The Linear Elastic Wedge Under a Tip Couple at the Critical Angle. Where Is the Paradox?

The classical problem of a two-dimensional infinite linear elastic wedge, with opening angle of 2α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2\alpha $\end{document}, subjected to the action of a concentrated couple at its tip, was considered by Carothers. The solution, which presents a quadratic singularity in the stress field, suffers a spurious behavior at the critical angle α=α∗≃0.715π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha = \alpha ^{*} \simeq \, 0.715 \,\pi $\end{document}, when the stress grows unboundedly within the body. This inconsistency is usually referred to as the “wedge paradox”. Here, relying on rather intuitive arguments, we present a representation for the stress field in the wedge which captures other states characterized by a quadratic singularity. This is first obtained by considering an auxiliary problem, in which the wedge is ideally split into three wedges under tip couples, whose values are determined by compatibility conditions in terms of stress and displacement on the common lines. Remarkably, we find that the state of stress varies continuously on α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha $\end{document}; at α=α∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha =\alpha ^{*}$\end{document} the stress does not vanish, although the action at the wedge tip has zero resultant and zero moment resultant. An alternative derivation of the elastic solution in the wedge, which relies upon the notion of nuclei of strain, suggests that, at the critical angle, the applied action is that of dipoles without moment at the tip point. We conclude that it is the form of Carothers’ solution that cannot account for all states of stress characterized by a quadratic singularity at the vertex. The paradox naturally disappears in the proposed representation.

Fig. 1 The plane linear elasticity problem for an infinite wedge of opening angle 2α loaded at the vertex by forces equipollent to a concentrated couple of moment M, with indication of the polar coordinate system

Introduction
The solution in linear elasticity theory for an infinite homogeneous and isotropic wedge with opening angle 2α in generalized plane stress or plane strain, loaded by actions concentrated at the vertex equipollent to a moment M, was first presented by Carothers [1] and later, independently, by Inglis [2]. Figure 1 shows a schematic representation of the problem: the wedge is loaded only by M at the tip point, while the rest of the boundary is traction free.
The pathological behavior of Carothers solution at the critical angle stimulated a longrunning discussion, which is still far from being over.Numerous studies [4][5][6] were dedicated to finding alternative solutions.
The elastic problem is generally faced following two different rationales: a truncated wedge loaded by tractions distributed on a arc of radius r 0 equipollent to the applied tip action, taking the limit as r 0 → 0; a wedge loaded by equipollent distributed tractions on its flanks, taking the limit as the length of the distribution vanishes.The first approach is the one followed by Carothers.The paradigm of the second approach is in the work by Sternberg and Koiter [3], who considered that on finite segments 0 ≤ r ≤ r of the lateral faces θ = ±α there is a distributed normal loading, antisymmetric with respect to the axis of the wedge and statically equipollent to a couple of moment M. Applying the Mellin integral transformation in the variable r to the governing bi-harmonic equation for the stress function, and by letting r → 0, the authors were able to find an alternative solution which presents an order of singularity different from the (r) −2 trend of (1.1a)-(1.1c),which was consistent also at the critical angle.Again using the Mellin transform, Bogy [7] investigated the dependence of the order of singularity on the opening angle and Lamé constants in a bi-material wedge subjected to surface tractions.
An alternative perspective was followed by Dunders and Markenscoff [8], who considered a wedge loaded by a concentrated couple in the interior that approaches the vertex from the inside.A more physical argument to solve the paradox was proposed by Neuber [9], who advocated the onset of plasticity.By assuming that the shear stress follows a plateau when the material yields, the concentrated couple could be replaced by an equipollent distribution of constant tangential stresses acting along a small circular cavity centered at the vertex of the wedge.In this way, it is possible to construct a general solution that does not suffer the spurious behavior at the critical angle, which reduces to the solution by Sternberg and Koiter [3] when some terms become negligible.Following a similar approach, based on the application of actions in cavities of various shapes, Villaggio [10] showed in several examples that the paradox is not restricted to wedges, but it also appears for unbounded regions that assume the shape of a rectilinear wedge at infinity, but can be very different in the neighborhood of the vertex where the concentrated couple is applied.
