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Fracture characterizations of V-notch tip in PMMA polymer material
Polymer Testing 23 (2004) 509–515 www.elsevier.com/locate/polytest
Test Method
Fracture characterizations of V-notch tip in PMMA polymer material W. Xu a, X.F. Yao a, , M.Q. Xu a, G.C. Jin a, H.Y. Yeh b b
a Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China Department of Mechanical and Aerospace Engineering, California State University, Long Beach, CA 90840-8305, USA
Received 23 September 2003; accepted 22 December 2003
Abstract In this paper, the fracture characterization of a V-notch tip in PMMA material is studied by means of an optical caustics method. The initial curve and caustic curve of the notch tip with different notch angles are simulated. Some typical stress concentration characterizations at the notch tip are analyzed. Fracture experiments of PMMA specimens with a V-notch subjected to three-point-bend loading are visualized using the optical pattern of caustics. The evolution of the stress intensity factor and the size of the caustic spot at the notch tip are analyzed. These results will be useful for the reliable design and strength evaluation of engineering structures with complex notch shapes. # 2003 Elsevier Ltd. All rights reserved. Keywords: PMMA material; Optical caustics; V-notch; Stress intensity factor; Fracture
1. Introduction It is well known that cutout geometrical features such as: cracks, holes and notches, etc., are frequently encountered in many engineering structures. Under external loading, the stress fields in the vicinity of these cutouts are very complicated. Therefore, it is important to study the stress information and fracture characteristics at the notch tip for evaluating the strength and predicting the service life of the structures. So far, the analytical solution of the notch tip stress field is based on the elastic mechanics theory and the finite element method. Experimentally, the evaluation of the complex stress intensity factor K at the apex of the sharp V-notch in homogeneous and isotropic elastic plates under symmetrical load in their plane can be studied using the optical method of reflective caustics [1–3]. Kondo [4] gave an efficient and simple strain Corresponding author. Tel.: +86-10-6278-5586; fax: +8610-6278-1824. E-mail address: [email protected] (X.F. Yao).
gauge method for determining the stress intensity of sharply-notched strips; the experimental results are in good agreement with theoretical analysis. Mahinfalah [5] studied the determination of mixed mode stress intensity factors for sharp re-entrant corners using photoelastic technology. In general, the caustics method is suitable for determining the local singular stress field at the crack tip, notch tip and hole edge, etc. The advantage of the caustics method lies in the simplified optical pattern, which can establish the relation between the stress field parameters and the characteristic size of the caustic curve. So far, the caustic optical method has been successfully applied in the field of engineering fracture mechanics [6–11]. In this paper, caustic analysis of a V-notch tip is firstly briefly reviewed. The fracture characteristics of the notch tip under mode-I loading are studied by optical caustics experiments and numerical modeling. Caustic patterns at the notch tip for different notch angles are simulated based on the caustic parameter equations of a V-notch tip. The fracture behavior of a three-point-bending beam with an edge notch is analyzed by means of the evolution of the caustic spot.
0142-9418/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2003.12.004
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W. Xu et al. / Polymer Testing 23 (2004) 509–515
2. Review of caustic analysis at V-notch tip For a V-notch tip subjected to symmetrical mode-I loading, the complex stress function QðzÞ for a notch in a homogeneous, isotropic and linearly elastic plate is expressed by: QðzÞ ¼ Kzk
ð1Þ
here z ¼ xþiy is the complex variable. k is the order of singularities at the apex of the wedge. K is the stress intensity factor at the notch tip. The relation between the sum of the principal stresses at any point of the specimen and the stress function can be expressed by: rx þ ry ¼ 2½QðzÞ þ QðzÞ
ð2Þ
The basic principle of the caustic curve at the Vnotch tip is shown in Fig. 1. According to the method of transmitted caustics, a parallel light beam illuminates the specimen in close proximity to the bottom of the notch and the transmitted rays are received on a reference plane parallel to the plane of the specimen. These rays are deviated in different directions according to the law of refraction and they are concentrated along a strongly illuminated curve, which is called a caustic. Here, the correspondence between the points P(X,Y) on the reference plane and p(x,y) on the specimen plane can be expressed as [1]: W ¼ X þ iY ¼ km ½z þ Ct gradðr1 þ r2 Þ
ð3Þ
where Ct ¼
z0 dct km
Here km is the magnification factor and z0 is the distance between the specimen plane and the reference plane. In transmission, z0 is negative if the reference plane is the back of the test piece, or positive if the reference plane is the front of the test piece. d is the specimen thickness. ct is the stress optical constant of the tested material. Because the caustic is a strongly illuminated singular curve, a mathematical singularity appears. The condition for existence of such a singularity is that the functional determinant Jacobian J equals
Fig. 1.
