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Stress/strain Relationship for Solids
This document discusses stress and strain concepts including: 1. Definitions of normal stress, normal strain, Poisson's ratio, shear stress, and shear strain. It also discusses tensile testing and stress-strain curves. 2. Stress-strain curves are shown for ductile and brittle materials. An example curve for low-carbon steel is described. 3. True stress and true strain are defined based on instantaneous cross-sectional area and gage length. Different regions of stress-strain curves are identified.
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Stress/strain Relationship for Solids
- 2. Definition of normalstress (axial stress) A F =σ
- 3. Definition of normalstrain 0L L∆ =ε
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- 5. Definition of shearstress 0A F =τ
- 6. Definition of shearstrain l x∆ == θγ tan
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- 12. Example: stress-strain curvefor low-carbon steel •1 - Ultimate Strength •2 - Yield Strength •3 - Rupture •4 - Strain hardening region •5 - Necking region Hooke's law is only valid for the portion of the curve between the origin and the yield point. http://en.wikipedia.org/wiki/Hooke's_law
- 13. σPL ⇒ ProportionalLimit - Stress above which stress is not longer proportional to strain. σEL ⇒ Elastic Limit - The maximum stress that can be applied without resulting in permanent deformation when unloaded. σYP ⇒ Yield Point - Stress at which there are large increases in strain with little or no increase in stress. Among common structural materials, only steel exhibits this type of response. σYS ⇒ Yield Strength - The maximum stress that can be applied without exceeding a specified value of permanent strain (typically .2% = .002 in/in). OPTI 222 Mechanical Design in Optical Engineering 21 σU ⇒ Ultimate Strength - The maximum stress the material can withstand (based on the original area)
- 14. True stress andtrue strain True stress and true strain are based upon instantaneous values of cross sectional area and gage length
- 15. The Region ofStress-Strain Curve Stress Strain Curve • Similar to Pressure-Volume Curve • Area = Work Volume Pressure Volume
- 16. Uni-axial Stress State Elasticanalysis
- 17. Stress-Strain Relationship εσ E= E-- Young’s modulus γτ G= G -- shear modulus Hooke’s Law:
- 18. Stresses on InclinedPlanes
- 19. Thermal Strain Straincaused bytemperature changes. α is a material characteristic called the coefficient of thermal expansion.
- 20. Strains caused bytemperature changes and strains caused by applied loads are essentially independent. Therefore, the total amount of strain may be expressed as follows:
- 21. Bi-axial state elasticanalysis
- 22. (1) Plane stress •State of plane stress occurs in a thin plate subjected to forces acting in the mid-plane of the plate • State of plane stress also occurs on the free surface of a structural element or machine component, i.e., at any point of the surface not subjected to an external force.
- 23. Transformation of PlaneStress
- 24. Mohr’s Circle (PlaneStress) http://www.tecgraf.puc-rio.br/etools/mohr/mohreng.html
- 25. Mohr’s Circle (PlaneStress)
- 26. Instruction to drawMohr’s Circle 1. Determine the point on the body in which the principal stresses are to be determined. 2. Treating the load cases independently and calculated the stresses for the point chosen. 3. Choose a set of x-y reference axes and draw a square element centered on the axes. 4. Identify the stresses σx, σy, and τxy = τyx and list them with the proper sign. 5. Draw a set of σ - τ coordinate axes with σ being positive to the right and τ being positive in the upward direction. Choose an appropriate scale for the each axis. 6. Using the rules on the previous page, plot the stresses on the x face of the element in this coordinate system (point V). Repeat the process for the y face (point H). 7. Draw a line between the two point V and H. The point where this line crosses the σ axis establishes the center of the circle. 8. Draw the complete circle. 9. The line from the center of the circle to point V identifies the x axis or reference axis for angle measurements (i.e. θ = 0). Note: The angle between the reference axis and the σ axis is equal to 2θp.
- 27. Mohr’s Circle (PlaneStress) http://www.egr.msu.edu/classes/me423/aloos/lectur e_notes/lecture_4.pdf
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- 30. http://www4.eas.asu.edu/concrete/elasticity2_95/sld001.htm Stress-Strain Relationship (Plane stress) ))(( 1 yxz E σσνε+−= −− = xy y x xy y x E γ ε ε ν ν ν ν τ σ σ 2 1 00 01 01 1 2
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- 32. Coordinate Transformation The transformationof strains with respect to the {x,y,z} coordinates to the strains with respect to {x',y',z'} is performed via the equations
- 33. Mohr's Circle (PlaneStrain) (εxx' - εavg )2 + ( γx'y' / 2 )2 = R2 εavg = εxx + εyy 2 http://www.shodor.org/~jingersoll/weave4/tutorial/tutorial.html
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- 36. Stress-Strain Relationship (Plane strain) + −+ =)( 211 yxz E εε ν ν ν σ − − − − −+ − = z y x z y x E ε ε ε ν ν ν ν ν ν νν ν σ σ σ )1(2 21 00 01 1 0 1 1 )21)(1( )1(
- 37. Tri-axial stress state elasticanalysis
- 38. 3D stress ata point three (3) normal stresses may act on faces of the cube, as well as, six (6) components of shear stress
- 39. Stress and straincomponents
- 40. The stress ona inclined plane ))(() 2 () 2 ( 3121 22322232 σσσσ σσ τ σσ σ −−+ − =+ + + lnn ))(() 2 () 2 ( 1232 22132213 σσσσ σσ τ σσ σ −−+ − =+ + + mnn ))(() 2 () 2 ( 2313 22212221 σσσσ σσ τ σσ σ −−+ − =+ + + nnn y x z (l, m, n) p 2σ 3σ 1σ nτ nσ
- 41. 3-D Mohr’s Circle *The 3 circles expressed by the 3 equations intersect in point D, and the value of coordinates of D is the stresses of the inclined plane D
- 42. Stress-Strain Relationship For isotropicmaterials Generalized Hooke’s Law: − ∆ − − − − − − − −+ = 0 0 0 1 1 1 21 2 21 00000 0 2 21 0000 00 2 21 000 0001 0001 0001 )21)(1( ν α γ γ γ ε ε ε ν ν ν ννν ννν ννν νν τ τ τ σ σ σ TEE zx yz xy z y x zx yz xy z y x