Materials Chapter 6 PDF

CHAPTER 6 Mechanical Properties of Metals - I 1  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Processing and Shaping of Metals Metal ingots are formed into functional shapes u...

CHAPTER 6 Mechanical Properties of Metals - I 1  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Processing and Shaping of Metals Metal ingots are formed into functional shapes using a variety of operations called Metal Forming Operations. Some of the operations used to form functional shapes are casting, forging, rolling, stamping, drawing, and extrusion. The functional shapes including sheets, plates, tubes, cylinders, rods, disks, wires, channels, and cups are then used in various industries including auto industries. 2 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Processing of Metals - Casting In the casting process, metals are first melted in a furnace. Alloying elements are then added and thoroughly mixed (magnesium is added to aluminum to produce a stronger and lighter alloy). Oxide impurities and unwanted gasses are removed. The melt is the poured into a mold and chilled to solidify. Both simple shapes such as ingots and complex shapes may be cast. 3 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Processing of Metals - Rolling Metal sheets and plates are produced using the rolling process. Very strong and hard rollers are used to reduce the thickness of the starting material such as an ingot. The process is performed in many passes and as the thickness is reduced, the length increases. The work materials may be at room temperature called cold rolling or at very high temperatures, below melting, called hot rolling. 4 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Processing of Metals – Hot Rolling In hot rolling, the work material is heated to high temperatures called recrystallization temp. This allows for an easier (less force) and greater reduction of thickness in a single pass. For steels, ingots are preheated to about 1200oC prior to rolling. Ingots reheating may be needed between passes if ingot cools and becomes hard to roll. Usually, a series of 4 high rolling mills are used to continuously reduce the thickness. 5 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Processing of Metals - Cold Rolling In cold rolling, the temperature of the work is significantly below recrystallization temperature. It is more difficult to cause plastic flow at lower temperatures. The cold rolled material strain hardens and becomes stronger (not so in hot rolling). The cold rolled materials also become more brittle. Reheating will improve ductility. Results in excellent surface finish. Less thickness reduction is possible in each pass at cold temperatures. Requires high power to cause plastic deformation in the work. 6 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Continuous Rolling Process Initial metal thickness – Final metal thickness % Cold work = x 100 Initial metal thickness 7 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Processing of Metals - Extrusion In extrusion, metal workpiece (billet) is placed under high pressure using a ram and forced through opening in a die. Normally performed at high temperatures. Metal There are both direct (left image) and indirect (right image) extrusion processes. Indirect extrusion experiences lower friction on the billet and therefore needs less power however has a limit on the applied load. Extrusion produces products such as cylindrical rods, tubes, and with some metals more irregular shapes. Lubricants such as molten glass may be required for extrusion of strong metals. 8 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Processing of Metals - Forging In the forging process, metal is hammered (dynamic repeated blows) or pressed (slowly increasing force) into desired shape; mostly in a hot condition – hot forging. Two types based on the die geometry: Open Die Geometries Ø Open die: Dies are either flat or simple in geometry * Products: Steel shafts Ø Closed die: Dies have upper and lower impression – more complex. * Products: Engine connecting rods Cold forging improves structural properties, removes porosity 9 and increases homogeneity. Closed Die Geometries Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Processing of Metals - Drawing In wire drawing, the starting rod or wire is drawn through several drawing dies to reduce diameter. Change in cross-sectional area % cold work = x 100 Original area Flat sheets of metal are deformed into cup like components. 10 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Stress and Strain in Metals Metals undergo deformation under force. In the case of an axial force, we first produce Elastic deformation: Atoms elongate resulting in overall elongation of the specimen. But return to their original dimensions after tensile force is removed, resulting in recoverable deformation. Plastic deformation: Atoms break bonds and slip on each other. Once slip occurs, atoms can not return to their original positions, resulting in permanent deformation. 11 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Engineering Stress and Strain – Normal Stress and Strain F (Average uniaxial tensile force) Engineering normal stress, σ = A0 (Original cross-sectional area) Units of Stress are lbf/in2 (psi) or N/m2 (Pa) 1 psi = 6.89 x 103 Pa Change in length Engineering normal strain, ε = Original length ℓ − ℓ0 Δℓ = = ℓ0 ℓ Units of strain are in/in or m/m (dimensionless) σ and ε are referred to as ‘normal’ engineering stress and strain, respectively, because they act perpendicular to the surface and cause change in length. 