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Исследование разрушения при нагружении смешанного типа I/II полиметилметакрилата с использованием критерия усредненной плотности энергии деформации
M.R.M. Aliha1, F. Berto23, A. Bahmani14, P. Gallo5
1 Иранский университет науки и технологии, Тегеран, 16846-13114, Иран 2 Падуанский университет, Виченца, 36100, Италия 3 Норвежский университет естественных и технических наук, Тронхейм, 7491, Норвегия 4 Университет Уотерлу, Уотерлу, ON N2L 3G1, Канада 5 Университет Аалто, Эспо, 02150, Финляндия
В статье на основе концепции локальной энергии проведен анализ результатов исследования сопротивления разрушению при нагружении смешанного типа I/II для образцов полиметилметакрилата с трещиной, подвергнутыж трехточечному изгибу. Испытания всех образцов проводили в одинаковом режиме, однако полученные огибающие кривые сопротивления разрушению поли-метилметакрилата при смешанном типе нагружения I/II различаются. Применен критерий усредненной плотности энергии деформации, ранее использованный для разныж типов образцов с надрезами (U-, V-, O-образными, ключевидными). Показано, что критерий плотности энергии деформации позволяет удовлетворительно предсказывать результаты по сопротивлению разрушению при смешанном нагружении полукруглых и треугольных образцов полиметилметакрилата с острыми надрезами по схеме «растяжение - плоский сдвиг».
Ключевые слова: нагружение смешанного типа I/II, полиметилметакрилат, критерий плотности энергии деформации, сопротивление разрушению, полукруглый и треугольный образец
Mixed mode I/II fracture investigation of Perspex based on the averaged strain
energy density criterion
M.R.M. Aliha1, F. Berto23, A. Bahmani14, and P. Gallo5
1 Welding and Joining Research Center, School of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran 2 Department of Management and Engineering, University of Padova, Vicenza, 36100, Italy
3 Department of Engineering Design and Materials, Norwegian University of Science and Technology, Trondheim, 7491, Norway 4 Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada 5 Department of Mechanical Engineering, Aalto University, Espoo, 02150, Finland
In this work, some recent mixed mode I/II fracture toughness results obtained from Perspex (or PMMA) with four simple cracked specimens subjected to the conventional three-point bend loading are reanalysed based on local energy concept. Although all the mentioned samples have been tested under the same and similar mode mixities, different fracture toughness envelopes were obtained for mixed mode I/II fracture of PMMA. The averaged strain energy density (SED) criterion has been applied in the past for different types of notched specimens (including U, V, O and key-hole notches). It is shown that the mixed mode tensile-in plane shear fracture toughness data obtained from the semi-circular and triangular crack type specimens are successfully predicted for sharp cracked PMMA samples using the SED criterion.