to zero, i.e. @ ðX ;Y Þ ¼0 J ¼ @ ðx;yÞ i.e. 00 4C Q ðzÞ ¼ 1 t
ð4Þ
ð5Þ
By using Eq. (5), the caustics eq. (3) can be written as: W ¼ km ½z þ 4Ct Q0 ðzÞ
ð6Þ
By combining Eq. (1) with Eq. (5), the radius r0 of the initial curve of the caustics is given by: 1=ð2 kÞ r0 ¼ Ct Kkðk 1Þ x x
h pþ p 2 2
ð7Þ
Here, x is the angle of the V-notch tip. The parameter equation of the caustic is also expressed as: 1 X ¼ kr0 cosh coshð1 kÞ ð8Þ jk 1j 1 sinhð1 kÞ ð9Þ Y ¼ kr0 sinh jk 1j Here, the positive and negative signs represent the real and virtual caustics, which has a direct relation with z0 . Finally, the polar radius q and the polar angle u of caustics on the reference plane can be defined by the following relation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ X 2 ðr;hÞ þ Y 2 ðr;hÞ ð10Þ tanu ¼
Y ðr;hÞ X ðr;hÞ
ð11Þ
3. Simulation of initial and caustics curve at V-notch tip From Eqs. (8) and (9), it can be concluded that the caustics situation at the V-notch tip in isotropic material has a direct relation to the material constant and the singularity order. The shape of the caustic and
Caustic formation of the notch tip.
W. Xu et al. / Polymer Testing 23 (2004) 509–515 Table 1 Elastic properties of PMMA material Material
E (GPa)
G (GPa)
m
ct 10 10 (m2/N)
PMMA
3.33
1.23
0.35
1.08
its initial curve depend on the mechanical properties of the isotropic material and the V-notch angle, while its size depends on the characteristics of the experimental set up and the applied load intensity. For the numerical simulation of the caustic and initial curve at the V-notch tip, the elastic constants of PMMA polymer material are listed in Table 1. The theoretically generated initial curves and caustic curves at the V-notch tip in PMMA material are shown in Figs. 2 and 3, respectively. Here, the angle of the v notch varied between 30–150 . The comprehensive effect of the material properties on the stress singularity of the notch tip is reflected by the shape and size of the initial and caustic curves. Once material properties are known, the caustic shape of the notch tip is also defined, the size changes proportionally with the amplitude of applied load, i.e. stress intensity factor. The positive symbol in Eqs. (8) and (9) represents the virtual image, while the negative symbol represents the real image. These phenomena can be observed in an experiment at the back of the specimen and in the front of specimen, respectively. The notch tips for different notch angles have different stress singular orders. However, the shape of caustics is of the same form, and the change of the size depends on material properties under the same conditions of optical setup and loading. For the sharper notch, the stress singularity is
511
low, while the stress singularity is high for the obtuse notch. When the V-notch angle becomes zero, the sharp notch becomes a crack. The detail of caustic analysis on mode–I crack can be found in [1]. Fig. 4 shows the geometrical configuration of the caustic curve at the V-notch tip. Some important geometrical characteristic angles for a V-notch are shown in Table 2. Here a1;2 represents the maximum polar angle of the caustic curve and b1;2 represents the position of the maximum transverse distance at the caustic curve. 4. Determination of stress intensity factor at V-notch tip The most important characteristic of the caustics method in the field of fracture mechanics lies in the fact that this optical method can be used to determine the stress intensity factor at the notch tip, and it is of great importance for estimating the strength of a material and the reliable design of structures. From the geometrical pattern of the caustic in Fig. 3, the maximum transverse diameter of the caustic may be used as the characteristic parameter for evaluating the stress intensity factor at the notch tip. In order to obtain the maximum transverse distance Dmax of the caustic as shown in Fig. 4, the extreme value of Eq. (9) can be obtained first by zeroing the derivative dY =dh. Thus, we have: dY =dh ¼ 0
ð12Þ
x
From the solutions of Eq. (12) for p 2 h
p þ x2 the two positions of h1 and h2 in the specimen plane corresponding to Ymax of the caustic curve can be determined. In the meantime, the two positions of b1 and b2 corresponding to Ymax of the caustic curve on the reference plane can also be obtained by Eq. (11). Values of b1 and b2 corresponding to Ymax of the caustic curve for several different notches are also shown in Table 2, which reveals that the caustic will be symmetrical to the initial notch midplane. Thus, Dmax can be written in the simplest form: 0 1=ð2 kÞ jC jK ð13Þ Dmax ¼ km d where d is a correction factor given by [1]: d¼
Fig. 2. Theoretical initial curves for different notch angles.
