DOCSLIB.ORG

[Physics.Class-Ph] 9 May 2020 the Incremental Bulk Modulus, Young's

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
[Physics.Class-Ph] 9 May 2020 the Incremental Bulk Modulus, Young's Mathematics and Mechanics of Solids (2007) 12, 526–542. doi: 10.1177/1081286506064719 Published 7 April 2006 Page 1 of 16 The incremental bulk modulus, Young’s modulus and Poisson’s ratio in nonlinear isotropic elasticity: physically reasonable response N.H. Scott∗ School of Mathematics, University of East Anglia, Norwich Research Park, Norwich NR4 7TJ, U.K. 12 May 2020 [Dedicated to Michael Hayes with esteem and gratitude] Abstract An incremental (or tangent) bulk modulus for finite isotropic elasticity is de- fined which compares an increment in hydrostatic pressure with the corresponding increment in relative volume. Its positivity provides a stringent criterion for phys- ically reasonable response involving the second derivatives of the strain energy function. Also, an average (or secant) bulk modulus is defined by comparing the current stress with the relative volume change. The positivity of this bulk modulus provides a physically reasonable response criterion less stringent than the former. The concept of incremental bulk modulus is extended to anisotropic elasticity. For states of uniaxial tension an incremental Poisson’s ratio and an incremental Young’s modulus are similarly defined for nonlinear isotropic elasticity and have properties similar to those of the incremental bulk modulus. The incremental Pois- son’s ratios for the isotropic constraints of incompressibility, Bell, Ericksen, and constant area are considered. The incremental moduli are all evaluated for a spe- cific example of the compressible neo-Hookean solid. Bounds on the ground state Lam´eelastic moduli, assumed positive, are given which are sufficient to guarantee the positivity of the incremental bulk and Young’s moduli for all strains. However, although the ground state Poisson’s ratio is positive we find that the incremental arXiv:2005.04212v1 [physics.class-ph] 9 May 2020 Poisson’s ratio becomes negative for large enough axial extensions. Keywords Nonlinear elasticity, incremental elastic moduli, compressible neo- Hookean material, constant-area constraint, superficial incompressibility MSC (2010) 74B20 ∗Email: [email protected] Page 2 of 16 N. H. Scott 1 Basic nonlinear and linear isotropic elasticity In this section we recount the basic theory of nonlinear and linear elasticity which we shall need before defining in Section 2 the incremental bulk and Young’s moduli and the incremental Poisson’s ratio and deriving some of their properties. In Section 3 we consider these moduli in detail for a specific example of the compressible neo-Hookean material. In terms of the deformation gradient F the left and right Cauchy-Green tensors are B = FFT and C = FTF, respectively. In terms of the three invariants −1 J1 = tr C, J2 = tr C , J = √det C (1.1) the Cauchy stress of an isotropic elastic material may be written −1 σ = β0I + β1B + β−1B (1.2) in which I denotes the unit tensor and βΓ (Γ=0, 1, 1) are functions only of the three − invariants J1, J2, J defined at (1.1). The response functions βΓ are often assumed to satisfy the empirical inequalities [1, (51.27)] β0 0, β1 > 0, β−1 0. (1.3) ≤ ≤ It seems reasonable to assume that these inequalities are satisfied for small enough strains of a non-linearly elastic material. In the theoretical discussion of Section 2 the inequalities (1.3) are assumed at the beginning of Sections 2.1 and 2.2 in order to show that equal principal stresses imply equal corresponding principal stretches. In fact, all that is required is the invertibility of the stress-strain law (1.2). For the specific example of a certain compressible neo-Hookean material considered in Section 3 it is shown that the empirical inequalities (1.3) are indeed satisfied for deformations for which the invariant J lies within a certain range of values, see (3.3) below. Other possible restrictions upon the response functions are considered in [1]. If the isotropic elastic material is hyperelastic there exists a strain-energy function W (J1,J2,J) such that the Cauchy stress (1.2) reduces to −1 −1 −1 σ = W I +2J W1B 2J W2B (1.4) J − in which W1 := ∂W/∂J1, W2 := ∂W/∂J2 and WJ := ∂W/∂J. Comparing with (1.2) we see that −1 −1 β0 = W , β1 =2J W1, β−1 = 2J W2 (1.5) J − and the empirical inequalities (1.3) become W 0, W1 > 0, W2 0. (1.6) J ≤ ≥ From (1.4) the principal Cauchy stresses σii (i = 1, 2, 3) are given in terms of the principal stretches λi (i =1, 2, 3) by −1 2 −1 −2 σ = W +2J W1λ 2J W2λ (i =1, 2, 3). (1.7) ii J i − i Incremental moduli in nonlinear elasticity Page 3 of 16 In terms of the Lam´emoduli λ0, µ0 of linear isotropic elasticity, the bulk modulus, Poisson’s ratio and Young’s modulus are given by 2 λ0 µ0(3λ0 +2µ0) K0 = λ0 + 3 µ0, ν0 = , E0 = , (1.8) 2(λ0 + µ0) λ0 + µ0 see [2, Table 3]. The positive definiteness of the strain energy of linear isotropic elasticity requires the inequalities µ0 > 0, K0 > 0 (1.9) to hold. In turn these inequalities imply the further inequalities 1 2 1 < ν0 < , E0 > 0, λ0 > µ0. (1.10) − 2 − 3 Commonly in linear isotropic elasticity it is found that inequalities more restrictive than (1.10) hold: 1 0 < ν0 < 2 , E0 > 0, λ0 > 0. (1.11) 2 Incremental elastic moduli 2.1 The incremental bulk modulus Consider an initially stress-free isotropic elastic material subject to a hydrostatic pressure p = σ11 = σ22 = σ33 = σpp/3 given by (1.2) and (1.7), employing the summation convention− on− repeated− suffixes.− By subtraction of the diagonal components of (1.2) and use of (1.3)2,3 it can be shown that the principal stretches must be equal, so that λ1 = λ2 = λ3 = λ, say, and therefore the strain resulting from the pure hydrostatic pressure σ = pI is the pure dilatation F = λI. After the application of the hydrostatic − 3 pressure p, an initial volume V0 becomes V = V0J = V0λ . A further increment δp 2 in pressure results in a further volume change δV = 3V0λ δλ. The incremental bulk modulus is defined by change in pressure Incremental bulk modulus = − relative change in volume δp δσ11 Jδσ11 λδσ11 = − = 3 3 = = . (2.1) δV/V δ(V0λ )/(V0λ ) δJ 3δλ Thus the incremental (or tangent) bulk modulus is defined in the limit δλ 0 by either of the equivalent forms → λ ∂σ11 K(λ) := , (2.2) 3 ∂λ λ1=λ2=λ3=λ or J ∂σ K(λ) := pp . (2.3) 3 ∂J 1/3 λ1=λ2=λ3=J Page 4 of 16 N. H. Scott 2 −2 3 For a pure dilatation of stretch λ we have J1 = 3λ , J2 = 3λ , J = λ and (1.7) reduces to −1 −5 σ11 = W +2λ W1 2λ W2. (2.4) J − Use of (2.4) in (2.2) shows that the incremental bulk modulus of isotropic hyperelasticity may be expressed in the form 2 −1 10 −5 K(λ)= λ W1 + λ W2 (2.5) − 3 3 3 −7 2 −2 −2 + λ W +4λW11 +4λ W22 +4λ W 1 4λ W 2 8λ W12 JJ J − J − 2 in which W12 denotes ∂ W/∂λ1∂λ2, etc. This is similar to the incremental bulk moduli of Rivlin and Beatty [3] in their analysis of the stability of a compressible unit cube under dead loading. Rivlin and Beatty [3] also define incremental Young’s moduli and Poisson’s ratios similar to those defined below. These incremental elastic moduli are also related to the generalized Lam´emoduli of Beatty [4, Appendix]. On physical grounds we might expect δp and δV to have opposite signs, so that, for example, a further increase in pressure results in a further decrease in volume. Then physically reasonable response would require K(λ) > 0 for 0 <λ< , (2.6) ∞ a criterion which may be shown to be equivalent to the P-C inequality of Truesdell and Noll [1, (51.3)]. However, for an empirically determined material it might not be possible to insist that K(λ) be positive for all λ. We might find that K(λ) can be positive only for a smaller λ-interval, provided that this interval includes the stress-free ground state λ = 1. Then the material model is physically realistic only for strains within this λ- interval. We find in the limit λ 1 of (2.2) that K(1) = K0, so that the ground state incremental bulk modulus is→ equal to the usual bulk modulus of linear isotropic elasticity. We may define an average (or secant) bulk modulus in which the current stress is compared with the relative volume change: σ11 K(λ) := . (2.7) λ3 1 − We find that K(1) = K0 = K(1), so that this alternative definition of bulk modulus agrees with (2.2) in