Non-Equilibrium Continuum Physics
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Foundations of Continuum Mechanics
Foundations of Continuum Mechanics • Concerned with material bodies (solids and fluids) which can change shape when loaded, and view is taken that bodies are continuous bodies. • Commonly known that matter is made up of discrete particles, and only known continuum is empty space. • Experience has shown that descriptions based on continuum modeling is useful, provided only that variation of field quantities on the scale of deformation mechanism is small in some sense. Goal of continuum mechanics is to solve BVP. The main steps are 1. Mathematical preliminaries (Tensor theory) 2. Kinematics 3. Balance laws and field equations 4. Constitutive laws (models) Mathematical structure adopted to ensure that results are coordinate invariant and observer invariant and are consistent with material symmetries. Vector and Tensor Theory Most tensors of interest in continuum mechanics are one of the following type: 1. Symmetric – have 3 real eigenvalues and orthogonal eigenvectors (eg. Stress) 2. Skew-Symmetric – is like a vector, has an associated axial vector (eg. Spin) 3. Orthogonal – describes a transformation of basis (eg. Rotation matrix) → 3 3 3 u = ∑uiei = uiei T = ∑∑Tijei ⊗ e j = Tijei ⊗ e j i=1 ~ ij==1 1 When the basis ei is changed, the components of tensors and vectors transform in a specific way. Certain quantities remain invariant – eg. trace, determinant. Gradient of an nth order tensor is a tensor of order n+1 and divergence of an nth order tensor is a tensor of order n-1. ∂ui ∂ui ∂Tij ∇ ⋅u = ∇ ⊗ u = ei ⊗ e j ∇ ⋅ T = e j ∂xi ∂x j ∂xi Integral (divergence) Theorems: T ∫∫RR∇ ⋅u dv = ∂ u⋅n da ∫∫RR∇ ⊗ u dv = ∂ u ⊗ n da ∫∫RR∇ ⋅ T dv = ∂ T n da Kinematics The tensor that plays the most important role in kinematics is the Deformation Gradient x(X) = X + u(X) F = ∇X ⊗ x(X) = I + ∇X ⊗ u(X) F can be used to determine 1. -
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. -
Pacific Northwest National Laboratory Operated by Battelie for the U.S
PNNL-11668 UC-2030 Pacific Northwest National Laboratory Operated by Battelie for the U.S. Department of Energy Seismic Event-Induced Waste Response and Gas Mobilization Predictions for Typical Hanf ord Waste Tank Configurations H. C. Reid J. E. Deibler 0C1 1 0 W97 OSTI September 1997 •-a Prepared for the US. Department of Energy under Contract DE-AC06-76RLO1830 1» DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government, Neither the United States Government nor any agency thereof, nor Battelle Memorial Institute, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or Battelle Memorial Institute. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. PACIFIC NORTHWEST NATIONAL LABORATORY operated by BATTELLE for the UNITED STATES DEPARTMENT OF ENERGY under Contract DE-AC06-76RLO 1830 Printed in the United States of America Available to DOE and DOE contractors from the Office of Scientific and Technical Information, P.O. Box 62, Oak Ridge, TN 37831; prices available from (615) 576-8401, Available to the public from the National Technical Information Service, U.S. -
Hydrostatic Forces on Surface
24/01/2017 Lecture 7 Hydrostatic Forces on Surface In the last class we discussed about the Hydrostatic Pressure Hydrostatic condition, etc. To retain water or other liquids, you need appropriate solid container or retaining structure like Water tanks Dams Or even vessels, bottles, etc. In static condition, forces due to hydrostatic pressure from water will act on the walls of the retaining structure. You have to design the walls appropriately so that it can withstand the hydrostatic forces. To derive hydrostatic force on one side of a plane Consider a purely arbitrary body shaped plane surface submerged in water. Arbitrary shaped plane surface This plane surface is normal to the plain of this paper and is kept inclined at an angle of θ from the horizontal water surface. Our objective is to find the hydrostatic force on one side of the plane surface that is submerged. (Source: Fluid Mechanics by F.M. White) Let ‘h’ be the depth from the free surface to any arbitrary element area ‘dA’ on the plane. Pressure at h(x,y) will be p = pa + ρgh where, pa = atmospheric pressure. For our convenience to make various points on the plane, we have taken the x-y coordinates accordingly. We also introduce a dummy variable ξ that show the inclined distance of the arbitrary element area dA from the free surface. The total hydrostatic force on one side of the plane is F pdA where, ‘A’ is the total area of the plane surface. A This hydrostatic force is similar to application of continuously varying load on the plane surface (recall solid mechanics). -