2k 2 kðk 1Þ p 1 pð1 kÞ k 2
sin þ sin 2 k jk 1j 2 k
ð14Þ
The factor d has a direct relation with the notch angle, which must be calculated for each notch separately. Finally, the relation between the stress intensity factor K and the maximum transverse distance Dmax of
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W. Xu et al. / Polymer Testing 23 (2004) 509–515
Fig. 3.
Theoretical caustic curves for different notch angles.
Table 2 Geometrical characterization of caustic curve for different V-notch angle V-notch angle v
Fig. 4. Geometrical characterization of caustic curve at the V-notch tip.
30 v 60 v 90 v 120 v 150
Virtual caustics
Real caustics
b1,2
a1,2
b1,2
a1,2
105.3 102.8 93.1 73.7 53.5
129 111.4 93.1 73.8 53.5
86.2 86 86.5 87 88.8
192.9 178.3 159.5 139.1 116.6
W. Xu et al. / Polymer Testing 23 (2004) 509–515
the caustics can be expressed as [1]: Dmax 2 k 1 K ¼ dk km jC 0 j
ð15Þ
Therefore, it is only required to measure the characteristic diameters of the notch tip caustics for the tested material in the experimental data analysis and the stress intensity factor at the notch tip may be determined by Eq. (15). 5. Caustic experiment on V-notch tip In this test, the amorphous glassy polymer polymethylmethacrylate (PMMA) was chosen because of its homogeneous and isotropic nature. The notched PMMA beam specimen with width w ¼ 45 mm and length l ¼ 210 mm was cut from a 5 mm thick plate. A v symmetrical edge V-notch of angle x ¼ 90 and notch tip radius q ¼ 0:1 mm was chosen and machined with a low speed cutting machine. The notch depth was 8 mm. The geometry of the specimen is also shown in Fig. 5(a). Mechanical properties of the PMMA material have been shown in Table 1. The optical experimental set-up of the caustics method is shown in Fig. 5(b), which consists of a 10 mW He-Ne laser, filter, two field lenses, CCD camera and personal computer. The specimen is loaded with a three point bending test machine and the load is measured with a 500 N force transducer. At room temperature, the caustic images for different load levels are stored in the computer, then the special image processing software is used to extract the maximum characteristic size of caustics at the notch tip.
Fig. 5.
