DOCSLIB.ORG

Ch.8. Plasticity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ch.8. Plasticity CH.8. PLASTICITY Multimedia Course on Continuum Mechanics Overview Introduction Lecture 1 Previous Notions Lecture 2 Lecture 3 Principal Stress Space Lecture 4 Normal and Shear Octahedral Stresses Stress Invariants Lecture 5 Effective Stress Rheological Friction Models Elastic Element Frictional Element Elastic-Frictional Model 2 Overview (cont’d) Rheological Friction Models (cont’d) Frictional Model with Hardening Elastic-Frictional Model with Hardening Phenomenological Behaviour Notion of Plastic Strain Notion of Hardening Lecture 6 Bauschinger Effect Elastoplastic Behaviour 1D Incremental Theory of Plasticity Additive Decomposition of Strain Lecture 7 Hardening Variable Yield Stress, Yield Function and Space of Admissible Stresses Lecture 8 Lecture 9 Constitutive Equation Lecture 9 Elastoplastic Tangent Modulus Lecture 10 Uniaxial Stress-Strain Curve 3 Overview (cont’d) 3D Incremental Theory of Plasticity Additive Decomposition of Strain Hardening Variable Lecture 11 Yield Function Loading - Unloading Conditions and Consistency Conditions Constitutive Equation Lecture 12 Elastoplastic Constitutive Tensor Yield Surfaces Lecture 13 Von Mises Criterion Tresca Criterion Mohr-Coulomb Criterion Lecture 14 Drucker-Prager Criterion 4 8.1 Introduction Ch.8. Plasticity 5 Introduction A material with plastic behavior is characterized by: A nonlinear stress-strain relationship. The existence of permanent (or plastic) strain during a loading/unloading cycle. Lack of unicity in the stress-strain relationship. Plasticity is seen in most materials, after an initial elastic state. 6 Previous Notions PRINCIPAL STRESSES Regardless of the state of stress, it is always possible to choose a special set of axes (principal axes of stress or principal stress directions) so that the shear stress components vanish when the stress components are referred to this system. The three planes perpendicular to the principle axes are the principal planes. The normal stress components in the principal planes are the principal σ stresses. σ 33 x 31 σ x 3 32 3 ′ σ x3 13 σ σ1 00 23 σ σ x′ σ σ = σ 11 12 σ 21 σ 1 3 [] 002 22 σ1 00σ 3 σ 2 x1 x1 x2 x2 x2′ 7 Previous Notions PRINCIPAL STRESSES The Cauchy stress tensor is a symmetric 2nd order tensor so it will diagonalize in an orthonormal basis and its eigenvalues are real numbers. Computing the eigenvalues λ and the corresponding eigenvectors v : σσ⋅=vλλ v[] −1 ⋅= v0 σ11 − λσ12 σ13 not det[]σσ−λ11= −= λ σ12 σ22 − λσ23 =0 σ σ σλ− INVARIANTS 13 23 33 σ σ 33 32 characteristic x 31 σ x λλ−I1 − II 23 λ −=0 3 32 3 equation x3′ σ13 σ 23 λσ≡ σ11 σ12 σ x1′ σ 11 21 σ 22 3 λσ≡ σ1 22 σ λσ33≡ 2 x1 x1 x2 x2 x2′ 8 Previous Notions STRESS INVARIANTS Principal stresses are invariants of the stress state. They are invariant w.r.t. rotation of the coordinate axes to which the stresses are referred. The principal stresses are combined to form the stress invariants I : I= Tr σ =σ =++ σσσ 1 () ii 123 REMARK 1 2 The I invariants are obtained II2=()σσ: −=−+1 (σσ12 σσ 13 + σσ 23) 2 from the characteristic equation I3 = det ()σ of the eigenvalue problem. These invariants are combined, in turn, to obtain the invariants J: JI11= = σ ii REMARK The J invariants can be 12 11 J2=() II 12 +=2:σσij ji =()σσ 2 22 expressed the unified form: 1 = σi ∈ 13 11Ji Tr() i {}1, 2, 3 J3=( I 1 +33 I 12 I + I 3) = Tr ()σσσ ⋅⋅ =σσij jk σ ki i 3 33 9 Previous Notions SPHERICAL AND DEVIATORIC PARTS OF THE STRESS TENSOR Given the Cauchy stress tensor σ and its principal stresses, the following is defined: Mean stress 1 11 σ=Tr ()σ =σ =() σσσ ++ m 3 33ii 123 REMARK Mean pressure In a hydrostatic state of stress, the 1 p =−=−++σm () σσσ123 stress tensor is isotropic and, thus, 3 its components are the same in any Cartesian coordinate system. A spherical or hydrostatic As a consequence, any direction state of stress : σ 00 is a principal direction and the σσσ= = 123 σ ≡=00σσ1 stress state (traction vector) is the 00σ same in any plane. 10 Previous Notions SPHERICAL AND DEVIATORIC PARTS OF THE STRESS TENSOR The Cauchy stress tensor σ can be split into: σ=σsph + σ′ The spherical stress tensor: Also named mean hydrostatic stress tensor or volumetric stress tensor or mean normal stress tensor. Is an isotropic tensor and defines a hydrostatic state of stress. Tends to change the volume of the stressed body 11 σσ:=σσ1 =Tr () 11 = sph m 33ii The stress deviatoric tensor: Is an indicator of how far from a hydrostatic state of stress the state is. Tends to distort the volume of the stressed body σ′ =dev σ = σ−σ m 1 11 Previous Notions STRESS INVARIANTS OF THE STRESS DEVIATORIC TENSOR The stress invariants of the stress deviatoric tensor: I1′′= Tr ()σ = 0 1 II′=σσ ′′: − 2 212 () 2221 I3′=det ()σ ′ =σσσ11 ′′′22 33 +2 σσσ 12 ′′′ 23 13 −−− σσ 12 ′′ 33 σσ 23′′ 11 σσ 13 ′′ 22 =() σσσij ′′′ jk ki 3 These correspond exactly with the invariants J of the same stress deviator tensor: JI11′′= = 0 1 1 JI′′= 2 +==2:II′ ′()σσ ′′ 212 ()222 1 3 11 JI31′′= + 3II12′′+3I33′ = I ′ = Tr ()σσσ ′′′ ⋅⋅ =()σσij ′′′ jk σ ki 3 ( ) 33 12 Previous Notions EFFECTIVE STRESS The effective stress or equivalent uniaxial stress σ is the scalar: 33 σ =3J ' = σσ′′ = σσ´: ´ 2 22ij ij It is an invariant value which measures the “intensity” of a 3D stress state in a terms of an (equivalent) 1D tensile stress state. It should be “consistent”: when applied to a real 1D tensile stress, should return the intensity of this stress. 13 Example Calculate the value of the equivalent uniaxial stress for an uniaxial state of stress defined by: E, ν y σ 00 uσ σ σ ≡ x x 0 00 σ u σ u 0 00 x z 14 σ u 00 σ ≡ 0 00 Example - Solution 0 00 1 σ σ =σ )= u σ u Mean stress: m Tr( 00 33 3 σ m 00 σ σ ≡=0σ 00u 0 Spherical and deviatoric parts sph m 3 00σ of the stress tensor: m σ 00 u 3 2 σ 00 3 u σσum− 00 1 σ′ =−≡ σσ 0 −σσ 00 = − 0 sph m 3 u 00−σ m 1 00− σ 3 u 3 3 411 32 σσ= σσ′′ =2 () + + = σσσ= 2ij ij 2u 9 9 9 23 u u 15 8.2 Principal Stress Space Ch.8. Plasticity 16 Principal Stress Space The principal stress space or Haigh–Westergaard stress space is the space defined by a system of Cartesian axes where the three spatial axes represent the three principal stresses for a body subject to stress: σσσ123≥≥ 17 Octahedral plane Any of the planes perpendicular to the hydrostatic stress axis is a octahedral plane. 