Necessary conditions for near-edge stick of complete contacts

Abstract

The conditions for slip within a contact and at the corner, together with separation conditions, are presented in a general sense and applied to two example finite complete contact problems. These conditions can be found directly from the signs of the relevant stress intensity factors. Furthermore, a new bound on the validity of the asymptotic solution is found and then applied to find a solution relating to separation conditions. Finally, this new bound is used to identify and classify cases in which a region of mode II dominance is seen from purely geometric considerations.

Introduction

Complete contacts are those in which the front (contacting) faces of the bodies being pressed together have the same shape in the unloaded state, that is, they ‘conform’, and the extent of the contact is defined by a discontinuity in the gradient of the profile, as exemplified by the rectangular block-on-half-plane arrangement shown in Figure 1. They have the property that the contact size is independent of the load carried and that a stress singularity will usually arise at the edges of the contact. Practically occurring contacts which are of this general form include splines, for example, in certain car transmissions and in the split shafts of a gas turbine.

Figure 1.

Rectangular block resting on a half-plane subject to four external loads.

The objective of the analysis presented here is to quantify the conditions for ensuring that the contact edges are stuck so that fretting damage is avoided, although this does leave the attendant risk of a notch-type failure. We shall assume that the block and the half-plane are made from the same material and also that the interface is, indeed, straight, as sketched, but note that the results to be found would apply equally well to a circular cylinder resting in a conforming groove, provided that the line of the interface was accounted for correctly.

A full analysis of the problem, including incorporating the presence of all features, calls for a finite element (FE) analysis, and indeed, the FE analysis must be used to calibrate the external loads to stress intensity factors to be defined, but our objective is to learn as much as possible about the properties of the contact from an analysis of the half-plane and contact corner modelled together, as a monolith, and use Williams’ analysis ¹ of a wedge as our starting point. The use of this classic asymptotic form, appropriate when the observation point is much nearer to the corner than any other features, adds considerable precision to our understanding of the local stress field.

This idea is one which we have used before. It enables the coefficient of friction needed to attain a fully stuck contact edge ² to be determined. But in that analysis only the dominant term in an eigenfunction expansion was used, and the question arises of what happens when we move slightly inwards from the edge: Is the coefficient of friction needed to achieve stick at the edge also sufficient to attain stick in the rest of the neighbourhood? We answer that question, here, together with looking at further forms of excitation of the local stress field.

One of the main topics of practical analysis in this laboratory is the study of fretting and fretting fatigue, and in pursuing these, we note that there are normally four sets of external loads which are relevant in describing the local contact edge stress field (Figure 1), namely, the normal load, P, the shear force, $Q$ , any moment carried by the contact-defining body, M, and the underlying surface tension in the counterbody, $\mathrm{\Delta }\sigma$ . All of these quantities become ‘coupled’ in the context of a complete contact, in the sense that all four excite all eigenmodes. Here, we will concentrate on only the first two terms in the infinite Williams’ series and investigate what these imply. This is a legitimate approximation to make as these are the only two terms in the series which are singular at the contact edge, all others terms being bounded. Hence, these terms dominate the solution.

Formulation

It is initially assumed that the two contacting bodies are in intimate contact everywhere, and fully adhered, giving a monolithic domain. We zoom in on the contact (wedge) corner. At the time of the initial consideration of this class of problem, we would write down the state of stress in the neighbourhood of the wedge (contact) corner ¹ in the form

$${\sigma }_{\mathrm{ij}}(r,\theta )=K_I{r}^{{\lambda }_I-1}{f}_{\mathrm{ij}}^I(\theta )+K_{\mathrm{II}}{r}^{{\lambda }_{\mathrm{II}}-1}{f}_{\mathrm{ij}}^{\mathrm{II}}(\theta )+\cdots$$

(1)

where $(r,\theta )$ are polar coordinates measured from the notch bisector (Figure 1), $(K_I,K_{\mathrm{II}})$ are the multipliers (generalised stress intensity factors) which match the eigensolution to the finite problem, $f_{\mathrm{ij}}^K(\theta )$ are the eigenvectors (see Appendix 1), and the eigenvalues $({\lambda }_I,{\lambda }_{\mathrm{II}})$ are given by the smallest, positive, non-trivial root of the following characteristic equation

$${\lambda }_I\sin (2\alpha )+\sin (2\alpha {\lambda }_I)=0\mathrm{and}{\lambda }_{\mathrm{II}}\sin (2\alpha )-\sin (2\alpha {\lambda }_{\mathrm{II}})=0$$

