Chapter 6 Lectures begin Feb. 26, 08 1 Chapter 6: Mechanical Properties ISSUES TO ADDRESS... Stress and strain: What are they and why are they used instead of load and deformation? Elastic behavior: Occurs for small loads. What materials deform least? Plastic behavior: At what point does permanent deformation occur? What materials are most resistant to permanent deformation? Toughness and ductility: What are they and how do we measure them? 2 Tensile Elastic Deformation 1. Initial 2. Small load bonds stretch return to initial 3. Unload F Elastic means reversible! F Linearelastic Non-Linearelastic 3 Tensile <a href="/keyword/plastic-deformation/" >plastic deformation</a> (Metals) 1. Initial 2. Small load bonds stretch &amp; planes shear elastic 3. Unload planes still sheared plastic F F Plastic means permanent! plastic F linear elastic ti las p c Note: The sample is still deformed elastically when it is being deformed plastically. The elastic deformation is not converted to <a href="/keyword/plastic-deformation/" >plastic deformation</a> . plastic linear elastic 4 Force Displacement Diagram Stress Strain Diagram = Stress (Pa) = force per unit area = Strain (unitless) = 5 normalized displacement Elastic region in - diagram Slope = Modulus of elasticity This is also called Young's modulus = E Units are force per unit area (N/m2 = Pa or pounds per square inch (psi)) Hooke's law = E 6 What happens to the material when we pull on it? Material lengthens in direction in which it is pulled Material shrinks in the perpendicular direction(s) 7 Pull a little harder get a little <a href="/keyword/plastic-deformation/" >plastic deformation</a> Yield point - Y Easy to define point where <a href="/keyword/plastic-deformation/" >plastic deformation</a> begins (i.e., where curvature begins) 0.002 offset yield point = Y Real yield point called the proportional limit This point is very difficult to determine Difficult to determine exact point where curvature (<a href="/keyword/plastic-deformation/" >plastic deformation</a> ) begins Practical way to determine yield point - use 0.002 strain offset (0.2%) to determine yield point Y0.002 = Y - we will use Y in class Construct line parallel to lower portion of - curve Begin line at = 0.002, = 0 MPa Draw line parallel to elastic region Y is where line hits - curve 8 Pull harder until the material breaks Tensile strength (TS) = ultimate tensile strength (UTS) = maximum stress on engineering stress-strain curve. TS = UTS y engineering stress F = fracture strength or breaking strength or rupture strength Typical response of a metal strain engineering strain Metals: occurs when noticeable necking starts. Polymers: occurs when polymer backbone chains are aligned and about to break. Neck acts as stress concentrator (and changes stress from uniaxial to triaxial) 9 True stress-strain curve True stress-strain curve True stress uses instantaneous area rather than initial area of sample True strain is adjusted through use of natural log True stress-strain curve never decreases 10 Summary Stress and strain: These are size-independent measures of load and displacement, respectively. Elastic behavior: This reversible behavior often shows a linear relation between stress and strain. To minimize deformation, select a material with a large elastic modulus. Plastic behavior: This permanent deformation behavior occurs when the tensile (or compressive) uniaxial stress reaches y. Toughness: The energy needed to break a unit volume of material = area under stress strain curve. Ductility: materials ability to undergo appreciable elongation before breaking express as % elongation or % reduction in area for a tensile test 11 Engineering Stress Tensile stress, : Shear stress, : Ft Area, A Area, A Ft F Fs Ft Ft lb f N = 2 or = in m2 Ao original area before loading Fs Fs = Ao F Ft Stress has units: N/m2 = Pa, or lbf/in2 12 Common States of Stress -1 Simple tension: cable F F = Ao M F A o = cross sectional area (when unloaded) Ski lift (<a href="/keyword/photo-courtesy/" >photo courtesy</a> P.M. Anderson) Torsion (a form of shear): drive shaft Ac M Fs Ao Fs = Ao 2R Note: = M/AcR here. 13 Common States of Stress -2 Simple compression: Ao Canyon Bridge, Los Alamos, NM (<a href="/keyword/photo-courtesy/" >photo courtesy</a> P.M. Anderson) (<a href="/keyword/photo-courtesy/" >photo courtesy</a> P.M. Anderson) Balanced Rock, Arches National Park F = Ao Note: compressive structure member ( &lt; 0 here). 