Strain-related Tensorial Properties: Elasticity, Piezoelectricity and Photoelasticity
|
|
- Grace Caldwell
- 5 years ago
- Views:
Transcription
1 Strain-related Tensorial Properties: Elasticity, Piezoelectricity and Photoelasticity Torino, Italy, September 4-9, 2016 Alessandro Erba Dipartimento di Chimica, Università di Torino (Italy)
2 Outline Tensorial Properties of Crystals with CRYSTAL14 and CRYSTAL17 Main Tensors related to Crystal Strain Elastic Fourth-rank Tensor and Related Properties Equation-of-State (EOS) Approach Piezoelectric Tensors Photoelastic and Piezo-optic Tensors DOI: /qua.24658
3 Outline Tensorial Properties of Crystals with CRYSTAL14 and CRYSTAL17 Main Tensors related to Crystal Strain Elastic Fourth-rank Tensor and Related Properties Equation-of-State (EOS) Approach Piezoelectric Tensors Photoelastic and Piezo-optic Tensors DOI: /qua.24658
4 Tensors in Crystals Many properties of anisotropic crystalline materials can be represented in terms of Cartesian tensors of different ranks.
5 Tensors with CRYSTAL14
6 Tensors with CRYSTAL14 Geometry Optimization
7 Tensors with CRYSTAL14 Dielectric Response
8 Tensors with CRYSTAL14 Spectroscopic Properties
9 Tensors with CRYSTAL14 Strain-related Properties
10 New Tensors in CRYSTAL17 - Piezo-optic Tensor (fourth-rank) - Force-responce Internal-strain tensor (second-rank) (mixed Hessian) - Second Harmonic Generation and Pockel's Tensors (third-rank) (electric field frequency-dependent first-hyper-polarizability tensor)
11 Outline Tensorial Properties of Crystals with CRYSTAL14 and CRYSTAL17 Main Tensors related to Crystal Strain Elastic Fourth-rank Tensor and Related Properties Equation-of-State (EOS) Approach Piezoelectric Tensors Photoelastic and Piezo-optic Tensors DOI: /qua.24658
12 Crystal Strain Several properties of crystals can be computed by finite differences over strained configurations. The second-rank strain tensor e can be decomposed into two parts: An antisymmetric part w that corresponds to a pure rotation A symmetric part ε that corresponds to a pure strain Then, if we are interested only in the pure strain part of the transformation, we can refer to the following pure strain tensor:
13 Crystal Strain When the strain tensor acts on the lattice parameters, they are transformed as: As only six components of the symmetric matrix are independent, a more compact oneindex notation can be adopted, according to Voigt s proposal. In this convention the strain tensor becomes: 1 = xx 2 = yy 3 = zz 4 = yz 5 = xz 6 = xy i,j,k = 1,...,3 Cartesian indices v,u = 1,...,6 Voigt's indices
14 Crystal Strain-related Properties Elastic Tensor 4 Piezoelectric Tensor 3 Photoelastic Tensor 4 Order of the Tensors Second derivatives of the total energy E with respect to a pair of strains, for a 3D crystal First derivative of the polarization P (computed through the Berry phase approach) with respect to the strain First derivative of the inverse dielectric tensor (difference with respect to the unstrained configuration) with respect to strain
15 Crystal Strain-related Properties Elastic Tensor Geometry definition ELASTCON [Optional keywords] END END Basis set definition END Comput. Parameters END Piezoelectric Tensor Photoelastic Tensor Geometry definition PIEZOCON [Optional keywords] END END Basis set definition END Comput. Parameters END Geometry definition PHOTOELA [Optional keywords] END END Basis set definition END Comput. Parameters END
16 Outline Tensorial Properties of Crystals with CRYSTAL14 and CRYSTAL17 Main Tensors related to Crystal Strain Elastic Fourth-rank Tensor and Related Properties Equation-of-State (EOS) Approach Piezoelectric Tensors Photoelastic and Piezo-optic Tensors DOI: /qua.24658
17 Hooke's Law Stress-strain relation: 81 components 36 as both stress and strain tensors are symmetric 21 as C itself is symmetric (second energy derivatives wrt strain)
18 Efect of Symmetry Fourth Rank Elastic Tensor Triclinic Cubic Hexagonal J. F. Nye, Oxford University Press, (1985)
19 Electronic and Nuclear Terms Strain-induced tensorial properties of solids can be formally decomposed into a purely electronic clamped-nuclei term and into a nuclear-relaxation term due to the rearrangment of atomic positions upon strain. The evaluation of the latter is generally much more computationally expensive than that of the former and can be achieved following two alternative approaches: (i) performing numerical geometry optimizations to relax atomic positions at actual strained lattice configurations; (ii) evaluating in a more analytical fashion the internal-strain tensor of energy secondderivatives with respect to atomic displacements and lattice deformations, as combined with the interatomic force constant Hessian matrix.
