Analytical solution for the stress field of hierarchical defects: multiscale framework and applications

Hierarchical defects are defined as adjacent defects at different length scales. Involved are the two scales where the stress field distribution is interrelated. Based on the complex variable method and conformal mapping, a multiscale framework for solving the problems of hierarchical defects is formulated. The separated representations of mapping function, the governing equations of potentials, and the stress field are subsequently obtained. The proposed multiscale framework can be used to solve a variety of simplified engineering problems. The case in point is the analytical solution of a macroscopic elliptic hole with a microscopic circular edge defect. The results indicate that the microscopic defect aggregates the stress concentration on the macroscopic defect and likely leads to global propagation and rupture. Multiple micro-defects have interactive effects on the distribution of the stress field. The level of stress concentration may be reduced by the coalescence of micro-defects. This work provides a unified method to analytically investigate the influence of edge micro-defects within the scope of multiscale hierarchy. The formulated multiscale approach can also be potentially applied to materials with hierarchical defects, such as additive manufacturing and bio-inspired materials.


Introduction
During the manufacturing process, microscopic edge defects emerge around non-ideal cracks such as elliptic cracks and inclusion. The adjacent defects at different length scales can be described as hierarchical defects. Usually, there are two scales involved. Presence of the microscopic defect has substantial influence over the stress level at the macroscopic defect, and possibly leads to global crack propagation and rupture [1][2] . The effects of the microscopic defect on the macroscopic crack propagation cannot be over-emphasized. Seeking the solution of the stress field at the crack tip is of great significance to engineering practice.
For non-ideal cracks, calculation of the stress field at the crack tip is relatively complex [3][4] . Analytical solutions and numerical methods stand out as the two main approaches. Pioneering research work includes the Westergaard stress function, the integral equation method, and the complex variable method [5][6] . These analytical solutions play an important role in the early stage of fracture mechanics. With the development of computational technique, numerical methods emerged, such as the finite element method [7][8] and the boundary element technique [9][10] . Through commercial softwares, the discretization method is generally adopted to approximately calculate the stress field distribution at the crack tip. Considering the complexity and variety of cracks, formulating a universal representation of the stress field at the crack tip is almost impossible. Tremendous calculation is needed, and the major effect factors are difficult to be extracted through the numerical methods. On the contrary, the major factors can be straightforwardly reflected in the analytical solutions. Although certain limitations cannot be avoided, accurate analytical solutions for the stress field at the crack tip remain prevalent yet more challenging among modern researchers.
A series of studies are initiated to seek the analytical solutions of notches and cracks [11][12][13] . Based on the analytical framework, the notch stress intensity factor (NSIF) is applied to the weld roots of welded joints with inclusion of the T-stress component [14] . It is noteworthy that the T-stress is determined from the finite element models evaluating the ligament stresses close to the pointed slit tip. Similar work can be found in Refs. [15] and [16]. Much progress has also been made to analytically solve the problem of edge dislocation in front of blunt crack for nanocrystalline materials [17][18][19] . Recent work witnesses that the use of digital image correlation method is combined with analytical analysis for determining the stress field at blunt V-notch neighborhood [20][21] . Despite the accumulating research on the analytical approach, it shall be emphasized that the multiscaling features of hierarchal defects are rarely addressed recently. A series of studies [16,[22][23] incorporated the multiscaling features in the crack growth model. Analytical representation of the stress intensity factor is derived for macroscopic crack possessing a micro/mesoscopic notch tip. The concept is further applied to the fatigue crack growth model [24][25][26] . Nevertheless, the model is stipulated to ideal cracks that are different from the hierarchical defects described here. To the best of the authors' knowledge, the research on the analytical solution for hierarchical defects is rather surprising within the framework of multiscaling.
The aim of this paper is to formulate a multiscale framework for the analytical stress field solution of hierarchical defects. The strategy is to employ the scale separation method, as well as the conformal mapping of complex variable method. Therefore, the analytical solution for the stress field at the crack tip is obtained. The present work focuses on the hierarchical defects where two scales are involved through the process. Subsequently, the proposed framework is applied to analytically investigate a macroscopic elliptic crack with microscopic circular edge defects. The framework is flexible to solve a variety of simplified engineering problems.
The general multiscale framework is formulated in Section 2, which is mainly based on the complex variable method and the scale separation method. The applications of the proposed framework are presented in Sections 3 and 4, with the emphasis put on single and multiple edge defects at the microscopic scale. Conclusions are briefly summarized in Section 5.
2 Complex variable method for solving the stress field of hierarchical defects: a multiscale framework

