Recent Advances in the Analytical Stress Field Solutions for Radiused Notches in Orthotropic Solids

The main aim of this work is to provide a brief overview of the analytical solutions available to describe the in-plane and out-of-plane stress fields in orthotropic solids with radiused notches. To this end, initially, a brief summary on the bases of complex potentials for orthotropic elasticity is presented, with reference to plane stress or strain and antiplane shear problems. Subsequently, the attention is moved to the relevant expressions for the notch stress fields, considering elliptical holes, symmetric hyperbolic notches, parabolic notches (blunt cracks), and radiused V-notches. Eventually, examples of applications are presented, comparing the presented analytical solutions with the results from numerical analyses carried out on relevant cases.


Introduction
Accounting for the effects of geometrical variations is an essential step in the design process of a mechanical component. Stress raisers, indeed, may severely hamper the static and fatigue strength of mechanical parts, and designers are often required to accurately assess the local stress fields in the stress concentration regions, either numerically or analytically.
Over the last 100 years and more, scientists and engineers devoted significant efforts to determining the stress fields around holes, notches, and cutouts, and the first steps in this direction can be dated back to the late 19th century or early 20th century [1][2][3][4]. Fundamental contributions in the field of linear elastic fracture and notch mechanics are those due to Williams [5], who described the stress field near sharp V-notches, and Irwin [6], who provided his renowned equation describing the stress fields near a sharp crack.
Moving the attention to radiused notches, namely notches with a finite tip radius, worth mentioning is the paper by Creager and Paris [7] and the thorough work by Neuber [8], who provided the stress concentration factors for a large variety of notch problems.
Many years later, Lazzarin and Tovo [9] provided a general expression for the mode 1 and 2 stress fields around blunt notches, demonstrating that Irwin, Williams, Creager and Paris, and Neuber's solutions could be obtained as particular cases of their more general solution. From the previously-mentioned works, several analytical solutions have been developed and are available to designers for predicting the stress fields of components where different stress raisers are present [10][11][12].
A large variety of mode 3 notch problems was addressed by Zappalorto and coworkers (see among the others, [12][13][14] and references reported therein).
All the above-mentioned solutions are valid for isotropic materials and, accordingly, cannot be used when dealing with materials characterized by an orthotropic or rectilinearly anisotropic elastic behavior, such as, for example, fiber-reinforced polymers, wood, or crystals.
Within the context of stress concentrations provoked by holes in bodies obeying an anisotropic elastic behavior, the contributions by Savin [15] and Lekhnitskii [16] are

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In Section 2, the fundamentals of complex potentials for orthotropic elasticity are discussed, with reference to plane stress or strain and antiplane shear problems. • Sections 3 to 6 present the analytical expressions for the mode 1, 2, and 3 stress fields related to different notch geometries, i.e., elliptical hole (Section 3), hyperbolic lateral notches (Section 4), parabolic notches, i.e., blunt cracks, (Section 5), and lateral radiused V-notches (Section 6). • Eventually, Section 7 reports some examples of application, comparing the presented analytical solutions to the results from numerical analyses related to relevant cases.

Mode 2 Loadings
Different from before, the pure mode 2 problem (plate subjected to in-plane shear), can be tackled by taking advantage of the following complex functions: where z 1 = x + µ 1 y, z 2 = x + µ 2 y, and τ g xy is the far applied shear stress. Substituting Equation (26) into Equation (2) gives the stress fields in the form: where r i , θ i , Θ i , Λ i , and Ω i are defined in Equations (24) and (25), and τ Max xy is the maximum shear stress occurring along the hole bisector line at a certain distance, x Max -a, from the notch tip. Moreover:

Mode 3 Loadings
Eventually, the pure mode 3 problem (out-of-plane stress fields) can be derived by taking advantage of the following complex function [48]: where k = a/(a + b), and the case of k tending to 1 represents the sharp crack case of length 2a, whereas for k tending to 0.5, a circular hole notch can be obtained. Accordingly, substituting Equation (30) into Equation (20) allows the stress components to be written as: where: (z 3 −ĉ) = r 31 e iθ 31 (z 3 +ĉ) = r 32 e iθ 32 (34) Stress components can also be re-written, invoking the maximum shear stress at the notch tip, τ tip zy = τ g zy 1 + 1 β 3 a ρ : Along the notch bisector line, Equation (35) simplifies to give:

Mode 1 Loadings
Consider a plate weakened by two symmetrical hyperbolic notches ( Figure 2), which can be obtained by invoking the following complex mapping [49]: where z = x + iy, ξ = u + iv 0 , and c is a constant. The case v 0 = 0 represents the deep crack case whereas, more generally, such a mapping allows two symmetric hyperbolic notches with foci at x = ±c to be described, with h = c · sin v 0 and ρ = c · cot v 0 · cos v 0 . The pure mode 1 stress problem (tension applied to the plate) can be determined using the following complex functions [16,45]: Figure 2. Symmetric hyperbolic notches in an infinite solid and reference system used for Equations (40) and (45). The pure mode 1 stress problem (tension applied to the plate) can be determined using the following complex functions [16,45]: where: σ g xx is the nominal stress on the net ligament, and ρ is the root radius at the notch tip. Substituting Equation (38) into Equation (2) results in: where:ρ Moreover: is the maximum normal stress at the notch tip (x = 0, y = h). Along the notch bisector line, x = 0, so thatŷ j =θ j = 0 and: Accordingly, the normal stresses become:

Mode 2 Loadings
The mode 2 problem can be addressed by taking advantage of the following complex potentials: Substituting Equation (46) into Equation (2) gives: where parameter A can be linked to the nominal shear stress on the net section, τ n xy , using the following expression: h 0 τ xy dy = τ n xy h (48) Substituting Equation (47) into Equation (48) allows one to obtain A = τ n xy h g , where: Eventually, stresses can be re-written taking advantage of the following variables: providing:

Mode 3 Loadings
Whenever the plate is subjected to antiplane shear, the following complex function guarantees that the required boundary conditions are satisfied: Invoking the following auxiliary variables: and substituting Equation (53) into Equation (20) allows the shear stress components to be determined in closed form as a function of the nominal shear stress on the net section: At the notch tip (x = 0 and y = h), the shear stress results to be: Accordingly, the theoretical stress concentration factor is: One should note that in the case of sharply curved notches (ρ/h small), Equation (57) simplifies into: The shear stress components can also be re-written as a function of the notch tip stress, τ

Mode 1 Problem
The problem of an orthotropic plate weakened by a parabolic notch ( Figure 3) can be addressed by taking advantage of the mapping function [8]: where z = x + iy and ξ = u 0 + iv.

Mode 1 Problem
The problem of an orthotropic plate weakened by a parabolic notch ( Figure 3) can be addressed by taking advantage of the mapping function [8]: where z = x + iy and ξ = u0 + iv.  The notch apex is at a distance equal to ρ/2 from the origin of the coordinate system, where ρ is the curvature radius at the tip (v = 0).
The mode 1 problem can be addressed using the following complex potentials: In Equation (61), A is a real quantity, while zj are complex variables defined as: where: The notch apex is at a distance equal to ρ/2 from the origin of the coordinate system, where ρ is the curvature radius at the tip (v = 0).
The mode 1 problem can be addressed using the following complex potentials: In Equation (61), A is a real quantity, while z j are complex variables defined as: where: Moreover: In Equations (63) and (64), x and y are the distances from the notch tip in the x and y directions, respectively. Substituting Equation (61) into Equation (2) gives: where σ tip yy is the maximum notch tip stress and: Along the notch bisector line (y = 0, x > 0): so that the following very simple equation can be found for the normal stress:

Mode 2 Problem
The mode 2 problem can be addressed using the following complex functions: where B is a real quantity. Substituting Equation (71) into Equation (2) gives the following stress field: where B is a constant to determine, depending on the nominal applied stress, the geometry of the notched body and its elastic properties.
Along the notch edge, the only non-vanishing stress is: There are several ways to define parameter B: 1.
As a function of the maximum shear stress along the bisector of the notch: where x represents the distance from the notch tip corresponding to the maximum value for the shear stress, τ xy .