the ground state. Physically reasonable response would require K(λ) > 0 for 0 <λ< . (2.8) ∞ However, this condition is less restrictive than (2.6) as it requires only that σ11 and λ 1 − have the same sign, not that δσ11 and δλ should have the same sign. By integrating (2.2) we find that, since the reference configuration λ = 1 is stress free, (2.7) may be written λ ′ 3 1 K(λ ) ′ K(λ)= 2 ′ dλ , (2.9) λ + λ +1 · λ 1 1 λ − Z Incremental moduli in nonlinear elasticity Page 5 of 16 clarifying the sense in which K(λ) may be regarded as an average of the incremental bulk modulus. It is clear from (2.9) that criterion (2.6) implies (2.8) but that the converse does not hold: it is possible for K(λ) to be negative with K(λ) remaining positive, see, for example, (3.6) and (3.9).
Load more

Recommended publications
  • Lecture 13: Earth Materials
    Earth Materials Lecture 13 Earth Materials GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Hooke’s law of elasticity Force Extension = E × Area Length Hooke’s law σn = E εn where E is material constant, the Young’s Modulus Units are force/area – N/m2 or Pa Robert Hooke (1635-1703) was a virtuoso scientist contributing to geology, σ = C ε palaeontology, biology as well as mechanics ij ijkl kl ß Constitutive equations These are relationships between forces and deformation in a continuum, which define the material behaviour. GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Shear modulus and bulk modulus Young’s or stiffness modulus: σ n = Eε n Shear or rigidity modulus: σ S = Gε S = µε s Bulk modulus (1/compressibility): Mt Shasta andesite − P = Kεv Can write the bulk modulus in terms of the Lamé parameters λ, µ: K = λ + 2µ/3 and write Hooke’s law as: σ = (λ +2µ) ε GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Young’s Modulus or stiffness modulus Young’s Modulus or stiffness modulus: σ n = Eε n Interatomic force Interatomic distance GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Shear Modulus or rigidity modulus Shear modulus or stiffness modulus: σ s = Gε s Interatomic force Interatomic distance GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Hooke’s Law σij and εkl are second-rank tensors so Cijkl is a fourth-rank tensor. For a general, anisotropic material there are 21 independent elastic moduli. In the isotropic case this tensor reduces to just two independent elastic constants, λ and µ.
    [Show full text]
  • Efficient Hydraulic Fluids Compressibility Is Still One of the Most Critically Important Factors
    WORLDWIDE R. David Whitby Efficient hydraulic fluids Compressibility is still one of the most critically important factors. ydraulic fluids transmit power as a hydraulic system con- while phosphate and vegetable-oil esters have compressibilities H verts mechanical energy into fluid energy and subsequently similar to that of water. Polyalphaolefins are more compressible to mechanical work. To achieve this conversion efficiently, hydrau- than mineral oils. Water-glycol fluids are intermediate between lic fluids need to be relatively incompressible. Hydraulic fluids mineral oils and water. must also minimize wear, reduce friction, provide cooling and pre- Mineral oils are relatively incompressible, but volume reduc- vent rust and corrosion, be compatible with system components tions can be approximately 0.5% for pressures ranging from 6,900 and help keep the system free of deposits. kPa (1,000 psi) up to 27,600 kPa (4,000 psi). Compressibility in- Compressibility measures the rela- creases with pressure and temperature tive change in volume of a fluid or solid and has significant effects on high-pres- as a response to a change in pressure and sure fluid systems. Hydraulic oils typi- is the reciprocal of the volume-elastic cally contain 6% to 10% of dissolved air, modulus or bulk modulus of elasticity. which has no measurable effect on bulk Bulk modulus defines the pressure in- modulus provided it stays in solution. crease needed to cause a given relative Bulk modulus is an inherent property decrease in volume. of the hydraulic fluid and, therefore, is The bulk modulus of a fluid is nonlin- an inherent inefficiency of the hydraulic ear.