Keywords: mixed mode I/II, PMMA, SED criterion, fracture toughness, semi-circular and triangular specimens
1. Introduction
Cracks may initiate inside the structure of brittle engi-
neering materials during their manufacturing process, as-
sembling and production stage or even during their service
life. Today, it is well established that the crack growth is one of the main reasons for the overall failure and brittle fracture in various engineering materials like brittle polymers, ceramics and rocks. Real cracked structures are often
© Aliha M.R.M., Berto F., Bahmani A., Gallo P., 2016
subjected to complex loading conditions. Therefore, the investigation of crack growth in brittle materials under any arbitrary combinations of tensile and shear (mixed mode I/II) loading is important for failure analysis in different engineering components and structures. Mixed mode brittle fracture is usually studied both experimentally and theoretically using suitable materials and test specimens. Perspex or polymethylmethacrylate (PMMA) has been recognized as a favourite material for evaluating the fracture characteristics of brittle materials. This is because of its certain advantages like: brittle fracture at room temperature, low cost, machinability and convenience of test sample preparation and optical transparency that allows direct observation of crack tip region and path of crack growth during the fracture processes. Moreover, as stated by Maccagno and Knott [1] a sharp crack can be introduced in the root of an artificially fabricated prenotch in PMMA by pressing a razor blade. Indeed, this simple method, eliminate time consuming and expensive process of fatigue precracking procedure which is applied for metals. Hence, it is reasonable to assume that the linear elastic stress field equations outlined by Irwin [2] or Williams [3] can provide good description for the stress/strain field near the tip of sharp precracks introduced in the laboratory scale PMMA test specimens. A review of literature indicates that several mixed mode I/II fracture experiments have been conducted on PMMA using various test specimens. PMMA plates containing inclined centre cracks with respect to the far field uniform tension loading [4, 5], rectangular plates of PMMA having an inclined edge crack and subjected to a far field tensile load or a bending moment [6], the compact tension shear specimen [7], the diagonally loaded square plate containing an inclined centre crack [8], the centrally cracked Brazilian disc specimen subjected to diametral compression [9, 10], the semi-circular and triangular bend specimen [1114] and the asymmetric three or four-point bend loading specimens [1, 15, 16] are some of the test configurations used by the researchers for investigating the mixed mode I/II fracture behaviour of PMMA. There have also been some attempts for analysing and predicting the experimental results (such as critical fracture loads, stress intensity factors, fracture initiation direction and the trajectory of fracture growth) obtained from the aforementioned test configurations using theoretical based fracture models.
Different approaches such as stress, strain and energy methods have been proposed for investigating the brittle fracture phenomenon under mixed mode I/II condition. For example, maximum tangential stress [4], minimum strain energy density [17], maximum energy release rate [18], cohesive zone model [19] and mean stress [20] are some of the frequently used mixed mode I/II fracture criteria. These criteria provide theoretical framework for predicting the ratio of Kj/Kjj for any mode mixity (or combinations of modes I and II). In a recent paper, Aliha and coworkers
[14] obtained a series of new fracture toughness data for PMMA by testing some edge cracked semi-circular and triangular specimens subjected to three-point bend loading. They determined magnitudes of Kj and Kjj for each specimen in the entire range of mode mixities from pure mode I to pure mode II and showed that different fracture envelopes are obtained depending on the shape and loading type of specimens.
A fracture model based on the strain energy density (SED) averaged over a control volume has been used successfully in the past by Lazzarin and Berto and coworkers [21-28] mainly for different types ofnotched samples containing U-, V-, O- and key-hole notches subjected to pure modes and also mixed mode I/III loading cases. In the present work, the application of SED criterion is examined for assessing the mixed mode I/II fracture results presented in Ref. [14] for different PMMA samples. Hence, in the upcoming sections of this paper, first the experimental activity performed by Aliha et al. [14] is briefly recalled and then the predictions of the averaged SED theory is presented for mixed mode I/II cracked PMMA specimens.
2. Mixed mode I/II PMMA fracture toughness data
Recently, Aliha et al. [14] conducted mixed mode I/II fracture toughness experiments on PMMA using simple test specimens in the shape of half disc and triangle containing an edge crack and subjected to conventional three-point bend loading. Figure 1 shows the geometry and loading configuration of the mentioned specimens. For easy understanding, the specimens are designated as SCB (semi-circular bend) and TB (triangular bend). Suffix "1" and "2" was used for inclined crack with symmetric supports and vertical crack with asymmetric loading supports, respectively.