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Fig. 6 shows a series of caustic patterns at the notch tip during the whole fracture process of the threepoint-bending beam, which exhibit the evolution of singularity at the crack tip. With increase of the applied load, the localized stress concentration at the notch increases resulting in the development of brittle fracture behavior. Because both the sample and the loading is symmetrical with respect to the symmetry plane of the V-notch, the notch is under the influence of a symmetric state of stress. The caustic of the notch tip is formed which continuously increases in size with increase of the applied load. The physical meaning of the caustic spot is to reflect the local stress concentration at the V-notch tip. The initial notch is symmetric to the longitudinal direction of the specimen and the V-notch is subject to Mode I loading. Hence, only the mode I stress intensity factor K is operative and the mode II stress intensity factor is zero. After measuring the maximum characteristic size of the caustics at the notch tip, the mode I stress intensity factor K is calculated by Eq. (15). Fig. 7 shows the trend of the maximum characteristic size of the caustic curve and the stress intensity factor at the V notch tip with different load levels. The theoretical value of the stress intensity factor for the three-point-bending specimen with a V-notch under mode-I loading was obtained from reference [12] as follows: 6P 1 k a K¼ w f2 x; ð16Þ wt w where P is the applied load, a is the notch depth, t is
Experimental details (a) Three-point-bending specimen with V-notch, (b) Experimental setup.
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W. Xu et al. / Polymer Testing 23 (2004) 509–515
Fig. 6.
Caustic patterns of V-notch tip under different load levels.
the specimen thickness, f2 x; wa is a dimensionless function which can be computed from reference [12]. The value of f2 x; wa is 2.106 for x ¼ 90v . According to Eq. (16), the theoretical prediction of stress intensity factor at the V-notch tip in three-point-bending specimen is also shown in Fig. 7. It is obviously seen that there is good agreement between the theoretical analysis and the experimental measurement below an applied load P ¼ 160 N. However, some differences between the test data and the predicted value exist from an applied load P ¼ 160 N to the critical fracture load P ¼ 384 N. The main reason is that the theoretical prediction is based on homogeneous linear elastic mechanics and plane stress, but the stress and deformation state of a V-notch tip is very complicated in the real
experimental study because the PMMA material shows some plastic behavior and the crack tip is under a 3 dimensional stress state. From the evolution of the caustic image at the notch tip, it is clear that the elastic strain energy stored in the notch tip gradually increased with the increment of the load level. Once the stress intensity factor at the notch tip reaches the fracture toughness of the PMMA material, or the elastic strain energy reaches the fracture energy of the PMMA material, a crack will initiate in the notch tip and propagate through the material. In this test, the caustic size at the notch tip has a maximum value of 4:87 10 3 m at the critical fracture point, which is shown at about 384 N in Fig. 7, and the corresponding maximum stress intensity factor is about 17:8 104 Nm 3=2 . 6. Conclusions
Fig. 7. Evolution on maximum transverse diameter of caustic spot and stress intensity factor at V-notch tip.
The optical caustics method was used to study the fracture of PMMA material with a V-notch under mode-I loading. Some valuable conclusions were obtained: 1) The singular characterization of a V-notch tip was simulated by means of the caustic principle and the elastic stress solution. Some typical characteristics of the stress singularity were analyzed. 2) The evolution of the V-notch tip under loading was visualized by the experimentally obtained caustic image. 3) Under different load levels, the maximum characteristic size and the stress intensity factor of the caustic at the notch tip were measured and calculated, which gave good agreement with the predicted theoretical stress intensity factor. These results will be useful for the reliable design and strength analysis for notched structures.
W. Xu et al. / Polymer Testing 23 (2004) 509–515
References [1] G.A. Papadopoulos, Fracture Mechanics, The Experimental Method of Caustics and the Det. Criterion of the Fracture, Springer-Verlag, London, 1993, 134–204. [2] P.S. Theocaris, G.A. Papadopoulos, The dynamic behavior of sharp V-notches under impact loading, International Journal of Solids and Structures 23 (1987) 1581–1600. [3] Y. Kawagishi, M. Shozu, Y. Hirose, Experimental evaluation of stress field around crack tip by caustic method, Mechanics of Materials 33 (2001) 741–757. [4] T. Kondo, M. Kobayashi, H. Sekine, Strain gauge method for determining stress intensities of sharp-notched strips, Experimental Mechanics 4 (2001) 1–7. [5] M. Mahinfalah, L. Zackery, Photoelastic determination of mixed mode stress intensity factors for sharp reentrant corners, Engineering Fracture Mechanics 52 (1995) 639–645. [6] J.F. Kalthoff, Shadow optical method of caustics, Chapter 9, in: A.S. Kobayashi (Ed.), Handbook on Experimental Mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1987, pp. 430–500.