    [Show full text]
  • Velocity, Density, Modulus of Hydrocarbon Fluids
    Velocity, Density and Modulus of Hydrocarbon Fluids --Data Measurement De-hua Han*, HARC; M. Batzle, CSM Summary measured with pressure up to 55.2 Mpa (8000 Psi) and temperature up to 100 °C. Density and ultrasonic velocity of numerous hydrocarbon fluids (oil, oil based mud filtrate, hydrocarbon gases and Velocity of Dead Oil miscible CO2-oil) were measured at in situ conditions of pressure up to 50 Mpa and temperatures up to100 °C. Initial measurements were on the gas-free or ‘dead’ oils at Dynamic moduli are derived from velocities and densities. pressure and temperature (see Fig. 1). We used the Newly measured data refine correlations of velocity and following model to fit data: density to API gravity, Gas Oil ratio (GOR), Gas gravity and in situ pressure and temperature. Gas in solution is Vp (m/s) = A – B * T + C * P + D * T * P (1) largely responsible for reducing the bulk modulus of the live oil. Phase changes, such as exsolving gas during Here A is a pseudo velocity at 0 °C and room pressure (0 production, can dramatically lower velocities and modulus, Mpa, gauge), B is temperature gradient, C is pressure but is dependent on pressure conditions. Distinguish gas gradient and D is coefficient of coupled temperature and from liquid phase may not be possible at a high pressure. pressure effects. Fluids are often supercritical. With increasing pressure, a gas-like fluid can begin to behave like a liquid Dead & live oil of #1 B.P. = 1900 psi at 60 C Introduction 1800 1600 Hydrocarbon fluids are the primary targets of the seismic exploration.
    [Show full text]
  • FORMULAS for CALCULATING the SPEED of SOUND Revision G
    FORMULAS FOR CALCULATING THE SPEED OF SOUND Revision G By Tom Irvine Email: [email protected] July 13, 2000 Introduction A sound wave is a longitudinal wave, which alternately pushes and pulls the material through which it propagates. The amplitude disturbance is thus parallel to the direction of propagation. Sound waves can propagate through the air, water, Earth, wood, metal rods, stretched strings, and any other physical substance. The purpose of this tutorial is to give formulas for calculating the speed of sound. Separate formulas are derived for a gas, liquid, and solid. General Formula for Fluids and Gases The speed of sound c is given by B c = (1) r o where B is the adiabatic bulk modulus, ro is the equilibrium mass density. Equation (1) is taken from equation (5.13) in Reference 1. The characteristics of the substance determine the appropriate formula for the bulk modulus. Gas or Fluid The bulk modulus is essentially a measure of stress divided by strain. The adiabatic bulk modulus B is defined in terms of hydrostatic pressure P and volume V as DP B = (2) - DV / V Equation (2) is taken from Table 2.1 in Reference 2. 1 An adiabatic process is one in which no energy transfer as heat occurs across the boundaries of the system. An alternate adiabatic bulk modulus equation is given in equation (5.5) in Reference 1. æ ¶P ö B = ro ç ÷ (3) è ¶r ø r o Note that æ ¶P ö P ç ÷ = g (4) è ¶r ø r where g is the ratio of specific heats.
    [Show full text]
  • Adiabatic Bulk Moduli
    8.03 at ESG Supplemental Notes Adiabatic Bulk Moduli To find the speed of sound in a gas, or any property of a gas involving elasticity (see the discussion of the Helmholtz oscillator, B&B page 22, or B&B problem 1.6, or French Pages 57-59.), we need the “bulk modulus” of the fluid. This will correspond to the “spring constant” of a spring, and will give the magnitude of the restoring agency (pressure for a gas, force for a spring) in terms of the change in physical dimension (volume for a gas, length for a spring). It turns out to be more useful to use an intensive quantity for the bulk modulus of a gas, so what we want is the change in pressure per fractional change in volume, so the bulk modulus, denoted as κ (the Greek “kappa” which, when written, has a great tendency to look like k, and in fact French uses “K”), is ∆p dp κ = − , or κ = −V . ∆V/V dV The minus sign indicates that for normal fluids (not all are!), a negative change in volume results in an increase in pressure. To find the bulk modulus, we need to know something about the gas and how it behaves. For our purposes, we will need three basic principles (actually 2 1/2) which we get from thermodynamics, or the kinetic theory of gasses. You might well have encountered these in previous classes, such as chemistry. A) The ideal gas law; pV = nRT , with the standard terminology. B) The first law of thermodynamics; dU = dQ − pdV,whereUis internal energy and Q is heat.
