Here, the parameter is the cylindrical coordinate, (a constant) is the radius, and the constant controls the rate of climb.
|
|
- Mabel Banks
- 5 years ago
- Views:
Transcription
1 Line integrals in 3D A line integral in space is often written in the form f( ) d. C To evaluate line integrals, you must describe the path parametrically in the form ( s). s a path in space Often, for example, is selected to equal the arc length along the line, but other interpretations might be more natural. Wednesday 10:30-12:00 Higher order Tensor Calculus s C EXAMPLE: Points on a helix may be described parametrically by ( θ) Rcosθẽ 1 + Rsinθẽ 2 + k θ ẽ 3 θ k Here, the parameter is the cylindrical coordinate, (a constant) is the radius, and the constant controls the rate of climb.. R f( ) d Once the path is parameterized, the integral is evaluated by. C s 2 s 1 f ( ) d ds ds JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 1
2 Often, the parameter is time so that f( ) d f( dt. ) C t 2 t 1 Wednesday 10:30-12:00 Higher order Tensor Calculus Higher-order line integrals In materials modeling, work is often written as W :dε This is a line integral in tensor space. The strain path must be defined parametrically. Usually, the parameter is time t, so the line integral is evaluated by W :ε dt JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 2
3 Exact differentials in three dimensions ux ( 1, x 2, x 3 ) du f 1 dx 1 + f 2 dx 2 + f 3 dx 3 Under what conditions does there exist a potential such that f k x 1 x 2 x 3 where the are each functions of,, and. If the potential exists, then (by the chain rule) f i u x i Wednesday 10:30-12:00 Higher order Tensor Calculus What does the notation ( fdx + gdy ) mean?!? Mixed partials are order independent: 2 u x i x j 2 u x j x i f i x j f j x i Which implies that, if a potential exists, then. Conversely, if the functions f k satisfy this equation, then a potential exists. In other words, for a potential to exist, the matrix H ij must be symmetric. f i x j ( f 1 dx 1 + f 2 dx 2 + f 3 dx 3 ) f d An integral is path independent only if the potential exists f d and its value is. du u final u initial JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 3
4 Integrability Consider a vector ỹ that is a proper function of another vector. y i Then dy i dx x j j y or dy i C ij dx j, where C i ij x j Wednesday 10:30-12:00 Higher order Tensor Calculus Order of second-partial derivatives may be swapped: 2 y i x j x k 2 y i x k x j. Therefore C ij x k C ik x j Inverse question: y Given a tensor C ij that varies with, under what conditions will a field ỹ exist for which C ij i? x j C ij C ik ANSWER: The field will exist if and only if. y i ỹ x k x j If so, C ij represents a set of nine coupled PDEs that may be solved to find the field ỹ. x j JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 4
5 Application: Compatibility For large deformations, the deformation gradient tensor is defined. Wednesday 10:30-12:00 Higher order Tensor Calculus F ij x i X j F A spatially varying field F ij F ij is physically realizable if and only if ik. X k X j Example: every layer in the following picture appears to stretch horizontally: initial candy deformed candy A horizontal stretch for a homogeneous deformation has a deformation gradient of the form [ ] F λ before 1 λ You might be (wrongly) tempted to say that the candy problem must have the same of a similar form: 1 after F [ ] F αx 2 + β (this is WRONG) This must be wrong, however, because F 11 F X 2 X 1 (Compatibility is violated!) JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 5
6 An allowable answer is Wednesday 10:30-12:00 Higher order Tensor Calculus [ ] F αx 2 +β αx The bad solution presumed that squares deformed to rectangles. This good solution lets squares deform to parallelograms. deformed candy JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 6
7 Preliminaries Multiple ways to analyze stresses on a windmill blade: Wednesday 15:00-16:30 Frame Indifference spatial frame g ṽ Ẽ 2 material frame ṽ g ẽ θ ẽ r Ẽ 1 g ṽ unrotated frame Ẽ 2 Ẽ 1 Suppose a deformation field is identical to one that you have analyzed in the past, except it is rigidly rotated and/or translated. QUESTION: Can you use the already-solved problem to help you solve the new problem? ANSWER: Yes, but you need to decide: space rotation or superimposed rotation? JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 7
8 Wednesday 15:00-16:30 Frame Indifference SPACE ROTATION PROBLEM (easy) previously solved (fiducial) deformed initial new problem all scalars: s new all free vectors: s old s new s old ṽ new ṽ old vnew i Q ip vold p all second-order tensors: T new old T T ij T new Q ip Q jq Told pq all third-order tensors: ζnew ijk Q ip Q iq Q ir ζold pqr JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 8
9 Wednesday 15:00-16:30 Frame Indifference SUPERIMPOSED ROTATION PROBLEM (harder) previously solved (fiducial) deformed initial new problem spatial (objective) scalars: s new spatial (objective) vectors: ṽ new s old reference scalars: s new s old ṽ old reference vectors: ṽ new ṽ old spatial (objective) tensors: T new old T reference tensors: T T new T old multi-point quantities: transform like neither spatial nor reference quantities JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 9
