Lecture - Chapter 6

Lecture - Chapter 6 - Mechanical Properties Chapter 6 Why...

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Unformatted text preview: Mechanical Properties Chapter 6 Why Study Mechanical Properties ? As engineers we need to know the mechanical properties of materials in order to choose the right material for a given application. If we know how these properties are measured, we can also make better decisions about which materials to select. Learning Objectives Define engineering stress, strain, Hooke's law and Poisson's ratio. Calculate modulus of elasticity, yield strength, tensile strength, percent elongation, ductility, flexural strength, hardness etc. Stress and Strain Stress () is the measure of how much force per unit area, or pressure, is acting on a particular material or member. = F / A Strain () is measure of how much a material or member deforms in response to the applied stress relative to its original dimensions. = lf l0 / l0 Stress and Strain cont. Stress is also defined as: = Load / Initial cross sectional area This stress is called engineering stress. Strain is also defined as: = Change in length ( l) / lo This strain is defined as engineering strain. Reference Figure 6.1 (a). Fig 6.1 (a) Schematic illustration of how a tensile load produces an elongation and positive linear strain. Dashed lines represent the shape before deformation. Tension Tests The first and most important test performed on a new alloy is the tension test, since several mechanical properties can be obtained from one test. Reference Figure 6.1 (a). For example: Modulus of Elasticity, Yield Strength, Ultimate Tensile Strength, Fracture Strength, Poisson's Ratio, and Ductility (Percent Elongation). Tension Tests cont. Usually a specimen with a rectangular or circular crosssection is pulled in tension until failure. Stress versus strain is then plotted. Reference Figures 6.2 & 6.3 Fig 6.2 A standard tensile specimen with circular cross section. Fig 6.3 Schematic representation of the apparatus used to conduct tensile stressstrain tests. Standard Tension Test Methods ASTM E8 Standard Test Methods for Tension Testing of Metallic Materials ASTM E101299 Standard Practice for Verification of Specimen Alignment Under Tensile Loading ASTM E34502 Standard Test Methods of Tension Testing of Metallic Foil Compression Tests This test is performed in a similar manner like tensile test but in compression, equations used to express stress and strain are same like in tensile stress and strain but with a negative sign. Reference Fig 6.1 (b). Fig 6.1 (b) Schematic illustration of how a compressive load produces contraction and a negative linear strain. Standard Compression Test Methods ASTM E9 Standard Test Methods for Compression Testing of Metallic Materials at Room Temperature Geometric consideration of the State of Stress Figure 6.4 Geometric Consideration of the State of Stress Consider a cylindrical tensile specimen of Figure 6.4 that is subjected to a tensile force F. Consider a plane (plane A1) in the specimen (plane pp') that is oriented as some arbitrary angle relative to the plane of the specimen end force (plane A). Geometric Consideration of the State of Stress The tensile stress, in the plane A is: =F/A The component of force normal to plane A1 = F Cos Normal (tensile stress) in plane A1 = F Cos / A1 But Cos = A/A1 or A1 = A Cos Geometric Consideration of the State of Stress Thus the normal stress in plane A1 = (F Cos )/(A/Cos ) = F/A Cos2 = Cos2 The component of force parallel to plane A1 = F Sin The shear stress in the plane A1 = F Sin / A1 = (F Sin ) / (A / Cos ) = F/A Sin Cos = Sin Cos Determination of Crystallographic Orientation ASTM E8201 Standard Test Method for Determining the Orientation of a Metal Crystal Elastic Modulus Elastic modulus is defined as the rate of change of bonding force between two atoms with respect to the inter atomic distance. Strong interatomic bonding will give a high elastic modulus (higher slope). Weak interatomic bonding will give a low modulus (low slope) (Figure 7.7). Fig 6.7 Force Vs inter atomic separation for weakly and strongly bonded atoms. Elastic Deformation Stress Strain Behavior Hooke's law states that stress, , is linearly proportional to strain, , or = E . E, the constant of proportionality, is defined as the modulus of elasticity or Young's modulus. Deformation in which stress is linearly proportional to strain is defined as elastic deformation. Elastic Deformation Stress Strain Behavior cont. The plot of stress versus strain will be a straight line in this case, as shown in Figure 6.5. Fig 6.5 Schematic stressstrain diagram showing linear elastic deformation for loading and unloading cycles. Elastic Deformation Stress Strain Behavior cont. E is the slope of this straight line or: E= d /d . The steeper the slope, the stiffer the material or the more stress which is required to deform the material. Under the elastic deformation, when the applied load is released, the strain becomes zero or the specimen goes back to its original shape. Non-Linear Elastic Deformation For some materials, stress strain curve is non linear (Fig 6.6). In this case either a tangent or a secant modulus is used. Slope of the stressstrain curve at some specified level of stress is the tangent modulus. Secant modulus is obtained from the slope of the straight line between zero stress and a specified stress 1. Fig 6.6 Schematic Stressstrain diagram showing nonlinear elastic behavior, and how