- Email: [email protected]
Photoelastic study on the effect of flow induced residual stresses on fracture parameters
Accepted Manuscript Photoelastic study on the Effect of Flow Induced Residual Stresses on Fracture Parameters M. Subramanyam Reddy, K. Ramesh PII: DOI: Reference:
S0167-8442(16)30032-5 http://dx.doi.org/10.1016/j.tafmec.2016.04.003 TAFMEC 1702
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
1 February 2016 8 April 2016 10 April 2016
Please cite this article as: M. Subramanyam Reddy, K. Ramesh, Photoelastic study on the Effect of Flow Induced Residual Stresses on Fracture Parameters, Theoretical and Applied Fracture Mechanics (2016), doi: http:// dx.doi.org/10.1016/j.tafmec.2016.04.003
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Photoelastic study on the Effect of Flow Induced Residual Stresses on Fracture Parameters Subramanyam Reddy Ma, Ramesh Ka a
Dept. of Applied Mechanics, IIT Madras, Chennai - 600036, India. Corresponding Author: Subramanyam Reddy M Dept. of Applied Mechanics, IIT Madras, Chennai - 600036, India. email: [email protected]
ABSTRACT Flow induced residual stresses, formed during manufacturing, influence the crack initiation and growth process in polymer products. The effect of these residual stresses on mode-I parameters is studied using photoelasticity. Polycarbonate (PC) sheets having only flow induced residual stresses are used for making the single edge notched (SEN) specimen. Residual stresses in the PC sheet are measured using carrier fringe method. Two crack configurations, where residual stress is parallel and perpendicular to the crack axis are considered. The study showed that flow induced residual stress introduces an additional crack tip constraint which depends on the direction and magnitude of the residual stress. A new parameter is defined to quantify the residual stress induced crack tip constraint. Key words: Flow Induced Residual Stress, Polycarbonate, T-stress, Crack Tip Constraint, Photoplasticity
1
1. Introduction Residual stresses often combine critically with external loads and preexisting defects (cracks) causing unexpected failure of structures. Thus, the role of residual stresses in failure assessment of structures with cracks has gained importance. Xu and Burdekin [1] used numerical simulations to study the effect of welding residual stresses on crack tip constraint and fracture behavior of wide plates. Pavier et. al. [2] have investigated the influence of compressive residual stresses around cold worked holes on fracture using finite element (FE) simulations. They found that in the presence of residual stresses around cold worked holes, J-integral became path dependent. Liu et. al. [3] suggested a three parameter approach (CTOD, Q and R) to characterize the crack tip stress field in the presence of residual stresses. So far, much of the work [1-5] carried out in this field have been numerical studies with main emphasis given to welding residual stresses. Use of polymer products for industrial applications is on the rise. In polymers structures, fracture is one of the common modes of failure. For example, in transportation industry, mechanical fastening is the most common method used to connect the polymer component with the metallic component. However, often presence of a hole leads to stress concentration and nucleation of a crack resulting in fracture. In addition, polymers are also extremely vulnerable to environmental stress corrosion cracking (ESCR), whose severity depends on the environment and mechanical loads acting on it. Flow induced residual stresses are formed in polymers, due to extensive plastic deformation, during their manufacturing process. In literature, much attention has been given to prediction [6, 7] of these residual stresses. One of the disadvantages of flow induced residual stresses is that their presence causes the polymer surface to be under
2
tensile strain, which can enhance crack initiation and growth [8]. However, no work has been carried out in the literature to quantify their influence on fracture. Thus, in this study, the effect of flow induced residual stresses on fracture is investigated using photoelasticity experiments. Commercial polycarbonate (PC) sheets, manufactured by extrusion process, are used for making the single edge notched (SEN) specimen. The magnitude of the flow induced residual stress is measured using carrier fringe method. A residual stress free sheet is obtained by annealing the commercial PC sheet using a modified annealing cycle developed in-house. Two crack configurations, one with the crack axis parallel and the other with the crack axis perpendicular to the residual stress direction are studied. The nature of the fringe field observed is analyzed to understand the effect of residual stresses on SIF and T-stress. 2. Residual Stress Distribution in the Polycarbonate Sheet Polycarbonate sheets are manufactured through extrusion process. In this process, at the beginning, the plastic raw material is heated using a twin screw mechanism. The hot plastic melt is then forced through a die to make the sheet of required thickness. As the plastic melt flows through the die, the polymer molecules are stretched in the flow direction. If the material is cooled before the molecules are fully relaxed, the molecular orientation is locked into the sheet, resulting in flow induced residual stresses. After coming out of the die, the plastic sheet is cooled to room temperature. If the sheet is cooled non-uniformly, then thermal induced residual stresses are formed. Thus, in general, residual stresses in a PC sheet are a combination of flow and thermal induced stresses. Broutman and Krishnakumar (1974) [9] have developed a simple method to distinguish between flow and thermal induced residual stresses. As polycarbonate is
3
birefringent, they made a thin saw-cut in the PC sheet and examined it using a polariscope. They observed that if the residual stresses are flow induced, the fringe patterns do not show any distortion whereas large distortions in the fringe patterns are observed if the residual stresses are introduced thermally. In the present study, commercially available polycarbonate sheet of thickness 1.2 mm is used. Figure 1(a) shows the dark field isochromatic fringe pattern observed using white light in one region of the sheet having a uniform residual stress indicated by the constant color of the isochromatics. Figure 1(b) shows the isochromatics after making a thin slit in the sheet. It is seen that the fringe pattern does not show any distortion even after a slit is cut across it. This demonstrates that the polycarbonate sheet used in the current study has only flow induced residual stresses. 3. Measurement of Residual Stresses in Polycarbonate Sheet using Carrier Fringe Method The SEN specimens, used in the experiment, display a uniform color when viewed in a circular polariscope with white light as a source. The residual stress corresponding to this residual isochromatics at any point in the specimen can be determined only by measuring the total fringe order at that point. Due to uniform nature of the residual isochromatics, it is not possible to find the residual stress using Tardy’s method of compensation or a digital polariscope using phase shifting. Even color code is not sufficient to identify the integer part of the fringe order. However, Babinet Soleil compensator can be used to find the total fringe order provided it is properly calibrated. Instead of a Babinet Soleil compensator, in this work the carrier fringe method [10] has been used advantageously to measure the residual stresses in the PC sheet. In this
4
method, the residual birefringence in the PC sheet is superimposed by carrier fringes resulting in the formation of composite fringes. The carrier fringes are introduced by a stress frozen C-specimen (Fig. 2) of 50 mm width and 5.7 mm thickness, which was subjected to uniaxial tension while stress freezing. The carrier fringes are vertical straight lines caused by the superposition of bending and tensile stresses. The optical arrangement for the measurement of residual stress of the polycarbonate sheet in a conventional circular polariscope using the carrier is shown in Fig. 2. Figure 3 shows the superposition of the residual stresses in a small region of the PC sheet with the carrier fringes. In Fig. 3, region A is the residual birefringence seen in the PC sheet, region B is the carrier fringes in the C-specimen and region C shows the composite fringes. Due to the residual birefringence, a horizontal shift is observed in the composite fringes which is constant and the major principal stress direction is horizontal. This shift in the fringes is proportional to the residual stresses in the PC sheet [10]. Since the retardation in the carrier is already known, the retardation in the PC sheet due to residual stresses can be measured. This is the principle behind the carrier fringe method and the methodology is discussed next. For a monochromatic light source, the intensity of light emerging from a dark field circular polariscope with superposition of the carrier fringes is given as [10], I Ib
Ia 1 sin 2 pc c cos 2 pc c cos 2 pc c cos 2 pc c 2
(1)
where Ia is the amplitude of light intensity, Ib accounts for the background intensity, δc and δpc are the retardations in the carrier and the PC sheet respectively and θc and θpc are the principal stress orientations in the carrier and the PC sheet.