20 Elastic Tensor: Numerical Approach
21 Elastic Tensor: Numerical Approach
22 Elastic Tensor: Numerical Approach
23 Independent Strains C11 C12 = C13 First deformation Cubic Lattice
24 Independent Strains C11 C12 = C13 First deformation C22 = C33 and C21 = C13 Cubic Lattice
25 Independent Strains C11 C12 = C13 First deformation C22 = C33 and C21 = C13 Fourth deformation C44 Cubic Lattice
26 Independent Strains C11 C12 = C13 First deformation C22 = C33 and C21 = C13 Fourth deformation C44 C66 = C55 = C44 Cubic Lattice
27 Elastic Tensor: Numerical Approach
28 Elastic Tensor: Internal-strain Approach The force-responce internal-strain tensor is defined as (second energy derivative wrt an atomic Cartesian displacement and a Cartesian lattice vector deformation): Let us express it in terms of the strain tensor components: By adopting Voigt's notation:
29 Elastic Tensor: Internal-strain Approach Let us introduce the displacement-responce internal-strain tensor, which describes firstorder atomic displacements as induced by a first-order strain: which combines the force-response internal-strain tensor with the interatomic forceconstant Hessian matrix:
30 Elastic Tensor: Internal-strain Approach The elastic tensor can be written as: Electronic Nuclear The diagonalization of the mass-weighted interatomic force-constant matrix allows for a physically meaningful partition of the nuclear relaxation contribution in terms of phonon normal modes p=1,...,3n: Vibrational frequencies
31 Elastic Tensor: Internal-strain Approach
32 Elastic Tensor: Internal-strain Approach
33 Geometry Optimizer Vs. Internal Strain ZnO Parallel mode on 16 CPUs Convergence tolerance of the geometry optimizer on the gradient
34 Elastic Tensor: The Evolution of the Implementation CRYSTAL03 and CRYSTAL06 CRYSTAL09 CRYSTAL14 CRYSTAL17 Many separate SCF calculations at strained configurations Results of different runs analysed by a script Elastic constants as numerical energy second-derivatives Geometry optimizations for nuclear-relaxed terms Fully-automated implementation (single calculation) Elastic constants as analytical gradients finite differences Generalization to 1D and 2D systems Elastic tensor under pressure via analytical stress-tensor Nuclear-relaxation term from analytical internal-strain tensor
35 Silicate Garnets X3Y2(SiO4)3 Silicate garnets are among the most important rockforming minerals of the Earth s lower crust, upper mantle and transition zone. They belong to the cubic space group Ia3d. Pyralspite Mg3Al2(SiO4)3 Pyrope Fe3Al2(SiO4)3 Mn3Al2(SiO4)3 Almandine Spessartine Ugrandite Ca3Al2(SiO4)3 Ca3Fe2(SiO4)3 Ca3Cr2(SiO4)3 Grossular Andradite Uvarovite 80 percell) cell (80atoms atoms per
36 Elastic Constants at P = 0
37 Elastic Constants at P = 0 EXPERIMENTAL THEORETICAL THIS STUDY
38 Elastic Constants at P = 0 EXPERIMENTAL THEORETICAL THIS STUDY
39 Elastic Constants at P = 0 EXPERIMENTAL THEORETICAL THIS STUDY
40 Elastic Constants at P = 0 EXPERIMENTAL THEORETICAL THIS STUDY
41 Elastic Properties at P = 0 Many elastic properties of isotropic polycrystalline aggregates can be computed from the elastic (C) and compliance (S=C -1) constants via the Voigt-Reuss-Hill averaging scheme (expressions for cubic crystals): Bulk modulus Shear modulus Young modulus Poisson's ratio Anisotropy index The average values of transverse (shear), vs, and longitudinal, vp, seismic wave velocities, for isotropic polycrystalline aggregates, can be computed as:
42 Elastic Properties at P = 0 Pyrope Almandine Spessartine Grossular Andradite Uvarovite
43 Seismic Wave Velocities at P = 0 From the elastic constants, through Christoffel's equation, seismic wave velocities can be computed: Vp Andradite Vs2 Vs1 Experimental data from single-crystal Brillouin scattering measurements by Jiang et al., J. Phys.: Condens. Matter, 16, S1041 (2004)
44 Seismic Wave Velocities at P = 0 Pyrope Almandine Spessartine Grossular Andradite Uvarovite
45 Seismic Wave Velocities at P = 0 Pyrope Almandine Elastic Anisotropy Spessartine Grossular Seismic wave velocity Andradite Uvarovite
46 Seismic Wave Velocities (AWESoMe) The AWESoMe program has been merged into CRYSTAL17 and can now be used as a keyword (AWESOME) of the ELASTCON input block. The code computes the phase and group velocities for all the possible propagation directions, as well as some related parameters such as the polarization vectors, the power flow angle and the enhancement factor. D. Munoz-Santiburcio, A. Hernandez-Laguna and J.I. Soto, Comp. Phys. Commun., 192, (2015)
47 Seismic Wave Velocities (AWESoMe) The AWESoMe program has been merged into CRYSTAL17 and can now be used as a keyword (AWESOME) of the ELASTCON input block. D. Munoz-Santiburcio, A. Hernandez-Laguna and J.I. Soto, Comp. Phys. Commun., 192, (2015)
