Microsoft Word - 2173 S. K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 47 (2019) 247-265; DOI: 10.3221/IGF-ESIS.47.19 247 Fracture and Structural Integrity: ten years of ‘Frattura ed Integrità Strutturale’ Analytical and numerical study of the stress field in a circular semi- ring under combined diametral compression and bending Stavros K. Kourkoulis, Ermioni D. Pasiou, Christos F. Markides National Technical University of Athens, School of Applied Mathematical and Physical Sciences, Department of Mechanics, Laboratory for Testing and Materials, 5, Heroes of Polytechnion Avenue, Zografou Campus, Theocaris Building, 157 73, Greece stakkour@central.ntua.gr, epasiou@teemail.gr, markidih@mail.ntua.gr ABSTRACT. The stress field developed in a circular semi-ring under the com- bined action of diametral compression and bending is explored both analytically and numerically. The analytic solution is implemented by means of the complex potentials technique as it was formulated by Muskhelishvili, while for the nu- merical study a finite element model, properly validated based on experi- mental data, is used. The analytic solution provided closed formulae for the stress field along strategic loci of the specimen, while the numerical model permitted thorough parametric investigation of the dependence of critical quantities on geometrical and loading factors. The idea behind the study is to assess the potentialities of the circular semi-ring as a possible substitute of the familiar Brazilian disc, in the direction of curing drawbacks of the latter. It was concluded that a circular semi-ring subjected to eccentric diametral com- pression provides reliable data for the tensile strength of very brittle materials, relieved from ambiguities characterizing the standardized Brazilian-disc test. KEYWORDS. Circular semi-ring; Brazilian-disc test; Tensile strength; Complex potentials; Finite element method; Stress field. Citation: Kourkoulis S.K., Pasiou E.D, Mar- kides Ch.F., Analytical and numerical study of the stress field in a circular semi-ring under combined diametral compression and bending, Frattura ed Integrità Strutturale, 47 (2019) 247-265. Received: 21.08.2018 Accepted: 28.10.2018 Published: 01.01.2019 Copyright: © 2019 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. INTRODUCTION he implementation of standardized uniaxial tensile tests, with specimens made of brittle materials (as for example various hard rocks and rock-like geomaterials), encounters insuperable difficulties, related mainly to local fragmen- tation of the specimens during gripping and, also, fracture of the specimens in the immediate vicinity of the gripping area (rendering the data of the tests unreliable, given that the stress field in these regions is not purely uniaxial). In an attempt to overcome the problem, the engineering community is long ago seeking for alternative configurations that could provide at least an estimation of the uniaxial tensile strength of very brittle structural materials. Among the alternative configurations proposed, the one most widely accepted worldwide, is that of a circular disc compressed by diametral forces, T http://www.gruppofrattura.it/VA/47/2173.mp4 S. K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 47 (2019) 247-265; DOI: 10.3221/IGF-ESIS.47.19 248 c -P Lower indenter Upper indenter CSR- specimen AΒ D E E΄ D΄ y ΝCSR z AΒ D E E΄ D΄ r θ-P Pc -Pc P y xO which is known as Brazilian-disc test, honouring the Brazilian engineer Fernando Carneiro [1] who was the first one that proposed the specific configuration as a substitute of the direct tension test, already since 1943 (almost simultaneously with the Japanese engineer Tsunei Akazawa [2]). In spite of its wide acceptance (mainly due to the simplicity of the geometry of the specimens and of the experimental set- up), the Brazilian-disc test was strictly criticized almost immediately after it was introduced. The main points raising concerns about the validity of its outcome (and the relation of the quantity obtained with the tensile strength determined from a uniaxial tension test) are related to the fact that the stress field at the center of the disc is biaxial rather than uniaxial and, also, to the fact that (under