Basic formulation of the complex variable method
According to the complex variable method [5] , the stress and displacement of an elastic continuum can be expressed in the form of functions of complex variables when the body force is ignored, which yields where E and ν are Young's modulus and Poisson's ratio of the material, respectively. ϕ 1 (z) and ψ 1 (z) are called the complex potentials which are both analytic or holomorphic functions inside the domain of Ω. Since the analytic function is equivalent to the holomorphic function in complex analysis, in what follows, these two terms are interchangeably used.
For the hole problems, technique of conformal mapping is jointly used with the above theory. If a transformation that could map the actual domain Ω (usually an infinite plane containing a hole/defect in the z-plane z = x + iy) onto a unit disk D in the ζ-plane (ζ = ξ + iη) is found as follows: the complex potentials are in the forms of where ϕ(ζ) = ϕ 1 ω(ζ) and ψ(ζ) = ψ 1 ω(ζ) are both analytic functions in the open unit disk, and are determined by the boundary conditions. The complex number X + iY is related to the boundary force applied on the inner hole. If the inner forces do not exist, the above equations could be simplified as Furthermore, ϕ 0 (ζ) and ψ 0 (ζ) are governed by the following integral equations: where Γ is the unit circle, and σ = ζ| Γ = e iθ , denoting that ζ is now on the boundary Γ. Also, we have where B and B − i C are determined by the remote stress status, corresponding to hydrostatic and shearing stress, respectively, in which γ is the included angle between the first principal stress σ 1 and the x-axis. In the condition of remotely uniaxial tension, assuming that the tensile stress q is applied in the direction of γ, the above equations could be further simplified as Then, denote The stress components described on the mapped curvilinear coordinates could be further written as [5] ⎧ ⎪ ⎪ ⎨ where ρ and θ are the directions mapped from the ξ-axis and the η-axis, respectively.

