ELASTICITY (MDM 10203)

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1 ELASTICITY () Lecture Module 3: Fundamental Stress and Strain University Tun Hussein Onn Malaysia

2 Normal Stress inconstant stress distribution σ= dp da P = da A dimensional Area of σ and A σ A 3 dimensional Volume of σ and A

3 Normal Stress Tensile and Compressive Compression Compressive stress P c Tension Tensile stress P t 3

4 Normal Strain Concept Normal strain ε is the elongation or contraction of a line segment per unit of length L = L Strain is a non-dimensional Quantity i.e. it has no units. It is simply a ratio of two quantities with the same unit. 4

5 Normal Strain Calculation F o = Ao Normal stress (engineering stress) Elongation = L L o Normal strain (engineering strain) L L o o= Lo Lo A Ao F L o= δ F L 1 Lo Volume of the bar must be the same before and after elongation Ao Lo =A L 5

6 True Stress and True Strain True stress is the stress at cross section A Since Ao Lo =A L A= Lo Ao L True stress F F L = = A Ao L o True strain dl d = L from o= L 1 Lo ln = F 1 o Ao = o 1 o L dl = L L =ln L Lo o L =ln o 1 Lo =ln 1 o 6

7 Stress Strain Test 7

8 Stress Strain Diagram 8

9 Elastic Behavior In this region the curve is a straight line, so that stress is proportional to the strain. It is linearly elastic The upper stress limit to this linear relationships is called the proportional limit, σpl If the load is released before the stress is below σpl then the material will return to its original shape. 9

10 Yielding A slight increase in stress above the elastic limit will cause the material to deform permanently, which is called yielding. The stress that causes yielding is called yield stress, σy When the material will continue to strain without any increase in load, it is often referred to as being perfectly plastic. 10

11 Strain Hardening When yielding has ended, a further load can be applied to the specimen, resulting in a curve that rises continuously until it reaches a maximum stress referred to as the ultimate stress, σu The rise in the curve is called strain hardening 11

12 Necking, Fracture At the ultimate stress, the cross sectional area begins to decrease in a localized region. A constriction or neck gradually tends to form in this region as the specimen elongates further. Necking phenomenon will be ended in a fracture of the specimen. The specimen breaks at the fracture stress, σu. 1

13 Typical Stress-Strain Diagram 13

14 Shear Strain Concept The tangent of the angle through which two adjacent sides rotate relative to their initial position is termed shear strain. x L tan = = x L 14

15 Poisson's Ratio The ratio of lateral strain to axial strain is a constant known as the Poisson's ratio. ratio Original shape Final shape ' L ' lat = r r L long = lat = long 15

16 General 3D Structure σy σxy σyz y σx σxz σz x z 16

17 Three-Dimensional Stress Strain Relationships Loading on x -direction x x =, y = x, z = x E x x x x =, y =, z = E E E 17

18 Three-Dimensional Stress Strain Relationships Loading on y -direction y y=, x = y, z = y E y y y y=, x =, z = E E E 18

19 Three-Dimensional Stress Strain Relationships Loading on z -direction z z =, y = z, x = z E z z z z =, y =, x = E E E 19

20 Three-Dimensional Stress Strain Relationships Loading on x,y, and z directions x x = y z E E y y = x z E E z z = x y E E {} [ ]{ } x 1 x 1 = y 1 y E 1 z z 0

21 Three-Dimensional Stress Strain Relationships Volumetric strain (dilatation): e= x y z e= when 1 x y z E x x = y z E E y y = x z E E z z = x y E E x = y = z = Bulk modulus: E = e 3 1 K= E 3 1 1

22 Three-Dimensional Shear stress Shear Strain Relationships xy xy = G yz yz = G Shear modulus (G): G= E 1 xz xz = G

23 Three-Dimensional Shear stress Shear Strain Relationships Bulk modulus (K): K= E 3 1 E= Shear modulus (G): G= E 1 9GK 3K G 3

24 Three-Dimensional Stress and Lame's constant Stress-strain relationships can be expressed: x =G x e y =G y e z =G z e Lame's constant: E = 1 1 4

25 Three-Dimensional (collection) relationships K, G, E and ν G 1 E K = G = = G= 1 3K 1 3KE E = = = 1 9K E 1 E= G 3 G 9GK = G 1 = 3K 1 = G 3K G = E 3K G 3K E = 1 = = G G 3K G 6K 5

26 Two-Dimensional Plane Stress vs Plane Strain Under certain conditions, the state of stresses and strains can be simplified. A general 3D structure can therefore be reduced to a -D analysis. PLANE STRESS PLANE STRAIN z= yz= zx =0 z = yz = zx =0 z 0 z 0 6

27 Two-Dimensional Plane Stress vs Plane Strain PLANE STRESS PLANE STRAIN 7

28 Stress Transformation y' y y y' xy x ' y ' x' x ' x x 8

29 Stress Transformation y' y y' x' y' A x' x ' A x Acos x A xy Acos xy Asin y Asin 9

30 Stress Transformation Applying F x ' =0 F y ' =0 x ' = x cos y sin xy sin cos y ' = x cos y sin xy sin cos x ' y ' = x y sin cos xy cos sin 30

