Mode I fracture in PMMA specimens with notches – Experimental and numerical studies

Accepted Manuscript Mode I fracture in PMMA specimens with notches – experimental and numerical studies Elżbieta Bura, Andrzej Seweryn PII: DOI: Reference:

S0167-8442(18)30243-X https://doi.org/10.1016/j.tafmec.2018.08.002 TAFMEC 2081

To appear in:

Theoretical and Applied Fracture Mechanics

Received Date: Revised Date: Accepted Date:

22 May 2018 6 July 2018 2 August 2018

Please cite this article as: E. Bura, A. Seweryn, Mode I fracture in PMMA specimens with notches – experimental and numerical studies, Theoretical and Applied Fracture Mechanics (2018), doi: https://doi.org/10.1016/j.tafmec. 2018.08.002

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1 Mode I fracture in PMMA specimens with notches – experimental and numerical studies Elżbieta Buraa*,, Andrzej Seweryn a a

Bialystok University of Technology, Faculty of Mechanical Engineering, Wiejska 45C, 15–351 Bialystok, Poland

*Corresponding author E-mail addresses: [email protected] (E. Bura); [email protected] (A. Seweryn) Abstract: This paper shows the results of experimental and numerical investigation of fracture in PMMA (Polymethyl-methacrylate) plate specimens under mode I. Samples had two thicknesses (9.7 and 18 mm), and they were weakened by two types of edge-notches: rounded V-notch with radius 0.5 mm and U-notch with radius 10 mm. During tests, critical load value, fracture initiation point and fracture initiation angle were obtained. Aditionally, the fracture process was recorded by a highspeed PHANTOM camera. Crack length growth and changes in crack tip velocity over time were analyzed. Stress and strain fields under the critical load value were calculated using Finite Element Method (FEM). Three-dimensional isoparametric elements were used and non-linearity of material and geometry was also included. A stress-strain criterion for notched specimens under tension was proposed. Key words: Crack propagation velocity, Brittle fracture, Ductile fracture, Mode I, V-notches, Unotches 1. Introduction The fracture is a complex phenomenon that depends on: element shape, material type, operating and loading conditions, et cetera. Experimental investigations make it possible to get to know the physical side of fracture and to build adequate numerical models for fracture predictions. Many articles show results of experimental fracture studies, in which PMMA specimens are often used. This polymer is typically brittle at low temperature or under dynamic loading conditions [1-3]. At room temperature, its stress-strain curve is non-linear. In experimental research, plate notched specimens were subjected to axial and biaxial loading. Tests under mode I [4,5], mode II, mixed mode I/II [6-8] and negative mode I [9-13] have been described. Specimens had the form of beams [14,15], compact tension beams weakened by a slot [4], circular discs with centered key-hole [9], semicircular discs [8], flat semicircular discs [10], diagonally loaded square plates [7], bars with edge-notches [6] and many others. These components were usually tested under tension and compression, less often under torsion. Cylindrical specimens were typically used during mode III or mixed mode I/III tests [16-18]. Experimental fracture investigations in plate specimens under torsion were described in [14,19,20]. Aliha et al. [21] described mixed mode I/III during three-point bending tests on a thick disc with a crack. In the cited papers, specimens were usually no more than 14 mm thick. Most of them were weakened by sharp or rounded notches with a small radius. These types of notches caused high stress concentration and quick destruction in the linear elastic range. As a result, the stress-strain curves were close to linear [12]. There is no publication that focuses on the effect of sample thickness. Numerical calculations are an important complement to experimental studies. Distributions of stress and strain fields can be obtained using Finite Element Method (FEM) or Boundary Element Method (BEM). To simplify FEM modeling, a linear elastic material model [13,22] and two-

