A Critical Review on the Complex Potentials in Linear Elastic Fracture Mechanics

Introducing a crack in an elastic plate is challenging from the mathematical point of view and relevant within an engineering context of evaluating strength and reliability of structures. Accordingly, a multitude of associated works is available to date, emanating from both applied mathematics and mechanics communities. Although considering the same problem, the given complex potentials prove to be different, revealing various inconsistencies in terms of resulting stresses and displacements. Essential information on crack near-tip fields and crack opening displacements is nonetheless available, while intuitive adaption is required to obtain the full-field solutions. Investigating the cause of prevailing deficiencies inevitably leads to a critical review of classical works by Muskhelishvili or Westergaard. Complex potentials of the mixed-mode loaded Griffith crack, sparing restrictive assumptions or limitations of validity, are finally provided, allowing for rigorous mathematical treatment. The entity of stresses and displacements in the whole plate is finally illustrated and the distributions in the crack plane are given explicitly.


Introduction
With the goal of calculating strains at every point in a beam, Airy introduced the now called Airy stress function [1]. With this function, the constitutive law of elasticity and compatibility conditions of strains a biharmonic equation is derived, finally providing stress solutions if an appropriate ansatz of the Airy stress function is chosen, satisfying boundary conditions. The first work on elastic crack problems is commonly attributed to Inglis [13], who adopted Love's solution of a plate with an elliptic hole [18] as a basis. Already six years before, Wieghardt [34], however, published a paper in German language, translated "On the cleavage and rupture of elastic bodies", where stresses at cracks and sharp corners are rigorously derived from Airy's stress function.
A more general and contemporary solution of modern fracture mechanics employs complex function theory and is given by two complex holomorphic potentials. Kolosov [15] was the first to obtain the displacements and stresses based on these two functions. Muskhelishvili [20] later extensively utilized these relations and comprehensively investigated the holomorphic functions, being the basis of a very general and powerful approach to determine the solution of various boundary value problems. In fracture mechanics, this approach flourished as it enables the calculation of stress and displacement fields in spite of noncontinuous and non-smooth boundary conditions along the crack plane, finally allowing for the calculation of the crack tip loading.
A less general but more common approach has subsequently been introduced by Westergaard [33] in terms of the Westergaard stress functions which is also applied to determine stress and displacement fields. However, the originally introduced approach was deemed incomplete, only solving a restricted class of crack problems, but was extended to resolve these restrictions [25,27,29]. For particular classes of problems, the two-complex-potential approach can be simplified to an approach requiring only one function, exploiting certain symmetries [27]. This single complex potential is related to the Westergaard stress function. The two methods are widespread in fracture mechanics featuring a multitude of references applied to various crack problems. A lot of them have been compiled, e.g., by Tada et al. [31].
However, Muskhelishvili's complex potentials (e.g., [4,6,8,10,12,16,19,20,23,26,27]) and the Westergaard stress functions as mostly employed (e.g., [2, 9, 12, 17, 22, 28, 30-32, 35, 37]) are just strictly valid in the positive (x > 0) half plane of the crack, see hatched area in Fig. 1. Applying them to the whole domain, obviously reveals discontinuities and unphysical point symmetry. Stresses and displacements in the negative half-plane are thus, inter alia, either obtained intuitively by reflection in the y-axis or by adapting the equations introducing an a priori case-by-case analysis which, however, is not mathematically motivated. While the limited validity in [20] is due to a transformation in the complex number plane which will be discussed in Sect. 4, the approach presented in [33] is basically valid on the whole domain, however, has been transferred erroneously to contemporary literature, basically giving rise to the same issue as in [20].
Following a brief presentation of essential fundamentals in Sect. 2, the complex potentials and stress functions provided by established literature as well as resulting stress and displacement fields are discussed in Sect. 3. Due to various conspicuous features in the results, works of Muskhelishvili [20] and Westergaard [33] and their transfer to contemporary literature are critically reviewed. In this context, Sect. 4 provides details on some mathematical issues to be considered. Subsequently, in Sect. 5, current approaches to overcome deficiencies and complement stress and displacement solutions are outlined, finally illuminating why this problem did not attract attention yet. Complex potentials for the whole elastic domain of the Griffith crack are eventually presented in Sect. 6, avoiding case-by-case analysis and incorporating rigid body rotation for the sake of generality, which is basically disregarded in literature. The crack fields are calculated sparing ineligible discontinuities or other irregularities. Being of major interest in fracture mechanics, all stresses and displacements in the crack plane are given explicitly.

