CE 335 - Elasticity PDF
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Purdue University
Mirian Velay-Lizancos
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This document contains lecture notes on elasticity, part of a civil engineering course at Purdue University. It covers various topics such as elastic behavior, stress-strain relationships, and toughness.
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CE 335 - Elasticity Prof. Velay CE 335 – Civil Engineering Materials Civil Engineering – Purdue University Prof. Mirian Velay-Lizancos CE 335 - Elasticity Prof. Velay Join Code :...
CE 335 - Elasticity Prof. Velay CE 335 – Civil Engineering Materials Civil Engineering – Purdue University Prof. Mirian Velay-Lizancos CE 335 - Elasticity Prof. Velay Join Code : https://join.iclicker.com/SNVY CODE: AA CE 335 - Elasticity Prof. Velay Elasticity CE 335 - Elasticity Prof. Velay 1. Elastic behavior Stress Upper yield point 𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳 𝝈𝝈 = 𝑨𝑨 Lower yield point Elastic Strain Yielding hardening Necking ∆𝐿𝐿 Strain 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = 𝐿𝐿0 CE 335 - Elasticity Prof. Velay 1. Elastic behavior Stress-Strain relationship is linear, it means the stress is directly proportional to the strain. Proportional limit (σpl) is the maximum stress limit in the elastic region. If you unload the material, it will return to the original shape -> No plastic deformation. Stress Strain CE 335 - Elasticity Prof. Velay 2. Elastic deformation vs plastic deformation Stress Elastic Stain Yielding hardening Necking Strain CE 335 - Elasticity Prof. Velay 3. Ductile and Brittle materials Brittle Ductile Stress Strain -> Plastic strain before failure CE 335 - Elasticity Prof. Velay 3. Ductile and Brittle materials Brittle Brittle Stress Stress Strain Elastic strain Plastic strain Strain Plastic strain Total strain CE 335 - Elasticity Prof. Velay Definition of toughness: 4. Toughness “toughness is the ability of a material to absorb energy and plastically deform without Brittle fracturing. “ Ductile “The material toughness is the amount of Stress energy per unit volume that a material can absorb before failure” Strain CE 335 - Elasticity Prof. Velay Definition of toughness: 4. Toughness “toughness is the ability of a material to absorb energy and plastically deform without Brittle fracturing. Ductile “The material toughness is the amount of Stress energy per unit volume that a material can absorb before failure” (J/m3) Strain CE 335 - Elasticity Prof. Velay 5. Elasticity and Hook’s Law x 2x 3x “Ut tensio, sic vis” F (As is the extension, so is the force) F = k.X F= Force K= Stiffness of the spring 2F X= Elongation 3F CE 335 - Elasticity Prof. Velay 5. Elasticity and Hook’s Law And more