Failure Theories

Total Page：16

File Type：pdf, Size：1020Kb

Failure Theories Predict capability of materials to withstand the infinite combination of non-standard loads Failure: Behavior of a member that renders it unsuitable for its intended function. Static loading (no impact, fatigue nor surface wear) How much distortion is too much ? World War II tanker broken in two by a brittle fracture, despite the normal ductility of the used steel. Impact loading, Sharp notches, Low temperatures, Can promote brittle fracture of usually ductile material Max. Normal-Stress-Theory (W.J.M. Rankine 1802-1872, engl. scientist & educator) simplest Failure occurs, when greatest tensile stress exceeds uniaxial tensile strength. +σ2 Sut +τ S +σ uc Sut +σ1 Suc Sut No Failure occurs within these bounds Suc No Failure occurs within this area Principle Mohr circles σ1 –σ2 plot Correlates well for brittle fracture Max. Shear-Stress-Theory (C.A. Coulomb 1736-1806, French scientist) oldest Also called: Tresca Theory or Guest’s Law Failure occurs, when maximum shear stress exceeds shear strength in uniaxial tension test. +σ 2 Sy +τ No Failure Sy Syt within + these bounds +σ σ1 σ1 − σ2 Sy = τmax = No Failure 2 2 within this area Principle Mohr circles σ1 –σ2 plot Correlates well for ductile yielding Max. Distortion-Energy-Theory (Maxwell 1856 Engl., Hueber 1904 Pol., best Mises 1913 & Hencky 1925 Ger./US) Also called: Max.-Octahedral-Shear Theory Failure occurs, when distortion-energy in unit volume (arbitrary load condition) equals distortion-energy in same volume for uniaxial yielding. P Energy stored in a spring U = ⋅ y 2 P P2 with spring rate k = U = y 2 ⋅ k 2 A ⋅E P ⋅L axially loaded bar: k = U = L 2A ⋅E U P2 per unit volume: u = u = ()V = A ⋅L 2A 2 ⋅E P σ2 stress: σ = u = A 2E δ ε ⋅ σ stress-strain rel.: E = u = ε 2 ε ⋅ σ ε ⋅ σ ε ⋅ σ 3-D: u = 1 1 + 2 2 + 3 3 2 2 2 Max. Distortion-Energy-Theory ε1 ⋅ σ1 ε2 ⋅ σ2 ε3 ⋅ σ3 δ νδ νδ u = + + ε = 1 − 2 − 3 σ 2 2 2 1 E E E δ νδ νδ with 3-D stress-strain rel. ε = 2 − 1 − 3 2 E E E 1 2 2 2 δ 3 νδ 1 νδ 2 uσ = []σ1 + σ2 + σ3 − 2ν()σ1σ2 + σ2σ3 + σ3σ1 ε 3 = − − 2E E E E actual average difference σ1 + σ2 + σ3 with 3-D averaged stress: σav = 2 3 3σ u = av []1− 2ν av 2E Distortion energy ud = uσ − uav ⎡ 2 2 2 ⎤ 1+ ν ()σ1 − σ2 + ()σ2 − σ3 + (σ3 − σ1) ud = ⎢ ⎥ 3E ⎣ 2 ⎦ Simple tension test: σ1 = Sy 1+ ν u = S 2 d 3E y ()σ − σ 2 + ()()σ − σ 2 + σ − σ 2 S > σ = 1 2 2 3 3 1 y e 2 Max. Distortion-Energy-Theory 3D: ()σ − σ 2 + ()()σ − σ 2 + σ − σ 2 S > σ = 1 2 2 3 3 1 y e 2 2D: (Juvinall p254) Homework Read Chapters 6.5 - 6.8 (6.0-6.1) Problem P6.28 (b) only Due: Monday 03/28/05 Exam Monday, 11 April Failure Theories Example: σ1 = 35 ksi Steel Sy=100 ksi σ2 = -25 ksi σ1= Sy= σ2= Load Point Shear Diagonal σmax<Sy 1.7 1.9 Load Line Max. Normal Sress Th. h. σ −σ <S T 1 2 y gy er 2.9 Max. Shear Stress nTh. En io rt to is . D ax M 2 2 σ1 +σ2 −σ1σ2 <Sy Sy=.

