Concepts of stress and strain

Transcription

1 Chapter 6: Mechanical properties of metals Outline Introduction Concepts of stress and strain Elastic deformation Stress-strain behavior Elastic properties of materials Plastic deformation Yield and yield strength Ductility Resilience Toughness Concepts of stress and strain Tension, compression, shear, and torsion 1

2 Stress-strain testing Tension tests engineering stress σ = A 0 engineering strain = l i l l 0 0 = Δl l 0 extensometer specimen Gauge length Compression tests Shear stress Shear and torsional tests Shear stress τ = Gγ A = 0 Shear strain γ = Δx/y = tan θ Δx y 90º 90º - θ Geometric considerations of the stress state 2

3 Elastic deformation 1. Initial 2. Small load 3. Unload bonds stretch δ Elastic means reversible! δ return to initial Linear elastic properties Modulus of Elasticity, E: (also known as Young's modulus) Hooke's Law: σ = E σ E Linearelastic Non-Linearelastic Linearelastic simple tension test 6 3

4 Stress-strain behavior Stress-strain for linear elastic deformation Stress-strain for non-linear elastic deformation Stress-strain behavior(continue) Modulus of elasticity E is proportional to (d/dr) r Slope of stress strain plot (which is proportional to the elastic modulus) depends on bond strength of metal 4

5 Other elastic properties Elastic Shear modulus, G: τ = G γ τ G γ M simple torsion test Elastic Bulk modulus, K: P = - Special relations for isotropic materials: G = K Δ V V o E 2(1 + ν) K = P K E 3(1 2ν) Δ V V o P M P P pressure test: Init. vol =V o. Vol chg. = ΔV Comparison of Young s moduli E(GPa) Tungsten Molybdenum Steel, Ni Tantalum Platinum Cu alloys Zinc, Ti Silver, Gold Aluminum Magnesium, Tin Diamond Si carbide Al oxide Si nitride <111> Si crystal <100> Glass -soda Concrete Graphite Polyester PET PS PC PP HDPE PTE Carbon fibers only CRE( fibers)* Aramid fibers only ARE( fibers)* Glass fibers only GRE( fibers)* GRE* CRE* GRE( fibers)* CRE( fibers) * ARE( fibers) * Epoxy only Wood( grain) 0.2 LDPE 5

6 Comparison of yield strength 2000 Metals/ Alloys Steel (4140)qt Graphite/ Ceramics/ Polymers Composites/ Semicond fibers Yield strength, σy (MPa) Ti (5Al-2.5Sn)a W(pure) Cu (71500)cw Mo (pure) Steel (4140)a Steel (1020)cd Al (6061)ag Steel (1020)hr Ti (pure)a Ta (pure) Cu (71500)hr Al (6061)a Hard to measure, since in tension, fracture usually occurs before yield. dry PC Nylon 6,6 PET humid PVC PP HDPE Hard to measure, in ceramic matrix and epoxy matrix composites, since in tension, fracture usually occurs before yield. 10 Tin (pure) LDPE Elastic properties of materials Poisson s ratio x ν = z y = z metals: ν ~ 0.33 ceramics: ν ~ 0.25 polymers: ν ~ 0.40 Units: E: [GPa] or [psi] ν: dimensionless 6

7 Plastic Deformation (Metals) 1. Initial 2. Small load 3. Unload bonds stretch planes & planes still shear sheared δelastic + plastic δplastic Plastic means permanent! linear elastic δplastic linear elastic δ Plastic deformation: yield and yield strength Yielding Proportional limit Yield strength 7

8 Tensile strength Tensile strength: the stress at the maximum on the engineering stress-strain curve Metals: occurs when noticeable necking starts Ceramics: occurs when crack propagation starts Polymers: occurs when polymer backbones are aligned and about to break TS σ y engineering stress Typical response of a metal engineering strain Plastic (permanent) deformation (At lower temperature: T< Tmelt/3) Simple tension test: engineering stress, σ Elastic+Plastic at larger stress Elastic initially permanent (plastic) after load is removed p engineering strain, plastic strain 8

9 Elastic and plastic deformations Stress-strain relations under uniaxial loading Ceramics Polymeric material Ductility Ductility: a measure of the degree of plastic deformation that has been sustained at fracture 9

10 Mechanic properties of typical metals Resilience Resilience: is the capacity of a material to absorb energy when it is deformed elastically and then, upon unloading, to have this energy recovered. Resilience 10

11 Toughness Energy to break a unit volume of material Approximate by the area under the stress-strain curve Engineering tensile stress, σ Adapted from ig. 6.13, Callister 7e. small toughness (ceramics) large toughness (metals) very small toughness (unreinforced polymers) Engineering tensile strain, Brittle fracture: elastic energy Ductile fracture: elastic + plastic energy 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 σ y. Toughness: The energy needed to break a unit volume of material. Ductility: The plastic strain at failure. 11