1 1 Its unit normal is n = 1 . 3 1 σσσ123≥≥ 18 Normal and Shear Octahedral Stresses Consider the principal stress space: The normal octahedral stress is defined as: 1/ 3 3σoct =OA = OP ⋅=n []σσσ123, , 1/ 3 = 1/ 3 3 =()σσσ ++ =3 σ 3 123 m I σσ= = 1 oct m 3 19 Normal and Shear Octahedral Stresses Consider the principal stress space: The shear or tangential octahedral stress is defined as: 3τ oct = AP Where the AP is calculated from: 2 2 22222 3τoct =AP = OP − OA =(σσσ123 ++) − 1 2 ' −()σσσ123 ++ =2J 2 3 Alternative forms of τ oct : 1/2 112 τ= σσσ222 ++− σσσ ++ oct 1 2 3() 123 3 3 1/2 12 1 222 2 τ= σσ − +− σσ +− σσ τ = []J′ oct ()()()1 2 23 13 oct 3 2 33 20 Normal and Shear Octahedral Stresses In a pure spherical stress state: 1 σ=σ11 →σ =3 σσ →σσ = = m 3 esf σ′ =−= σσesf 0 J2′ = 0 τ = A pure spherical stress state is oct 0 located on the hydrostatic stress axis. In a pure deviator stress state: ′ σ = σσ= σ m =Tr()σσ = Tr (′ ) = 0 oct 0 A pure deviator stress state is located on the octahedral plane containing the origin of the principal stress space 21 Stress Invariants Any point in space is unambiguously defined by the three invariants: The first stress invariant I 1 characterizes the distance from the origin to the octahedral plane containing the point. The second deviator stress invariant J 2′ characterizes the radius of the cylinder containing the point and with the hydrostatic stress axis as axis. The third deviator stress invariantJ 3′ characterizes the position of the point on the circle obtained from the intersection of the octahedral plane and the cylinder. It defines an angle θ () J 3′ . 22 Projection on the Octahedral Plane The projection of the principal stress space on the octahedral plane results in the division of the plane into six “sectors”: These are characterized by the different principal stress orders. Election of a criterion, FEASIBLE e.g.: σσσ123≥≥ WORK SPACE 23 8.4 Phenomenological Behaviour Ch.8. Plasticity 38 Notion of Plastic Strain PLASTIC STRAIN εε=ep + ε elastic limit: σ e LINEAR ELASTIC BEHAVIOUR σε= E e 39 Bauschinger Effect Also known as kinematic hardening. σ f σεe + K K σ e K −+σεe K K −σ e 41 Elastoplastic Behaviour Considering the phenomenological behaviour observed, elastoplastic materials are characterized by: Lack of unicity in the stress-strain relationship. The stress value depends on the actual strain and the previous loading history. A nonlinear stress-strain relationship. There may be certain phases in the deformation process with incremental linearity. The existence of permanent (or plastic) strain during a loading / unloading cycle.
