DOCSLIB.ORG

Mechanical Properties Stress and Strain

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mechanical Properties Stress and Strain Chapter 7: Mechanical Properties Chapter 7: Mechanical Properties Why mechanical properties? ISSUES TO ADDRESS... • Need to design materials that will withstand • Stress and strain: Normalized force and displacements. applied load and in-service uses for… Engineering : σ e = Fi / A0 εe = Δl / l0 Bridges for autos and people MEMS devices True : σT = Fi / Ai εT = ln(lf / l0 ) • Elastic behavior: When loads are small. Young 's Modulus : E [GPa] Canyon Bridge, Los Alamos, NM • Plastic behavior: dislocations and permanent deformation skyscrapers Yield Strength : σYS [MPa] (permanent deformation) Ulitmate Tensile Strength : σ [MPa] (fracture) TS • Toughness, ductility, resilience, toughness, and hardness: Define and how do we measure? Space exploration Space elevator? • Mechanical behavior of the various classes of materials. 1 2 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 Stress and Strain Pure Tension Pure Compression F stress σ = normal e A Stress: Force per unit area arising from applied load. o l − lo strain εe = Tension, compression, shear, torsion or any combination. lo σ = Eε Elastic e Stress = σ = force/area response F Pure Shear stress τ = shear e A Strain: physical deformation response of a o material to stress, e.g., elongation. strain γ = tanθ Elastic τ = Gγ e response Pure Torsional Shear 3 4 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 1 Engineering Stress Common States of Stress • Tensile stress, s: • Shear stress, t: • Simple tension: cable F σ = Ao Ski lift (photo courtesy P.M. Anderson) F • Simple shear: drive shaft σ = t Ao original area Fs = before loading Stress has units: N/m2 (or lb/in2 ) τ Ao Note: τ = M/AcR here. 5 6 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 Common States of Stress Common States of Stress • Simple compression: • Bi-axial tension: • Hydrostatic compression: Ao Canyon Bridge, Los Alamos, NM (photo courtesy P.M. Anderson) Pressurized tank Fish under water (photo courtesy Note: compressive (photo courtesy P.M. Anderson) Balanced Rock, Arches structure member (σ < 0). P.M. Anderson) National Park σθ > 0 (photo courtesy P.M. Anderson) σz > 0 σ h < 0 7 8 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 2 Engineering Strain Elastic Deformation • Tensile strain: • Lateral (width) strain: 1. Initial 2. Small load 3. Unload δ/2 bonds Lo wo stretch δ/2 return to δL/2 δL/2 initial • Shear strain: δ θ/2 F F Linear- γ = tan θ Strain is always elastic π/2 - θ dimensionless. Elastic means reversible! Non-Linear- elastic π/2 θ/2 δ 9 10 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 Plastic Deformation of Metals Strain Testing 1. Initial 2. Small load 3. Unload • Tensile specimen • Tensile test machine bonds planes stretch Often 12.8 mm x 60 mm & planes still sheared Adapted from Fig. 7.2, shear Callister & Rethwisch 3e. extensometer specimen δ δelastic + plastic plastic gauge F length F linear linear • Other types: Plastic means permanent! elastic elastic δ -compression: brittle materials (e.g., concrete) δplastic -torsion: cylindrical tubes, shafts. δelastic 11 12 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 3 Linear Elasticity Example: Hooke’s Law • Hooke's Law: σ = E ε (linear elastic behavior) • Modulus of Elasticity, E: Units: E [GPa] or [psi] (also known as Young's modulus) Copper sample (305 mm long) is pulled in tension with stress of 276 MPa. If deformation is elastic, what is elongation? • Hooke's Law: σ = E ε For Cu, E = 110 GPa. σ Axial strain Axial strain Δl σ l0 E σ = Eε = E ⇒ Δl = l0 E ε (276MPa)(305mm) Δl = = 0.77mm Linear- Width strain 110x103 MPa Width strain elastic Hooke’s law involves axial (parallel to applied tensile load) elastic deformation. 