2 dimensional finite elements [23] were used. More complicated models (containing grips) were built using three-dimensional elements [14,19,27]. Generalized stress intensity factors were usually determined for brittle fracture predictions [24-26]. Numerical calculations and their results are very useful for the creation and verification of fracture criteria. Brittle fracture criteria are used to determine safe working conditions for PMMA. Most of them are based on stress values. Experimental verification of fracture criteria has been described in numerous studies. In [28], the best match between experimental and numerical results for elements with notches under mixed mode I/II was obtained by a modified McClintock’s Stress Criterion [29,30] and the Non-local Stress Criterion by Seweryn and Mróz [31]. The experimental data from [28] was used by Yosibash et al. [32] to verify the Leguillon fracture criterion for a mixed mode loading. Authors received a very good agreement between experimental and theoretical results for V-notched samples. Next, this criterion has been extended to rounded V-notched specimens under a mixed mode loading [33]. Point Stress Criterion and Mean Stress Criterion [34] can be used to predict fracture in plate specimens with a U-notch [19]. In [14], the Maximum Tangential Stress Criterion [35] was used to predict the fracture process in PMMA specimens under mixed mode I/III. In the present paper, the results of experimental and numerical studies of fracture in PMMA plate specimens under tensile load are described. Specimens were weakened by V- and U-notches and were 9.7 and 18 mm in thickness. Mechanical properties of the material were determined during monotonic quasi-static tensile tests on smooth specimens. Special attention was paid to stress-strain curves, which were non-linear (at room temperature). As a result of experimental investigation, the authors obtained the value of critical load, location of the fracture initiation point and fracture initiation angle. The crack propagation process was recorded by a PHANTOM high-speed camera. Changes in crack front velocity over time were analyzed. Furthermore, estimated values of bifurcations in each time step were shown. Distribution of stress and strain fields in specimens were obtained by FEM. Three-dimensional elements and a non-linear material model were used. Maximum values of normal stress and plastic strain under critical loading conditions were determined. A new stress-strain criterion for PMMA specimens was presented. This approach made it possible to predict fracture in PMMA samples, using real characteristic of this material without any replacement concepts. Numerical calculations were based only on the true hardening material curve obtained for the smooth specimen. Assuming this criterion, there is a possibility to investigate a whole set of elements weakened by different notches, without any information about the stress-strain curve for each particular specimen. It should be underlined, that the value of a stress intensity factor KIc is also not required. 2. Test stand and specimens Monotonic tensile tests were carried out on an MTS 809.10 dynamic biaxial testing machine (Fig.1). Specimens were fixed in hydraulic holders with knurled jaws. Tests were conducted under displacement control, which was measured by an INSTRON 2620.601 axial extensometer with measuring base 25±5 mm for notched and 15±5 mm for smooth specimens. Each test was monitored by a PHANTOM v1610/96 monochromatic high-speed camera. The maximum frame rate (1 million frames per second) can be reached in resolution: 128x16 pixels. For better illumination of the working area, a light panel was used. Experiments were conducted at room temperature (20°C). The experimental investigation was carried out using 9.7 and 18 mm thick specimens. Elements were weakened by two types of edge-notches: rounded V-notch with a radius 0.5 mm and U-notch with a radius 10 mm, such that the first was inscribed in the second (Fig. 2). Specimens were manufactured in few steps. Firstly, a water jet machine was used to cut out the blanks from the cast material sheet. Secondly, notches were made on a milling machine. Finally, notch surfaces were polished to remove unnecessary roughness.

3 Samples were made of Polymethyl-methacrylate. It is an isotropic and thermoplastic polymer. Its transparency makes crack observation easier. It is possible to observe the crack propagation process, fracture surfaces, fracture initiation angle and initiation point. Material sheets are obtained using two methods: by extrusion or by pouring molten polymer. This amorphous material is also very sensitive to temperature changes and deformation rate. It is typically brittle at -60° [1] and also during dynamic testing at a deformation rate of approximately 18.61 1/s [2,36]. At a higher temperature and lower rate, non-linear behavior is observed. It changes from brittle to ductile. 3. Experimental studies of fracture in notched specimens Material properties (Tab. 1) were determined during quasi-static tensile tests on a smooth specimen (Fig. 2c). Experiments were conducted under displacement control at a rate of 0.02 mm/s [37]. Also, unloading tests were performed to exclude the PMMA’s non-linear elasticity. The Poisson’s ratio was determined using the ARAMIS 3D 4M non-contact vision measuring system. Elongation was measured in two mutually perpendicular directions. Fig. 3a shows the stress-strain curve. Each specimen was loaded under displacement at a rate of 0.02 mm/s, until the final fracture. Critical load values are given in Tab. 2. Fig. 3 illustrates all stress-strain curves. For V-notched specimens, they are close to linear. Maximum elongation was no more than 0.3 mm (Fig. 3c,d). For specimens with U-notches, the authors obtained non-linear stress-strain curves, for which the maximum elongation was equal to around 1.2 mm. The critical load value was approximately the same as for smooth specimens (Fig. 3a,b). Tab. 1. PMMA mechanical properties. Material property Young’s modulus, E [MPa] Yield stress, R0,2 [MPa] Elastic limit, R0,05 [MPa] Tensile strength, Rm [MPa] Poisson’s ratio, v

Value 3325 45 25 75 0.38

V-notch R=0.5 mm

Tab. 2. The critical load value for smooth and notched specimens. Notch Sample Real dimensions of Nominal sample Critical load type cross-section [mm2] thickness [mm] value [kN] 1-1 20.08×9.87 9.70 14.47 1-2 20.08×9.87 14.31 1-3 20.08×9.87 14.29 4-1 20.00×18.68 18.00 24.60 4-2 20.00×18.68 22.69 4-3 20.00×18.68 26.02 2-1 20.20×9.70 9.70 6.17 2-2 20.20×9.70 5.97 2-3 20.18×9.68 6.27 2-4 20.26×9.72 6.36 5-1 20.45×18.00 18.00 10.06 5-2 20.45×18.02 10.41 5-3 20.23×18.00 10.29