5
From Eq. (1), it is evident that the light intensity depends on two parameters, retardation δ and isoclinic angle θ. Under special conditions like θpc − θc = 0 ± 90°, the intensity of light depends on the retardation only and is given as, I Ib
Ia 1 cos 2 pc c 2
(2)
The total retardation of the composite fringes is given as,
t pc c
(3)
In the current investigation, θpc − θc = ± 90° as the PC sheet and the carrier are prealigned that way. For the reference directions shown in Fig. 3, the stresses in the carrier vary linearly along ‘x’ direction but is constant in the ‘y’ direction. Since the retardation is proportional to the stresses in the carrier, it is a function of ‘x’ only. In the PC sheet, as seen in region A of Fig. 3, the retardation is constant. Hence, the total retardation in Eq. (3) can be written as
t x pc c x
(4)
The pitch ‘p’ of the carrier is defined as the distance between two adjacent integral fringe orders. The retardation at a point x on the carrier (δc), with respect to an ith carrier fringe can be written as,
c x ci
x xi p
(5)
Now considering the point at which δt(x) = δci on the composite fringes, Eq. (4) becomes,
t x ci pc ci
x xi p
(6)
Solving Eq. (6), the retardation in the PC sheet can be obtained as,
6
pc
x xi p
(7)
where (x − xi) is the deviation of the fringes at an ordinate y as shown in Fig. 3. Thus, using Eq. (7), the retardation in the PC sheet, δpc is evaluated to be 2.08 fringe orders. The material stress fringe value of the PC sheet is 6.48 N/mm/fringe. From the stress optic law, the principal stress difference, (σ1 - σ2) is evaluated to be 11.25 MPa. In section 2, it has been established that the residual stresses in the PC sheet are a resultant of the stretching of the polymer molecules in the flow direction. This stretching results in the formation of tensile stresses in the flow direction only. Thus, the residual stress measured in the PC sheet is a tensile stress in the extrusion direction of the PC sheet with a magnitude of σ1 = 11.25 MPa. 4. Annealing of Polycarbonate Sheet Several thermal cycles [11, 12] are available in the literature for annealing polycarbonate sheets. Shreesudha and Ramesh [11] have done a comparative study of the cycles and have proposed a new thermal cycle (shown in Fig. 4(a)) to anneal the 1.2 mm thick PC sheet used in this study. It is a modification of the cycle proposed by Carmine Pappalettere [12]. The new cycle successfully prevented the viscous flow that was observed in the cycle proposed in [12]. In the new cycle, the sheet is initially thermo formed by heating it to 160oC at the rate of 15oC/hr, soaking at 160oC for 10 hrs and cooling to 70oC at the rate of 5oC/hr. It is then annealed by heating it to 155oC at the rate of 20oC/hr, soaking at 155oC for 10 hrs, cooling to 100oC at the rate of 2oC/hr and further cooling it to 70oC at the rate of 5oC/hr. The annealed PC sheet is then used to make a stress free SEN specimen. Figure 4(b) shows the bright field isochromatics of the annealed PC sheet. The sheet is fully transparent without any tinge of color indicating it to be free of any residual stress. 7
5. Experiments 5.1 Specimen Preparation The flow induced residual stress in the entire PC sheet is not uniform. The SEN specimens with dimensions of 150×50×1.2 mm and a crack length of 10 mm, are cut such that the residual stresses are uniform over the specimen size and are parallel and perpendicular to the crack axis. The specimens are named based on the direction they are cut from the PC sheet. If the longitudinal axis of the SEN specimen is in the direction of extrusion, it is labeled as SENL and if it is perpendicular, the specimen is labeled as SENT. In addition to these, a specimen is made from the annealed PC sheet, which is named as SENA. The residual stresses measured using carrier fringe method in all these specimens are shown in Table1. A slight difference in the residual stresses of SENL and SENT specimens is observed. 5.2 Observations from the Isochromatic Fringe Patterns of the SEN Specimen The three SEN specimens are subjected to mode-I loading and viewed in a circular polariscope set to dark field with a monochromatic illumination. The specimen is loaded in such a way that the ratio of applied stress (σ) to yield stress (σys) is varied from 0 to 0.3124 in steps of 0.0195. The yield stress (σys) of Polycarbonate is 67.5 MPa. The isochromatic fringe patterns for SENA, SENL, and SENT specimen at different selected σ/σys ratios are shown in Fig. 5. Only for the case σ/σys = 0, the dark field isochromatics recorded in white light are shown. The black color in Fig. 5(a) indicates absence of any residual stress in the specimen. The colored contours in Figs. 5(b) and (c) indicate the presence of residual stresses in the specimen. The fringes in the SENA and SENL specimen are forward
8
tilted for all the values of σ/σys ratio. The fringes in the SENT specimen are initially backward tilted and as the σ/σys ratio increases they become forward tilted. Observing the fringe order variation in Fig. 5, it can be seen that for SENL specimen, residual stresses and external loads add up while for SENT specimen, they try to cancel each other. This is clearly pictured in the fringe patterns for the loading condition of σ/σys = 0.0781. For the SENL specimen, compared to the SENA specimen, the fringe orders increase by 2 fringes. For the SENT specimen, fringe order ‘0’ is observed at a position where maximum stress is usually seen in an SEN specimen. This is because the external loads and residual stresses cancel each other. As fringe order at a point is related to the principal stress difference (Stress – Optic law), one can conclude from Fig. 5 that the stress field near the crack tip is effected by the presence of residual stresses. 6. Effect of Flow Induced Residual Stresses on Fracture Parameters 6.1 Evaluation of the Crack Tip Stress Field Parameters The crack tip stress field parameters, SIF and T-stress, are evaluated using over deterministic least squares technique in conjunction with corrected Atluri-Kobayashi stress field equation [13] which is given by,
σ σ
9
(8) where
I
I
,
II
-
II
and
.