13 14 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 Elastic Deformation Mechanical Properties 1. Initial 2. Small load 3. Unload • Recall: Bonding Energy vs distance plots bonds stretch return to initial δ F F Linear- elastic tension Elastic means reversible! Non-Linear- compression elastic Adapted from Fig. 2.8 δ Callister & Rethwisch 3e. 15 16 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 4 Mechanical Properties Elasticity of Ceramics • Recall: Slope of stress strain plot (proportional to the E) • Elastic Behavior And Effects of Porosity 2 depends on bond strength of metal E= E0(1 - 1.9P + 0.9 P ) E larger E smaller Al2O3 Adapted from Fig. 7.7, Callister & Rethwisch 3e. Neither Glass or Alumina experience plastic deformation before fracture! 17 18 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 Comparison of Elastic Moduli Polymers: Tangent and Secant Modulus • Tangent Modulus is experienced in service. • Secant Modulus is effective modulus at 2% strain. - grey cast iron is also an example • Modulus of polymer changes with time and strain-rate. - must report strain-rate dε/dt for polymers. - must report fracture strain εf before fracture. Silicon (single xtal) 120-190 (depends on crystallographic direction) initial E Glass (pyrex) 70 Stress (MPa) SiC (fused or sintered) 207-483 secant E Graphite (molded) ~12 tangent E High modulus C-fiber 400 Carbon Nanotubes ~1000 Normalize by density, 20x steel wire. strength normalized by density is 56x wire. %strain 1 2 3 4 5 ….. 19 20 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 5 Young’s Modulus, E Poisson's ratio, ν Graphite Metals Composites Ceramics Polymers Alloys /fibers Semicond • Poisson's ratio, ν: Units: ν dimensionless 1200 1000 800 Diamond 600 Si carbide width strain Δw / w εL 400 Tungsten Al oxide Carbon fibers only Molybdenum Si nitride ν = − = − = − Steel, Ni CFRE(|| fibers)* E(GPa) 200 Tantalum <111> axial strain Δ / ε ε Platinum Si crystal l l L Cu alloys <100> Aramid fibers only 100 Zinc, Ti 80 Silver, Gold Glass-soda AFRE(|| fibers)* 60 Aluminum Glass fibers only Magnesium, 40 Tin GFRE(|| fibers)* Concrete Axial strain 9 20 GFRE* 10 Pa CFRE* GFRE( fibers)* 10 Graphite ε 8 CFRE( fibers)* 6 AFRE( fibers)* Based on data in Table B2, Callister 6e. Polyester 4 PET Composite data based on metals: ν ~ 0.33 PS reinforced epoxy with 60 vol% - Epoxy only ν 2 PC of aligned carbon (CFRE), ceramics: ν ~ 0.25 PP aramid (AFRE), or glass (GFRE) fibers. 1 HDPE 0.8 Width strain polymers: ν ~ 0.40 0.6 Wood( grain) PTFE 0.4 Why does ν have minus sign? 0.2 LDPE 21 22 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 Limits of the Poisson Ratio Poisson Ratio: materials specific • Poisson Ratio has a range –1 ≤ ν ≤ 1/2 Metals: Ir W Ni Cu Al Ag Au 0.26 0.29 0.31 0.34 0.34 0.38 0.42 Look at extremes generic value ~ 1/3 • No change in aspect ratio: Δ w /w = Δl /l Solid Argon: 0.25 Δw / w ν = − = −1 Covalent Solids: Si Ge Al2O3 TiC Δl / l 0.27 0.28 0.23 0.19 generic value ~ 1/4 € • Volume (V = AL) remains constant: ΔV =0. Ionic Solids: MgO 0.19 Hence, ΔV = (L ΔA+A ΔL) = 0. So, ΔA / A = −ΔL /L Silica Glass: 0.20 In terms of width, A = w2, then ΔA/A = 2 w Δw/w2 = 2Δw/w = –ΔL/L. Polymers: Network (Bakelite) 0.49 Chain (PE) 0.40 ~generic value Δw / w (− 1 Δ / ) Elastomer: Hard Rubber (Ebonite) 0.39 (Natural) 0.49 Hence, ν = − = − 2 l l = 1/ 2 Incompressible solid. Δ / €Δ / Water (almost). l l l l 23 24 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 MatSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004/2006-2008 6 Example: Poisson Effect Other Elastic Properties Tensile stress is applied along cylindrical brass rod (10 mm M diameter). Poisson ratio is ν = 0.34 and E = 97 GPa. • Elastic Shear τ G simple –3 modulus, G: • Determine load needed for 2.5x10 mm change in diameter if 1 γ Torsion test the deformation is entirely elastic? τ = G γ Width strain: (note reduction in diameter) M • Elastic Bulk ε = Δd/d = –(2.5x10–3 mm)/(10 mm) = –2.5x10–4 P x modulus, K: Axial strain: Given Poisson ratio P P Pressure test: ε = –ε /ν = –(–2.5x10–4)/0.34 = +7.35x10–4 z x Init. vol = Vo. Vol chg. = ΔV 3 –4 • Special relations for isotropic materials: Axial Stress: σz = Eεz = (97x10 MPa)(7.35x10 ) = 71.3 MPa. E E So, only 2 independent elastic Required Load: F = σ A = (71.3 MPa)π(5 mm)2 = 5600 N. G = K = z 0 2(1 ) 3(1 2 ) constants for isotropic media + ν − ν 25 26 MatSE 280: Introduction to Engineering Materials ©D.D.
