Compressive Brittle Fracture Prediction in Blunt V-Notched PMMA Specimens by Means of the Strain Energy Density Approach

The paper aims to examine the suitability of the strain energy density criterion in predicting the fracture behavior of blunt V-notched specimens under compression load. Recent studies on local stress fields have shown that the strain energy density, averaged over a specific control volume which embraces the notch round border, could be a robust parameter in the brittle fracture assessment of several materials. A set of experimental results recently published in the literature on compressive brittle fracture of V-notched specimens made out of polymethyl methacrylate has here been considered. Finite element analyses have been performed on plane strain condition and experimental data have been summarized by means of the SED criterion. It has been shown that the proposed criterion permits a satisfactory evaluation of the fracture load of polymethyl methacrylate specimens weakened by notches having different opening angles and radii.


INTRODUCTION
The two main groups of stress concentrators in components and structures are cracks and notches. A literature survey indicates that brittle fracture investigations in components containing stress raisers under compression are limited to notched members. This is probably because the closure phenomenon takes place in cracked members under compression and, the faces coming in contact with each other, little chance of crack propagation is allowed. Conversely, in a notched member, particularly in V-and U-shaped notches, the faces may or may not come in contact with each other, depending on the gap value and the maximum face displacement at the onset of failure, and the compressive damage may nucleate and lead to the final fracture.
Studies on notched components under tensile and shear loading conditions have been widely performed during the past two decades, see a comprehensive list of references in Refs. [1,2]. However, it is of crusial importance to develop appropriate failure criteria for structures containing various shaped notches, such as V-, U-and O-notches, under compressive loading conditions to bridge the current gap on the topic and prevent the onset of sudden fracture.
The first use of the notch fracture mechanics principles in compressive brittle fracture prediction of notched components has probably been made in [3]. They have recently investigated the fracture behavior of graphite specimens weakened by double V-notches with end holes (VO-notches) both experimentally and theoretically by means of the average strain energy density (SED) criterion. Torabi and Ayatollahi [4] have successfully predicted the test results reported in [3] by means of the two brittle fracture criteria, known as the point-stress (PS) criterion and mean-stress (MS) criterion. The most recent works on compressive brittle COMPRESSIVE BRITTLE .  [5,6] in which the PS and MS fracture criteria have been successfully employed for predicting the fracture loads of two new Vnotched test specimens, called the V-notch stepped cottage (VSC) and the flattened V-notched semidisk (.VSD) specimens, made of polymethylmethacrylate (PMMA), for various notch angles and different notch radii.
Since the extension of the brittle fracture criteria under tension and shear loading conditions to compressive loading conditions has taken place in the most recent years, their validity should strongly be verified by means of various experimental results on different brittle materials and test specimens. The attempt made here is to examine the suitability of the average strain energy density (SED) criterion in predicting the fracture load of PMMA specimens. The criterion states that brittle fracture occurs when the strain energy density, averaged in a specific control volume, which includes a crack or notch tip, reaches a critical value dependent on the material. Good agreement has been found between numerical predictions and the experimental results, demonstrating that the proposed criterion works well not only on compressive brittle frac-ture of VO-notched graphite specimens, but also on that of blunt V-notched PMMA specimens.

EXPERIMENTAL RESULTS .ROM THE LITERATURE
A set of new experimental results have been recently published by Ayatollahi et al. [5] dealing with brittle fracture in blunt V-notches under compression. The details about specimens and test procedures have been reported in the following subsections.

Material
Polymethylmethacrylate (PMMA) has been used for conducting fracture experiments at room temperature on blunt V-shaped notches under compression. The essential mechanical properties of the adopted PMMA at room temperature have been taken from the experiments performed earlier by Ayatollahi et al. [5], according to the standard ASTM D695-10, and are shown in Table 1. In .ig. 1 the true stress-strain curve of the PMMA sample tested under compression is depicted.

Specimen
The adopted specimen, named V-notch stepped cottage (VSC) specimen, is schematically shown in .ig. 2. It has firstly been proposed and utilized in Ref. [5] for conducting compressive fracture experiments on blunt V-notches. In .ig. 2, the parameters 2α, ρ, a, b, c, d, e, L, t and P are respectively the notch opening angle, notch radius, notch length, flattened length, step height, step width, specimen height, specimen length, specimen thickness and the compressive load.
Due to the bending effects, tensile stresses are unavoidably generated at some points far enough from the notch tip. By making a step on the top and bottom of the specimen and applying the compressive load to .ig. 1. A sample true stress-strain curve of the PMMA under compression according to the results of [5]. the remaining edge, the notch surroundings experience pure compressive deformations which could be significantly higher compared to tensile ones. The particular shape of the proposed specimen is then a consequence of the need to make sure to have compressive fracture from the notch border before tensile fracture from other locations of the specimen. Because the compressive strength of brittle materials is usually two to three times greater than the tensile one, it was necessary to adopt VSC specimens having dimensions so that the level of the compressive stresses at the notch tip was several times greater than the maximum tensile stress in the whole specimen. The parameters above introduced have been chosen as 2α = 30°, 60° and 90°; ρ = 0.5, 1.0 and 2.0 mm; a = 10 and 25 mm; b = 30 mm; c = 20 mm; d = 5 mm; e = 85 mm; L = 110 mm and t = 6 mm. Note that these dimensions have also been chosen in order to make sure that buckling would not take place during the experiments. Tested specimens have been fabricated by laser cutting from a PMMA plate of 6 mm thick. The cut surfaces were accurately polished by means of appropriate abrasive papers to remove possible defects due to the cutting process. The strain rate in the tests was set to 0.000075 1/s providing quasistatic monotonic loading conditions. Three specimens have been tested for each geometry in order to check the repeatability of the test. Totally, 54 fracture experiments have been carried out.  Table 2 summarizes the experimentally recorded fracture loads. It is possible to point out that all the fracture loads are less than 18 kN and most of them are less than 13 kN. No evidence of buckling has been observed.

