Chapter One - Part Three - Strain and Deformation.pdf
- 1. By : Behailu Mamo March 18, 2025 Strength of Materials I Strain and Deformation
- 2. Outline Deformation and Strain Type of Strain Normal strain Shear strain Stress - Strain Diagram
- 3. Deformation So far, we've focused on the STRESS within structural elements. When you apply stress to an object, it deforms. Think of a rubber band: you pull on it, and it gets longer – it stretches. Deformation is a measure of how much an object is stretched, and strain is the ratio between the deformation and the original length. Think of strain as percent elongation – how much bigger (or smaller) is the object upon loading it.
- 4. Deformation In Engineering, deformation is a change in the shape or size of an object due to: an applied force (the deformation energy in this case is transferred through work) or a change in temperature (the deformation energy in this case is transferred through heat). The first case can be a result of tensile (pulling) forces, compressive (pushing) forces, shear, bending or torsion (twisting). Deformation is often described as strain. Strain is the measure of deformation
- 5. Cont… As deformation occurs, internal inter- molecular forces arise that oppose the applied force. If the applied force is not too large these forces may be sufficient to completely resist the applied force, allowing the object to assume a new equilibrium state and to return to its original state when the load is removed. A larger applied force may lead to a permanent deformation of the object or even to its structural failure.
- 6. Strain Type of strain: Normal Strain :- strain which is due to normal force. Shearing Strain :- strain which is due to shearing (tangential) force.
- 7. Normal Strain Consider a prismatic bar of a homogeneous material is subjected to an axial force P, the bar of length L shows a change in length by δ (deformation). The change in length per unit length is defined a strain, and is denoted by the Greek letter ε (epsilon).
- 8. Cont… • If the bar is in tension, the strain is called a tensile strain representing an elongation of the material. • If the bar is in compression, the strain is a compressive strain, the bar shortens. • Sign convention: Tensile strain is taken as positive and compressive strain as negative. • It is a dimensionless quantity.
- 9. Shear Strain A shear strain results from shear stress. It is a strain computed from relative displacements that are measured parallel to two reference planes. Shear strains measure the relative parallel movement of one reference plane with respect to another.
- 10. Shear Strain The symbol for the shear strain is usually the lowercase Greek symbol gamma (γ). The unit of the shear stress is frequently expressed in radians
- 11. Stress - Strain Diagram Tension or Compression Test - Property of Materials
- 12. Strength of a material can only be determined by experiment One test used by engineers is the tension or compression test This test is used primarily to determine the relationship between the average normal stress and average normal strain in common engineering materials, such as metals, ceramics, polymers and composites TENSION & COMPRESSION TEST
- 13. Performing the tension or compression test Specimen of material is made into “standard” shape and size Before testing, 2 small punch marks identified along specimen’s length Measurements are taken of both specimen’s initial x-sectional area A0 and gauge-length distance L0; between the two marks Seat the specimen into a testing machine shown below TENSION & COMPRESSION TEST
- 14. Performing the tension or compression test TENSION & COMPRESSION TEST The machine will stretch specimen at slow constant rate until breaking point At frequent intervals during test, data is recorded of the applied load P.
- 15. Performing the tension or compression test Elongation δ = L − L0 is measured using either a caliper or an extensometer δ is used to calculate the normal strain in the specimen Sometimes, strain can also be read directly using an electrical- resistance strain gauge TENSION & COMPRESSION TEST
- 16. A stress-strain diagram is obtained by plotting the various values of the stress and corresponding strain in the specimen Conventional stress-strain diagram Using recorded data, we can determine nominal or engineering stress by STRESS-STRAIN DIAGRAM P A0 σ = Assumption: Stress is constant over the x-section and throughout region between gauge points
- 17. Conventional Stress-Strain Diagram Likewise, nominal or engineering strain is found directly from strain gauge reading, or by STRESS-STRAIN DIAGRAM δ L0 = Assumption: Strain is constant throughout region between gauge points By plotting σ (ordinate) against (abscissa), we get a conventional stress-strain diagram
- 18. Conventional stress-strain diagram Figure shows the characteristic stress-strain diagram for steel, a commonly used material for structural members and mechanical elements STRESS-STRAIN DIAGRAM
- 19. Conventional stress-strain diagram STRESS-STRAIN DIAGRAM Elastic behavior A straight line Stress is proportional to strain, i.e., linearly elastic Upper stress limit, or proportional limit; σpl If load is removed upon reaching elastic limit, specimen will return to its original shape
- 20. Conventional stress-strain diagram STRESS-STRAIN DIAGRAM Figure 3-4 Yielding Material deforms permanently; yielding; plastic deformation Yield stress, σY Once yield point reached, specimen continues to elongate (strain) without any increase in load Material is referred to as being perfectly plastic
- 21. Conventional stress-strain diagram STRESS-STRAIN DIAGRAM Figure 3-4 Strain hardening Ultimate stress, σu While specimen is elongating, its x-sectional area will decrease Decrease in area is fairly uniform over entire gauge length
- 22. Conventional stress-strain diagram STRESS-STRAIN DIAGRAM Figure 3-4 Necking At ultimate stress, x-sectional area begins to decrease in a localized region As a result, a constriction or “neck” tends to form in this region as specimen elongates further Specimen finally breaks at fracture stress, σf
- 23. Conventional stress-strain diagram STRESS-STRAIN DIAGRAM Figure 3-4 Necking Specimen finally breaks at fracture stress, σf
- 24. True stress-strain diagram STRESS-STRAIN DIAGRAM Instead of using original cross-sectional area and length, we can use the actual cross-sectional area and length at the instant the load is measured Values of stress and strain thus calculated are called true stress and true strain, and a plot of their values is the true stress- strain diagram
- 25. True stress-strain diagram STRESS-STRAIN DIAGRAM In strain-hardening range, conventional σ- diagram shows specimen supporting decreasing load While true σ- diagram shows material to be sustaining increasing stress
- 26. True stress-strain diagram STRESS-STRAIN DIAGRAM Although both diagrams are different, most engineering design is done within elastic range provided 1. Material is “stiff,” like most metals 2. Strain to elastic limit remains small 3. Error in using engineering values of σ and is very small (0.1 %) compared to true values
- 27. E represents the constant of proportionality, also called the modulus of elasticity or Young’s modulus E has units of stress, i.e., pascals, MPa or GPa. HOOKE’S LAW Most engineering materials exhibit a linear relationship between stress and strain with the elastic region Discovered by Robert Hooke in 1676 using springs, known as Hooke’s law σ = E
- 28. As shown above, most grades of steel have same modulus of elasticity, Est = 200 GPa Modulus of elasticity is a mechanical property that indicates the stiffness of a material Materials that are still have large E values, while spongy materials (vulcanized rubber) have low values HOOKE’S LAW
- 29. IMPORTANT • Modulus of elasticity E, can be used only if a material has linear-elastic behavior. • Also, if stress in material is greater than the proportional limit, the stress-strain diagram ceases to be a straight line and the equation is not valid HOOKE’S LAW
- 30. Thank you for your attention !