Engineering Stress-Strain Curve Lecture Notes

19-01-2026 Engineering stress strain curve Stress UTS: Ultimate tensile strength Material volume remains constant Necking 𝒍𝒐 𝑨𝒐 =𝒍𝒊 𝑨𝒊 ① ③ ② Lecture 2 Hooke’s Law • Mechanical Behavior of Materials by Thomas H Courtney, 2nd edition, Chapter 1 • Materials Science and Engineering: An Introduction by David G. Rethwisch and William Callister, Chapter 6 What are the possibilities when we apply an external force on a material? Elastic vs. Plastic deformation F Hooke’s Law: 𝒍 − 𝒍𝒐 = 𝜹𝒍 ~ 𝑭 F 𝜹𝒍 𝑭 ~ 𝒍𝒐 𝑨𝒐 𝜺= 𝝈 𝑬 𝒍𝒐 𝑨𝒐 F 𝒍𝒐 + 𝜹𝒍 𝒍𝒐 𝟐𝑨𝒐 𝜹𝒍 𝒍𝒐 + 𝟐 𝟐𝒍𝒐 𝑨𝒐 𝟐𝒍𝒐 + 2𝜹𝒍 𝜺: Strain E: Young’s Modulus Measure of stiffness Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials 2 1 19-01-2026 Shear Modulus 𝜹𝒍 𝜏 𝛾 𝒍𝒐 𝜏 𝑻𝒂𝒏 𝜸 ~ 𝜸 = 𝝉 𝑮 W.D. Callister, D.G. Rethwisch, Materials Science and Engineering: An Introduction Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials 3 Nonlinear elasticity H.R.C. Screen, P.D. Clegg, C.T. Thorpe, in: Tendon Regeneration, Elsevier, 2025, pp. 41–84 Non linear and time-dependent elasticity Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials 4 2 19-01-2026 Effect of temperature W.D. Callister, D.G. Rethwisch, Materials Science and Engineering: An Introduction Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials 5 Stress Engineering stress strain curve Strain Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials 6 3 19-01-2026 Universal testing machine ASTM standards Force is measured with a load cell Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials https://encyclopedia.pub/entry/36096 Effect of Grip section on elongation Location of the tensile specimen 7 Measuring strain Strain Gauge Clip-on Video non-contact www.epsilontech.com www.shimadzu.com/ www.michsci.com https://www.instron.com/en/products/testing-accessories/extensometers/ Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials 8 4 19-01-2026 Measuring strain Video non-contact: Digital image correlation www.lavision.de https://www.dantecdynamics.com/ Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials 9 Engineering stress strain curve Yield point phenomenon UTS: Ultimate tensile strength Stress Ductility = Material volume remains constant Necking ② 𝒍𝒐 Yield strength Ductility = Percentage area reduction P: proportional limit Toughness: Area under the curve 𝒍𝒐 𝑨𝒐 =𝒍𝒊 𝑨𝒊 ① 𝒍𝒇𝒓𝒂𝒄 𝒍𝒐 Slope: Young’s Modulus ③ Strain Resilience is the capacity of a material to absorb energy when it is deformed elastically and then, upon unloading, to have this energy recovered. Toughness is the ability of a material to absorb energy and