Copy Link

Recommended publications

Tech Spotlight
impact.qxd 3/12/04 11:25 AM Page 1 TECH SPOTLIGHT Impact Testing notched side by the moving striker by the metrology of the parts that are Wayne Hayward pendulum, and the energy needed to subject to the most wear and tear, and Tinius Olsen Testing Machine Co. break off the free end is the measure some machines were manufactured Horsham, Pennsylvania of the impact strength. This article before strict adherence to the specifi- discusses the equipment, calibration, cations in ASTM E23 was deemed to and procedures for Charpy testing. be critical. Accurate results are not pos- he purpose of impact testing is to sible if the equipment is not operating Tdetermine the toughness of a ma- Impact testing equipment according to the specification; there- terial by measuring the amount of en- Instruments for impact testing of fore, it is strongly recommended that ergy absorbed by a specimen as it frac- metals have been manufactured com- the impact tester be calibrated and/or tures while being struck by a striker mercially since the 1900’s, and several verified by an accredited calibration pendulum moving at high speed. The manufacturers of testing equipment service at regular intervals. impact strength is defined as the max- meet the requirements of ASTM E23, imum amount of energy that can be “Standard Test Methods for Notched Instrument calibration absorbed by the specimen without Bar Impact Testing of Metallic Mate- A calibration swing is performed to fracture. rials.” The standard provides dimen- ensure that energy losses due to In the Charpy impact test, a notch sion and tolerance requirements for windage and friction in the bearings is placed in the specimen.
[Show full text]
Lecture 1: Introduction
Lecture 1: Introduction E. J. Hinch Non-Newtonian fluids occur commonly in our world. These fluids, such as toothpaste, saliva, oils, mud and lava, exhibit a number of behaviors that are different from Newtonian fluids and have a number of additional material properties. In general, these differences arise because the fluid has a microstructure that influences the flow. In section 2, we will present a collection of some of the interesting phenomena arising from flow nonlinearities, the inhibition of stretching, elastic effects and normal stresses. In section 3 we will discuss a variety of devices for measuring material properties, a process known as rheometry. 1 Fluid Mechanical Preliminaries The equations of motion for an incompressible fluid of unit density are (for details and derivation see any text on fluid mechanics, e.g. [1]) @u + (u · r) u = r · S + F (1) @t r · u = 0 (2) where u is the velocity, S is the total stress tensor and F are the body forces. It is customary to divide the total stress into an isotropic part and a deviatoric part as in S = −pI + σ (3) where tr σ = 0. These equations are closed only if we can relate the deviatoric stress to the velocity field (the pressure field satisfies the incompressibility condition). It is common to look for local models where the stress depends only on the local gradients of the flow: σ = σ (E) where E is the rate of strain tensor 1 E = ru + ruT ; (4) 2 the symmetric part of the the velocity gradient tensor. The trace-free requirement on σ and the physical requirement of symmetry σ = σT means that there are only 5 independent components of the deviatoric stress: 3 shear stresses (the off-diagonal elements) and 2 normal stress differences (the diagonal elements constrained to sum to 0).