(2)

where $2\alpha$ is the total wedge angle, equal to $\pi +\varphi$ , where $\varphi$ is the internal angle of the block, equal to $\pi /2$ in Figure 1 but to be generalised later. The angles are measured from the bisector as the eigensolutions uncouple along that line, and we also use this angular position to define the values of the generalised stress intensity factors, that is

$$K_I=\underset{r\to 0}{lim}{\sigma }_{\theta \theta }(r,0)r^{1-{\lambda }_I},f_{r\theta }^I(r,0)=0,\mathrm{and}{f}_{\theta \theta }^I(r,0)=1$$

(3)

and

$$K_{\mathrm{II}}=\underset{r\to 0}{lim}{\sigma }_{r\theta }(r,0)r^{1-{\lambda }_{\mathrm{II}}},f_{\theta \theta }^{\mathrm{II}}(r,0)=0,\mathrm{and}{f}_{r\theta }^{\mathrm{II}}(r,0)=1$$

(4)

Before introducing an alternative representation of this solution, we will consider the physical requirements for the solution to be developed to be appropriate when applied to contact problems. These are (a) that the two bodies be in intimate contact, implied by ${\sigma }_{\theta \theta }(r,\psi )<0$ , where $\psi =1/2(\pi -\varphi )$ is the angle to the interface, and (b) stick must prevail which requires $-|{\sigma }_{r\theta }(r,\psi )|/{\sigma }_{\theta \theta }(r,\psi )<f$ , where $f$ is the coefficient of friction, for all points $r$ on the interface. Some local violations of these inequalities may be tolerated without invalidating the whole solution, but first considering conditions where they are satisfied everywhere.

For the time being, focus attention wholly on the square block (Figure 1) so that $2\alpha =3\pi /2$ and $\psi =\pi /4$ . For this example, the eigenvectors $(f_{\mathrm{ij}}^I(\theta ),f_{\mathrm{ij}}^{\mathrm{II}}(\theta ))$ are given in Appendix 1. We note that $f_{\theta \theta }^I(\pi /4)\approx 0.7$ so that it follows, because we can always find points sufficiently close to the contact corner for the mode I contribution to dominate the solution; a requirement for the contact edge to remain closed is that $K_I<0$ . If we turn our attention, now, to the second term in the series, it has been shown in Appendix 1 that $f_{\theta \theta }^{\mathrm{II}}(\pi /4)\approx -1.2$ so that if $K_{\mathrm{II}}>0$ we could be confident that the penetration of intimate contact inwards from the contact edge will last to at least the point where the contributions to the overall solution from mode I and II loading were equal. The question that arises is, ‘Physically, where is the point along the interface where mode I domination of the solution gives way to a contribution from the second solution which is of at least comparable magnitude?’ To answer, we may re-write the first two terms of equation (1), which brings out the internal length scale, $d_0$ , and introduce this quantity and a further quantity, $G_0$ , which has dimensions of stress, defined by ³

$$d_0={\left|\frac{K_{\mathrm{II}}}{K_I}\right|}^{\frac{1}{{\lambda }_I-{\lambda }_{\mathrm{II}}}}$$

(5)

$$G_o={\left|K_I\right|}^{\frac{{\lambda }_{\mathrm{II}}-1}{{\lambda }_{\mathrm{II}}-{\lambda }_I}}{\left|K_{\mathrm{II}}\right|}^{\frac{{\lambda }_I-1}{{\lambda }_{\mathrm{II}}-{\lambda }_I}}$$

(6)

So equation (1) may be re-written as

$$\frac{{\sigma }_{\mathrm{ij}}(r,\theta )}{G_0}={\left(\frac{r}{d_0}\right)}^{{\lambda }_I-1}{f}_{\mathrm{ij}}^I(\theta )-{\left(\frac{r}{d_0}\right)}^{{\lambda }_{\mathrm{II}}-1}{f}_{\mathrm{ij}}^{\mathrm{II}}(\theta )$$

(7)

This form is extremely useful for studying notch-corner plasticity ³ and highlights that $d_0$ may be viewed as the notional radial distance which represents the partition between mode $I$ domination $(r<d_0)$ and higher order term domination $(r>d_0)$ . It may also be used in our present study: although it is not possible to state unequivocally the distance from the contact corner where higher order terms in a series dominate the solution, it is possible to define the distance where the mode I term loses it significance, namely, $d_0$ . This quantity is defined wholly by mode mixity (equation (5)), and in cases where this exceeds the distance to the nearest physically relevant feature (e.g. the contact half-width, Figure 1), we can be assured that the first term controls the state of stress until the observation point reaches this feature.