14 Common States of Stress -3 Bi-axial tension: Hydrostatic compression: Snare drum http://billyblastdrums.easystor ecreator.com/images/12Blue%20Pearl%20Snare%20 Drum%201.jpg Fish under water y&gt; 0 x&gt; 0 z = 0 (<a href="/keyword/photo-courtesy/" >photo courtesy</a> P.M. Anderson) h&lt; 0 15 Engineering Strain - Tension w0 l0 L w0 l0 Tensile strain () Lateral strain (L) = l0 Strain is always dimensionless. -L L = w0 16 Engineering Strain - Shear x y 90 90 - y 90 - Shear strain () = x/y = tan Strain is always dimensionless. 17 Stress-Strain Testing Typical tensile test machine extensometer specimen Typical tensile specimen gauge length 18 Linear Elastic Properties Modulus of Elasticity, E: (also known as Young's modulus) Hooke's Law: =E Tension F E Compression Linearelastic F simple tension test 19 Young's modulus and bond strength Slope of stress-strain plot (= Young's modulus E) depends on bond strength of material dE F= dr 20 Young's Moduli: Comparison Metals Alloys 1200 1000 800 600 400 Graphite Composites Ceramics Polymers /fibers Semicond Diamond Si carbide Al oxide Si nitride Si crystal &lt;100&gt; &lt;111&gt; E(GPa) 200 100 80 60 40 Tungsten Molybdenum Steel, Ni Tantalum Platinum Cu alloys Zinc, Ti Silver, Gold Aluminum Magnesium, Tin Carbon fibers only CFRE(|| fibers)* Aramid fibers only Glass -soda AFRE(|| fibers)* Glass fibers only GFRE(|| fibers)* Concrete GFRE* Graphite CFRE* GFRE( fibers)* CFRE( fibers) * AFRE( fibers) * Based on data in Table B2, Callister 7e. Composite data based on reinforced epoxy with 60 vol% of aligned carbon (CFRE), aramid (AFRE), or glass (GFRE) fibers. 109 Pa 20 10 8 6 4 2 1 0.8 0.6 0.4 0.2 Polyester PET PS PC PP HDPE PTFE LDPE Epoxy only This table shows the great strength of the covalent and metallic bonds 21 Wood( grain) Poisson's ratio, L w0 l0 = l0 -L L = w0 Note: L &lt;0 Poisson's ratio () -L = Units: and have no units Compression metals: ~ 0.33 ceramics: ~ 0.25 polymers: ~ 0.40 Tension 22 Shear and Bulk moduli Elastic Shear modulus, G: M G M simple torsion test =G Elastic Bulk modulus, K: P K V P Vo P P pressure test: Init. vol =Vo Vol chg. = V (&lt;0) V P = -K Vo Special relations for isotropic materials: E G= 2(1 + ) E K= 3(1 - 2) 23 Useful Linear Elastic Relationships Simple tension: Simple torsion: = Fl o = - Fw o L EA o EA o F Ao /2 lo = 2Ml o 4 r o G M = moment = angle of twist wo lo 2ro L /2 Material, geometric, and loading parameters all contribute to deflection. Larger elastic moduli minimize elastic deflection. 24 Plastic (Permanent) Deformation Simple tension test: engineering stress, Elastic+Plastic at larger stress Starts Elastic permanent (plastic) after load is removed engineering strain, plastic strain 25 p Yield Strength, y Stress at which observable <a href="/keyword/plastic-deformation/" >plastic deformation</a> has occurred often defined as the proof stress p = 0.002 tensile stress, y Note the observable definition p will include a small amount of <a href="/keyword/plastic-deformation/" >plastic deformation</a> (p = 0.002) This measure of yield is an engineering approximation 26 engineering strain, p = 0.002 Yield Strength : Comparison Metals/ Alloys 2000 Steel (4140) qt Graphite/ Ceramics/ Semicond Polymers Composites/ fibers in ceramic matrix and epoxy matrix composites, since in tension, fracture usually occurs aaaaaaaaaat or before yield. Yield strength, y (MPa) 700 600 500 400 300 200 Ti (5Al-2.5Sn) a W (pure) Cu (71500) cw Mo (pure) Steel (4140) a Steel (1020) cd since in tension, fracture usually occurs at or before yield. 