10 TERMINOLOGY fiducial previously solved problem. star Wednesday 15:00-16:30 Frame Indifference new problem with same initial state, and deformed state identical to the fiducial problem but with superimposed rotation. spatial (objective): a quantity that transforms like the space rotation problem. reference: a quantity that is unaffected by superimposed rotation multi-point: a quantity that is neither objective nor reference The superimposed rotation problem µ A material fiber is a vector connecting two points in a body. µ 0 0 µ * 0 µ Any initial material fiber is a reference quantity ( ). µ * µ The deformed material fiber is a spatial quantity ( ). The deformation gradient is a two-point tensor. F F Let deformation gradient for the fiducial problem. Let deformation gradient for the star problem F F (two-point) JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 10
11 Likewise, the polar rotation tensor becomes, after superimposed rotation, R R (two-point) Wednesday 15:00-16:30 Frame Indifference Cauchy stress is force per current area, not force per initial area. Its definition requires no knowledge of the initial configuration. T (spatial, objective) Ũ The right polar stretch tensor is defined as material stretch that occurs before rotation. It is therefore unaffected by superimposed rotation: Ũ Ũ (reference) Ṽ The left polar stretch tensor is the stretch after material rotation. Superposed rigid rotation will re-orient the eigenvectors without changing the eigenvalues. Ṽ T Ṽ (spatial, objective) The principle of material frame indifference (PMFI) is the very intuitive requirement that superimposed rigid motion of a material will require the applied forces and stresses to co-rotate and co-translate in a manner consistent with their definition. Regardless of whether you apply a material model for the fiducial problem or for the star problem, the results must be consistent with these objectivity relations JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 11
12 Mathematics of frame indifference (introduction) Wednesday 15:00-16:30 Frame Indifference Spring Example: Translational frame indifference f o spring constant change in length kδ Force in spring:. undeformed o undeformed Fiducial deformation Starred deformation o Let current location of the tip of the spring. Let original (unstressed) location of the tip of the spring. First (BAD) attempt at a vector spring equation: f k( o) (violates translational PMFI) JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 12
13 Consider Fiducial deformation: stretch the spring by some known amount. Wednesday 15:00-16:30 Frame Indifference Starred deformation: stretch the spring by the same amount and also translate it by an amount. Both scenarios involve the same starting tip location, so o o. (1) The star deformation has the same final tip location plus extra translation : + c. (2) c c The problems differ only by translation, so they both involve the same spring elongation. They should therefore involve the same spring force. f desired f. f k( o) Does the model give this desired result? For starred deformation, this (bad) spring model predicts f predicted k( o ) Substitute (1) and (2) into (3) to obtain f predicted k( + ) c f f predicted desired f o. desired + kc (NOT translationally invariant!) JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 13 (3)
14 Spring Example: Rotational frame indifference. Wednesday 15:00-16:30 Frame Indifference To satisfy translation invariance, consider a revised spring equation f better, but still bad. (here, tip tail ) k( o) Because represents the difference between two points on the spring, it will be invariant under a superimposed translation. Under translation, o o f f As before,. Therefore, under translation, as desired. This model permits force to co-translate with the spring. That s good. However... Another requirement of PMFI: Force must co-rotate with the spring. This model FAILS in this respect JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 14
15 Wednesday 15:00-16:30 Frame Indifference o undeformed undeformed o PROPOSED MODEL f k( o) fiducial: starred: fiducial deformation stretch the spring without rotation f starred deformation f predicted same deformation as fiducial, but also rigidly rotate the spring. o f desired MODEL PREDICTION f k( o ) o o f desired. f From the sketch, it s clear that f desired (NOT rotationally invariant!) predicted f In fact, f predicted desired k(. PMFI requires this to be, which happens only if. f o o ) 0 Ĩ JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 15
16 At last... a spring model that satisfies PMFI. To satisfy both translational and rotational frame indifference, consider Wednesday 15:00-16:30 Frame Indifference f kδñ δ o ñ where, and This one satisfies PMFI! The only difference between the fiducial and starred deformations is a rigid motion. Therefore, deformed lengths are the same in both cases. Hence,. o As before, the initial state is the same for both cases, so. o Consequently, δ δ Under a superimposed rigid motion,. Therefore, ñ ñ. Thus, f predicted kδ ñ ñ ( kδñ) f f desired kδ JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 16 YAHOO! Success at last!.
