Fracture mechanics of circular discs with a V-notch subjected to wedging

Abstract

Objective

A method proposed for determining the fracture toughness (FT) of dental materials involves a ‘roller’ wedging open a V-notch in a cylindrical specimen. There are a number of problems with the design of this test and its mechanical analysis, and thus with the validity of the results obtained, were it to be used. Firstly, friction is ignored in calculating the horizontal wedging force. Secondly, the test specimen does not make use of a pre-crack at the notch tip. The aim of this study was to evaluate the effects of these factors on the FT calculated.

Methods

An analytical solution for the mode-I stress intensity factor (K I ) of the compact tension specimen, which bears some similarities, is taken to be applicable. The mechanics of the specimen has been reanalysed, with a finite-element study of the resultant stresses, and compared with the compact-tension test.

Results

The assumed analytical solution can provide accurate estimates for K I for the V notched specimen. However, the apparent agreement is due to the fortuitous combination of an overestimated horizontal wedging force and an underestimated stress singularity at the crack tip. In any case, ignoring friction will lead to an overestimate of FT.

Significance

It is concluded that the test as presented is invalid.

Introduction

The mechanical behaviour of materials in service in the mouth is of fundamental importance, with the implicit (idealized) requirement that devices survive the load challenges of mastication and the exigencies of that service. Primarily, this has been addressed through various measures of strength – in tension, or bending, but often in compression. Such tests are apparently simple to execute, although there are a number of problems that require careful consideration in order to avoid uninterpretable results [ , ]. There is, however, a school of thought that argues that fracture toughness (FT) is the more important measure, based as it is on the concepts of stress intensity factor ( K ) and strain-energy release rate [ ]. Measuring FT is considerably more difficult than measuring ‘strength’, and there is an extensive literature devoted to the topic [ ], with a variety of approaches in use. The situation was succinctly summarized by Üçtaşli et al. [ ], who went on to propose what would appear to be a novel approach for the determination of FT, alias K Ic , where the ‘I’ refers to mode I crack opening ( i.e. in tension) at the critical (‘ c ’) value of the stress intensity factor. This method has been employed a number of times, both directly [ ] and as a test of bond strength [ , ]. The stated motivation of wishing to mimic the clinical situation of a cusp in an occlusal notch is perhaps laudable in itself, as most mechanical tests used in dentistry bear no relation to actual service conditions, although in this case it is doubtful whether an equivalent circumstance could arise in practice. However, there are a number of problems with the analysis presented [ ], an analysis that seems to have gone unexamined subsequently. Accordingly, we present a reconsideration of the approach, some numerical results from finite-element modelling, and a comparison with the so-called ‘compact-tension test’ for FT.

The essence of the method is illustrated in Fig. 1 . A cylindrical specimen with a sectoral notch is subjected to the wedging action of a cylindrical bar (the “roller”) such that a crack is expected to propagate from the root of the notch. The first thing to notice is that the sharpness of this notch is undefined, it simply being noted first that it represents a stress concentration. This must be contrasted with the well-understood need for a very sharp pre-crack to be present for valid fracture toughness test results to be obtained. No such preparation is involved for the V-notch test. Thus, although the importance of the precondition was acknowledged, it was said that the merely moulded notch represents a “preformed sharp crack” — which it is not. (In passing, it was claimed – falsely – that “the specimen becomes aligned centrally under the influence of the roller”, i.e. generally. This can only happen under one very particular set of conditions. 1

1 When the test piece is in contact with the roller on both sides of the notch, any rotation from its perfectly aligned position will result in the roller being nearer the platen than in that perfect alignment. Restoration to perfect alignment under loading is therefore not possible as this would require raising the roller against the downward motion of loading. Only when the test piece is in contact with the roller on one side of the notch only, and for one exact combination of lateral offset and rotation, could the test piece be brought to exact alignment. The probability for this is vanishingly small.

Otherwise, the tendency is to make the malalignment greater. Further, because there is non-zero friction between all moving parts, there are necessarily unquantifiable asymmetric parasitic stresses present, complicating the interpretation of results, but surely adding to the scatter.)

Fig. 1
(a) V-notched disc test proposed for fracture toughness. (b) Compact-tension disc test with similar mechanics.

The next step is to calculate the “torque” supposedly acting at the notch root, based on the action causing the reaction F R and the distance from that point of contact to the notch root. In fact, this is not a torque because there is no (axial) separation of force vectors even though it may be construed as a lever — there is no moment couple. However, this so-called torque is then considered to be acting over the area radius × length (of the test piece cylinder, that is) to give the “torque per unit area”, which is inappropriate as it is not a uniform measure over that radial area. It is unclear what this supposed measure of fracture resistance was actually meant to (or can) represent, but the assertion of a “new concept known as ‘torque to initiate fracture’” appears to be erroneous and based on a misapprehension.

Following this, the value of K Ic is to be found using an erroneous equation for the horizontal, crack-opening force, but in any case ignoring friction. They employed the equation for the tensile test piece of a (now superseded and withdrawn) British Standard (BS 5447: 1977) which was meant to be applied to metals in which a fatigue-generated “starter crack” had been prepared (not less than 1.25 mm long). The absence of the starter crack in the notched-cylinder test piece is noteworthy, as is the absence of the required result validity checking of the standard.