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[7] A.J. Rosakis, Analysis of the optical method of caustics for dynamic crack propagation, Engineering Fracture Mechanics 13 (1980) 331–347. [8] K. Kawakawa, T. Mada, K. Takahashi, Correlations among dynamic stress intensity factor, crack velocity and acceleration in brittle fracture, International Journal of Fracture Mechanics 105 (2000) 311–320. [9] D. Semenski, Method of caustics in fracture mechanics of mechanically anisotropic materials, Engineering Fracture Mechanics 58 (1997) 1–10. [10] X.Y. Yao, G.C. Jin, K. Arakawa, K. Takahashi, Experimental studies on dynamic fracture behavior of thin plates with parallel single edge cracks, Polymer Testing 21 (2002) 933–940. [11] X.F. Yao, W. Xu, M.Q. Xu, K. Arawaka, T. Mada, K. Takahashi, Experimental study of dynamic fracture behavior of PMMA with overlapping offset-parallel cracks, Polymer Testing 22 (2003) 663–670. [12] M. Strandberg, Fracture at V-notches with contained plasticity, Engineering Fracture Mechanics 69 (2002) 403–415.
Test Method
Fracture characterizations of V-notch tip in PMMA polymer material W. Xu a, X.F. Yao a, , M.Q. Xu a, G.C. Jin a, H.Y. Yeh b b
a Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China Department of Mechanical and Aerospace Engineering, California State University, Long Beach, CA 90840-8305, USA
Received 23 September 2003; accepted 22 December 2003
Abstract In this paper, the fracture characterization of a V-notch tip in PMMA material is studied by means of an optical caustics method. The initial curve and caustic curve of the notch tip with different notch angles are simulated. Some typical stress concentration characterizations at the notch tip are analyzed. Fracture experiments of PMMA specimens with a V-notch subjected to three-point-bend loading are visualized using the optical pattern of caustics. The evolution of the stress intensity factor and the size of the caustic spot at the notch tip are analyzed. These results will be useful for the reliable design and strength evaluation of engineering structures with complex notch shapes. # 2003 Elsevier Ltd. All rights reserved. Keywords: PMMA material; Optical caustics; V-notch; Stress intensity factor; Fracture
1. Introduction It is well known that cutout geometrical features such as: cracks, holes and notches, etc., are frequently encountered in many engineering structures. Under external loading, the stress fields in the vicinity of these cutouts are very complicated. Therefore, it is important to study the stress information and fracture characteristics at the notch tip for evaluating the strength and predicting the service life of the structures. So far, the analytical solution of the notch tip stress field is based on the elastic mechanics theory and the finite element method. Experimentally, the evaluation of the complex stress intensity factor K at the apex of the sharp V-notch in homogeneous and isotropic elastic plates under symmetrical load in their plane can be studied using the optical method of reflective caustics [1–3]. Kondo [4] gave an efficient and simple strain Corresponding author. Tel.: +86-10-6278-5586; fax: +8610-6278-1824. E-mail address: [email protected] (X.F. Yao).
gauge method for determining the stress intensity of sharply-notched strips; the experimental results are in good agreement with theoretical analysis. Mahinfalah [5] studied the determination of mixed mode stress intensity factors for sharp re-entrant corners using photoelastic technology. In general, the caustics method is suitable for determining the local singular stress field at the crack tip, notch tip and hole edge, etc. The advantage of the caustics method lies in the simplified optical pattern, which can establish the relation between the stress field parameters and the characteristic size of the caustic curve. So far, the caustic optical method has been successfully applied in the field of engineering fracture mechanics [6–11]. In this paper, caustic analysis of a V-notch tip is firstly briefly reviewed. The fracture characteristics of the notch tip under mode-I loading are studied by optical caustics experiments and numerical modeling. Caustic patterns at the notch tip for different notch angles are simulated based on the caustic parameter equations of a V-notch tip. The fracture behavior of a three-point-bending beam with an edge notch is analyzed by means of the evolution of the caustic spot.