17 PMFI in material constitutive models Wednesday 15:00-16:30 Frame Indifference Any material model must be invariant under a basis change (space rotation problem). This requirement is necessary, but unrelated to PMFI. PMFI requires consistency of predictions when comparing deformations that differ by superimpose rotation. You can work out in advance how variables should transform under superimposed rotation. Input (independent variables) sent to your constitutive model must convey sufficient information to allow model predictions (output dependent variables) to transform correctly under superimposed rotation. If you apply your model to a fiducial problem, then you obtain a certain fiducial prediction. If you apply the model to a star problem that is identical to the fiducial problem except that the input corresponds to a superimposed rotation, then the output needs correspond to the superimposed rotation. Suppose the model takes the stretch as its only input, and returns the Cauchy stress as output. Ṽ When a star stretch T is sent as input, Ṽ Ṽ PMFI demands that the model must return T as output JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 17
18 Here s why a model that takes only left stretch V is deficient. Wednesday 15:00-16:30 Frame Indifference Fiber composite [ F] 10 [ F] [ F] [ F] [ R] 10 [ R] [ R] [ R] [ V] 10 [ V] [ V] [ V] JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 18
19 PMFI in general materials modeling Thursday 13:00-14:30 Stress rates and isotropy When testing for frame indifference, you must first note what variables are involved. You must use physical reasoning along with rigorous mathematical consistency analysis to assert how those quantities are supposed to change upon a superimposed rigid motion. PMFI analysis for linear elasticity. Consider a constitutive model in which Cauchy stress is regarded to depend linearly on some spatial strain. ε σ ij E ijkl ε kl Ẽ :ε, or (1) Since Cauchy stress is spatial, this constitutive model satisfies PMFI only if T Ẽ ε ( T) for all orthogonal : Indicial form, recalling Eq. (1), Q pi ( E ijkl ε kl )Q qj E pqmn ( Q mk ε kl Q nl ) This must hold for all strains, and therefore, PMFI requires that Q pi ( E ijkl )Q qj E pqmn ( Q mk Q nl ) which may be rearranged to give E pqmn ( Q pi Q qj Q mk Q nl ) E ijkl This says that must be isotropic. If your material Ẽ is anisotropic, then you must receive more than just spatial strain as input to satisfy PMFI JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 19
20 Thursday 13:00-14:30 Stress rates and isotropy Satisfying PMFI does not per se make your model valid for large deformations. A model that satisfies PMFI is merely self-consistent for arbitrarily large rotations. PMFI says nothing about whether your model is any good for large material distortions (i.e., problems where material fibers significantly rotate and stretch relative to other fibers). Alternative configurations and alternative stress/strain measures are often used to get better results under large distortions. Finding appropriate representations for distortion of material symmetry directions is primarily the subject of physics, not PMFI JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 20
21 Stress Rates Thursday 13:00-14:30 Stress rates and isotropy PMFI in rate forms of constitutive equations Consider a constitutive law σ ij E ijkl ε kl, or Ẽ :ε in which the stiffness tensor is constant and the stress and strain are both spatial. To satisfy PMFI, let s presume that the stiffness tensor is also isotropic. Taking the time rate of both sides gives σ ij E ijkl ε kl, or Ẽ :ε This rate constitutive equation satisfies PMFI because it was obtained by differentiating a frame indifferent equation. Here s why JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 21
22 Thursday 13:00-14:30 Stress rates and isotropy For for isotropic linear elasticity, the stress deviator S ij is related to the strain deviator γ ij by G S 2Gγ, where is the shear modulus (a constant). Under superimposed rotation, this becomes T S 2G γ T. In rate form, T T T + + 2G[ T + T + T ] S S S γ γ γ S 2Gγ T 2G[ T] S γ Recalling that, the RED terms involving all cancel, leaving only which is self consistent with S 2Gγ PMFI is satisfied for the rate equation.. In general, whenever a non-rate constitutive relation satisfies PMFI, then its rate form will also satisfy PMFI JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 22
23 Thursday 13:00-14:30 Stress rates and isotropy REPLACING A GENUINE STRAIN RATE WITH SOME APPROXIMATION TO STRAIN RATE HAS DISTRESSING PMFI RAMIFICATIONS! The so-called rate of deformation tensor is just the symmetric part of the velocity gradient: D 1 v -- ( + T) L 2 ij i x j, where. If the principal directions of stretch never change, it equals the rate of the logarithmic strain. Otherwise, it is not a true rate. D The rate of logarithmic strain,, is very difficult to compute, whereas is very easy to compute. For this reason, many people take a perfectly valid (PMFI-conforming) constitutive model in rate form and mess it up by replacing it with g( ) ε g( ) D ε [satisfies PMFI if the material is isotropic]. [violates PMFI] D JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 23
24 Thursday 13:00-14:30 Stress rates and isotropy g( ) ε The original material model satisfied PMFI because T T + + T 2G T [ + T + T ] S S S γ γ γ These terms cancel with those on right-hand side The new (bad) material model violates PMFI. The tensor is spatial, so the PMFI requirement is D g( ) D T T + + T 2G[ dev T] S S S D These terms no longer go away D Replacement of a genuine strain rate with is going to require replacing with a pseudo stress rate (often more graciously called an objective stress rate) that subtracts away the offending terms JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 24
