Revisiting classical concepts of Linear Elastic Fracture Mechanics Part III: The stress field in a double-edge notched finite strip by means of the "stress-neutralization" technique

This is the third part of a short series of papers revisiting classical concepts of Linear Elastic Fracture Mechanics (LEFM) in the frame of alternative analytic approaches, based mainly on the complex potential technique, as it was introduced and formulated by Kolosov [1] and Muskhelishvili [2]. The objective of the series is to provide analytical expressions for the stress field around particular geometric discontinuities in elastic media, under specific loading conditions. In this context, Part-I of the series [3] provided analytical formulae for the Stress Intensity Factors (SIFs) at the tip of a natural (i.e., "mathematical") crack (namely, a crack with zero distance between its lips), in an infinite plate, loaded in its plane, for both opening and closing (contacting each other) crack lips. For the latter case an alternative way was presented for the solution of the familiar problem of 'overlapping' lips [4-6]. In that paper, it was also quantitatively verified that the dimensions of the plate may indeed be comparable to the crack length without significantly affecting the final results (as it was, in fact, tacitly implied by Muskhelishvili [2]), thus relaxing somehow the assumption of the infinite-plate.
Along a similar line of thought, Part-II of the series [7] dealt with the stretching of a strip with finite dimensions, weakened by a single edge notch of parabolic shape, approximating the rounded V-notch configuration. The specific issue concerns the engineering community already from the end of the ${19}^{\text{th}}$${19}^{\text{th }}$19^("th ")19^{\text {th }}${19}^{\text{th }}$ century, given that the presence of geometrical discontinuities of any kind was found to strongly amplify the local stress field, as it was highlighted by Kirsch [8] and Irwin [9] for circular and elliptical holes, respectively, in infinite domains. Obviously, this amplification of the stress field is crucial for the integrity and safety of any engineering structure, justifying the huge effort devoted since then for the accurate description of the stress field around various types of discontinuities. Initially the efforts were mainly focused on the quantification of the respective Stress Concentration Factors (SCFs), starting with Neuber's pioneering works [10] already from the middle thirties of the ${20}^{\text{th}}$${20}^{\text{th }}$20^("th ")20^{\text {th }}${20}^{\text{th }}$ century, and, also, the works by Hardrath and Ohman [11] and that by Lin [12] for a finite strip weakened by a pair of symmetrical notches, under the simplifying assumption that the notches were of semicircular shape (although the latter was criticized by Peterson [13] as providing values for the SCF exceeding the ones provided by Neuber).
The attempts for the determination of full-field solutions for the stresses around discontinuities were facilitated after Irwin [14] introduced his seminal solution for the stress field developed in the immediate vicinity of a "mathematical" crack, based on the ideas presented earlier by Westergaard [15]. It sounds perhaps strange, however the solution for the stress field around discontinuities in the form of open 'cracks', i.e., discontinuities in the form of V- or U-shaped notches, was proven much more complicated compared to the respective solution for the stress field around the tips of "mathematical" cracks. It was Williams [16] who presented such a solution for V-shaped notches by means of eigenfunction series expansions. Later on, solutions for the stress field around discontinuities with non-zero tip radius were proposed, also, by Creager and Paris [17] and Glinka and Newport [18], although the latter was found to suffer from insensitivity of its results to the variations of the opening angle of the notch [19].

In spite of the intensive efforts devoted to the determination of full-field solutions for the stresses around V - and U -shaped notches, the issue is by no means closed and it concerns continuously the scientific community due to its paramount practical importance. The relative studies are implemented either numerically or experimentally or adopting hybrid approaches [20, 21]. In this direction it is worth mentioning the milestone contributions by the team of late Professor Lazzarin, who provided analytical solutions for either sharp or rounded V-notches in 1996 [19]. Shortly afterwords they improved their solution for bodies of finite dimensions and then they revised their approach in order to increase its accuracy in case of rounded Vshaped notches with a large opening angle, by "... adding a component in the polynomial arrangement of potential functions", adopting the configuration of Fig. 1 [22]. Further on, the team contributed significantly in the development of the "generalized stress intensity factor" for either rounded V- or U-shaped notches, focusing attention on the fracture of notched structural elements, by means of proper either energy- or stress-based fracture criteria [23-25].

