Crack onset in stretched open hole PMMA plates considering linear and non-linear elastic behaviours
Introduction
Stress raisers (either stress concentrations or singularities) in components or joints are always object of special attention due to the risk of premature failure by the nucleation of a crack there. The failure can be originated by static, dynamic or fatigue loadings and can take place in a brittle or ductile manner.
The increasing use of polymers in the last decades requires developing reliable criteria for crack onset at stress risers, like V- or U-notches and holes. Currently there is no widely accepted general criterion for crack onset at such stress risers in polymers, in spite of many works dealing with this problem using, e.g., the criteria based on concepts related to the strain energy [1], [2], [3], [4], [5], [6], [7], [8], [9], the Cohesive Zone Models (CZM) [10], [11], [12], [13], [14], [15], and the so-called Theory of Critical Distances (TCD) recently revisited and thoroughly presented by Taylor [16]. The TCD is not only one method but a group of methods that enables the prediction of brittle fracture and fatigue failure from plain specimens (without any stress concentrations) to components containing any kind of stress risers, including cracks but also notches, holes and pores. References to other works using TCD approach can be found in [17], [18], [19].
A promising approach towards such a widely accepted general criterion is the Coupled Criterion of Finite Fracture Mechanics (CCFFM) introduced by Leguillon [20] and Cornetti et al. [21] to predict crack onset in brittle materials at stress raisers. Leguillon et al. [22] also applied CCFFM to PMMA specimens with blunt notches of a very small tip radius (from 1 to 100 m), tested in Three Point Bending (TPB) specimens, and also to holed plates made of rock, tested in compression. The CCFFM was later used by Carrère et al. [23], Martin et al. [24] and Camanho et al. [25], among others, for the prediction of failure of holed specimens made of carbon fibre/epoxy under tension. A comprehensive analysis of crack onset in holed specimens under a general biaxial loading was recently developed applying the CCFFM by Sapora and Cornetti [26].
Polymers present several challenges versus traditional metallic materials: they can have Non-linear Elastic (NLE) behaviour and are quite sensible to strain rate, having viscous behaviour. Additionally, it is well known that polymers, and especially PMMA, in general have considerable dispersion in published material properties, namely in tensile strength and fracture energy (fracture toughness).
Examples of tensile strength values for PMMA found in the literature are quoted in the following. Low tensile strength, between 70 and 75 MPa were obtained in [10], [16], [17], [27], [28], [29] using the standard tension specimens. Higher values were indicated by Seweryn [1] who obtained a value of 104.9 MPa by using TPB specimens and Compact Tension (CT) specimens, both having a semicircular notch. Seweryn et al. [30] obtained MPa and Seweryn and Łukaszewicz [2] obtained a value of 102.8 MPa both using tensile specimens with semicircular notches. They argued that by using specimens with semicircular notches they obtained less scatter than by using the dogbone specimens. Dunn et al. [31] obtained 124 MPa by testing unnotched TPB specimens. Taylor [16], [28] obtained a more than double value than that of plain tensile specimen, with a MPa, for PMMA in the presence of blunt type notches like is the case of holes. Berto et al. [5] obtained higher MPa after calculating the fracture stress by means of linear Hooke’s law with the rupture strain and tangential Young’s modulus, from a previous work [17].
Regarding the fracture toughness of PMMA, defined as the critical Stress Intensity factor (SIF), , Berto et al. [5] obtained a high value of MPa and also Taylor [16], [28] obtained a very high value of 2.23 MPa . Seweryn [1] obtained MPa . Seweryn et al. [30] obtained MPa and Seweryn and Łukaszewicz [2] obtained a mean value of 1.202 MPa , using notched specimens in tension with notch angles 20, 40 and 60. A lower average value of 1.02 MPa was obtained by Dunn et al. [31] using Single Edge Notch Bending (SENB) specimens with the initial crack made with a sharp razor blade. Seldén [32] measured fracture toughness using CT specimens, concluding that the fracture toughness of PMMA 1.5 MPa is approximately independent of its thickness. Li and Zhang [27] and Gomez et al. [10] obtained for PMMA also a rather low value of MPa . All these lower values are in accordance with the published values in bibliography, e.g. [33], that indicates values of PMMA fracture toughness ranging from 0.7 to 1.6 MPa .