Load more

Recommended publications
  • 10-1 CHAPTER 10 DEFORMATION 10.1 Stress-Strain Diagrams And
    EN380 Naval Materials Science and Engineering Course Notes, U.S. Naval Academy CHAPTER 10 DEFORMATION 10.1 Stress-Strain Diagrams and Material Behavior 10.2 Material Characteristics 10.3 Elastic-Plastic Response of Metals 10.4 True stress and strain measures 10.5 Yielding of a Ductile Metal under a General Stress State - Mises Yield Condition. 10.6 Maximum shear stress condition 10.7 Creep Consider the bar in figure 1 subjected to a simple tension loading F. Figure 1: Bar in Tension Engineering Stress () is the quotient of load (F) and area (A). The units of stress are normally pounds per square inch (psi). = F A where: is the stress (psi) F is the force that is loading the object (lb) A is the cross sectional area of the object (in2) When stress is applied to a material, the material will deform. Elongation is defined as the difference between loaded and unloaded length ∆푙 = L - Lo where: ∆푙 is the elongation (ft) L is the loaded length of the cable (ft) Lo is the unloaded (original) length of the cable (ft) 10-1 EN380 Naval Materials Science and Engineering Course Notes, U.S. Naval Academy Strain is the concept used to compare the elongation of a material to its original, undeformed length. Strain () is the quotient of elongation (e) and original length (L0). Engineering Strain has no units but is often given the units of in/in or ft/ft. ∆푙 휀 = 퐿 where: is the strain in the cable (ft/ft) ∆푙 is the elongation (ft) Lo is the unloaded (original) length of the cable (ft) Example Find the strain in a 75 foot cable experiencing an elongation of one inch.
    [Show full text]
  • Transportation Vehicle Light-Weighting with Polymeric Glazing and Mouldings
    GCEP Final Report for Advanced Transportation Transportation Vehicle Light-Weighting with Polymeric Glazing and Mouldings Investigators Reinhold H. Dauskardt, Professor; Yichuan Ding, Graduate Researcher; Siming Dong, Graduate Researcher; Dongjing He, Summer student; Material Science and Engineering, Stanford University. Abstract Polymeric glazing and mouldings are an extremely high “want” from the transportation community, enabling more creative designs as well as improved part consolidation。 However, plastic windows and mouldings must have high-performance and low-cost protective coating systems with lifetimes in excess of 10 years. Current polymeric glazings do not meet durability/performance requirements for near-term implementation. Our project targets new coating system manufacturing to address durability and cost issues necessary to meet or exceed transportation engineering requirements. Atmospheric plasma deposition (APD) is an emerging technique that enables plasma deposition of coatings on large and/or complex geometry substrates in ambient air without the need for expensive vacuum or inert manufacturing platforms. It is an environmentally friendly and solvent-free technique, minimizing chemical waste throughout the process as well as greenhouse gas emissions when compared to current wet chemistry aqueous sol-gel manufacturing techniques. Low deposition temperatures (<50°C) allows the deposition on plastic and organic substrates. Using our state-of-the-art APD coating capabilities, we have demonstrated the ability to deposit highly transparent bilayer organosilicate coatings with superior combinations of elastic modulus and adhesion compared to commercial sol-gel coatings. The bilayer is deposited on large substrates by atmospheric plasma, in ambient air, at room temperature, in a one-step process, using a single inexpensive precursor. The significantly improved elastic modulus translates into improved durability and resistance to scratching and environmental degradation.