In the frame of the above concepts, the approach that was introduced in Part-II of this short series of papers dealt with the single-edge notched finite strip under tension, providing a simple and flexible expression for the respective SCFs, along with the compact expressions for the stress field all over the strip. The results of that study were found in relatively satisfactory agreement to the ones obtained by the well-established approach (analytical and numerical) by Filippi et al. [22], revealing the potentialities of the procedure proposed. The present Part-III of the series completes Part-II by considering instead of a single-edge notched strip, a finite strip weakened by two symmetric, parabolic, stress-free, edge notches, the common axis of which is perpendicular to the direction of the external loading. Though relatively perplex, the solution of this problem might be proven more useful, compared to that for a single-edge notched strip, since it corresponds to a configuration which, due to its symmetry, diminishes bending effects and it is easier to be managed in experimental protocols.
In this direction, using Muskhelishvili's complex potentials technique [2] in combination with a novel procedure for "stress neutralization" of specific areas of the loaded strip, a full-field approximate solution is obtained here in closed form for the problem of a stretched, double-edge notched, linear elastic and isotropic finite strip (Fig. 2a). The notches are of parabolic shape. The radius of curvature at the bases of the parabolas can substitute quite efficiently the radius of the respective rounded V-shaped notches (Fig. 2b), thus approximating according to a very satisfactory manner the configuration of a Vshaped notch with rounded tip. It is mentioned from the very beginning that the role of the exact geometry of the notch (and especially of its tip) is of critical importance for the overall approach to the problem, as it has been highlighted by Lazzarin and Tovo [19]. This aspect of the analysis will be addressed in the "Discussion and Conclusions" section. The basic assumption adopted in the present study is that the ratio of the notch depth over the width of the strip is small, i.e., the notches are assumed to be "shallow". This assumption simplifies significantly the algebraic manipulations.
In this context, the present solution is obtained by means of the solution recently presented in Part-II of this series [7], for the single-edge parabolic notch taking, in addition, advantage of the superposition principle and the "stress -neutralization" concept. The validity of the solution is then proven by the fulfilment of the stress boundary conditions along the sides of the strip and, also, along the periphery of the notches. Moreover, the outcomes of the present solution are considered in juxtaposition to the ones of the solution presented by Filippi et al. [22]. Emphasis is given to the stress field around the base ("crown" or "tip") of the notches and, also, along the axis of symmetry of the strip normal to the loading direction, where the non-zero tensile stress component is given explicitly in terms of a convenient and flexible formula. Taking advantage of this formula, easy-to-use expressions are obtained for the respective SCF, k (for the case of blunt notches), as well as for the respective SIF, ${\mathrm{K}}_{\mathrm{I}}$${\mathrm{K}}_{\mathrm{I}}$K_(I)\mathrm{K}_{\mathrm{I}}${\mathrm{K}}_{\mathrm{I}}$ (for the case of "mathematical" edge cracks).

In this study an attempt is made to obtain the stress field in a strip of length 2 b and width 2 h , stretched by a uniform stress ${\sigma }_{\mathrm{o}}$${\sigma }_{\mathrm{o}}$sigma_(o)\sigma_{\mathrm{o}}${\sigma }_{\mathrm{o}}$, and weakened by two symmetric rounded V-shaped notches (Fig. 2a). For mathematical convenience, the rounded V -notches are described as parabolic cavities, for which the curvature at the base of the parabola approximates that of the rounded $V$$V$VV$V$-notch (Fig. 2b). The strip lies in the complex $z=x+iy$$z=x+iy$z=x+iyz=x+i y$z=x+iy$ plane. The origin of the coordinate system is the focus of the upper parabolic cavity/notch. The two notches have the same length (depth) equal to $\mathrm{d}=\mathrm{c}+{\alpha }^2$$\mathrm{d}=\mathrm{c}+{\alpha }^2$d=c+alpha^(2)\mathrm{d}=\mathrm{c}+\alpha^{2}$\mathrm{d}=\mathrm{c}+{\alpha }^2$, where $\alpha$$\alpha$alpha\alpha$\alpha$ in the equations of the parabolas (Fig. 2a) dictates the sharpness of the cavities; namely, as $\alpha$$\alpha$alpha\alpha$\alpha$ tends to zero the notches tend to the edge "mathematical" cracks. The material of the strip is assumed isotropic and linearly elastic and Muskhelishvili's complex potentials technique [2] is used to solve the problem. As it was mentioned the notches are assumed "shallow" and therefore the stress state at the center of the strip corresponds to that of uniaxial tension, i.e., ${\sigma }_{xx}={\sigma }_o,{\sigma }_{yy}={\sigma }_{xy}=0$${\sigma }_{xx}={\sigma }_o,{\sigma }_{yy}={\sigma }_{xy}=0$sigma_(xx)=sigma_(o),sigma_(yy)=sigma_(xy)=0\sigma_{x x}=\sigma_{o}, \sigma_{y y}=\sigma_{x y}=0${\sigma }_{xx}={\sigma }_o,{\sigma }_{yy}={\sigma }_{xy}=0$.