    [Show full text]
  • Course: US01CPHY01 UNIT – 1 ELASTICITY – I Introduction
    Course: US01CPHY01 UNIT – 1 ELASTICITY – I Introduction: If the distance between any two points in a body remains invariable, the body is said to be a rigid body. In practice it is not possible to have a perfectly rigid body. The deformations are (i) There may be change in length (ii) There is a change of volume but no change in shape (iii) There is a change in shape with no change in volume All bodies get deformed under the action of force. The size and shape of the body will change on application of force. There is a tendency of body to recover its original size and shape on removal of this force. Elasticity: The property of a material body to regain its original condition on the removal of deforming forces, is called elasticity. Quartz fibre is considered to be the perfectly elastic body. Plasticity: The bodies which do not show any tendency to recover their original condition on the removal of deforming forces are called plasticity. Putty is considered to be the perfectly plastic body. Load: The load is the combination of external forces acting on a body and its effect is to change the form or the dimensions of the body. Any kind of deforming force is known as Load. When a body is subjected to a force or a system of forces it undergoes a change in size or shape or both. Elastic bodies offer appreciable resistance to the deforming forces. As a result, work has to be done to deform them. This amount of work is stored in body as elastic potential energy.
    [Show full text]
  • The Mechanical Properties and Elastic Anisotropies of Cubic Ni3al from First Principles Calculations
    crystals Article The Mechanical Properties and Elastic Anisotropies of Cubic Ni3Al from First Principles Calculations Xinghe Luan 1,2, Hongbo Qin 1,2,* ID , Fengmei Liu 1,*, Zongbei Dai 1, Yaoyong Yi 1 and Qi Li 1 1 Guangdong Provincial Key Laboratory of Advanced Welding Technology, Guangdong Welding Institute (China-Ukraine E.O. Paton Institute of Welding), Guangzhou 541630, China; [email protected] (X.L.); [email protected] (Z.D.); [email protected] (Y.Y.); [email protected] (Q.L.) 2 Key Laboratory of Guangxi Manufacturing System and Advanced Manufacturing Technology, Guilin University of Electronic Technology, Guilin 541004, China * Correspondence: [email protected] (H.Q.); [email protected] (F.L.) Received: 21 June 2018; Accepted: 22 July 2018; Published: 25 July 2018 Abstract: Ni3Al-based superalloys have excellent mechanical properties which have been widely used in civilian and military fields. In this study, the mechanical properties of the face-centred cubic structure Ni3Al were investigated by a first principles study based on density functional theory (DFT), and the generalized gradient approximation (GGA) was used as the exchange-correlation function. The bulk modulus, Young’s modulus, shear modulus and Poisson’s ratio of Ni3Al polycrystal were calculated by Voigt-Reuss approximation method, which are in good agreement with the existing experimental values. Moreover, directional dependences of bulk modulus, Young’s modulus, shear modulus and Poisson’s ratio of Ni3Al single crystal were explored. In addition, the thermodynamic properties (e.g., Debye temperature) of Ni3Al were investigated based on the calculated elastic constants, indicating an improved accuracy in this study, verified with a small deviation from the previous experimental value.