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 ....................................................................................... -
Ch. 8 Deflections Due to Bending
446.201A (Solid Mechanics) Professor Youn, Byeng Dong CH. 8 DEFLECTIONS DUE TO BENDING Ch. 8 Deflections due to bending 1 / 27 446.201A (Solid Mechanics) Professor Youn, Byeng Dong 8.1 Introduction i) We consider the deflections of slender members which transmit bending moments. ii) We shall treat statically indeterminate beams which require simultaneous consideration of all three of the steps (2.1) iii) We study mechanisms of plastic collapse for statically indeterminate beams. iv) The calculation of the deflections is very important way to analyze statically indeterminate beams and confirm whether the deflections exceed the maximum allowance or not. 8.2 The Moment – Curvature Relation ▶ From Ch.7 à When a symmetrical, linearly elastic beam element is subjected to pure bending, as shown in Fig. 8.1, the curvature of the neutral axis is related to the applied bending moment by the equation. ∆ = = = = (8.1) ∆→ ∆ For simplification, → ▶ Simplification i) When is not a constant, the effect on the overall deflection by the shear force can be ignored. ii) Assume that although M is not a constant the expressions defined from pure bending can be applied. Ch. 8 Deflections due to bending 2 / 27 446.201A (Solid Mechanics) Professor Youn, Byeng Dong ▶ Differential equations between the curvature and the deflection 1▷ The case of the large deflection The slope of the neutral axis in Fig. 8.2 (a) is = Next, differentiation with respect to arc length s gives = ( ) ∴ = → = (a) From Fig. 8.2 (b) () = () + () → = 1 + Ch. 8 Deflections due to bending 3 / 27 446.201A (Solid Mechanics) Professor Youn, Byeng Dong → = (b) (/) & = = (c) [(/)]/ If substitutng (b) and (c) into the (a), / = = = (8.2) [(/)]/ [()]/ ∴ = = [()]/ When the slope angle shown in Fig. -
Chapter 14 Solids and Fluids Matter Is Usually Classified Into One of Four States Or Phases: Solid, Liquid, Gas, Or Plasma
Chapter 14 Solids and Fluids Matter is usually classified into one of four states or phases: solid, liquid, gas, or plasma. Shape: A solid has a fixed shape, whereas fluids (liquid and gas) have no fixed shape. Compressibility: The atoms in a solid or a liquid are quite closely packed, which makes them almost incompressible. On the other hand, atoms or molecules in gas are far apart, thus gases are compressible in general. The distinction between these states is not always clear-cut. Such complicated behaviors called phase transition will be discussed later on. 1 14.1 Density At some time in the third century B.C., Archimedes was asked to find a way of determining whether or not the gold had been mixed with silver, which led him to discover a useful concept, density. m ρ = V The specific gravity of a substance is the ratio of its density to that of water at 4oC, which is 1000 kg/m3=1 g/cm3. Specific gravity is a dimensionless quantity. 2 14.2 Elastic Moduli A force applied to an object can change its shape. The response of a material to a given type of deforming force is characterized by an elastic modulus, Stress Elastic modulus = Strain Stress: force per unit area in general Strain: fractional change in dimension or volume. Three elastic moduli will be discussed: Young’s modulus for solids, the shear modulus for solids, and the bulk modulus for solids and fluids. 3 Young’s Modulus Young’s modulus is a measure of the resistance of a solid to a change in its length when a force is applied perpendicular to a face. -
Evaluating High Temperature Elastic Modulus of Ceramic Coatings by Relative Method Guanglin NIE, Yiwang BAO*, Detian WAN, Yuan TIAN