Outlining the method of solution - The "stress-neutralization" technique
In obtaining the solution of the doubly notched stretched strip, use is made of the solution of the problem of the intact stretched strip which will be denoted as "Problem 1" (Fig. 3a). Namely, at any point of the intact stretched strip, and, therefore, along the points of its two internal parabolic loci shown in Fig.3a, the stress state on normal sections will be that of simple uniaxial tension, namely, ${\sigma }_{xx}={\sigma }_0,{\sigma }_{yy}={\sigma }_{xy}=0$${\sigma }_{xx}={\sigma }_0,{\sigma }_{yy}={\sigma }_{xy}=0$sigma_(xx)=sigma_(0),sigma_(yy)=sigma_(xy)=0\sigma_{x x}=\sigma_{0}, \sigma_{y y}=\sigma_{x y}=0${\sigma }_{xx}={\sigma }_0,{\sigma }_{yy}={\sigma }_{xy}=0$. By adding to the points of these two parabolic loci the opposite stress field ${\sigma }_{\mathrm{xx}}=-{\sigma }_{\mathrm{o}}$${\sigma }_{\mathrm{xx}}=-{\sigma }_{\mathrm{o}}$sigma_(xx)=-sigma_(o)\sigma_{\mathrm{xx}}=-\sigma_{\mathrm{o}}${\sigma }_{\mathrm{xx}}=-{\sigma }_{\mathrm{o}}$, without interfering to the boundary conditions on the outer boundaries of the stretched strip (Fig. 3b), the shaded parabolic regions of the stretched strip (Fig. 3c) become stress free. That is equivalent to stress neutralizing, thus removing these shaded regions from the strip, transforming it into the doubly notched stretched strip in question (Fig. 3d).

In order to apply the above-described procedure, an auxiliary problem, denoted from here on as "Problem 2" (Fig. 4b), is superposed to "Problem 1" (Fig. 4a). The boundary conditions for "Problem 2" consist of stresses ${\sigma }_{\eta \eta }$${\sigma }_{\eta \eta }$sigma_(eta eta)\sigma_{\eta \eta}${\sigma }_{\eta \eta }$, ${\sigma }_{\xi \eta }$${\sigma }_{\xi \eta }$sigma_(xi eta)\sigma_{\xi \eta}${\sigma }_{\xi \eta }$ along the two parabolic notches corresponding to the $-{\sigma }_o$$-{\sigma }_o$-sigma_(o)-\sigma_{o}$-{\sigma }_o$ stress field, i.e., the opposite of that occurring in the respective parabolic loci of the intact strip in "Problem 1" (Figs. 4b, 4c). Thus, after superposition of "Problem 2" to "Problem 1", the stretched double notched strip with stress-free notches in question will be obtained (Fig. 4d). The superposition of the notched strip on to the intact one is topologically justified by the fact that the $-{\sigma }_o$$-{\sigma }_o$-sigma_(o)-\sigma_{o}$-{\sigma }_o$ stress field of "Problem 2" upon superimposed to the intact strip automatically renders it a notched strip as well, since it stress-neutralizes the shadow parabolic regions, discussed previously with regard to Fig. 3. Concerning the sides of the strip, it is seen from the final results that "Problem 2" leaves more or less unaffected the stress state at the strip sides of "Problem 1".
In this context, denoting the complex potentials solving "Problem 1" and "Problem 2" by ${\mathrm{\Phi }}_1,{\mathrm{\Psi }}_1$${\mathrm{\Phi }}_1,{\mathrm{\Psi }}_1$Phi_(1),Psi_(1)\Phi_{1}, \Psi_{1}${\mathrm{\Phi }}_1,{\mathrm{\Psi }}_1$ and ${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$Phi_(2),Psi_(2)\Phi_{2}, \Psi_{2}${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$, respectively, the solution of the double-notched strip in question (Fig. 4d) will read as:

The solution of the (trivial) "Problem 1" reads as:

The complex potentials ${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$Phi_(2),Psi_(2)\Phi_{2}, \Psi_{2}${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$, solving the auxiliary "Problem 2" are subjected to determination, according to the method that will be described analytically in next section (The solution of the auxiliary "Problem 2"), by means of the superposition of the solutions of two additional auxiliary elementary problems, denoted as "Problem 3" and "Problem 4".

The complex potentials ${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$Phi_(2),Psi_(2)\Phi_{2}, \Psi_{2}${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$, solving the auxiliary "Problem 2" (Fig. 4b), are here obtained under the assumption of small length $d=c+{\alpha }^2$$d=c+{\alpha }^2$d=c+alpha^(2)d=c+\alpha^{2}$d=c+{\alpha }^2$ of the notches. In this way ${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$Phi_(2),Psi_(2)\Phi_{2}, \Psi_{2}${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$ upon superposed to ${\mathrm{\Phi }}_1,{\mathrm{\Psi }}_1$${\mathrm{\Phi }}_1,{\mathrm{\Psi }}_1$Phi_(1),Psi_(1)\Phi_{1}, \Psi_{1}${\mathrm{\Phi }}_1,{\mathrm{\Psi }}_1$ will provide the solution $\mathrm{\Phi },\mathrm{\Psi }$$\mathrm{\Phi },\mathrm{\Psi }$Phi,Psi\Phi, \Psi$\mathrm{\Phi },\mathrm{\Psi }$ of the overall problem in question, Fig. 4d, leaving, in the limit, the stress field at the center of the strip unaffected, so that the stress state at the center of the strip for the overall problem (Fig. 4d) to be that of simple tension ${\sigma }_{xx}={\sigma }_o,{\sigma }_{yy}={\sigma }_{xy}=0$${\sigma }_{xx}={\sigma }_o,{\sigma }_{yy}={\sigma }_{xy}=0$sigma_(xx)=sigma_(o),sigma_(yy)=sigma_(xy)=0\sigma_{x x}=\sigma_{o}, \sigma_{y y}=\sigma_{x y}=0${\sigma }_{xx}={\sigma }_o,{\sigma }_{yy}={\sigma }_{xy}=0$ (holding in the case of the intact strip - "Problem 1"). In this case the two notches will not influence one another, and therefore ${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$Phi_(2),Psi_(2)\Phi_{2}, \Psi_{2}${\mathrm{\Phi }}_2,{\mathrm{\Psi }}_2$ can be obtained by simply superposing the solutions of "Problem 3" and "Problem 4" shown in Fig. 5, as:

The complex potentials ${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$Phi_(3),Psi_(3)\Phi_{3}, \Psi_{3}${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$ solving "Problem 3", have already been obtained in a recent work of the authors [7], and they can be written as:

The complex potentials ${\mathrm{\Phi }}_4,{\mathrm{\Psi }}_4$${\mathrm{\Phi }}_4,{\mathrm{\Psi }}_4$Phi_(4),Psi_(4)\Phi_{4}, \Psi_{4}${\mathrm{\Phi }}_4,{\mathrm{\Psi }}_4$, which solve "Problem 4", in turn, are easily obtained from ${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$Phi_(3),Psi_(3)\Phi_{3}, \Psi_{3}${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$ (by means of the noninteracting notches assumption), by modifying ${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$Phi_(3),Psi_(3)\Phi_{3}, \Psi_{3}${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$, by changing properly the coordinate axes (Fig. 6). Namely, starting from the solution ${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$Phi_(3),Psi_(3)\Phi_{3}, \Psi_{3}${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$, of "Problem 3" in Fig. 5a (or equivalently in Fig. 6a), then by a ${180}^{\circ }$${180}^{\circ }$180^(@)180^{\circ}${180}^{\circ }$ clockwise rotation, and an upwards translation of the coordinate axes by an amount $2(\mathrm{h}-\mathrm{c})$$2(\mathrm{h}-\mathrm{c})$2(h-c)2(\mathrm{~h}-\mathrm{c})$2(\mathrm{h}-\mathrm{c})$, one obtains the solution ${\mathrm{\Phi }}_4,{\mathrm{\Psi }}_4$${\mathrm{\Phi }}_4,{\mathrm{\Psi }}_4$Phi_(4),Psi_(4)\Phi_{4}, \Psi_{4}${\mathrm{\Phi }}_4,{\mathrm{\Psi }}_4$ of "Problem 4" (Fig. 6c) in terms of ${\mathrm{\Phi }}_3$${\mathrm{\Phi }}_3$Phi_(3)\Phi_{3}${\mathrm{\Phi }}_3$ and ${\mathrm{\Psi }}_3$${\mathrm{\Psi }}_3$Psi_(3)\Psi_{3}${\mathrm{\Psi }}_3$ as:

where prime in ${\mathrm{\Phi }}_3$${\mathrm{\Phi }}_3$Phi_(3)\Phi_{3}${\mathrm{\Phi }}_3$ denotes the first order derivative with respect to $z$$z$zz$z$. Clearly, the right-hand sides of Eqns.(6) are directly obtained from Eqns.(4) and (5) substituting in the latter $z$$z$zz$z$ by $-z-i2(h-c)$$-z-i2(h-c)$-z-i2(h-c)-z-i 2(h-c)$-z-i2(h-c)$.

Thus, taking advantage of Eqns.(3) and (6), the solution of the auxiliary "Problem 2" (Fig. 4b), reads as:

where the general expressions of ${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$Phi_(3),Psi_(3)\Phi_{3}, \Psi_{3}${\mathrm{\Phi }}_3,{\mathrm{\Psi }}_3$, are given by Eqns.(4) and (5).

Substituting from Eqns.(2) and (7) into Eqns.(1), the complex potentials $\mathrm{\Phi }(z)$$\mathrm{\Phi }(z)$Phi(z)\Phi(z)$\mathrm{\Phi }(z)$ and $\mathrm{\Psi }(z)$$\mathrm{\Psi }(z)$Psi(z)\Psi(z)$\mathrm{\Psi }(z)$ solving the overall problem (Fig. 4d) in question, i.e., that of stretching a finite strip weakened by two "shallow" edge notches, are written after some algebra as:

Introducing the complex potentials $\mathrm{\Phi }(z)$$\mathrm{\Phi }(z)$Phi(z)\Phi(z)$\mathrm{\Phi }(z)$ and $\mathrm{\Psi }(z)$$\mathrm{\Psi }(z)$Psi(z)\Psi(z)$\mathrm{\Psi }(z)$ from Eqns.(8) and (9) in Muskhelishvili's well-known formulae (see next Eqns.(10), where $\mathrm{\Re }$$\mathrm{\Re }$ℜ\Re$\mathrm{\Re }$ denotes the real part and over-bar the conjugate complex value), the Cartesian components of the stress field are obtained all over the strip.