[Show full text]
Navier-Stokes-Equation
Math 613 * Fall 2018 * Victor Matveev Derivation of the Navier-Stokes Equation 1. Relationship between force (stress), stress tensor, and strain: Consider any sub-volume inside the fluid, with variable unit normal n to the surface of this sub-volume. Definition: Force per area at each point along the surface of this sub-volume is called the stress vector T. When fluid is not in motion, T is pointing parallel to the outward normal n, and its magnitude equals pressure p: T = p n. However, if there is shear flow, the two are not parallel to each other, so we need a marix (a tensor), called the stress-tensor , to express the force direction relative to the normal direction, defined as follows: T Tn or Tnkjjk As we will see below, σ is a symmetric matrix, so we can also write Tn or Tnkkjj The difference in directions of T and n is due to the non-diagonal “deviatoric” part of the stress tensor, jk, which makes the force deviate from the normal: jkp jk jk where p is the usual (scalar) pressure From general considerations, it is clear that the only source of such “skew” / ”deviatoric” force in fluid is the shear component of the flow, described by the shear (non-diagonal) part of the “strain rate” tensor e kj: 2 1 jk2ee jk mm jk where euujk j k k j (strain rate tensro) 3 2 Note: the funny construct 2/3 guarantees that the part of proportional to has a zero trace. The two terms above represent the most general (and the only possible) mathematical expression that depends on first-order velocity derivatives and is invariant under coordinate transformations like rotations.
[Show full text]
Leonhard Euler Moriam Yarrow
Leonhard Euler Moriam Yarrow Euler's Life Leonhard Euler was one of the greatest mathematician and phsysicist of all time for his many contributions to mathematics. His works have inspired and are the foundation for modern mathe- matics. Euler was born in Basel, Switzerland on April 15, 1707 AD by Paul Euler and Marguerite Brucker. He is the oldest of ﬁve children. Once, Euler was born his family moved from Basel to Riehen, where most of his childhood took place. From a very young age Euler had a niche for math because his father taught him the subject. At the age of thirteen he was sent to live with his grandmother, where he attended the University of Basel to receive his Master of Philosphy in 1723. While he attended the Universirty of Basel, he studied greek in hebrew to satisfy his father. His father wanted to prepare him for a career in the ﬁeld of theology in order to become a pastor, but his friend Johann Bernouilli convinced Euler's father to allow his son to pursue a career in mathematics. Bernoulli saw the potentional in Euler after giving him lessons. Euler received a position at the Academy at Saint Petersburg as a professor from his friend, Daniel Bernoulli. He rose through the ranks very quickly. Once Daniel Bernoulli decided to leave his position as the director of the mathmatical department, Euler was promoted. While in Russia, Euler was greeted/ introduced to Christian Goldbach, who sparked Euler's interest in number theory. Euler was a man of many talents because in Russia he was learning russian, executed studies on navigation and ship design, cartography, and an examiner for the military cadet corps.
[Show full text]
Euler and Chebyshev: from the Sphere to the Plane and Backwards Athanase Papadopoulos
Euler and Chebyshev: From the sphere to the plane and backwards Athanase Papadopoulos To cite this version: Athanase Papadopoulos. Euler and Chebyshev: From the sphere to the plane and backwards. 2016. hal-01352229 HAL Id: hal-01352229 https://hal.archives-ouvertes.fr/hal-01352229 Preprint submitted on 6 Aug 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. EULER AND CHEBYSHEV: FROM THE SPHERE TO THE PLANE AND BACKWARDS ATHANASE PAPADOPOULOS Abstract. We report on the works of Euler and Chebyshev on the drawing of geographical maps. We point out relations with questions about the fitting of garments that were studied by Chebyshev. This paper will appear in the Proceedings in Cybernetics, a volume dedicated to the 70th anniversary of Academician Vladimir Betelin. Keywords: Chebyshev, Euler, surfaces, conformal mappings, cartography, fitting of garments, linkages. AMS classification: 30C20, 91D20, 01A55, 01A50, 53-03, 53-02, 53A05, 53C42, 53A25. 1. Introduction Euler and Chebyshev were both interested in almost all problems in pure and applied mathematics and in engineering, including the conception of industrial ma- chines and technological devices. In this paper, we report on the problem of drawing geographical maps on which they both worked.
