Crack onset in stretched open hole PMMA plates considering linear and non-linear elastic behaviours

https://doi.org/10.1016/j.tafmec.2021.102931Get rights and content

Highlights

  • Experimental data on tensile, three point bending (TPB) and holed plates are provided.

  • A numerical code based on the Coupled Criterion of Finite Fracture Mechanics (CCFFM) is developed using Linear and Non-Linear material laws.

  • The effect of small but highly stressed volume is studied using TPB data.

  • This code is successfully applied to the crack onset in PMMA holed plates.

  • For the holed plates failure characterization, a simple inverse problem is solved.

Abstract

Crack onset in PMMA holed plates subjected to tensile stresses is studied experimentally and by the coupled stress and energy criterion of the Finite Fracture Mechanics (CCFFM). The elastic, strength and fracture properties of PMMA are determined by the standard tests, a clearly nonlinear stress-strain relation being identified in the tensile tests. Thus, a novel numerical implementation of the CCFFM considering a non-linear elastic (NLE) material model, using the Ramberg-Osgood approximation, in addition to the usually used linear elastic (LE) model, is developed. Testing of plates with different hole sizes shows a hole size effect in the nominal failure load as expected. For a better fitting of the experimental results, higher strength values obtained by three point bending (TPB) flatwise and edgewise coupons (without any notch), for these material models, are used, apparently for the first time, in the CCFFM predictions. This approach reflects the observation that the strength values associated to smaller but highly stressed volumes, like those located at stress maxima in the holed plates and TPB specimens, are higher. For finite-width holed plates and both material behaviours, suitable FEM models are developed to implement the CCFFM for both LE and NLE models, considering plane stress state. Moreover, an inverse procedure is devised, using the experimental data for holed plates and predictions by CCFFM, to estimate the strength and fracture properties to be used in both material models, providing very good correlations of the CCFFM predictions with the experimental results.

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 μm), 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 σc=115 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 σu=146 MPa, for PMMA in the presence of blunt type notches like is the case of holes. Berto et al. [5] obtained higher σc=138 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), KIc, Berto et al. [5] obtained a high value of KIc=2.04 MPa m and also Taylor [16], [28] obtained a very high value of 2.23 MPa m. Seweryn [1] obtained KIc=1.86 MPa m. Seweryn et al. [30] obtained KIc=1.37 MPa m and Seweryn and Łukaszewicz [2] obtained a mean value of 1.202 MPa m, using notched specimens in tension with notch angles 20°, 40° and 60°. A lower average value of 1.02 MPa m 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 m is approximately independent of its thickness. Li and Zhang [27] and Gomez et al. [10] obtained for PMMA also a rather low value of KIc=1.0 MPa m. 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 m.

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 σy near the hole corresponding to the maximum value of unom, 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 Kt=3 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 σc and Gc 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)

  • P. Cornetti et al.

    Finite Fracture Mechanics: A coupled stress and energy failure criterion

    Eng. Fract. Mech.

    (2006)
  • D. Leguillon et al.

    Prediction of crack initiation at blunt notches and cavities - size effects

    Eng. Fract. Mech.

    (2007)
  • E. Martin et al.

    A coupled strength and toughness criterion for the prediction of the open hole tensile strength of a composite plate

    Int. J. Solids Struct.

    (2012)
  • P. Camanho et al.

    A Finite Fracture Mechanics model for the prediction of the open-hole strength of composite laminates

    Compos. Part A: Appl. Sci. Manuf.

    (2012)
  • J. Li et al.

    A criterion study for non-singular stress concentrations in brittle or quasi-brittle materials

    Eng. Fract. Mech.

    (2006)
  • D. Taylor et al.

    The effect of stress concentrations on the fracture strength of polymethylmethacrylate

    Mater. Sci. Eng. A

    (2004)
  • M. Dunn et al.

    Fracture initiation at sharp notches: Correlation using critical stress intensities

    Int. J. Solids Struct.

    (1997)
  • R. Seldén

    Fracture energy measurements in polycarbonate and PMMA

    Polym. Testing

    (1987)
  • E. Bura et al.

    Mode I fracture in PMMA specimens with notches - experimental and numerical studies

    Theoret. Appl. Fract. Mech.

    (2018)
  • A. Doitrand et al.

    Nonlinear implementation of Finite Fracture Mechanics: A case study on notched Brazilian disk samples

    Int. J. Non-Linear Mech.

    (2020)
  • F. Zhou et al.

    A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials

    Eng. Fract. Mech.

    (2005)
  • V. Mantič

    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.

    (2009)
  • B. Fiedler et al.

    Failure behavior of an epoxy matrix under different kinds of static loading

    Compos. Sci. Technol.

    (2001)
  • T. Hobbiebrunken et al.

    Experimental determination of the true epoxy resin strength using micro-scaled specimens

    Compos. Part A: Appl. Sci. Manuf.

    (2007)
  • E. Martin et al.

    Understanding the tensile strength of ceramics in the presence of small critical flaws

    Eng. Fract. Mech.

    (2018)
  • P. Lazzarin et al.

    A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches

    Int. J. Fract.

    (2001)
  • Z. Yosibash et al.

    Failure criteria for brittle elastic materials

    Int. J. Fract.

    (2004)
  • P. Lazzarin et al.

    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.

    (2009)
  • Cited by (11)

    • Crack impinging on a curved weak interface: Penetration or deflection?

      2023, Journal of the Mechanics and Physics of Solids
    • Strength-based regularization length in phase field fracture

      2023, Theoretical and Applied Fracture Mechanics
    View all citing articles on Scopus
    View full text