The crack tip stress state can be changed in these specimens by changing either crack inclination angle a or the loading support distance S 2. Therefore, the stress intensity factors Kj and Kjj are function of specimen geometry and its loading conditions (i.e. parameters such as applied load P, crack length a, disc radius R, base length of triangle W, thickness t, crack angle a and bottom loading roller distances from the crack S, Sj and S2). Thus, the mode I and mode II stress intensity factors of the mentioned specimens can be obtained by the following equations:
P^fna ,
K j =
2Rt P^fna
2Rt P^fna
2Wt P^fna
2Wt
Yj( a/R, S/R, a) for SCBJ, Yj( aR, S/R, SjR) for SCB2, Yj( a/R, S/R, a) for TBJ, Yj(a/R,S/R, SjR) for TB2,
(1)
2 W 2 W
Fig. 1. Geometry and loading conditions of TBI, TB2, SCB1 and SCB2 specimens
K
Pa/to
2Rt P4%a
2Rt pjm
2Wt Pyfrn
2Wt
Yn( a/R, S/R, a) for SCB1, Yn(a/R, S/R, S2/R) for SCB2, Yn(a/R,S/R,a) for TBI, Yn( a/R, S/R, SJ R ) for TB2,
(2)
in which YI and Yn are geometry factors that are functions of a/R, a/W, S/R, R, 5JR and a, SCB1, SCB2, TBI, TB2 are specimens. Six values Me (0.0, 0.2, 0.4, 0.6, 0.8 and 1.0) were used in Ref. [14] to obtain the mixedmode fracture toughness values, where Me is defined as:
M e = 2/ n arctg( K J K n).
The overall dimensions of both SCB and TB specimen types were equal to the following values in the experiments of Ref. [14]: R = W = 50 mm, S = S1 = 20 mm, a = 15 mm and t = 5 mm. Other parameters such as a and S2 were considered as variable in [14]. Loading specifications and geometry factors of the tested specimens for different mode mixities have been presented in Table 1.
The corresponding values of mode I and mode II fracture toughness values for the considered mode mixities were obtained by recording the fracture load of the SCB 1, SCB2, TB1 and TB2 specimens made of PMMA using Eqs. (1) and (2). A servohydraulic compression test machine (as shown in Fig. 2) was used for testing the specimens.
All the test samples were fractured from the tip of crack and, as it is obvious, from Fig. 3 the load-displacement curves obtained from the test samples demonstrated linear
Table 1
Loading specifications and corresponding geometry factors YI and YII for the investigated SCB and TB specimens
M e
Speci- a S2, mm Y Yi
men 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 0.0
TB2 20 12.5 9.5 8 6.5 5 3.4 3.07 2.03 1.26 0.62 0 0 0.10 0.23 0.39 0.87 1.62
TBI 0° 10° 26° 38° 45.5° 52.5° 3.4 3.24 2.79 2.11 1.35 0 0 0.48 0.88 1.11 1.13 0.86
SCB2 20 14 11.5 9 7.5 6 1.8 1.64 1.44 1.22 0.68 0 0 0.09 0.20 0.34 0.76 1.41
SCB1 0° 12° 23° 31° 37° 42.5° 1.8 1.66 1.27 0.73 0.16 0 0 0.40 0.70 0.84 0.98 0.85
0
2500 2000 ¡-1500
hooo
500
-TBI
---SCB 1
/
/ /
X /
X /
/ /
/ /
/ /
y/
500
1000
1500
2000 2500
Deformation, |um
Fig. 3. Typical load-displacement curves obtained for the SCB and TB specimens
Fig. 2. Test setup for testing SCB (a) and TB (b) specimens made of PMMA
elastic fracture behavior, with negligible nonlinear deformation for PMMA at room temperature.
The obtained mixed mode I/II fracture toughness results have been presented in Fig. 4 in a K J K Ic - K n/ KIc diagram, in which KIc is the pure mode I fracture toughness of PMMA. KI and Kn are the critical values of stress intensity factors at the onset of fractures and corresponds to the fracture load of each tested specimen. The average KIc obtained from the four specimens is 1.87 MPa-m05.
This value is in the range of 1-2 MPa-m05 reported in previous papers for fracture toughness of PMMA [8, 1113, 29-31]. Figure 4 reveals that depending on the specimen shape and its loading condition, different fracture toughness results are obtained for a same material. These results are evaluated theoretically in the next section using the averaged SED criterion.