Other methods rely on the seminal work by Williams [11] who, by using separation of variables in the Airy stress function, obtained eigenfunctions for singular stress states which could be superimposed to define solutions for the elastic wedge with free edges.The limits of the form of the stress functions proposed by Williams [11], were indicated by Dempsy [12], who extended the results for a wedge with surfaces under uniform pressure and derived the stress states for all possible values of α, including the critical value α * .Leguillon [13] obtained a solution that provided the desired equipollence with the tip couple at the critical angle, but was deficient since it provided infinite moment resultant on arcs of arbitrary length at infinite, tending to a definite limit only when the total wedge angle was considered.Interestingly, the author observed that eigenfunctions entailing a stress singularity of order (r) −2 had to be disregarded because they produced a tip couple of zero moment at the critical angle.
From a pure mathematical point of view, the strange (paradoxical) behavior of the solution by Carothers represents an intriguing problem, for which a wide spectrum of alternative methods have been used.The solutions in linear elasticity are strongly characterized by the order of the singularity of the stress field at the wedge tip, as a function of geometry and boundary conditions.The use of the Mellin transform is a useful method for the asymptotic analysis of the stress state [3,[14][15][16][17].Another approach, comprehensively employed in the study of singularity problems, relies on asymptotic series [18,19].Using a bi-harmonic Green's function for an infinite strip domain and then generalizing the results to a wedge problem, Gregory [20] retrieved the eigenfunctions represented by Williams [11].Green and Zerna [21] used conformal transformation in the complex plane to map the wedge to an infinite strip region and considered the problem via Fourier transformation.
In the practice, the wedge loaded at the vertex by a concentrated couple is a paradigmatic case-study for numerous applications.In fact, it represents a mathematical model problem that can be used in a wide range of engineering cases related to cracks, anti-cracks, notches, bi-material interfaces, re-entrant corners, for which failure initiation is usually associated with the coefficient of the singular part of the stress fields.Therefore, the interpretation of the singularities is of paramount importance not only from a theoretical, but also from a practical point of view.
Here, we do not propose a solution for the wedge loaded by a nonzero tip couple at the critical angle.Rather, our contribution consists in the discussion of the stress states that provide a singularity of order (r) −2 , which correspond to the eigenfunctions overlooked by Leguillon [13] because, at the critical angle, they were associated with tip actions with zero resultant and zero moment resultant.Indeed, we do not use sophisticated methods of analysis, but an intuitive approach, which provides an insight to visualize stress states of this type.We show, in fact, that at the α = α * , the state of stress cannot be expressed in the form (1.1a)-(1.1c),since it corresponds to the action of a system of dipoles with zero moment resultant, concentrated at the vertex of the wedge.We then motivate a generalized representation that does not present any spurious behavior and is valid for any value of the opening angle α.Carothers' solution can be recovered when α = α * .
To present our result, we consider in Sect. 2 an auxiliary problem, in which the elastic wedge of width 2α, with π/2 < α < π, is ideally divided into three sub-wedges loaded by concentrated couples.Compatibility conditions at the common interface lines define the ratio between the couples applied to the wedges and their moment resultant: remarkably this is zero when α = α * , although the state of stress is not null.The derivation of the compatibility conditions in terms of displacement is obtained via Cesaro's line integral representation, following the same arguments by Fosdick and Royer-Carfagni [22,23], as detailed in the Appendix.In Sect. 3 we present an alternative derivation of the generalized solution for the wedge problem by considering the nuclei of strain, according to the definition given by Love [24], associated with the effect of two double forces (dipoles) with moment.We verify that, at the critical angle, the moment resultant of the dipoles shall be zero.

Generalized Solution of the Wedge Problem
Consider an infinite two-dimensional wedge with total opening angle 2α, with π < 2α < 2π , as represented in Fig. 2.