    [Show full text]
  • Atwood's Machine? (5 Points)
    rev 09/2019 Atwood’s Machine Equipment Qty Equipment Part Number 1 Mass and Hanger Set ME‐8979 1 Photogate with Pully ME‐6838A 1 Universal Table Clamp ME‐9376B 1 Large Rod ME‐8736 1 Small Rod ME‐8977 1 Double rod Clamp ME‐9873 1 String Background Newton’s 2nd Law (NSL) states that the acceleration a mass experiences is proportional to the net force applied to it, and inversely proportional to its inertial mass ( ). An Atwood’s Machine is a simple device consisting of a pulley, with two masses connected by a string that runs over the pulley. For an ‘ideal Atwood’s Machine’ we assume the pulley is massless, and frictionless, that the string is unstretchable, therefore a constant length, and also massless. Consider the following diagram of an ideal Atwood’s machine. One of the standard ways to apply NSL is to draw Free Body Diagrams for the masses in the system, then write Force Summation Equations for each Free Body Diagram. We will use the standard practice of labeling masses from smallest to largest, therefore m2 > m1. For an Atwood’s Machine there are only forces acting on the masses in the vertical direction so we will only need to write Force Summation Equations for the y‐direction. We obtain the following Free Body Diagrams for the two masses. Each of the masses have two forces acting on it. Each has its own weight (m1g, or m2g) pointing downwards, and each has the tension (T) in the string pointing upwards. By the assumption of an ideal string the tension is the same throughout the string.
    [Show full text]
  • Impulse and Momentum
    Impulse and Momentum All particles with mass experience the effects of impulse and momentum. Momentum and inertia are similar concepts that describe an objects motion, however inertia describes an objects resistance to change in its velocity, and momentum refers to the magnitude and direction of it's motion. Momentum is an important parameter to consider in many situations such as braking in a car or playing a game of billiards. An object can experience both linear momentum and angular momentum. The nature of linear momentum will be explored in this module. This section will discuss momentum and impulse and the interconnection between them. We will explore how energy lost in an impact is accounted for and the relationship of momentum to collisions between two bodies. This section aims to provide a better understanding of the fundamental concept of momentum. Understanding Momentum Any body that is in motion has momentum. A force acting on a body will change its momentum. The momentum of a particle is defined as the product of the mass multiplied by the velocity of the motion. Let the variable represent momentum. ... Eq. (1) The Principle of Momentum Recall Newton's second law of motion. ... Eq. (2) This can be rewritten with accelleration as the derivate of velocity with respect to time. ... Eq. (3) If this is integrated from time to ... Eq. (4) Moving the initial momentum to the other side of the equation yields ... Eq. (5) Here, the integral in the equation is the impulse of the system; it is the force acting on the mass over a period of time to .
    [Show full text]
  • Correlation of Hardness Values to Tensile Strength
    Correlation of Hardness Values to Tensile Strength Semih Genculu, P.E. Various procedures and approaches are utilized to determine if a given material is suitable for a certain application. The material may be tested for its ability to deform satisfactorily during a forming operation, or perhaps for its ability to operate under a certain stress level at high temperatures. For technological purposes, economy and ease of testing are important factors. Hardness tests: In many cases it is possible to substitute for the relatively time consuming and costly tensile test with a more convenient test of the plastic deformation behavior of metals, a hardness test. Hardness is defined as resistance of a material to penetration of its surface, and the majority of commercial hardness testers force a small penetrator (indenter) into the metal by means of an applied load. A definite value is obtained as the hardness of the metal, and this number can be related to the tensile strength of the metal. In the Rockwell test, hardness is measured by the depth to which the penetrator moves under a fixed load. The elastic component of the deformation is subtracted from the total movement. In the Brinell and Vickers/Knoop scales, on the other hand, the hardness is measured by dividing the load by the area of an indentation formed by pressing the corresponding indenters into the metal. Therefore while the Rockwell number is read directly from a gage, which is part of the tester, the Brinell and Vickers/Knoop require optical measurements of the diameters or diagonals, respectively. While all indentation hardness tests may be thought to serve the same purpose, each one has definite advantages with some being more applicable to certain types of materials and size and shape parts than the others.