3. Strain energy density averaged over a control volume: the fracture criterion
With the aim to assess the fracture load in notched PMMA components, an appropriate fracture criterion is required which has to be based on the mechanical behavior
of material around the crack or the notch tip. In this section, a criterion proposed by Lazzarin and co-authors [21] based on the strain energy density (SED) is briefly described.
The averaged strain energy density criterion (SED) as presented in Refs. [21-28] states that brittle failure occurs when the mean value of the strain energy density over a given control volume is equal to a critical value Wc. This critical value is a material characteristic parameter and does not depend on the notch geometry and sharpness. Dealing with mode I loading, the control volume is considered to be dependent on the ultimate tensile strength at and the fracture toughness KIc in the case of brittle or quasi-brittle materials subjected to static loading.
The method based on the averaged SED was formalized and applied first to sharp (zero radius) V-notches under mode I and mixed mode I/II loading [21] and later extended to blunt U- and V-notches [22-24]. When dealing with cracks, as in the present research, the control volume is a circle of radius Rc centered at the crack tip (Fig. 5, a). Under plane strain conditions, the radius Rc can be evaluated according to the following expression:
R1c =
(1 + v)(5 - 8v)
4k
K,,
(3)
where v is the Poisson's ratio and at is the ultimate tensile stress of a flat specimen.
Fig. 4. Mixed mode I/II fracture toughness results obtained from SCB1, SCB2, TB1 and TB2 specimens
2a = 0
= 00
Fig. 5. Control volume for crack (a), sharp V-notch (6)
For a sharp V-notch, the critical volume becomes a circular sector of radius Rc centred at the notch tip (Fig. 5, b).
Dealing here with sharp notches under mode II loading, the control radius R2c can be estimated by means of the following equation first proposed in Ref. [27]: 2
R2c -
9 - 8 v
8k
K
IIc
(4)
where KIIc is the mode II critical notch stress intensity factor and xt is the ultimate shear strength of the unnotched material and can be calculated approximately as
xt -CTt/V2(1 + v). (5)
4. SED approach in fracture analysis of the tested PMMA specimens
The fracture criterion described in the previous section is employed here to estimate the fracture loads obtained from the experiments conducted on the PMMA specimens.
As originally thought for pure modes of loading the averaged strain energy density criterion states that failure occurs when the mean value of the strain energy density over a control volume W reaches a critical value Wc, which depends on the material but not on the notch geometry.
Under tension loads, this critical value can be determined from the ultimate tensile strength a t according to Beltrami's expression for the uncracked material:
at
2E
(6)
By using the values of at - 70 MPa and E = 2950 MPa, the critical SED for the tested PMMA is determined as Wlc - 0.83 MJ/m3 [31].