0142-9418/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2003.12.004
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W. Xu et al. / Polymer Testing 23 (2004) 509–515
2. Review of caustic analysis at V-notch tip For a V-notch tip subjected to symmetrical mode-I loading, the complex stress function QðzÞ for a notch in a homogeneous, isotropic and linearly elastic plate is expressed by: QðzÞ ¼ Kzk
ð1Þ
here z ¼ xþiy is the complex variable. k is the order of singularities at the apex of the wedge. K is the stress intensity factor at the notch tip. The relation between the sum of the principal stresses at any point of the specimen and the stress function can be expressed by: rx þ ry ¼ 2½QðzÞ þ QðzÞ
ð2Þ
The basic principle of the caustic curve at the Vnotch tip is shown in Fig. 1. According to the method of transmitted caustics, a parallel light beam illuminates the specimen in close proximity to the bottom of the notch and the transmitted rays are received on a reference plane parallel to the plane of the specimen. These rays are deviated in different directions according to the law of refraction and they are concentrated along a strongly illuminated curve, which is called a caustic. Here, the correspondence between the points P(X,Y) on the reference plane and p(x,y) on the specimen plane can be expressed as [1]: W ¼ X þ iY ¼ km ½z þ Ct gradðr1 þ r2 Þ
ð3Þ
where Ct ¼
z0 dct km
Here km is the magnification factor and z0 is the distance between the specimen plane and the reference plane. In transmission, z0 is negative if the reference plane is the back of the test piece, or positive if the reference plane is the front of the test piece. d is the specimen thickness. ct is the stress optical constant of the tested material. Because the caustic is a strongly illuminated singular curve, a mathematical singularity appears. The condition for existence of such a singularity is that the functional determinant Jacobian J equals
Fig. 1.
to zero, i.e. @ ðX ;Y Þ ¼0 J ¼ @ ðx;yÞ i.e. 00 4C Q ðzÞ ¼ 1 t
ð4Þ
ð5Þ
By using Eq. (5), the caustics eq. (3) can be written as: W ¼ km ½z þ 4Ct Q0 ðzÞ
ð6Þ
By combining Eq. (1) with Eq. (5), the radius r0 of the initial curve of the caustics is given by: 1=ð2 kÞ r0 ¼ Ct Kkðk 1Þ x x
h pþ p 2 2
ð7Þ
Here, x is the angle of the V-notch tip. The parameter equation of the caustic is also expressed as: 1 X ¼ kr0 cosh coshð1 kÞ ð8Þ jk 1j 1 sinhð1 kÞ ð9Þ Y ¼ kr0 sinh jk 1j Here, the positive and negative signs represent the real and virtual caustics, which has a direct relation with z0 . Finally, the polar radius q and the polar angle u of caustics on the reference plane can be defined by the following relation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ X 2 ðr;hÞ þ Y 2 ðr;hÞ ð10Þ tanu ¼
Y ðr;hÞ X ðr;hÞ
ð11Þ
3. Simulation of initial and caustics curve at V-notch tip From Eqs. (8) and (9), it can be concluded that the caustics situation at the V-notch tip in isotropic material has a direct relation to the material constant and the singularity order. The shape of the caustic and
Caustic formation of the notch tip.
