Chapter 6: Behavior Of Material Under Mechanical Loads = Mechanical Properties. Stress and strain: What are they and why are they used instead of load and deformation Elastic behavior: Recoverable Deformation of small magnitude Plastic behavior: Permanent deformation We must consider which materials are most resistant to permanent deformation? Toughness and ductility: Defining how much energy that a material can take before failure. How do we measure them? Hardness: How we measure hardness and its relationship to material strength

Elastic Deformation d F F d 1. Initial 2. Small load 3. Unload bonds stretch return to initial F d Linear- elastic Non-Linear- Elastic means reversible!

Plastic Deformation (Metals) 1. Initial 2. Small load 3. Unload p lanes still sheared F d elastic + plastic bonds stretch & planes shear plastic F d linear elastic plastic Plastic means permanent!

Comparison of Units: SI and Engineering Common Eng. Common Force Newton (N) Pound-force (lbf) Area mm2 or m2 in2 Stress Pascal (N/m2) or MPa (106 pascals) psi (lbf/in2) or Ksi (1000 lbf/in2) Strain (Unitless!) mm/mm or m/m in/in Conversion Factors SI to Eng. Common Eng. Common to SI N*4.448 = lbf Lbf*0.2248 = N Area I mm2*645.16 = in2 in2 *1.55x10-3 = mm2 Area II m2 *1550 = in2 in2* 6.452x10-4 = m2 Stress I - a Pascal * 1.450x10-4 = psi psi * 6894.76 = Pascal Stress I - b Pascal * 1.450x10-7 = Ksi Ksi * 6.894 x106 = Pascal Stress II - a MPa * 145.03 = psi psi * 6.89x 10-3 = MPa Stress II - b MPa * 1.4503 x 10-1= Ksi Ksi * 6.89 = MPa One other conversion: 1 GPa = 103 MPa

we can also see the symbol ‘s’ used for engineering stress • Tensile stress, s: original area before loading Area, A F t s = A o 2 f m N or in lb • Shear stress, t: Area, A F t s = A o  Stress has units: N/m2 (Mpa) or lbf/in2 we can also see the symbol ‘s’ used for engineering stress

Geometric Considerations of the Stress State The four types of forces act either parallel or perpendicular to the planar faces of the bodies. Stress state is a function of the orientations of the planes upon which the stresses are taken to act.

Common States of Stress • Simple tension: cable A o = cross sectional area (when unloaded) F o s = F A s Ski lift (photo courtesy P.M. Anderson) • Torsion (a form of shear): drive shaft M A o 2R F s c o t = F s A Note: t = M/AcR here. Where M is the “Moment” Ac shaft area & R shaft radius

OTHER COMMON STRESS STATES (1) • Simple compression: A o Balanced Rock, Arches National Park (photo courtesy P.M. Anderson) Canyon Bridge, Los Alamos, NM o s = F A Note: compressive structure member (s < 0 here). (photo courtesy P.M. Anderson)

OTHER COMMON STRESS STATES (2) • Bi-axial tension: • Hydrostatic compression: Fish under water Pressurized tank (photo courtesy P.M. Anderson) (photo courtesy P.M. Anderson) s z > 0 q s < 0 h

Engineering Strain: w e = d L - d e = w q q g = Dx/y = tan • Tensile strain: d /2 L o w • Lateral strain: e = d L o - d e L = w o Here: The Black Outline is Original, Green is after application of load • Shear strain: Adapted from Fig. 6.1 (a) and (c), Callister 7e. q Strain is always Dimensionless! x q g = Dx/y = tan y 90º - q 90º We often see the symbol ‘e’ used for engineering strain

Stress-Strain: Testing Uses Standardized methods developed by ASTM for Tensile Tests it is ASTM E8 • Typical tensile test machine • Typical tensile specimen (ASTM A-bar) Adapted from Fig. 6.2, Callister 7e. gauge length specimen extensometer Adapted from Fig. 6.3, Callister 7e. (Fig. 6.3 is taken from H.W. Hayden, W.G. Moffatt, and J. Wulff, The Structure and Properties of Materials, Vol. III, Mechanical Behavior, p. 2, John Wiley and Sons, New York, 1965.)