2.
It can be linked to the maximum value of the normal stress, σ vv , along the notch boundary: where: and θ is the solution of the following equation: 3. It can be linked to a generalized stress intensity factor, K 2ρ . Indeed, at a proper distance from the notch tip: where K 2ρ is the mode 2 generalized stress intensity factor for the orthotropic blunt crack [50].

Mode 3 Problem
The mode 3 problem associated to a parabolic notch in an orthotropic plate can be addressed by taking advantage of the following complex function: where B is a real constant, whereas z 3 is a complex variable defined as: where: In Equation (81), x and y are the distances from the notch tip in the x and y directions, respectively.
Stress components can be determined by substituting Equation (79) into Equation (20), leading to the following expressions: where τ tip zy is the maximum shear stress at the notch tip.

Mode 1 Loadings
The edge of a blunt notch with a generic opening angle and curvature radius at its tip can be described using the following mapping function [8,9] z = ξ q (see also Figure 4), where z = x + iy and ξ = u + iv, and the notch edge is described by the equation u = u 0 . Moreover: where tip zy  is the maximum shear stress at the notch tip.

Mode 1 Loadings
The edge of a blunt notch with a generic opening angle and curvature radius at its tip can be described using the following mapping function [8,9] q z ξ  (see also Figure 4), where z x iy   and ξ u iv   , and the notch edge is described by the equation With reference to the coordinate system shown in Figure 4, the solution for this notch problem can be sought using a series formulation for the complex potentials in the form However, in order to obtain manageable expressions for the stress fields, the series expansion can be truncated to a finite number of terms, with a tradeoff between the simplicity and accuracy of the associated solution.
Dealing with the mode 1 problem, Zappalorto and Carraro [45] used a one-term-based solution, obtaining simple yet accurate expressions which were found to be in satisfactory agreement with numerical results, in particular near the notch tip and along the bisector line.
Some years later, with the aim to improve this last-mentioned solution, and to obtain very accurate stress fields both along and outside of the notch bisector line, Pastrello et al. [46] used the following enriched forms for complex potentials: where j iθ j j j j z =x +iy =r e and: With reference to the coordinate system shown in Figure 4, the solution for this notch problem can be sought using a series formulation for the complex potentials in the form However, in order to obtain manageable expressions for the stress fields, the series expansion can be truncated to a finite number of terms, with a tradeoff between the simplicity and accuracy of the associated solution.
Dealing with the mode 1 problem, Zappalorto and Carraro [45] used a one-termbased solution, obtaining simple yet accurate expressions which were found to be in satisfactory agreement with numerical results, in particular near the notch tip and along the bisector line.
Some years later, with the aim to improve this last-mentioned solution, and to obtain very accurate stress fields both along and outside of the notch bisector line, Pastrello et al. [46] used the following enriched forms for complex potentials: where z j = x j +iy j = r j e iθ j and: In Equation (86), x = x − r 0 and y = y represent the distances from the apex of the notch; instead, A 1 , B 1 , C 1 , D 1 , E 1 , λ 1 , µ 1 , ζ 1 , and t 1 are real constants to be determined with proper boundary conditions, under the hypothesis that 1 < λ 1 < µ 1 < ζ 1 .