    [Show full text]
  • SMALL DEFORMATION RHEOLOGY for CHARACTERIZATION of ANHYDROUS MILK FAT/RAPESEED OIL SAMPLES STINE RØNHOLT1,3*, KELL MORTENSEN2 and JES C
    bs_bs_banner A journal to advance the fundamental understanding of food texture and sensory perception Journal of Texture Studies ISSN 1745-4603 SMALL DEFORMATION RHEOLOGY FOR CHARACTERIZATION OF ANHYDROUS MILK FAT/RAPESEED OIL SAMPLES STINE RØNHOLT1,3*, KELL MORTENSEN2 and JES C. KNUDSEN1 1Department of Food Science, University of Copenhagen, Rolighedsvej 30, DK-1958 Frederiksberg C, Denmark 2Niels Bohr Institute, University of Copenhagen, Copenhagen Ø, Denmark KEYWORDS ABSTRACT Method optimization, milk fat, physical properties, rapeseed oil, rheology, structural Samples of anhydrous milk fat and rapeseed oil were characterized by small analysis, texture evaluation amplitude oscillatory shear rheology using nine different instrumental geometri- cal combinations to monitor elastic modulus (G′) and relative deformation 3 + Corresponding author. TEL: ( 45)-2398-3044; (strain) at fracture. First, G′ was continuously recorded during crystallization in a FAX: (+45)-3533-3190; EMAIL: fluted cup at 5C. Second, crystallization of the blends occurred for 24 h, at 5C, in [email protected] *Present Address: Department of Pharmacy, external containers. Samples were gently cut into disks or filled in the rheometer University of Copenhagen, Universitetsparken prior to analysis. Among the geometries tested, corrugated parallel plates with top 2, 2100 Copenhagen Ø, Denmark. and bottom temperature control are most suitable due to reproducibility and dependence on shear and strain. Similar levels for G′ were obtained for samples Received for Publication May 14, 2013 measured with parallel plate setup and identical samples crystallized in situ in the Accepted for Publication August 5, 2013 geometry. Samples measured with other geometries have G′ orders of magnitude lower than identical samples crystallized in situ.
    [Show full text]
  • What Is Hooke's Law? 16 February 2015, by Matt Williams
    What is Hooke's Law? 16 February 2015, by Matt Williams Like so many other devices invented over the centuries, a basic understanding of the mechanics is required before it can so widely used. In terms of springs, this means understanding the laws of elasticity, torsion and force that come into play – which together are known as Hooke's Law. Hooke's Law is a principle of physics that states that the that the force needed to extend or compress a spring by some distance is proportional to that distance. The law is named after 17th century British physicist Robert Hooke, who sought to demonstrate the relationship between the forces applied to a spring and its elasticity. He first stated the law in 1660 as a Latin anagram, and then published the solution in 1678 as ut tensio, sic vis – which translated, means "as the extension, so the force" or "the extension is proportional to the force"). This can be expressed mathematically as F= -kX, where F is the force applied to the spring (either in the form of strain or stress); X is the displacement A historical reconstruction of what Robert Hooke looked of the spring, with a negative value demonstrating like, painted in 2004 by Rita Greer. Credit: that the displacement of the spring once it is Wikipedia/Rita Greer/FAL stretched; and k is the spring constant and details just how stiff it is. Hooke's law is the first classical example of an The spring is a marvel of human engineering and explanation of elasticity – which is the property of creativity.