48 Towards Geophysical Conditions 2900 km depth In The Earth's Mantle: Pressures up to ~140 GPa Temperatures between ~800 K and 1200 K
49 Towards Geophysical Conditions Ab initio characterization of structural and elastic properties of minerals at geophysical conditions. Compositional models for the Earth's deep interior are based on seismic information obtained by analysing earthquakes. In order to correctly interpret seismological data, the elastic response properties of the constituents of the mantle have to be fully characterized in terms of: - elastic constants - bulk modulus - seismic wave propagation velocity - elastic anisotropy - P,T dependence
50 Elastic Tensor under Pressure Elastic constants computed at V(P) Eulerian strain correction The P-V relation must be determined (EoS approach, analytical stress tensor approach)
51 P-V Relation An analytical approach is used to determine the P-V relation that is based on the stress tensor: exprerssed in terms of analytical cell gradients, wrt deformed lattice parameters: By adding a pre-stress in the form of a hydrostatic pressure the expression of the pressure-constrained cell gradients is obtained:
52 P-V Relation
53 Elastic Constants at P > 0 A. Erba, A. Mahmoud, D. Belmonte and R. Dovesi, J. Chem. Phys., 140, (2014)
54 Directional Seismic Velocities under Pressure
55 Directional Seismic Velocities under Pressure Forsterite Mg2SiO4
56 Outline Tensorial Properties of Crystals with CRYSTAL14 and CRYSTAL17 Main Tensors related to Crystal Strain Elastic Fourth-rank Tensor and Related Properties Equation-of-State (EOS) Approach Piezoelectric Tensors Photoelastic and Piezo-optic Tensors DOI: /qua.24658
57 Equation of State It is a pressure-volume (or energy-volume) relation describing the behavior of a solid under compression or expansion. It is given an analytical (approximated) expression. Many functional forms have been proposed. Universal expressions are not specifically parametrized for a given system or family of systems. It embodies information on the bulk modulus and its dependence on pressure. In Solid State Physics and Chemistry, it is commonly used to extrapolate at high pressure from low pressure data. In Solid State Quantum Chemistry, E-V data are usually fitted to get the P-V relation.
58 Equation of State at Work The energy of the system is computed at different volumes (compression and expansion). For each volume, the most stable configuration is found by performing a V-constrained optimization. Energy Volume
59 Equation of State at Work The energy of the system is computed at different volumes (compression and expansion). For each volume, the most stable configuration is found by performing a V-constrained optimization. Energy E-V points are fitted to a particular EOS Murnaghan, order 3 (1944): Volume
60 Bulk Modulus Energy Volume K >> K <<
61 Equation of State at Work The energy of the system is computed at different volumes (compression and expansion). For each volume, the most stable configuration is found by performing a V-constrained optimization. Energy E-V points are fitted to a particular EOS Murnaghan, order 3 (1944): By taking the derivative wrt the volume, one gets the pressure: Volume
62 Equation of State at Work The energy of the system is computed at different volumes (compression and expansion). For each volume, the most stable configuration is found by performing a V-constrained optimization. Energy E-V points are fitted to a particular EOS Murnaghan, order 3 (1944): By taking the derivative wrt the volume, one gets the pressure: Commonly used to extrapolate at high pressure from (few) low-p data Volume
63 Equation of State at Work The energy of the system is computed at different volumes (compression and expansion). For each volume, the most stable configuration is found by performing a V-constrained optimization. Energy E-V points are fitted to a particular EOS Murnaghan, order 3 (1944): By taking the derivative wrt the volume, one gets the pressure: Commonly used to extrapolate at high pressure from (few) low-p data Volume
64 Equation of State at Work Vinet exponential EOS: T The exponential "universal" Vinet's equation of state, published in 1987, reads:
65 Equation of State at Work Poirier-Tarantola logarithmic EOS: The third-order Poirier-Tarantola logarithmic equation of state (derived from the natural strain), proposed in 1998, is:
66 Equation of State at Work In CRYSTAL, one can compute the EOS with a fully-automated procedure: In the output:
67 Equation of State at Work
68 Outline Tensorial Properties of Crystals with CRYSTAL14 and CRYSTAL17 Main Tensors related to Crystal Strain Elastic Fourth-rank Tensor and Related Properties Equation-of-State (EOS) Approach Piezoelectric Tensors Photoelastic and Piezo-optic Tensors