12 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Engineering Stress and Strain – Shear Stress and Strain S (Shear force) Engineering shear stress, τ = A (Area of application) Engineering shear strain, γ = Amount of shear displacement, a = tanθ Distance ‘h’ over which shear acts *For small angles θ = tanθ τ and γ are referred to as engineering ‘shear’ stress and strain, respectively; note that τ is parallel to the surface and γ represents deviation from a 90o angle or change in shape. 13 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display The Tensile Test Important properties of materials may be determined by performing a tensile test and drawing the engineering stress– strain (σ-ε) diagram. Load Cell Specimen Extensometer Force data is obtained from load cell Elongation data is obtained from extensometer. 14 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display The Tensile Test, Cont. Standard cylindrical (left) and flat (right) dog-bone specimens used in tensile testing The tensile tester applies displacement to the specimen. The specimen responds by developing stress (σ) and strain (ε). σ is determined by dividing force, F, (measured by the load cell) by the original cross-sectional area of the specimen, Ao. ε is determined by dividing the elongation (Δℓ ) by the initial gage length, ℓ o. At each increment the corresponding σ-ε pairs are plotted. The final engineering stress-strain diagram is produced. Typical engineering stress-strain curve for a ductile metal 15 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display The Engineering Stress-Strain (σ-ε) Diagram The σ-ε diagram has many key features: Ultimate tensile strength Fracture point 1. Linear-elastic range: Plastic range Ø Linear means σ and ε are related in a linear fashion; σ=Eε, Hooke’s Law. Yield point Ø Elastic means the generated strain is recoverable: remove the load and strain disappears. Ø E is the modulus of elasticity or the proportionality constant in the linear range. 2. Yield point: Ø The point on the curve when the elastic Linear elastic range range ends. Ø After this point, any additional deformation is not recoverable and is permanent. Typical engineering Ø Plastic range starts past yield point and stress-strain curve for a ends with the fracture point. 16 ductile metal Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display The Engineering Stress-Strain (σ-ε) Diagram The σ-ε diagram has many key features: Ultimate tensile Fracture point 3. Ultimate tensile strength strength Ø This is the peak pint in the σ-ε curve. Plastic range Ø It represents the largest stress the Yield point specimen can take before fracture. 4. Fracture point: Ø The point on the curve where the specimen fails. The σ-ε curve can be used to extract information about the behavior of the tested materials and the respective mechanical Linear elastic range property data. Typical engineering stress-strain curve for a 17 ductile metal Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Mechanical Properties Obtained from the Tensile Test – Modulus of Elasticity, E Modulus of Elasticity (E) : Stress and strain are linearly related in the elastic region. (Hooke’s law) Stress σ (Stress) Δσ E= E= Δε ε (Strain) Δσ Δε Strain Higher the bonding strength, Linear portion of the higher is the modulus of elasticity. stress strain curve Examples: Modulus of Elasticity of steel is 207 Gpa. Modulus of elasticity of Aluminum is 76 Gpa This means that steel is stiffer than aluminum and therefore deforms less (elastically) under the same load. 18 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Mechanical Properties Obtained from the Tensile Test – Poisson’s Ratio, ν As a specimen is elastically elongated under a tensile load, its diameter must decrease. This produces the Poisson’s ratio, v. ε (lateral ) εy Poisson’s ratio, ν =− =− ε (longitudinal ) εz w ( w − wo ) / wo ν =− ℓ0 w0 w (ℓ − ℓ 0 ) / ℓ 0 ℓ Poisson’s ratio is always positive in isotropic materials and ranges from 0.25 to 0.4. Example: Stainless steel 0.28 Copper 0.33 19 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Mechanical Properties Obtained from the Tensile Test – Yield Strength, σy Yield strength, σy, is the stress at which a material begins to experience permanent or plastic deformation. The location of the yield point is not always clear or precise. The 0.2% offset yield strength standard is used to identify the yield point. Construction line: starting at 0.2% strain (0.002) a line parallel to elastic region is drawn. The stress at which this line intersects the σ-ε curve, is the 0.2% offset yield strength. 20 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Mechanical Properties Obtained from the Tensile Test – Tensile Strength, σu Ultimate tensile strength (UTS), σu, is the maximum stress reached in the engineering stress strain curve. Necking become prominent around Stress, Mpa the UTS point and is largest prior to Al 2024-Tempered fracture. Necking Point Al 2024-Annealed The more ductile the metal is, the more enhanced is the necking before failure. Stress increases until failure. The drop Strain σ-ε curves of in stress strain curve is due to stress Al 2024 with two different calculation based on original area. heat treatments. 21 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Mechanical Properties Obtained from the Tensile Test – % Elongation at Fracture, εf % elongation at fracture, εf, is a measure of ductility of a material. It is the elongation of the metal before fracture expressed as percentage of original length. final length – initial length % Elongation at fracture = initial length % reduction in area will give the same information about the ductility of the tested materials. Example: % elongation of pure aluminum is 35% while for 7076-T6 aluminum alloy it is 11%. 