I
The formulation of the least squares problem is described in ref. [13]. The fringe order data and their corresponding positional coordinates at integral fringe orders are collected from the thinned dark field images. An in-house developed VC++ software, with the least squares formulation implemented, is used. Since the number of parameters, n, required to characterize the fringe field is not known a priori, the iteration is initiated with minimum number of parameters (n = 2) and then increased progressively as needed. The iteration is stopped once the convergence criteria is satisfied. The convergence criteria [13] is based on the difference of theoretical and experimental fringe orders and is given as,
N
theory
N exp
total no. of data points
convergence error
(9)
In general, when the iteration is initiated with n = 2, the convergence is not achieved unless a large convergence error of the order of 0.5 is specified. The converged solution at this stage does not accurately simulate the stress field near the crack tip. Thus, the number of parameters is then increased by one and the iterations are initiated with a less convergence error. The process is repeated until a convergence error of 0.05 or less is achieved. The parameters thus evaluated are used to reconstruct the fringe field analytically to verify the convergence.
10
In the evaluation of the crack tip stress field parameters, the crack tip location plays an important role. Even a variation of 1 or 2 pixels in the crack tip location can vary the parameters significantly. In the current study, the crack tip is interactively picked up by the user and is prone to error. For correcting the crack tip error, the simple semi-automatic methodology proposed by Neethi and Ramesh [14] is used. In this method, once the number of parameters required to simulate the crack tip stress field is frozen, a 5×5 pixel mask surrounding the initial estimated crack tip is considered. The convergence errors are evaluated at each position of the crack tip in the 5×5 pixel mask. The next location of crack tip is considered to be one having the least convergence error. The above procedure is then repeated for the new crack tip until the chosen crack tip has the least convergence in the 5×5 pixel mask surrounding it. Figures 6(a), (b) and (c) show the experimental fringe fields for the SENA, SENL and SENT specimen respectively for the σ/σys ratio of 0.117. Figures 6(d), (e) and (f) show the reconstructed fringe fields for the SENA, SENL and SENT specimen respectively for the σ/σys ratio of 0.117. The reconstructed fringe fields match well with the experimental images for all the three specimens. Similarly the stress field parameters are evaluated and reconstructed fringe fields are compared for other σ/σys ratios up to the σ/σys ratio of 0.1562. Beyond this, yielding is observed in the specimen and the conditions of small scale yielding (SSY) are no longer valid. 6.2 Results and Discussion Figure 7(a) shows the variation of KI with σ/σys ratio for the three SEN specimens. The difference between SIF values for the SENL and SENA specimens is 1.24% - 7.08% whereas for the SENT and SENA specimens, it is 0.2% - 6.09%. Overall, these variations
11
are small and SIF values are considered to be unaffected by the presence of residual stresses. Figure 7(b) shows the variation of T/4 or AI2 parameter with σ/σys ratio for the three SEN specimens. In the presence of tensile residual stresses perpendicular to the crack axis, as in the case of the SENL specimen, T-stress is negative for all σ/σys ratios. In the presence of tensile residual stresses parallel to the crack axis, as in the case of the SENT specimen, Tstress is initially positive and as the σ/σys ratio increases, it decreases. It should be understood that the T-stress for the SENT specimen eventually changes sign from positive to negative as evidenced by the formation of forward tilted fringes at higher σ/σys ratios in Fig.5. In failure assessment methods, T-stress is used as a parameter to describe the level of constraint near the crack tip. Ayatollahi and Safari [15] have related the T-stress term to the crack tip constraint using photoelasticity. They have observed that the crack tip constraint changes with change in the sign of the T-stress term. From Fig. 7(b), it is clear that flow induced residual stresses effect the constraint near the crack tip. However, Tstress is the crack tip constraint due to the geometry and loading of the specimen only. Thus to quantify the constraint due to flow induced residual stresses, a new parameter ‘R’ is introduced. It is defined as, Ri
Ti T 4 i
(10) L, T
From Fig. 7(b), it can be inferred that RL is negative (as TL < T) and RT is positive (as TT > T). Figure 7(c) shows the variation of |Ri| with σ/σys ratio. In general, it is observed that R is higher for SENT specimen compared to SENL specimen. The average value of R for SENT and SENL specimen is 0.253 and -0.237 respectively. This indicates that the crack tip 12
constraint due to flow induced residual stresses depends on the direction and magnitude of the residual stresses. Adding the parameter that quantifies constraint due to residual stress, the crack tip stress field equation becomes,
σ σ
q.