W. Xu et al. / Polymer Testing 23 (2004) 509–515 Table 1 Elastic properties of PMMA material Material
E (GPa)
G (GPa)
m
ct 10 10 (m2/N)
PMMA
3.33
1.23
0.35
1.08
its initial curve depend on the mechanical properties of the isotropic material and the V-notch angle, while its size depends on the characteristics of the experimental set up and the applied load intensity. For the numerical simulation of the caustic and initial curve at the V-notch tip, the elastic constants of PMMA polymer material are listed in Table 1. The theoretically generated initial curves and caustic curves at the V-notch tip in PMMA material are shown in Figs. 2 and 3, respectively. Here, the angle of the v notch varied between 30–150 . The comprehensive effect of the material properties on the stress singularity of the notch tip is reflected by the shape and size of the initial and caustic curves. Once material properties are known, the caustic shape of the notch tip is also defined, the size changes proportionally with the amplitude of applied load, i.e. stress intensity factor. The positive symbol in Eqs. (8) and (9) represents the virtual image, while the negative symbol represents the real image. These phenomena can be observed in an experiment at the back of the specimen and in the front of specimen, respectively. The notch tips for different notch angles have different stress singular orders. However, the shape of caustics is of the same form, and the change of the size depends on material properties under the same conditions of optical setup and loading. For the sharper notch, the stress singularity is
511
low, while the stress singularity is high for the obtuse notch. When the V-notch angle becomes zero, the sharp notch becomes a crack. The detail of caustic analysis on mode–I crack can be found in [1]. Fig. 4 shows the geometrical configuration of the caustic curve at the V-notch tip. Some important geometrical characteristic angles for a V-notch are shown in Table 2. Here a1;2 represents the maximum polar angle of the caustic curve and b1;2 represents the position of the maximum transverse distance at the caustic curve. 4. Determination of stress intensity factor at V-notch tip The most important characteristic of the caustics method in the field of fracture mechanics lies in the fact that this optical method can be used to determine the stress intensity factor at the notch tip, and it is of great importance for estimating the strength of a material and the reliable design of structures. From the geometrical pattern of the caustic in Fig. 3, the maximum transverse diameter of the caustic may be used as the characteristic parameter for evaluating the stress intensity factor at the notch tip. In order to obtain the maximum transverse distance Dmax of the caustic as shown in Fig. 4, the extreme value of Eq. (9) can be obtained first by zeroing the derivative dY =dh. Thus, we have: dY =dh ¼ 0
ð12Þ
x
From the solutions of Eq. (12) for p 2 h
p þ x2 the two positions of h1 and h2 in the specimen plane corresponding to Ymax of the caustic curve can be determined. In the meantime, the two positions of b1 and b2 corresponding to Ymax of the caustic curve on the reference plane can also be obtained by Eq. (11). Values of b1 and b2 corresponding to Ymax of the caustic curve for several different notches are also shown in Table 2, which reveals that the caustic will be symmetrical to the initial notch midplane. Thus, Dmax can be written in the simplest form: 0 1=ð2 kÞ jC jK ð13Þ Dmax ¼ km d where d is a correction factor given by [1]: d¼
Fig. 2. Theoretical initial curves for different notch angles.
2k 2 kðk 1Þ p 1 pð1 kÞ k 2
sin þ sin 2 k jk 1j 2 k
ð14Þ
The factor d has a direct relation with the notch angle, which must be calculated for each notch separately. Finally, the relation between the stress intensity factor K and the maximum transverse distance Dmax of
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W. Xu et al. / Polymer Testing 23 (2004) 509–515
Fig. 3.
Theoretical caustic curves for different notch angles.
Table 2 Geometrical characterization of caustic curve for different V-notch angle V-notch angle v
Fig. 4. Geometrical characterization of caustic curve at the V-notch tip.
30 v 60 v 90 v 120 v 150
Virtual caustics
Real caustics
b1,2
a1,2
b1,2
a1,2
105.3 102.8 93.1 73.7 53.5
129 111.4 93.1 73.8 53.5
86.2 86 86.5 87 88.8
192.9 178.3 159.5 139.1 116.6
W. Xu et al. / Polymer Testing 23 (2004) 509–515
the caustics can be expressed as [1]: Dmax 2 k 1 K ¼ dk km jC 0 j
ð15Þ
Therefore, it is only required to measure the characteristic diameters of the notch tip caustics for the tested material in the experimental data analysis and the stress intensity factor at the notch tip may be determined by Eq. (15). 5. Caustic experiment on V-notch tip In this test, the amorphous glassy polymer polymethylmethacrylate (PMMA) was chosen because of its homogeneous and isotropic nature. The notched PMMA beam specimen with width w ¼ 45 mm and length l ¼ 210 mm was cut from a 5 mm thick plate. A v symmetrical edge V-notch of angle x ¼ 90 and notch tip radius q ¼ 0:1 mm was chosen and machined with a low speed cutting machine. The notch depth was 8 mm. The geometry of the specimen is also shown in Fig. 5(a). Mechanical properties of the PMMA material have been shown in Table 1. The optical experimental set-up of the caustics method is shown in Fig. 5(b), which consists of a 10 mW He-Ne laser, filter, two field lenses, CCD camera and personal computer. The specimen is loaded with a three point bending test machine and the load is measured with a 500 N force transducer. At room temperature, the caustic images for different load levels are stored in the computer, then the special image processing software is used to extract the maximum characteristic size of caustics at the notch tip.