specific conditions) fracture may originate from the immediate vicinity of the load application area (the vicinity of the points where the load is imposed) rather than from the center of the disc (rendering the validity of solutions [3-9] providing the “tensile strength” questionable [10-14]). In this direction, a variety of alternative configurations were gradually introduced, each one with its own pro and cons. Among them one should mention the ring test [15, 16], the flattened Brazilian-disc test [17] and the semi-circular bend test [18]. In the present study an alternative configuration is introduced, curing some drawbacks of previous attempts. The con- figuration proposed is that of a circular semi-ring (outer radius R2, inner radius R1, thickness 2h), which is loaded under compression, as it is shown in Fig. 1a. The respective test will be denoted from here on as the Circular Semi-Ring test (CSR test). The main advantage of the specific test is that the stress field at the critical point A (i.e., the point at which fracture is expected to start) includes a single tensile component. In addition, the ratio between the maximum tensile stress (developed at point A) and the respective compressive one (developed at point B) is relatively easily controlled by the ratio ρ=R2/R1. Moreover, the force required to cause fracture of the specimens is relatively low (compared to other more compact configurations). For the sake of generality, a non-zero eccentricity c of the applied load P, with respect to the vertical y-axis, has been considered (obviously, one can always assume c=0, simplifying significantly the analytic calculations). The similarity of the CSR-configuration to the familiar arc-shaped notched tension specimen, proposed by ASTM [19] for the standardized determination of Mode-I fracture toughness (ASTM E399-90 standard), is to be highlighted. (a) (b) Figure 1: (a) The configuration proposed for the experimental implementation of the CSR-test; (b) The NCSR configuration considered in the analytic solution of the problem. ANALYTICAL CONSIDERATIONS he stress field in the CSR will be here obtained analytically by adopting Muskhelishvili’s solution for a curved beam [20], assuming that the material of the CSR is homogeneous, isotropic and linearly elastic. It is emphasized from the very beginning that, the configuration considered in the analytic solution of the problem (shown in Fig. 1b) is somehow simplified compared to that of Fig. 1a, which is, in fact, proposed for the laboratory implementation of the test. The configuration of Fig. 1b will be denoted from here on as Net Circular Semi-Ring (NCSR), in order to be distin- guished from that of Fig. 1a. In this context, and taking into account the eccentricity c of the externally applied loading with respect to y-axis (which, for practical reasons, is very difficult to be avoided during the laboratory implementation of the experiments), the analytic solution deals with an NCSR, simultaneously subjected to bending by transverse forces –P, P and couples Pc, –Pc applied to its straight edges ED and E΄D΄ (Fig. 1b). T S. K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 47 (2019) 247-265; DOI: 10.3221/IGF-ESIS.47.19 249 In spite of the above mentioned difference between the CSR-specimen and the NCSR-configuration, the theoretically pre- dicted stresses in the vicinity of the critical cross-section AB (which is in fact the area of major interest for engineering ap- plications) are expected to approach well the respective ones developed in the CSR-specimen, provided that the width (w= R2-R1) of the NCSR is relatively small and that the resultant force and moment considered at its straight edges can efficiently replace the shaded parts of the CSR-specimen (Saint Venant’s principle). As it will be seen by the comparative consideration of the results of the numerical and the analytical approaches, the above assumptions are well justified even in case the inner radius, R1, of the CSR-specimen is equal to half of its outer one, R2. The analytic solution described in next sections is deduced from the respective one of the circular ring (CR) (in fact the NCSR is here considered as part of the CR [20]), under the admission of multi-valued