69 Piezoelectric Efects
70 Efect of Symmetry Third Rank Piezoelectric Tensor Triclinic Hexagonal Cubic J. F. Nye, Oxford University Press, (1985)
71 Piezoelectric Tensors in CRYSTAL In the CRYSTAL program, the direct piezoelectric tensor is explicitly computed, whereas the converse piezoelectric tensor is obtained by combining the direct one with the elastic compliance tensor: In CRYSTAL, there are two possible approaches to the calculation of the direct piezoelectric tensor: NUMERICAL (from CRYSTAL06) (QUASI)ANALYTICAL (from CRYSTAL17) Berry-phase approach for electronic term CPHF/KS approach for electronic term Geometry optimizations for nuclear-relaxation term Internal-strain tensor approach for nuclear-relaxation term
72 Numerical Approach In CRYSTAL, the direct piezoelectric effect is computed: In the theory of polarization of 3D crystals, the Berry Phase along the l-th crystallographic axis, for a given strain state of the system, is given by: So that the polarization (by inverting the previous expression) reads: and the piezoelectric constants can be expressed as:
73 Numerical Approach: The Algorithm Geometry optimization and calculation of the cell gradients of the reference structure Berry phase calculation Full symmetry analysis and definition of minimal set of strains Application of each strain, geometry optimization of atomic positions and calculation of cell gradients and Berry phase for different strain amplitudes Piezoelectric constants are obtained by numerical fitting with respect to the strain
74 Analytical Approach In CRYSTAL17 we have recently developed an analytical approach to the calculation of the direct piezoelectric tensor of 3D periodic systems, based on the Coupled-Perturbed-Hartree-Fock/Kohn-Sham (CPHF/KS) scheme and on the internal-strain tensor: where Electronic Nuclear
75 Analytical Approach In CRYSTAL17 we have recently developed an analytical approach to the calculation of the direct piezoelectric tensor of 3D periodic systems, based on the Coupled-Perturbed-Hartree-Fock/Kohn-Sham (CPHF/KS) scheme and on the internal-strain tensor: where Electronic Nuclear
76 Numerical vs. Analytical Approach We already discussed the comparison for the nuclear-relaxation term: ZnO Parallel mode on 16 CPUs Convergence tolerance of the geometry optimizer on the gradient
77 Numerical vs. Analytical Approach Now for the electronic term:
78 Numerical vs. Analytical Approach Now for the electronic term:
79 Piezoelectricity of 1D and 2D Systems The numerical approach to the evaluation of the piezoelectric response has also been generalized to low-dimensional 1D and 2D systems. Induced piezoelectricity in Graphene: The in-plane response is dominated by the electronic term and tends to a common limit value as the defect concentration decreases Kh. E. El-Kelany, Ph. Carbonnière, A. Erba and M. Rérat, J. Phys. Chem. C 119, (2015)
80 Piezoelectricity of 1D and 2D Systems The numerical approach to the evaluation of the piezoelectric response has also been generalized to low-dimensional 1D and 2D systems. Induced piezoelectricity in Graphene: The out-of-plane response is dominated by the nuclear term, tends to zero as the defect concentration decreases and is large when a soft IR-active phonon mode is found. Kh. E. El-Kelany, Ph. Carbonnière, A. Erba, J.-M. Sotiropoulos and M. Rérat, J. Phys. Chem. C, 120, (2016)
81 Outline Tensorial Properties of Crystals with CRYSTAL14 and CRYSTAL17 Main Tensors related to Crystal Strain Elastic Fourth-rank Tensor and Related Properties Equation-of-State (EOS) Approach Piezoelectric Tensors Photoelastic and Piezo-optic Tensors
82 Photo-elastic and Piezo-optic Tensors With the CRYSTAL program, the photoelastic tensor is computed numerically by evaluating the dielectric tensor (as computed via the CPHF/KS approach) at different strained configurations: The piezo-optic tensor can then be computed by combining the photo-elastic one with the elastic compliance one:
83 Photoelastic Tensor: The Algorithm Geometry optimization and calculation of the cell gradients of the reference structure Dielectric tensor calculation through CPHF/KS Full symmetry analysis and definition of minimal set of strains Application of each strain, relaxation of atomic positions and calculation of cell gradients and the dielectric tensor for different strain amplitudes Photoelastic constants are obtained by numerical fitting with respect to the strain
84 Photoelastic Tensor: Validation We have computed the photoelastic properties of several simple crystals for which accurate experimental data exist: MgO, NaCL, LiF, KCl, Silicon, Diamond, Quartz, Rutile.
