 13 Orthorhombic 9 Tetragonal (1) 7 Tetragonal (2) 6 Rhombohedral (1) 7 Rhombohedral (2) 6 Hexagonal 5 Cubic 3 N.B. The extra categories (1) and (2) are associated with different point groups.

31 Classical elasticity theory In classical elasticity theory, we use the notation e e n n n 2, n, n, n. We also reduce the subscript on e to e, for = 1,...,6 where xx 1, yy 1, zz 1, yz 4, zx 5, xy 6.

32 Classical elasticity theory The integral for the potential energy then becomes U 1 3 e Ce d 2 r, where C c snt and the double subscripts s and tn are replaced by single subscripts and respectively in the same way as previously given.

33 Classical elasticity theory This classical expression has wave solutions of the form where p is the polarisation vector, satisfying the eigenvalue equation 2 w p ikrwt, t pe, u r t sn c snt k s k n p This is of the same form as the solutions to the discrete harmonic mode model in the long wavelength limit. t.

34 Cubic system For the cubic system, we have only three independent components C 11 c xxxx c yyyy c zzzz, C 12 c xxyy c yyzz c zzxx, C 44 c xyxy c yzyz c zxzx.

35 Cubic semiconductors Material C 11 (GPa) C 12 (GPa) C 44 (GPa) Ref C [1] Si [2] Ge [3] 3C-SiC [4] BN* [5] GaN* [6] AlAs [7] GaAs [8] InAs [9] *zinc blende

36 Waves in elastic media

37 Waves in cubic, elastic media For a cubic system, the eigenvalue equation 2 w p t sn c snt k s k n p t. reduces to 2 w p x C 11 k C C k k p C C k k p, 12 2 x C k x 2 y C y y 44 k 2 z p 12 x 44 x z z etc.

38 Waves in cubic, elastic media For a longitudinal wave along the k x, k y or k z direction (i.e. the [100] direction), we would then have and hence v L C11 w 1/ 2 k x C , 1/ 2

39 Waves in cubic, elastic media For a transverse wave along the k y direction polarised in the x direction, we have w C44 1/ 2 k y and similarly for the k x and k z directions. Hence v T C44 1/

40 Strain in elastic media

41 Strain in cubic systems bulk modulus Previously, we saw that the bulk modulus is defined as the resistance of a material to compression. In terms of the pressure P and volume V, the bulk modulus B was given as B V P V.

42 Strain in cubic systems bulk modulus Alternatively, we may describe the bulk modulus in terms of internal energy. For a hydrostatic pressure P, the change in energy with volume is given by U V P. Substituting this into the previous expression B V 2 U 2 V.

43 Strain in cubic systems dilation We may calculate this by defining the dilation V, V where V is the change in the crystal volume V. For a uniform dilation, this is given by D 3 D xx yy zz.

44 Strain in cubic systems dilation Now, we require the energy density, which in the primitive cell is given by U 1 2 nst stn so we shall now put V. For the uniform dilation, this expression reduces to s c tn, U 1 2 n c nn nn.

45 Strain in cubic systems cubical dilation For a cubic crystal, we therefore have U D 6 C 2C. We now differentiate with respect to volume to obtain 2 U C 2C, D where, since is the initial volume, it is taken to be constant.

46 Strain in cubic systems cubical dilation The change in is given by so D, 1 D. Making this substitution, we have 2 U D D 6 C 2C.

47 Strain in cubic systems cubical dilation Hence However, since 2 U 2 B 1 3 C 2C U 2 So, multiplying the RHS by -1 to obtain a positive solution, B 1 3, 12 C 11 2C 12.

48 Strain In a cubic system, the strain is given by s s s s s s xx yy zz xy yz zx C C C C C C C C C C C C xx yy zz xy yz zx.

49 Strain in cubic systems shear modulus We saw previously that the shear modulus G is defined as the ratio of shear stress to shear strain. Hence s xy G xy. From the previous matrix equation for the stress, for a cubic system, we therefore have G C 44.