A remark on separation

It is well known that we may always find a point, sufficiently close to the corner, where the stress state is totally dominated by the mode I term. Its sign is determined by the sign of the stress intensity factor $K_I$ . Hence, the lift-off or intimate contact of the corner itself is determined by the sign of this term alone, which is both loading and geometry dependent.

We now suppose that the edge of the contact is intimately closed but consider the possibility of lift-off at a point within the body, according to the asymptotic results. For this to occur, and to be observable within the asymptote, we require a turning point in the equation

$${\sigma }_{\theta \theta }=K_I{f}_{\theta \theta }^I{r}^{{\lambda }_I-1}+K_{\mathrm{II}}{f}_{\theta \theta }^{\mathrm{II}}{r}^{{\lambda }_{\mathrm{II}}-1}$$

(8)

found by $d{\sigma }_{\theta \theta }/\mathrm{dr}=0$ . This is shown in Figure 2, where in the central sketch we see a case where such a turning point exists and how this may lead to an internal lift-off point.

Figure 2.

Sketch of different possible shapes for ${\sigma }_{\theta \theta }$ .

The location of this point is given by

$$r={\frac{-K_I}{K_{\mathrm{II}}}}^{\frac{{\lambda }_I-2}{{\lambda }_{\mathrm{II}}-2}}{\left(\frac{f_{\theta \theta }^I({\lambda }_I-1)}{f_{\theta \theta }^{\mathrm{II}}({\lambda }_{\mathrm{II}}-1)}\right)}^{\frac{{\lambda }_I-2}{{\lambda }_{\mathrm{II}}-2}}$$

(9)

Note that the quantity within parentheses depends only on $\alpha$ , so the behaviour of the function is governed by the ratio $K_I/K_{\mathrm{II}}$ . Therefore, for a given geometry, the behaviour and/or existence of such a turning point is purely loading dependent.

Application to the 90° block

Retaining a focus on the square block resting on a half-plane, we aim to improve on and refine the conditions set out by Churchman and Hills ⁴ for separation at the edge of the contact and within the body, as far as it is possible with only asymptotic considerations.

To do this, we use an FE analysis of the problem to calibrate the external loading with the stress intensity factors. This is done by modelling the problems and then applying each of the four external loads individually. The stresses along the angle bisector are then found from FE, and the limits are taken as described above in equations (3) and (4). The results of this for the square block are shown in equation (10). However, we will first concentrate on excitation by only a normal force $P$ and a shear force applied along the line of the interface $Q$

$$\left[\begin{array}{c}{K}_I{a}^{{\lambda }_{I-1}}\hfill \\ \multicolumn{1}{c}{K_{\mathrm{II}}{a}^{{\lambda }_{\mathrm{II}-1}}}\end{array}\right]=\left[\begin{array}{cccc}-0.2265\hfill & 0.2443\hfill & -0.3890\hfill & 0.2462\hfill \\ \multicolumn{1}{c}{0.1118}& 0.2536\hfill & 0.3639\hfill & 0.1483\hfill \end{array}\right]\left[\begin{array}{c}\frac{P}{2a}\hfill \\ \multicolumn{1}{c}{\frac{Q}{2a}}\\ \multicolumn{1}{c}{\frac{M}{2a^2}}\\ \multicolumn{1}{c}{\frac{\mathrm{\Delta }\sigma }{2}}\end{array}\right]$$

(10)

From the calibration, we see that the value of $Q/P$ required to cause lift-off at the corner is $Q/P=0.927$ . This differs slightly from the value found by Churchman and Hills, ⁴ but this is purely due to differences in the calibrations from FE analysis. The details of this analysis are discussed in Appendix 2.

We now intend to see how the location of the turning point (equation (9)) changes with changes in normal and shear load. We can see from Figure 3(a) that the ratio $K_I/K_{\mathrm{II}}$ is negative for values of $Q/P$ in the range $-0.44<Q/P<0.927$ . This implies an internal lift-off point cannot be found once the loading ratio is within the range needed to maintain intimate contact at the corner, as the location of this turning point, as given by equation (9), is negative and hence outside the contact patch.