displacements and the concept of dis- location, shortly recapitulated in the next paragraph. It is here recalled that the specific way of approaching this family of problems was first introduced by Golovin [21] (together with a series of solutions for the problem of curved beams under various loading schemes). Characteristics of the dislocation In this section a brief outline of the concepts of multi-valued displacements and dislocation will be given (together with a short description of the method adopted for obtaining the stress field in the NCSR through the solution of the respective CR problem), as they were analyzed in Muskhelishvili’s milestone book [20]. In this context, a homogeneous, isotropic and linearly elastic CR, of inner and outer radii R1 and R2, respectively, is considered in equilibrium under an arbitrary in-plane loading scheme. Assuming that the cross-section of the CR lies in the z=x+iy=reiθ plane, with its centre at the origin of the Cartesian refer- ence system, Muskhelishvili’s general solution for the first fundamental problem for the CR is written as [20]: ( ) log , ( )k kk kz A z a z z a z           (1) The demand for the displacements to be single valued in the CR reads as [20]: 0, 0u u v v       (2) where u and v are the horizontal and vertical components of the displacement field, respectively, with (+), (–) indicating the two sides of a cut joining the outer and inner perimeters of the CR, converting it into a simply connected region (Figs. 2 and 3). Constants ak, ak΄ in Eqs.(1) (excluding the imaginary part of a0 that remains arbitrary) are determined from the fulfillment of the boundary conditions for stresses and the following conditions, resulting from Eqs.(2) [20]: 1 1 2 [(1 )(1 2 )], plane strain3 0, 0 , (1 ), plane stress E A a a E                         (3) In Eqs.(3), E is the Young’s modulus, ν the Poisson’s ratio and μ the shear modulus. Over-bar denotes the complex con- jugate value. In case, now, multi-valued displacements are permitted in the cut CR, then instead of Eqns. (2) it holds that [20]: ,u u y v v x             (4) In this context, the additional expressions, besides the boundary conditions, for obtaining ak, ak΄ of Eqns. (1) (except the imaginary part of a0) are given (instead by Eqns. (3)) by the following expressions [20]: 1 1 (1 ) , ( i ) i A a a                (5) In Eqs.(4) and (5), ε (with dimensions of angle), and α, β (with dimensions of length), are arbitrary, infinitesimal, real con- stants, expressing the relative angle of rigid body rotation of the two sides of the cut about the origin, and the relative rigid S. K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 47 (2019) 247-265; DOI: 10.3221/IGF-ESIS.47.19 250 +β<0 y xO CR cut along the negative x-axis Overlapped parallel cut sides – -P P β>0 y xO CR cut along the negative x-axis Gap between the parallel cut sides + – P -P ε>0 – + y xO CR cut along the negative x-axis Overlapped radially cut sides Pc -Pc y xO CR cut along the negative x-axis Gap between the radially cut sides + –ε<0 -Pc Pc body translation of the two sides of the cut along x- and y-axis, respectively (Figs. 2 and 3). The difference from the previous case (Eqs.(2, 3)) is that by admitting the above multi-valued displacements in the cut CR, gaps or/and overlaps will appear (a) (b) Figure 2: The nature of the β-dislocation and the resulting transverse forces for (a) negative and (b) positive values. (a) (b) Figure 3: The nature of the ε-dislocation and the resulting couples for (a) positive and (b) negative values. between the sides of the cut, dictated by the values of ε, α and β and stresses will be developed within the cut CR, even in case of zero external forces. These stresses, due to ε, α and β (and not due to the application of some external loading scheme), are of particular interest in this study and will actually provide the stress field in the NCSR-configuration and in turn in the CSR-specimen in question, at least approximately. Before dealing with these stresses (as it is thoroughly described in next section), a few comments must be made for clarity on the nature of multi-valued displacements within the cut CR, which are due to Muskhelishvili [20]. Namely, it is mentioned that ε, α