Conformal mapping and scale separation for problems of hierarchical defects
In this section, we start from a normalized case and further generalize the problem. Finally, the universal form of separated mapping functions at multiple scales is derived.
First, we consider an infinite plate with a circular hole and a semicircular micro-defect on its edge (specifically at z = 1), as shown in Fig. 1(a). The function, which transforms the unit disk (the ζ-plane) into an infinite plane with a unit circular hole (the z 1 -plane), can be expressed as Assume that the analytic function of complex variable mapping the hierarchical-defects domain (see Fig. 1(a) in the z-plane) onto the unit disk (see Fig. 1(b) in the ζ-plane) has been found as z = ω(ζ). Then, the deviation from the original mapping function ω 0 could be defined as Generally, δ(ζ) maps the unit disk to a very small defect in the order of ρ 1, as shown in Fig. 1(c). The δ(ζ) function, which though may have branch points on the boundary of |ζ| = 1, is holomorphic inside the open unit disk. Similar to a compactly supported function, its value falls in the order of ρ in the very neighborhood of the micro-defect (i.e., ζ = 1 here), and dramatically approaches zero when ζ is far from the defect. Detailed discussion on this function could be found in Subsection 3.1. It should be noted that this paper just illustrates an example where the microscopic edge defect is modeled as a semicircular hole. In fact, the micro-defect could be a crack or other shapes in the multiscale framework.
Then, Eq. (14) could transform ζ = e −iα to a micro-defect. Now, a new mapping function ω * (ζ, α) that maps an ideal unit disk (the ζ-plane) to the micro-defected unit disk on its edge (here described in the ζ 1 -plane), as shown in Fig. 1(d), could be constructed by combining Eqs. (13) and (14), From Eq. (15), a general form of conformal mapping that treats the arbitrary shape of hole could be derived. Assuming that the original function which maps the unit disk onto an infinite plate with this hole (without any edge defect) is known, where c n are complex constants. Substituting the variable ζ 1 in Eq. (15) into ζ in Eq. (16), the required function ω(ζ) is obtained. It could successfully map the unit disk to an infinite plate containing a general hole (the primary defect) as well as a microscopic edge defect (the secondary defect).
Figures 2(a) and 2(b) show two cases of elliptic and square holes, respectively, using Eq. (17), where the original mapping functions ω 0 (ζ) could be found in Ref. [5]. Since the value of δ(ζ, α) is far less than 1, the mapping function of Eq. (15) could be asymptotically expanded as follows: Ignoring higher-order terms over O(δ), Eq. (17) turns out to be It is noticed that δ(ζ) is a compactly supported function when ζ is far from e −iα , δ(ζ) approaches zero rapidly. The above equation could also be written as Obviously, ζ 2 ω 0 (ζ) = 0 on the contour |ζ| = 1. Equation (18) is the asymptotic form that transforms a unit disk in the ζ-plane onto the plate with an arbitrary shape of the hole and a microscopic edge defect. To be specific, the location of the micro-defect is at ζ = e −iα in the ζ-plane, ζ 1 = e iα in the ζ 1 -plane, and/or z = ω 0 (e −iα ) in the z-plane.
Defining that Eq. (18) could be rewritten as or more directly, be written in the scale separated form as Here, ω macro is ω 0 , and ω micro = kδ(ζ). To facilitate formula writing, the notation of ω 0 would be kept later.
In Eq. (20), the first term is the mapping function for the primary defect (the hole), and the second term is the perturbation due to the existence of the secondary defect (the microscale edge defect). The parameter k defined in Eq. (19) is a scaling factor for micro-defect due to the transformation process, which is related to ω 0 (ζ), the defect location, and the defect size.
It is found from Eq. (4) that the complex variables of B and B + i C are both scale independent. Since the mapping function ω(ζ) has been written in a separated form as shown in Eq. (20) or (21), we now focus on separating ϕ 0,micro and ψ 0,micro in Eq. (5), i.e., First of all, f 0 in the right-side integral of Eq. (5) could be simply divided into two parts. According to Eqs. (6) and (7), we have where Now, we focus on the integral on the left side of Eq. (5), According to Eq. (17), the derivative of ω(ζ) could be written as Since ω 0 (ζ) and ω * (ζ) are both conformal transformations that are reversible and singlevalued, ω 0 (ζ) has no zeros inside the open unit disk |ζ| < 1, which also applies to dζ 1 /dζ. It should be noted that in some cases, the derivative may equal zero on the boundary Γ. A typical example is a crack, where one geometric point on the crack surface may correspond to two material points. At this time, ω 0 (ζ) = 0 occurs at the crack tip (the z-plane), corresponding to one certain point on the boundary of |ζ| = 1 in the ζ-plane. Overall, there is no zero point inside the unit circle for ω (ζ).
Considering σ on the boundary Γ, we have σ = 1/σ and Since ω (σ) has no zeros and σ 2 ω(σ) is bounded, the above expression is analytic in the open disk, and thus could be written in the form of Taylor series, where h(σ) is analytic inside the unit circle, and is bounded at the infinite, i.e., h(∞) = C. Recalling Eq. (5), ϕ 0 (σ) can be divided into into two parts, i.e., Here, ϕ 0,macro (σ) and ϕ 0,micro (σ) are both analytic inside the unit circle, and thus could be expressed in polynomial series that ϕ 0,macro (σ) = ∞ n=1 a n σ n and ϕ 0, respectively. We have Together with Eq. (27), the integrand in Eq. (5) could be written as It could further be simplified through Cauchy's integral formula, which yields . It is known that H 1 (σ) and H 2 (σ) are analytic in the domain outside the unit circle, including the infinity. Thus, σ = ∞ is a removable singular point, and Based on the above results, the first governing equation in Eq. (5) could thus be calculated as For a cluster of primary (macro) holes that the highest order in the polynomial expansion of the transformation function ω 0 (ζ) is no more than 1, named as the simple hole here, c j = 0 (m equals zero). The circle (ω 0 = 1/ζ) and the ellipse (ω 0 = 1/ζ + mζ) both fall into this category. The above expression could easily be constituted as For simple-hole cases, the governing equations at macroscopic and microscopic scales could be completely decoupled.
Substituting it into Eq. (5), it could be achieved that By the definition of ϕ 0,macro (ζ), we have and the governing equation for ϕ 0,micro (ζ) could thus be extracted, Factors that play roles in determining the potential function at the microscopic scale could be identified from Eq. (36). First, on the left side, the integral kernel has been modified, introducing the deviation of mapping function due to the microscopic edge defect. Also, the macroscopic potential of ϕ 0,macro should be passed into the governing equation at the microscopic level, which could also be treated as a modification to f 0,micro . Obviously, when ω micro (σ) = 0, i.e., δ(σ, α) = 0, the complex potential ϕ 0,micro (ζ) degenerates to zero.