Test Results
A sample measured load-displacement curve is shown in .ig. 3. A linear trend has been found up to a load peak, after which a sudden fall to zero occurs, confirming the predominantly linear elastic behavior of the specimens up to the final failure. .or this reason, the experimental results can potentially be predicted by means of brittle fracture criteria based on linear elastic notch mechanics principles, e.g. the averaged SED criterion.
In the forthcoming sections, the SED averaged over a specified control volume which embraces the notch border is presented and adopted to predict the measured fracture loads summarized in Table 2.

AVERAGED STRAIN ENERGY DENSITY APPROACH
The averaged strain energy density criterion (SED), according to [7], states that brittle failure occurs when the mean value of the strain energy density over a given control volume , W reaches the critical value c . W In agreement with [8], named c W the ultimate compression strength under elastic stress field conditions and E the Youngs modulus of the material, the critical value of the total strain energy can be determined by the following: This critical value varies from material to material but it does not depend on the notch geometry and sharpness.
Under plane strain condition and tension loading, the following expression has been proposed by Lazzarin and Zambardi [7] to evaluate the control volume dimension 0 : R where Ic K is the material fracture toughness, ν is the Poissons ratio and t σ is the ultimate tensile strength of an unnotched specimen. Under compressive load, Eq. (2) cannot be used because there is no definition for fracture toughness. An empirical approach can then be a good alternative for determining 0 R for notched components under compression, as described in the following section.

.INITE ELEMENT MODEL
Linear static finite element analyses were conducted in order to numerically evaluate the averaged SED. Analyses were performed in 2D plane strain condition, by means of ANSYS® software and adopting quadratic finite elements. The mesh insensitivity of the SED approach has already been proved by Lazzarin et al. [9] and it is a consequence of the finite element method, in which the elastic strain energy is computed .ig. 3. A sample load-displacement curve of the VSC specimen according to the results of [5].   directly from the nodal displacements, without involving stresses and strains. .igure 4a shows the schematic representation of the finite element model and a sample of the coarse mesh adopted.
The external load was introduced as uniformly distributed along the top flat line of the specimen and the nodes belonging to the same line were constrained so that they could move only along the loading direction. The nodes belonging to the corresponding bottom line of the specimen were completely fixed. A contour plot of the main principal stress component 1 σ is shown in .ig. 4b for the case 2α = 30°, ρ = 0.5 mm and a = 25 mm. According to the SED approach, for rounded V-notches the critical volume is represented by a portion of circular sector. The shape of the volume is shown in .ig. 5a while the contour lines of the SED are depicted in .ig. 5b for the same case presented in .ig. 4.
On the notch side the control volume boundary is represented by the circular notch edge with radius ρ, on the other side the boundary is a circle arc with radius 0 0 . R r + The center of the notch radius is geometrically given while the center of the radius 0 0 R r + coincides with the center of the polar coordinate system in the analytical solution for the blunt V-notch problem in which 0 ( 1) , r q q = ρ − with (2 2 ) q = π− α π according to [10].

RESULTS
By considering five different values of 0 , R ranging from 0.5 to 1.5 mm, and by plotting the computed averaged SED as a function of 0 , R it has been found that assuming 0 R = 1 mm the specimens having ρ = 0.5 mm, 2α = 30° and ρ = 2 mm, 2α = 90° experience the same averaged SED value at failure. This means that at the critical load the specimens are characterized by the same SED value averaged in a control volume having radius equal to 1 mm, which is independent of the notch shape and sharpness. This value also corresponds to the critical value that can be obtained by Eq. (1) using as compressive strength c σ = 230 MPa, according to [28], and E = 2018 MPa. .or the herein considered PMMA the critical SED has been found equal to 13.1 MJ/m 3 . .igure 6 shows a summary of the experimental data in terms of the square root value of the SED averaged on the control volume of radius 0 R = 1 mm, normalised with respect to the constant value of the critical SED (13.1 MJ/m 3 ). Indeed, this parameter is proportional to the fracture load. The aim was to investigate the range of accuracy of the SEDbased fracture assessment for the PMMA specimens under compression. It has been found that about all experimental data fit in a narrow scatter band, whose limits are drawn here with an engineering judgement from 0.75 to 1.25. However, it should be pointed out that the majority of the results falls inside a band ranging from 0.8 to 1.1. It can be concluded that the scatter of the data is quite limited and substantially independent of the notch geometry and, in the present author opinion, the averaged SED criterion appears suitable for the fracture strength assessment of V-notched PMMA specimens under compression.

CONCLUSION
Brittle fracture in V-notched PMMA specimens was investigated under compression loading both experi-.ig. 6. SED based summary from V-notched PMMA specimens under compression.
.ig. 5. Shape of the control volume under mode I loading and SED contour lines for the case of 2α = 30°, ρ = 0.5 mm and a = 25 mm. Control volume for blunt V-notches (a), sample of SED contour lines (b). mentally and numerically. .racture tests conducted on prismatic specimens and reported in the recent literature have been reanalyzed in this paper by means of the averaged strain energy density criterion. Being the experimental results in good agreement with the numerical assessments, it has been shown that the proposed method is suitable for predicting PMMA brittle failure under compressive loading condition and in the presence of notches having a different opening angle and sharpness.