25 The violation of PMFI can be repaired by replacing the bad constitutive model with, Thursday 13:00-14:30 Stress rates and isotropy where denotes a special co-rotational rate that effectively eliminates the part of the stress rate caused by rotation rates. g ( D ) The co-rotational rate is (must be) a spatial tensor to satisfy PMFI for isotropic materials. Objective co-rotational rates (Convected, Jaumann, Polar) Co-rotational stress rates are often defined in the form Λ T Λ Λ + T, ϒ Λ where ϒ The tensor is required only to transform under superimposed rotation so that Λ T This will ensure that the pseudo (co-rotational) stress rate is spatial (as needed by PMFI): T * JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 25
26 Thursday 13:00-14:30 Stress rates and isotropy Possible choices for the nebulous spin-like tensor Velocity gradient ( ) Λ If then L ij v i x j Convected rate T Λ Vorticity tensor ( ). Being skew symmetric, T. Λ W skw W 1 -- ( T) W 2 + W If then W Jaumann rate W Λ The the nebulous tensor can be taken as the polar spin tensor Λ Ω R R If T then Polar rate + Ω Ω All of these choices ensure that T In other words, the co-rotational rate is a spatial tensor! Consequently, the repaired constitutive model g( ) g D satisfies PMFI so long as the constitutive function is isotropic. Anisotropy can be accommodated by introducing additional material orientation input variables that transform spatially JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 26
27 Thursday 13:00-14:30 Stress rates and isotropy Oscillatory stresses under simple shear. REMEMBER! satisfying PMFI will not ensure the model will give accurate or even reasonable results. To demonstrate that satisfying PMFI is merely a consistency condition that is necessary but not sufficient to obtain sensible constitutive model predictions, Dienes 1 considered simple shear with isotropic linear-elasticity in the form Ẽ :D τ γ τ Polar Jaumann γ Dienes demonstrated that the Jaumann rate predicts anomalous oscillatory shear stresses whereas the polar rate predicts intuitively more appealing monotonically increasing shear stress. The Jaumann rate performs so poorly because it uses vorticity as the material spin used to subtract away the rotational part of the stress rate. W W However, for simple shear, vorticity is constant throughout time. The Jaumann objective rate effectively presumes that a material element tumbles unceasingly. Drawing a single material element as it deforms demonstrates that such thinking is flawed. 1. DIENES, J.K. (1979) Acta Mechanica 32, pp Incidentally, this effect was reported in the rheology literature earlier than Dienes paper. Dienes influenced the solid mechanics community JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 27
28 Lie derivatives and reference configurations Thursday 13:00-14:30 Stress rates and isotropy Lie derivatives justify the concept of reference constitutive models by making the connection with co-rotational (objective) rates clear. Let denote any tensor that transforms under a rigid superimposed rotation according to Some choices... (two point tensor) co-convected:, where is the deformation gradient tensor. F F contra-convected: T, where J. JF detf polar:, where is the polar rotation tensor. R R Consider any spatial tensor à à 1 à à The basis expansion of is T. Define a generalized overbar operation as à 1 ẽ i T 1 ẽ i 1 ẽ j A ij ( )( ẽ j ) A ij ( )( ), If a tensor is a spatial, then T, and therefore is a reference tensor à à à à à à JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 28
29 To take rates of d dt ---- Thursday 13:00-14:30 Stress rates and isotropy T, you need the following helper identity for the rate of an inverse: 1 Ã Ã Λ Λ, where 1. Using this, you can show that à which may be written à 1 à à à à T T, where. Λ Ã Ã Λ The ordinary rate of the bared tensor is the same as the bar operation acting on the co-rotational rate. Most textbooks write this Lie derivative in an equivalent (but more confusing) way: à d ---- ( 1 dt à T) T JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 29
30 WHY ARE LIE DERIVATIVES USEFUL? Thursday 13:00-14:30 Stress rates and isotropy Lie derivatives let you apply a constitutive model in a reference configuration where rates are genuine true rates, not pseudo rates, and you can prove that the result will be identical to a more complicated model that uses pseudo (co-rotational) rates in a spatial formulation. You can write a material model that presumes no rotation (which is often the case for available experimental data). Once you match non-rotational data adequately, you can upgrade your non-rotational computer model to satisfy PMFI by merely adding a pre-conditioning wrapper that un-transforms all spatial inputs to the reference state prior to calling the model. Results are then transformed in this wrapper back to the spatial frame and sent back to the host finite element code JUNE 2004, DELFT RESEARCH SCHOOL STRUCTURAL ENGINEERING PAGE 30
Caveats concerning conjugate stress and strain measures for frame indifferent anisotropic elasticity
This is a modified version of the paper by the same title that appears in Acta Mech, Vol. 129, No. 1 2, 1998. In this version, the tensor order of a variable equals the number of underlines beneath it.