Determination of stress components
First, we denote the differentiation of ϕ 1 (z) as Φ(ζ), It could be separated in the form of where Similarly, the differentiation of ψ 1 (z) could also be separated where The stress could thus be calculated as follows: and It could be expanded as In order to simplify the above formula, we define The above equation could be written as Equations (45) and (48) could be split into two parts, namely, where the macro part is naturally defined as Substituting the above equation into Eqs. (45) and (48), the micro level of stress expressions could be extracted, Combining Eqs. (50) and (51), the stress field is distinctively solved, where the micro part could be seen as a deviation (or perturbation) from the macro part due to the existence of microscopic edge effect, making it possible to evaluate its effect on global behaviors. Also, the macro part of stress would turn zero when ω micro (ζ) = 0, which coincides with the actual situation.

Multiscale framework application: an elliptic hole with single edge defect
In engineering practice, inclusions or certain defects can be simplified as elliptic holes. The problem stated here is an elliptic hole (the primary defect) with an semicircular edge defect (the secondary defect). The adjacent primary and secondary defects are thus referred to as hierarchical defects. It should be noted that the framework is applicable for any other shapes of primary and secondary defects.

Mapping functions for semicircular micro-defect
First, an infinite plate containing a circular hole (unit size) and a semicircular edge defect (radius of ρ 1) at z = 1 is considered, as shown in Fig. 1(a) or Fig. A1. The transformation that maps the above domain Ω (the z-plane) into a unit disk D (the ζ-plane) is written below. The details could be found in Appendix A.
where ω(ζ) is a multi-valued function, and the branch of √ −1 = i is chosen here. It could be verified that when ρ = 0, the above equation will be degenerated to 1/ζ, which is exactly the mapping function for the classical solution of circular hole.
The deviation of mapping function could thus be obtained through Eq. (13), The micro-defect location could also be taken into account by Eq. (14), allowing δ(ζ) to be generalized as δ(ζ, α) has two branch points on the unit circle around the location of e −iα , denoted as ζ 1 and ζ 1 , respectively.
As seen in Fig. 1(c), the short arc linking the two branch points (the ζ-plane) is mapped to the semicircular arc in the z-plane. This ideal problem could be directly solved using the formulae presented in Subsection 2.1, without using the multiscale framework presented in this article. Here, we briefly give the solution for comparison. It is assumed that the plate is subject to a uniformly remote force of q parallel to the y-axis, which allows Eqs. (8) and (9) to be

H(σ) of Eq. (27) could be calculated as
It could be verified that the integral on the left-side of Eq. (5) equals zero, which suggests that ω(ζ) is analytic in the punctured unit disk D \ {0}, and has a simple pole at 0. Therefore, it could be expanded by the Laurent series in the disk, where b n are complex constants. According to the Cauchy integral formula, Eq. (5) is reduced to and Through further computation from Eq. (11), the SCF for this problem could be achieved,