While the statistical analysis is inappropriate, inadequate and ill-reported (ignoring the scalar variables in favour of unordered categories, using multiple one-way analyses of variance instead of full designs, no check of normality of distribution; sample size is not stated), it is clear that there are effects that vitiate the required condition that the test outcome is scale-invariant if the values of material properties are to be determined. The theoretical expectation is not met.

Deformation of the notch surface must occur under the roller. This will have two key effects: firstly, the contact area must increase as the material conforms to the roller, modifying the geometry (and thus the basis of the calculation by reducing the effective crack length as well as the horizontal load) and secondly, modifying the frictional interaction. Since both effects depend on the properties of the test material, there is an unquantifiable feedback such that the calculated properties are unreliable. This is reminiscent of the problems with axial compression tests in that the outcome is controlled by the contact conditions and specimen deformation beyond the simplistic constant-geometry assumption [ ]. Indeed, the deformation of the notched specimen at the contact with the (lower) platen also changes both the geometry and the stress field in a material-dependent fashion. Additionally, the behaviour must be modified by the depth of the roller in the notch ( Fig. 1 a has been drawn, and subsequent calculations made, for contact at the lip, but the local deformation would depend on the support of material at a greater radius from the notch root) and therefore on the notch angle as well. A further worry would be that the sliding contact of the so-called roller in the notch on the obviously abrasive material (for the group of products tested then) would change the interaction over use, much as the platens in uniaxial compression tests suffer wear. This is usually ignored, even if noticed, but the resultant dishing violates the assumed geometry and certainly changes the parasitic stresses. Were there any grounds for retaining the notched cylinder test, a high-pressure lubricant ( e.g. MoS 2 ) might be required as well as regular replacement or refacing of the contact surfaces of the test rig to ensure reproducibility, as in fact should be routine in other testing. The use of lubricants has its own problems [ ].

Methods

Analytical solution

In this section, we revisit the analytical solution for the mode I stress intensity factor (K I ) for the V-notched disc specimen as a function of the vertical load. The specimen is assumed to behave in a similar fashion as the compact-tension specimen ( Fig. 1 ).

W is the vertical distance of the horizontal wedging force, F H , from the base; a is the effective crack length; a c is the initial crack length at the notch tip; <SPAN role=presentation tabIndex=0 id=MathJax-Element-1-Frame class=MathJax style="POSITION: relative" data-mathml='φV’>𝜑𝑉φV
φ V
and <SPAN role=presentation tabIndex=0 id=MathJax-Element-2-Frame class=MathJax style="POSITION: relative" data-mathml='φCT’>𝜑𝐶𝑇φCT
φ C T
are the notch half-angles of the V-notched disc and compact–tension disc, respectively; D V and D CT are the diameters of the V-notched disc and compact–tension disc, respectively; D R is the diameter of the roller; F R and F F are the normal and frictional reaction forces, respectively, at the point of contact between the roller and the disc; and F V is the vertical wedging force.

Consider first the equilibrium of the loading roller. Vertically,

<SPAN role=presentation tabIndex=0 id=MathJax-Element-3-Frame class=MathJax style="POSITION: relative" data-mathml='FV2=FFcosφV+FRsinφV’>𝐹𝑉2=𝐹𝐹𝑐𝑜𝑠𝜑𝑉+𝐹𝑅𝑠𝑖𝑛𝜑𝑉FV2=FFcosφV+FRsinφV
FV2=FFcosφV+FRsinφV

with the frictional force F F being related to the reaction force F R by:

<SPAN role=presentation tabIndex=0 id=MathJax-Element-4-Frame class=MathJax style="POSITION: relative" data-mathml='FF=μFR’>𝐹𝐹=𝜇𝐹𝑅FF=μFR
FF=μFR

where μ is the frictional coefficient. Substituting (2) into (1) gives:

<SPAN role=presentation tabIndex=0 id=MathJax-Element-5-Frame class=MathJax style="POSITION: relative" data-mathml='FV2=FRμcosφV+sinφV.’>𝐹𝑉2=𝐹𝑅(𝜇𝑐𝑜𝑠𝜑𝑉+𝑠𝑖𝑛𝜑𝑉).FV2=FRμcosφV+sinφV.
FV2=FRμcosφV+sinφV.

The horizontal wedging force exerted by the roller on each side of the disc is derived from the sum of the horizontal components of F F and <SPAN role=presentation tabIndex=0 id=MathJax-Element-6-Frame class=MathJax style="POSITION: relative" data-mathml='FR’>𝐹𝑅FR
F R
, i.e.

<SPAN role=presentation tabIndex=0 id=MathJax-Element-7-Frame class=MathJax style="POSITION: relative" data-mathml='FH=FRcosφV-FFsinφV=FRcosφV-μsinφV.’>𝐹𝐻=𝐹𝑅𝑐𝑜𝑠𝜑𝑉𝐹𝐹𝑠𝑖𝑛𝜑𝑉=𝐹𝑅(𝑐𝑜𝑠𝜑𝑉𝜇𝑠𝑖𝑛𝜑𝑉).FH=FRcosφV-FFsinφV=FRcosφV-μsinφV.
FH=FRcosφV-FFsinφV=FRcosφV-μsinφV.
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Apr 27, 2020 | Posted by in Dental Materials | Comments Off on Fracture mechanics of circular discs with a V-notch subjected to wedging
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