50 Strain in cubic systems shear modulus In practice, this result may not be accurate due to the deformation of the material and some kind of averaging scheme may be required. Two schemes used are the Voigt average, giving G V and the Reuss average G R C 5 4C 11 3C 44 C 44 C C 12 C44 3C C

51 Young s modulus and Poisson s ratio From the linear theory of elasticity, we have the relations Y for Young s modulus and v 9BG 3B G 3B 2 2G, 3B G. for Poisson s ratio. Hence, if we know how to choose G we can approximate these quantities in terms of the elastic stiffness constants C ij.

52 Cubic semiconductors (references) References: [1] McSkimin, H. J. and P. Andreatch, J. Appl. Phys. 43, (1972) [2] McSkimin, H. J., J. Appl. Phys. 24, (1953) [3] Nikanorov S. P. and B. K. Kardashev, Moscow, "Nauka" Publ. House (1985) [4] Gmelins Handbuch der Anorganischen Chemie, 8th edition, Silicium, Part B, Weinheim, Verlag Chemie, GmbH (1959) [5] Grimsditch, M., E.S. Zouboulis, J. Appl. Phys. 76, (1994) [6] Wright, A.F., J. Appl. Phys (1997) [7] S.Adachi, J. Appl. Phys. 58, R1-R29 (1985). [8] Burenkov, Yu. A., Yu. M. Burdukov, S. Yu. Davidov, and S. P. Nikanorov, Sov. Phys. Solid State 15, (1973) [9] Burenkov, Yu. A., S. Yu. Davydov, and S. P. Nikanorov, Sov. Phys. Solid State 17, (1975)

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

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

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

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

Lecture contents. Stress and strain Deformation potential. NNSE 618 Lecture #23

Lecture contents. Stress and strain Deformation potential. NNSE 618 Lecture #23 1 Lecture contents Stress and strain Deformation potential Few concepts from linear elasticity theory : Stress and Strain 6 independent components 2 Stress = force/area ( 3x3 symmetric tensor! ) ij ji

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

Effective mass: from Newton s law. Effective mass. I.2. Bandgap of semiconductors: the «Physicist s approach» - k.p method

Effective mass: from Newton s law. Effective mass. I.2. Bandgap of semiconductors: the «Physicist s approach» - k.p method Lecture 4 1/10/011 Effectie mass I.. Bandgap of semiconductors: the «Physicist s approach» - k.p method I.3. Effectie mass approximation - Electrons - Holes I.4. train effect on band structure - Introduction:

More information

PEAT SEISMOLOGY Lecture 2: Continuum mechanics

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

More information

Mechanics PhD Preliminary Spring 2017

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

More information

[5] Stress and Strain

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

More information

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

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

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

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

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

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

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

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

Lecture 8: Tissue Mechanics

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

More information

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

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

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

More information

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

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

6.730 Physics for Solid State Applications

6.730 Physics for Solid State Applications 6.730 Physics for Solid State Applications Lecture 5: Specific Heat of Lattice Waves Outline Review Lecture 4 3-D Elastic Continuum 3-D Lattice Waves Lattice Density of Modes Specific Heat of Lattice Specific

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

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

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

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

Supporting Information:

Supporting Information: Supporting Information: Strain Induced Optimization of Nanoelectromechanical Energy Harvesting and Nanopiezotronic Response in MoS 2 Monolayer Nanosheet Nityasagar Jena, Dimple, Shounak Dhananjay Behere,

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

ELASTICITY (MDM 10203)

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

More information

Unit IV State of stress in Three Dimensions

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

More information

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

THREE-DIMENSIONAL SIMULATION OF THERMAL OXIDATION AND THE INFLUENCE OF STRESS

THREE-DIMENSIONAL SIMULATION OF THERMAL OXIDATION AND THE INFLUENCE OF STRESS THREE-DIMENSIONAL SIMULATION OF THERMAL OXIDATION AND THE INFLUENCE OF STRESS Christian Hollauer, Hajdin Ceric, and Siegfried Selberherr Institute for Microelectronics, Technical University Vienna Gußhausstraße