More informationEngineering Sciences 241 Advanced Elasticity, Spring Distributed Thursday 8 February.
Engineering Sciences 241 Advanced Elasticity, Spring 2001 J. R. Rice Homework Problems / Class Notes Mechanics of finite deformation (list of references at end) Distributed Thursday 8 February. Problems
More informationKinematics: The mathematics of deformation. 1. What is deformation?
March 0, 003 7:5 pm What is deformation? D R A F T Ω o Kinematics: The mathematics of deformation. What is deformation? Let denote the initial configuration of a physical body (e.g., an airplane part,
More informationSymmetry 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 informationJuly 30, :15 pm. Geometric Insight into Return Mapping Plasticity Algorithms. Rebecca M. Brannon ABSTRACT
July 30, 2002 9:15 pm DRAFT Geometric Insight into Return Mapping Plasticity Algorithms Rebecca M. Brannon ABSTRACT Return mapping algorithms are probably the most popular means of numerically solving
More informationBy 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 information3 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 informationMechanics 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 informationRotational motion of a rigid body spinning around a rotational axis ˆn;
Physics 106a, Caltech 15 November, 2018 Lecture 14: Rotations The motion of solid bodies So far, we have been studying the motion of point particles, which are essentially just translational. Bodies with
More informationFundamentals 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 informationMathematical 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 informationExercise: 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 informationContinuum 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 informationChapter 2. Rubber Elasticity:
Chapter. Rubber Elasticity: The mechanical behavior of a rubber band, at first glance, might appear to be Hookean in that strain is close to 100% recoverable. However, the stress strain curve for a rubber
More informationIntroduction to Seismology Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 1.510 Introduction to Seismology Spring 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 1.510 Introduction to
More informationA 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(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 informationIntroduction 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 information2.073 Notes on finite-deformation theory of isotropic elastic-viscoplastic solids
2.073 Notes on finite-deformation theory of isotropic elastic-viscoplastic solids Lallit Anand Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA 02139, USA April
More informationContinuum 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 informationAdd-on unidirectional elastic metamaterial plate cloak
Add-on unidirectional elastic metamaterial plate cloak Min Kyung Lee *a and Yoon Young Kim **a,b a Department of Mechanical and Aerospace Engineering, Seoul National University, Gwanak-ro, Gwanak-gu, Seoul,
More informationPEAT 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 information2.20 Fall 2018 Math Review
2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more
More informationChapter 0. Preliminaries. 0.1 Things you should already know
Chapter 0 Preliminaries These notes cover the course MATH45061 (Continuum Mechanics) and are intended to supplement the lectures. The course does not follow any particular text, so you do not need to buy
More informationMath 575-Lecture Viscous Newtonian fluid and the Navier-Stokes equations
Math 575-Lecture 13 In 1845, tokes extended Newton s original idea to find a constitutive law which relates the Cauchy stress tensor to the velocity gradient, and then derived a system of equations. The
More informationElements 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 informationMeasurement 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 informationA SPECT reconstruction method for extending parallel to non-parallel geometries Junhai Wen and Zhengrong Liang
This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: 148.251.232.83 This content was downloaded on 03/04/2019 at 09:09 Please note that
More informationMechanics 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 informationContinuum 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 informationElastic Wave Theory. LeRoy Dorman Room 436 Ritter Hall Tel: Based on notes by C. R. Bentley. Version 1.