Mode 2 Loadings
The mode 2 problem can be addressed using the following complex potentials [47]: where z j = x j +iy j = r j e iθ j and: In Equation (93), x = x − r 0 and y = y represent the distances from the apex of the notch. A 2 , B 2 , C 2 , D 2 , E 2 , λ 2 , µ 2 , ζ 2 , and t 2 are real constants to be determined with proper boundary conditions, under the hypothesis that 1 < λ 2 < µ 2 < ζ 2 .
Equation (92) provides the following expression for the mode 2 stress field: Here, λ 2 is a linear elastic eigenvalue to be determined by solving the following equation: where γ = π − α. Parameters t 2 , µ 2 , ζ 2 , χ 12 , χ 21 , χ 22 , χ 23 depend on the notch geometry and material properties and can be determined according to the procedure proposed in ref. [47].
As mentioned before for the parabolic notch, the generic parameter A in Equations (93)-(95) can be expressed as a function of the maximum shear stress along the notch bisector, the maximum principal stress along the notch edge, or as a function of a mode 2 Generalized Stress Intensity Factor.

Mode 3 Loadings
In the case of mode 3 loadings, the solution for the stress field can be determined by taking advantage of the following complex potential function: where A 3 is a real constant, whereas z 3 is a complex variable defined as: In Equation (98): and x and y are the distances from the notch tip in the x and y directions, respectively. Substituting Equation (97) into Equation (21) allows the shear stress components to be determined: where: and It is worth noting that in the case of an isotropic material (β 3 = 1), Equation (101) turns out to be: in agreement with the exact solution [13,14,51].

Examples of Application
In this section, several examples of application for the solutions described in this paper are presented, considering several materials and different geometries. In particular, Figures 5-7 contain new data derived from numerical analyses specifically carried out within this work. Conversely, Figures 8-16 contain numerical data taken from the literature, and the original references are reported in their captions.

Examples of Application
In this section, several examples of application for the solutions described in this paper are presented, considering several materials and different geometries. In particular, Figures 5-7 contain new data derived from numerical analyses specifically carried out within this work. Conversely, Figures 8-16 contain numerical data taken from the literature, and the original references are reported in their captions.                 The results related to mode 1 and mode 2 loadings have been obtained using the following material systems (under the hypothesis that the x-direction corresponds to the notch bisector direction):
In particular, we have chosen Material 1 and Material 2 since they can be regarded as limiting cases within the context of composites materials, whereas Material 3 has been chosen as an intermediate case between Material 1 and 2.
Differently, several G xz /G yz were used to obtain results related to mode 3 loadings. The stress distributions in plates with an elliptical hole under tension, in-plane shear and out-of-plane shear are presented in Figures 5-7, respectively, considering three different materials. In particular, in Figure 5, the normalized stress components σ yy /σ tip yy and σ xx /σ tip yy along the horizontal axis, derived from Equation (23), are compared with the results from the numerical analyses carried out on orthotropic finite size (150 mm × 150 mm) plates under pure tension. As evident, the accuracy of Equation (23) is noteworthy also in the presence of a finite-size solid.
In Figure 6, the numerical results from finite-size orthotropic plates (150 mm × 150 mm) under shear are compared with the analytical solution reported in Section 3. In this case, the maximum principal stress has been evaluated along the notch edge and compared with the following analytical expression (obtained from Equation (27)): In Figure 7, instead, the attention is focused on out-of-plane shear stresses, evaluated along the bisector line of elliptical holes in orthotropic plates subjected to Mode 3 loadings. Numerical results were obtained by considering three different materials (G xz /G yz = 0.1; 1; 10) and compared with the predictions based on Equation (36).
The results related to symmetric hyperbolic notches in plates under Mode 1, 2, and 3 loadings are shown in Figures 8, 9 and 10, respectively. Moreover, in this case, the analytical solutions, theoretically valid for infinitely deep notches, are able to describe with great accuracy the numerical results from finite-size solids.
Eventually, the results for blunt V-notches with different notch opening angles are presented in Figures 11-16.
In more detail, the results for the stress fields arising in plates with parabolic notches (blunt cracks) under tension, in-plane shear, and out-of-plane shear are summarized in Figures 11, 12 and 13, respectively. Furthermore, for this case, Equations (65), (72) and (83), exact in the case of deep blunt cracks, can be effectively used to characterize the local stress fields arising in finite-size solids.
Eventually, the attention is moved to lateral radiused notches with a notch opening angle different from zero (2α = 90 • ), documenting once again the accuracy of the equations proposed in this work (see .
Based on the results reported in this section, the following main comments can be drawn in relation to the features of the stress fields:

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With regards to Mode 1 loadings, Figures 5, 8, 11 and 14 make it evident that when the material is very stiff along the loading direction (Material 2), the stress gradient is high, and the distribution of the maximum principal stress along the notch bisector line (i.e., the normal stress orthogonal to the notch bisector line) is very steep. Conversely, in the case of a material very stiff along the notch bisector line (Material 1), the stress gradient is much lower, and the distribution of the maximum principal stress along the notch bisector line (i.e., the normal stress orthogonal to the notch bisector line) is mild. As evident, this behavior is general and does not depend on the particular notch geometry under investigation.

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With regards to Mode 2 loadings, from Figures 6,9,12 and 15, it is evident that when the material is very stiff along the direction normal to the bisector line (Material 2), the stress gradient is high, and the point, along the notch bisector line, exhibiting the maximum shear stress is very close to the notch tip. This behavior is general and does not depend on the particular notch geometry under investigation. • Eventually, with reference to Mode 3 loadings, from Figures 7, 10, 13 and 16, it is evident that when G iz is much larger than G jz , where j is the direction of the notch bisector line, the stress gradient is high, and the distribution of the maximum antiplane shear stress along the notch bisector line is very steep, independent of the considered notched geometry. Vice versa, when G iz is smaller than G jz , the stress gradient is mild.
In order to conclude this section, it is possible to state that the equations and solutions reported in this paper, either exact or approximated, are characterized by a very satisfactory accuracy and represent useful tools to assess the notch stress fields in orthotropic solids weakened by a large variety of geometrical variations.

Conclusions and Final Remarks
In this work, a brief overview of the analytical solutions available to describe the in-plane and out-of-plane stress fields in orthotropic solids with radiused notches has been presented, and their accuracy discussed versus a number of numerical results.
In more detail, initially, a brief summary of the fundamentals of complex potentials for orthotropic elasticity was presented, with reference to plane stress or strain and antiplane shear problems.
Subsequently, the attention was moved to the relevant expressions for the notch stress fields, considering elliptical holes, symmetric hyperbolic notches, parabolic notches, blunt cracks, and radiused V-notches.
Eventually, examples of application were presented, comparing the presented analytical solutions to the results from numerical analyses carried out on relevant cases.
Based on the cases analyzed, the following main comments can be drawn in relation to the effect of the elastic material properties, independent of the particular notch geometry considered: • With regards to Mode 1 loadings, when the material is very stiff along the traction direction, the stress gradient is high, and the distribution of the maximum principal stress along the notch bisector line (i.e., the normal stress orthogonal to the notch bisector line) is very steep. Conversely, in the case of a material very stiff along the notch bisector line, the stress gradient is much lower, and the distribution of the maximum principal stress along the notch bisector line is mild.

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With regards to Mode 2 loadings, when the material is very stiff along the direction normal to the bisector line, the stress gradient is high, and the point, along the notch bisector line, exhibiting the maximum shear stress is very close to the notch tip. • With reference to Mode 3 loadings, when G iz is much larger than G jz , where j is the direction of the notch bisector line, the stress gradient is high, and the distribution of the maximum antiplane shear stress along the notch bisector line is very steep. Vice versa, when G iz is smaller than G jz , the stress gradient is mild.
As a major conclusion of this work, it can be stated that the equations and solutions reported in this paper, either exact or approximated, are characterized by a very satisfactory accuracy and represent useful tools to assess the notch stress fields in orthotropic solids weakened by a large variety of geometrical variations.
A final remark concerns the choice of employing an analytical solution or a numerical one (e.g., FEA) in front of a real problem. All the reviewed solutions showed an excellence accuracy when compared with FEA. It means that at an up-front cost of implementing the equations in a spreadsheet or in some programming language (e.g., Python), very accurate solutions at low computational cost could be obtained, saving the cost of running FE simulations.