Averaged critical load value [kN] 14.36

24.44

6.19

10.17

U-notch R=10 mm

4 5-4 3-1 3-2 3-3 6-1 6-2 6-3

20.30×18.01 19.98×9.71 19.98×9.69 19.98×9.70 20.02×17.99 20.02×17.99 20.02×17.99

9.70

18.00

9.90 13.86 13.79 13.87 26.71 27.49 25.90

13.84

26.70

Fig. 4 shows examples of destroyed specimens. A small fracture area is observed for V-notched components. The fracture surface is smooth with slight bifurcations which occur near the second notch. U-notched specimens have a significant fracture area with many more bifurcations. During observations of fracture surface, there is an opportunity to unequivocally point to the place of initiation, even if the specimen is strongly damaged. A view from the top of the fracture surface is presented in Fig. 5. It should be noted that the fracture always initiates near the middle of the sample’s thickness. It occurs once in the left and once in the right notch, but always at its root. The fracture initiation point is clearly visible – in Fig. 5 (a small smooth surface) it is marked with a red circle. On SEM images, it can be seen that the place of a fracture initiation is a semi-circular surface which includes a notch root. In the V-notched sample a quasi-regular structure can be observed. The fanshaped scheme of ribs appears near the fracture origin and it continues along almost the whole crosssection (Fig. 5a,b). Only in the vicinity of the opposite notch, the structure changed (Fig. 5c). Not only cleavage but also torn edges can be observed. The increase of a crack propagation rate provokes these changes. A different microstructure is associated with U-notched samples. The fracture starts from an initiation surface which is surrounded by beach marks (Fig. 5d). The remaining part of a cross-section represents an irregular structure, typically ductile. It includes many of torn edges, unordered ribs and dimples (Fig. 5e,f). In [38] authors noted that when a crack develops at higher rates, the temperature increases and softening process can be observed. Figs. 6-7 show the crack propagation process step by step. It should be noted that the fracture initiation angle (the angle between the crack and the horizontal surface of symmetry) is equal to 0°. This is confirmed by the first frame (Figs. 6-7). Obtained recordings provide an opportunity to describe the crack propagation process even if it runs at high speed. The crack in specimens with semi-round notches developed on a curved track. The first significant bifurcation can be observed on the second frame (after about 10 µs), before the crack tip crossed half of the distance between notches. The quite large angle between bifurcations caused enlargement of the fracture area. The fracture process in V-notched specimens developed in a different way. First, bifurcations usually appeared at the end of the process, and they were oriented at an acute angle to the main crack. These differences arise from the energy balance. The U-notch is not a strong stress concentrator, so specimens could work under a much higher value of load. As a result, a greater decrease in potential energy and development of the fracture surface were observed. Three configurations of the PHANTOM camera were used. The angle between the lens axis and normal to the specimen face was equal to: 0°, 10° and -15°. The last two angles permitted observation of the crack propagation process over the specimen’s thickness (Figs. 6b, 7b). Pictures show the fracture initiation point precisely. It starts from the middle of the element’s thickness and then develops simultaneously along and across the thickness (with a characteristic arch Fig. 7b). After around 10 µs, the fracture surface spreads throughout the specimen’s entire thickness and from that moment, it propagates evenly. It must be said that a one photo was taken by the PAHTOM camera before the start of the loading process. This was really useful for picture scaling. The process of preparing photographs is shown in Fig. 8. ImageJ open-source software was used. First of all, the unit was changed from pixels to millimeters. Secondly, picture quality was improved by maximum contrast. Next, the image was converted to a binary form, and lastly, the main crack

5 was revealed by the skeleton option. Coordinates of pixels were pointed out in photographs like one in Fig. 8e. In every set, points which corresponded to the main crack were distinguished (Fig. 9). For recording the fracture process in V-notched specimens, a frequency equal to 221 052 frames per second was used. Therefore, the time between successive frames was equal to 4.52 µs. For Unotched specimens, pictures were taken at a rate of 200 000 frames per second, therefore the time between consecutive frames was equal to 5 µs. The crack’s length (Fig. 10) and its propagation velocity (Fig. 11) over time were described. The graph of crack length growth can be approximated by a second-degree polynomial. It is important to underline that the increase in crack length is faster over time. Due to the fact that the start of recording was not synced to the initiation of the crack, the first and the last frame were eliminated. The section of the curve which corresponded to the first frame was shown with a broken line. The time delay was obtained by extrapolation. Based on the behavior of the curve in later frames, the value of the first time step was estimated. Averaged crack tip velocity was calculated (Fig. 11). The value of the crack tip velocity in every time step was represented by only one point, which was placed in the middle of the time section. The crack tip velocity over time for V-notched specimens is shown in Figs. 11a,b. Moreover, these figures show the estimated number of bifurcations in every time step. It is clear that they can cause a drop in fracture speed. The crack in V-notched specimens propagated at a speed of about 670 m/s. The maximum value of velocity was reached at the end of the process, and it was equal to about 850 m/s (9.7 mm in thickness) and 800 m/s (18 mm in thickness). Much higher speeds can be observed for U-notched specimens (Fig. 11c,d). The averaged crack tip velocity is equal to 870 m/s. This is much more than for V-notches (around 200 m/s). For elements with a thickness of 9.7 mm, the crack tip velocity reached a value between: 950-1105 m/s, and for 18 mm in thickness between: 800-1120 m/s. Thus, it can be concluded that as the notch radius increases, the load value is higher and more potential energy is also allocated to the development of the fracture surfaces. It should be noted that values of velocity for thin specimens were lower than for thick elements. In addition, it should be underlined that, for specimens with V-notches there are no or a small number of bifurcations on three first images. A different situation was observed for U-notched samples. The first bifurcation appears at the first time step. Tab. 3 presents estimated numbers of bifurcations for the first three steps. Tab. 3. The average number of bifurcations at the first three time intervals. Time interval Average number of bifurcations V-notched specimens U-notched specimens 0 – 5 µs 0-1 1-2 5 - 10 µs 1-2 3-5 10 – 15 µs 1-4 3-8 4. Numerical studies of stress and strain fields distribution Numerical calculations of stress and strain fields were conducted by the Finite Element Method (MSC MARC software). Only 1/8 of the specimen was used because of symmetry of loading and geometry. Boundary conditions are shown in Fig. 12. In the FEM model, grips were not included. the authors conducted numerical calculations on a finite element model which consisted of the specimen and holders. These calculations showed a negligible impact of grips on the distribution of stress and strain fields near the notch root. This fact made it possible to simplify calculations. The specimen model was loaded by using a reference link (multi-point constraints) connecting nodes from the top surface of the specimen to one point (Fig. 17). Only 1/4 of the experimental critical load value was used. Firstly, average stress was counted accounting for the real dimensions of the element. Next,