Under torsion loads, this critical value can be determined from the ultimate shear strength xt according to Beltrami's expression for the uncracked material:
W2c 2c 2G
(7)
Table 2
Experimental results. Overview of the data by using the proposed SED method for the SCB1 specimens
a Wl5 MJ/m3 W2, MJ/m3 WJ W1c + + W2I W2c W W1c + + W2I W>c)°.5
0.0° 0.712 0.000 0.858 1.016
0.0° 0.793 0.000 0.956 1.073
0.0° 0.879 0.000 1.059 1.129
12.0° 0.665 0.087 0.904 1.044
12.0° 0.744 0.099 1.015 1.106
12.0° 0.736 0.095 1.000 1.097
23.0° 0.501 0.355 1.026 1.112
23.0° 0.494 0.355 1.018 1.107
23.0° 0.501 0.355 1.026 1.112
31.0° 0.190 0.594 0.936 1.062
31.0° 0.211 0.649 1.027 1.113
31.0° 0.194 0.594 0.941 1.065
37.0° 0.013 1.105 1.331 1.266
37.0° 0.012 1.033 1.243 1.224
37.0° 0.013 0.963 1.161 1.183
42.2° 0.000 0.707 0.842 1.007
42.2° 0.000 0.594 0.707 0.923
42.2° 0.000 0.830 0.988 1.091
Table 3
Experimental results. Overview of the data by using the proposed SED method for the TB1 specimens
a Wl5 MJ/m3 W2, MJ/m3 Wi/Wlc + + W2I W2c (Wi/Wlc + + W2/W2c)°.5
0.0° 0.673 0.000 0.811 0.988
0.0° 0.712 0.000 0.858 1.016
0.0° 0.793 0.000 0.956 1.073
10.0° 0.914 0.079 1.195 1.200
10.0° 0.941 0.079 1.228 1.216
10.0° 0.905 0.079 1.184 1.195
26.0° 0.835 0.314 1.381 1.290
26.0° 0.776 0.314 1.310 1.256
26.0° 0.879 0.314 1.433 1.314
38.0° 0.613 0.541 1.383 1.291
38.0° 0.598 0.594 1.428 1.312
38.0° 0.681 0.541 1.464 1.328
45.5° 0.301 1.033 1.592 1.385
45.5° 0.343 0.963 1.559 1.371
45.5° 0.332 0.895 1.466 1.329
52.5° 0.127 0.649 0.926 1.056
52.5° 0.130 0.541 0.801 0.983
52.5° 0.134 0.767 1.075 1.138
By using the values of xt = 43 MPa and G = 1100 MPa, the critical SED for the tested PMMA is W2c = 0.84 MJ/m3.
In parallel, the control volume definition via the control radius Rf needs the knowledge of the mode I and mode II critical stress intensity factor KI and KII and the Poisson's ratio v, see Eqs. (3) and (4). For the considered material the resulting values are KIc = 1.87 MPa-m05 and KIIc = = 1.31 MPa-m05 (average values of data reported in [32]) which provide the control radii Rjc = 0.18 mm and R2c = = 0.23 mm, under pure tension and pure shear, respectively. In the absence of precise indications or shear tests, it is possible to approximately estimate the mode II fracture toughness KIIc as a function of KIc, according to Richard et al. [33]
S
K IIc =— K Ic-
(8)
The approach proposed here is a reminiscent of the work by Gough and Pollard [34] who proposed a stress-based expression able to summarize together the results obtained from bending and tension. The criterion was extended in terms of the local SED to V-notches under fatigue loading
in the presence of combined tension and torsion [35] and recently under static loading [36].
In agreement with [35, 36] and extending the method to the static case, the following expression:
W W2
Wie W
= 1
(9)
2c
is obtained. In Eq. (6) and (7) Wjc and W2c are the critical values of SED under pure tension and pure shear. For the considered PMMA, Wjc = 0.83 MJ/m3 and W2c = = 0.84 MJ/m3. Each specimen reaches its critical energy when the sum of the weighted contributions of mode I and mode II is equal to 1, which represents the complete damage of the specimen.