and β are the so-called “characteristics of the dislocation” (as named by Love [22]) and it was Timpe [23] who gave first a physical interpretation to these quantities in the case of the CR (later Volterra [24] dealt with the general case of multiply connected regions). Timpe [23] stated that even allowing multi-valued displacements, deformation can still be compatible assuming that one adds to (or/and removes from) the cut CR, strips of the same material, of dimensions ε, α and β and then rejoin the cut CR in one part. Obviously, when ε, α and β are to cause a gap between the sides of the cut (Figs. (2b, 3b)), a strip should be added between the sides of the cut and joined to them, whereas when ε, α and β are to cause overlapping of the sides of the cut (Figs. (2a, 3a)), a strip should be removed from the sides of the cut and the gen- erated new free sides of the cut should be brought into contact and joined together. In this way, the resulting CR will remain continuous after any actual deformation due to a non-zero external loading scheme, without gaps or overlaps, since the two sides of the cut will move as a single unit. The stress field in the NCSR due to the characteristics of the dislocation The next step is to return back to the initial problem, in order to describe the way the “characteristics of the dislocation” ε, α and β can provide the stress field in the CSR-specimen. As it was mentioned in previous section, the stress field in the CSR-specimen may (under certain conditions meeting Saint Venant’s Principle) be provided approximately from the solution of the problem of the NCSR subjected simultaneously to transverse forces –P, P and couples Pc, –Pc at its straight S. K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 47 (2019) 247-265; DOI: 10.3221/IGF-ESIS.47.19 251 edges ED and E΄D΄ (Fig. 1b). On the other hand, as it will be shown below, suitable combinations of ε, α and β lead to effective solutions to the problems of the NCSR under transverse forces and couples at its straight edges, separately. In this context, two independent problems are considered next for the NCSR: (a) the problem of bending by transverse forces of magnitude P applied to its straight edges ED, E΄D΄ and (b) the problem of bending by couples of magnitude Pc applied to its straight edges ED, E΄D΄. Then within the framework of linear elasticity, the solution of the whole problem for the NCSR, approximating that of the CSR, will be obtained by just superposing the solutions of these two problems. (a) Bending of the NCSR by transverse forces applied to its straight edges Assume that the cut CR is free from external forces and that ε=α=0 and β≠0. Then the deformed cut CR would attain one of the configurations of Fig. 2, depending on the sign of β, whereas stresses would occur in the cut CR due to β, given as [20]: 2 2 1 2 2 2 2 2 3 1 2 1 2 ( ) ( ) 1 1 cos ( 2 ) r R Rr r R R R R r                   (6) 2 2 1 2 2 2 2 2 3 1 2 1 2 ( ) ( ) 1 3 1 cos ( 2 ) R Rr r R R R R r                    (7) 2 2 1 2 2 2 2 2 3 1 2 1 2 ( ) ( ) 1 1 sin ( 2 ) r R Rr r R R R R r                    (8) Setting in Eqns. (6-8) θ=±π/2, it is seen that the only non-zero stress-component on the cross-sections ED and E΄D΄of the cut CR is τrθ(β). Moreover, integrating τrθ(β)2hdr over ED and E΄D΄, one can determine the transverse force acting on ED and E΄D΄ as follows [20]:    2 1 2 2 2 22 1 2 2 1 1 2 2 1 2 ( ) log 2 ( ) 2 2 d ( 2 ) R r R R R R R R h R P h r R R                      (9) Then keeping from the CR the part between θ=±π/2, one obtains the NCSR configuration under transverse forces of magnitude P (P>0) at its straight edges ED and E΄D΄. Moreover, as the fractions on the right-hand side of Eqn.(9) are always positive, the sign of β will dictate the direction of P, and it is easily seen that for β<0, i.e., for (using Eqn.