σ 0 0
(11)
Using the fracture parameters of SENA specimen and average values of Ri in Eq. (11), the isochromatic fringe fields of the SENL and SENT specimen are reconstructed. Figures 8(c) and (d) show these reconstructed fringe fields with echoed data points for the SEN L and SENT specimen respectively at the σ/σys ratio of 0.117. The reconstructed fringe field compares well with the experimental images. This shows that the crack tip stress field is affected by the direction and magnitude of flow induced residual stress near the crack tip. 7. Effect of flow induced residual stresses on post yielding behavior of the crack As the load on the SEN specimen is increased, the region near the crack tip begins to yield. This region undergoing yielding is seen as dark bands (shear bands) ahead of the crack tip in the captured images. These bands originate from the crack tip at an angle to the crack axis and become parallel to the crack as they propagate. Due to the formation of shear bands, the fringe contours ahead of the crack tip get distorted. Figure 9 shows the shear bands formed in the three SEN specimens for the σ/σys ratio of 0.3124. By qualitative comparison of the length of the shear bands, it can be said that the presence of residual stresses has accelerated the yielding of the region ahead of the crack tip, irrespective of the direction of the residual stress.
13
8. Conclusion The influence of flow induced residual stress on fracture parameters is studied using photoelasticity experiments. The residual stresses in the PC sheet are measured using the carrier fringe method. In the presence of residual stresses, no considerable change is observed in mode-I SIF values while the T-stress changed significantly. A new parameter ‘R’ is defined to quantify the crack tip constraint due to flow induced residual stresses. Residual stresses parallel to the crack axis introduced a higher crack tip constraint than residual stresses perpendicular to the crack axis. Irrespective of their direction, presence of residual stresses has accelerated the yielding of the region ahead of the crack tip.
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References [1] W. G. Xu and F. M. Burdekin, Effects of residual stresses on constraint and fracture behavior of wide plates, Proc. R. Soc. Lond. 454 (1998) 2505-2528. [2] M. J. Pavier, C. G. C. Poussard and D. J. Smith, Effect of residual stress around cold worked holes on fracture under superimposed mechanical load, Engg. Frac. Mech. 63 (1999) 751-773. [3] J. Liu, Z. L. Zhang, and B. Nyhus, Residual stress induced crack tip constraint, Engg. Frac. Mech. 75 (2008) 4151-4166. [4] T. L. Panontin and M. R. Hill, The effect of residual stresses on brittle and ductile
[5] [6] [7] [8]
fracture initiation predicted by micromechanical models, Int. Journal of Fracture 82 (1996) 317-333. Y. Lei, N.P. O’Dowd and G.A. Webster, Fracture mechanics analysis of a crack in a residual stress field, Int. Journal of Fracture 106 (2000) 195-216. T. W. David Chan and L. James Lee, Stress Development in Plastic Sheet Extrusion, Polymer Engg. And Science 29 (1989) 731-739. E P.T. Baaijens, Calculation of residual stresses in injection molded products, Rheologica Acta 30 (1991) 284-299. K. M. B. Jansen, Residual stresses in quenched and injection moulded products, Int. Polymer Processing IX (1994).
[9] L. J. Broutman and S. M. Krishnakumar, Cold Rolling of Polymers 2. Toughness Enhancement in Amorphous Polycarbonates, Polymer Engg. and Science 14 (1974) 249-259. [10] K. Ramesh, R. Vivek, P. Tarkes Dora, and Dipayan Sanyal, A Simple Approach to Photoelastic Calibration of Glass using Digital Photoelasticity, J. Non-Crystalline Solids 378 (2013) 7-14. [11] B. Shreesudha, and K. Ramesh, Annealing of Polycarbonate for Photoplastic Analysis, 9th International Symposium on Advanced Science and Technology in Experimental Mechanics (2014) New Delhi, India. [12] Carmine Pappalettere, Annealing polycarbonate sheets, Strain 20 (1984) 179-180. [13] K. Ramesh, S. Gupta, and A. A. Kelkar, Evaluation of stress field parameters in fracture mechanics by Photoelasticity – revisited, Engg. Frac. Mech. 56 (1997) 2545.
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[14] B. Neethi Simon, and K. Ramesh, Effect of error in crack-tip identification on the photoelastic evaluation of SIFs of interface cracks, Fourth International Conference on Experimental Mechanics (ICEM 2009), Singapore, Proc. SPIE, 7522, 75220D, DOI:10.1117/12.852519. [15] M. R. Ayatollahi, and H. Safari, Evaluation of Crack Tip Constraint using Photoelasticity, Int. Journal of Pressure Vessels and Piping 80 (2003) 665-670.
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Figure Captions Fig. 1 Isochromatic fringe patterns in (a) Original PC sheet (b) After making the slit Fig. 2 Schematic diagram of the circular polariscope arrangement used in the carrier fringe method Fig. 3 Measurement of residual stresses in PC sheet using carrier fringe method Fig. 4(a) Thermal cycle used for annealing the PC sheet [11] (b) Bright field image of the annealed PC sheet Fig. 5 (a) to (c) Dark field residual fringe pattern in the three specimens seen in white light at σ/σys = 0, (d) to (o) Dark field isochromatics for the three specimens under monochromatic light at different selected σ/σys ratios Fig. 6 (a), (b) and (c) Isochromatic fringe contours from experiments and (d), (e) and (f) Reconstructed isochromatic fringe contours of SENA, SENL and SENT specimen respectively for σ/σys ratio of 0.117 Fig. 7 Variation of (a) mode-I SIF (b) T-stress and (c) |Ri| with σ/σys ratio Fig.8 (a), (b) Isochromatic fringe contours from experiments and (c), (d) reconstructed isochromatic fringe contours of SENL, SENT specimen using 4-parameter solution of SENA and average RL and RT respectively for σ/σys ratio of 0.117 Fig. 9 Yield region near the crack tip for (a) SENA (b) SENL and (c) SENT specimen for σ/σys ratio of 0.3124
17
18
19
20
21
22
23
24
25
26
Table-1 Residual stresses measured in the SEN specimen Specimen
Residual Stress (in fringe orders)
Residual Stress (in MPa)
SENL
2.08
11.25
SENT
1.85
9.99
SENA
0
0
27
Highlights Flow induced residual stresses in PC sheet are measured using carrier fringe method
The influence of flow induced residual stress on fracture parameters is studied using photoelasticity experiments
In the presence of residual stresses, no considerable change is observed in mode-I SIF values while the T-stress changed significantly
A new parameter ‘R’ is defined to quantify the crack tip constraint introduced due to flow induced residual stresses
Residual stresses parallel to the crack axis introduce a higher crack tip constraint than residual stresses perpendicular to the crack axis
28
S0167-8442(16)30032-5 http://dx.doi.org/10.1016/j.tafmec.2016.04.003 TAFMEC 1702
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
1 February 2016 8 April 2016 10 April 2016
Please cite this article as: M. Subramanyam Reddy, K. Ramesh, Photoelastic study on the Effect of Flow Induced Residual Stresses on Fracture Parameters, Theoretical and Applied Fracture Mechanics (2016), doi: http:// dx.doi.org/10.1016/j.tafmec.2016.04.003
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Photoelastic study on the Effect of Flow Induced Residual Stresses on Fracture Parameters Subramanyam Reddy Ma, Ramesh Ka a
Dept. of Applied Mechanics, IIT Madras, Chennai - 600036, India. Corresponding Author: Subramanyam Reddy M Dept. of Applied Mechanics, IIT Madras, Chennai - 600036, India. email: [email protected]