    [Show full text]
  • Effects of Pore Fluid Properties at High Pressure and Temperature On
    Effects of pore fluid properties at high pressure and temperature on seismic response Joel Walls*, Rock Solid Images and Jack Dvorkin, Rock Solid Images and Stanford University Summary means that 50 MPa occurs at approximately 5 km TVD. In overpressured formations, the pressure may be higher even We use software from the National Institute of at shallower depths. Also, tremendous amounts of Standards and Testing (NIST) to assess the adiabatic bulk domestic natural gas (55 Tcf offshore, according to MMS, modulus and density of natural gas and brine at pressures o and 135 Tcf onshore, according to USGS) may be available up to 200 MPa and temperatures up to 200 C. The at depths below 15,000 ft (about 5 km TVD) and as deep as calculations are based on equations of state which are 25,000 ft (about 7.5 km). This promising domestic gas calibrated and verified by many experimental potential calls for improvements in the interpretation of measurements. The results indicate that as pressure very deep seismic events and, as part of this technical task, increases from the normal range of 20 to 50 MPa to the valid estimates for the bulk modulus and density of the pore very high range of 150 to 200 MPa, the bulk modulus of fluid, especially gas, in deep reservoirs at very high methane may increase tenfold, from about 0.1 to about 1.0 pressure. GPa. The latter values are comparable to those for oil. For heavier hydrocarbon gases (ethane, propane, butane, and Comparison to Batzle-Wang (1992) their mixtures) the modulus will be even higher.
    [Show full text]
  • Physics 140 Sound
    Physics 140 Sound Chapter 12 Sound waves Sound is composed of longitudinal pressure waves. wave propagaon Compression è when Compression Compression parBcles come together RarefacBon RarefacBon RarefacBon è when parBcles are farthest apart 2 Physics 140, Prof. M. Nikolic Sound waves Described by the gauge pressure – difference between the pressure at a given point and average pressure 3 Physics 140, Prof. M. Nikolic The speed of sound waves T Let’s remember Chapter 11 and the speed of string waves v = µ The speed of sound B waves v = B is the bulk modulus of the medium ρ and ρ its density. The speed of sound Y v = waves in thin rods Y is the Young’s modulus of the ρ medium and ρ its density. All other equaons from Chapter 11 sBll apply. 2π v = f λ k = 4 Physics 140, Prof. M. Nikolic λ Conceptual quesBon – Speed of sound Q1 Given the informaon in the table below, which substance would have the highest speed of sound? Substance Bulk Modulus Density Speed (GPa) (kg/m3) (m/s) Aluminum 70 2700 Brass 61 8400 Copper 14 8900 Mercury 27 13600 A. aluminum B B. brass v = C. copper ρ D. mercury 5 Physics 140, Prof. M. Nikolic Conceptual quesBon – Speed of sound Q1 Given the informaon in the table below, which substance would have the highest speed of sound? Substance Bulk Modulus Density Speed (GPa) (kg/m3) (m/s) Aluminum 70 2700 5.1 x 103 Brass 61 8400 2.7 x 103 Copper 14 8900 1.25 x 103 Mercury 27 13600 1.4 x 103 A.
    [Show full text]
  • CHAPTER 13 FLUIDS • Density ! Bulk Modulus ! Compressibility
    Liquids CHAPTER 13 FLUIDS FLUIDS Gases • Density To begin with ... some important definitions ... ! Bulk modulus ! Compressibility Mass m DENSITY: , i.e., ρ = • Pressure in a fluid Volume V ! Hydraulic lift [M] ! Hydrostatic paradox Dimension: ⇒ 3 [ L] • Measurement of pressure −3 Units: ⇒ kg ⋅ m ! Manometers and barometers • Buoyancy and Archimedes Principle ! Upthrust Force F ! Apparent weight PRESSURE: , i.e., P = Area A • Fluids in motion [M] ! Continuity Dimension: ⇒ [L][T]2 ! Bernoulli’s equation Units: ⇒ N ⋅ m−2 ⇒ Pascals (Pa) Mass = Volume × Density Question 13.1: What, approximately, is the mass of air Volume of room ≈ 15 m ×10 m × 3 m −3 in this room if the density of air is 1 .29 kg ⋅ m ? 3 −3 ∴ M ≈ 450 m ×1.29kg ⋅ m = 580 kg (over half-a-ton!) The weight of a medium apple is ~ 1 N, so the mass of a medium apple is ~ 0.1 kg. 3 A typical refrigerator has a capacity of ~ 18 ft . 3 3 3 ∴ 18 ft ⇒ 18 × (0.305 m) = 0.51 m . 3 But 1 m of air has a mass of 1 .3 kg, so the mass of air in the refrigerator is Question 13.2: How does the mass of a medium sized 0 .51×1.3 kg = 0.66 kg, apples compare with the mass of air in a typical i.e., approximately the mass of 6 apples! 3 refrigerator? the density of cold air is ~ 1.3 kg/m . We don’t notice the weight of air because we are immersed in air ... you wouldn’t notice the weight of a bag of water if it was handed to you underwater would you? definitions (continued) ..