[Show full text]
Galling and Galling Resistance of Stainless Steels
Stainless Steel Advisory Service Tel: 0114 267 1265 Fax 0114 266 1252 A service provided by the SSAS Information Sheet No.5.60 th British Stainless Steel Association Issue 02 12 March 2001 Page 1 of 1 Galling and Galling Resistance of Stainless Steels Introduction Galling, or cold welding as it is sometimes referred to as, is a form of severe adhesive wear. Adhesive wear occurs between two metal surfaces that are in relative motion to each other and under sufficient load to permit the transfers of material between themselves. This is a solid-phase welding process. The load must be sufficient, during relative motion, to disrupt the protective oxide layer covering surface asperities of the metal and permit metal to metal contact. Under high stress and poor lubrication conditions, stronger bonds may form over a larger surface area. Large fragments or surface protrusions may be formed and the result is galling of the surfaces. Severe galling can result in the seizure of metal components. Materials which are highly ductile or which possess low work-hardening rates tend to be prone to galling. Austenitic stainless steels show a tendency to gall under certain conditions. Factors Affecting Wear and Galling The factors affecting wear and galling are related to design, lubrication, environment, and steel properties. The material properties relevant to the issue of galling of stainless steels include: - • design tolerances and surface finish of the steel • surface properties (hardness) and structure of the steel • applied load • contact area and degree of movement. Design tolerances should provide sufficient clearance. The contact load on sliding components should be kept to a minimum, while the contact area should be maximised.
[Show full text]
On the Geometric Character of Stress in Continuum Mechanics
Z. angew. Math. Phys. 58 (2007) 1–14 0044-2275/07/050001-14 DOI 10.1007/s00033-007-6141-8 Zeitschrift für angewandte c 2007 Birkhäuser Verlag, Basel Mathematik und Physik ZAMP On the geometric character of stress in continuum mechanics Eva Kanso, Marino Arroyo, Yiying Tong, Arash Yavari, Jerrold E. Marsden1 and Mathieu Desbrun Abstract. This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other funda- mental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented. Mathematics Subject Classification (2000). Keywords. Continuum mechanics, elasticity, stress tensor, differential forms. 1. Motivation This paper proposes a reformulation of classical continuum mechanics in terms of bundle-valued exterior forms. Our motivation is to provide a geometric description of force in continuum mechanics, which leads to an elegant geometric theory and, at the same time, may enable the development of space-time integration algorithms that respect the underlying geometric structure at the discrete level. In classical mechanics the traditional approach is to define all the kinematic and kinetic quantities using vector and tensor fields. For example, velocity and traction are both viewed as vector fields and power is defined as their inner product, which is induced from an appropriately defined Riemannian metric. On the other hand, it has long been appreciated in geometric mechanics that force should not be viewed as a vector, but rather a one-form.
[Show full text]
Dealing with Stress! Anurag Gupta, IIT Kanpur Augustus Edward Hough Love (1863‐1940) Cliffor D Aabmbrose Tdlltruesdell III (1919‐2000)