85 Photoelastic Tensor: Frequency dependence We have computed the photoelastic properties of several simple crystals for which accurate experimental data exist: MgO, NaCL, LiF, KCl, Silicon, Diamond, Quartz, Rutile.
86 Piezo-Optic Tensor 57 measurements on 16 properly cut samples To determine the 36 independent piezo-optic constants of a triclinic system
87 Piezo-Optic Tensor
88 Piezo-Optic Tensor of CaWO4
89 Piezo-Optic Tensor of PbMoO4
90 Thank you for your kind attention Alessandro Erba Dipartimento di Chimica, Università di Torino, Italy
properties Michele Catti Dipartimento di Scienza dei Materiali Università di Milano Bicocca, Italy
Elastic and piezoelectric tensorial properties Michele Catti Dipartimento di Scienza dei Materiali Università di Milano Bicocca, Italy (catti@mater.unimib.it) 1 Tensorial physical properties of crystals
More informationThermodynamics of Solids: Harmonic and Quasi-harmonic Approximations
Thermodynamics of Solids: Harmonic and Quasi-harmonic Approximations, USA, July 9-14, 2017 Alessandro Erba Dipartimento di Chimica, Università di Torino (Italy) alessandro.erba@unito.it 2017 Outline -
More informationVibrational frequencies in solids: tools and tricks
Vibrational frequencies in solids: tools and tricks Roberto Dovesi Gruppo di Chimica Teorica Università di Torino Torino, 4-9 September 2016 This morning 3 lectures: R. Dovesi Generalities on vibrations
More informationCrystal 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 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 informationMacroscopic 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 informationSolid 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 informationIntroduction to Condensed Matter Physics
Introduction to Condensed Matter Physics Elasticity M.P. Vaughan Overview Overview of elasticity Classical description of elasticity Speed of sound Strain Stress Young s modulus Shear modulus Poisson ratio
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 informationStructure and Dynamics : An Atomic View of Materials
Structure and Dynamics : An Atomic View of Materials MARTIN T. DOVE Department ofearth Sciences University of Cambridge OXFORD UNIVERSITY PRESS Contents 1 Introduction 1 1.1 Observations 1 1.1.1 Microscopic
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 informationStructural Calculations phase stability, surfaces, interfaces etc
Structural Calculations phase stability, surfaces, interfaces etc Keith Refson STFC Rutherford Appleton Laboratory September 19, 2007 Phase Equilibrium 2 Energy-Volume curves..................................................................
More informationELASTICITY AND CONSTITUTION OF THE EARTH'S INTERIOR*
JOURNAL OF GEOPHYSICAL RESEARCH VOLUME 57, NO. 2 JUNE, 1952 ELASTICITY AND CONSTITUTION OF THE EARTH'S INTERIOR* BY FRANCIS BIRCtt Harvard University, Cambridge, Massachusetts (Received January 18, 1952)
More informationPiezo-optic tensor of crystals from quantum-mechanical calculations
THE JOURNAL OF CHEMICAL PHYSICS 143, 144504 (2015) Piezo-optic tensor of crystals from quantum-mechanical calculations A. Erba, 1,a) M. T. Ruggiero, 2 T. M. Korter, 2 and R. Dovesi 1 1 Dipartimento di
More informationLecture 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 information6.730 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 informationNDT&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 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 informationLecture 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 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 informationINTRODUCTION TO THE PHYSICS OF THE EARTH S INTERIOR
INTRODUCTION TO THE PHYSICS OF THE EARTH S INTERIOR SECOND EDITION JEAN-PAULPOIRIER Institut de Physique du Globe de Paris PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building,
More informationFinite 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 informationLindgren 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 informationTensorial 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 informationCalculating anisotropic physical properties from texture data using the MTEX open source package
Calculating anisotropic physical properties from texture data using the MTEX open source package David Mainprice 1, Ralf Hielscher 2, Helmut Schaeben 3 February 10, 2011 1 Geosciences Montpellier UMR CNRS
More informationEquations of State. Tiziana Boffa Ballaran
Equations o State iziana Boa Ballaran Why EoS? he Earth s interior is divided globally into layers having distinct seismic properties Speed with which body waves travel through the Earth s interior are
More informationPressure Volume Temperature (P-V-T) Relationships and Thermo elastic Properties of Geophysical Minerals
Pressure Volume Temperature (P-V-T) Relationships and Thermo elastic Properties of Geophysical Minerals A PROPOSAL FOR Ph.D PROGRAMME BY MONIKA PANWAR UNDER THE SUPERVISION OF DR SANJAY PANWAR ASSISTANT
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 information3D 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 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 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 informationASSESSMENT OF DFT METHODS FOR SOLIDS
MSSC2009 - Ab Initio Modeling in Solid State Chemistry ASSESSMENT OF DFT METHODS FOR SOLIDS Raffaella Demichelis Università di Torino Dipartimento di Chimica IFM 1 MSSC2009 - September, 10 th 2009 Table