Some Fundamentals of Elastic Crack Solutions
An arbitrarily loaded straight crack of the length 2a in an infinitely large sheet of isotropic linear-elastic material, i.e., a Griffith crack, is depicted in Fig. 1. The origin of the complex plane is located in the center of the crack, introducing the complex coordinate z = x + iy. The positions of the crack faces are defined as: assuming traction-free boundaries, i.e., where the superscript ± indicates the positive (+) or negative (−) crack face. The stresses at infinity are controlled by uniaxial tension σ ∞ 22 and in-plane shear loading σ ∞ 12 : The calculation of the crack fields is achieved by Kolosov's equations [15] where and are holomorphic complex potentials on C \ [−a, a] with their respective spatial derivatives , and . The stresses are denoted as σ ij and u i are the displacements, where μ is the shear modulus and κ is Kolosov's parameter. Bars on quantities denote complex conjugates, e.g.,z = x −iy. For holomorphic functions, the Cauchy-Riemann equations are satisfied and employing the Wirtinger derivatives [36], the displacement gradients are calculated as: Exploiting the axial and skew symmetries for the mode I and II, respectively, loading of the Griffith crack, the two complex potentials can be reduced to one, requiring a relation between and [27]. According to [25,29], functions A(z) and B(z) have to be introduced for the sake of generality, whereupon with real constants A and B is provided as special case in [27,29] being valid for the considered problem. The subscript I or I I indicates the affiliation to the respective crack opening mode. Furthermore, the reduction to one complex potential enables a relation to the Westergaard stress functions [33] Z I (z), Z I I (z) according to [25,29]: Applying the superposition principle, e.g., = I + I I , and inserting Eq. (7) into Eq. (4) yields with Z I/I I being the spatial derivative ofẐ I/I I . The constant B has an impact just on the displacements and A does not contribute to singular crack tip stresses, accounting for, inter alia, homogeneous stress in case of tension or compression loading in x-direction.

Commonly Applied Complex Potentials
The complex potentials commonly employed for the Griffith crack are basically given in [4,16,23,27], whereas [6,8,10,12,26] only cover mode I. Providing just and or and , and sometimes only or bearing on Eq. (6), some calculus is required to obtain the whole set of functions to be inserted into Eqs. (4) and (5). Evolving integration constants represent rigid body translation and may be canceled in a local crack related coordinate system, which is mentioned briefly only in [6]. Rigid body rotation, being included in the derivatives of the potentials, is basically disregarded, thus depriving the solution of some generality. In [19,20] this issue is taken into account, however, skipped in an early stage of derivation. Following [27], in which and are given, and assuming σ ∞ 11 = 0 the complex potentials read: In [16] a different fraction +iσ ∞ 12 /4 is found for the constant mode II term in , in [23] −σ ∞ 22 /2 is given for the mode I term. Inserted into Eq. (4), Eq. (9) provides the crack fields depicted in Figs. 2 and 3. The figures obviously reveal inconsistencies, e.g., the crack tip stress singularities of σ 22 and σ 11 in mode I and σ 12 in mode II exhibit point symmetry and the displacement field is discontinuous (u 2 of mode I, u 1 of mode II) on C \ [−a, a]. Accordingly, the equations of Eq. (9), commonly denoted as holomorphic functions, do not meet the requirements of this specification.
Complex potentials were extensively elaborated by Muskhelishvili [20], originally introducing and according to In general holds, where ε ∞ is the rigid body rotation at infinity which is set to zero in the derivation of Eq. (10) in [20]. Furthermore, σ I/I I are the principal stresses at infinity and α is the angle between the σ I -and the x-axis. Equations (10) and (11) are cited by, e.g., [7,32]. For the case of biaxial tensile and in-plane shear stress loading, and 2 read inserted into Eq. (10) resulting in the functions of Eq. (9), for σ ∞ 11 = 0. It is noted that in the original [20] 2 is denoted as . However, a majority of authors applies stress functions according to Westergaard's approach in terms of e.g., [8,12,17,22,28,32,35,37] for mode I and [2,9,30,31] for mixed mode crack opening, basically raising the same problems as the complex potentials of Eq. (9) in terms of symmetry and discontinuity. In the original paper by Westergaard [33], just the function Z I (z) is introduced, however, being dissimilar from the one in Eq. (14), actually reading While Eqs. (14) and (15) seem to be equivalent at the first glance, in fact they are not. This issue being one source of problems, the following section provides some detailed discussion.