31 Stress Transformation Further working by substituting sin =sin cos 1 cos sin = 1 cos cos = x y x y x ' = cos xy sin x y x y y'= cos xy sin x y x ' y ' = sin xy cos 31

32 Mohr's circle construction 1. Create coordinate system, see the positive directions below 3

33 Mohr's circle construction. Locate the center of the circle x y avg = 33

34 Mohr's circle construction 3. Locate a point for which the x' axis coincides with the x axis x y x y x 34

35 Mohr's circle construction 4. Now you can draw the circle, the radius is CA x y x y C p x A 35

36 Mohr's circle construction Identifying Principal Stress 1= OC CA 1 = OC CA x y x y P O P1 C x A x y OC = CA= x y xy 36

37 Mohr's circle construction Stresses on Arbitrary Plane If you want to know the stresses at orientation θ x ' y' O x' y' C x' A x xy 37

38 Strain Transformation y' x x = dx y y y= dy x = x dx dy y = y dy xy x' xy dx x x positive shear dx=dx ' cos dx=dx ' sin dy dx ' dx 38

39 Strain Transformation from x ' = x dx cos y '= x dx sin x ' dx ' dx x dx x x dx 39 x

40 Strain Transformation from x ' = y dy sin y '= y dy cos y dy dy dx ' x ' x 40 y

41 Strain Transformation xy from y '= xy dy sin xy dy x ' = xy dy cos xy dy dy dx ' x ' x 41

42 Strain Transformation combining x y xy x ' = x dx cos y dy sin xy dy cos y '= x dx sin y dy cos xy dy sin finally x y x y xy x ' = cos cos x y x y xy y ' = cos cos x ' y ' x y xy = sin cos 4

43 Principal Strains An element can be oriented, so that the element's deformation is caused ONLY by NORMAL STRAINS with NO SHEAR STRAIN. The NORMAL STRAINS are referred to as PRINCIPAL STRAINS. STRAINS xy tan P = x y x y 1 = x y = x y xy x y xy 43

44 Mohr's circle construction 1. Create coordinate system, see the positive directions below 44

45 Mohr's circle construction. Locate the center of the circle x y avg = 45

46 Mohr's circle construction 3. Locate a point for which the x' axis coincides with the x axis x y x y xy x 46

47 Mohr's circle construction 4. Now you can draw the circle, the radius is CA x y x y C xy x A 47

48 Mohr's circle construction Identifying Principal Strains 1= OC CA 1 = OC CA x y x y C O xy x A x y OC = CA= x y xy 48

49 Mohr's circle construction Orientation Principal Strains xy tan p = x y xy 1 p = atan x y x y O C p xy A 49

50 Mohr's circle construction Physical Orientation Principal Strains O C p A x and x' coincide (θ=0) 50

51 Mohr's circle construction Physical Orientation Principal Strains y' y O C p A x' x and x' coincide (θ=0) 51

52 Mohr's circle construction Physical Orientation Principal Strains Conveniently drawn in x and y as horizontal and vertical lines y' y dy ' 1 dx ' x' p x 5

53 Mohr's circle construction In-Plane Shear Strain s =90o p O C s A x and x' coincide (θ=0) 53

54 Mohr's circle construction In-Plane Shear Strain s =90o p O C s A x and x' coincide (θ=0) 54

55 Mohr's circle construction In-Plane Shear Strain Conveniently drawn in x and y as horizontal and vertical lines y y' avg dy ' x y avg = avg dx ' x p x' 55

56 Mohr's circle construction Strains on Arbitrary Plane If you want to know the strains at orientation θ x ' y ' O C xy x' A x 56

57 Mohr's circle construction Strains on Arbitrary Plane Conveniently drawn in x and y as horizontal and vertical lines y' y y ' dy ' x' x x ' dx ' 57

58 Measuring strain A strain gauge is a device used to measure the strain of an object. Invented by Edward E. Simmons and Arthur C. Ruge in 1938, the most common type of strain gauge consists of an insulating flexible backing which supports a metallic foil pattern. The gauge is attached to the object by a suitable adhesive, such as cyanoacrylate. As the object is deformed, the foil is deformed, causing its electrical resistance to change 58

59 Measuring strain 59

60 Strain Rossetes Since the strain gauge measures only in one direction, several strain gauges can be Used to measure other direction of strains. Three gauges are required to measure the strains. b a a= x cos a y sin a xy sin a cos a a b c b= x cos b y sin b xy sin b cos b c = x cos c y sin c xy sin c cos c c From the three equations, x y xy can be found 60

61 Strain Rossetes b a {}[ ]{ } cos sin a sin a cos a a a x b = cos b sin b sin b cos b y c cos c sin c sin c cos c xy a b c c { }[ cos a sin a sin a cos a x y = cos b sin b sin b cos b xy cos c sin c sin c cos c ]{} 1 a b c 61

62 Strain Rossete 45 c b x = a y = c 45o 45 xy = b a c o a 6

63 Strain Rossete 60 b c x = a 1 y = b c a 3 60o 60o a xy = b c 3 63

64 64