plastically deform before fracturing Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials W.D. Callister, D.G. Rethwisch, Materials Science and Engineering: An Introduction 10 5 19-01-2026 Brittle vs. Ductile materials Resilience 𝑈 = Toughness 𝜎𝑑𝜀 𝑈 = 1 𝜎 𝜀 2 𝑈 = 𝜎 2𝐸 Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials W.D. Callister, D.G. Rethwisch, Materials Science and Engineering: An Introduction 11 True stress & True Strain As 𝒍𝒊 ↑, 𝑨𝒊 ↓ UTS: Ultimate tensile strength Stress 𝝈𝑻 = Material volume remains constant 𝑭 𝑭 𝒍𝒊 𝒍𝒊 = × =𝝈× 𝑨 𝒊 𝑨 𝒐 𝒍𝒐 𝒍𝒐 ② 𝑭 𝑨𝒐 Engineering strain 𝝈𝑻 = 𝝈 × 𝟏 + 𝜹𝒍 𝒍𝒐 𝑭 𝑨𝒊 𝜹𝒍 𝒍𝒐 𝝈𝑻 = 𝝈 × 𝟏 + 𝜺 ≠ Instantaneous strain Instantaneous strain = ③ 𝝈𝑻 = Tension vs. Compression | 𝝈𝑻 vs. 𝝈 Necking 𝒍𝒐 𝑨𝒐 =𝒍𝒊 𝑨𝒊 ① 𝝈= 𝜺𝑻 = 𝜹𝒍 𝒍𝒊 𝒍𝟏 = 𝒍𝒐 + 𝜹𝒍 𝒍𝟐 = 𝒍𝟏 + 𝜹𝒍 𝒍𝟑 = 𝒍𝟐 + 𝜹𝒍 𝜹𝒍 𝜹𝒍 𝜹𝒍 𝜹𝒍 𝜹𝒍 𝜹𝒍 + + + + + ⋯ = ∑𝒊 𝒍𝒐 𝒍𝟏 𝒍𝟐 𝒍𝟑 𝒍𝟒 𝒍𝒊 Strain 𝒅𝜺𝑻 = 𝜺𝑻 𝒍𝒊 𝒅𝜺𝑻 = 𝟎 𝒅𝒍 𝒍 𝒅𝒍 𝒍𝒐 𝒍 𝜺𝑻 = 𝒍𝒏 𝒍𝒊 𝒍𝒐 𝜺𝑻 = 𝒍𝒏 𝟏 + 𝜺 𝒍𝒏 𝟏 + 𝒙 < 𝒙 Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials W.D. Callister, D.G. Rethwisch, Materials Science and Engineering: An Introduction 12 6 19-01-2026 Work hardening Stress UTS: Ultimate tensile strength Material volume remains constant What is happening to the rate of work hardening? Necking Progressively more difficult to work harden an incipient instability sufficiently to remove it 𝒍𝒐 𝑨𝒐 =𝒍𝒊 𝑨𝒊 ① Validity of Ductility beyond 𝜺𝑬𝒖 ? ③ ② 𝜺𝑬𝒖 Strain Work hardening (Strain hardening): The positive slope of the stress-strain curve indicates that the material is made more resistant to plastic deformation by virtue of the deformation itself; that is, the material work hardens Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials W.D. Callister, D.G. Rethwisch, Materials Science and Engineering: An Introduction 13 Considère criterion UTS: Ultimate tensile strength Work hardening rate 𝑭 = 𝑨𝒊 𝝈𝑻 𝒅𝑭 = 𝒅𝑨𝒊 𝝈𝑻 + 𝑨𝒊 𝒅𝝈𝑻 = 𝟎 Material volume remains constant Necking 𝒍𝒐 𝑨𝒐 =𝒍𝒊 𝑨𝒊 ① ② 𝒍𝒐 𝑨𝒐 (𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕)=𝒍𝒊 𝑨𝒊 𝒅𝒍𝒊 𝑨𝒊 + 𝒍𝒊 𝒅𝑨𝒊 = 𝟎 𝒅𝒍𝒊 𝒅𝑨𝒊 =− 𝒍𝒊 𝑨𝒊 ③ Strain Criterion for necking True stress 𝒅𝝈𝑻 𝒅𝒍𝒊 = = 𝒅𝜺𝑻 𝝈𝑻 𝒍𝒊 True Stress Stress 𝒅𝝈𝑻 𝒅𝑨𝒊 =− 𝝈𝑻 𝑨𝒊 𝒅𝝈𝑻 = 𝝈𝑻 𝒅𝜺𝑻 Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials True Strain 14 7 19-01-2026 Hollomon equation 𝝈𝑻 = 𝑲 𝜺𝑻 𝒏 n is the strain-hardening exponent (measure of work hardening behaviour) <1 K is the strength coefficient Strain rate sensitivity 𝝈𝑻 = 𝑲′ 𝜺𝑻̇ 𝒎 𝜺𝑻̇ is the true strain rate m is the strain rate sensitivity Rama Srinivas Varanasi | rama@msme.iith.ac.in | Mechanical Behaviour of Materials 15 8

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