The values of Wj and W2 have, instead, to be calculated as a function of the geometry and of the applied mode ratio
W =
w2 =—
2 E
K 2
ei R5 _
1RC
(10)
(11)
where KI and Kn represent the mode I and mode II SIF ranges, R1c and R2c are the radii of the control volume
Table 4
Experimental results. Overview of the data by using the proposed SED method for the SCB2 specimens
Table 5
Experimental results. Overview of the data by using
W1, MJ/m3 W2, MJ/m3 Wi/Wie + + Wi/Wie (Wi/Wie + the proposed SED method for the TB2 specimens
S, mm + Wi/ Wie)05 S, mm Wi, MJ/m3 W2, MJ/m3 WJ Wie + (Wi/Wie +
20.0 0.793 0.000 0.956 1.073 + Wi/Wie + Wi/Wie)
20.0 0.879 0.000 1.059 1.129 20.0 0.712 0.000 0.858 1.016
20.0 0.969 0.000 1.167 1.186 20.0 0.752 0.000 0.906 1.045
14.0 0.760 0.036 0.958 1.075 20.0 0.879 0.000 1.059 1.129
14.0 0.712 0.031 0.894 1.038 12.5 0.635 0.020 0.788 0.975
14.0 0.793 0.044 1.008 1.102 12.5 0.712 0.031 0.894 1.038
11.5 0.431 0.149 0.696 0.915 12.5 0.673 0.024 0.839 1.005
11.5 0.475 0.123 0.718 0.930 9.5 0.343 0.087 0.517 0.789
11.5 0.494 0.177 0.806 0.985 9.5 0.316 0.079 0.475 0.756
9.0 0.233 0.398 0.754 0.953 9.5 0.316 0.083 0.479 0.760
9.0 0.261 0.491 0.899 1.041 8.0 0.141 0.220 0.432 0.721
9.0 0.281 0.594 1.045 1.122 8.0 0.178 0.276 0.543 0.809
7.5 0.027 0.830 1.020 1.109 8.0 0.159 0.241 0.478 0.759
7.5 0.035 0.963 1.188 1.196 6.5 0.020 0.594 0.731 0.939
7.5 0.055 1.105 1.382 1.290 6.5 0.014 0.491 0.601 0.851
6.0 0.000 1.105 1.315 1.259 6.5 0.027 0.767 0.946 1.068
6.0 0.000 0.963 1.146 1.175 5.0 0.000 1.033 1.229 1.217
6.0 0.000 1.257 1.497 1.343 5.0 0.000 0.963 1.146 1.175
20.0 0.793 0.000 0.956 1.073 5.0 0.000 0.895 1.065 1.133
Fig. 6. Synthesis of the data obtained for PMMA by means of the SED approach
related to mode I and mode III loadings, while e1 and e2 are equal to 0.1186 and 0.3332, respectively.
The detailed calculations employing the proposed criterion are reported in Tables 2-5. The square root of the left-hand side term of Eq. (9), which is, in fact, proportional to the critical load, is given in the last column of these tables.
A synthesis in terms of the square root value of the considered parameter, that is the sum of the weighted energy contributions related to mode I and mode II loading, is shown in Fig. 6 as a function of the critical load. Many of the results are inside a scatter band ranging from °.8 to 1.2 with only few exceptions.
The fracture models proposed in this paper can be used for predicting the onset of brittle fracture in cracked PMMA components which are subjected to a combination of tension and shear loadings. The critical radii evaluated under tension and shear loadings have very close values confirming previously proposed simplified approaches [22, 23] in which the control volume size has been considered the same under tension and shear. The size of control volume is related to the size of critical or the length of the localized damage zone in PMMA which is often considered as a constants material independent of its geometry, loading condition or even mode combinations. The size of control volume determined for the PMMA material of this research was in the typical range of 0.05 to 2.00 mm reported by the researchers for PMMA [8, 11, 12, 37].
5. Conclusions
Pure mode I, pure mode II and different intermediate mixed mode I/II fracture toughness values of commercial PMMA material were obtained experimentally using cracked specimens characterized by four different shapes. All the mixed mode PMMA fracture toughness data were predicted successfully using the SED fracture criterion. In fact, the SED approach has been found a very suitable tool to summarize all the data in a narrow scatter band. The critical radii under pure mode I and pure mode II loadings have been found very close each other confirming the engineering proposal of local mode I developed in previous contributions and used in combination with the SED approach.
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Mohammad Reza Mohammad Aliha, Dr. PhD, Assist. Prof., IUST, Iran, [email protected], [email protected] Filippo Berto, PhD, Prof., University of Padova, Italy, [email protected], [email protected] Aram Bahmani, PhD Stud., IUST, Iran, [email protected] Pasquale Gallo, Prof., Res. Assist., Aalto University, Finland, [email protected]