Solutions of complex potentials and stresses
First, it has been known that the transformation for mapping an ideally elliptic-holed infinite plane into a unit disk is Here, R = a+b 2 , and m = a−b a+b , where 2a and 2b are the width and the height of the ellipse, respectively.
Following Eq. (17) or (18), the overall mapping function could be constructed as follows: where δ(ζ, α) could be found in Eq. (55). Due to the compactly supported property of δ function (at ζ = e −iα ), the above transformation could also be approximately written as where k = R(1 − me −2iα ) is the scaling factor. It is found that the actual radius of the micro-defect is Figure 3 shows an example of the transformation. The branch points located in the neighborhood of ζ = e −iα are ζ 1,2 = e −iα (1 − 2ρ 2 ± 2 ρ 4 − ρ 2 ), which are mapped to the two points where the microscale circle intersects with the ellipse, i.e., Points B and C.
To exactly locate the micro-defect, the relationship of the radial angles between the two coordinate systems is provided, where β is the angle between the x-axis and the center of the micro-defect (the z-plane). Based on the above preparations, the potentials and stresses are ready to be solved. Since the elliptic hole is a simple hole, Eq. (31) is adopted.
Here, δ(ζ, α) is analytic in the punctured unit disk, and ζ = 0 is a removable pole. Hence, it could be expressed in the form of Taylor series within the open unit disk, where c n are complex constants related to α. Using the Cauchy integral formula, we have Through similar integral operations, ψ 0 (ζ) could be obtained as follows: The macroscale part of the stress components in the curvilinear coordinates could be calculated through Eq. (50) as follows: where ρ 1 = |ζ| is the radial coordinate in the ζ-plane. Equation (76) is exactly the same as the classical solutions. The microscale part of the stress which is seen as a deviation from the macroscopic part could be calculated from Eq. (51), Substituting the expressions of ϕ macro (ζ) and ϕ micro (ζ) into the above equation, we have According to Eq. (47), the micro part of the stress field in Eq. (52) could be calculated as follows:

Results and discussion
As an example, Fig. 4 shows the contours of the micro part of the circumferential stress σ micro θ and the global part with the parameters of a = 3 mm, b = 1 mm, α = 0, γ = π/2, ρ = 0.1, and q = 1 MPa. It is graphically displayed that the micro part of the stress field σ micro θ is appreciable around the micro-defect, while in the area afar, the stress can be ignored, indicating that the influence of micro-defect is more local but not negligible. It is noteworthy that σ micro θ at the corner between the elliptic hole and circular micro-defect is negative, which guarantees the corner stress be zero in overall (the macro part of stress at the corner is obviously positive). The level of stress concentration around the primary hole is significantly affected by the presence of micro-defect. Here, we focus on investigating the situation of micro-defect located at the long axis (the x-axis) of the elliptic hole, i.e., α = 0. Here, a = 3 mm, b = 1 mm, and the actual size of micro-defect is equal to the parameter of ρ (in the ζ-plane), r ≈ |k| ρ = bρ = ρ. Figure 5 shows the stress concentrations in front of the micro-defect with the changes of the remote tensile direction γ and the defect radius ρ. To investigate the influence of micro effect, an additional SCF caused by the microscale stress field is especially extracted and plotted in Fig. 5 with dashed lines.
Besides that, the original value of the SCF without considering the micro-defect is denoted as K 0 . It actually equals K macro , if it is defined in a similar way that K macro = σ macro θ,front /q. K 0 , as plotted with solid lines in Fig. 5, proves the independence of the micro-defect existence. The overall SCF is the summation of K 0 and K micro , Here, the curve of K 0 (the SCF in the case of no micro-defect) coincides with the classic solution for elliptic holes [27] . Furthermore, it could be seen that the smaller the micro-defect, the larger stress concentration it contributes. When ρ = 0.01, K micro contributed by the micro-defect overtakes the original K 0 . Also, the stress at the front of the defect increases with the angle γ of the remote force, reaching its maximum when the remote stress is parallel to the y-axis, or perpendicular to the long axis. In the condition of γ parallel to the long axis, K micro is shown to be zero, indicating that the defect has no extra contribution to the original stress concentration. Here, the attention is paid to a special condition of γ = 90 • for it has the greatest SCF among all directions of remote forces. Through further calculation, the relationship between the defect size r and the overall SCF K could be obtained as follows: In the special case of a = b = 1, it agrees well with Eq. (64). When r is sufficiently small, the SCF approaches 3 + 4a/b. In comparison to K 0 = 1 + 2a/b (Inglis's solution [28] ), we suggest a modified formula to estimate the stress concentrations when it comes to evaluate the possible effect of microscopic edge defect on the safety of holed structures As shown in Fig. 6, the effect of micro-defect location on the SCF is also investigated. K micro reaches its maximum when micro-defect occurs on the long axis, i.e., β = 0 • . It decreases with the increase in β, gradually falls to zero when β = 90 • . It shall also be noted that the maximum influence on the SCF does not exceed 8, even the micro-defect might be very small, which could be validated from Eq. (82). The affecting area of micro-defect on the stress field is also investigated. Figure 7 provides the change of the normalized circumferential stress σ micro θ /q along with the radial distance to the micro-defect front. It clearly shows that the stress decreases rapidly with the increase in the distance, suggesting that the influence along the radial direction is fairly local. Also, Fig. 8 shows the normalized circumferential stress on the hole edge, where the micro-defect is located at α = 45 • (i.e., β = 18.4 • ) and ρ = 0.1. It is graphically displayed that in the circumferential direction (described by its polar angle β in the z-plane), the influence area is mainly between 0 • and 50 • .
The scale effect cannot be overemphasized in the proposed multiscale framework. At this stage, we mainly focus on solving the designated problems with the features of hierarchical defects. It is stipulated that the size of secondary defect is far smaller than the primary defect. We investigate the relative defect sizes varying from 0.01 to 0.2. It could be verified that under these situations, the separated form of conforming mapping, as shown from Eqs. (16)- (18), agrees well with the exact defect boundary. Since the conformal mapping is the key procedure in solving holed problems, using the complex variable method, the exact stress field could be guaranteed through the exact mapping function.
It is undeniable that quantitatively studying the size range of micro-defect within the multiscale framework is rather challenging. First of all, if the lower bound of the micro-defect  17) and (18), a smaller size of micro-defect naturally leads to a more accurate approximation (see Eq. (18)), while for the upper bound of micro-defect size, several factors may influence the accuracy of the approximation, including the shape of the macro-hole that is expressed in the mapping function of Eq. (16), the relative size of micro-defect expressed in the asymptotical expansion next to Eq. (17), and the local curvature of the macro-defect boundary at the micro-defect location (as expressed in Eq. (8)). Quantitatively defining an upper bound of the micro-defect size is not within the scope of the present framework, which will be left for our forthcoming work. It is also noted that for the micro-defect size, approaching zero and being zero are two different cases in terms of the stress distribution. The presence of micro-defects causes the stress concentration, regardless of the micro-defect size. When r is sufficiently small, the overall SCF approaches 3 + 4a/b, as shown in Eq. (82). If r is directly set to be zero, Eqs. (18) and (53) (the conformal mapping), as well as Eqs. (51) and (52) (the stress solution), are automatically degenerated to the case of no edge defect, meaning that no extra stress concentration occurs.
A similar situation is the classical problem of stress concentration of a holed continuum. Taking the axial tension for example, if a circular hole is small enough, the SCF is constantly K = 3, even its size will approach zero. However, in the limit state of no hole, there is obviously no stress concentration at all.