6 using the final size of the numerical model, the numerical value of critical load was obtained (Tab. 4). Three-dimensional and isoparametric HEX-8 finite elements were used. Finite element meshes were respectively condensed near the notch root (Fig. 13). To describe the material’s behavior, an elastic-plastic material model with isotropic hardening and Huber-von Mises condition of plasticity was used. The material model was set up according to the actual hardening curve, which was determined by a hybrid method (experimental-numerical) [39]. It should be added that the real stress-strain curve was obtained by FEM calculations based on the experimental relationship between load and elongation. The actual hardening curve of the material was determined during iterative calculations. Originally, the nominal hardening curve was predicted, and then the value of load and displacement were counted. Secondly, these values were compared with experimental data. Next, the curve was modified, and calculations were not terminated until the numerical load-displacement curve was fitted to the real hardening curve. The above activities made it possible to calculate maximum values of the principal stress and principal plastic strain (σ1, ) as well as the equivalent stress and equivalent plastic strain (σeq, ) in accordance with the Huber-Von Mises hypothesis. Based on the engineering stress-strain curve for a smooth specimen, the yield stress R0.2 and elastic limit R0.05 were obtained (Tab. 1). In the case of the approach assuming that plastic strains appear beyond the yield point (R0.2=45 MPa), major discrepancies between numerical and experimental results were obtained, especially for notched specimens. Therefore, the approach assuming that plastic strains can appear instantly after the elastic limit (R0.05=25 MPa) was adopted. A good agreement between numerical and experimental results was reached (Fig. 3). Tab. 4. Load F [kN] values for MES calculations. Nominal sample thickness [mm] 9.70 14.55 18.00 27.00

Notch type V-notch 5.82 10.80

U-notch 14.16 26.28

Based on the FEM results, distributions of the maximum principal stress σ1 (Fig. 14), equivalent stress σeq (Fig. 15), maximum principal plastic strain (Fig. 16) and equivalent plastic strain (Fig. 17) under critical loading conditions were obtained. Maximum values of these quantities are given in Tables 5-6. It should be noted that, as the notch radius decrease, stress concentration is higher. In the V-notch, the maximum value of normal stress σ1 is equal to 100 MPa, but in the Unotch, it is equal to around 83-86 MPa (close to the maximum value of normal stress in the smooth specimen). As the notch radius grows, the area of plasticity also expands (big fracture zones in real elements). Plastic strain reaches a maximum value of 3.9% for V-notched and 4.2% for U-notched specimens. The biggest area of plasticity was observed in a smooth specimen, where plastic strain is equal to around 6%. Figs. 14-17 present stress-strain field distributions over specimen thickness g. It should be underlined, that their maximum values are observed in a plane of symmetry. Tab. 5. The location of point with the maximum stress value σ1, σeq and corresponding values of plastic strain , at fracture initiation moment. Notch Specimen Measuring point Stress [MPa] Plastic strain [%] type thickness [mm] location σ1 σeq smooth V-notch

9.70 18.00 9.70 18.00

symmetry axis 0.13 mm from the notch root

82.40 84.80 98.97 99.36

81.70 81.16 78.54 78.46

5.96 5.66 1.72 1.65

5.96 5.66 4.20 4.15

7 U-notch

9.70 18.00

1.62 mm from the notch root

83.08 86.56

78.62 78.26

2.71 2.53

4.24 4.04

Tab. 6. The location of point with the maximum plastic strain value , and corresponding values of stress σ1, σeq at fracture initiation moment. Notch Specimen Measuring point Stress [MPa] Plastic strain [%] type thickness [mm] location σ1 σeq 81.70 smooth 9.70 82.40 5.96 5.96 symmetry axis 18.00 84.80 81.16 5.66 5.66 V-notch 9.70 92.53 78.54 3.92 4.20 the notch root 18.00 92.63 78.46 3.83 4.15 U-notch 9.70 80.04 78.62 4.24 4.24 the notch root 18.00 83.48 78.26 4.01 4.04 Fig. 18a presents the normal stress as a function of the distance from the notch root (measurements were taken at the middle of the specimen’s thickness). It was found that, as the notch radius increased, the point of maximum stress value σ1 was closer to the specimen’s axis of symmetry. The distance between the measuring point and the notch root is approximately equal to: 0.13 mm for Vnotch and 1.62 mm for U-notch. In specimens without notches, normal stress σ1 reaches its maximum value exactly at the plane of symmetry, and it is close to the normal stress value for U-notched specimens. In V-notched elements, a sharp increase of the normal stress value can be observed. It appears near the notch root, while in the rest of the area, the stress value reaches around 20 MPa (Fig. 18a). The highest equivalent stress σeq occurs in the notch root (in notched specimens). For smooth samples, equivalent stress σeq is constant (Fig. 18b). For every type of specimen, equivalent stress σeq in the notch root is equal to around 80 MPa. Fig. 18c shows the plastic strain distribution as a function of the distance from the notch root. For notched specimens, the maximum value of plastic strain always occurs in the notch tip (around 4%), but for smooth specimens, it occurs on their axis (around 6%). The same trend was observed for both specimen thicknesses. In V-notched specimens, the average stress value is close to the yield point. Plastic strains only appear at the notch root. Maximum differences between the two element thicknesses do not exceed 3% for strain and 5% for stress . In V-notched specimens, these differences are insignificant. Based on FEM results, it was observed that distributions of maximum principal plastic strain and equivalent plastic strain are the same (Figs.18c,d). Distributions of maximum principal stress σ1 and equivalent stress σeq across the specimen’s thickness are presented in Figs. 19-20. The r/g ratio is the quotient of the distance from the specimen’s edge and its thickness. All quantities were measured at the notch root (Fig. 19) and also at the point of the maximum normal stress value σ1 (Fig. 20). It was observed that maximum values of stress σ1 and strain are always located at the middle of the specimen’s thickness. It should be noted that, as the sample’s thickness increases, the value of plastic strain decreases (maximum relative difference is around 5%). For the smaller notch radii, higher plastic strain gradients near the notch root were observed. Maximum principal plastic strain and equivalent plastic strain at the center of the sample’s thickness are equal to around 4%. Normal stress σ1 reaches a constant value at coefficients: r/g>0.15 (V-notched samples) r/g>0.3 (smooth and U-notched samples). Linear or close to linear curve sections can correspond to real fracture initiation points. In Fig. 19, fracture initiation points obtained in the experiment are marked. The distribution of stress and strain presented in Fig. 20 corresponds to the point of the maximum normal stress value σ1. Measurements were taken at a certain distance from the notch root, which