Elastic Wave Theory LeRoy Dorman Room 436 Ritter Hall Tel: 4-2406 email: ldorman@ucsd.edu Based on notes by C. R. Bentley. Version 1.1, 20090323 1 Chapter 1 Tensors Elastic wave theory is concerned with
More informationContravariant and Covariant as Transforms
Contravariant and Covariant as Transforms There is a lot more behind the concepts of contravariant and covariant tensors (of any rank) than the fact that their basis vectors are mutually orthogonal to
More informationLecture notes Models of Mechanics
Lecture notes Models of Mechanics Anders Klarbring Division of Mechanics, Linköping University, Sweden Lecture 7: Small deformation theories Klarbring (Mechanics, LiU) Lecture notes Linköping 2012 1 /
More informationMechanical Design in Optical Engineering
Torsion Torsion: Torsion refers to the twisting of a structural member that is loaded by couples (torque) that produce rotation about the member s longitudinal axis. In other words, the member is loaded
More informationRock 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 informationGetting started: CFD notation
PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =
More informationSymmetries 2 - Rotations in Space
Symmetries 2 - Rotations in Space This symmetry is about the isotropy of space, i.e. space is the same in all orientations. Thus, if we continuously rotated an entire system in space, we expect the system
More information16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations
6.2 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive quations Constitutive quations For elastic materials: If the relation is linear: Û σ ij = σ ij (ɛ) = ρ () ɛ ij σ ij =
More information3. The linear 3-D elasticity mathematical model
3. The linear 3-D elasticity mathematical model In Chapter we examined some fundamental conditions that should be satisfied in the modeling of all deformable solids and structures. The study of truss structures
More informationCE-570 Advanced Structural Mechanics - Arun Prakash
Ch1-Intro Page 1 CE-570 Advanced Structural Mechanics - Arun Prakash The BIG Picture What is Mechanics? Mechanics is study of how things work: how anything works, how the world works! People ask: "Do you
More informationUnderstand basic stress-strain response of engineering materials.
Module 3 Constitutive quations Learning Objectives Understand basic stress-strain response of engineering materials. Quantify the linear elastic stress-strain response in terms of tensorial quantities
More informationMATH45061: SOLUTION SHEET 1 II
MATH456: SOLUTION SHEET II. The deformation gradient tensor has Cartesian components given by F IJ R I / r J ; and so F R e x, F R, F 3 R, r r r 3 F R r, F R r, F 3 R r 3, F 3 R 3 r, F 3 R 3 r, F 33 R
More information1. 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 informationINTRODUCTION TO STRAIN
SIMPLE STRAIN INTRODUCTION TO STRAIN In general terms, Strain is a geometric quantity that measures the deformation of a body. There are two types of strain: normal strain: characterizes dimensional changes,
More informationCVEN 5161 Advanced Mechanics of Materials I
CVEN 5161 Advanced Mechanics of Materials I Instructor: Kaspar J. Willam Revised Version of Class Notes Fall 2003 Chapter 1 Preliminaries The mathematical tools behind stress and strain are housed in Linear
More informationProfessor George C. Johnson. ME185 - Introduction to Continuum Mechanics. Midterm Exam II. ) (1) x
Spring, 997 ME85 - Introduction to Continuum Mechanics Midterm Exam II roblem. (+ points) (a) Let ρ be the mass density, v be the velocity vector, be the Cauchy stress tensor, and b be the body force per
More informationME185 Introduction to Continuum Mechanics
Fall, 0 ME85 Introduction to Continuum Mechanics The attached pages contain four previous midterm exams for this course. Each midterm consists of two pages. As you may notice, many of the problems are
More informationChapter 7. Kinematics. 7.1 Tensor fields
Chapter 7 Kinematics 7.1 Tensor fields In fluid mechanics, the fluid flow is described in terms of vector fields or tensor fields such as velocity, stress, pressure, etc. It is important, at the outset,
More informationBasic concepts to start Mechanics of Materials
Basic concepts to start Mechanics of Materials Georges Cailletaud Centre des Matériaux Ecole des Mines de Paris/CNRS Notations Notations (maths) (1/2) A vector v (element of a vectorial space) can be seen
More informationIntegrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61
Integrals D. DeTurck University of Pennsylvania January 1, 2018 D. DeTurck Math 104 002 2018A: Integrals 1 / 61 Integrals Start with dx this means a little bit of x or a little change in x If we add up
More informationKinematics of fluid motion
Chapter 4 Kinematics of fluid motion 4.1 Elementary flow patterns Recall the discussion of flow patterns in Chapter 1. The equations for particle paths in a three-dimensional, steady fluid flow are dx
More informationStress, 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 informationCH.3. COMPATIBILITY EQUATIONS. Multimedia Course on Continuum Mechanics
CH.3. COMPATIBILITY EQUATIONS Multimedia Course on Continuum Mechanics Overview Introduction Lecture 1 Compatibility Conditions Lecture Compatibility Equations of a Potential Vector Field Lecture 3 Compatibility
More informationMathematical 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 informationCourse Syllabus: Continuum Mechanics - ME 212A
Course Syllabus: Continuum Mechanics - ME 212A Division Course Number Course Title Academic Semester Physical Science and Engineering Division ME 212A Continuum Mechanics Fall Academic Year 2017/2018 Semester
More informationCHAPTER 11: REYNOLDS-STRESS AND RELATED MODELS. Turbulent Flows. Stephen B. Pope Cambridge University Press, 2000 c Stephen B. Pope y + < 1.