The CCFFM was applied to predict crack onset in PMMA holed plates under tension by several authors using Linear Elastic (LE) material model [27], [29], [34], [35], [36], [37] providing somewhat contradictory predictions. On the one hand, the CCFFM predictions in [27], [37] significantly underestimate the observed failure loads, the predictions in [37] being closer to the experimental data due to a higher value of fracture energy used. On the other hand, a better agreement of predictions and experimental results was obtained in [29] by employing a very high value of fracture toughness and using a different test velocity (five times lower than the ones used by [27], [37]), thus not allowing a direct comparison of results with the previous works [27], [37].
The present work is a further development of the previous studies of crack onset in holed PMMA plates under tension [27], [37] using the CCFFM, considering both Linear Elastic (LE) and Non-linear Elastic (NLE) material models. A schematic of the geometry and loading in uncracked and cracked holed specimens is shown in Fig. 1, the notation shown will be explained in detail in Section 4.
The effect of the highly stressed volume of a polymer material on the measured strength value is well-known, see e.g. [16], [28]. Especially in structural configurations with stress concentrations, where only a small volume is subjected to the highest stresses, the measured strength values can be much higher than in configurations with uniform stress distributions. This is due to the fact that the probability of large defects is much smaller in a small highly stressed volume, in the former, than in the whole volume of structure, in the latter.
In view of this, in the present work, the strength properties are evaluated, in addition to the tensile test, also by TPB test. The highly stressed zone is given by a small neighbourhood of a line where the maximum tensile stress is achieved, in the bending coupon, instead of a large volume in the dogbone coupon. Recall that in an open hole plate, the highly stressed zone at the hole border is also given by a small neighbourhood of a line. One of the aims of the present TPB tests is also to check if the obtained strength in PMMA material is also significantly higher when considering a NLE material model.
Section snippets
Material properties
The material employed is an amorphous thermoplastic polymer, Polymethyl Metacrylate (PMMA). First, the determination of elastic and strength properties and fracture properties, respectively, is described in Section 2.1 and Section 2.2.
Open-hole tensile tests
Tensile tests are carried out for plates with circular holes to study onset of cracks at stress concentration points at the hole border.
The Coupled Criterion (CC) of FFM for holed plate in tension
The Coupled Criterion of FFM (CCFFM) [20], [21], [51] is used to predict onset of two transverse symmetrical cracks at the hole border of a stretched PMMA plate, for both LE and NLE models. The two configurations before and after the crack onset are shown in Fig. 1. The CCFFM predicts the remote failure stress and the size of crack after its onset. Note that, in opposite to the remote failure stress, the crack size after its onset can hardly be observed experimentally because the crack onset is
Stresses near the hole
The distributions of near the hole corresponding to the maximum value of , for diameters 0.5 mm and 10 mm, in the case of plane stress, are represented in Fig. 10 for both LE and NLE cases. In Fig. 10(a), corresponding to the smallest hole case ( 0.5 mm), the well-known stress distribution with concentration factor near the hole and nearly uniform stresses in the zone away from the hole are observed in the LE case, whereas in the NLE case it is seen that the stress
Improving the representativity of strength and fracture properties of the material: comparison with experimental results
The predictions by CCFFM, using the values of material properties and measured in the present work by the standard tests (see Section 2), do not fit well with the experimentally obtained failure stresses, as can be seen in Fig. 16. The authors consider that this fact may be due to the level of representativity of the critical values obtained, associated with the different volumes of material potentially involved in the mechanism of failure. In this sense, an inverse procedure is proposed
Concluding remarks
The original motivation of this research was to achieve a satisfactory agreement between the size effect on the remote failure stress in holed PMMA plates under tension observed in the new experimental campaign carried out and the Coupled Criterion of Finite Fracture Mechanics (CCFFM) [20], [21]. The use of CCFFM to predict failure in blunt notches and holes in materials which are not linear elastic, like the PMMA used in the present experiments, is not yet successfully proved. In fact, Li and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was partially supported by the PhD grant (SFRH/BD/49279/2008) of the first author by FCT - Fundação para a Ciência e Tecnologia, Portugal, and also by the Spanish Ministry of Economy and Competitiveness and European Regional Development Fund (Project MAT2015-71036-P), and the Consejería de Economía y Conocimiento de la Junta de Andalucía and European Regional Development Fund (Project P18-FR-1928). The first author acknowledges all the support given by the Group of Elasticity and
References (56)
Brittle fracture criterion for structures with sharp notches
Eng. Fract. Mech.