8 was equal to: 0.13 mm for V-notches, 1.62 mm for U-notches and 10 mm for smooth specimen. The highest values of stress and strain always occur at the center of the specimen’s thickness. At a certain value of the r/g ratio, all quantities become constant (in the case of strain , and equivalent stress σeq at: r/g>0.1 and in case of normal stress σ1: r/g>0.3). 5. Fracture criteria In the case of an approach assuming that PMMA is a typically brittle material with a linear stressstrain curve, the fracture initiation should be predicted by using traditional brittle fracture criteria. Novozhilov [34] proposed one of the most important criterion called the Mean Stress (MS). According to this formula, the brittle fracture occurs when the average normal stress over some distance reaches a critical value. This criterion was used to predict the fracture in samples with sharp Vnotches [5], blunt V-notches [40] and also U-notches [41]. The Maximum Tangential Stress (MTS) criterion is also a well-known. In its first form [35] it was used to predict the brittle fracture in components with sharp cracks. According to the MTS, the brittle fracture occurs when the tangential stress over some distance reaches a critical value. Modification of this criterion allows to use it for fracture predictions in rounded V-notched samples [23]. The problems occur when a stress-strain material curve is a non-linear and plastic strain appears. There is many of criteria for materials which fail under elastic-plastic conditions but only in few works they were used for notched polymers like PMMA. In [42] Torabi proposed a new method called The Equivalent Material Concept (EMC). It is based on a virtual brittle material which is described by a perfectly linear stress-strain curve. According to this theory, the stress energy density is equal for both of material models: ductile and brittle, which were described by the same value of elastic modulus. This concept allows to predict the ductile fracture without non-linear numerical analysis. The EMC was connected with MTS and MS criterion and then was applied to ductile polymeric materials with U-notches [43] and also to the ductile polymer-based nanocomposite weakened by V-notches [44]. The main issue using these criteria is to estimate critical distances. Assuming the MTS criterion [45]: (1) and MS criterion [40]: (2) The main problem of these two criteria above is that, the value of the stress intensity factor KIc is necessary. There is an inevitability to conduct special experimental tests during which KIc will be determined. Papers [40-45] do not give any tips for situations when the value of the stress intensity factor KIc is unknown. This is a reason why EMC-MTS and EMC-MS criteria can not be used to predict a fracture in elements which were described in this paper. A problem with unknown KIc was solved in [46]. In this paper authors described the procedure of an implementation a Non-local Criterion by Seweryn and Mróz [31] to graphite samples under tension and torsion. It is assumed that the fracture initiate when the normal stress value on some special distance reach a critical value. The calculation should be carried out using a linear elastic material model. This special distance is a size of fracture zone and it can be determined using: (3) Authors given an opportunity to find d0 in other – numerical way, described in [46]. It is really important that this criterion is also useful even if value of KIc is unknown.