Multiscale framework application: a problem with multiple edge defects
One of the advantages of the multiscale framework is to treat the hierarchical problem with multiple edge defects. Under this situation, the mapping function is hard to be exactly constructed via ordinary methods. However, starting from the proposed multiscale framework, micro-defects could be superimposed, which simplifies the process of finding the mapping function and the forthcoming solutions of the stress field.

Conformal mapping and governing equations
Assuming that there are n different defects existing on the edge of the primary defect, denoted as ζ = e iαn (n = 1, 2, · · · ), the transformation for multiple edge defects could be constructed by extending Eq. (17) to a general form, Ignoring the terms higher than the order of 2, the above function is expanded as z = ω 0 (ζ) + k 1 δ 1 (ζ, α 1 ) + k 2 δ 2 (ζ, α 2 ) + · · · + k n δ n (ζ, α n ) where k n = −e −2iαn ω 0 (e −iαn ). Figure 2(c) shows an example of the transformation. It indicates that the above transformation could successfully map the complicated domain into a unit disk. In the neighborhood of each defect center, there might be two branch points located on the unit circle. Since the function is not differentiable at the branch point, the transformation is not conformal anymore at the point. These points should not overlap with each other in order to guarantee the accuracy of transformation.
Scale separation for multiple defect problems is similar to that of single micro-defect, except for replacing the expressions related to ω micro with the formula defined in Eq. (85) or (84).
First, H(σ) could be written as