    [Show full text]
  • Guide to Copper Beryllium Wire Bar Tube Plate
    strip rod Guide to Copper Beryllium wire bar tube plate Brush Wellman is the leading worldwide supplier of High Performance Copper Alloys, including Copper Beryllium. We provide manufacturing excellence in the form of high reliability products and services to satisfy our customers’ most demanding applications. We provide these services in a culture of local support and global teamwork. © 2002 Brush Wellman Inc. Cleveland, Ohio Product Guide - Strip Content Alloy Guide . 3 Wrought Alloys. 4 Wrought Products . 5 Physical Properties . 6 Product Guide . 7 Strip . 8 Temper Designations . 9 Mechanical and Electrical Properties. 10 Forming . 12 Stress Relaxation . 13 Wire. 14 Rod, Bar and Tube . 16 Plate and Rolled Bar. 18 Forgings and Extrusions . 20 Drill String Products . 21 Other Products and Services . 22 Engineering Guide. 23 Heat Treatment Fundamentals. 24 Phase Diagrams . 24 Cold Work Response . 25 Age Hardening . 26 Microstructures . 29 Cleaning and Finishing. 30 Joining-Soldering, Brazing and Welding. 31 Machining. 32 Hardness . 33 Fatigue Strength . 35 Corrosion Resistance . 36 Other Attributes . 37 Your Supplier . 39 This is Brush Wellman . 40 Company History. 40 Corporate Profile. 40 Mining and Manufacturing.. 41 Product Distribution . 42 Customer Service . 43 Quality . 43 Safe Handling . 44 2 Alloy Guide Wrought Alloys 4 Wrought Products 5 Physical Properties 6 The copper beryllium alloys commonly supplied in wrought product form are highlighted in this section. Wrought products are those in which final shape is achieved by working rather than by casting. Cast alloys are described in separate Brush Wellman publications. Although the alloys in this guide are foremost in the line that has established Brush Wellman’s worldwide reputation for quality, they are not the only possibilities.
    [Show full text]
  • Elastic Properties of Fe−C and Fe−N Martensites
    Elastic properties of Fe−C and Fe−N martensites SOUISSI Maaouia a, b and NUMAKURA Hiroshi a, b * a Department of Materials Science, Osaka Prefecture University, Naka-ku, Sakai 599-8531, Japan b JST-CREST, Gobancho 7, Chiyoda-ku, Tokyo 102-0076, Japan * Corresponding author. E-mail: [email protected] Postal address: Department of Materials Science, Graduate School of Engineering, Osaka Prefecture University, Gakuen-cho 1-1, Naka-ku, Sakai 599-8531, Japan Telephone: +81 72 254 9310 Fax: +81 72 254 9912 1 / 40 SYNOPSIS Single-crystal elastic constants of bcc iron and bct Fe–C and Fe–N alloys (martensites) have been evaluated by ab initio calculations based on the density-functional theory. The energy of a strained crystal has been computed using the supercell method at several values of the strain intensity, and the stiffness coefficient has been determined from the slope of the energy versus square-of-strain relation. Some of the third-order elastic constants have also been evaluated. The absolute magnitudes of the calculated values for bcc iron are in fair agreement with experiment, including the third-order constants, although the computed elastic anisotropy is much weaker than measured. The tetragonally distorted dilute Fe–C and Fe–N alloys exhibit lower stiffness than bcc iron, particularly in the tensor component C33, while the elastic anisotropy is virtually the same. Average values of elastic moduli for polycrystalline aggregates are also computed. Young’s modulus and the rigidity modulus, as well as the bulk modulus, are decreased by about 10 % by the addition of C or N to 3.7 atomic per cent, which agrees with the experimental data for Fe–C martensite.