ABSTRACT Flow induced residual stresses, formed during manufacturing, influence the crack initiation and growth process in polymer products. The effect of these residual stresses on mode-I parameters is studied using photoelasticity. Polycarbonate (PC) sheets having only flow induced residual stresses are used for making the single edge notched (SEN) specimen. Residual stresses in the PC sheet are measured using carrier fringe method. Two crack configurations, where residual stress is parallel and perpendicular to the crack axis are considered. The study showed that flow induced residual stress introduces an additional crack tip constraint which depends on the direction and magnitude of the residual stress. A new parameter is defined to quantify the residual stress induced crack tip constraint. Key words: Flow Induced Residual Stress, Polycarbonate, T-stress, Crack Tip Constraint, Photoplasticity
1
1. Introduction Residual stresses often combine critically with external loads and preexisting defects (cracks) causing unexpected failure of structures. Thus, the role of residual stresses in failure assessment of structures with cracks has gained importance. Xu and Burdekin [1] used numerical simulations to study the effect of welding residual stresses on crack tip constraint and fracture behavior of wide plates. Pavier et. al. [2] have investigated the influence of compressive residual stresses around cold worked holes on fracture using finite element (FE) simulations. They found that in the presence of residual stresses around cold worked holes, J-integral became path dependent. Liu et. al. [3] suggested a three parameter approach (CTOD, Q and R) to characterize the crack tip stress field in the presence of residual stresses. So far, much of the work [1-5] carried out in this field have been numerical studies with main emphasis given to welding residual stresses. Use of polymer products for industrial applications is on the rise. In polymers structures, fracture is one of the common modes of failure. For example, in transportation industry, mechanical fastening is the most common method used to connect the polymer component with the metallic component. However, often presence of a hole leads to stress concentration and nucleation of a crack resulting in fracture. In addition, polymers are also extremely vulnerable to environmental stress corrosion cracking (ESCR), whose severity depends on the environment and mechanical loads acting on it. Flow induced residual stresses are formed in polymers, due to extensive plastic deformation, during their manufacturing process. In literature, much attention has been given to prediction [6, 7] of these residual stresses. One of the disadvantages of flow induced residual stresses is that their presence causes the polymer surface to be under
2
tensile strain, which can enhance crack initiation and growth [8]. However, no work has been carried out in the literature to quantify their influence on fracture. Thus, in this study, the effect of flow induced residual stresses on fracture is investigated using photoelasticity experiments. Commercial polycarbonate (PC) sheets, manufactured by extrusion process, are used for making the single edge notched (SEN) specimen. The magnitude of the flow induced residual stress is measured using carrier fringe method. A residual stress free sheet is obtained by annealing the commercial PC sheet using a modified annealing cycle developed in-house. Two crack configurations, one with the crack axis parallel and the other with the crack axis perpendicular to the residual stress direction are studied. The nature of the fringe field observed is analyzed to understand the effect of residual stresses on SIF and T-stress. 2. Residual Stress Distribution in the Polycarbonate Sheet Polycarbonate sheets are manufactured through extrusion process. In this process, at the beginning, the plastic raw material is heated using a twin screw mechanism. The hot plastic melt is then forced through a die to make the sheet of required thickness. As the plastic melt flows through the die, the polymer molecules are stretched in the flow direction. If the material is cooled before the molecules are fully relaxed, the molecular orientation is locked into the sheet, resulting in flow induced residual stresses. After coming out of the die, the plastic sheet is cooled to room temperature. If the sheet is cooled non-uniformly, then thermal induced residual stresses are formed. Thus, in general, residual stresses in a PC sheet are a combination of flow and thermal induced stresses. Broutman and Krishnakumar (1974) [9] have developed a simple method to distinguish between flow and thermal induced residual stresses. As polycarbonate is
3
birefringent, they made a thin saw-cut in the PC sheet and examined it using a polariscope. They observed that if the residual stresses are flow induced, the fringe patterns do not show any distortion whereas large distortions in the fringe patterns are observed if the residual stresses are introduced thermally. In the present study, commercially available polycarbonate sheet of thickness 1.2 mm is used. Figure 1(a) shows the dark field isochromatic fringe pattern observed using white light in one region of the sheet having a uniform residual stress indicated by the constant color of the isochromatics. Figure 1(b) shows the isochromatics after making a thin slit in the sheet. It is seen that the fringe pattern does not show any distortion even after a slit is cut across it. This demonstrates that the polycarbonate sheet used in the current study has only flow induced residual stresses. 3. Measurement of Residual Stresses in Polycarbonate Sheet using Carrier Fringe Method The SEN specimens, used in the experiment, display a uniform color when viewed in a circular polariscope with white light as a source. The residual stress corresponding to this residual isochromatics at any point in the specimen can be determined only by measuring the total fringe order at that point. Due to uniform nature of the residual isochromatics, it is not possible to find the residual stress using Tardy’s method of compensation or a digital polariscope using phase shifting. Even color code is not sufficient to identify the integer part of the fringe order. However, Babinet Soleil compensator can be used to find the total fringe order provided it is properly calibrated. Instead of a Babinet Soleil compensator, in this work the carrier fringe method [10] has been used advantageously to measure the residual stresses in the PC sheet. In this
4
method, the residual birefringence in the PC sheet is superimposed by carrier fringes resulting in the formation of composite fringes. The carrier fringes are introduced by a stress frozen C-specimen (Fig. 2) of 50 mm width and 5.7 mm thickness, which was subjected to uniaxial tension while stress freezing. The carrier fringes are vertical straight lines caused by the superposition of bending and tensile stresses. The optical arrangement for the measurement of residual stress of the polycarbonate sheet in a conventional circular polariscope using the carrier is shown in Fig. 2. Figure 3 shows the superposition of the residual stresses in a small region of the PC sheet with the carrier fringes. In Fig. 3, region A is the residual birefringence seen in the PC sheet, region B is the carrier fringes in the C-specimen and region C shows the composite fringes. Due to the residual birefringence, a horizontal shift is observed in the composite fringes which is constant and the major principal stress direction is horizontal. This shift in the fringes is proportional to the residual stresses in the PC sheet [10]. Since the retardation in the carrier is already known, the retardation in the PC sheet due to residual stresses can be measured. This is the principle behind the carrier fringe method and the methodology is discussed next. For a monochromatic light source, the intensity of light emerging from a dark field circular polariscope with superposition of the carrier fringes is given as [10], I Ib
Ia 1 sin 2 pc c cos 2 pc c cos 2 pc c cos 2 pc c 2
(1)
where Ia is the amplitude of light intensity, Ib accounts for the background intensity, δc and δpc are the retardations in the carrier and the PC sheet respectively and θc and θpc are the principal stress orientations in the carrier and the PC sheet.