Load more

Recommended publications
  • Mechanics of Materials Plasticity
    Provided for non-commercial research and educational use. Not for reproduction, distribution or commercial use. This article was originally published in the Reference Module in Materials Science and Materials Engineering, published by Elsevier, and the attached copy is provided by Elsevier for the author’s benefit and for the benefit of the author’s institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to specific colleagues who you know, and providing a copy to your institution’s administrator. All other uses, reproduction and distribution, including without limitation commercial reprints, selling or licensing copies or access, or posting on open internet sites, your personal or institution’s website or repository, are prohibited. For exceptions, permission may be sought for such use through Elsevier’s permissions site at: http://www.elsevier.com/locate/permissionusematerial Lubarda V.A., Mechanics of Materials: Plasticity. In: Saleem Hashmi (editor-in-chief), Reference Module in Materials Science and Materials Engineering. Oxford: Elsevier; 2016. pp. 1-24. ISBN: 978-0-12-803581-8 Copyright © 2016 Elsevier Inc. unless otherwise stated. All rights reserved. Author's personal copy Mechanics of Materials: Plasticity$ VA Lubarda, University of California, San Diego, CA, USA r 2016 Elsevier Inc. All rights reserved. 1 Yield Surface 1 1.1 Yield Surface in Strain Space 2 1.2 Yield Surface in Stress Space 2 2 Plasticity Postulates, Normality and Convexity
    [Show full text]
  • An Effective Stress Equation for Unsaturated Granular Media in Pendular Regime
    University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2014-04-28 An Effective Stress Equation for Unsaturated Granular Media in Pendular Regime Khosravani, Sarah Khosravani, S. (2014). An Effective Stress Equation for Unsaturated Granular Media in Pendular Regime (Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/24849 http://hdl.handle.net/11023/1443 master thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY An Effective Stress Equation for Unsaturated Granular Media in Pendular Regime by Sarah Khosravani A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF CIVIL ENGINEERING CALGARY, ALBERTA April, 2014 c Sarah Khosravani 2014 Abstract The mechanical behaviour of a wet granular material is investigated through a microme- chanical analysis of force transport between interacting particles with a given packing and distribution of capillary liquid bridges. A single effective stress tensor, characterizing the ten- sorial contribution of the matric suction and encapsulating evolving liquid bridges, packing, interfaces, and water saturation, is derived micromechanically. The physical significance of the effective stress parameter (χ) as originally introduced in Bishop’s equation is examined and it turns out that Bishop’s equation is incomplete.
    [Show full text]
  • A Plasticity Model with Yield Surface Distortion for Non Proportional Loading
    page 1 A plasticity model with yield surface distortion for non proportional loading. Int. J. Plast., 17(5), pages :703–718 (2001) M.L.M. FRANÇOIS LMT Cachan (ENS de Cachan /CNRS URA 860/Université Paris 6) ABSTRACT In order to enhance the modeling of metallic materials behavior in non proportional loadings, a modification of the classical elastic-plastic models including distortion of the yield surface is proposed. The new yield criterion uses the same norm as in the classical von Mises based criteria, and a "distorted stress" Sd replacing the usual stress deviator S. The obtained yield surface is then “egg-shaped” similar to those experimentally observed and depends on only one new material parameter. The theory is built in such a way as to recover the classical one for proportional loading. An identification procedure is proposed to obtain the material parameters. Simulations and experiments are compared for a 2024 T4 aluminum alloy for both proportional and nonproportional tension-torsion loading paths. KEYWORDS constitutive behavior; elastic-plastic material. SHORTENED TITLE A model for the distortion of yield surfaces. ADDRESS Marc François, LMT, 61 avenue du Président Wilson, 94235 Cachan CEDEX, France. E-MAIL [email protected] INTRODUCTION Most of the phenomenological plasticity models for metals use von Mises based yield criteria. This implies that the yield surface is a hypersphere whose center is the back stress X in the 5-dimensional space of stress deviator tensors. For proportional loadings, the exact shape of the yield surface has no importance since all the deviators governing plasticity remain colinear tensors.
    [Show full text]
  • Identification of Rock Mass Properties in Elasto-Plasticity D
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Archive Ouverte en Sciences de l'Information et de la Communication Identification of rock mass properties in elasto-plasticity D. Deng, D. Nguyen Minh To cite this version: D. Deng, D. Nguyen Minh. Identification of rock mass properties in elasto-plasticity. Computers and Geotechnics, Elsevier, 2003, 30, pp.27-40. 10.1016/S0266-352X(02)00033-2. hal-00111371 HAL Id: hal-00111371 https://hal.archives-ouvertes.fr/hal-00111371 Submitted on 26 Sep 2019 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. Identification of rock mass properties in elasto-plasticity Desheng Deng*, Duc Nguyen-Minh Laboratoire de Me´canique des Solides, E´cole Polytechnique, 91128 Palaiseau, France Abstract A simple and effective back analysis method has been proposed on the basis of a new cri-terion of identification, the minimization of error on the virtual work principle. This method works for both linear elastic and nonlinear elasto-plastic problems. The elasto-plastic rock mass properties for different criteria of plasticity can be well identified based on field measurements.