The Problem of Complex Roots
The Eqs. (9), (10), (14) and (15) include roots with complex radicands. The common root function x → √ x is defined for real x ≥ 0 and results in non-negative values. In order to define the root function on C completely, the common polar form z = |z|e iφ (16) is considered, with the argument φ ∈ (−π, π], and the root being Equation (17) is called the principal root and ω 2 = z with unknown ω has the solutions ω = ± √ z. Moreover, in Cartesian coordinates holds. Note that z → √ z is holomorphic in the slit plane C \ (−∞, 0] and discontinuous in the real interval (−∞, 0). In general the principal branch is defined as z r := |z| r e riφ (19) for r ∈ R. It is straightforwardly shown that the derivation rule d dz (z r ) = rz r−1 (20) and the law of exponentiation z r 1 z r 2 = z r 1 +r 2 (21) hold in C. In particular need to be considered in fracture mechanics problems. Note that with Eqs. (18) and (22), the complex potentials in Eq. (9) are formulated in Cartesian coordinates instead of employing polar coordinates. The irregularities in the complex potentials of the Griffith crack in Eqs. (9), (10) and (14) arise, because in general z r ω r does not equal (zω) r . For the sake of clarification, the function appearing in these equations, will be compared to the function with its derivative Furthermore, holds on C \ [−a, a], which is needed for the calculation of in Eq. (10). Looking at reveals that the functions do not coincide, even though they are treated equally in literature, starting with the cutting-edge work of Muskhelishvili [20], who replaces the initially introduced g(z) by f (z) in his derivation of Eq. (10). The function f (z) even has discontinuities on the imaginary axis, which can be shown by the example It is hence not holomorphic in the relevant domain C \ [−a, a]. In contrast, the function g(z) of Eq. (24) is continuous on the imaginary axis and is equivalently represented by which appears in Eq. (15). In order to prove this equivalence, the real arguments of g(z) are examined.
Step 3: Equation (29) shows that g(z) is odd and holomorphic on C \ [−a, a]. Finally, the behavior of g(x), according to Eq. (24), is examined on the crack faces, i.e., x ∈ [−a, a], where and lim ε 0 hold. The equivalence of Eqs. (24) and (29) is also valid for x ∈ (0, a] where is satisfied. For x ∈ (−a, 0] the equivalence is not given, as g(x) in Eq. (24) is even on [−a, a], whereas x → x 1 − a 2 /x 2 is odd. It is noteworthy that f (z) and g(z), Eqs. (23) and (24), are equal on [−a, a] if y = 0 is set without employing a limit. To summarize, the relations are crucial for the deduction of complex potentials and holomorphic functions, respectively, of linear elastic crack problems. They emphasize that the function √ z 2 − a 2 must not be involved, which has been disregarded in Eqs. (9), (10) and (14), eventually giving rise to the problems depicted in Figs. 2 and 3. Equation (15), on the other hand, is correct. Equations (31) and (32) further demonstrate that approaching y = 0 on the crack faces, necessarily involves taking an upper or lower limit. Fig. 4 Comparison of normalized stresses in the crack plane, i.e., y = 0, for mode I loading from different sources

Current Approaches Encountering the Deficiencies
Despite of the deficiencies of the solutions illustrated in Figs. 2 and 3, stresses and displacements turn out to be correct in the positive half of the boundary value problem according to Fig. 1. To obtain crack fields on the whole domain from complex potentials being valid only for Re[z] > 0, a case distinction is commonly applied in literature. This is most suitably demonstrated taking the stress σ 22 in the crack plane, i.e., y = 0, for a single mode I loading, which is given directly in [10,16,24,38] and indirectly via Westergaard's stress functions according to Eq. (14) in [2, 9, 12, 17, 22, 28, 30-32, 35, 37]: σ 22 is obviously positive for x > a and negative for x < −a. Therefore, the following formulation is intuitively employed actually lacking mathematical rigorousness. In, e.g., [4,21] the stress is given monolithically as however, the associated complex potentials are either not provided [21] or do not result in Eq. (37) [4]. Equation (15), yielding a mode I stress according to is depicted in [14,31], whereat [31], however, concurrently provide the stress function Z I (z) of Eq. (14). The stresses of the Eqs. (35) to (38) are illustrated in Fig. 4.
In order to obtain Eq. (36), the complex potentials need to be formulated likewise by separating ranges of validity, for the example of reading For the sake of generality, an integration constant C = C 1 + iC 2 in (z) accounts for linear rigid body motion and the last term in (z) incorporates rigid body rotation. While translational motion vanishes in the local crack related coordinate system, trivially satisfying the condition and thus C = 0, the rotational motion does not basically vanish, depending on the boundary conditions. In [19,20] it is introduced in "general formulae", however, disregarded in the specific boundary value problem. The rotational term introduced in Eq. (41) has to satisfy the condition of rigid body rotation at infinity, i.e., which, with the displacement gradients of Eq. (5), leads to −2iμε ∞ /(1 + κ) in . According to Eq. (4), the stress fields are not affected by the additional term. Inserting Eq. (41) into Eq. (4), with σ ∞ 11 = 0 and zero rigid body rotation, the crack fields depicted in Figs. 5 and 6 are obtained, consistently employing the principal complex root. Due to expected symmetries as well as continuities and discontinuities, the plots illustrate the plausibility of the results. Equations (24) and (29) being equivalent for z ∈ C \ (−a, 0], the complex potentials alternatively are formulated as follows: In the crack plane (y → ±0), being of major interest in fracture mechanics, the holomorphic functions of Eqs. (41) or (44) for σ ∞ 11 = 0 and ε ∞ = 0 yield the following results for mode I, |x| ≥ a, and for mode II |x| ≥ a, x − a and the ligament of just one crack tip. Secondly, a rigorous mathematical derivation is not required to obtain a comprehensive solution, but can be replaced by intuitive continuation of a partially valid solution. Finally it is noteworthy, that the addressed problems of the complex potentials or Westergaard stress functions do not only arise for the Griffith crack, but probably for various other crack problems, e.g., the Dugdale crack model [3,5,11,31].
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Conflict of Interest
The authors declare that they have no conflict of interest.
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