22 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Mechanical Properties Obtained from the Tensile Test – Modulus of Resilience, Ur The modulus of resilience, Ur, is a measure of the amount of energy needed to cause yield in the material. This energy is almost fully recovered once the load is removed. Ur is measured by calculating the area under the linear elastic range of the σ-ε curve. 1 U r = σ yε y 2 Toughness is the amount of energy required to fracture the material. It is a measure of combination of strength and ductility. The units for both Ur and It is measured by determining the area under toughness are J/m3 (lbf.in/ in3) or N/m2 (lbf/in2). the full curve. 23 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Comparison of Properties Among Different Metals Engineering stress-strain curves of different metals. Can you compare and contrast the properties of various metal? Which one the strongest? Which one is most ductile? Which one is stiffest? Which one is toughest? 24 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display True Stress – True Strain Curve True stress and true strain are determined based on instantaneous cross-sectional area and length of the specimen. This is in contrast to engineering stress that is determined based on the original cross-sectional area. F True Stress = σ t = A i ℓi dℓ l A εt = ∫ = ln i = ln 0 True Strain = ℓ0 ℓ l0 Ai True stress is always greater than engineering stress because it considers the instantaneous area (which is always smaller). However, this difference is not significant in the elastic range. 25 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Hardness and Hardness Testing of Materials Hardness is a measure of the resistance of a metal to localized plastic deformation or indentation. Hardness is important in applications where one component articulates on another and cause wear. Hardness test indentations on a gear tooth Harder materials wear less. Rockwell hardness tester and steps. The hardest material in the nature is diamond. The indentations on the surface and inside the gear tooth have different sizes. What does this tell you? 26 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Hardness Tests and Scales 27 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Plastic Deformation in Single Crystals Plastic deformation in a single crystal results in formation of bands called slip bands. Atoms on specific crystallographic planes (slip planes) and in specific directions slip to form these bands. Slip bands in ductile metals are uniform (occurs in many slip planes). Slip bands Slip bands are about 200A thick and are offset by about 2000A. 28 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Slip Mechanism Slip is caused by application of shear stress on the grain. The mechanism by which slip occurs is by formation of dislocations and their movement. The movement of a dislocation is similar to the movement of a ripple in a rug. As the ripple moves in a rug, so does the dislocation in a grain. Once the dislocation reaches the end of the grain, it creates a unit step of slip. 29 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Slip Preference Slip occurs more readily on close packed planes and along close packed directions on those planes. τ τ τ τ Close packed plane Less densely packed plane A smaller shear stress is required for slip to occur in densely packed planes because the unit step slip (distance between consecutive red markers) is smaller in close-packed planes. Less energy is required to move atoms along denser planes. If slip is restricted in close planes, then less dense planes become operative. 30 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Slip Systems Therefore, metals have a preferred slip system: a combination of a densely packed plane and a densely packed 4 (111) type planes and direction. 3 type directions. Each crystal has a number of 4 x 3 = 12 slip systems. characteristic slip systems based on its structure. For example, in FCC crystal, slip takes place in {111} octahedral planes and directions. Note BCC crystals also have 12 slip systems but HCP crystals have only 3 slip systems under normal conditions. What does this tell you? 31 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Schmid’s Law The relationship between uniaxial stress action on a single cylinder of pure metal single crystal and resulting resolved shear stress, τr, produced on a slip system is give by F F.Cos λ F τr = r = = Cos λ.Cos Φ A1 A0 / Cos λ A0 F σ= A0 τ r = σCosλCosΦ Schmid’s Law 32 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Critical Resolved Shear Stress, τc Critical resolved shear stress, τc, is the shear stress required to cause slip in a single crystal. Depends upon Ø Crystal Structure Ø Atomic bonding characteristics Ø Temperature Ø Orientation of slip planes relative to shear stress Slip begins when shear stress in the slip plane and the slip direction reaches critical resolved shear stress. This is equivalent to yield strength. Example : Zn HCP 99.999% pure 0.18MPa Ti HCP 99.99% pure 13.7 MPa Ti HCP 99.9% pure 90.1 MPa 33 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Twinning In a twin, a part of atomic lattice is deformed and forms mirror image of lattice next to it. Distance moved by atoms is proportional to their distance from twinning plane. Deformation from twinning is small. Twins reorient the slip system. Twining is important in deformation in HCP crystals due to lower number of slip systems. The figure shows twins in pure titanium. 