LOAD vs. EXTENSION PLOTS During Tensile Testing, Instantaneous load and displacement is measured LOAD vs. EXTENSION PLOTS These load / extension graphs depend on the size of the specimen. E.g. if we carry out a tensile test on a specimen having a cross-sectional area twice that of another, you will require twice the load to produce the same elongation. The Force .vs. Displacement plot will be the same shape as the Eng. Stress vs. Eng. Strain plot

The Engineering Stress - Strain curve Divided into 2 regions ELASTIC PLASTIC

Linear: Elastic Properties Units: E: [GPa] or [psi] : in [Mpa] or [psi] : [m/m or mm/mm] or [in/in] • Modulus of Elasticity, E: (also known as Young's modulus) • Hooke's Law: s = E e F A o d /2 L Lo w s Linear- elastic E e Here: The Black Outline is Original, Green is after application of load

Typical Engineering Issue in Mechanical Testing: Aluminum tensile specimen, square X-Section (16.5 mm on a side) and 150 mm long Pulled in tension to a load of 66700 N Experiences elongation of: 0.43 mm Determine Young’s Modulus if all the deformation is recoverable

Solving:

Poisson's ratio, n eL e - n e n = - • Poisson's ratio, n: Units: metals: n ~ 0.33 ceramics: n ~ 0.25 polymers: n ~ 0.40 Units: n: dimensionless  > 0.50 density increases  < 0.50 density decreases (voids form)

Other Elastic Properties simple torsion test M t G g • Elastic Shear modulus, G: t = G g • Elastic Bulk modulus, K: pressure test: Init. vol =Vo. Vol chg. = DV P P = - K D V o • Special relations for isotropic materials: 2(1 + n) E G = 3(1 - 2n) K E is Modulus of Elasticity  is Poisson’s Ratio

Looking at Aluminum and the earlier problem:

Young’s Moduli: Comparison Graphite Ceramics Semicond Metals Alloys Composites /fibers Polymers 0.2 8 0.6 1 Magnesium, Aluminum Platinum Silver, Gold Tantalum Zinc, Ti Steel, Ni Molybdenum G raphite Si crystal Glass - soda Concrete Si nitride Al oxide PC Wood( grain) AFRE( fibers) * CFRE GFRE* Glass fibers only Carbon fibers only A ramid fibers only Epoxy only 0.4 0.8 2 4 6 10 00 1200 Tin Cu alloys Tungsten <100> <111> Si carbide Diamond PTF E HDP LDPE PP Polyester PS PET C FRE( fibers) FRE( fibers)* FRE(|| fibers)* E(GPa) 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

Useful Linear Elastic Relationships • Simple tension: • Simple torsion: a = 2 ML o p r 4 G M = moment = angle of twist 2ro Lo d = FL o E A d L = - n Fw o E A F A o d /2 L Lo w • Material, geometric, and loading parameters all contribute to deflection. • Larger elastic moduli minimize elastic deflection.

Resilience, Ur Ability of a material to store (elastic) energy Energy stored best in elastic region If we assume a linear stress-strain curve this simplifies to y r 2 1 U e s @ Adapted from Fig. 6.15, Callister 7e.

Plastic (Permanent) Deformation (at lower temperatures, i.e. T < Tmelt/3) • Simple tension test: Elastic+Plastic engineering stress, s at larger stress Elastic initially permanent (plastic) after load is removed ep engineering strain, e plastic strain Adapted from Fig. 6.10 (a), Callister 7e.

Yield Strength, sy y = yield strength sy • Stress at which noticeable plastic deformation has occurred. when ep  0.002 tensile stress, s engineering strain, e sy p = 0.002 y = yield strength Note: for 2 inch sample  = 0.002 = z/z  z = 0.004 in Adapted from Fig. 6.10 (a), Callister 7e.