More information3.22 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 information16.20 HANDOUT #2 Fall, 2002 Review of General Elasticity
6.20 HANDOUT #2 Fall, 2002 Review of General Elasticity NOTATION REVIEW (e.g., for strain) Engineering Contracted Engineering Tensor Tensor ε x = ε = ε xx = ε ε y = ε 2 = ε yy = ε 22 ε z = ε 3 = ε zz =
More informationElasticity 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 informationQuasi-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 informationInternational Journal of Quantum Chemistry
International Journal of Quantum Chemistry First-principles calculation of second-order elastic constants and equations of state for Lithium Azide, LiN, and Lead Azide, Pb(N ) Journal: International Journal
More informationPHYSICAL PROPERTIES OF CRYSTALS
PHYSICAL PROPERTIES OF CRYSTALS THEIR REPRESENTATION TENSORS AND MATRICES BY By J. F. NYE, F.R.S. CLARENDON PRESS OXFORD NOTATION INTRODUCTION xiii xv PART 1. GENERAL PRINCIPLES I. THE GROUNDWORK OF CRYSTAL
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 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 informationELASTICITY (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 informationLecture 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 informationMechanics 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 informationChapter 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 informationConstitutive Equations
Constitutive quations David Roylance Department of Materials Science and ngineering Massachusetts Institute of Technology Cambridge, MA 0239 October 4, 2000 Introduction The modules on kinematics (Module
More 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 informationBasic Concepts of Strain and Tilt. Evelyn Roeloffs, USGS June 2008
Basic Concepts of Strain and Tilt Evelyn Roeloffs, USGS June 2008 1 Coordinates Right-handed coordinate system, with positions along the three axes specified by x,y,z. x,y will usually be horizontal, and
More informationStress equilibrium in southern California from Maxwell stress function models fit to both earthquake data and a quasi-static dynamic simulation
Stress equilibrium in southern California from Maxwell stress function models fit to both earthquake data and a quasi-static dynamic simulation Peter Bird Dept. of Earth, Planetary, and Space Sciences
More informationa) b) t h d e 40 N 20 N 60 E 80 E 100 E 60 E 80 E 100 E avearge number of paths in a 2 o x2 o cell
Supporting Online Material Thinning and ow of Tibetan crust constrained by seismic anisotropy Nikolai M. Shapiro, Michael H. Ritzwoller, Peter Molnar 2, and Vadim Levin 3 Department ofphysics, University
More information3.091 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 informationExercise: 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 informationSecond harmonic generation in silicon waveguides strained by silicon nitride
Second harmonic generation in silicon waveguides strained by silicon nitride M. Cazzanelli, 1 F. Bianco, 1 E. Borga, 1 G. Pucker, 2 M. Ghulinyan, 2 E. Degoli, 3 E. Luppi, 4,7 V. Véniard, 4 S. Ossicini,
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 informationTensors and Anisotropic physical properties of rocks Part I : Tensor representation of the physical properties of single crystals
Tensors and Anisotropic physical properties of rocks Part I : Tensor representation of the physical properties of single crystals David Mainprice Lab. Tectonophysique, Université Montpellier II France
More informationWe briefly discuss two examples for solving wave propagation type problems with finite differences, the acoustic and the seismic problem.
Excerpt from GEOL557 Numerical Modeling of Earth Systems by Becker and Kaus 2016 1 Wave propagation Figure 1: Finite difference discretization of the 2D acoustic problem. We briefly discuss two examples
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 informationElasticity of single-crystal aragonite by Brillouin spectroscopy
Phys Chem Minerals (2005) 32: 97 102 DOI 10.1007/s00269-005-0454-y ORIGINAL PAPERS Lin-gun Liu Æ Chien-chih Chen Æ Chung-Cherng Lin Yi-jong Yang Elasticity of single-crystal aragonite by Brillouin spectroscopy
More informationPhysics 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 informationContinuum 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 informationElastic and piezoelectric fields in substrates GaAs 001 and GaAs 111 due to a buried quantum dot
JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 10 15 MAY 2002 Elastic and piezoelectric fields in substrates GaAs 001 and GaAs 111 due to a buried quantum dot E. Pan a) Structures Technology Incorporated,
More informationUNIVERSITY 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 informationComments on the characteristics of incommensurate modulation in quartz: discussion about a neutron scattering experiment
65 Acta Cryst. (1999). A55, 65±69 Comments on the characteristics of incommensurate modulation in quartz: discussion about a neutron scattering experiment T. A. Aslanyan,² T. Shigenari* and K. Abe Department
More informationRotational Raman Spectroscopy
Rotational Raman Spectroscopy If EM radiation falls upon an atom or molecule, it may be absorbed if the energy of the radiation corresponds to the separation of two energy levels of the atoms or molecules.