    [Show full text]
  • Module 3 Constitutive Equations
    Module 3 Constitutive Equations Learning Objectives • Understand basic stress-strain response of engineering materials. • Quantify the linear elastic stress-strain response in terms of tensorial quantities and in particular the fourth-order elasticity or stiffness tensor describing Hooke's Law. • Understand the relation between internal material symmetries and macroscopic anisotropy, as well as the implications on the structure of the stiffness tensor. • Quantify the response of anisotropic materials to loadings aligned as well as rotated with respect to the material principal axes with emphasis on orthotropic and transversely- isotropic materials. • Understand the nature of temperature effects as a source of thermal expansion strains. • Quantify the linear elastic stress and strain tensors from experimental strain-gauge measurements. • Quantify the linear elastic stress and strain tensors resulting from special material loading conditions. 3.1 Linear elasticity and Hooke's Law Readings: Reddy 3.4.1 3.4.2 BC 2.6 Consider the stress strain curve σ = f() of a linear elastic material subjected to uni-axial stress loading conditions (Figure 3.1). 45 46 MODULE 3. CONSTITUTIVE EQUATIONS σ ˆ 1 2 ψ = 2 Eǫ E 1 ǫ Figure 3.1: Stress-strain curve for a linear elastic material subject to uni-axial stress σ (Note that this is not uni-axial strain due to Poisson effect) In this expression, E is Young's modulus. Strain Energy Density For a given value of the strain , the strain energy density (per unit volume) = ^(), is defined as the area under the curve. In this case, 1 () = E2 2 We note, that according to this definition, @ ^ σ = = E @ In general, for (possibly non-linear) elastic materials: @ ^ σij = σij() = (3.1) @ij Generalized Hooke's Law Defines the most general linear relation among all the components of the stress and strain tensor σij = Cijklkl (3.2) In this expression: Cijkl are the components of the fourth-order stiffness tensor of material properties or Elastic moduli.
    [Show full text]
  • Speed of Sound
    An Internet Book on Fluid Dynamics Speed of Sound The next step in our treatment of compressible flows is to develop relations for and understanding of the speed of sound in a compressible fluid. Sound consists of small amplitude pressure waves propagating through the fluid and, in most of the analysis which follows we will retain only the mathematical terms that are linear in the pressure perturbations while neglecting all terms that are quadratic or higher order in those perturbations. We begin with the following thought experiment. Consider a rigid tube of cross-sectional area, A, that contains a piston that is initially at rest. The tube is filled with gas at rest with pressure, p, temperature, T , and density, ρ, but at time t = 0 it is suddenly given a small velocity, Δu, directed into the gas as sketched in Figure 1. This generates a wave that travels down the pipe with velocity, V ,and Figure 1: Tube with piston set in motion at t = 0, with small velocity, Δu. leaves behind it gas that now has a velocity, Δu, and increased pressure, p +Δp, temperature, T +ΔT , and density, ρ +Δρ. We seek to determine the wave velocity, V , by applying the basic conservation laws Figure 2: Flow with wave at rest. that apply to the gas flow. To do so we examine the flow in a frame fixed in the wave as sketched in Figure 2 by applying a Galilean transformation to the flow in Figure 1. Conservation of mass across the wave then requires that A(ρ +Δρ)(Δu − V )=−ρV A (Bod1) so that V Δρ Δu = (Bod2) (ρ +Δρ) In addition, applying the momentum equation to the wave yields AΔp = AρV 2 − A(ρ +Δρ)(Δu − V )2 (Bod3) that, using the continuity result (Bod2), leads to Δp Δρ V 2 = 1+ (Bod4) Δρ ρ The speed of sound is defined as the speed of the wave for very small perturbations or V → c when Δu → 0(andsoΔp, ΔT,Δρ → 0) (Bod5) and therefore dp c2 = (Bod6) dρ This result holds for any substance whether solid, liquid or gas and, at various points throughout this book we will use it for many different substances.
    [Show full text]