    [Show full text]
  • Crystal Plasticity Model with Back Stress Evolution. Wei Huang Louisiana State University and Agricultural & Mechanical College
    Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1996 Crystal Plasticity Model With Back Stress Evolution. Wei Huang Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Huang, Wei, "Crystal Plasticity Model With Back Stress Evolution." (1996). LSU Historical Dissertations and Theses. 6154. https://digitalcommons.lsu.edu/gradschool_disstheses/6154 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type o f computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps.
    [Show full text]
  • Yield Surface Effects on Stablity and Failure
    XXIV ICTAM, 21-26 August 2016, Montreal, Canada YIELD SURFACE EFFECTS ON STABLITY AND FAILURE William M. Scherzinger * Solid Mechanics, Sandia National Laboratories, Albuquerque, New Mexico, USA Summary This work looks at the effects of yield surface description on the stability and failure of metal structures. We consider a number of different yield surface descriptions, both isotropic and anisotropic, that are available for modeling material behavior. These yield descriptions can all reproduce the same nominal stress-strain response for a limited set of load paths. However, for more general load paths the models can give significantly different results that can have a large effect on stability and failure. Results are shown for the internal pressurization of a cylinder and the differences in the predictions are quantified. INTRODUCTION Stability and failure in structures is an important field of study that provides solutions to many practical problems in solid mechanics. The boundary value problems require careful consideration of the geometry, the boundary conditions, and the material description. In metal structures the material is usually described with an elastic-plastic constitutive model, which can account for many effects observed in plastic deformation. Crystal plasticity models can account for many microstructural effects, but these models are often too fine scaled for production modeling and simulation. In this work we limit the study to rate independent continuum plasticity models. When using a continuum plasticity model, the hardening behavior is important and much attention is focused on it. Equally important, but less well understood, is the effect of the yield surface description. For classical continuum metal plasticity models that assume associated flow the yield surface provides a definition for the elastic region of stress space along with the direction of plastic flow when the stress state is on the yield surface.
    [Show full text]
  • Durability of Adhesively Bonded Joints in Aerospace Structures
    Durability of adhesive bonded joints in aerospace structures Washington State University Date: 11/08/2017 AMTAS Fall meeting Seattle, WA Durability of Bonded Aircraft Structure • Principal Investigators & Researchers • Lloyd Smith • Preetam Mohapatra, Yi Chen, Trevor Charest • FAA Technical Monitor • Ahmet Oztekin • Other FAA Personnel Involved • Larry Ilcewicz • Industry Participation • Boeing: Will Grace, Peter VanVoast, Kay Blohowiak 2 Durability of adhesive bonded joints in aerospace structures 1. Influence of yield criteria Yield criterion 2. Biaxial tests (Arcan) Plasticity 3. Cyclic tests Hardening rule 4. FEA model Nonlinearity in Bonded Joints 1. Creep, non-linear response Tension (closed form) 2. Ratcheting, experiment Time Dependence 3. Creep, model development Shear (FEA) 4. Ratcheting, model application Approach: Plasticity vPrimary Research Aim: Modeling of adhesive plasticity to describe nonlinear stress-strain response. ØSub Task: 1. Identify the influence of yield criterion and hardening rule on bonded joints (complete) 2. Characterizing hardening rule (complete Dec 2017) 3. Characterizing yield criterion (complete Dec 2017) 4. Numerically combining hardening rule and yield criterion (complete Feb 2017) Approach: Plasticity 1. Identify yield criterion and hardening rule of bonded joints • Input properties: Bulk and Thin film adhesive properties in Tension. 60 Characterization of Tensile properties 55 50 45 40 35 30 25 Bulk Tension: Tough Adheisve 20 Tensile Stress [MPa] Bulk Tension: Standard Adhesive 15 10 Thin film Tension: Tough Adhesive 5 Thin film Tension: Standard Adhesive 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Axial Strain Approach: Plasticity 1. Identify the influence of yield criterion and hardening rule on bonded joints • Background: Investigate yield criterion and hardening rule of bonded joints.