Along the periphery of the notches use can be made, also, of the curvilinear stress components ${\sigma }_{\xi \xi },{\sigma }_{\eta \eta },{\sigma }_{\xi \eta }($${\sigma }_{\xi \xi },{\sigma }_{\eta \eta },{\sigma }_{\xi \eta }($sigma_(xi xi),sigma_(eta eta),sigma_(xi eta)(\sigma_{\xi \xi}, \sigma_{\eta \eta}, \sigma_{\xi \eta}(${\sigma }_{\xi \xi },{\sigma }_{\eta \eta },{\sigma }_{\xi \eta }($ see Fig. 4 c$)$$)$))$)$ as they are obtained by employing the familiar transformations:

Substituting from Eqns.(8, 9, 10) in Eqns.(11) it can be seen that along the notches only the ${\sigma }_{\xi \eta }$${\sigma }_{\xi \eta }$sigma_(xi eta)\sigma_{\xi \eta}${\sigma }_{\xi \eta }$ component is non-zero, while ${\sigma }_{\eta \eta }={\sigma }_{\xi \eta }=0$${\sigma }_{\eta \eta }={\sigma }_{\xi \eta }=0$sigma_(eta eta)=sigma_(xi eta)=0\sigma_{\eta \eta}=\sigma_{\xi \eta}=0${\sigma }_{\eta \eta }={\sigma }_{\xi \eta }=0$, fulfilling the boundary condition for stress free notches.

In order to highlight the potentialities of the solution introduced, the above stress formulae will be applied for a specific strip of length $2b=30\mathrm{cm}$$2b=30\mathrm{cm}$2b=30cm2 b=30 \mathrm{~cm}$2b=30\mathrm{cm}$, width $2h=20\mathrm{cm}$$2h=20\mathrm{cm}$2h=20cm2 h=20 \mathrm{~cm}$2h=20\mathrm{cm}$, with two short notches of length $d=c+{\alpha }^2=0.5+{0.5}^2=0.75\mathrm{cm}$$d=c+{\alpha }^2=0.5+{0.5}^2=0.75\mathrm{cm}$d=c+alpha^(2)=0.5+0.5^(2)=0.75cmd=c+\alpha^{2}=0.5+0.5^{2}=0.75 \mathrm{~cm}$d=c+{\alpha }^2=0.5+{0.5}^2=0.75\mathrm{cm}$ and span 1.73 cm (Figs. 7a, 7b). The notches are free from stresses while the edges of the strip are stretched by a uniform tensile stress ${\sigma }_{\mathrm{xx}}$${\sigma }_{\mathrm{xx}}$sigma_(xx)\sigma_{\mathrm{xx}}${\sigma }_{\mathrm{xx}}$ equal to ${\sigma }_{xx}={\sigma }_o=10\mathrm{MPa}$${\sigma }_{xx}={\sigma }_o=10\mathrm{MPa}$sigma_(xx)=sigma_(o)=10MPa\sigma_{x x}=\sigma_{o}=10 \mathrm{MPa}${\sigma }_{xx}={\sigma }_o=10\mathrm{MPa}$.