Dealing with Stress! Anurag Gupta, IIT Kanpur Augustus Edward Hough Love (1863‐1940) Cliffor d AbAmbrose TdllTruesdell III (1919‐2000) (Portrait by Joseph Sheppard) What is Stress? “The notion (of stress) is simply that of mutual action between two bodies in contact, or between two parts of the same body separated by an imagined surface…” ∂Ω ∂Ω ∂Ω Ω Ω “…the physical reality of such modes of action is, in this view, admitted as part of the conceptual scheme” (Quoted from Love) Cauchy’s stress principle Upon the separang surface ∂Ω, there exists an integrable field equivalent in effect to the acon exerted by the maer outside ∂Ω to tha t whic h is iidinside ∂Ω and contiguous to it. This field is given by the traction vector t. 30th September 1822 t n ∂Ω Ω Augustin Louis Cauchy (1789‐1857) Elementary examples (i) Bar under tension A: area of the c.s. s1= F/A s2= F/A√2 (ii) Hydrostatic pressure Styrofoam cups after experiencing deep‐sea hydrostatic pressure http://www.expeditions.udel.edu/ Contact action Fc(Ω,B\ Ω) = ∫∂Ωt dA net force through contact action Fd(Ω) = ∫Ωρb dV net force through distant action Total force acting on Ω F(Ω) = Fc(Ω,B\ Ω) + Fd(Ω) Ω Mc(Ω,B\ Ω, c) = ∫∂Ω(X –c) x t dA moment due to contact action B Md(Ω, c) = ∫Ω(X – c) x ρb dV moment due to distant action Total moment acting on Ω M(Ω) = Mc(Ω,B\ Ω) + Md(Ω) Linear momentum of Ω L(Ω) = ∫Ωρv dV Moment of momentum of Ω G(Ω, c) = ∫Ω (X –c) x ρv dV Euler’s laws F = dL /dt M = dG /dt Euler, 1752, 1776 Or eqqyuivalently ∫∂Ωt dA + ∫Ωρb dV = d/dt(∫Ωρv dV) ∫∂Ω(X –c) x t dA + ∫Ω(X –c) x ρb dV = d/dt(∫Ω(X –c) x ρv dV) Leonhard Euler (1707‐1783) Kirchhoff, 1876 portrait by Emanuel Handmann Upon using mass balance these can be rewritten as ∫∂Ωt dA + ∫Ωρb dV = ∫Ωρa dV ∫∂Ω(X –c) x t dA + ∫Ω(X –c) x ρb dV = ∫Ω(X –c) x ρa dV Emergence of stress (A) Cauchy’s hypothesis At a fixed point, traction depends on the surface of interaction only through the normal, i.e.
[Show full text]
Hydrostatic Stress Effect on the Yield Behavior of Inconel 100
5q H HYDROSTATIC STRESS EFFECT ON THE YIELD BEHAVIOR OF INCONEL 100 Phillip A. AHen and Christopher D. Wilson Departmem of Mechanical Engineering Tennessee Technological University P.O. Box 5014 Cookeville, TN 38505-0001 931-372-3102 voice 931-372-6340 fax paa5443 @tmectLedu ABSTRACT Classical metal plasticity theory assumes that hydrostatic stress has no effect on the yield and postyield behavior of metals. Recent reexaminations of classical theory have revealed a significant effect of hydrostatic stress on the yield behavior of notched geometries. New experiments and nonlinear finite element analyses (FEA) of Inconel 100 (IN 100) equal-arm bend and double-edge notch tension (DENT) test specimens have revealed the effect of internal hydrostatic tensile stresses on yielding. Nonlinear FEA using the von Mises (yielding is independent of hydrostatic stress) and the Drucker-Prager (yielding is linearly dependent on hydrostatic stress) yield functions was performed. In all test cases, the yon Mises constitutive model, which is independent of hydrostatic pressure, overestimated the load for a given displacement or strain. Considering the failure displacements or strains, the Drucker-Prager FEM's predicted loads that were 3% to 5% lower than the von Mises values. For the failure loads, the Drucker Prager FEM's predicted strains that were 20% to 35% greater than the yon Mises values. The Drucker-Prager yield function seems to more accurately predict the overall specimen response of geometries with significant internal hydrostatic stress influence. Phillip A. Allen 1 INTRODUCTION Since the 1940's, many have considered Bridgman's experiments on the effects of hydrostatic pressure on rr_tals the definitive study.