(9)):   2 2 1 2 2 2 2 22 1 2 1 2 1 ( 2 ) 0 2 ( ) log R RP Rh R R R R R                 (10) one arrives at the first problem in question for the NCSR under bending by transverse forces –P, P applied to its straight edges. (b) Bending of the NCSR by couples applied to its straight edges Assume that the cut CR is free from external forces and that ε≠0, α=β=0. In that case, the deformed cut CR would attain one of the configurations of Fig. 3, depending on the sign of ε and stresses would appear in the cut CR due to ε, given as [20]: 2 2 2 2 1 2 2 2 2 1 1 2 2 2 2 2 12 1 2 1 ( ) ( ) log log1log log ( 2 ) r R R R R R R R r Rr R R R R                   (11) 2 2 2 2 1 2 2 2 2 1 1 2 2 2 2 2 12 1 2 1 ( ) ( ) log log1log log 1 ( 2 ) R R R R R R R r Rr R R R R                     (12) ( ) 0r   (13) S. K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 47 (2019) 247-265; DOI: 10.3221/IGF-ESIS.47.19 252 From Eqns. (11-13), it is immediately seen that by considering any isolated part of the cut CR, one has a curved beam, the only non-zero stress-component on the straight edges of which is σθ(ε). The distribution of that σθ(ε), attaining its extrema at the beam’s peripheries, is statically equivalent to counterbalancing couples at the beam’s straight edges of magnitude [20]:   2 1 2 22 2 2 2 2 2 1 1 2 1 2 2 2 1 ( ) 4 log ( ) 2 d 2 ( 2 ) R R R R R R R Rh h r r R R                     (14) Taking into account that according to the assumptions adopted this magnitude should be equal to Pc, then equating Eqn. (14) to Pc and solving for ε, one obtains:   2 2 2 1 2 22 2 2 2 2 2 1 1 2 1 ( 2 )2 0 ( ) 4 log R RPc h R R R R R R                     (15) It is seen that for the above positive value of ε, and by considering the part of the cut CR between θ=±π/2, one obtains the second problem in question for the NCSR under bending by couples –Pc, Pc applied to its straight edges ED and E΄D΄. (c) The overall problem of the NCSR for bending by transverse forces and couples applied to its straight edges The overall stress field in the NCSR under transverse forces –P, P and couples Pc, –Pc at its straight edges ED and E΄D΄, approaching that of the CSR-specimen, is given by superposing the respective stress components: σr=σr(β)+σr(ε) (Eqns. (6, 11)), σθ=σθ(β)+σθ(ε) (Eqns. (7, 12)) and τrθ=τrθ(β)+τrθ(ε) (Eqns. (8, 13)), with β and ε given by Eqs.(10, 15), respectively. It is mentioned that these stresses depend on the magnitude P of the transverse force, the eccentricity c, the ratio ρ=R2/R1, the thickness 2h of the CSR, the loading conditions adopted (namely plane stress or plain strain), and, also, on the material (appearing in the expressions for the dislocations β, ε). It must be, also, mentioned that assigning to the parameters of the analysis arithmetic data corresponding to very brittle materials, the values calculated for β and ε (depending on the values of P and c considered) are extremely small, satisfying the principal assumptions governing the present study, i.e., the assumption of linear elasticity. Effectiveness of the analytic solution The effectiveness of the analytic solution was assessed in terms of the data for the displacement field, as it was obtained from a recent experimental protocol [25]. CSR-specimens made of PMMA (E=3.2 GPa, ν=0.36) with 2h=10 mm, R1=25 mm, R2=50 mm, and c=6.25 mm were tested according to the scheme shown in Fig. 1a. A 10 kN electromechanical frame was used to load the specimens and the displacement field was measured with the aid of the Digital Image Correlation (DIC) technique. The experimental set-up is shown in Fig. 4a. All tests were quasi-static, implemented under displacement-control conditions, at a constant rate of 0.1 mm/min. A typical fractured specimen is shown in Fig. 4b. During loading, series of successive photos of the specimens were taken permitting the determination of the displacement field by means of the software of the DIC system. The experimental data for the displacements were then compared with the ones provided by the analytic solution (as it is analytically described by Markides et al. [25]), which read as: (a) Displacements due to bending of the NCSR by transverse forces applied to its straight edges   2 ( ) 2 2 1 2 2 2 2 1 2 2 2 22 2 1 21 2 ( ) 3 1 3 log 1 4 ( 2 ) 1 1 2 cos 2 2 r u r R R R R r r R RR R                                        (16) S. K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 47 (2019) 247-265; DOI: 10.3221/IGF-ESIS.47.19 253 0.9 1.0 1.1 1.2 0 10 20 30 40 50 D is p la ce m en t a lo n g lo cu s A B , u x [m m ] r [mm] Analytical solution DIC data   2 22 ( ) 1 2 2 2 22 2 1 2 1 2 ( ) 3 1 3 1 1 1 sin 2 4 ( 2 ) 