5
From Eq. (1), it is evident that the light intensity depends on two parameters, retardation δ and isoclinic angle θ. Under special conditions like θpc − θc = 0 ± 90°, the intensity of light depends on the retardation only and is given as, I Ib
Ia 1 cos 2 pc c 2
(2)
The total retardation of the composite fringes is given as,
t pc c
(3)
In the current investigation, θpc − θc = ± 90° as the PC sheet and the carrier are prealigned that way. For the reference directions shown in Fig. 3, the stresses in the carrier vary linearly along ‘x’ direction but is constant in the ‘y’ direction. Since the retardation is proportional to the stresses in the carrier, it is a function of ‘x’ only. In the PC sheet, as seen in region A of Fig. 3, the retardation is constant. Hence, the total retardation in Eq. (3) can be written as
t x pc c x
(4)
The pitch ‘p’ of the carrier is defined as the distance between two adjacent integral fringe orders. The retardation at a point x on the carrier (δc), with respect to an ith carrier fringe can be written as,
c x ci
x xi p
(5)
Now considering the point at which δt(x) = δci on the composite fringes, Eq. (4) becomes,
t x ci pc ci
x xi p
(6)
Solving Eq. (6), the retardation in the PC sheet can be obtained as,
6
pc
x xi p
(7)
where (x − xi) is the deviation of the fringes at an ordinate y as shown in Fig. 3. Thus, using Eq. (7), the retardation in the PC sheet, δpc is evaluated to be 2.08 fringe orders. The material stress fringe value of the PC sheet is 6.48 N/mm/fringe. From the stress optic law, the principal stress difference, (σ1 - σ2) is evaluated to be 11.25 MPa. In section 2, it has been established that the residual stresses in the PC sheet are a resultant of the stretching of the polymer molecules in the flow direction. This stretching results in the formation of tensile stresses in the flow direction only. Thus, the residual stress measured in the PC sheet is a tensile stress in the extrusion direction of the PC sheet with a magnitude of σ1 = 11.25 MPa. 4. Annealing of Polycarbonate Sheet Several thermal cycles [11, 12] are available in the literature for annealing polycarbonate sheets. Shreesudha and Ramesh [11] have done a comparative study of the cycles and have proposed a new thermal cycle (shown in Fig. 4(a)) to anneal the 1.2 mm thick PC sheet used in this study. It is a modification of the cycle proposed by Carmine Pappalettere [12]. The new cycle successfully prevented the viscous flow that was observed in the cycle proposed in [12]. In the new cycle, the sheet is initially thermo formed by heating it to 160oC at the rate of 15oC/hr, soaking at 160oC for 10 hrs and cooling to 70oC at the rate of 5oC/hr. It is then annealed by heating it to 155oC at the rate of 20oC/hr, soaking at 155oC for 10 hrs, cooling to 100oC at the rate of 2oC/hr and further cooling it to 70oC at the rate of 5oC/hr. The annealed PC sheet is then used to make a stress free SEN specimen. Figure 4(b) shows the bright field isochromatics of the annealed PC sheet. The sheet is fully transparent without any tinge of color indicating it to be free of any residual stress. 7
5. Experiments 5.1 Specimen Preparation The flow induced residual stress in the entire PC sheet is not uniform. The SEN specimens with dimensions of 150×50×1.2 mm and a crack length of 10 mm, are cut such that the residual stresses are uniform over the specimen size and are parallel and perpendicular to the crack axis. The specimens are named based on the direction they are cut from the PC sheet. If the longitudinal axis of the SEN specimen is in the direction of extrusion, it is labeled as SENL and if it is perpendicular, the specimen is labeled as SENT. In addition to these, a specimen is made from the annealed PC sheet, which is named as SENA. The residual stresses measured using carrier fringe method in all these specimens are shown in Table1. A slight difference in the residual stresses of SENL and SENT specimens is observed. 5.2 Observations from the Isochromatic Fringe Patterns of the SEN Specimen The three SEN specimens are subjected to mode-I loading and viewed in a circular polariscope set to dark field with a monochromatic illumination. The specimen is loaded in such a way that the ratio of applied stress (σ) to yield stress (σys) is varied from 0 to 0.3124 in steps of 0.0195. The yield stress (σys) of Polycarbonate is 67.5 MPa. The isochromatic fringe patterns for SENA, SENL, and SENT specimen at different selected σ/σys ratios are shown in Fig. 5. Only for the case σ/σys = 0, the dark field isochromatics recorded in white light are shown. The black color in Fig. 5(a) indicates absence of any residual stress in the specimen. The colored contours in Figs. 5(b) and (c) indicate the presence of residual stresses in the specimen. The fringes in the SENA and SENL specimen are forward
8
tilted for all the values of σ/σys ratio. The fringes in the SENT specimen are initially backward tilted and as the σ/σys ratio increases they become forward tilted. Observing the fringe order variation in Fig. 5, it can be seen that for SENL specimen, residual stresses and external loads add up while for SENT specimen, they try to cancel each other. This is clearly pictured in the fringe patterns for the loading condition of σ/σys = 0.0781. For the SENL specimen, compared to the SENA specimen, the fringe orders increase by 2 fringes. For the SENT specimen, fringe order ‘0’ is observed at a position where maximum stress is usually seen in an SEN specimen. This is because the external loads and residual stresses cancel each other. As fringe order at a point is related to the principal stress difference (Stress – Optic law), one can conclude from Fig. 5 that the stress field near the crack tip is effected by the presence of residual stresses. 6. Effect of Flow Induced Residual Stresses on Fracture Parameters 6.1 Evaluation of the Crack Tip Stress Field Parameters The crack tip stress field parameters, SIF and T-stress, are evaluated using over deterministic least squares technique in conjunction with corrected Atluri-Kobayashi stress field equation [13] which is given by,
σ σ
9
(8) where
I
I
,
II
-
II
and
.