Figure 3.

The variation of stress intensity factors: (a) relevant constants and (b) with varying $Q/P$ .

The positive value of the ratio $K_I/K_{\mathrm{II}}$ for negative values of $Q/P$ would appear to imply that the corner also would lift off for values of $Q/P<-0.44$ . However, looking at Figure 3(b), we see that for values, roughly, $Q/P<0.2$ , the ratio $d_0/a>1$ , that is, the mode II term dominates the mode I term only at a point beyond the centre of the contact, beyond where the asymptotic solution can be assumed to apply. Therefore, all points along the interface are mode I dominated, so the stress may be described by ${\sigma }_{\theta \theta }=K_I{f}_{\theta \theta }^I{r}^{{\lambda }_I-1}$ alone, which displays no turning point, and hence intimate contact is maintained within the asymptote for this load region as $K_I<0$ giving ${\sigma }_{\theta \theta }<0$ which is compressive.

So, for the 90° block, at least, we cannot improve on the statement presented by Churchman by the use of a two-term solution. We conclude that lift-off only occurs in the near edge when $K_I$ is positive. Further inferences on the effect of other methods of loading are made in section ‘Further forms of loading’, together with observations on their effect on slip conditions.

Slip

As laid out at the start of this article, one of the objectives of this analysis is to determine the conditions required for the corners of the contact to remain stuck to avoid fretting damage. Equally important is ensuring that there is no region of slip within the body, which would equally lead to fretting damage.

Recall from Hills and Dini ³ that the corner will slip for a sufficiently low coefficient of friction. Hence, we must determine under what conditions slip will occur within the body, and within the asymptote, or at the corner. As already discussed in section ‘A remark on separation’, there are conditions under which we get lift-off at the corners of the block. This will necessarily lead to some partial slip. We deduce this because the direct traction $({\sigma }_{\theta \theta })$ changes sign, so it must fall smoothly to zero in the region just interior to the true point of separation. Recall, further, that it was noted earlier that a shear force had to be present to induce separation. The shear traction $({\sigma }_{r\theta })$ will be non-zero local to the point of separation. So, there must exist a point where the equation ${\sigma }_{r\theta }=f{\sigma }_{\theta \theta }$ is satisfied. Hence, this point will be slipping.

The turning point

To determine whether an interior point is the point of first slip, let us first consider the ratio ${\sigma }_{r\theta }/{\sigma }_{\theta \theta }$ . If this ratio is greater than the coefficient of friction over some distance, slip will occur there; if not, it will remain stuck. So, we will find the turning point of this ratio as either the value near to this point or the contact edge value, which will give the coefficient of friction required to prevent all slip.

The turning point can be found by

$$\frac{d}{\mathrm{dr}}\left(\frac{{\sigma }_{r\theta }}{{\sigma }_{\theta \theta }}\right)=-\frac{(f_{r\theta }^{\mathrm{II}}{f}_{\theta \theta }^I-f_{r\theta }^I{f}_{\theta \theta }^{\mathrm{II}})({\lambda }_I-{\lambda }_{\mathrm{II}})r^{{\lambda }_I+{\lambda }_{\mathrm{II}}-1}{K}_I{K}_{\mathrm{II}}}{{(f_{\theta \theta }^I{K}_I{r}^{{\lambda }_I}+f_{\theta \theta }^{\mathrm{II}}{K}_{\mathrm{II}}{r}^{{\lambda }_{\mathrm{II}}})}^2}=0$$

(11)

Provided that the denominator is non-zero, we may first deduce that we can in fact solve the equation

$$-(f_{r\theta }^{\mathrm{II}}{f}_{\theta \theta }^I-f_{r\theta }^I{f}_{\theta \theta }^{\mathrm{II}})({\lambda }_I-{\lambda }_{\mathrm{II}})r^{{\lambda }_I+{\lambda }_{\mathrm{II}}-1}{K}_I{K}_{\mathrm{II}}=0$$

(12)

Now set

$$A=f_{r\theta }^{\mathrm{II}}{f}_{\theta \theta }^I-f_{r\theta }^I{f}_{\theta \theta }^{\mathrm{II}},B=-({\lambda }_I-{\lambda }_{\mathrm{II}})\mathrm{and}C=r^{{\lambda }_I+{\lambda }_{\mathrm{II}}-1}$$

(13)

So equation (12) becomes

$$\mathrm{ABC}{K}_I{K}_{\mathrm{II}}=0$$

From Williams’ solution, we have ${\lambda }_{\mathrm{II}}>{\lambda }_I$ , so we find that $B>0$ . Second, we consider positive $r$ only, that is, within the block, so $C>0$ . Finally, we cannot easily see algebraically, but on plotting $A$ we find that $A>0$ (Figure 4). Hence, we have that the solutions to equation (11) are given by $K_I=0$ or $K_{\mathrm{II}}=0$ .