In Fig. 7a, the variation of the ${\sigma }_{\mathrm{xx}}$${\sigma }_{\mathrm{xx}}$sigma_(xx)\sigma_{\mathrm{xx}}${\sigma }_{\mathrm{xx}}$ stress, as obtained from Eqns.(10), is plotted along the bisector of the notches (namely, along $y$$y$yy$y$-axis), as well as along the sides of the half (due to the symmetry) upper notch. As it is seen, at the midpoint of the bisector of the notches it holds that ${\sigma }_{\mathrm{xx}}=10.09\mathrm{MPa}$${\sigma }_{\mathrm{xx}}=10.09\mathrm{MPa}$sigma_(xx)=10.09MPa\sigma_{\mathrm{xx}}=10.09 \mathrm{MPa}${\sigma }_{\mathrm{xx}}=10.09\mathrm{MPa}$, a value almost identical to ${\sigma }_{\mathrm{o}}$${\sigma }_{\mathrm{o}}$sigma_(o)\sigma_{\mathrm{o}}${\sigma }_{\mathrm{o}}$ holding in the case of the intact strip ("Problem 1"), thus approximating quite satisfactory the demand of non-interacting edge notches, set as a prerequisite while obtaining the present solution. At the bases (tips) of the notches ${\sigma }_{\mathrm{xx}}$${\sigma }_{\mathrm{xx}}$sigma_(xx)\sigma_{\mathrm{xx}}${\sigma }_{\mathrm{xx}}$ attains (as it is expected) its maximum value equal to ${\sigma }_{\mathrm{xx}}=32.07\mathrm{MPa}$${\sigma }_{\mathrm{xx}}=32.07\mathrm{MPa}$sigma_(xx)=32.07MPa\sigma_{\mathrm{xx}}=32.07 \mathrm{MPa}${\sigma }_{\mathrm{xx}}=32.07\mathrm{MPa}$, a reasonable stress amplification (equal to about 3 times ${\sigma }_{\mathrm{o}}$${\sigma }_{\mathrm{o}}$sigma_(o)\sigma_{\mathrm{o}}${\sigma }_{\mathrm{o}}$ ), taking into account the particular geometry of the notch (i.e., that of an almost semi-circular cavity). In addition, ${\sigma }_{xx}$${\sigma }_{xx}$sigma_(xx)\sigma_{x x}${\sigma }_{xx}$ is around 10 MPa all along the sides of the strip. These results show that the short notches assumption suffices both insignificant stress disturbance along the sides of the strip, and, also, effective predictions about the stress concentration at the tips of the notches. The value ${\sigma }_{xx}=-3.17\mathrm{MPa}$${\sigma }_{xx}=-3.17\mathrm{MPa}$sigma_(xx)=-3.17MPa\sigma_{x x}=-3.17 \mathrm{MPa}${\sigma }_{xx}=-3.17\mathrm{MPa}$ at the end points of the notches is to suffice stress equilibrium at the particular points.
The only non-zero curvilinear stress component $\sigma \xi$$\sigma \xi$sigma xi\sigma \xi$\sigma \xi$ is, also, plotted in Fig. 7a, along the upper half notch (green color). Its maximum value ${\sigma }_{\xi \xi }=32.07\mathrm{MPa}$${\sigma }_{\xi \xi }=32.07\mathrm{MPa}$sigma_(xi xi)=32.07MPa\sigma_{\xi \xi}=32.07 \mathrm{MPa}${\sigma }_{\xi \xi }=32.07\mathrm{MPa}$, attained at the notch base, obviously coincides with the maximum value of ${\sigma }_{\mathrm{xx}}$${\sigma }_{\mathrm{xx}}$sigma_(xx)\sigma_{\mathrm{xx}}${\sigma }_{\mathrm{xx}}$ at the same point (where ${\sigma }_{\mathrm{xx}}=\sigma \xi \xi$${\sigma }_{\mathrm{xx}}=\sigma \xi \xi$sigma_(xx)=sigma xi xi\sigma_{\mathrm{xx}}=\sigma \xi \xi${\sigma }_{\mathrm{xx}}=\sigma \xi \xi$ ). Similarly, the variation of the other two Cartesian stress components ${\sigma }_{\mathrm{yy}}$${\sigma }_{\mathrm{yy}}$sigma_(yy)\sigma_{\mathrm{yy}}${\sigma }_{\mathrm{yy}}$ and ${\sigma }_{\mathrm{xy}}$${\sigma }_{\mathrm{xy}}$sigma_(xy)\sigma_{\mathrm{xy}}${\sigma }_{\mathrm{xy}}$ are plotted in Figs. 7c and 7d, respectively. The very small value of ${\sigma }_{yy}$${\sigma }_{yy}$sigma_(yy)\sigma_{y y}${\sigma }_{yy}$ (Fig. 7c) at the midpoint of the notches' bisector ( 0.16 MPa ) and the almost zero value attained along the sides of the strip guarantee further the validity of the solution, from the point of view of satisfying the boundary conditions of the problem.

Similar conclusions can be drawn from the variation of the ${\sigma }_{\mathrm{xy}}$${\sigma }_{\mathrm{xy}}$sigma_(xy)\sigma_{\mathrm{xy}}${\sigma }_{\mathrm{xy}}$ stress component (Fig. 7d). The non-zero values of ${\sigma }_{\mathrm{yy}}$${\sigma }_{\mathrm{yy}}$sigma_(yy)\sigma_{\mathrm{yy}}${\sigma }_{\mathrm{yy}}$ and ${\sigma }_{\mathrm{xy}}$${\sigma }_{\mathrm{xy}}$sigma_(xy)\sigma_{\mathrm{xy}}${\sigma }_{\mathrm{xy}}$ along the half upper notch (referring to the interior of the strip) are to suffice stress equilibrium there.