The wedge is subjected to actions equipollent to a concentrated couple with moment resultant M applied at its tip, while the boundaries OA and OB are traction-free.Suppose, for the moment, that α = α * .From (1.1c), we observe that which is satisfied for θ = ±α and θ = ±(π − α).The lines θ = ±α represent the tractionfree faces of the wedge, whereas θ = ±(π − α) correspond to the complementary lines OC and OD, indicated in Fig. 2. All these lines are traction-free because, from (1.1b), σ θθ (r, θ ) = 0 everywhere.The aforementioned observation suggests to consider a complementary problem in which the original body of Fig. 2 is split into three sub-wedges, as shown in Fig. 3, defined by the traction-free lines of the original problem, i.e., such that β = π − α.As indicated in the Figure, a concentrated couple of moment M 1 is applied on the vertex of the wedge I of width 2β, whereas the two wedges II, with opening angle π − 2β, are loaded by tip couples M 2 .
As indicated in Fig. 4, consider two radial paths, similar in type to that represented in Fig. 8, tending to the lines OC and OC (a similar argument can be repeated in a neighborhood of the lines OD and OD ).Let ε (1)  rr (r, θ 1 ), ε (1)  rθ (r, θ 1 ), ε (1)  θθ (r, θ 1 ), and ε (2)  rr (r, θ 2 ), ε (2)  rθ (r, θ 2 ), ε (2)  θθ (r, θ 2 ) represent the strain field components for wedges I and II, respectively.As derived in Appendix, the displacement field on the borders of each one of the three wedges can be calculated from the expressions (A.5), for the radial component υ r , and (A.8) for the tangential component υ θ .In these expressions, only the radial component of strain ε rr and its derivative ε rr,θ are present.Observe that since σ θθ = 0, one has that ε rr = σ rr /E , where E = E in generalized plane stress and E = E/(1 − ν 2 ) in plane strain, being E and ν the Young's modulus and the Poisson's ratio of the material, respectively.Therefore, one Fig. 4 The two radial paths for Cesaro's integral representation, tending to the common lines OC and OC , on wedges II and I , respectively obtains that continuity of the displacement components υ r and υ θ , respectively imply σ (1)  rr (r, β − ) = σ (2)  rr (r, −(π/2 − β) + ) , (2.5a) Taking into account the expressions (2.2a)-(2.2c)and (2.3a)-(2.3c),from (2.5a) one obtains the condition which provides the relationship between M 2 and M 1 in the form It can be directly verified that this choice also satisfies (2.5b).
The moment resultant of the couples applied at the tip of the three wedges reads However, it is of paramount importance to notice that at β = β * the resultant of the applied couples vanishes, as per (2.8).In other words, one has that M 1 = 2M 2 .
Using (2.7), the equations (2.2a)-(2.2c)and (2.3a)-(2.3c)can be merged in a more convenient form.Setting β = π − α, θ 1 = 0 in (2.2a)-(2.2c)and θ 2 = θ − π/2 in (2.3a)-(2.3c),one obtains an expression for the stress field, which is valid for −α ≤ θ ≤ α, in the form (2.9a) (2.9c) which is well defined for π/2 < α < π.Of course, here M 1 does not represent the tip couple, but the couple applied to the sub-wedge I , acccording to the construction of Figure (3).We conclude that, if one considers the wedge problem of width 2α, with π/2 < α < π, via the auxiliary problem of the three wedges, as per Fig. 3, no spurious behavior is detected whatever the angle α is.In fact, the state of stress is given by (2.2a)-(2.2c)and (2.3a)-(2.3c),where M 1 and M 2 have to satisfy (2.7) or, equivalently, (2.9a)-(2.9c).We have to remark, however, that at the critical angle α * the tip couple vanishes, although the state of stress is still well defined and different from zero.We anticipate that this condition, as it will be illustrated later on in Sect.3, corresponds to a wedge loaded by two double forces (dipoles) without moment, concentrated at the tip.