More informationMECH 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 informationMcMAT 2007 Micromechanics of Materials Austin, Texas, June 3 7, 2007
McMAT 2007 Micromechanics of Materials Austin, Texas, June 3 7, 2007 RANDOM POLYCRYSTALS OF GRAINS WITH CRACKS: MODEL OF ELASTIC BEHAVIOR FOR FRACTURED SYSTEMS James G. Berryman Earth Sciences Division
More informationMATERIAL MECHANICS, SE2126 COMPUTER LAB 4 MICRO MECHANICS. E E v E E E E E v E E + + = m f f. f f
MATRIAL MCHANICS, S226 COMPUTR LAB 4 MICRO MCHANICS 2 2 2 f m f f m T m f m f f m v v + + = + PART A SPHRICAL PARTICL INCLUSION Consider a solid granular material, a so called particle composite, shown
More informationElectronic Supplementary Information
Electronic Supplementary Material (ESI) for CrystEngComm. This journal is The Royal Society of Chemistry 2014 Electronic Supplementary Information Configurational and energetical study of the (100) and
More informationQuadratic and cubic monocrystalline and polycrystalline materials: their stability and mechanical properties
Journal of Physics: Conference Series Quadratic and cubic monocrystalline and polycrystalline materials: their stability and mechanical properties To cite this article: C Jasiukiewicz et al 010 J. Phys.:
More informationEOS-FIT V6.0 R.J. ANGEL
EOS-FIT V6. R.J. AGEL Crystallography Laboratory, Dept. Geological Sciences, Virginia Tech, Blacksburg, VA46, USA http://www.geol.vt.edu/profs/rja/ ITRODUCTIO EosFit started as a program to fit equations
More informationMechanics 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 informationSupporting Information
Electronic Supplementary Material (ESI) for Nanoscale. This journal is The Royal Society of Chemistry 2015 Supporting Information Single Layer Lead Iodide: Computational Exploration of Structural, Electronic
More informationIAP 2006: From nano to macro: Introduction to atomistic modeling techniques and application in a case study of modeling fracture of copper (1.
IAP 2006: From nano to macro: Introduction to atomistic modeling techniques and application in a case study of modeling fracture of copper (1.978 PDF) http://web.mit.edu/mbuehler/www/teaching/iap2006/intro.htm
More informationElasticity, the fourth-rank tensor defining the strain of crystalline
Elasticity of MgO and a primary pressure scale to 55 GPa Chang-Sheng Zha*, Ho-kwang Mao, and Russell J. Hemley Geophysical Laboratory and Center for High Pressure Research, Carnegie Institution of Washington,
More informationResearch Article Dispersion of Love Waves in a Composite Layer Resting on Monoclinic Half-Space
Applied Mathematics Volume 011, Article ID 71349, 9 pages doi:10.1155/011/71349 Research Article Dispersion of Love Waves in a Composite Layer Resting on Monoclinic Half-Space Sukumar Saha BAS Division,
More informationAccuracy and transferability of GAP models for tungsten
Accuracy and transferability of GAP models for tungsten Wojciech J. Szlachta Albert P. Bartók Gábor Csányi Engineering Laboratory University of Cambridge 5 November 214 Motivation Number of atoms 1 1 2
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 information20. 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 informationTABLE OF CONTENTS. Mechanics of Composite Materials, Second Edition Autar K Kaw University of South Florida, Tampa, USA
Mechanics of Composite Materials, Second Edition Autar K Kaw University of South Florida, Tampa, USA TABLE OF CONTENTS 1. INTRODUCTION TO COMPOSITE MATERIALS 1.1 Introduction... 1.2 Classification... 1.2.1
More informationTinselenidene: 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 informationA NANOSCALE SIMULATION STUDY OF ELASTIC PROPERTIES OF GASPEITE
Studia Geotechnica et Mechanica, Vol. XXXVI, No., 014 DOI: 10.478/sgem-014-0015 A NANOSCALE SIMULATION STUDY OF ELASTIC PROPERTIES OF GASPEITE BRAHIM-KHALIL BENAZZOUZ Laboratoire des Fluides Complexes
More informationPart II Materials Science and Metallurgy TENSOR PROPERTIES SYNOPSIS
Part II Materials Science and Metallurgy TENSOR PROPERTIES Course C4 Dr P A Midgley 1 lectures + 1 examples class Introduction (1 1 / lectures) SYNOPSIS Reasons for using tensors. Tensor quantities and
More informationSupporting 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 informationGeometry Optimisation