2 R Rr v R R rR R                                   (17) (b) Displacements due to bending of the NCSR by couples applied to its straight edges 2 2 ( ) 2 2 1 1 2 2 2 1 2 2 1 2 2 2 2 12 1 ( ) log log3 1 1 log cos 4 ( 2 ) 2 23 3 cos 1 sin cos log R R R R u r r R R R R R r r R rR R                                                     (18) 2 2 ( ) 2 2 1 1 2 2 2 1 2 2 1 2 2 2 2 12 1 ( ) log log3 1 1 log sin 4 ( 2 ) 2 23 3 sin 1 cos sin log R R R R v r r R R R R R r r R rR R                                                     (19) The results of the comparison are plotted in Fig. 4c, in which the displacement ux≡u developed along the AB locus (see Fig. 1a), is plotted according to both the analytic and the experimental data. The agreement is satisfactory almost all along the locus considered (which is, in fact the most crucial for practical applications), excluding a weak maximum recorded by the DIC-system in the immediate vicinity of point B, which is not actually predicted by the analytic solution. (a) (b) (c) Figure 4: (a) An overview of the experimental set-up; (b) A fractured CSR specimen; (c) Analytic versus experimental data for the dis- placement component developed along the critical locus AB. The stress field in the NCSR at some strategic points and along some critical loci of the CSR In this paragraph the formulae obtained for the stress field in the NCSR are applied along some loci of particular practical (and engineering) interest. However, for a direct comparison between the data for the CSR- and the NCSR-configurations to be feasible, some preliminary remarks should be made. Initially, one should consider the conditions related to the ec- centric application of the compressive forces (Fig. 1a) through the two cylindrical rods. Let Ro be the radius of the rods and assume that half of their profile is inserted into the respective congruent hollow cavity of the CSR-specimen, at a given distance c on the left of y-axis (Fig. 5). Then, the points of contact of the CSR with the indenters, corresponding to any specific eccentricity c, will be described as: S. K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 47 (2019) 247-265; DOI: 10.3221/IGF-ESIS.47.19 254 -P Upper indenter CSR- specimen y D E Ro θo θo c xO i / 2( )  cΘ cR e ο ο 2 ο ο R R R θ θ -1 sintan - cos Rosinθo i (π/2 )e c± +c Θz = R , 2 2 -1 -12 2 2 2 sin cos tan sin cos o o c o o o c o o o o R θ c R = R + R - 2R R θ , Θ =θ + , θ = R - R θ R       (20) Figure 5: The procedure adopted to specify the contact point i ( /2 )e ccz R   . Moreover, it is noted that during the experimental implementation of the CSR-test (as well as in the numerical model that will be described in next sections, which reproduces accurately the experimental procedure) both rods are restricted con- cerning their motion in the horizontal direction. In addition, the lower rod is fixed also concerning its vertical motion (i.e., it is assumed completely fixed, without any degree of freedom) and a vertical displacement vP is imposed to the CSR-specimen, with the aid of the upper rod. On the contrary, in the analytic study, diametral compressive forces P and couples Pc are imposed on both vertical edges of the NCSR. Obviously, for comparison reasons, the magnitude of these forces and couples must produce a vertical displacement equal to (1/2)vP to the points where they are applied. In this direction, advantage is taken of the above mentioned formulae for the displacement field in the NCSR. More specifically, the NCSR is ideally extended to the CSR-configuration with the supporting parts shown shaded in Fig. 1a. The contact points i ( /2 )e ccz R    , mentioned previously, are assumed lying on the shadowed parts (which is always true since NCSR is part of the whole cut CR, to which the analytic solutions obtained before refer to). Then, in accordance with the experimental procedure (and the respective numerical models), point i ( /2 )e ccz R    must be completely fixed while point +i ( /2 )e ccz R   must be fixed only in the x-direction. For this to be achieved one should subtract from the analytically determined displacement field the respective displacements of the points i ( /2 )e ccz R    , considering them as rigid-body ones. As a next step, the experimental/numerical value of vP must be introduced in the expression for the vertical displacement v of point +i ( /2 )e ccz R   providing the magnitude of the force P, corresponding to the specific value of vP. Then, inserting this value of P (as well as the proper value of c, i.e., the value matching the experimental/numerical configuration), into Eqs.