9 5.1. The new stress-strain fracture criterion for plate PMMA specimens witch notches under tension The fracture initiation point was clearly identified at the notch root. The results of numerical studies showed that the maximum value of normal stress σ1 was not reached there, but at a certain distance from the notch tip. In this case, predicting the fracture process using local criteria is unjustified. Plastic strain is a significant factor which determines fracture behavior and causes damage initiation. In Fig. 21, the relationship between the critical value of normal stress σ1 (at the initiation moment) and the maximum value of plastic strain is shown. Values from two different measuring points were taken into account: ○ point of the maximum normal stress value σ1 and ● point of the maximum plastic strain value or . It can be concluded that, for predicting the fracture process, the criterion based on the stress-strain relationship should be used (Fig. 21). The stress immediately causes fracture and plastic strain, driving down the material’s fracture resistance. The stress-strain criterion can be presented in a form similar to the Kachanov [47] proposition: (4) where σc is the failure stress in undamaged material (ω=0), and x0 is a vector defining the place of crack initiation. The damage variable ω can be dependent on the maximum principal plastic strain value, that is: (5) where – critical value of plastic strain, or it can be also dependent on the Huber-von Mises equivalent plastic strain value: (6) This stress-strain fracture criterion Eq. (4)-(6) is based on the assumption that fracture initiation takes place when the maximum normal stress reaches a critical value at one of two points: 1- point at the notch tip which corresponds to the maximum plastic strain value or 2- point at a certain distance from the notch tip which corresponds to the maximum normal stress value. This critical value is dependent on the maximum principal or equivalent plastic strain. It is clear that knowledge about stress and strain field distributions is necessary. They should be calculated during non-linear numerical studies which include an approach assuming that plastic strain occurs above the elastic limit R0.05. Critical values of normal stress and plastic strain for three types of specimens (V-notched, Unotched and smooth) were calculated based on empirical and numerical data (Fig. 21a,b). Obtained results:  for equation (4)-(5): σc=98.89 MPa, =0.349;  for equation (4)-(6): σc=98.69 MPa, =0.360. In order to predict the fracture initiation in V-, U-notched or smooth specimens made of PMMA using a new stress-strain fracture criterion, several steps should be done. Firstly, the true stress-strain material curve must be determined by the assumption that, the yield stress is equal to the elastic limit R0.05 for this material. Next, the numerical analysis must be carried out using an elastic-plastic material model and an arbitrary load value. The stress-strain field distributions should be obtained. Lastly, two critical points should be checked. Values of the maximum normal stress and the plastic strain (or an equivalent plastic strain) from the notch bisector points: 1- at the notch root and 2- at a certain distance from the notch root (which corresponds with maximum value of a normal stress) should be applied to Eq. (4). The worst case is significant.

10 It should be noted that similar values were obtained during ductile fracture prediction in cylindrical EN-AW 2024 aluminum specimens with notches, which are presented in papers [48,49]. PMMA is a completely different material than aluminum. In these two cases, the proposed stress-strain criterion in the form of Eq. (4)-(5) or Eq. (4)-(6) gives good results (under axial and monotonic tensile loading). Under critical load values, higher values of normal stress σ1 in V-notched specimens can be observed. Under the same conditions in U-notched elements, a higher plastic strain ( , ) was reached. Table 7 shows the results of experimental verification of the proposed criterion. The maximum value of a relative error was reach for U-notched specimens. It does not exceed 9%. For the rest of examined components, the results stayed with a good agreement with the experimental data (0.55.7%). As it was mentioned earlier, the biggest problem using a well-known fracture criterion like MTS or MS criterion, was an unknown value of KIc. Authors decided to used a Non-local Stress Criterion in the form proposed in [46]. The distribution of stress and strain fields was obtained using a linear elastic material model (a mesh model was the same like in the Section 4). The d0 parameter was determined for V-notched specimen according to a procedure proposed in [46] and the values of σc was taken from smooth samples. The obtained results were stored in Tab. 7. It should be noted, that this criterion predicts really good a fracture in smooth or V-notched samples, the relative error does not exceed 3%. Definitely, it can not be used for fracture predictions in U-notched samples (error around 40%). These results shows that non-local criteria are dedicated for specimens with small plastic zones. In case, when the area of a plasticity is significant (like for U-notched specimen) the process of averaging normal stress over some distance has no use. Tab. 7. The maximum values of normal stress and relative errors. Specimen

smooth

Thickness [mm]

9.70

18

U-notched 9.70

V-notched

18

9.70

18

New stress-strain criterion

98.89 MPa 0.349

Eq. (4)-(5)

[%]

point 2

99.37

101.21

90.07

93.33

104.10 104.29

point 1

99.37

101.21

91.11

94.32

104.24 104.05

point 2

0.49

2.35

8.90

5.62

5.27

5.46

point 1

0.49

2.36

7.89

4.62

5.41

5.23

Non-local stress criterion [46]