1/3 η 1C 2C, axi 1/6 2C y + < 1 axi, ξ > 0 y + 7 axi, ξ < 0 log-law region iso ξ -1/6 0 1/6 1/3 Figure 11.1: The Lumley triangle on the plane of the invariants ξ and η of the Reynolds-stress anisotropy
More information2.1 Strain energy functions for incompressible materials
Chapter 2 Strain energy functions The aims of constitutive theories are to develop mathematical models for representing the real behavior of matter, to determine the material response and in general, to
More informationLecture #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 informationElements 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 informationFig. 1. Circular fiber and interphase between the fiber and the matrix.
Finite element unit cell model based on ABAQUS for fiber reinforced composites Tian Tang Composites Manufacturing & Simulation Center, Purdue University West Lafayette, IN 47906 1. Problem Statement In
More informationExercise solutions: concepts from chapter 5
1) Stud the oöids depicted in Figure 1a and 1b. a) Assume that the thin sections of Figure 1 lie in a principal plane of the deformation. Measure and record the lengths and orientations of the principal
More informationMechanical Design in Optical Engineering. For a prismatic bar of length L in tension by axial forces P we have determined:
Deformation of Axial Members For a prismatic bar of length L in tension by axial forces P we have determined: σ = P A δ ε = L It is important to recall that the load P must act on the centroid of the cross
More informationGeneral tensors. Three definitions of the term V V. q times. A j 1...j p. k 1...k q
General tensors Three definitions of the term Definition 1: A tensor of order (p,q) [hence of rank p+q] is a multilinear function A:V V }{{ V V R. }}{{} p times q times (Multilinear means linear in each
More informationLecture 2: Constitutive Relations
Lecture 2: Constitutive Relations E. J. Hinch 1 Introduction This lecture discusses equations of motion for non-newtonian fluids. Any fluid must satisfy conservation of momentum ρ Du = p + σ + ρg (1) Dt
More informationMASSACHUSETTS 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 informationMath 426H (Differential Geometry) Final Exam April 24, 2006.
Math 426H Differential Geometry Final Exam April 24, 6. 8 8 8 6 1. Let M be a surface and let : [0, 1] M be a smooth loop. Let φ be a 1-form on M. a Suppose φ is exact i.e. φ = df for some f : M R. Show
More informationTensors, and differential forms - Lecture 2
Tensors, and differential forms - Lecture 2 1 Introduction The concept of a tensor is derived from considering the properties of a function under a transformation of the coordinate system. A description
More informationLent 2019 VECTOR CALCULUS EXAMPLE SHEET 1 G. Taylor
Lent 29 ECTOR CALCULU EXAMPLE HEET G. Taylor. (a) The curve defined parametrically by x(t) = (a cos 3 t, a sin 3 t) with t 2π is called an astroid. ketch it, and find its length. (b) The curve defined
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Thursday 1 June 2006 1.30 to 4.30 PAPER 76 NONLINEAR CONTINUUM MECHANICS Attempt FOUR questions. There are SIX questions in total. The questions carry equal weight. STATIONERY
More information3D 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 informationElasticity 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 informationClassical Mechanics. Luis Anchordoqui
1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body
More information3.2 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 informationIntroduction to Finite Element Method
Introduction to Finite Element Method Dr. Rakesh K Kapania Aerospace and Ocean Engineering Department Virginia Polytechnic Institute and State University, Blacksburg, VA AOE 524, Vehicle Structures Summer,
More informationPhysics 110. Electricity and Magnetism. Professor Dine. Spring, Handout: Vectors and Tensors: Everything You Need to Know
Physics 110. Electricity and Magnetism. Professor Dine Spring, 2008. Handout: Vectors and Tensors: Everything You Need to Know What makes E&M hard, more than anything else, is the problem that the electric
More informationGame 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 informationBasic 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 informationPart 5 ACOUSTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA
Part 5 ACOUSTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA Review of Fundamentals displacement-strain relation stress-strain relation balance of momentum (deformation) (constitutive equation) (Newton's Law)
More informationCVEN 7511 Computational Mechanics of Solids and Structures
CVEN 7511 Computational Mechanics of Solids and Structures Instructor: Kaspar J. Willam Original Version of Class Notes Chishen T. Lin Fall 1990 Chapter 1 Fundamentals of Continuum Mechanics Abstract In
More informationMODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam. Isotropic Elastic Models: Invariant vs Principal Formulations
MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar Willam University of Colorado at Boulder Class Meeting #2: Nonlinear Elastic Models Isotropic Elastic Models: Invariant vs Principal Formulations Elastic
More informationUNIVERSITY OF DUBLIN
UNIVERSITY OF DUBLIN TRINITY COLLEGE JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Faculty of Engineering, Mathematics and Science school of mathematics Trinity Term 2015 Module MA3429
More information3. BEAMS: STRAIN, STRESS, DEFLECTIONS
3. BEAMS: STRAIN, STRESS, DEFLECTIONS The beam, or flexural member, is frequently encountered in structures and machines, and its elementary stress analysis constitutes one of the more interesting facets
More informationKINEMATICS 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 informationHOMEWORK 2 SOLUTIONS
HOMEWORK SOLUTIONS MA11: ADVANCED CALCULUS, HILARY 17 (1) Find parametric equations for the tangent line of the graph of r(t) = (t, t + 1, /t) when t = 1. Solution: A point on this line is r(1) = (1,,
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 06
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 06 In the last lecture, we have seen a boundary value problem, using the formal
More informationAdvanced Structural Analysis EGF Section Properties and Bending
Advanced Structural Analysis EGF316 3. Section Properties and Bending 3.1 Loads in beams When we analyse beams, we need to consider various types of loads acting on them, for example, axial forces, shear
More informationAN INTRODUCTION TO CURVILINEAR ORTHOGONAL COORDINATES
AN INTRODUCTION TO CURVILINEAR ORTHOGONAL COORDINATES Overview Throughout the first few weeks of the semester, we have studied vector calculus using almost exclusively the familiar Cartesian x,y,z coordinate
More informationContinuum Mechanics and Theory of Materials
Peter Haupt Continuum Mechanics and Theory of Materials Translated from German by Joan A. Kurth Second Edition With 91 Figures, Springer Contents Introduction 1 1 Kinematics 7 1. 1 Material Bodies / 7
More informationStress of a spatially uniform dislocation density field
Stress of a spatially uniform dislocation density field Amit Acharya November 19, 2018 Abstract It can be shown that the stress produced by a spatially uniform dislocation density field in a body comprising
More informationÁdx 1 + Áu 1 + u 1 ˆ
Muddy Card Responses Lecture M16. The mud marked * is from the equivalent lecture (also M16) in 2000. I seem to have done a better job of explaining some of the derivations, as there were fewer questions
More information3D 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 informationCHAPTER 5 KINEMATICS OF FLUID MOTION
CHAPTER 5 KINEMATICS OF FLUID MOTION 5. ELEMENTARY FLOW PATTERNS Recall the discussion of flow patterns in Chapter. The equations for particle paths in a three-dimensional, steady fluid flow are dx -----
More information1 Vectors and Tensors
PART 1: MATHEMATICAL PRELIMINARIES 1 Vectors and Tensors This chapter and the next are concerned with establishing some basic properties of vectors and tensors in real spaces. The first of these is specifically
More information1 Gauss integral theorem for tensors
Non-Equilibrium Continuum Physics TA session #1 TA: Yohai Bar Sinai 16.3.216 Index Gymnastics: Gauss Theorem, Isotropic Tensors, NS Equations The purpose of today s TA session is to mess a bit with tensors
More informationMetrics and Curvature
Metrics and Curvature How to measure curvature? Metrics Euclidian/Minkowski Curved spaces General 4 dimensional space Cosmological principle Homogeneity and isotropy: evidence Robertson-Walker metrics
More informationCourse Notes Math 275 Boise State University. Shari Ultman
Course Notes Math 275 Boise State University Shari Ultman Fall 2017 Contents 1 Vectors 1 1.1 Introduction to 3-Space & Vectors.............. 3 1.2 Working With Vectors.................... 7 1.3 Introduction
More informationQuaternion Curvatures
Quaternion Curvatures Kurt Nalty September 7, 2011 Abstract Four dimensional curves can be parameterized along pathlength by three scalars using the Frenet-Serret formulas. Using quaternion notation, these
More informationENGN 2340 Final Project Report. Optimization of Mechanical Isotropy of Soft Network Material
ENGN 2340 Final Project Report Optimization of Mechanical Isotropy of Soft Network Material Enrui Zhang 12/15/2017 1. Introduction of the Problem This project deals with the stress-strain response of a
More informationIn this section, thermoelasticity is considered. By definition, the constitutive relations for Gradθ. This general case
Section.. Thermoelasticity In this section, thermoelasticity is considered. By definition, the constitutive relations for F, θ, Gradθ. This general case such a material depend only on the set of field
More informationTexture, Microstructure & Anisotropy A.D. (Tony) Rollett
1 Carnegie Mellon MRSEC 27-750 Texture, Microstructure & Anisotropy A.D. (Tony) Rollett Last revised: 5 th Sep. 2011 2 Show how to convert from a description of a crystal orientation based on Miller indices
More information