10 00 Room temperature values Based on data in Table B4, Callister 7e. a = annealed hr = hot rolled ag = aged cd = cold drawn cw = cold worked qt = quenched &amp; tempered Hard to measure , Al (6061) ag Steel (1020) hr Ti (pure) a Ta (pure) Cu (71500) hr 100 70 60 50 40 30 20 Tin (pure) Al (6061) a dry PC Nylon 6,6 PET PVC humid PP HDPE Hard to measure, Note the enormous difference between metals and polymers and ceramics etc 27 LDPE 10 Tensile Strength, TS Maximum stress on engineering stress-strain curve. TS engineering stress y F = fracture or ultimate strength Typical response of a metal strain engineering strain Metals: occurs when noticeable necking starts. Polymers: occurs when polymer backbone chains are aligned and about to break. Neck acts as stress concentrator (and changes stress from uniaxial to triaxial) 28 Ductility Plastic tensile strain at failure: smaller %EL Engineering tensile stress, Lf - Lo x 100 %EL = Lo larger %EL Lo Ao Af Lf Engineering tensile strain, Reduction of area at the neck is another ductility measure: Ao - Af %RA = x 100 Ao 29 Toughness Energy to break a unit volume of material Approximate by the area under the stress-strain curve. Engineering tensile stress, small toughness (ceramics) large toughness (metals) very small toughness (unreinforced polymers) Engineering tensile strain, Brittle fracture: only elastic energy appears to be used Ductile fracture: elastic + significant plastic energy 30 Resilience, Ur Ability of a material to store energy Energy stored efficiently in elastic region Ur = 0 y d If we assume a linear stress-strain curve this simplifies to 1 Ur y y 2 31 Elastic Strain Recovery 32 Hardness Resistance to permanently indenting the surface. Large hardness means: --resistance to <a href="/keyword/plastic-deformation/" >plastic deformation</a> or cracking in compression. --better wear properties. apply known force e.g., 10 mm sphere measure size (width or depth) of indent after removing load D most plastics brasses Al alloys d easy to machine steels file hard Smaller indents mean larger hardness. cutting tools nitrided steels diamond increasing hardness 33 Hardness Measurement Rockwell indentation by ball or diamond Small indent ~0.5mm Each scale runs to 130 but only useful in range 20-100. Minor load 10 kg Major load 60 (A), 100 (B) &amp; 150 (C) kg A = diamond, B = 1/16 in. ball, C = diamond 34 Hardness: Measurement Table 6.5 35 True Stress &amp; Strain Note: cross-section changes when sample stretched, especially after peak load True stress True Strain T = ln(l i l o ) T = F Ai T = (1 + ) T = ln(1 + ) 36 Work hardening occurs in plastic flow An increase in y due to <a href="/keyword/plastic-deformation/" >plastic deformation</a> . T y 1 y large hardening small hardening 0 T Curve fit to the stress-strain response valid from y to UTS measured on engineering - curve: T = K T &quot;true&quot; stress (F/A) ( ) n hardening exponent: n = 0.15 (some steels) to n = 0.5 (some coppers) &quot;true&quot; strain: ln i l li 37 Variability in Material Properties Elastic modulus is atomic bonding material property Plastic properties often are controlled by sample defect populations with significant sample to sample variability. Statistical approach often needed Mean xn x= n 2 n (x i - x ) s= n -1 n 1 2 Standard Deviation where n is the number of data points 38 Design or Safety Factors Design uncertainties mean we do not push the limit. Factor of safety, N Often N is working = y N between 1.2 and 4 Example: Calculate a diameter, d, to ensure that yield does not occur in the 1045 carbon steel rod below. Use a factor of safety of 5. working = 220,000N d2 / 4 y N d Lo ( ) 5 1045 plain carbon steel: y = 310 MPa TS = 565 MPa F = 220,000N d = 0.067 m = 6.7 cm 39