More information

Soft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies

Soft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies Soft-Body Physics Soft Bodies Realistic objects are not purely rigid. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies Deformed

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

CONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS

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

More information

Tensorial and physical properties of crystals

Tensorial and physical properties of crystals Tensorial and physical properties of crystals Michele Catti Dipartimento di Scienza dei Materiali, Universita di Milano Bicocca, Milano, Italy (catti@mater.unimib.it) MaThCryst Nancy 2005 International

More information

Smart Materials, Adaptive Structures, and Intelligent Mechanical Systems

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

More information

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

Tensor Visualization. CSC 7443: Scientific Information Visualization

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

More information

Crystal Relaxation, Elasticity, and Lattice Dynamics

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

More information

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

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

More information

Strain-related Tensorial Properties: Elasticity, Piezoelectricity and Photoelasticity

Strain-related Tensorial Properties: Elasticity, Piezoelectricity and Photoelasticity Strain-related Tensorial Properties: Elasticity, Piezoelectricity and Photoelasticity Torino, Italy, September 4-9, 2016 Alessandro Erba Dipartimento di Chimica, Università di Torino (Italy) alessandro.erba@unito.it

More information

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. 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

More information

Two-dimensional ternary locally resonant phononic crystals with a comblike coating

Two-dimensional ternary locally resonant phononic crystals with a comblike coating Two-dimensional ternary locally resonant phononic crystals with a comblike coating Yan-Feng Wang, Yue-Sheng Wang,*, and Litian Wang Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing,

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

Symmetry Crystallography

Symmetry Crystallography Crystallography Motif: the fundamental part of a symmetric design that, when repeated, creates the whole pattern In 3-D, translation defines operations which move the motif into infinitely repeating patterns

More information

EE C247B ME C218 Introduction to MEMS Design Spring 2017

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

More information

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

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

More information

F-Praktikum Physikalisches Institut (PI) Stark korrelierte Elektronen und Spins. Report

F-Praktikum Physikalisches Institut (PI) Stark korrelierte Elektronen und Spins. Report F-Praktikum Physikalisches Institut (PI) Stark korrelierte Elektronen und Spins Report Experiment Schallausbreitung in Kristallen (Sound Propagation in Crystals) Intsar Bangwi Physics bachelor 5. term

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

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

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

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

Tinselenidene: a Two-dimensional Auxetic Material with Ultralow Lattice Thermal Conductivity and Ultrahigh Hole Mobility

Tinselenidene: a Two-dimensional Auxetic Material with Ultralow Lattice Thermal Conductivity and Ultrahigh Hole Mobility Tinselenidene: a Two-dimensional Auxetic Material with Ultralow Lattice Thermal Conductivity and Ultrahigh Hole Mobility Li-Chuan Zhang, Guangzhao Qin, Wu-Zhang Fang, Hui-Juan Cui, Qing-Rong Zheng, Qing-Bo

More information

FINITE ELEMENT AND EXPERIMENTAL STUDY OF NOVEL CONCEPT OF 3D FIBRE CELL STRUCTURE

FINITE ELEMENT AND EXPERIMENTAL STUDY OF NOVEL CONCEPT OF 3D FIBRE CELL STRUCTURE FINITE ELEMENT AND EXPERIMENTAL STUDY OF NOVEL CONCEPT OF 3D FIBRE CELL STRUCTURE M. Růžička, V. Kulíšek 2, J. Had, O. Prejzek Department of Mechanics, Biomechanics and Mechatronics, Faculty of Mechanical

More information

Elasticity Constants of Clay Minerals Using Molecular Mechanics Simulations

Elasticity Constants of Clay Minerals Using Molecular Mechanics Simulations Elasticity Constants of Clay Minerals Using Molecular Mechanics Simulations Jin-ming Xu, Cheng-liang Wu and Da-yong Huang Abstract The purpose of this paper is to obtain the elasticity constants (including