    [Show full text]
  • Infinitesimal Strain
    GG303 Lecture 15 8/24/03 1 FINITE STRAIN AND INFINITESIMAL STRAIN I Main Topics (on infinitesimal strain) A The finite strain tensor [E] B Deformation paths for finite strain C Infinitesimal strain and the infinitesimal strain tensor ε II The finite strain tensor [E] A Used to find the changes in the squares of lengths of line segments in a deformed body. B Definition of [E] in terms of the deformation gradient tensor [F] Recall the coordinate transformation equations: x′ abx 1 = or [] X′ = [][] F X y′ cdy dx′ abdx 2 = or [] dX′ = [][] F dX dy′ cddy dx T If = []dX, then [ dx dy ]= [] dX ; transposing a matrix is switching dy its rows and columns 222 dx TT 3 ()ds= () dx+ () dy= [] dx dy = [][]dX dX= [][][] dX I dX , dy 10 where I = is the identity matrix. 01 222 dx′ T 4 ()ds′ = () dx′ + () dy′ = [] dx′′ dy = []dX′ [] dX′ dy′ Now dX’ can be expressed as [F][dX] (see eq. II.B.2). Making this substitution into eq. (4) and proceeding with the algebra 2 T TT 5 ()ds′ = [][][] F dX[][][] F dX= [][] dX F[][] F dX 22 TT T 6 ()ds′ − () ds′ = [][] dX F[][] F dX− [][][] dX I dX 22 TT 7 ()ds′ − () ds′ = [][] dX[] F[] F− I[] dX 1 221 TT T 8 ()ds′ − () ds′ = [][] dX F[] F− I[] dX≡ [][][] dX E dX 2{}2 [] 1 T 9 []EF≡ [][]FI− = finite strain tensor 2[] Stephen Martel 15-1 University of Hawaii GG303 Lecture 15 8/24/03 2 IIIDeformation paths Consider two different finite strains described by the following two coordinate transformation equations: ′ ab ax by x1 11x 11+ = = = []FX1 [] y ′ cdy cx+ dy A 1 11 11 Deformation 1 ′ ab ax by
    [Show full text]
  • 1 Introduction to Hardness Testing
    Copyright © 1999 ASM International ® Hardness Testing, 2nd Edition, 06671G All rights reserved. Harry Chandler, editor www.asminternational.org 1 Introduction to Hardness Testing Hardness has a variety of meanings. To the metals industry, it may be thought of as resistance to permanent deformation. To the metallurgist, it means resistance to penetration. To the lubrication engineer, it means resis- tance to wear. To the design engineer, it is a measure of flow stress. To the mineralogist, it means resistance to scratching, and to the machinist, it means resistance to machining. Hardness may also be referred to as mean contact pressure. All of these characteristics are related to the plastic flow stress of materials. Measuring Hardness Hardness is indicated in a variety of ways, as indicated by the names of the tests that follow: • Static indentation tests: A ball, cone, or pyramid is forced into the sur- face of the metal being tested. The relationship of load to the area or depth of indentation is the measure of hardness, such as in Brinell, Knoop, Rockwell, and Vickers hardness tests. • Rebound tests: An object of standard mass and dimensions is bounced from the surface of the workpiece being tested, and the height of rebound is the measure of hardness. The Scleroscope and Leeb tests are exam- ples. • Scratch file tests: The idea is that one material is capable of scratching another. The Mohs and file hardness tests are examples of this type. • Plowing tests: A blunt element (usually diamond) is moved across the surface of the workpiece being tested under controlled conditions of load Copyright © 1999 ASM International ® Hardness Testing, 2nd Edition, 06671G All rights reserved.
    [Show full text]
  • Chapter 7 – Geomechanics
    Chapter 7 GEOMECHANICS GEOTECHNICAL DESIGN MANUAL January 2019 Geotechnical Design Manual GEOMECHANICS Table of Contents Section Page 7.1 Introduction ....................................................................................................... 7-1 7.2 Geotechnical Design Approach......................................................................... 7-1 7.3 Geotechnical Engineering Quality Control ........................................................ 7-2 7.4 Development Of Subsurface Profiles ................................................................ 7-2 7.5 Site Variability ................................................................................................... 7-2 7.6 Preliminary Geotechnical Subsurface Exploration............................................. 7-3 7.7 Final Geotechnical Subsurface Exploration ...................................................... 7-4 7.8 Field Data Corrections and Normalization ......................................................... 7-4 7.8.1 SPT Corrections .................................................................................... 7-4 7.8.2 CPTu Corrections .................................................................................. 7-7 7.8.3 Correlations for Relative Density From SPT and CPTu ....................... 7-10 7.8.4 Dilatometer Correlation Parameters .................................................... 7-11 7.9 Soil Loading Conditions And Soil Shear Strength Selection ............................ 7-12 7.9.1 Soil Loading .......................................................................................
    [Show full text]