It should be noticed that the state of stress defined by (2.9a)-(2.9c)corresponds, as discussed in [13], to eigenfunctions associated with Airy's stress function of the form (r, θ, α) = (θ, α), characterized by a singularity in the stress field of the order (r) −2 .The three-wedge problem provides a convenient and intuitive representation, which depends smoothly on the opening angle α, with no spurious behavior at α = α * .The form (1.1a)-(1.1c) of Carothers' solution is misleading at the critical angle because M = 0, hence the well-known paradox.We recall that our argument does solve the problem of the wedge with critical angle loaded by a non-zero tip couple.For this, it would be necessary to consider different-in-type types of solutions, such as those proposed by Sternberg and Koiter [3], entailing stress singularities that are not quadratic.
We are not even touching here the issues of the uniqueness of the solution and convergence of the energy integral.If one considered a truncated wedge bounded by an arc of radius r 0 , various systems of distributed tractions on the arc could be found having resultant force zero and resultant moment M = 0.In the limit r 0 → 0, they would correspond to diverse states of stress.Our goal has been only to propose a somehow "physical" representation of those stress states that correspond to the particular eigenfunction enforcing a quadratic singularity at the wedge tip.
We conclude this Section by observing that our construction relies upon the ideal separation of the wedge of Fig. 2 into the three wedges of Fig. 3, which was suggested by the form of Carothers solution, in particular by the condition (2.1).One may wonder if there exist other possible solutions of this type by changing the orientation of the cutting line, identified by the angle β.To do so, let β be an arbitrary angle, with 0 < β < π/2, as indicated in Fig. 5a.Consider again the ideal splitting represented in Fig. 5b, where now the wedge I has width of 2β and the opening angle of the wedges II is α − β.The analysis follows the same arguments outlined above.
The stress state is still given by (2.2a)-(2.2c)for wedge I .Expression analogous to (2.3a)-(2.3c),modulo substitution of π − 2β with α − β, characterize the stress in the wedges II.Traction continuity at the inclined lines automatically follows from the counterpart, for this case, of conditions (2.4a), (2.4b) where, again, (α − β)/2 takes the place of π/2 − β.Using Cesaro's integral representation for radial lines approaching the free boundary of the wedges, the compatibility conditions in terms of displacements on the lines OC and OC (a similar argument holds for the lines OD and OD ) are satisfied if σ (1)  rr (r, β − ) = σ (2)  rr (r, −(α/2 − β/2) + ) , (2.10a) From these, respectively obtains These equations are satisfied provided that Taking into account that, for the problem at hand, 0 < β < π/2 and π/2 < α < π, with α − β < π, this condition yields Therefore, one finds that β = π − α, so that we are back to the case of Fig. 3.This result shows that the only possible splitting of the original problem of Fig. 2 into the three-wedge problem is the one represented in Fig. 3.This result corroborates the proposed generalized solution for the wedge problem under a tip couple, provided by (2.2a)-(2.2c),(2.3a)-(2.3c),(2.7) and (2.8) or, equivalently, (2.9a)-(2.9c).

Stress State Derived from Nuclei of Strain
We now present an alternative derivation of the generalized solution for the wedge problem proposed in Sect. 2. This approach considers possible sources of singularities with different strengths that can be found in linear elastic bodies, referred to by Love [24] as nuclei of strain.
For an infinite three-dimensional linear elastic isotropic solid, the fundamental solution by Kelvin [26] defines the singular displacement field due to a single force applied at a point.Love [24] investigated the stress state in some plane strain problems, for which the stresses tend to become infinite in some specific points inside cavities within a body, outside or on its boundaries.Here, we refer to the analysis contained in [24], Sect.152, for the investigation of the solution of the equations of plane strain that present singularities.