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

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

3.091 Introduction to Solid State Chemistry. Lecture Notes No. 5a ELASTIC BEHAVIOR OF SOLIDS

3.091 Introduction to Solid State Chemistry. Lecture Notes No. 5a ELASTIC BEHAVIOR OF SOLIDS 3.091 Introduction to Solid State Chemistry Lecture Notes No. 5a ELASTIC BEHAVIOR OF SOLIDS 1. INTRODUCTION Crystals are held together by interatomic or intermolecular bonds. The bonds can be covalent,

More information

Lindgren CRYSTAL SYMMETRY AND ELASTIC CONSTANTS MICHAEL WANDZILAK. S.B., Massachusetts Institute of Technology (196'7)

Lindgren CRYSTAL SYMMETRY AND ELASTIC CONSTANTS MICHAEL WANDZILAK. S.B., Massachusetts Institute of Technology (196'7) CRYSTAL SYMMETRY AND ELASTIC CONSTANTS by MICHAEL WANDZILAK S.B., Massachusetts Institute of Technology (196'7) Submitted in partial fulfillment of the requirements for the degree of Master of Science

More information

Two-dimensional Phosphorus Carbide as Promising Anode Materials for Lithium-ion Batteries

Two-dimensional Phosphorus Carbide as Promising Anode Materials for Lithium-ion Batteries Electronic Supplementary Material (ESI) for Journal of Materials Chemistry A. This journal is The Royal Society of Chemistry 2018 Supplementary Material for Two-dimensional Phosphorus Carbide as Promising

More information

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

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

More information

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

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

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

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

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

More information

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

2 Introduction to mechanics

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

More information

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

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

More information

Supporting Information. Potential semiconducting and superconducting metastable Si 3 C. structures under pressure

Supporting Information. Potential semiconducting and superconducting metastable Si 3 C. structures under pressure Supporting Information Potential semiconducting and superconducting metastable Si 3 C structures under pressure Guoying Gao 1,3,* Xiaowei Liang, 1 Neil W. Ashcroft 2 and Roald Hoffmann 3,* 1 State Key

More information

Strain distributions in group IV and III-V semiconductor quantum dots

Strain distributions in group IV and III-V semiconductor quantum dots International Letters of Chemistry, Physics and Astronomy Online: 2013-09-21 ISSN: 2299-3843, Vol. 7, pp 36-48 doi:10.18052/www.scipress.com/ilcpa.7.36 2013 SciPress Ltd., Switzerland Strain distributions

More information

6.730 Physics for Solid State Applications

6.730 Physics for Solid State Applications 6.730 Physics for Solid State Applications Lecture 29: Electron-phonon Scattering Outline Bloch Electron Scattering Deformation Potential Scattering LCAO Estimation of Deformation Potential Matrix Element

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

Assessment of phonon mode characteristics via infrared spectroscopic ellipsometry on a-plane GaN

Assessment of phonon mode characteristics via infrared spectroscopic ellipsometry on a-plane GaN phys. stat. sol. (b) 243, No. 7, 1594 1598 (2006) / DOI 10.1002/pssb.200565400 Assessment of phonon mode characteristics via infrared spectroscopic ellipsometry on a-plane GaN V. Darakchieva *, 1, T. Paskova

More information

A short review of continuum mechanics

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

More information

5 Symmetries and point group in a nut shell

5 Symmetries and point group in a nut shell 30 Phys520.nb 5 Symmetries and point group in a nut shell 5.1. Basic ideas: 5.1.1. Symmetry operations Symmetry: A system remains invariant under certain operation. These operations are called symmetry

More information

Technische Universität Graz. Institute of Solid State Physics. 22. Crystal Physics

Technische Universität Graz. Institute of Solid State Physics. 22. Crystal Physics Technische Universität Graz Institute of Solid State Physics 22. Crystal Physics Jan. 7, 2018 Hall effect / Nerst effect Technische Universität Graz Institute of Solid State Physics Crystal Physics Crystal

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

Mechanics of Biomaterials

Mechanics of Biomaterials Mechanics of Biomaterials Lecture 7 Presented by Andrian Sue AMME498/998 Semester, 206 The University of Sydney Slide Mechanics Models The University of Sydney Slide 2 Last Week Using motion to find forces

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

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

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

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

Chapter 2: Elasticity

Chapter 2: Elasticity OHP 1 Mechanical Properties of Materials Chapter 2: lasticity Prof. Wenjea J. Tseng ( 曾文甲 ) Department of Materials ngineering National Chung Hsing University wenjea@dragon.nchu.edu.tw Reference: W.F.