I
The formulation of the least squares problem is described in ref. [13]. The fringe order data and their corresponding positional coordinates at integral fringe orders are collected from the thinned dark field images. An in-house developed VC++ software, with the least squares formulation implemented, is used. Since the number of parameters, n, required to characterize the fringe field is not known a priori, the iteration is initiated with minimum number of parameters (n = 2) and then increased progressively as needed. The iteration is stopped once the convergence criteria is satisfied. The convergence criteria [13] is based on the difference of theoretical and experimental fringe orders and is given as,
N
theory
N exp
total no. of data points
convergence error
(9)
In general, when the iteration is initiated with n = 2, the convergence is not achieved unless a large convergence error of the order of 0.5 is specified. The converged solution at this stage does not accurately simulate the stress field near the crack tip. Thus, the number of parameters is then increased by one and the iterations are initiated with a less convergence error. The process is repeated until a convergence error of 0.05 or less is achieved. The parameters thus evaluated are used to reconstruct the fringe field analytically to verify the convergence.
10
In the evaluation of the crack tip stress field parameters, the crack tip location plays an important role. Even a variation of 1 or 2 pixels in the crack tip location can vary the parameters significantly. In the current study, the crack tip is interactively picked up by the user and is prone to error. For correcting the crack tip error, the simple semi-automatic methodology proposed by Neethi and Ramesh [14] is used. In this method, once the number of parameters required to simulate the crack tip stress field is frozen, a 5×5 pixel mask surrounding the initial estimated crack tip is considered. The convergence errors are evaluated at each position of the crack tip in the 5×5 pixel mask. The next location of crack tip is considered to be one having the least convergence error. The above procedure is then repeated for the new crack tip until the chosen crack tip has the least convergence in the 5×5 pixel mask surrounding it. Figures 6(a), (b) and (c) show the experimental fringe fields for the SENA, SENL and SENT specimen respectively for the σ/σys ratio of 0.117. Figures 6(d), (e) and (f) show the reconstructed fringe fields for the SENA, SENL and SENT specimen respectively for the σ/σys ratio of 0.117. The reconstructed fringe fields match well with the experimental images for all the three specimens. Similarly the stress field parameters are evaluated and reconstructed fringe fields are compared for other σ/σys ratios up to the σ/σys ratio of 0.1562. Beyond this, yielding is observed in the specimen and the conditions of small scale yielding (SSY) are no longer valid. 6.2 Results and Discussion Figure 7(a) shows the variation of KI with σ/σys ratio for the three SEN specimens. The difference between SIF values for the SENL and SENA specimens is 1.24% - 7.08% whereas for the SENT and SENA specimens, it is 0.2% - 6.09%. Overall, these variations
11
are small and SIF values are considered to be unaffected by the presence of residual stresses. Figure 7(b) shows the variation of T/4 or AI2 parameter with σ/σys ratio for the three SEN specimens. In the presence of tensile residual stresses perpendicular to the crack axis, as in the case of the SENL specimen, T-stress is negative for all σ/σys ratios. In the presence of tensile residual stresses parallel to the crack axis, as in the case of the SENT specimen, Tstress is initially positive and as the σ/σys ratio increases, it decreases. It should be understood that the T-stress for the SENT specimen eventually changes sign from positive to negative as evidenced by the formation of forward tilted fringes at higher σ/σys ratios in Fig.5. In failure assessment methods, T-stress is used as a parameter to describe the level of constraint near the crack tip. Ayatollahi and Safari [15] have related the T-stress term to the crack tip constraint using photoelasticity. They have observed that the crack tip constraint changes with change in the sign of the T-stress term. From Fig. 7(b), it is clear that flow induced residual stresses effect the constraint near the crack tip. However, Tstress is the crack tip constraint due to the geometry and loading of the specimen only. Thus to quantify the constraint due to flow induced residual stresses, a new parameter ‘R’ is introduced. It is defined as, Ri
Ti T 4 i
(10) L, T
From Fig. 7(b), it can be inferred that RL is negative (as TL < T) and RT is positive (as TT > T). Figure 7(c) shows the variation of |Ri| with σ/σys ratio. In general, it is observed that R is higher for SENT specimen compared to SENL specimen. The average value of R for SENT and SENL specimen is 0.253 and -0.237 respectively. This indicates that the crack tip 12
constraint due to flow induced residual stresses depends on the direction and magnitude of the residual stresses. Adding the parameter that quantifies constraint due to residual stress, the crack tip stress field equation becomes,
σ σ
q.