Figure 4.

Value of $f_{r\theta }^{\mathrm{II}}{f}_{\theta \theta }^I-f_{r\theta }^I{f}_{\theta \theta }^{\mathrm{II}}$ with $\psi$ .

So a turning point can exist only if $K_I=0$ or $K_{\mathrm{II}}=0$ . In these cases, the ratio ${\sigma }_{r\theta }/{\sigma }_{\theta \theta }$ is equal to $f_{r\theta }^{\mathrm{II}}/f_{\theta \theta }^{\mathrm{II}}$ and $f_{r\theta }^I/f_{\theta \theta }^I$ , respectively, and is independent of $r$ . Hence, all we can say is that in the first case $(K_1=0)$ we will always see slip at the corner and in the second, we will see slip at the corner if $f<f_{r\theta }^I/f_{\theta \theta }^I$ , provided that $K_I<0$ . The independence of these ratios from $r$ means that we cannot say what happens at any point other than the corner.

For all other cases, $K_I\ne 0$ and $K_{\mathrm{II}}\ne 0$ ; we can say that there is no turning point within the asymptote, so either slip will start at the corner or within the body but beyond the asymptote. We can determine which of these cases apply by looking at the gradient of the ratio ${\sigma }_{r\theta }/{\sigma }_{\theta \theta }$ . This could be found by taking the limit of the expression in equation (11) as $r\to 0$ , but this limit does not exist. So, we find the sign of the expression in the same way the conditions for the existence of a turning point were found.

Recall that

$$\mathrm{sign}(-(f_{r\theta }^{\mathrm{II}}{f}_{\theta \theta }^I-f_{r\theta }^I{f}_{\theta \theta }^{\mathrm{II}})({\lambda }_I-{\lambda }_{\mathrm{II}})r^{{\lambda }_I+{\lambda }_{\mathrm{II}}-1}{K}_I{K}_{\mathrm{II}})=\mathrm{sign}(K_I{K}_{\mathrm{II}})$$

(14)

$$\mathrm{sign}((f_{\theta \theta }^I{K}_I{r}^{{\lambda }_I}+f_{\theta \theta }^{\mathrm{II}}{K}_{\mathrm{II}}{r}^{{\lambda }_{\mathrm{II}}}{)}^2)\ne -1$$

(15)

Hence

$$\mathrm{sign}\left(\frac{d}{\mathrm{dr}}\frac{{\sigma }_{r\theta }}{{\sigma }_{\theta \theta }}\right)=\mathrm{sign}(K_I{K}_{\mathrm{II}})$$

whenever the gradient is determinable.

The results here are interesting. We know that if $d({\sigma }_{r\theta }/{\sigma }_{\theta \theta })/\mathrm{dr}$ is positive, then slip will begin within the body, at some point beyond the validity of the asymptote. However, in the case where $K_I$ is positive, the corner will also be slipping, giving a large region of slip. Whereas if $K_I$ is negative, slip will originate at a point within the body with $f>f_{r\theta }^I/f_{\theta \theta }^I$ , but for some coefficients of friction, given by Hills and Dini, ³ stick will persist at the corner. The reverse is true if the gradient is negative, and slip will always originate at the corner provided $K_I<0$ for $f<f_{r\theta }^I/f_{\theta \theta }^I$ . If $K_I$ is positive, then, as already outlined, slip will start at the corner but stick will persist within the body and, for sufficiently large coefficients of friction, within the asymptote. These results are summarised in Table 1.

Table 1.

Matrix of inferred result from stress intensity factor signs.

		$K_I$
	Sign of $K_I$	Positive	Negative
$K_{\mathrm{II}}$	Positive	Separation and entire asymptote slip (A)	Slip starts at corner (B)
	Negative	Separation and small slip region (C)	Slip from within (D)

Application to the 90° block

Returning focus to the 90° block, we may distinguish the regions of the load space where each of these forms of behaviour occurs. The results of this analysis are shown in Figure 5. The letters shown correspond to those defined in Table 1. Note that, under these loading conditions, there is no region in which $K_I>0$ and $K_{\mathrm{II}}<0$ .