In Cartesian coordinates (x, y), setting r 2 = x 2 + y 2 , the fundamental solution in terms of stress is given by with η = 2(λ + μ)/(λ + 2μ), γ = μ/(λ + 2μ), being λ and μ the Lamé's constants.This state corresponds to a single force1 applied at the origin of an infinite plane elastic body, of magnitude 2πA and acting in the direction of −x.
Fig. 6 Nuclei of strain in an infinite plane body.In the limit (A, B) → +∞ and h → 0 while Ah and Bh remain constant and one obtains two double forces with moment, respectively acting in the direction of the x axis and the y axis Suppose that, in addition to this force, another force with the same magnitude of 2πA but in the positive direction of x acts at (0, h), as indicated in Fig. 6.We now shall let A → +∞ and h → 0 in such a way that the product Ah remains constant and finite.The resulting state of stress ( [24], Sect.152), i.e., ) corresponds to a double force with moment acting in the direction of the x axis.Consider now another nuclei of strain, derived from double forces of magnitude 2πB acting in the direction of the y axis, also represented in Fig. 6.The stress state for the fundamental problem of the force acting at the origin, can be obtained from (3.1a)-(3.1c)with the substitution A → B and interchanging x and y.Taking again the limit for B → +∞, h → 0 such that Bh remains finite, one obtains ) ) ) ) This state of stress corresponds to the limit of the full load condition represented in Fig. 6, which will be referred to [24] as two double forces with moment.
The state of stress given by (3.4a)-(3.4c)can be expressed in the polar coordinate system indicated in Fig. 6, in the form ) define the stress state in the infinite plane.One can use this solution in order to recover the state of stress in an elastic wedge of width 2α, with π/2 < α < π.
Recalling the discussion about Fig. 2 and the conditions (2.1), it is sufficient to choose T A and T B such that σ rθ (r, ±α) = 0, σ rθ (r, ±(π − α)) = 0 . ( From (3.5c), one finds that this requirement is satisfied provided that In this case, one can rewrite the stress state of (3.5a)-(3.5c)as ) ) In conclusion, the state of stress defined by (3.8a)-(3.8c)can be consistently attained in an elastic wedge of width 2α, with π/2 < α < π, with stress-free borders: the action is concentrated at the wedge vertex.This state presents the same order of singularity of Carothers solution (1.1a)-(1.1c)and it coincides with it provided that There is however a substantial and fundamental difference, because the form (3.9) is perfectly consistent whatever the opening angle α is.
Fig. 7 The wedge of width 2α, from which a circular cavity of radius r 0 , concentric with respect to the center, has been cut On the other hand, the representation (2.9a)-(2.9c)for the state of stress, derived in Sect. 2 from three wedge problem, can be recovered from (3.8a)-(3.8c)by setting, for π/2 < α < π, In order to characterize the action applied at the tip, following [24] we consider the geometry of Fig. 7, which represents an elastic wedge of width 2α, from which a circular cavity of small radius r 0 has been cut in a neighborhood of the vertex.It is immediate to verify that the force resultant of the surface stresses acting on the boundary of such cavity is zero.The moment M of the surface stresses with respect to the origin can be calculated as It is important to observe that, only when sin 2α − 2α cos 2α = 0, one can find from this equation T A + T B as a function of M, and substitute it in (3.8a)-(3.8c) to recover Carothers solution (1.1a)-(1.1c).When sin 2α − 2α cos 2α = 0, this substitution cannot be made.
Observe that, in general, T A + T B can take any value, independently of α, since these quantities represent the moments of the dipoles applied to the infinite plane, from which the elastic solution for the wedge problem has been derived.Therefore, one can regard the state of stress defined by (3.8a)-(3.8c)as that occurring in an elastic wedge under the actions of dipoles applied at the vertex.The dipoles have zero force resultant, while the moment resultant M is provided by (3.11).We find that M = 0 at α = α * .We conclude that, at the critical angle, the expressions (3.8a)-(3.8c)represent the state of stress in the wedge loaded by dipoles without moment.The representation by Carothers "hides" this possible static state, because it can be derived only when T A + T B can be found as a function of M from (3.11).