σ 0 0
(11)
Using the fracture parameters of SENA specimen and average values of Ri in Eq. (11), the isochromatic fringe fields of the SENL and SENT specimen are reconstructed. Figures 8(c) and (d) show these reconstructed fringe fields with echoed data points for the SEN L and SENT specimen respectively at the σ/σys ratio of 0.117. The reconstructed fringe field compares well with the experimental images. This shows that the crack tip stress field is affected by the direction and magnitude of flow induced residual stress near the crack tip. 7. Effect of flow induced residual stresses on post yielding behavior of the crack As the load on the SEN specimen is increased, the region near the crack tip begins to yield. This region undergoing yielding is seen as dark bands (shear bands) ahead of the crack tip in the captured images. These bands originate from the crack tip at an angle to the crack axis and become parallel to the crack as they propagate. Due to the formation of shear bands, the fringe contours ahead of the crack tip get distorted. Figure 9 shows the shear bands formed in the three SEN specimens for the σ/σys ratio of 0.3124. By qualitative comparison of the length of the shear bands, it can be said that the presence of residual stresses has accelerated the yielding of the region ahead of the crack tip, irrespective of the direction of the residual stress.
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8. Conclusion The influence of flow induced residual stress on fracture parameters is studied using photoelasticity experiments. The residual stresses in the PC sheet are measured using the carrier fringe method. In the presence of residual stresses, no considerable change is observed in mode-I SIF values while the T-stress changed significantly. A new parameter ‘R’ is defined to quantify the crack tip constraint due to flow induced residual stresses. Residual stresses parallel to the crack axis introduced a higher crack tip constraint than residual stresses perpendicular to the crack axis. Irrespective of their direction, presence of residual stresses has accelerated the yielding of the region ahead of the crack tip.
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References [1] W. G. Xu and F. M. Burdekin, Effects of residual stresses on constraint and fracture behavior of wide plates, Proc. R. Soc. Lond. 454 (1998) 2505-2528. [2] M. J. Pavier, C. G. C. Poussard and D. J. Smith, Effect of residual stress around cold worked holes on fracture under superimposed mechanical load, Engg. Frac. Mech. 63 (1999) 751-773. [3] J. Liu, Z. L. Zhang, and B. Nyhus, Residual stress induced crack tip constraint, Engg. Frac. Mech. 75 (2008) 4151-4166. [4] T. L. Panontin and M. R. Hill, The effect of residual stresses on brittle and ductile
[5] [6] [7] [8]
fracture initiation predicted by micromechanical models, Int. Journal of Fracture 82 (1996) 317-333. Y. Lei, N.P. O’Dowd and G.A. Webster, Fracture mechanics analysis of a crack in a residual stress field, Int. Journal of Fracture 106 (2000) 195-216. T. W. David Chan and L. James Lee, Stress Development in Plastic Sheet Extrusion, Polymer Engg. And Science 29 (1989) 731-739. E P.T. Baaijens, Calculation of residual stresses in injection molded products, Rheologica Acta 30 (1991) 284-299. K. M. B. Jansen, Residual stresses in quenched and injection moulded products, Int. Polymer Processing IX (1994).
[9] L. J. Broutman and S. M. Krishnakumar, Cold Rolling of Polymers 2. Toughness Enhancement in Amorphous Polycarbonates, Polymer Engg. and Science 14 (1974) 249-259. [10] K. Ramesh, R. Vivek, P. Tarkes Dora, and Dipayan Sanyal, A Simple Approach to Photoelastic Calibration of Glass using Digital Photoelasticity, J. Non-Crystalline Solids 378 (2013) 7-14. [11] B. Shreesudha, and K. Ramesh, Annealing of Polycarbonate for Photoplastic Analysis, 9th International Symposium on Advanced Science and Technology in Experimental Mechanics (2014) New Delhi, India. [12] Carmine Pappalettere, Annealing polycarbonate sheets, Strain 20 (1984) 179-180. [13] K. Ramesh, S. Gupta, and A. A. Kelkar, Evaluation of stress field parameters in fracture mechanics by Photoelasticity – revisited, Engg. Frac. Mech. 56 (1997) 2545.
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[14] B. Neethi Simon, and K. Ramesh, Effect of error in crack-tip identification on the photoelastic evaluation of SIFs of interface cracks, Fourth International Conference on Experimental Mechanics (ICEM 2009), Singapore, Proc. SPIE, 7522, 75220D, DOI:10.1117/12.852519. [15] M. R. Ayatollahi, and H. Safari, Evaluation of Crack Tip Constraint using Photoelasticity, Int. Journal of Pressure Vessels and Piping 80 (2003) 665-670.
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Figure Captions Fig. 1 Isochromatic fringe patterns in (a) Original PC sheet (b) After making the slit Fig. 2 Schematic diagram of the circular polariscope arrangement used in the carrier fringe method Fig. 3 Measurement of residual stresses in PC sheet using carrier fringe method Fig. 4(a) Thermal cycle used for annealing the PC sheet [11] (b) Bright field image of the annealed PC sheet Fig. 5 (a) to (c) Dark field residual fringe pattern in the three specimens seen in white light at σ/σys = 0, (d) to (o) Dark field isochromatics for the three specimens under monochromatic light at different selected σ/σys ratios Fig. 6 (a), (b) and (c) Isochromatic fringe contours from experiments and (d), (e) and (f) Reconstructed isochromatic fringe contours of SENA, SENL and SENT specimen respectively for σ/σys ratio of 0.117 Fig. 7 Variation of (a) mode-I SIF (b) T-stress and (c) |Ri| with σ/σys ratio Fig.8 (a), (b) Isochromatic fringe contours from experiments and (c), (d) reconstructed isochromatic fringe contours of SENL, SENT specimen using 4-parameter solution of SENA and average RL and RT respectively for σ/σys ratio of 0.117 Fig. 9 Yield region near the crack tip for (a) SENA (b) SENL and (c) SENT specimen for σ/σys ratio of 0.3124
17
18
19
20
21
22
23
24
25
26
Table-1 Residual stresses measured in the SEN specimen Specimen
Residual Stress (in fringe orders)
Residual Stress (in MPa)
SENL
2.08
11.25
SENT
1.85
9.99
SENA
0
0
27
Highlights Flow induced residual stresses in PC sheet are measured using carrier fringe method
The influence of flow induced residual stress on fracture parameters is studied using photoelasticity experiments
In the presence of residual stresses, no considerable change is observed in mode-I SIF values while the T-stress changed significantly
A new parameter ‘R’ is defined to quantify the crack tip constraint introduced due to flow induced residual stresses
Residual stresses parallel to the crack axis introduce a higher crack tip constraint than residual stresses perpendicular to the crack axis
28