Tensile properties Yielding Strength Some steels (lo-C) Proof stress Most structures operate in elastic region, therefore need to know when it ends Yield strength -- Generally quoted Proportional Limit Some steels (lo-C) Proof stress

Yield Strength : Comparison Graphite/ Ceramics/ Semicond Metals/ Alloys Composites/ fibers Polymers Yield strength, s y (MPa) PVC Hard to measure , since in tension, fracture usually occurs before yield. Nylon 6,6 LDPE 70 20 40 60 50 100 10 30 2 00 3 4 5 6 7 Tin (pure) Al (6061) a ag Cu (71500) hr Ta (pure) Ti Steel (1020) cd (4140) qt (5Al-2.5Sn) W Mo (pure) cw Hard to measure, in ceramic matrix and epoxy matrix composites, since in tension, fracture usually occurs before yield. H DPE PP humid dry PC PET ¨ Room Temp. values Based on data in Table B4, Callister 7e. a = annealed hr = hot rolled ag = aged cd = cold drawn cw = cold worked qt = quenched & tempered

Tensile Strength, TS TS engineering stress strain engineering strain • TS is Maximum stress on engineering stress-strain curve. y strain Typical response of a metal F = fracture or ultimate strength Neck – acts as stress concentrator engineering TS stress engineering strain Adapted from Fig. 6.11, Callister 7e. • Metals: occurs when noticeable necking starts. • Polymers: occurs when polymer backbone chains are aligned and about to break.

Tensile Strength : Comparison Si crystal <100> Graphite/ Ceramics/ Semicond Metals/ Alloys Composites/ fibers Polymers Tensile strength, TS (MPa) PVC Nylon 6,6 10 100 200 300 1000 Al (6061) a ag Cu (71500) hr Ta (pure) Ti Steel (1020) (4140) qt (5Al-2.5Sn) W cw L DPE PP PC PET 20 30 40 2000 3000 5000 Graphite Al oxide Concrete Diamond Glass-soda Si nitride H wood ( fiber) wood(|| fiber) 1 GFRE (|| fiber) ( fiber) C FRE A FRE( fiber) E-glass fib Aramid fib Room Temp. values Based on data in Table B4, Callister 7e. a = annealed hr = hot rolled ag = aged cd = cold drawn cw = cold worked qt = quenched & tempered AFRE, GFRE, & CFRE = aramid, glass, & carbon fiber-reinforced epoxy composites, with 60 vol% fibers.

Ductility x 100 L EL % - = • Plastic tensile strain at failure: e s Lf o f - = • Plastic tensile strain at failure: Engineering tensile strain, e E ngineering tensile stress, s smaller %EL larger %EL Lf Ao Af Lo Adapted from Fig. 6.13, Callister 7e. • Another ductility measure: 100 x A RA % o f - =

Lets Try one (like Problem 6.29) Load (N) len. (mm) len. (m) l 50.8 0.0508 12700 50.825 0.050825 2.5E-05 25400 50.851 0.050851 5.1E-05 38100 50.876 0.050876 7.6E-05 50800 50.902 0.050902 0.000102 76200 50.952 0.050952 0.000152 89100 51.003 0.051003 0.000203 92700 51.054 0.051054 0.000254 102500 51.181 0.051181 0.000381 107800 51.308 0.051308 0.000508 119400 51.562 0.051562 0.000762 128300 51.816 0.051816 0.001016 149700 52.832 0.052832 0.002032 159000 53.848 0.053848 0.003048 160400 54.356 0.054356 0.003556 159500 54.864 0.054864 0.004064 151500 55.88 0.05588 0.00508 124700 56.642 0.056642 0.005842 GIVENS:

Leads to the following computed Stress/Strains: e stress (Pa) e str (MPa) e. strain 98694715.7 98.694716 0.000492 197389431 197.38943 0.001004 296084147 296.08415 0.001496 394778863 394.77886 0.002008 592168294 592.16829 0.002992 692417257 692.41726 0.003996 720393712 720.39371 0.005 796551839 796.55184 0.0075 837739398 837.7394 0.01 927885752 927.88575 0.015 997049766 997.04977 0.02 1163354247 1163.3542 0.04 1235626755 1235.6268 0.06 1246506488 1246.5065 0.07 1239512374 1239.5124 0.08 1177342475 1177.3425 0.1 969073311 969.07331 0.115

Leads to the Eng. Stress/Strain Curve: T. Str.  1245 MPa F. Str  970 MPa Y. Str.  742 MPa %el  11.5% E  195 GPa (by regression)

TOUGHNESS Is a measure of the ability of a material to absorb energy up to fracture High toughness = High yield strength and ductility Important Factors in determining Toughness: 1. Specimen Geometry & 2. Method of load application Dynamic (high strain rate) loading condition (Impact test) 1. Specimen with notch- Notch toughness 2. Specimen with crack- Fracture toughness Static (low strain rate) loading condition (tensile stress-strain test) 1. Area under stress vs strain curve up to the point of fracture.