Geometry Optimisation Matt Probert Condensed Matter Dynamics Group Department of Physics, University of York, UK http://www.cmt.york.ac.uk/cmd http://www.castep.org Motivation Overview of Talk Background
More informationPotentials, periodicity
Potentials, periodicity Lecture 2 1/23/18 1 Survey responses 2 Topic requests DFT (10), Molecular dynamics (7), Monte Carlo (5) Machine Learning (4), High-throughput, Databases (4) NEB, phonons, Non-equilibrium
More informationTUTORIAL 8: PHONONS, LATTICE EXPANSION, AND BAND-GAP RENORMALIZATION
TUTORIAL 8: PHONONS, LATTICE EXPANSION, AND BAND-GAP RENORMALIZATION 1 INVESTIGATED SYSTEM: Silicon, diamond structure Electronic and 0K properties see W. Huhn, Tutorial 2, Wednesday August 2 2 THE HARMONIC
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 informationPiezo materials. Actuators Sensors Generators Transducers. Piezoelectric materials may be used to produce e.g.: Piezo materials Ver1404
Noliac Group develops and manufactures piezoelectric materials based on modified lead zirconate titanate (PZT) of high quality and tailored for custom specifications. Piezoelectric materials may be used
More informationCellular solid structures with unbounded thermal expansion. Roderic Lakes. Journal of Materials Science Letters, 15, (1996).
1 Cellular solid structures with unbounded thermal expansion Roderic Lakes Journal of Materials Science Letters, 15, 475-477 (1996). Abstract Material microstructures are presented which can exhibit coefficients
More informationarxiv: v1 [physics.ins-det] 4 Jun 2018
Noname manuscript No. (will be inserted by the editor) Simulation of force-insensitive optical cavities in cubic spacers Eugen Wiens Stephan Schiller arxiv:186.2176v1 [physics.ins-det] 4 Jun 218 Received:
More informationGarnet yield strength at high pressures and implications for upper mantle and transition zone rheology
Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi:10.1029/2007jb004931, 2007 Garnet yield strength at high pressures and implications for upper mantle and transition zone rheology
More informationDYNAMIC RESPONSE OF SYNTACTIC FOAM CORE SANDWICH USING A MULTIPLE SCALES BASED ASYMPTOTIC METHOD
ECCM6-6 TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS, Seville, Spain, -6 June 4 DYNAMIC RESPONSE OF SYNTACTIC FOAM CORE SANDWICH USING A MULTIPLE SCALES BASED ASYMPTOTIC METHOD K. V. Nagendra Gopal a*,
More informationA Visualization System for Mineral Elasticity
A Visualization System for Mineral Elasticity Bijaya B. Karki and Ravi Chennamsetty Department of Computer Science, Louisiana State University, Baton Rouge, Louisiana 70803 E-mail: karki@bit.csc.lsu.edu,
More informationGeometry optimization of solids
The Minnesota Workshop on ab Initio Modeling in Solid State Chemistry with CRYSTAL Minneapolis, MN(U.S.A.) 9-14 July 2017 7 1 Geometry optimization of solids Bartolomeo Civalleri Dip. di Chimica IFM, Via
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 informationConstitutive Relations
Constitutive Relations Dr. Andri Andriyana Centre de Mise en Forme des Matériaux, CEMEF UMR CNRS 7635 École des Mines de Paris, 06904 Sophia Antipolis, France Spring, 2008 Outline Outline 1 Review of field
More informationLecture 11 - Phonons II - Thermal Prop. Continued
Phonons II - hermal Properties - Continued (Kittel Ch. 5) Low High Outline Anharmonicity Crucial for hermal expansion other changes with pressure temperature Gruneisen Constant hermal Heat ransport Phonon
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 informationThe tensors useful for geophysics and estimation of anisotropic polycrystalline physical properties
The tensors useful for geophysics and estimation of anisotropic polycrystalline physical properties David Mainprice Laboratoire de Tectonophysique, ISTEEM, CNRS UMR 5558, Université Montpellier II, 34095
More informationEE 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 informationResource reading (If you would like to brush up on stress, strain, and elasticity):
12.108 Struct. & Prop s. of Earth Mat. Lecture 9 1 Single-crystal elasticity Assigned Reading: Nye JF (1957) Physical Properties of Crystals. Oxford University Press, Oxford, UK (Chapters 5 and 6). Nye,
More information