Fig. 5.
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Fig. 6 shows a series of caustic patterns at the notch tip during the whole fracture process of the threepoint-bending beam, which exhibit the evolution of singularity at the crack tip. With increase of the applied load, the localized stress concentration at the notch increases resulting in the development of brittle fracture behavior. Because both the sample and the loading is symmetrical with respect to the symmetry plane of the V-notch, the notch is under the influence of a symmetric state of stress. The caustic of the notch tip is formed which continuously increases in size with increase of the applied load. The physical meaning of the caustic spot is to reflect the local stress concentration at the V-notch tip. The initial notch is symmetric to the longitudinal direction of the specimen and the V-notch is subject to Mode I loading. Hence, only the mode I stress intensity factor K is operative and the mode II stress intensity factor is zero. After measuring the maximum characteristic size of the caustics at the notch tip, the mode I stress intensity factor K is calculated by Eq. (15). Fig. 7 shows the trend of the maximum characteristic size of the caustic curve and the stress intensity factor at the V notch tip with different load levels. The theoretical value of the stress intensity factor for the three-point-bending specimen with a V-notch under mode-I loading was obtained from reference [12] as follows: 6P 1 k a K¼ w f2 x; ð16Þ wt w where P is the applied load, a is the notch depth, t is
Experimental details (a) Three-point-bending specimen with V-notch, (b) Experimental setup.
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Fig. 6.
Caustic patterns of V-notch tip under different load levels.
the specimen thickness, f2 x; wa is a dimensionless function which can be computed from reference [12]. The value of f2 x; wa is 2.106 for x ¼ 90v . According to Eq. (16), the theoretical prediction of stress intensity factor at the V-notch tip in three-point-bending specimen is also shown in Fig. 7. It is obviously seen that there is good agreement between the theoretical analysis and the experimental measurement below an applied load P ¼ 160 N. However, some differences between the test data and the predicted value exist from an applied load P ¼ 160 N to the critical fracture load P ¼ 384 N. The main reason is that the theoretical prediction is based on homogeneous linear elastic mechanics and plane stress, but the stress and deformation state of a V-notch tip is very complicated in the real
experimental study because the PMMA material shows some plastic behavior and the crack tip is under a 3 dimensional stress state. From the evolution of the caustic image at the notch tip, it is clear that the elastic strain energy stored in the notch tip gradually increased with the increment of the load level. Once the stress intensity factor at the notch tip reaches the fracture toughness of the PMMA material, or the elastic strain energy reaches the fracture energy of the PMMA material, a crack will initiate in the notch tip and propagate through the material. In this test, the caustic size at the notch tip has a maximum value of 4:87 10 3 m at the critical fracture point, which is shown at about 384 N in Fig. 7, and the corresponding maximum stress intensity factor is about 17:8 104 Nm 3=2 . 6. Conclusions
Fig. 7. Evolution on maximum transverse diameter of caustic spot and stress intensity factor at V-notch tip.
The optical caustics method was used to study the fracture of PMMA material with a V-notch under mode-I loading. Some valuable conclusions were obtained: 1) The singular characterization of a V-notch tip was simulated by means of the caustic principle and the elastic stress solution. Some typical characteristics of the stress singularity were analyzed. 2) The evolution of the V-notch tip under loading was visualized by the experimentally obtained caustic image. 3) Under different load levels, the maximum characteristic size and the stress intensity factor of the caustic at the notch tip were measured and calculated, which gave good agreement with the predicted theoretical stress intensity factor. These results will be useful for the reliable design and strength analysis for notched structures.
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