CE 335 - Elasticity PDF

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