    [Show full text]
  • Tuning the Performance of Metallic Auxetic Metamaterials by Using Buckling and Plasticity
    materials Article Tuning the Performance of Metallic Auxetic Metamaterials by Using Buckling and Plasticity Arash Ghaedizadeh 1, Jianhu Shen 1, Xin Ren 1,2 and Yi Min Xie 1,* Received: 28 November 2015; Accepted: 8 January 2016; Published: 18 January 2016 Academic Editor: Geminiano Mancusi 1 Centre for Innovative Structures and Materials, School of Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Australia; [email protected] (A.G.); [email protected] (J.S.); [email protected] (X.R.) 2 Key Laboratory of Traffic Safety on Track, School of Traffic & Transportation Engineering, Central South University, Changsha 410075, China * Correspondence: [email protected]; Tel.: +61-3-9925-3655 Abstract: Metallic auxetic metamaterials are of great potential to be used in many applications because of their superior mechanical performance to elastomer-based auxetic materials. Due to the limited knowledge on this new type of materials under large plastic deformation, the implementation of such materials in practical applications remains elusive. In contrast to the elastomer-based metamaterials, metallic ones possess new features as a result of the nonlinear deformation of their metallic microstructures under large deformation. The loss of auxetic behavior in metallic metamaterials led us to carry out a numerical and experimental study to investigate the mechanism of the observed phenomenon. A general approach was proposed to tune the performance of auxetic metallic metamaterials undergoing large plastic deformation using buckling behavior and the plasticity of base material. Both experiments and finite element simulations were used to verify the effectiveness of the developed approach. By employing this approach, a 2D auxetic metamaterial was derived from a regular square lattice.
    [Show full text]
  • Constitutive Behavior of Granitic Rock at the Brittle-Ductile Transition
    CONSTITUTIVE BEHAVIOR OF GRANITIC ROCK AT THE BRITTLE-DUCTILE TRANSITION Josie Nevitt, David Pollard, and Jessica Warren Department of Geological and Environmental Sciences, Stanford University, Stanford, CA 94305 e-mail: [email protected] fracture to ductile flow with increasing depth in the Abstract earth’s crust. This so-called “brittle-ductile transition” has important implications for a variety of geological Although significant geological and geophysical and geophysical phenomena, including the mechanics processes, including earthquake nucleation and of earthquake rupture nucleation and propagation propagation, occur in the brittle-ductile transition, (Hobbs et al., 1986; Li and Rice, 1987; Scholz, 1988; earth scientists struggle to identify appropriate Tse and Rice, 1986). Despite the importance of constitutive laws for brittle-ductile deformation. This deformation within this lithospheric interval, significant paper investigates outcrops from the Bear Creek field gaps exist in our ability to accurately characterize area that record deformation at approximately 4-15 km brittle-ductile deformation. Such gaps are strongly depth and 400-500ºC. We focus on the Seven Gables related to the uncertainty in choosing the constitutive outcrop, which contains a ~10cm thick aplite dike that law(s) that govern(s) the rheology of the brittle-ductile is displaced ~45 cm through a contractional step transition. between two sub-parallel left-lateral faults. Stretching Rheology describes the response of a material to an and rotation of the aplite dike, in addition to local applied force and is defined through a set of foliation development in the granodiorite, provides an constitutive equations. In addition to continuum excellent measure of the strain within the step.