(10, 15), ε and β and in turn the theoretical stresses (Eqs.(6-8) and Eqs.(11-13)) in the NCSR are obtained, in a manner comparable to the experimental and numerical ones for the CSR-specimen. In this context, and in order for some features of the analytical solution to be quantitatively enlightened, an NCSR made of PMMA (E=3.2 GPa, ν=0.36) is here considered with c=10 mm and vP=3 mm (corresponding to the data of the experi- mental protocol discussed previously and to the numerical models that will be described next). The force corresponding to these data (i.e., the force that has to be introduced in the analytic formulae for the stress field) is determined equal to P=1188.3 N for plane strain conditions and to P=994.5 N for plane stress ones. S. K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 47 (2019) 247-265; DOI: 10.3221/IGF-ESIS.47.19 255 -80 -40 0 40 80 0 10 20 30 40 50 σ θ [M P a] r [mm] 0.7 0.5 0.3 0.1 R2/R1 = 1.43 2.00 3.33 10.0 A B AB Adopting the above values the stress components at the two critical points of the configuration (namely points A(R2,0) and B(R1,0)) are calculated equal to: ‐ σθ(R2,0)=39.97 MPa, σθ(R1,0)= –75.98 MPa, for plane strain, ‐ σθ(R2,0)=33.45 MPa, σθ(R1,0)= –63.59 MPa, for plane stress. The ratio of the two stresses is equal to:     1 2 2 1 , 0 1.9, for 2 , 0 R R R R         (21) In other words, the ratio of the maximum compressive stress developed over the respective maximum tensile one, in a CSR with ρ=2, is less than 2. Compared to the respective ratio of the Brazilian disc test, which at the center of the disc is equal to 3, one of the major advantages of CSR-specimen is highlighted, since the CSR-test can be used in a broader variety of materials concerning the ratio of compressive over tensile strength. Regarding now the contribution of each one of the two constituent components (transverse forces and bending moments) of the overall stress field, it can be determined that for plane strain conditions it holds that: - Contribution of transverse forces, –P, P (β-dislocation): σθ(β)(R2,0)=30.62 MPa, σθ(β)(R1,0)= –61.24 MPa, ‐ Contribution of couples, –Pc, Pc (ε-dislocation): σθ(ε)(R2,0)= 9.35 MPa, σθ(ε)(R1,0)= –14.74 MPa, while for plane stress it holds that: ‐ Contribution of transverse forces, –P, P (β-dislocation): σθ(β)(R2,0)=25.62 MPa, σθ(β)(R1,0)= –51.25 MPa, ‐ Contribution of couples, –Pc, Pc (ε-dislocation): σθ(ε)(R2,0)= 7.82 MPa, σθ(ε)(R1,0)= –12.34 MPa. In both cases (plane strain - plane stress), it is determined that: - Point A: σθ(β)(R2,0) =0.77σθ(R2,0), σθ(ε)(R2,0)=0.23 σθ(R2,0), - Point B: σθ(β)(R2,0) =0.81σθ(R2,0), σθ(ε)(R2,0)=0.19 σθ(R2,0). In addition, for either plane strain or plane stress, it holds that: - Bending by transverse forces, –P, P (β-dislocation):     ( ) 1 2 ( ) 12 , 0 2 , 0 R R RR           , - Bending by couples, –Pc, Pc (ε-dislocation):     2 2 2 2 ( ) 2 1 2 1 1 ( ) 2 2 2 22 2 1 1 1 2 log , 0 1.58 , 0 2 log R R R R R R RR R R R R             . Figure 6: The distribution of the transverse normal stress along the axis of symmetry (locus AB) of the CSR for various ρ-values. S. K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 47 (2019) 247-265; DOI: 10.3221/IGF-ESIS.47.19 256 As it has been already mentioned, the ratio of the maximum compressive stress developed in a CSR over the respective maximum tensile one is controllable by a geometric parameter, namely the ratio ρ=R2/R1 of the specimen’s radii. In order to quantitatively explore this issue, the variation of the transverse stress, σθ, along the locus AB, is plotted in Fig. 6, for a series of values of the ratio ρ, within the 1.43<ρ<10.00 range. It is seen from Fig. 6 that, the transverse stress is tensile (as it is expected) at the outermost point A of the locus AB and its value gradually decreases as one moves towards the innermost point B. 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