75.59 MPa – 9.7 mm 75.42 MPa – 18 mm (d0=2.17 mm - 9.7 mm) (d0=2.33 mm - 18 mm) [%]

73.26

72.91

105.08 107.26

76.02

74.51

3.08

2.36

39.01

0.57

0.21

43.65

11 6. Conclusions In this paper, the results of experimental investigation of the fracture in plate PMMA specimens with two types of notches and also with two thicknesses are presented. As a result, stress and strain field distributions were obtained. Summarizing the result of experimental and numerical investigations, the following conclusions can be listed: 1. Experimental studies of the fracture in plate PMMA specimens with two types of notches showed that Polymethyl-methacrylate has a non-linear behavior at room temperature. In notched specimens under tension, the fracture initiation point was indicated on the real fracture surface. It always occurred in the notch root, near the middle of the specimen’s thickness. The fracture initiation surface corresponds to the horizontal symmetry plane of notches. 2. Recording of the fracture process using a high-speed PHANTOM camera made it possible to determinate the crack tip velocity, its trajectory and the total number of bifurcations. The crack propagation speed is dependent on the amount of potential energy, which is expended to create new fracture surfaces. In specimens with a small notch radius, the crack propagated at a rate of around 670 m/s. In elements with a bigger notch radius, the crack developed at higher velocities – the average value was around 870 m/s. A significant drop in potential energy provokes an increase of surface energy. This caused a large value of bifurcations and a big damage area. In Vnotched specimens, the first significant bifurcation appeared near the second notch. It should be noted that crack propagation velocity was also dependent on the specimen’s thickness. In thick elements, the fracture process developed more slower. 3. Finite Element analysis allowed for determination of the effect of specimen thickness. In order to achieve high consistency between experimental and numerical results, the non-linearity of PMMA was taken into account. A new approach was taken, based on the assumption that plastic strains appear not above the yield stress but above the elastic limit. It was mentioned that as the notch radius gets smaller, the value of the maximum principal stress σ1 increases. The highest value of the plastic strain always occurred at the notch root, but the maximum value of stress - at different points, which are located at a certain distance from the notch root. It was: 0.13 mm for V-notched specimens and 1.62 mm for U-notched specimens. It should be noted that, when notch radius increases, the influence of thickness become more obvious. The biggest differences in stress and strain values, which are caused by different specimen thicknesses, are observed in smooth samples. 4. Presented results of experimental and numerical fracture investigations in plate notched PMMA specimens under tension show that both the maximum normal stress and plastic strain have a strong influence on this process. A stress-strain fracture criterion was proposed. It is based on the assumption that crack initiation takes place when the maximum normal stress reaches a critical value, which is dependent on the maximum principal or equivalent plastic strain. 5. It is necessary to carry out experimental and numerical studies of fracture in plate notched specimens with different thicknesses under biaxial loading conditions (mixed mode I/II or mixed mode I/III). This will allow for making the proposed fracture criterion more accurate and carrying out detailed experimental verification. Acknowledgements The numerical and experimental studies described in this paper are a part of research project No. S/WM/4/2017 at a Bialystok University of Technology. References

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13 [22] M.R. Ayatollahi, A.R. Torabi, M. Firoozabadi, Theoretical and experimental investigation of brittle fracture in V-notched PMMA specimens under compressive loading, Eng. Fract. Mech. 135 (2015) 187-205. [23] M.R. Ayatollahi, A.R. Torabi, Investigation of mixed mode brittle fracture in rounded-tip Vnotched components, Eng. Fract. Mech. 77 (2010) 3087-3104. [24] A. Seweryn, J. Zwoliński, Solution for the stress and displacement fields in the vicinity of a Vnotch of negative wedge angle in plane problems of elasticity, Eng. Fract. Mech. 44 (1993) 275281. [25] M.P. Savruk, A. Kazberuk, Problems of fracture mechanics of solid bodies with V-sharped notches, Mat. Sci. 45 (2009) 162-180. [26] M.P. Savruk, A. Kazberuk, Antisymmetric stress distribution in an elastic body with a sharp or a rounded V-sharped notch, Mat. Sci. 46 (2011) 711-722. [27] M.R. Ayatollahi, Karo Sedighiani, A T-stress controlled specimen for mixed mode fracture experiments on brittle materials, Eur. J. Mech. A-Solid. 36 (2012) 83-93. [28] A. Seweryn, A. Łukaszewicz, Verification of brittle fracture criteria for elements with V-shaped notches, Eng. Fract. Mech. 69 (2002) 1487-1510. [29] F.A. McClintock, Ductile fracture instability in shear, J. Appl. Mech. 25 (1958) 582-587. [30] R.O. Ritchie, J.F. Knott, J.R. Rice, On the relation between critical tensile stress and fracture toughness in mild steel, J. Mech. Phys. Solids, 21 (1973) 395-410. [31] A. Seweryn, Z. Mróz, A non-local stress failure condition for structural elements under multiaxial loading, Eng. Fract. Mech. 51 (1995) 955-973. [32] Z. Yosibash, E. Priel, D. Leguillon, A failure criterion for brittle elastic materials under mixedmode loading, Int. J. Fract. 141 (2006) 291-312. [33] E. Priel, Z. Yosibash, D. Leguillon, Failure initiation at a blunt V-notches tip under mixed mode loading, Int. J. Fract. 149 (2008) 143-173. [34] V. Novozhilov, On a necessary and sufficient criterion for brittle strength, J. Appl. Math. Mech. 33 (1969) 212-222. [35] F. Erdogan, G. Sih, On the crack extension in plates under plane loading and transverse shear, J. Basic. Eng. 85 (1963) 519-527. [36] Hengyi Wu, Gang Ma, Yuanming Xia, Experimental study od tensile properties of PMMA at intermediate strain rate, Mat. Lett. 58 (2004) 3681-3685. [37] ISO 527-1:1993, Plastics - Determination of tensile properties, British Standard. [38] E. Ghorbel, I. Hadriche, G. Casalino, N. Masmoudi, Characterization of Thermo-Mechanical and Fracture Behaviors of Thermoplastic Polymers, Materials 7 (2014) 375-398. [39] L. Derpenski, A. Seweryn, Experimental research into fracture of EN-AW 2024 and EW-AW 2007 aluminum alloy specimens with notches subjected to tension, Exp. Mech. 51 (2010) 10751094. [40] M.R. Ayatollahi, A.R. Torabi, Brittle fracture in rounded-tip V-shaped notches, Mater. Des. 31 (2010) 60-67. [41] F.J. Gomez, G.V. Guinea, M. Elices, Failure criteria for linear elastic material with U-notches, Int. J. Fract. 122 (2006) 1-21. [42] A.R. Torabi, Estimation of tensile load-bearing capacity of ductile metallic materials weakened by a V-notch: the equivalent material concept, Mater. Sci. Eng. A. 536 (2012) 249-255. [43] A.R. Torabi, A.S. Rahimi, M.R. Ayatollahi, Tensile fracture analysis of a ductile polymeric material weakened by U-notches, Polym. Test. 64 (2017) 117-126. [44] A.R. Torabi, A.S. Rahimi, M.R. Ayatollahi, Fracture study of a ductile polymer-based nanocomposite weakened by blunt V-notches under mode I loading: Application of the Equivalent Material Concept, Theor. Appl. Fract. Mech. 94 (2018) 26-33. [45] M.R. Ayatollahi, M.R.M. Aliha, Mixed mode fracture analysis of polycrystalline graphite - a modified MTS criterion, Carbon 46 (2009) 1883-1896.