Journal of Advanced Ceramics 2017, 6(4): 288–303 ISSN 2226-4108 https://doi.org/10.1007/s40145-017-0241-5 CN 10-1154/TQ Research Article Evaluating high temperature elastic modulus of ceramic coatings by relative method Guanglin NIE, Yiwang BAO*, Detian WAN, Yuan TIAN State Key Laboratory of Green Building Materials, China Building Materials Academy, Beijing 100024, China Received: June 16, 2017; Revised: August 07, 2017; Accepted: August 09, 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract: The accurate evaluation of the elastic modulus of ceramic coatings at high temperature (HT) is of high significance for industrial application, yet it is not easy to get the practical modulus at HT due to the difficulty of the deformation measurement and coating separation from the composite samples. This work presented a simple approach in which relative method was used twice to solve this problem indirectly. Given a single-face or double-face coated beam sample, the relative method was firstly used to determine the real mid-span deflection of the three-point bending piece at HT, and secondly to derive the analytical relation among the HT moduli of the coating, the coated and uncoated samples. Thus the HT modulus of the coatings on beam samples is determined uniquely via the measured HT moduli of the samples with and without coatings. For a ring sample (from tube with outer-side, inner-side, and double-side coating), the relative method was used firstly to determine the real compression deformation of a split ring sample at HT, secondly to derive the relationship among the slope of load-deformation curve of the coated ring, the HT modulus of the coating and substrate. -
Topics in Solid Mechanics: Elasticity, Plasticity, Damage, Nano and Biomechanics
Topics in Solid Mechanics: Elasticity, Plasticity, Damage, Nano and Biomechanics Vlado A. Lubarda Montenegrin Academy of Sciences and Arts Division of Natural Sciencies OPN, CANU 2012 Contents Preface xi Part 1. LINEAR ELASTICITY 1 Chapter 1. Anisotropic Nonuniform Lam´eProblem 2 1.1. Elastic Anisotropy 2 1.2. Radial Nonuniformity 3 1.3. Governing Differential Equations 4 1.4. Stress and Displacement Expressions 5 1.5. Traction Boundary Conditions 6 1.6. Applied External Pressure 7 1.7. Plane Stress Approximation 8 1.8. Generalized Plane Stress 10 References 11 Chapter 2. Stretching of a Hollow Circular Membrane 12 2.1. Introduction 12 2.2. Basic Equations 12 2.3. Traction Boundary Conditions 13 2.4. Mixed Boundary Conditions 16 References 18 Chapter 3. Energy of Circular Inclusion with Sliding Interface 21 3.1. Energy Expressions for Sliding Inclusion 21 3.2. Energies due to Eigenstrain 22 3.3. Energies due to Remote Loading 23 3.4. Inhomogeneity under Remote Loading 25 References 26 Chapter 4. Eigenstrain Problem for Nonellipsoidal Inclusions 27 4.1. Eshelby Property 27 4.2. Displacement Expression 27 4.3. The Shape of an Inclusion 28 4.4. Other Shapes of Inclusions 30 References 32 Chapter 5. Circular Loads on the Surface of a Half-Space 33 5.1. Displacements due to Vertical Ring Load 33 5.2. Alternative Displacement Expressions 34 iii iv CONTENTS 5.3. Tangential Line Load 37 5.4. Alternative Expressions 39 5.5. Reciprocal Properties 43 References 45 Chapter 6. Elasticity Tensors of Anisotropic Materials 46 6.1. Elastic Moduli of Transversely Isotropic Materials 46 6.2. -
Chapter 6: Bending
Chapter 6: Bending Chapter Objectives Determine the internal moment at a section of a beam Determine the stress in a beam member caused by bending Determine the stresses in composite beams Bending analysis This wood ruler is held flat against the table at the left, and fingers are poised to press against it. When the fingers apply forces, the ruler deflects, primarily up or down. Whenever a part deforms in this way, we say that it acts like a “beam.” In this chapter, we learn to determine the stresses produced by the forces and how they depend on the beam cross-section, length, and material properties. Support types: Load types: Concentrated loads Distributed loads Concentrated moments Sign conventions: How do we define whether the internal shear force and bending moment are positive or negative? Shear and moment diagrams Given: Find: = 2 kN , , Shear- = 1 m Bending moment diagram along the beam axis Example: cantilever beam Given: Find: = 4 kN , , Shear- = 1.5 kN/m Bending moment = 1 m diagram along the beam axis 2 Relations Among Load, Shear and Bending Moments Relationship between load and shear: Relationship between shear and bending moment: Wherever there is an external concentrated force, or a concentrated moment, there will be a change (jump) in shear or moment respectively. Concentrated force F acting upwards: Concentrated Moment acting clockwise Shear and moment diagrams Given: Use the graphical = 2 kN method the sketch = 1 m diagrams for V(x) and M(x) Example: cantilever beam Given: Use the graphical = 4 kN method the sketch = 1.5 kN/m diagrams for V(x) and = 1 m M(x) 2 Pure bending Take a flexible strip, such as a thin ruler, and apply equal forces with your fingers as shown. -
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 . -
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.