(1994)- et al.
Verification of brittle fracture criteria for elements with V-shaped notches
Eng. Fract. Mech.
(2002) - et al.
A synthesis of Polymethylmethacrylate data from U-notched specimens and V-notches with end holes by means of local energy
Mater. Des.
(2013) - et al.
Fracture assessment of Brazilian disc specimens weakened by blunt V-notches under mixed mode loading by means of local energy
Mater. Des.
(2011) - et al.
Fracture behaviour of notched round bars made of PMMA subjected to torsion at -60°C
Eng. Fract. Mech.
(2013) - et al.
Fracture behaviour of notched round bars made of PMMA subjected to torsion at room temperature
Eng. Fract. Mech.
(2012) - et al.
Fracture of U-notched specimens under mixed mode: Experimental results and numerical predictions
Eng. Fract. Mech.
(2009) - et al.
The cohesive crack concept: application to PMMA at -60°C
Eng. Fract. Mech.
(2005) - et al.
Analysis of notch effect in PMMA using the theory of critical distances
Eng. Fract. Mech.
(2012) Strength or toughness? A criterion for crack onset at a notch
Eur. J. Mech. A/Solids
(2002)
Finite Fracture Mechanics: A coupled stress and energy failure criterion
Eng. Fract. Mech.
Prediction of crack initiation at blunt notches and cavities - size effects
Eng. Fract. Mech.
A coupled strength and toughness criterion for the prediction of the open hole tensile strength of a composite plate
Int. J. Solids Struct.
A Finite Fracture Mechanics model for the prediction of the open-hole strength of composite laminates
Compos. Part A: Appl. Sci. Manuf.
A criterion study for non-singular stress concentrations in brittle or quasi-brittle materials
Eng. Fract. Mech.
The effect of stress concentrations on the fracture strength of polymethylmethacrylate
Mater. Sci. Eng. A
Fracture initiation at sharp notches: Correlation using critical stress intensities
Int. J. Solids Struct.
Fracture energy measurements in polycarbonate and PMMA
Polym. Testing
Mode I fracture in PMMA specimens with notches - experimental and numerical studies
Theoret. Appl. Fract. Mech.
Nonlinear implementation of Finite Fracture Mechanics: A case study on notched Brazilian disk samples
Int. J. Non-Linear Mech.
A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials
Eng. Fract. Mech.
Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion
Int. J. Solids Struct.
Failure behavior of an epoxy matrix under different kinds of static loading
Compos. Sci. Technol.
Experimental determination of the true epoxy resin strength using micro-scaled specimens
Compos. Part A: Appl. Sci. Manuf.
Understanding the tensile strength of ceramics in the presence of small critical flaws
Eng. Fract. Mech.
A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches
Int. J. Fract.
Failure criteria for brittle elastic materials
Int. J. Fract.
Brittle failures from U- and V-notches in mode I and mixed, I + II, mode: a synthesis based on the strain energy density averaged on finite-size volumes
Fatigue Fract. Eng. Mater. Struct.
Cited by (11)
Influence of debonding and substrate plasticity on thin film multicracking
2024, Theoretical and Applied Fracture MechanicsThe elasto-plastic numerical study of crack initiation in notched PMMA specimens under uniaxial loading conditions – Tension and torsion
2023, Theoretical and Applied Fracture MechanicsCrack impinging on a curved weak interface: Penetration or deflection?
2023, Journal of the Mechanics and Physics of SolidsThe fracture behaviour of notched PMMA specimens under simple loading conditions – Tension and torsion experimental tests
2023, Engineering Failure AnalysisCrack initiation in PMMA plates with circular holes considering kinetic energy and nonlinear elastic material behavior
2023, Theoretical and Applied Fracture MechanicsStrength-based regularization length in phase field fracture
2023, Theoretical and Applied Fracture Mechanics