14 [46] L. Derpenski, A. Seweryn, F. Berto, Brittle fracture of axisymmetric specimen with notched made of graphite EG0022A, Theor. Appl. Fract. Mech. 89 (2017) 45-51. [47] L.M. Kachanov, Introduction to Continuum Damage Mechanics, Martinus Nijhoff Publishers, Dordrecht 1986. [48] L. Derpenski, A. Seweryn, Ductile fracture of EN-AW 2024 aluminum alloy specimens with notches under biaxial loading. Part-2 Numerical research and ductile fracture criterion, Theor. Appl. Fract. Mech. 84 (2016) 203-214. [49] L. Derpenski, A. Seweryn, Ductile fracture criterion for specimens with notches made of aluminum ally EN-AW 2024, J. Theor. Appl. Mech. 54 (2016) 1079-1093.

15

Fig. 1. Testing stand configurations: a) grips with a specimen; b) testing machine with camera; c) testing machine with ARAMIS 3D 4M. Sings: 1- specimen; 2- MTS 809.10 testing machine; 3PHANTOM camera V1610/93; 4- axial extensometer INSTRON 2620.60; 5- light panel; 6- noncontact measuring system ARAMIS 3D 4M.

Fig. 2. Specimens a) V-notched, b) U-notched, c) smooth. Dark areas correspond to volume fixed in grips.

16

Fig. 3. Experimental and numerical stress-strain curves: smooth specimens a) 9.7 mm b) 18 mm thick; V-notched specimens c) 9.7 mm d) 18 mm thick and U-notched specimens e) 9.7 mm f) 18 mm thick.

17

Fig. 4. Specimens with notches after tests (signs like in the Tab. 2).

Fig. 5. Fracture surfaces: V-notched (1-2) and U-notched (3-4) specimens. SEM images of the material fracture surface structure, taken at the selected points: a, b, c (V-notched sample) and d, e, f (Unotched sample).

18

Fig. 6. Crack propagation in U-notched specimens: a) 18 mm thick, recording angle 0° b) 9.7 mm thick recording angle -15°.

19

Fig. 7. Crack propagation in V-notched specimens: a) 18 mm thick, recording angle 0° b) 9.7 mm thick, recording angle -15°.

20

Fig. 8. The process of preparing pictures: a) input image; b) unit conversion; c) picture quality correction; d) binary conversion; e) skeleton implementation.

Fig. 9. An example of a main crack revelation process (V-notched specimen - 9.7 mm).

Fig. 10. The crack length increase over time for V-notched specimens: a) 9.7 mm b) 18 mm and for U-notched specimens: c) 9.7 mm d) 18 mm.

21

Fig. 11. The crack propagation velocity over time and the total number of bifurcations for V-notched specimens: a) 9.7 mm b) 18 mm and for U-notched specimens: c) 9.7 mm d) 18 mm.

Fig. 12. Boundary conditions for the V-notched specimen.

22

Fig. 13. Loading conditions and finite elements meshes.

Fig. 14. The distribution of the maximum principal stress σ1 (smooth specimens – a, b; U-notched specimens – c, d; V-notched specimens – e, f).

23

Fig. 15. The distribution of the equivalent stress σeq (smooth specimens – a, b; U-notched specimens – c, d; V-notched specimens – e, f).

Fig. 16. The distribution of the maximum principal plastic strain notched specimens – c, d; V-notched specimens – e, f).

(smooth specimens – a, b; U-

24

Fig. 17. The distribution of the equivalent plastic strain specimens – c, d; V-notched specimens – e, f).

(smooth specimens – a, b; U-notched

Fig. 18. The distribution of maximum principal stress σ1 (a), equivalent stress σeq (b), maximum principal plastic strain (c) and equivalent plastic strain (d) as a function of the distance from the notch root.

25

Fig. 19. The distribution of maximum principal stress σ1 (a), equivalent stress σeq (b), maximum principal plastic strain (c) and equivalent plastic strain (d) as a function of the specimen thickness. It was measured at the notch root and at the specimen’s edge (smooth element).

26

Fig. 20. The distribution of maximum principal stress σ1 (a), equivalent stress σeq (b), maximum principal plastic strain (c) and equivalent plastic strain (d) as a function of the specimen thickness. It was measured at the point of the maximum normal stress value σ1.

27

Fig. 21. The maximum value of normal stress σ1 as a function of the maximum principal plastic strain (a) or equivalent plastic strain (b).

28 Highlights:  The experimental investigation of fracture in PMMA (Polymethyl-methacrylate) plate specimens under mode I was carried out.  The stress and strain field distribution under the critical load value was calculated using Finite Element Method (FEM).  The stress-strain criterion for notched specimens under tension was proposed.