ME 201Chapter_3.pptx,all abot stress,strain in 3 d

Stress strain relation and related curves

1. Stress-Strain Relations Dr. Indrasen Singh Principal investigator Computational Solid Mechanics Lab Discipline of Mechanical Engineering, IIT Indore

2. To establish relation between stress and strain for a material, we have to performed tension, compression, shear (torsion) experiments. Tensile Test gage marks gage length Specimen for tension experiments Specimen for Compression Tests

3. Tensile Test The test specimen is placed in a testing machine, and then specimen is stretched by machine either in load control or displacement control . The following quantities are generally recorded during a tension experiment: load P on the specimen, The distance L between the two gage marks, The change in diameter of specimen. The elongation is determined as for each value of P. The change in cross-sectional area is determine as, , where and are current and initial x-sectional area of the specimen, and and are current and initial diameter of the specimen. From load, the (Average) engineering stress and (Average) engineering strain are determined as:  

4. Tensile Stress-Strain Curves Mild Steel Typical Materials Brittle Tool Steel/ concrete Aluminum Alloy Rubber Materials capable of withstanding large strains before failure are ductile materials, otherwise they are brittle materials The terminal point of curve represent complete failure (rupture) of specimen.

5. Nominal Stress versus nominal strain curves Ductile metals Homogeneous deformation Heterogeneous deformation Point Stress-strain relation is linear (can be represented by a line). : Nonlinear response though deformations are purely elastic. Point plastic yielding commences. Dislocation (carriers of plasticity) begins to move leading to permanent deformation. Beyond material exhibit plastic or permanent deformation. Strain hardening occurs and geometrical softening. Strain hardening (due to dislocations pile up) is more dominant. Strain softening (due to reduction in cross-sectional area) is weaker at beginning of yielding, but its effect continues to increase. At point , geometrical softening is strong enough to counter strain hardening resulting in saturation in stress-strain curve. Beyond point , plastic deformation gets localised in the form of necks. fracture point.  

6. Nominal Stress versus nominal strain curve Ductile metals Linear Elastic If load is removed gradually on or before pl , specimen will come back to original shape. There is no permanent from O to pl . This is known as elastic deformation. Since curve is linear from O-pl , the behavior is known as linear elastic response.   From O-pl:   is known as Young’s Modulus  

7. Nominal Stress versus nominal strain curve Ductile metals Linear Elastic Non-Linear Elastic From : no permanent deformation- elastic response. Since curve is between pl- el , the behavior is known as nonlinear elastic response.   From pl- el :  

8. Nominal Stress versus nominal strain curve Ductile metals Linear Elastic Non-Linear Elastic Strain Hardening   Instantaneous tangent Modulus     From y-u: Materials yields=> Plastic deformation starts=> there will be a permanent deformation in the materials after removal of the applied load. Slope of unloading curve = E. Total strain at point C = , Plastic strain at P         C

9. Nominal Stress versus nominal strain curve Ductile metals Linear Elastic Non-Linear Elastic Strain Hardening   Instantaneous tangent Modulus           C If loaded again after unloading from C, material will yield at point C. Thus yield strength increases . This behavior is known as strain hardening .

10. Nominal Stress versus nominal strain curve Ductile metals Linear Elastic Non-Linear Elastic Strain Hardening Geometrical Softening Necking start at point u X-section changes non-uniformly. X-section changes uniformly.

11. Nominal Stress versus nominal strain curve Ductile metals Homogeneous deformation Linear Elastic Non-Linear Elastic Strain Hardening Geometrical Softening

12. Nominal Stress versus nominal strain curve Ductile metals Homogeneous deformation Heterogeneous deformation Linear Elastic Non-Linear Elastic Strain Hardening Geometrical Softening

13. Nominal Stress versus nominal strain curve Ductile metals : yield strength of material : Ultimate tensile strength Fracture strength Note: Slope of in elastic regime = Young’s Modulus, Slope of line Nominal fracture strength .   Homogeneous deformation Heterogeneous deformation

14. Measure of Ductility A standard measure of the ductility of a material is its percent elongation which is defined as: Percentage elongation = Here, and are final length of specimen at rupture and initial length of the specimen. If percentage of elongation is greater than 5%, material is ductile otherwise it is brittle. Another measure of ductility that is sometimes used is the percent reduction in area, and is given by: Percentage reduction in area = and are final X-section of the specimen rupture and initial X-section. For structural steel, percent reductions in area of 60 to 70 percent are common .  