Conclusions
have proposed a generalized representation for the stress state in a linear elastic plane wedge with stress-free borders, which presents a singularity of order (r) −2 .The actions are only applied at the vertex of the wedges and, in general, they are equipollent to a tip couple of moment M. The new form of the solution is intuitively visualized by considering an auxiliary problem, in which the wedge is split into three sub-wedges loaded by tip couples, whose relative value is determined by compatibility conditions in terms of stress and displacement.This allows to consider configurations for which the resulting moment M is zero, although the state of stress does not vanish inside the body.The proposed representation does not suffer the spurious behavior that is observed in the classical solution proposed by Carothers at the critical wedge opening-angle, a geometry for which the stress apparently grows unboundedly within the whole body.On the contrary, the form of the solution we have presented is valid whatever the wedge opening angle is.Remarkably, when the opening angle approaches the critical value, the resulting moment M results to be zero.Our analysis, based on an alternative intuitive derivation of the stress state in the wedge using the notion of nuclei of strain, suggests that, at the critical angle, the action is that of dipoles without moment, acting at the tip point.
Our study does not aim at solving the problem of a wedge loaded by a nonzero concentrated tip couple at the critical opening angle.As indicated by many authors, we repute that a solution associated with an eigenfunction entailing a stress singularity of order (r) −2 cannot be found in this case.Other types of solutions, with a different strength of the singularity, should be considered, but this is another story.According to the classical characterization given by Quine [27], a paradox is an apparently successful argument having as its conclusion a statement or proposition that seems false or absurd.The representation of the stress state in the form provided by Carothers is apparently successful, but it becomes absurd in one particular case.Can this spurious behavior spoil the soundness of the classical solution to this famous problem in linear elasticity theory?Our study suggests that, perhaps, it is the statement of the problem that should be modified.Instead of referring to "a linear elastic wedge under the action of a concentrated couple applied at its vertex", we should formulate the problem as that of "a linear elastic wedge under the action of dipoles at its vertex".In this way, the paradox naturally disappears.

Appendix: Displacement Field from Cesaro's Representation
Cesaro's line integral representation [25] permits a direct integration of the linearized straindisplacement equations.This is used here to state the compatibility conditions in terms of displacement on the common lines in the three-wedge problem of Fig. 3 in Sect. 2. Although the displacement field for the wedge problem loaded by a tip couple is well known [8], we are interested in using the argument proposed in [22,23] by Fosdick and Royer-Carfagni, whose major advantage is that the compatibility conditions can be directly stated in terms of stress components.For the wedge problem, consider a radial path of the type represented in Fig. 8, comprised between the points r = r 1 and r = r 2 , defined by x = x(s) = (r 2 − s) e r , s ∈ [0, r 2 − r 1 ] and θ = const . (A. 2) The path is directed from the periphery towards the vertex because, on the one hand, there is a tip singularity in the wedge problem and the path should tend towards this point; on the other hand, we expect that the displacement tends to zero as r → ∞.Since s = r 2 − r, one obtains x (s) ds = e r dr, (A.3a) x (r 2 − r 1 ) − x(0) = − (r 2 − r 1 ) e r , (A.3b) x (r 2 − r 1 ) − x(s) = − (r − r 1 ) e r , (A.3c)

Fig. 5
Fig. 5 Alternative ideal splitting of the wedge of width 2α loaded by tip couple M. (a) Arbitrary angle 0 < β < π/2 for the cutting line.(b) The auxiliary problem where the wedge I of opening angle 2β is loaded by the tip couple M 1 and the two wedges I I , of width α − β are loaded by M 2 .3c) Let us define T A := 2πAh and T B =: 2πBh.Superposition of the two states represented by (3.2a)-(3.2c)and (3.3a)-(3.3c),provides

Fig. 8
Fig. 8 Radial path for the application of Cesaro's line integral representation, for the wedge problem