Figure 5.

Plot showing the regions of behaviour for the 90° block.

Further forms of loading

So far, we have considered excitation only by a normal and shearing force, and we have assumed that these are not sufficient to cause sliding, that is, $|Q|/\mathrm{fP}<1$ . However, excitation by other means is possible, and the appropriate calibrations for the 90° block were included in equation (10).

We can divide the load space to characterise behaviour under general loading as we did with just two loading types. Figure 6 shows the results of these where, in each, one of the loading methods is not excited. Consequently, we can see the load regions where each of these types of behaviour occurs; again the letters refer to those in Table 1.

Figure 6.

Characterisation of regions of the four-method load space (see Table 1).

A further bound on asymptotic validity

From section ‘A remark on separation’, we can see that, provided that the corner is in intimate contact, the whole interface, within the region which can be approximated by the asymptotic solution, must remain in intimate contact – in other words, throughout the region of asymptotic validity, ${\sigma }_{\theta \theta }<0$ provided that conditions are such that the corner itself is in intimate contact. A tidier way to locate the point where the interfacial direct traction changes sign is to find a solution to ${\sigma }_{\theta \theta }=0$ . This point is itself beyond the limit of the validity of the asymptotic solution, but can provide an upper bound.

So, we solve the equation evaluated on $\theta =\psi$

$$K_I{f}_{\theta \theta }^I(\psi )r^{{\lambda }_I-1}+K_{\mathrm{II}}{f}_{\theta \theta }^{\mathrm{II}}(\psi )r^{{\lambda }_{\mathrm{II}}-1}=0$$

(16)

to find the value of $r$ which satisfies this. We denote this value by $d_r$ ; its value is given by

$$d_r={\left(\frac{-f_{\theta \theta }^{\mathrm{II}}(\psi )}{f_{\theta \theta }^I(\psi )}\right)}^{\frac{1}{{\lambda }_I-{\lambda }_{\mathrm{II}}}}{\left(\frac{K_{\mathrm{II}}}{K_I}\right)}^{\frac{1}{{\lambda }_I-{\lambda }_{\mathrm{II}}}}$$

(17)

So we can state, using this bound, that for the stresses at a point $s$ along the interface to be approximated by the asymptotic solution, it is a necessary but not sufficient condition that $s/d_r<1$ .

Consideration of other angles

So far, throughout this article, our examples have been limited to a square block resting on half-plane; however, it is not necessary to do this, and the general results presented are equally applicable to any complete contact. An obvious extension is to take the internal angle of the block as a parameter, $\psi$ . In order to apply these results to a new example, it is necessary to perform calibrations by FE analysis to find the stress intensity factors. Otherwise, however, we may proceed as for the 90° block. The details of this analysis are discussed in Appendix 2. Several such calibrations have been done, and included here is such a calibration, whose internal angle is $\arctan (-3/4)\approx {143}^{°}$ . This is chosen so as to display a mode II dominant region, as explained in section ‘A bound on the existence of mode II–dominated regions’

$$\left[\begin{array}{c}{K}_I{a}^{{\lambda }_{I-1}}\hfill \\ \multicolumn{1}{c}{K_{\mathrm{II}}{a}^{{\lambda }_{\mathrm{II}-1}}}\end{array}\right]=\left[\begin{array}{cccc}-0.2260\hfill & 0.1070\hfill & -0.4415\hfill & 0.1516\hfill \\ \multicolumn{1}{c}{0.0369}& 0.2711\hfill & 0.0930\hfill & 0.1094\hfill \end{array}\right]\left[\begin{array}{c}\frac{P}{2a}\hfill \\ \multicolumn{1}{c}{\frac{Q}{2a}}\\ \multicolumn{1}{c}{\frac{M}{2a^2}}\\ \multicolumn{1}{c}{\frac{\mathrm{\Delta }\sigma }{2}}\end{array}\right]$$

(18)

Use of this calibration leads us to conclude that under no combination of pure $P\mathrm{and}Q$ loading does this block lift off at the corner (see Figure 7), but it can be made to do so by using other methods of loading (as in Figure 8), however only for large values of the ratios $a\mathrm{\Delta }\sigma /P$ and $M/\mathrm{aP}$ . The argument against an interior lift-off point that applied to the 90° block applies again here, as can be seen in Figure 9.