15. True Stress and True Strain True stress True strain or logarithmic strain   Nominal/Engineering stress versus Nominal/Engineering strain True Stress-True strain True stress continually increases. True fracture stress is greater than the true ultimate stress. Note: During Plastic deformation, volume of the metal remains constant, hence  

16. Compressive Stress-Strain Curve For a ductile material, the stress-strain curve for compressive loading is essentially the same within linear elastic regime. Young′s Modulus in compression = Young′s Modulus in Tension For steel, the yield strength is the same in both tension and compression. For larger values of the strain, the tension and compression stress-strain curves diverge, and necking does not occur in compression. For most brittle materials, the ultimate strength in compression is much larger than in tension  

17. Shear Stress-Strain Curve Shear stress, -shear strain, curve are obtained from torsion tests. Within linear elastic regime,            

18. 1D Hook’s Law- Stress-Strain Relations We have from tension and shear experiments that slope of stress-strain curve is constants up to proportionality limit. Thus, stress and strain can be related within linear elastic regime as: and, While writing above relations, material is assumed to be homogeneous and isotropic. Above relations are also known as Hook’s law. Hook’s Law is valid only up to proportional limit or within linear elastic limit and itis not valid for non-linear elastic material.  

19. POISSON’S RATIO In all engineering materials, the elongation (or contraction) produced by an axial force P in the direction of the force is accompanied by a contraction (or elongation) in other two transverse directions . Assume that materials are homogeneous (i.e., mechanical properties, E, G etc. are independent of position) and isotropic (i.e., mechanical properties, E, G etc. are independent of direction).         For homogeneous and isotropic material, the same value of strains for any transverse direction should be equal. For stretching along x axis, . The Poisson’s ratio, is defined as:    

20. POISSON’S RATIO Thus, for axial loading along x axis:           Note that is an elastic constant, a material properties governing the elastic response of material.   The value of fluctuates for different materials over a relatively narrow range, generally, it is on the order of 0.25 to 0.35. for some concretes) and for occur. is attained by materials during plastic deformation due to the constancy of volume during plastic flow. Poisson's effect exhibited by materials causes no additional stresses unless the deformation or displacement along transverse directions is constrained.  

21. POISSON’S RATIO Example: A 500 mm long, 16-mm-diameter rod made of a homogenous, isotropic material is observed to increase in length by 300 m, and to decrease in diameter by 2.4 m when subjected to an axial 12 kN load. Determine the modulus of elasticity and Poisson’s ratio of the material.  

22. 3D stress-strain relations Consider a material subjected to multiaxial loading. If stresses are within linear elastic regime, the strains at point due to combined loading can be obtained by employing superposition theorem.                   Only normal stresses Only shear stresses Normal + Shear stresses                   (a) (b)             (c)          

23. 3D stress-strain relations           (b)   (b1)   (b2)     (b3)              

24. 3D stress-strain relations                       (c)     (c1)     (c2)       (c3)        

25. 3D stress-strain relations                             (a) (b)             (c)          

26. 3D stress-strain relations   The generalized stress-strain relation with linear elastic regime for a homogeneous isotropic material is given by:   Above relation is known as Inverse Hook’s Law. In above relation, three elastic constants, E, G and appear, but out of three only two are independent. The deformation behavior of homogeneous isotropic linear elastic solid is governed by only two elastic constants  

27. Deformation under uniform pressure Consider a body subjected to following stress state. Determine the strain and the ratio tr [ ] and tr [ ].   Solution: From the generalized stress-strain relations, The strain matrix:        

28. Dilatation or Volumetric Strain         d     Consider a cubical body with edges subjected to normal stresses, and Lets and are effective strain along x y and z axis due to applied stresses. Size of the edges of cube after deformation: Along x axis: Along y axis: Along x axis: Volume of the cube before deformation:   Before Deformation After Deformation Volume of the cube after deformation: . . Change in volume Volumetric strain  

29. Dilatation or Volumetric Strain Volumetric strain represents the change in volume per unit volume. is Bulk Modulus