secant and tangent moduli are determined. Temperature Effects on Elastic Modulus As temperature increases the elastic modulus decreases, reference Figure 6.8. Some materials are more temperature sensitive than others, aluminum for example, but the temperature at which a material is used always has an impact on its strength. Fig 6.8 Plot of modulus of elasticity versus temperature for tungsten, steel, and aluminum. Shear Tests In shear tests, the force is applied parallel to the crosssectional area Shear stress = = F /Ao Shear strain = = tan (Angle of deformation) Reference Figure 6.1 (c). Fig 6.1 (c) Schematic representation of shear strain , where = tan . Torsional Tests In torsional tests, the structural member is twisted under an applied force or torque, in the manner shown in Figure 6.1 (d). Fig 6.1 (d) Schematic representation of Torsional deformation (i.e angle of twist produced by an applied torque T. Shear stress and strain Shear stress , , and shear strain, , are also linearly proportional to each other = G where G is the shear modulus. Poisson's Ratio Poisson's ratio, , is defined as the ratio of the lateral strain to axial strain when the applied stress is along the axial direction, where the stress is applied along the Z direction. z = axial strain x, y = lateral strain( strain in the perpendicular direction) = * z Reference Figure 6.9. Fig 6.9 Axial (Z) elongation (positive strain) and lateral (X and Y) contractions (negative strains) in response to an imposed tensile stress. Relationship Between Shear Stress and the Elastic Modulus Shear and elastic moduli are related to each other through Poisson's ratio E = 2G (1+) Plastic Deformation The stressstrain behavior of the metals is nonlinear as shown in Figure 6.10. After unloading a permanent strain stays in the sample or it is permanently deformed. At the atomic level, the inter atomic bonds are broken and a large number of atoms move to a new position leading to a permanent deformation. Tensile Properties Proportional limit The point at which stressstrain curve deviates from linearity is called proportional limit. Yield Strength The point at which a line offset by 0.002 strain, which is parallel to the slope of the elastic curve intersects the stress vs. strain curve. Reference Figure 6.10. Fig 6.10 (a) Typical stressstrain behavior for a metal showing elastic and plastic deformations, the proportional limit P, and the yield strength y as determined using the 0.002 strain offset method. Yield Point Phenomenon For some materials, there is a more obvious yield point, which does not require calculation by the 0.2% offset method (reference Fig 6.10 b). Upper Yield Point The maximum stress reached while exhibiting elastic behavior. Lower Yield Point The minimum stress after yielding which continues to produce strain for some before rising again with further increases strain. Fig 6.10 (b) Representative stressstrain behavior found for some steels demonstrating the yield point phenomenon. Tensile Properties Yield Strength If there is no marked yield point, the stress corresponding to 0.2% offset strain secant line is defined as the yield strength of the metal. The stressstrain curve of steel has marked yield points (upper and lower yield points). The stress corresponding to the lower yield point is defined as the yield strength. Tensile Strength The maximum point in the stressstrain curve is defined as tensile strength, T.S., or ultimate tensile strength, U.T.S. Reference Figure 6.11. Fracture Strength The stress at which the sample breaks is defined as the fracture strength, F.S. (Figure 6.11). The sample starts to neck down after the tensile strength is applied and the fracture occurs at the necking region. Fig 6.11 Typical engineering stressstrain behavior to fracture, point F. The tensile strength TS is indicated at point M. Tensile Strength Fracture Strength Fig 6.12 The stressstrain behavior for the brass specimen discussed in example problem 6.3. Ductility Ductility is defined s percentage elongation or percentage reduction in area before fracture. % Elongation = % El =[( lf lo) / lo] *100 Where: lo = initial length lf = length at the fracture Ductility % Reduction in Area = % RA %RA = [(Af Ao) / Ao] * 100% Where: Ao = original cross sectional area Af = cross sectional area at the fracture. Ductility Ductile metals have large % elongation or % reduction in area or large plastic deformation before fracture. Brittle materials, on the other hand, have very little plastic deformation before fracture. Resilience Resilience is the capacity of the material to absorb energy when it is deformed elastically. Modulus of resilience, Ur, which is the strain energy per unit volume required to stress a material from zero stress to yield strength. It is the area under the stressstrain curve up to the yield strength, and is expressed by: Ur = 2y / 2E Fig 6.15 Schematic representation showing how modulus of resilience (corresponding to the shaded area) is determined from the tensile stressstrain behavior of a material. Fig 6.14 Engineering stressstrain behavior for iron at three temperatures. Toughness Toughness is the ability of the material to absorb energy up to fracture. f Toughness = d 0 Or the entire area under the stressstrain curve. Fig 6.13 Schematic representations of tensile stressstrain behavior for brittle and ductile loaded to fracture. True Stress and True Strain t = F/Ai True Stress t , is defined as F = applied load Ai = instantaneous area. t = ln (li / lo) True Strain, t , defined as li = instantaneous length lo = original length Relation Between True Stress and Strain And Engineering Stress and Strain t = F/Ai = (F/Ao)*( Ao/ Ai) Ao = original cross sectional area Since the total volume is conserved. Ao lo= Ai li or Ao /Ai = li /lo Engineering Strain = = ( li lo) / lo = ( li / lo)1 Or ( li / lo) = (1+ ) Thus t = (F/ Ao) x (li / lo) F/ Ao= =Engineering Stress or t = (1+ ) t = ln (li / lo) = ln(1+) True Stress and True Strain Beyond the yield point the relation between true stress and true strain is t = K tn where n and K are constants. n is called strain hardening exponent and K is called strength coefficient. Fig 6.16 A comparison of typical tensile engineering stress strain and true stressstrain behaviors. Strain Hardening Strain hardening is the strengthening (or hardening) of a material in response to plastic deformation. Most metals exhibit this to one degree or another, but especially high strength and other engineering alloys. As deformation occurs additional dislocations are generated. Strain Hardening cont. The increased dislocation density, instead of causing more deformation does the opposite (reference Figure 6.16). High dislocation density inhibits the motion of dislocations, similar to the way the motion of cars is restricted in a traffic jam. It also limits strain recovery, contributing to permanent plastic deformation (reference Fig 6.17). Fig 6.17 Schematic tensile stressstrain diagram showing the phenomena of elastic strain recovery and strain hardening. Hardness Hardness is a measure of material's resistance to localized plastic deformation ( i.e. a small dent or scratch). To measure hardness a sphere, diamond pyramid or a diamond cone of a very hard material is pressed on the surface of the sample to make a permanent indentation. Table 6.4 Hardness Testing Techniques Hardness The hardness is determined from the amount of load and the shape and size of the indenter and the indentation. Several hardness tests can be used to measure hardness of metals as described below. Rockwell Hardness Tests Rockwell hardness test can be performed to measure hardness of any metals and some polymers. Spherical balls made of steel and conical diamond are used as an indenter. ASTM (American Society for Testing Materials) standard method for measuring hardness is used. An initial minor load and a larger major load are used. Table 6.5a Rockwell Hardness Scales Rockwell Hardness Tests There are two types of test Rockwell and Superficial Rockwell. Superficial Rockwell test is performed on thin samples. For Rockwell test, minor load used is 10 Kg and Major load used are 60,100 and 150 Kg. Depending on the indenter used, each scale is indented by a letter. Thus 80 HRB means a Rockwell hardness of 80 on the B scale. For superficial test 3 kg is the minor load 15, 30 and 45 kg are the possible major load values. 60 HR 30 W indicates a superficial hardness of 60 in a 30 W scale. Table 6.5b Superficial Rockwell Hardness Scales Standard Rockwell Hardness Test Methods ASTM E1802 Standard Test Methods for Rockwell Hardness and Rockwell Superficial Hardness of Metallic Materials Brinell Hardness Test In the Brinell hardness tests, a hard spherical indenter made of steel or tungsten carbide and of diameter 10.00 mm is pressed on the surface of the indenter. The Brinell hardness number is then calculated from the amount of load, the diameter of the indenter and the diameter of the indentation (see table 6.4 for equation). Fig 6.18 Comparison of several hardness scales. Fig 6.19 Representation between hardness and tensile strength for steel, brass, and cast iron. Knoop and Vickers Micro Hardness Tests These two micro hardness tests are done on smaller samples with smaller loads (between 1 and 1000 g). Knoop micro hardness test is done on ceramic materials. Standard Vickers Hardness Test Methods ASTM E92 Standard Test Method for Vickers Hardness of Metallic Materials Standard Microindentation Hardness Test Methods ASTM E38499 Standard Test Method for Microindentation Hardness of Materials Hardness Conversion Comparison of several hardness scale is shown in Figure 6.18 . Relation between hardness and tensile strength. TS (Mpa) =3.45 * HB HB = Brinell hardness TS (psi) = 500 * HB Standard Hardness Conversion Tables ASTM E14002 Standard Hardness Conversion Tables for Metals Relationship Among Brinell Hardness, Vickers Hardness, Rockwell Hardness, Superficial Hardness, Knoop Hardness, and Scleroscope Hardness Fig 6.20 (a) Tensile strength data associated with example problem 6.6. (b) The manner in which these data could be plotted. An average value is obtained by dividing the sum of all measured values by the number of measurements taken. Where n is the number of observations or measurements and x, is the value of a discrete measurement. In mathematical terms, the average of some parameter x is: Computation of Average and Standard Deviation Computation of Average and Standard Deviation Furthermore, the standard deviation s is determined using the following expression: Where x , and n are defined above. A large value of the standard deviation corresponds to a high degree of scatter. Factor of Safety Safe stress W is used as: W = y / N Where: y = yield stress N = factor of safety Problems Group Discussion Problem 6.17 Group Discussion Problem 6.29 Hint: Use Equations 6.2, 6.8 & 6.9 HW #5 6.5, 6.9, 6.16, 6.28, 6.31, 6.33, 6.52 Due 3/7/05 ...
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