    [Show full text]
  • Plasticity, Localization, and Friction in Porous Materials
    RICE UNIVERSITY Plasticity, Localization, and Friction in Porous Materials by Daniel Christopher Badders A T h e s is S u b m i t t e d in P a r t ia l F u l f il l m e n t o f t h e R equirements f o r t h e D e g r e e Doctor of Philosophy A p p r o v e d , T h e s is C o m m i t t e e : Michael M. Carroll, Chairman Dean of the George R. Brown School of Engineering and Burton J. and Ann M. McMurtry Professor of Engineering Mary F. Wbjeeler and Applied Mathematics Yves C. Angel Associate Professor of Mechanical Engineering and Materials Science Houston, Texas May, 1995 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted Ihus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted Also, if unauthorized copyright material had to be removed anote will indicate the deletion.
    [Show full text]
  • On the Path-Dependence of the J-Integral Near a Stationary Crack in an Elastic-Plastic Material
    On the Path-Dependence of the J-integral Near a Stationary Crack in an Elastic-Plastic Material Dorinamaria Carka and Chad M. Landis∗ The University of Texas at Austin, Department of Aerospace Engineering and Engineering Mechanics, 210 East 24th Street, C0600, Austin, TX 78712-0235 Abstract The path-dependence of the J-integral is investigated numerically, via the finite element method, for a range of loadings, Poisson's ratios, and hardening exponents within the context of J2-flow plasticity. Small-scale yielding assumptions are employed using Dirichlet-to-Neumann map boundary conditions on a circular boundary that encloses the plastic zone. This construct allows for a dense finite element mesh within the plastic zone and accurate far-field boundary conditions. Details of the crack tip field that have been computed previously by others, including the existence of an elastic sector in Mode I loading, are confirmed. The somewhat unexpected result is that J for a contour approaching zero radius around the crack tip is approximately 18% lower than the far-field value for Mode I loading for Poisson’s ratios characteristic of metals. In contrast, practically no path-dependence is found for Mode II. The applications of T or S stresses, whether applied proportionally with the K-field or prior to K, have only a modest effect on the path-dependence. Keywords: elasto-plastic fracture mechanics, small scale yielding, path-dependence of the J-integral, finite element methods 1. Introduction The J-integral as introduced by Eshelby [1,2] and Rice [3] is perhaps the most useful quantity for the analysis of the mechanical fields near crack tips in both linear elastic and non-linear elastic materials.
    [Show full text]
  • Introduction to FINITE STRAIN THEORY for CONTINUUM ELASTO
    RED BOX RULES ARE FOR PROOF STAGE ONLY. DELETE BEFORE FINAL PRINTING. WILEY SERIES IN COMPUTATIONAL MECHANICS HASHIGUCHI WILEY SERIES IN COMPUTATIONAL MECHANICS YAMAKAWA Introduction to for to Introduction FINITE STRAIN THEORY for CONTINUUM ELASTO-PLASTICITY CONTINUUM ELASTO-PLASTICITY KOICHI HASHIGUCHI, Kyushu University, Japan Introduction to YUKI YAMAKAWA, Tohoku University, Japan Elasto-plastic deformation is frequently observed in machines and structures, hence its prediction is an important consideration at the design stage. Elasto-plasticity theories will FINITE STRAIN THEORY be increasingly required in the future in response to the development of new and improved industrial technologies. Although various books for elasto-plasticity have been published to date, they focus on infi nitesimal elasto-plastic deformation theory. However, modern computational THEORY STRAIN FINITE for CONTINUUM techniques employ an advanced approach to solve problems in this fi eld and much research has taken place in recent years into fi nite strain elasto-plasticity. This book describes this approach and aims to improve mechanical design techniques in mechanical, civil, structural and aeronautical engineering through the accurate analysis of fi nite elasto-plastic deformation. ELASTO-PLASTICITY Introduction to Finite Strain Theory for Continuum Elasto-Plasticity presents introductory explanations that can be easily understood by readers with only a basic knowledge of elasto-plasticity, showing physical backgrounds of concepts in detail and derivation processes
    [Show full text]