Figure 7.

Plot showing the regions of behaviour for the 143° block.

Figure 8.

Characterisation of slip behaviours of the 143° block in four load space.

Figure 9.

The variation of stress intensity factors and resultant constants with varying $Q/P$ for the 143° block.

Slip consideration can also be made, and the letters shown in Figures 7 and 8 relate again to Table 1, which is still applicable here. Common throughout all of these is that, for moderate loading and largely for positive valued loading (as defined by convention in Figure 1), slip will originate from the corner, regions marked ‘B’, and hence the conditions necessary to prevent slip at the corner are sufficient to prevent interfacial slip throughout the interface.

A bound on the existence of mode II–dominated regions

Throughout this analysis, we have utilised a two-term analysis. We have already seen that it is always possible to find a point where the state of stress is dominated by the first term in our expansion. However, we now ask whether the same can be said for the second term. We already know that these points, should they exist, would be at a point $r>d_0$ . We also know that for them to be valid for consideration with asymptotic methods, we require $r<d_r$ . So it is natural to conclude that we require the ratio $d_r/d_0>1$ for asymptotic consideration to be appropriate. A comparison of the forms of these values leads to the ratio

$$\frac{d_r}{d_0}={\left(\frac{-f_{\theta \theta }^{\mathrm{II}}(\psi )}{f_{\theta \theta }^I(\psi )}\right)}^{\frac{1}{{\lambda }_I-{\lambda }_{\mathrm{II}}}}$$

(19)

Plotting this, we find that, as shown in Figure 10, the value $d_r/d_0=1$ is achieved at approximately $\alpha ={\alpha }_0=0.89\pi$ or alternatively $\psi ={\psi }_0={139}^{°}$ . Consequently, we cannot find a point where the mode II term dominates the mode I term and which is within the valid range of the asymptote, for values of $\alpha <0.89\pi$ . So we cannot see a region of mode II dominance for an indenter of internal angle less than 139°.

Figure 10.

The value of the ratio $d_r/d_0$ with varying total internal angle $\alpha$ .

A new normalised form of the solution

Finally, we may use $d_r$ to re-write the equation for ${\sigma }_{\mathrm{ij}}$ in a different way

$$\frac{{\sigma }_{\mathrm{ij}}(r,\psi )}{G_r}={\left(\frac{r}{d_r}\right)}^{{\lambda }_I-1}-{\left(\frac{r}{d_r}\right)}^{{\lambda }_{\mathrm{II}}-1}$$

(20)

where

$$G_r={\left(\frac{K_I}{f_{\mathrm{ij}}^I(\psi )}\right)}^{\frac{{\lambda }_{\mathrm{II}}-1}{{\lambda }_I-{\lambda }_{\mathrm{II}}}}(-K_{\mathrm{II}}{f}_{\mathrm{ij}}^{\mathrm{II}}(\psi ){)}^{\frac{{\lambda }_I-1}{{\lambda }_I-{\lambda }_{\mathrm{II}}}}$$

(21)

$$d_r={\left(\frac{-f_{\mathrm{ij}}^{\mathrm{II}}(\psi )}{f_{\mathrm{ij}}^I(\psi )}\right)}^{\frac{1}{{\lambda }_I-{\lambda }_{\mathrm{II}}}}{\left(\frac{K_{\mathrm{II}}}{K_I}\right)}^{\frac{1}{{\lambda }_I-{\lambda }_{\mathrm{II}}}}$$

(22)

Conclusion

The objective of this analysis was to gain extra information about the form of solution from a two-term asymptotic consideration, compared to that found using a single-term approximation. This has proven to be a difficult task. However, several general conclusions can be drawn from the work presented.

A set of conditions which categorise slip behaviour solely from the signs of the relevant stress intensity factors have been devised. We find that in cases with positive $K_{\mathrm{II}}$ , the coefficient of friction required to prevent slip at the corner is sufficient to prevent any slip within the asymptote. The same is not true when $K_I$ is negative, with slip starting within the body. This allows, with appropriate FE calibrations, for the categorisation of four load space into four regions of distinct behaviour covering the entire space.

Furthermore, a bound on the validity of asymptotic solutions was established, and from this, a new normalisation of Williams’ solution is possible. Also resulting is a bound on indenter angle necessary for a mode II dominant region to be visible by asymptotic methods.