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

Deformable Materials 2 Adrien Treuille. source: Müller, Stam, James, Thürey. Real-Time Physics Class Notes.

Deformable Materials 2 Adrien Treuille. source: Müller, Stam, James, Thürey. Real-Time Physics Class Notes. Deformable Materials 2 Adrien Treuille source: Müller, Stam, James, Thürey. Real-Time Physics Class Notes. Goal Overview Strain (Recap) Stress From Strain to Stress Discretization Simulation Overview Strain

More information

Colloque National sur les Techniques de Modélisation et de Simulation en Science des Matériaux, Sidi Bel-Abbès Novembre 2009

Colloque National sur les Techniques de Modélisation et de Simulation en Science des Matériaux, Sidi Bel-Abbès Novembre 2009 Colloque National sur les Techniques de Modélisation et de Simulation en Science des Matériaux, Sidi Bel-Abbès. 23-24 Novembre 2009 Elastic, electronic and optical properties of SiGe 2N 4 under pressure

More information

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

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

More information

SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION In the format provided by the authors and unedited. DOI: 10.1038/NPHYS3962 A global inversion-symmetry-broken phase inside the pseudogap region of YBa 2 Cu 3 O y Contents: S1. Simulated RA patterns for

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

MSE 201A Thermodynamics and Phase Transformations Fall, 2008 Problem Set No. 7

MSE 201A Thermodynamics and Phase Transformations Fall, 2008 Problem Set No. 7 MSE 21A Thermodynamics and Phase Transformations Fall, 28 Problem Set No. 7 Problem 1: (a) Show that if the point group of a material contains 2 perpendicular 2-fold axes then a second-order tensor property

More information

Unit 15 Shearing and Torsion (and Bending) of Shell Beams

Unit 15 Shearing and Torsion (and Bending) of Shell Beams Unit 15 Shearing and Torsion (and Bending) of Shell Beams Readings: Rivello Ch. 9, section 8.7 (again), section 7.6 T & G 126, 127 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering

More information

Strain-Induced Band Profile of Stacked InAs/GaAs Quantum Dots

Strain-Induced Band Profile of Stacked InAs/GaAs Quantum Dots Engineering and Physical Sciences * Department of Physics, Faculty of Science, Ubon Ratchathani University, Warinchamrab, Ubon Ratchathani 490, Thailand ( * Corresponding author s e-mail: w.sukkabot@gmail.com)

More information

Motivation. Confined acoustics phonons. Modification of phonon lifetimes Antisymmetric Bulk. Symmetric. 10 nm

Motivation. Confined acoustics phonons. Modification of phonon lifetimes Antisymmetric Bulk. Symmetric. 10 nm Motivation Confined acoustics phonons Modification of phonon lifetimes 0 0 Symmetric Antisymmetric Bulk 0 nm A. Balandin et al, PRB 58(998) 544 Effect of native oxide on dispersion relation Heat transport

More information

N = Shear stress / Shear strain

N = Shear stress / Shear strain UNIT - I 1. What is meant by factor of safety? [A/M-15] It is the ratio between ultimate stress to the working stress. Factor of safety = Ultimate stress Permissible stress 2. Define Resilience. [A/M-15]

More information

HIGHER-ORDER THEORIES

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

More information

Solutions for Homework 4

Solutions for Homework 4 Solutions for Homework 4 October 6, 2006 1 Kittel 3.8 - Young s modulus and Poison ratio As shown in the figure stretching a cubic crystal in the x direction with a stress Xx causes a strain e xx = δl/l

More information

20. Rheology & Linear Elasticity

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

More information