Following previous studies [7], the variation of the polar stress components ${\sigma }_{\mathrm{rr}},{\sigma }_{\theta \theta },{\sigma }_{\mathrm{r}\theta }$${\sigma }_{\mathrm{rr}},{\sigma }_{\theta \theta },{\sigma }_{\mathrm{r}\theta }$sigma_(rr),sigma_(theta theta),sigma_(rtheta)\sigma_{\mathrm{rr}}, \sigma_{\theta \theta}, \sigma_{\mathrm{r} \theta}${\sigma }_{\mathrm{rr}},{\sigma }_{\theta \theta },{\sigma }_{\mathrm{r}\theta }$, along a circular locus centered at the origin of the reference system, and extended from the upper notch boundary to the bisector of the notches ( $y$$y$yy$y$-axis), is plotted in Fig. 8, for the numerical values of the parameters that were considered in previous paragraph. The radius of the circular locus (colored red in the figure) was considered equal to $\mathrm{r}=2.5{\alpha }^2$$\mathrm{r}=2.5{\alpha }^2$r=2.5alpha^(2)\mathrm{r}=2.5 \alpha^{2}$\mathrm{r}=2.5{\alpha }^2$. It is easily seen from Fig. 8 that for the specific value of $r$$r$rr$r$ the locus corresponds to an angle varying in the interval ${11.54}^{\circ }\le \theta \le -{90}^{\circ }$${11.54}^{\circ }\le \theta \le -{90}^{\circ }$11.54^(@) <= theta <= -90^(@)11.54^{\circ} \leq \theta \leq-90^{\circ}${11.54}^{\circ }\le \theta \le -{90}^{\circ }$. The variations of ${\sigma }_{\mathrm{rr}},{\sigma }_{\theta \theta },{\sigma }_{\mathrm{r}\theta }$${\sigma }_{\mathrm{rr}},{\sigma }_{\theta \theta },{\sigma }_{\mathrm{r}\theta }$sigma_(rr),sigma_(theta theta),sigma_(rtheta)\sigma_{\mathrm{rr}}, \sigma_{\theta \theta}, \sigma_{\mathrm{r} \theta}${\sigma }_{\mathrm{rr}},{\sigma }_{\theta \theta },{\sigma }_{\mathrm{r}\theta }$, closely resemble qualitatively the ones described in ref. [7], highlighting further the potentialities of the present solution.

The critical stress at the bases (crowns or tips) of the notches, and the respective Stress Concentration Factor $k$$k$kk$k$

Adopting the Cartesian representation of stress, the critical stress component at the base of the upper and the lower notch, i.e., the component quantifying the severity of the stress field, is ${\sigma }_{\mathrm{xx},\mathrm{cr}}={\sigma }_{\mathrm{xx}}(0,-{\alpha }^2)={\sigma }_{\mathrm{xx}}(0,-2(\mathrm{h}-\mathrm{c})+{\alpha }^2)$${\sigma }_{\mathrm{xx},\mathrm{cr}}={\sigma }_{\mathrm{xx}}\left(0,-{\alpha }^2\right)={\sigma }_{\mathrm{xx}}\left(0,-2(\mathrm{h}-\mathrm{c})+{\alpha }^2\right)$sigma_(xx,cr)=sigma_(xx)(0,-alpha^(2))=sigma_(xx)(0,-2((h)-c)+alpha^(2))\sigma_{\mathrm{xx}, \mathrm{cr}}=\sigma_{\mathrm{xx}}\left(0,-\alpha^{2}\right)=\sigma_{\mathrm{xx}}\left(0,-2(\mathrm{~h}-\mathrm{c})+\alpha^{2}\right)${\sigma }_{\mathrm{xx},\mathrm{cr}}={\sigma }_{\mathrm{xx}}(0,-{\alpha }^2)={\sigma }_{\mathrm{xx}}(0,-2(\mathrm{h}-\mathrm{c})+{\alpha }^2)$. Combining Eqns.(8, 9, 10), it is obtained after some algebra that: