CHAPTER 6 PLASTIC ZONE SIZE

Transcription

1 96 CHAPTER 6 PLASTIC ZONE SIZE This chapter elaborates on conventional methods and finite element methods to evaluate the plastic zone size at failure. An elastic-plastic finite element analysis procedure for determination of plastic zone size and its shape is also presented. A failure criterion is also suggested based on the plastic zone size. Linear elastic stress analysis of sharp cracks predicts infinite stresses at the crack tip. It is very evident from the Equation (1.1) that in case of a sharp crack, when0, these equations result in infinite stresses which is known as stress singularity in the field of linear elastic fracture mechanics (LEFM). In real materials, however, stresses at the crack tip are finite because the crack-tip radius must be finite. Since structural materials deform plastically above the yield stress, in reality, there will be a plastic zone surrounding the crack tip which leads to further relaxation of crack-tip stresses. When yielding occurs, stresses must be redistributed in order to satisfy equilibrium. The elastic stress analysis becomes increasingly inaccurate as the inelastic region at the crack tip grows. As rigorous analysis is complex, closed form solutions are not available. Hence, approximate solutions had been developed by various researchers.

2 APPROXIMATE PLASTIC ZONE SHAPE AND SIZE Before applying yield criterion, it is useful to determine the principal stresses. Consider a case of mode I, is a principal stress because for plane problems. It needs to rotate axes in plane to determine the principal stresses and. Shear stress x R Normal stress x 2 Applied Shear stress Figure 6.1 Mohr circle to determine principal stress = = = These equations determine principal stresses and as(anderson 2005), = + = (6.1) = = (6.2)

3 98 The third principal stress becomes, = 0, for plane stress = ( + ) = 2, for plane strain Plastic Zone Shape for Plane Stress Two widely used yield criteria, Mises and Tresca, are applied to Mode I so as to determine the plastic zone size for plane stress cases(anderson 2005). All the three principal stresses and associated Mohr circles are shown in Figure 6.1. To ensure the yielding of the material, the von Mises criterion states that, ( ) + ( ) + ( ) 2 (6.3) Where, is the yield stress. Substituting and in the equation and using the symbol r pz in place of r for the equality sign of the Mises criterion, we obtain = 2 This is simplified to = sin + cos (6.4) The shape of the plastic zone is shown in Figure 6.2 The shape of plastic zone size is slightly different if the Tresca yield criterion is invoked. In order to invoke yielding, the Tresca yield criterion states that,

4 99 2 For plane stress, which is between and. Thus at 0 2 = 2 Substituting, the approximate shape of plastic zone is obtained as = (6.5) The shape of the plastic zone by Tresca yield criterion is shown in Figure Plastic Zone Shape for Plane Strain The third principal stress is no longer zero and, therefore, it influences the yielding considerably. To find the shape of the plastic zone using the Mises yield criterion,, and are substituted and the resulting equation (Anderson 2005) simplifies to = (1 2) (1 + ) (6.6) The shape of the plastic zone size by plane strain von Mises criterion is shown in Figure 6.2. If Tresca yield criterion is applied, it is found is always larger than and. Subtracting we get,

5 100 = [( )(1 + )] 2 For small 38.9 = 1 3, is the smallest principal stress and the yielding is governed by and, we have Leading to, = (6.7) For large value of 38.9 = 1 3 is the smallest principal stress and the yielding is governed by and. Then is obtained by using, = 2 (6.8) The resulting shape of plastic zone in terms of non-dimensional distance for plane strain and Tresca yield criterion is shown in Figure 6.3,

6 Plane stress plane (von Mises) stress(vonmises) Plane strain plane (von Mises) strain(vonmises) Figure 6.2 von Mises plane stress and strain (General equation) Plane stress (Tresca) Plane strain (1,3 Tresca) Plane stress(tresca) plane strain(1,3 tresca) plane Plane strain(1,2 tresca) (1,2 Tresca) Figure 6.3 Tresca plane stress and strain (General equation)

7 EFFECTIVE CRACK LENGTH The appearance of the plastic zone at the tip does not allow its material to bear high stresses predicted by elastic analysis. Owing to the presence of plastic zone, the stiffness of the component decreases or the compliance increases. Consequently, the crack is equivalent to a length that is longer than actual length. The size of the plastic zone in front of the crack tip determines the effective crack length (Anderson 2005). There are various approaches to determine the effective crack length which are elaborated below Approximate Approach One of the simplest expressions for the plastic zone size (Anderson 2005) is found from curve for mode I. The length of the plastic zone r * along (Figure 6.4) direction is then obtained from the relation Figure 6.4 Approximate plastic zone size r *

8 103 Where k depends on whether the case is of plane stress or plane strain. Substituting ( ) and solving r *, we get = 2 (6.9) Evaluate k for both plane stress and plane strain. Due to the symmetry, vanishes on the =0 plane. The principal stresses as, =0, for plane stress ( )= 2, for plane strain Substituting and then comparing resulting expression, we get, =1, for plane stress = 1/(1 ), for plane strain value of k, we get, In fact, k=3 for = 1/3 in the case of plane strain. Substituting the For plane stress, = 1 2 (6.10) For plane strain( = 1/3),

9 104 = 1 18 (6.11) Irwin Plastic Zone Correction Irwin suggested a vastly improved expression (Anderson 2005) for the plastic zone size along the -axis (Figure 6.5) through a model which accounts for the absence of high stresses within the yield zone. Consider a case of plane stress with k=1. The tip of the effective crack is located somewhere inside the plastic zone. There should be an appropriate criterion to find the length of the effective crack. Figure 6.5 Irwin plastic zone correction The overall plastic zone size becomes, = 2 = (6.12) The effective crack length a eff is given by, +

10 Plastic Zone Size through Dugdale Approach Dugdale determined the plastic zone size (Dugdale 1960) through a different approach. He considered the effective crack to be of length where is plastic zone size as shown in Figure 6.6 a (Anderson 2005). Figure 6.6 (a) plastic zone size through Dugdale s approach and (b) nullifying the singularity using Green s function approach In the Dugdale approach, singularity at the tip of the effective crack is nullified by a uniform pressure equal to yield stress on the portion of the crack, as shown in Figure 6.6a. in fact, to determine we employ the criterion that is the length on which pressure exactly nullifies the singularity. If is the singularity due to pressure and due to external load on the effective crack, we have =0

11 106 An approximate but a simpler solution may be obtained as, = 8 (6.13) 6.3 NON-LINEAR FINITE ELEMENT ANALYSIS In this work, non-linear finite element analysis is performed to observe the elastic-plastic behaviour around the crack tip and hence to find out the plastic zone size. The elastic-plastic process requires continuous assessment of stress and plastic strain at all points of the structure, as the applied load increases. Hence the load is applied in a sequence of relatively small increments and within each step a check on equilibrium and stress is made. As loading starts, the program starts to iterate the stress above the yield stress to consider the plastic effects. In order to run the non-linear analysis, stress-strain data of the material is to be supplied in addition to the Young's modulus and Poisson's ratio. The stress-strain relation is established using inverse Ramberg- Osgood relation [Equation (4.58)] in which the material constants are; = , n = 1.11, Young s modulus, E = 2 10 MPa and Poisson ratio, = 0.3. The whole non-linear curve is considered to consist of a number of straight lines, each being designated as a load step. From the finite element analysis, at the load corresponding to the failure, the distance along crack path where stress in the direction of applied load is greater than or equal to the yield strength is noted as the plastic zone size.

12 RESULTS AND DISCUSSION For determining the plastic zone size and shape analytically and numerically, the results of the experiments carried out by Hall & Finger (1976) using SS304 is utilized. Table 5.1 shows the detail used for this analysis. The surface crack is made equivalent to through crack using the procedure as explained in section 5.6. The size of the equivalent through crack is given in Table 5.2. Plastic zone size and shape is analytically determined for the equivalent through crack of centre crack tensile (CCT) specimen. As the analytical methods are based on elastic assumptions, the plastic zone size and shape are determined at lower far field stress, 100 MPa, in linear finite element analysis as shown in Figure MPa mm 100 MPa mm Figure 6.7 Centre Crack Tension Specimen (Full view)

13 108 The properties of stainless steel 304 are, yield stress, = 328 and ultimate strength, = 612. The radius of plastic zone determined for Irwin s approach, Dugdale approach, Approximate approach and FEA are presented in Table 6.1 and the first set of results are shown in bar chart as shown in Figure 6.8. The plastic zone determined from FEA is almost in line with plastic zone size obtained from Approximate approach. As the strip yield condition of Dugdale is not incorporated in the FEA, the observation from FEA is compared to the approximate solution. Approximate approach and results obtained from FEA are compared as shown in Table 6.1 and the standard error is Figure 6.9 shows the plastic zone size comparison between results obtained from FEA and theoretical approach (Approximate approach). The shape of the plastic zone is determined for von Mises criterion and Tresca s criterion and shown in Figures 6.10 and In plane strain conditions, plane 1, 3 and plane 1, 2 mean that they are governed by principal stresses 1 & 3 and 1 & 2 respectively (Figure 6.11) mm Dugdale model Irwin approach Plane stress, approximate approach FEA Figure 6.8 Radius of Plastic zone (First set in Table 6.1)

14 109 Table 6.1 Radius of plastic zone in various approaches Sl. No Irwin correction (mm) Dugdale Approach (mm) Approximate Approach (mm) FEA (mm) Relative Error (%) Standard error = Length, m Length, m Figure 6.9 Plastic zone comparison (First set)

15 Plane stress (von Mises)-Steel plane stress- vonmises-steel plane strain-vonmises-steel Plane strain (von Mises)-Steel Figure 6.10 Shape of plastic zone in von Mises plane stress and strain for SS 304

16 plane Plane stress-tresca-steel (Tresca)-Steel Plane strain (1,3 Tresca)-Steel plane strain-tresca-1,3-steel plane strain-tresca-1,2-steel Plane strain (1,2 Tresca)-Steel Figure 6.11 Shape of plastic zone in Tresca plane stress and strain for SS 304 The plastic zone size computed through non-linear finite element analysis matches well with the theoretical models at lower applied stresses. However they do not agree well at higher loads since the theoretical models are based on elastic assumptions. Hence for failure analysis, one has to rely on non-linear finite element analysis in order to estimate plastic zone size. The non-linear finite element analysis process automatically terminates at a load corresponding to a phenomenon known as Global plastic deformation (GPD). Global Plastic Deformation is marked by the von Mises stress reaching a value above yield strength throughout the section. The failure criterion is proposed as the load at GPD which takes the net section to a von Mises stress value between yield strength and ultimate strength as

17 112 shown in Figure This criterion will be used for analysing the failure of pressure vessel/piping in Chapter 7. Plastic zone contour at various loadings in non-linear finite element analysis is shown in Figure ult ys Von Mises stress, MPa Applied Stress, MPa Figure 6.12 Variation of von Mises Stress with Applied Stress

18 113 Crack Crack (a) (b) (c) (d) (e) (f) Figure 6.13 Plastic zone contours at various loadings

19 CONCLUDING REMARKS The computational methods are found to be useful for determination of stress intensity factor as well as plastic zone size. The plastic zone size to be used as a variable for fracture assessment has been demonstrated. The result so obtained may be correlated to the failure of actual structure such as pipes and pressure vessels through appropriate fracture model. The finite element analysis procedure can also be extended to actual structures like conical pipes and spherical pressure vessels of power plants and process industries.