Chapter 6: Mechanical Properties II Outline • Elastic recovery during plastic deformation • Compressive, shear, and torsional deformation • Hardness • Variability of material properties • Design/safety factors Chapter 6 - 1 Concepts of stress and strain • Tension tests – engineering stress – engineering strain • Compression tests Chapter 6 - Elastic Deformation 1. Initial 2. Small load 3. Unload bonds stretch return to initial δ F F Linear- elastic Elastic means reversible! Non-Linear- elastic δ Chapter 6 - 3 Plastic Deformation (Metals) 1. Initial 2. Small load 3. Unload bonds stretch p lanes & planes still shear sheared δ δ elastic + plastic plastic F F Plastic means permanent! linear linear elastic elastic δ δ plastic Chapter 6 - 4 Elastic Strain Recovery Adapted from Fig. 6.17, Callister 7e. Chapter 6 - 5 Other Elastic Properties M • Elastic Shear τ modulus, G: G simple γ torsion τ = G γ test M • Elastic Bulk P P modulus, K: P P ΔV ΔV P = -K V o pressure V K o test: Init. vol =Vo. • Special relations for isotropic materials: Vol chg. = ΔV E E G = K = 2(1 + ν) 3(1 - 2ν) Chapter 6 - 6 Useful Linear Elastic Relationships • Simple tension: • Simple torsion: 2ML δ = FL o δ = - ν Fw o o L α = 4 E A o E A o π r o G F M = moment δ /2 α = angle of twist A o L o Lo wo 2ro δ L /2 • Material, geometric, and loading parameters all contribute to deflection. • Larger elastic moduli minimize elastic deflection. Chapter 6 - 7 Hardness • Resistance to permanently indenting the surface. • Large hardness means: --resistance to plastic deformation or cracking in compression. --better wear properties. apply known force measure size e.g., of indent after 10 mm sphere removing load Smaller indents D d mean larger hardness. most brasses easy to machine cutting nitrided plastics Al alloys steels file hard tools steels diamond increasing hardness Chapter 6 - 8 Hardness: Measurement • Rockwell – No major sample damage – Each scale runs to 130 but only useful in range 20-100. – Minor load 10 kg – Major load 60 (A), 100 (B) & 150 (C) kg • A = diamond, B = 1/16 in. ball, C = diamond • HB = Brinell Hardness – TS (psia) = 500 x HB – TS (MPa) = 3.45 x HB Chapter 6 - 9 Hardness: Measurement Table 6.5 Chapter 6 - 10 Hardness tests (continued) • Rockwell and superficial rockwell • 20<hardness<100 Chapter 6 - Hardness tests • Brinell: 10-mm sphere of steel or tungsten carbide • Knoop and Vickers microhardness • Hardness conversion Chapter 6 - Correlation between hardness and tensile strength • Relations between hardness and tensile strength for steel, brass, and cast iron. • For most steels: TS (MPa)=3.45xHB TS (psi)=500xHB Chapter 6 - Example • Estimate the Brinell and Rockwell hardness for brass Chapter 6 - Errors and Measurements (lab) Error in a scientific measurement usually does not mean a mistake or blunder. Instead, the terms "error" and "uncertainty" both refer to unavoidable imprecision in measurements. Error analysis may seem tedious; however, without proper error analysis, no valid scientific conclusions can be drawn. http://phys.columbia.edu/ ~tutorial/ Chapter 6 - 15 Variability in Material Properties • Elastic modulus is material property • Critical properties depend largely on sample flaws (defects, etc.). Large sample to sample variability. • Statistics – Mean – Standard Deviation where n is the number of data points Chapter 6 - 16 Example • Determine the average and standard deviation of tensile strength Chapter 6 - Design or Safety Factors • Design uncertainties mean we do not push the limit. • Factor of safety, N Often N is 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. d 1045 plain carbon steel: L σy = 310 MPa o 5 TS = 565 MPa F = 220,000N d = 0.067 m = 6.7 cm Chapter 6 - 18 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 σy. • Toughness: The energy needed to break a unit volume of material. • Ductility: The plastic strain at failure. • Hardness: Resistance to permanently indenting the surface. • Safety: Design uncertainties mean we do not push the limit Chapter 6 - 19 .