34 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Effects of Grain Boundaries on the Strength of a Metal Grain boundaries stop dislocation movement and therefore strengthen the metals. See dislocation pile up in figure (left). Fine grain size is desirable for uniform properties and high strength at room temperature, and therefore metals are produced with finer grains. See comparison in copper (middle). In polycrystalline metals, slip bands are parallel inside the grain but change direction from grain to grain due to orientation of the grain (right) 35 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Hall Petch Equation The finer the grains in a metal, the more superior are the strength properties (at room temperature). Hall-Petch equation is an empirical equation that relates strength to grain size (diameter). σy = σo + k / (d)1/2 σy = Yield strength d = average grain diameter σo and k are constants for a metal. σo = 70 Mpa and k = 0.74 Mpam1/2 for mild steel. In addition to improvements in strength, finer grains result in Ø more uniform and isotropic properties Ø less resistant to corrosion and creep 36 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Effects of Plastic Deformation Plastic deformation results in shearing of grains relative to each other. The grains elongate in the rolling direction. The grain distortion makes the metal stronger. 37 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Effect of Cold Work on Tensile Strength Number of dislocations are increased by increasing cold work. Dislocation movements are hindered by both grain boundaries and other dislocations Strain Hardening 1018-Cold Rolled 1018-Annealed Stress-Strain curves of 1018 steel 38 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Solid Solution Strengthening In solid solutions, addition of one or more impurity (solute) to the host metal can increase the strength of metals. Solute atoms create stress fields around themselves and hinder the dislocation movement. This strengthens the host metal. Distortion of lattice and clustering of like atoms also impede dislocation movement. In short, impurity atoms prevent propagation of slip. Example: Solid solution of 70 wt % Cu & 30 wt % Zn (cartridge brass) has tensile strength of 500 MPa. Tensile strength of unalloyed copper is 330 MPa 39 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Annealing: Recovery and Recrystallization and Grain Growth Annealing is a heat treatment 85% CW Al alloy: process applied to a strain- Ø elongated grains Ø Strain hardened hardened or cold-worked metal to remove residual stresses, grow new equiaxed Above alloy stress relieved grains and soften the metal. at 302oC for 1 hr: Ø Some recrystallization The three stages of the Ø Reduced residual stress annealing process are: recovery, recrystallization and grain growth. Above alloy annealed Annealing takes place above at 316oC for 1 hr: Ø Major recrystallization the recrystallization Ø Major grain growth temperature, TR, for one hour. 40 Foundations (Adapted from Z.D. Jastrzebski, of Materials “The Nature Science andof and Properties Engineering, 6th Edn. Engineering Smith and2d Materials,” Hashemi ed., Wiley, 1976, p.228.)  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Annealing: Recovery and Recrystallization and Grain Growth During the recovery stage: Ø Dislocations rearrange themselves to form lower energy configurations (polygonization) Ø Residual stresses decrease significantly Ø Strength and ductility change very little During the recrystallization stage: Ø Diffusion dissolves dislocation Ø Strength and hardness decrease significantly Ø Ductility increases significantly Ø New grains start to grow During the grain growth stage: Impact of temperature on recovery, Ø Growth of equiaxed grains increases Recrystallization, and grain growth significantly 41 Foundations (Adapted from Z.D. Jastrzebski, of Materials “The Nature Science andof and Properties Engineering, 6th Edn. Engineering Smith and2d Materials,” Hashemi ed., Wiley, 1976, p.228.)  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Effect of Annealing on Mechanical Properties Annealing decreases tensile strength, increases ductility. Example: 85% Cu & 15% Zn 50% cold Annealed 1 h rolled 4000C Tensile strength Tensile strength 75 KSI 45 KSI Ductility 3% Ductility 38 % Factors affecting recrystallization: Ø Amount of prior deformation Ø Temperature Ø Time Ø Initial grain size Ø Composition of metal 42 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Facts About Recrystallization A minimum amount of deformation is needed to start the process. The smaller the degree of deformation, the higher the recrystallization temperature required. The higher the temperature, the lower the time required. The greater the degree of deformation, the smaller are the recrystallized grains. The larger the original grain size, the greater the amount of deformation required to produce equivalent Continuous annealing temperature. Recrystallization temperature increases with purity of metals. 43 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Superplasticity in Metals At elevated temperature and slow loading, some alloys deform 2000%. Annealed Ti alloy Ø Elongates 12% at room temperature Ø Elongates up to 1170% at 870oC and 1.3x10-4/s loading rate. Superplasticity of zn-Al alloy Conditions favorable for superplasticity: *very fine grain size (5-10 microns *highly strain sensitive *temperature above 0.5 melt temperature *Slow strain rate 44 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi  Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display Nanocrystalline Metals Average grain diameter < 100 nm Results in very high strength and hardness, and Superplasticity. If grain diameter reduces from 10 microns to 10 nm, yield strength of copper increases 31 times. Very difficult to produce nanocrystalline metals. If d < 5 nm, elastic modulus drops as more atoms are in grain boundary. Hall-Petch equation is invalid in lower nanocrystalline range. Negative Hall-Petch effect might take place 45 Foundations of Materials Science and Engineering, 6th Edn. Smith and Hashemi