than one hundred years later… “Ut tensio, sic vis” (As is the extension, so is the force) F = k.X F= Force K= Stiffness of the spring X= Elongation CE 335 - Elasticity Prof. Velay 5. Elasticity and Hook’s Law Robert Hook Thomas Young “Ut tensio, sic vis” “Stress is proportional to strain” (As is the extension, so is the force) F= Force F = k.X 𝝈𝝈 = 𝑬𝑬 𝜺𝜺 𝝈𝝈 = Stress (MPa or psi) K= Stiffness of the spring E= Young’s Modulus (MPa or psi) X= Elongation 𝜺𝜺 = Strain (dimensionless) CE 335 - Elasticity Prof. Velay 6. Linear elasticity and Hook’s law L0 Stress 𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳 L0 Δ𝐿𝐿1 𝝈𝝈 = 𝑨𝑨 F1 L0 Δ𝐿𝐿2 F2 L0 Δ𝐿𝐿2 Strain F2 ∆𝐿𝐿 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = 𝐿𝐿0 CE 335 - Elasticity Prof. Velay 6. Linear elasticity and Hook’s law L0 Stress 𝑳𝑳𝑳𝑳𝑳𝑳𝑳𝑳 L0 Δ𝐿𝐿1 𝝈𝝈 = 𝑨𝑨 F1 E 1 L0 Δ𝐿𝐿2 F2 L0 Strain ∆𝐿𝐿 𝐿𝐿𝑓𝑓 − 𝐿𝐿0 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = = 𝐿𝐿0 𝐿𝐿0 CE 335 - Elasticity Prof. Velay 7. Deformation and Young’s Modulus L0 𝝈𝝈 = 𝑬𝑬 𝜺𝜺 L0 𝛿𝛿 = Δ𝐿𝐿 𝑷𝑷 = 𝑬𝑬 𝜺𝜺 𝑨𝑨 P 𝜹𝜹 = Δ𝑳𝑳 𝑷𝑷 𝜹𝜹 = 𝑬𝑬 𝑨𝑨 𝑳𝑳𝟎𝟎 Remove the load.. 𝟎𝟎𝟎𝟎 𝑷𝑷𝑳𝑳 𝑷𝑷𝑳𝑳𝟎𝟎 𝜹𝜹 = 𝜹𝜹 = 𝜹𝜹 = 𝑨𝑨𝑬𝑬 𝑨𝑨𝑬𝑬 𝑨𝑨𝑬𝑬 CE 335 - Elasticity Prof. Velay 7. Deformation and Young’s Modulus Stress-Strain relationship is linear, it means the stress is directly proportional to the strain. Proportional limit (σpl) is the maximum stress limit in the elastic region. If you unload the material, it will return to the original shape -> No plastic deformation. Stress E 1 𝝈𝝈 = 𝑬𝑬 𝜺𝜺 Strain CE 335 - Elasticity Prof. Velay 8. Unloading and deformation F F 𝛿𝛿 𝛿𝛿 CE 335 - Elasticity Prof. Velay 9. Strain energy “Internal energy stored in the material due to its deformation” CE 335 - Elasticity Prof. Velay 9. Strain energy “Internal energy stored in the material due to its deformation” Resilience -> Stored elastic strain energy at yield strength Stress 1 Y.S. Area = 𝑏𝑏 ℎ 2 1 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 𝜀𝜀 𝜎𝜎 2 𝑒𝑒𝑒𝑒 𝑦𝑦 𝜎𝜎 = 𝐸𝐸 𝜀𝜀 1 𝜎𝜎𝑦𝑦 𝜎𝜎𝑦𝑦 = 𝐸𝐸 𝜀𝜀 el 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 𝜎𝜎𝑦𝑦 2 𝐸𝐸 𝜎𝜎𝑦𝑦 1 𝜎𝜎𝑦𝑦2 𝜀𝜀 el= 𝑢𝑢𝑅𝑅 = [J/m3] 𝐸𝐸 2 𝐸𝐸 𝜀𝜀𝑒𝑒𝑒𝑒 Strain 0.002 (0.2%) CE 335 - Elasticity Prof. Velay 9. Strain energy “Internal energy stored in the material due to its deformation” 1 𝜎𝜎𝑦𝑦2 Resilience -> Stored elastic strain energy at yield strength 𝑢𝑢𝑅𝑅 = 2 𝐸𝐸 1 𝜎𝜎2 Strain energy density -> 𝑢𝑢 = (If elastic) 2 𝐸𝐸 1 𝜎𝜎2 1 𝜎𝜎 𝜎𝜎 1 𝜎𝜎 𝐸𝐸 𝜀𝜀 Strain energy -> 𝑈𝑈 = 𝑉𝑉𝑉𝑉𝑉𝑉 𝑈𝑈 = 𝑉𝑉𝑉𝑉𝑉𝑉 𝑈𝑈 = 𝑉𝑉𝑉𝑉𝑉𝑉 2 𝐸𝐸 2 𝐸𝐸 2 𝐸𝐸 1 𝑈𝑈 = 𝜎𝜎 𝜀𝜀 𝑉𝑉𝑉𝑉𝑉𝑉 2 1 ∆𝑈𝑈 = 𝜎𝜎 𝜀𝜀 ∆𝑉𝑉𝑉𝑉𝑉𝑉 2 CE 335 - Elasticity Prof. Velay 10. Poisson’s ratio L’f L’i ∆L’/2 ∆𝐿𝐿′ /𝐿𝐿𝐿 ∆L/2 𝜗𝜗 = − ∆𝐿𝐿/𝐿𝐿 Li Lf ∆𝐿𝐿 = 𝐿𝐿𝑓𝑓 − 𝐿𝐿i CE 335 - Elasticity Prof. Velay 10. Poisson’s ratio 5 cm ∆𝐿𝐿′ /𝐿𝐿𝐿 5.5 cm 𝜗𝜗 = − ∆𝐿𝐿/𝐿𝐿 7 cm 10 cm ∆𝐿𝐿 = 𝐿𝐿𝑓𝑓 − 𝐿𝐿i CE 335 - Elasticity Prof. Velay Thank you