Hierarchical defects: an elliptic hole with multiple semicircular defects
In this subsection, we consider the problem of an infinite plate with an elliptic hole and multiple semicircular defects on its edge. For semicircular micro-defect, the polar angle between the two branch points (see Fig. 3) could be estimated from Eq. (56), where i denotes the ith micro-defect, and ρ i could be calculated by Eq. (68). Thus, the polar angle between the neighbored defect centers should satisfy the following requirement to avoid overlapping of branch points: Assuming that there are n micro-defects on the elliptic hole, the mapping function (see Eq. (85)) turns to be where ρ i could be calculated from the actual defect size r i via the formula r i = |k i | ρ i . Here, we also assume the plate is subject to a uniformly remote force of q parallel to the y-axis, which is the same as that in Subsection 3.2. Using Eqs. (87)-(91), the macro part as well as the micro part of the stress field could be calculated.
As for the macro part, the expressions are the same, To make a distinction from the definition of r i in Eq. (94), we use R 1 = |ζ| as the notation of radial coordinate in the ζ-plane, making the above formula slightly different from Eq. (76). The micro part of potentials could be calculated through similar operations described in Subsection 3.2. First, we have where ϕ 0,macro (ζ) is the same as Eq. (73), Thus, we have Finally, we can obtain Re and where Δ 5 is defined as As an example, we calculate a dual micro-defects problem with α 1 = 0 (i.e., β = 0 • ), ρ 1 = 0.1 mm, α 2 = π/4 (i.e., β = 18.4 • ), and ρ 2 = 0.01 mm. Other parameters are a = 3 mm, b = 1 mm, γ = π/2, and q = 1 MPa. The circumferential stress along the hole edge is plotted in Fig. 9. It shows that the micro part of σ θ oscillates in the neighborhood of α = 0 • and 20 • , and rapidly approaches zero afar. The curve has two stress peaks, which correspond to the locations of the two micro-defects, respectively. The interactive effect between multiple edge defects is also investigated. Let the first micro-defect at β = 0 • with r = 0.1 mm be fixed. The location and size of the second micro-defect vary. In this case, a = 3 m, b = 2 mm, γ = π/2, and q = 1 MPa. The stress concentrations at the front tip of the first defect K micro are shown in Fig. 10. It could be seen that the multiple micro-defects have an interactive effect on the stress field, especially when the distance is close. However, if the distance is large enough, the interaction could be ignored as all curves retreat to its original SCF value caused solely by the first micro-defect (here approximately at 4.1). Generally, the coalescence of micro-defects may release the stress concentration, which is understandable if noticing that a smaller radius of micro-defect leads to a higher stress concentration in the previous section. It can also be seen from Fig. 10 that a larger radius of the second defect has a greater effect on the first defect, and certainly has a larger area of influence. An extreme case is that when the size of the second crack is 0.01 mm, its effect is negligible.

Conclusions
A multiscale framework for the analytical solution of the stress field of hierarchical defects is formulated. Emphasis is laid on the hierarchical defects at different scales. Within the general multiscale framework, this mathematical model can be flexibly applied to a variety of engineering problems. The accurate analytical solution is also capable of capturing the factors that significantly affect the process of crack propagation. The main findings are summarized as follows: (i) Representation of the conformal mapping, the governing equations of potentials, and the stress field at two scales is analytically achieved within the formulated multiscale framework.
(ii) The influence of the secondary micro-defect is local but cannot be ignored under certain circumstances. The present work provides a general tool to exactly investigate the influence. A modified formula K u = 1 + 2K 0 is proposed to estimate the stress concentrations.
(iii) One of the benefits of the multi-scale framework is to solve the problems of multiple edge defects. The results show that multiple micro-defects have an interactive effect on the stress filed. Coalescence of micro-defects may release the stress concentration.
The proposed framework can be extended to more general problems, such as primary defect modeled as a square hole or a blunt crack, secondary defect modeled as a micro edge crack, even solution of three levels of hierarchical defects is feasible. Nevertheless, it shall be noted that the formulated multiscale framework is stipulated to elastic materials. Last but not least, the approach can be also applied to additive manufacturing materials, bio-inspired materials that have the features of hierarchical defects [29][30][31] . They are left for our forthcoming work.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International
License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.