Toughness • Energy to break a unit volume of material • Approximate by the area under the stress-strain curve. very small toughness (unreinforced polymers) Engineering tensile strain, e E ngineering tensile stress, s small toughness (ceramics) large toughness (metals) Adapted from Fig. 6.13, Callister 7e. Brittle fracture: elastic energy Ductile fracture: elastic + plastic energy

Elastic Strain Recovery – After Plastic Deformation It is an important factor in forming products (especially sheet metal and spring making) Adapted from Fig. 6.17, Callister 7e.

True Stress & Strain Note: Stressed Area changes when sample is deformed (stretched) True stress True Strain Adapted from Fig. 6.16, Callister 7e.

Strain Hardening • An increase in sy due to continuing plastic deformation. s e large Strain hardening small Strain hardening y 1 • Curve fit to the stress-strain response: s T = K e ( ) n “true” stress (F/Ai) “true” strain: ln(Li /Lo) hardening exponent: n = 0.15 (some steels) 0.5 (some coppers)

Earlier Example: True Properties T. Stress T. Strain 98.74328592 0.000492 197.5875979 0.001003 296.5271076 0.001495 395.571529 0.002006 593.9401363 0.002988 695.1842003 0.003988 723.9956807 0.004988 802.525978 0.007472 846.1167917 0.00995 941.8040385 0.014889 1016.990761 0.019803 1209.888417 0.039221 1309.764361 0.058269 1333.761942 0.067659 1338.673364 0.076961 1295.076722 0.09531 1080.516741 0.108854 Necking Began

We Compute: T = KTn to complete our True Stress vs. True Strain plot (plastic data to necking) Take Logs of both T and T ‘Regress’ the values from Yielding to Necking Gives a value for n (slope of line) and K (its T when T = 1) Plot as a continuation line beyond necking start

Stress Strain Plot w/ True Values K  2617 MPa

Variability in Material Properties Critical properties depend largely on sample flaws (defects, etc.). Most can exhibit Large sample to sample variability. Real (reported) Values, thus, are Statistical measures usually mean values. Mean Standard Deviation All samples have same value Because of large variability must have safety margin in engineering specifications where n is the number of data points (tests) that are performed

Design or Safety Factors • Design uncertainties mean we do not push the limit! • Typically as engineering designers, we introduce a Factor of safety, N Often N is between 1.5 and 4 • Example: Calculate a diameter, d, to ensure that yield does not occur in the 1045 carbon steel rod below subjected to a working load of 220,000 N. Use a factor of safety of 5. 1045 plain carbon steel: s y = 310 MPa TS = 565 MPa F = 220,000N d L o

Solving:

Hardness • Resistance to permanently (plastically) indenting the surface of a product. • Large hardness means: --resistance to plastic deformation or cracking in compression. --better wear properties. e.g., Hardened 10 mm sphere apply known force measure size of indentation after removing load d D Smaller indents mean larger hardness. increasing hardness most plastics brasses Al alloys easy to machine steels file hard cutting tools nitrided diamond

Hardness: Common Measurement Systems Callister Table 6.5

Comparing Hardness Scales:

Inaccuracies in Rockwell (Brinell) hardness measurements may occur due to: An indentation is made too near a specimen edge. Two indentations are made too close to one another. Specimen thickness should be at least ten times the indentation depth. Allowance of at least three indentation diameters between the center on one indentation and the specimen edge, or to the center of a second indentation. Testing of specimens stacked one on top of another is not recommended. Indentation should be made into a smooth flat surface.

Correlation Between Hardness and Tensile Strength Both measures the resistance to plastic deformation of a material.

HB = Brinell Hardness TS (psia) = 500 x HB TS (MPa) = 3.45 x HB

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 (E or G). • Plastic behavior: This permanent deformation behavior occurs when the tensile (or compressive) uniaxial stress reaches sy. • Toughness: The energy needed to break a unit volume of material. • Ductility: The plastic strain at failure.