[Show full text]
Equation of Motion for Viscous Fluids
1 2.25 Equation of Motion for Viscous Fluids Ain A. Sonin Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 2001 (8th edition) Contents 1. Surface Stress …………………………………………………………. 2 2. The Stress Tensor ……………………………………………………… 3 3. Symmetry of the Stress Tensor …………………………………………8 4. Equation of Motion in terms of the Stress Tensor ………………………11 5. Stress Tensor for Newtonian Fluids …………………………………… 13 The shear stresses and ordinary viscosity …………………………. 14 The normal stresses ……………………………………………….. 15 General form of the stress tensor; the second viscosity …………… 20 6. The Navier-Stokes Equation …………………………………………… 25 7. Boundary Conditions ………………………………………………….. 26 Appendix A: Viscous Flow Equations in Cylindrical Coordinates ………… 28 ã Ain A. Sonin 2001 2 1 Surface Stress So far we have been dealing with quantities like density and velocity, which at a given instant have specific values at every point in the fluid or other continuously distributed material. The density (rv ,t) is a scalar field in the sense that it has a scalar value at every point, while the velocity v (rv ,t) is a vector field, since it has a direction as well as a magnitude at every point. Fig. 1: A surface element at a point in a continuum. The surface stress is a more complicated type of quantity. The reason for this is that one cannot talk of the stress at a point without first defining the particular surface through v that point on which the stress acts. A small fluid surface element centered at the point r is defined by its area A (the prefix indicates an infinitesimal quantity) and by its outward v v unit normal vector n .
[Show full text]
Stress and Momentum in a Capacitor That Moves with Constant Velocity
StressandMomentuminaCapacitor That Moves with Constant Velocity Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (April 21, 1984; updated July 29, 2017) 1Problem Consider a parallel-plate capacitor whose plates are held apart by a nonconducting slab of unit (relative) dielectric constant and unit (relative) magnetic permeability.1 Discuss the energy, momentum and stress in this (isolated) system when at rest and when moving with constant velocity parallel or perpendicular to the electric field. Does the system contain hidden momentum, Phidden, defined for a subsystem by, Phidden ≡ P − Mvcm − (x − xcm)(p − ρvb) · dArea, (1) boundary where P is the total momentum of the subsystem, M = U/c2 is its total “mass,” U is its total energy, c is the speed of light in vacuum, xcm is its center of mass/energy, vcm = dxcm/dt, 2 p is its momentum density, ρ = u/c is its “mass” density, u is its energy density, and vb is the velocity (field) of its boundary?2 Fringe-field effects can be ignored. The velocity can be large or small compared to the speed of light. 2Solution This problem is concerned with the relativistic transformation of properties of the capacitor. It represents a macroscopic application of the concepts of “Poincaré stresses” [3] that were introduced into classical models of the electron. Versions of this problem have also appeared in[4,5,6,7]. We suppose that the capacitor supports a uniform electric field E = E zˆ between its plates, in its rest frame. Taking the (relative) dielectric constant of the material between the plates to be unity, the electric displacement is given by D = E (in Gaussian units).
[Show full text]
2 Review of Stress, Linear Strain and Elastic Stress- Strain Relations
2 Review of Stress, Linear Strain and Elastic Stress- Strain Relations 2.1 Introduction In metal forming and machining processes, the work piece is subjected to external forces in order to achieve a certain desired shape. Under the action of these forces, the work piece undergoes displacements and deformation and develops internal forces. A measure of deformation is defined as strain. The intensity of internal forces is called as stress. The displacements, strains and stresses in a deformable body are interlinked. Additionally, they all depend on the geometry and material of the work piece, external forces and supports. Therefore, to estimate the external forces required for achieving the desired shape, one needs to determine the displacements, strains and stresses in the work piece. This involves solving the following set of governing equations : (i) strain-displacement relations, (ii) stress- strain relations and (iii) equations of motion. In this chapter, we develop the governing equations for the case of small deformation of linearly elastic materials. While developing these equations, we disregard the molecular structure of the material and assume the body to be a continuum. This enables us to define the displacements, strains and stresses at every point of the body. We begin our discussion on governing equations with the concept of stress at a point. Then, we carry out the analysis of stress at a point to develop the ideas of stress invariants, principal stresses, maximum shear stress, octahedral stresses and the hydrostatic and deviatoric parts of stress. These ideas will be used in the next chapter to develop the theory of plasticity.
[Show full text]