Lecture Triaxial Stress and Yield Criteria. When does yielding occurs in multi-axial stress states?

Transcription

1 Lecture 5.11 Triaial Stress and Yield Criteria When does ielding occurs in multi-aial stress states?

2 Representing stress as a tensor

3 operational stress sstem

4 Compressive stress sstems Triaial stresses: In the oblique zone of a three-roll rotar plug piercing mill, in the closed forging die, near the die throat in etrusion of bar, and under the punch in tube etrusion. Biaial stress: between the rolls of a longitudinal rolling mill with no front and/or back tension, and in the upsetting, open dies.

5 Process Rolling State of Stress in Main Part During Forming Bi-aial compression Forging Tri-aial compression

6 Etrusion Tri-aial compression Wire and tube drawing Bi-aial compression, tension.

7 Straight bending At bend, bi-aial compression and bi-aial tension

8 Stress State: depending on the operation generall comple multiaial stresses develop * Tensile compressive sstems (rolling, deep drawing, wire and bar drawing) 1. Biaial tension uniaial compression. Uniaial tension uniaial compression 3. Uniaial tension biaial compression * Compressive stress sstems (forging, tube etrusion, rolling) 1. Triaial stresses. Biaial stresses * Tensile stress sstems (stretch forming, bulging) 1. Biaial stress * Shear stress sstems (blanking, punch piercing) 1. Shear and possibl tension/compression

9 Uniaial tension and uniaial compression: (1) between the rolls in roll forming, and Uniaial tensiodbiaial compression: (1) in the drawing die. Compressive stress sstems Triaial stresses: (1) in the closed forging die, () near the die throat in etrusion of bar, and Biaial stress: (1) between the rolls of a longitudinal rolling mill () in the upsetting, open dies. Tensile stress sstem Biaial stress: (1) stretch forming

10

11 3- Mohr Circle The previous equations are combined to ield parametric equations for a circle, where ave ave R R

12 Plane Stress and Coordinate Transformations Consider the conditions for equilibrium of a prismatic element with faces perpendicular to the,, and aes. F F 0 0 A The equations ma be rewritten to ield Acos cos Acos Asin sin Asin cos A Acos sin Acos Asin cos Asin sin sin cos cos cos sin cos sin sin

13 cos sin sin cos sin cos

14 CONSTRUCTION OF MOHR S CIRCLE FOR A GIVEN STRESS ELEMENT 1. The ais sstem of Mohr circle is ais. Mark the point X with a coordinate and along the coordinate sstem. 3. Mark another point Y with the coordinate and 4. Join the XY line. Let at point C, XY intersects the horizontal ais. The point C denotes the average normal stress. The line CX denotes X ais and line CY denotes Y ais. 5. Note CX and CY are making angle with each other, whereas in realit the X and Y aes are at an angle RULE: ALL ANGLES IN MOHR S CIRCLE IS TWICE THE REAL ANGLE. 7. Use C as the center, and draw a circle with XY as the diameter. 8. To find stress in a direction U-V, which is angle CW from X-Y ais, draw a line UV through C at an angle CW from XY line. 9. The coordinate values of U & V denote the normal and shear stress in UV direction. Shear stress ais ( uv Y v X u Y(,-) avg + )/ C X uv Normal stress ais (

15 Principal stresses Principal stresses occur on the principal planes of stress with zero shearing stresses. ma,min tan p two angles separated b 90 o

16 Maimum Shearing Stress Maimum shearing stress occurs for ave ma R tan s Note :defines two angles separated b 90 offset from ave p b 45 o o and

17 Find the element orientation for the principal stresses from tan p p 53.1, MPa 10MPa 40MPa p 6.6, Determine the principal stresses from ma,min ma min 70MPa 30MPa

18 Y 4,000 psi ma (8k,13k) 0,000 psi 0,000 psi 5,000 psi 4,000 psi X o 5k X (0k,5k) 0k -5k 4k 8k 1k 5k Y(-4k,-5k) Y avg 0k 4k 8k psi 0 4 R 5 13k psi tan tan R k psi 1 ma avg R k psi avg R 13k psi (8k,-13k) tan , 33.7

19 the trace of an n n square matri σ is defined to be the sum of the elements on the main diagonal I1(A) = tr(a) I1(σ) = tr(σ) = σ +σ + σzz

20 I I I 1 3 I I I z z z z z 3 z z z z z z The three roots of the characteristic equation are the eigenvalues of the stress tensor σij. Within the contet of solid mechanics, such eigenvalues are known as the principal stresses

21 Mohr s Circle for Triaial Stress

22 Determine the maimum principal stresses and the maimum shear stress for the following triaial stress state MPa

23 z z z z z MPa I I I 1 z = = 40 MPa z z z z 3 z z z z z z = MPa 3 I I I = -305 MPa

24 3 65.3MPa 6.5MPa MPa ma 1/ ( ) 58.5MPa

25 Yield criteria and stress-strain relations Yielding in unidirectional tension test takes place when the stress σ = F/A reaches the critical value. Yielding in multiaial stress states is not dependent on a single stress but on a combination of all stresses. Von Mises ield criterion (Distortion energ criterion) Tresca ield criterion (maimum shear stress)

26 Maimum-Shear-Stress or Tresca Criterion This ield criterion assumes that ielding occurs when the maimum shear stress in a comple state of stress equals the maimum shear stress at the onset of flow in uniaial-tension. From the maimum shear stress is given b: ma Where 1 is the algebraicall largest and is the algebraicall smallest principal stress.

27 For uniaial tension, shearing ield stress Y 0 Substituting in Eq. (8.3), we have 0, and the maimum is given b: 1 Y, 3 0 ma Y / Therefore, the maimum-shear-stress criterion is given b: 1 3 Y

28 Since uniform thermal strains occur in all directions in isotropic material, Hooke s law for 3-D z E 1 z E 1 z z E 1

29 Eample An element is subjected to the following stresses: = 70 MPa = 10 MPa = 60 MPa Determine the ielding for both Tresca s and von Mises criteria, given that Y = 150 MPa (the ield stress).

30 Solution Hence,, 1 1 = 160 MPa; = 30 MPa; z = 0 According to Tresca, ma = (160-0)/ = 80 MPa For ielding in uniaial tension: Y/ = 75 MPa Since the 80 MPa > 75 MPa, Tresca criterion would be unsafe.

31 The von Mises criterion can be invoked from Eq o = 175 MPa. 1 ( ) ( ) ( 1/ ) The Von misses predicts that the material will not ield. the Tresca criterion is more conservative than the von Mises criterion in predicting failure.

32

33 The interior of this ellipse defines the region of combined biaial stress where the material is safe against ielding under static loading.

34

35 Von Mises ield criterion (Distortion energ criterion Yielding when (second invariant of stress deviator) = critical value in terms of principal stress components Y in terms of the stress components in the,, z coordinate sstem

36 purel hdrostatic stress σ 1 = σ = σ 3 =σ H will lie along the vector [111] in principal stress space. For an point on this line, there can be no ielding, since in metals, it is found eperimentall that hdrostatic stress does not induce plastic deformation The 'hdrostatic line'

37 Yield Criteria for Ductile Materials Under Plane Stress Maimum distortion energ criteria: Structural component is safe as long as the distortion energ per unit volume is less than that occurring in a tensile test specimen at ield.

38 Fracture Criteria for Brittle Materials Under Plane Stress Brittle materials fail suddenl through rupture or fracture in a tensile test. The failure condition is characterized b the ultimate strength U. Maimum normal stress criteria: Structural component is safe as long as the maimum normal stress is less than the ultimate strength of a tensile test specimen. a b U U