Stresses and displacements in a circular ring under parabolic diametral compression

International Journal of Rock Mechanics & Mining Sciences 71 (2014) 272–292

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Stresses and displacements in a circular ring under parabolic diametral compression S.K. Kourkoulis n, Ch.F. Markides National Technical University of Athens, School of Applied Mathematical and Physical Sciences, Department of Mechanics, 5 Heroes of Polytechnion Avenue, Theocaris Building, Zografou Campus, 157 73 Athens, Greece

art ic l e i nf o

a b s t r a c t

Article history: Received 4 April 2013 Received in revised form 28 February 2014 Accepted 23 July 2014

The response of a linear elastic circular ring to a parabolic distribution of radial pressure, imposed along two symmetric arcs of its outer perimeter, is explored analytically. Muskhelishvili's complex potentials technique is employed and both the stress- and displacement-fields are obtained in series form. For small holes the ring's loaded arc is assumed identical with that provided by the contact problem of a solid disc squeezed under the same load between the jaws of the ISRM's suggested device for the implementation of the Brazilian-disc test. This assumption is experimentally assessed by properly adapting the Reflected Caustics technique. It is concluded that for inner ring's radius smaller than about thirty per cent of the outer radius, the agreement for the length of the loaded arc between the ring and the solid disc is quite satisfactory. For bigger holes the formulae obtained are still valid assuming however that the boundary conditions are properly modified considering a predefined loaded arc. The variation of stresses and displacements along critical loci of the ring is investigated and is compared with pre-existing solutions for rings under point-load or uniform pressure over arbitrarily chosen loaded arcs. It is concluded that for small holes the type of pressure imposed is not crucial as long as attention is focused at the critical points of the ring where the maximum tensile stress is developed. On the contrary, near the loaded arc the stress field strongly depends on the kind of the externally imposed load which influences also the results at the ring's critical points in case the inner radius increases approaching the outer one. The ratio between the tensile stress at the centre of a solid disc (Brazilian-disc test) and the analogous one at the critical points of a ring loaded under identical conditions is also investigated thoroughly. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Circular ring Brazilian-disc test Stress field Displacement field Indirect tensile strength Reflected Caustics

1. Introduction The configuration of a circular ring subjected to either equal compressive line-forces along diametrically opposed generatrices of its lateral surface or to uniform radial pressure along two finite arcs of its periphery (symmetric with respect to the ring's centre), is widely used in various fields of both theoretical and experimental mechanics. Although the respective stress field was studied already since 1910 by Timoshenko [1] and a few years later by Filon [2], various aspects of the problem remain still open. For example, the analytic solution for the displacement field in case a thin ring is subjected to uniform pressure was presented in a convenient easyto-use form only a few years ago by Tokovyy et al. [3]. The interest of the engineering community on the subject is continuous and undiminished. It has been used, for example, for the analysis of finite strain [4], assessing hot workability [5],

n

Corresponding author. Tel.: þ 30 210 7721263; fax: þ 30 210 7721302. E-mail address: [email protected] (S.K. Kourkoulis).

http://dx.doi.org/10.1016/j.ijrmms.2014.07.009 1365-1609/& 2014 Elsevier Ltd. All rights reserved.

determining the collapse load of mild steel rings [6], the stress concentration index of sea-ice [7] etc. Moreover, it was recently proposed as a convenient mean for the determination of fracture toughness without using fatigue pre-cracked specimens [8]. Among the most interesting applications of the ring test is the indirect determination of the tensile strength of brittle materials, especially concrete and various rock-like geomaterials. In fact, the use of ring-shaped specimens instead of solid discs was proposed immediately after the Brazilian-disc test had been introduced by Carneiro [9] and Akazawa [10]. This was done in an attempt to cure some drawbacks of the Brazilian-disc test related to the inevitable stress concentration along the disc–jaw interface which can cause premature local fractures deteriorating the validity of the results obtained by Hondros' classic analysis [11]. The origin of this technique is found in a paper by Ripperger and Davis [12] dated back to 1946. Bortz and Lund [13] and Addinall and Hackett [14] studied the role of the ratio ρ¼ R1/R2 (i.e. the ratio of the inner over the outer radii of the ring). Around the same period it was Hobbs [15] who clearly stated that the results of the Brazilian-disc test “…are suspect because wedge

S.K. Kourkoulis, Ch.F. Markides / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 272–292

Nomenclature A Ak

the initial curve functions of angle ϑ determined from the boundary conditions a, b the end points of the caustic curve bj, b0 j the coefficients of the infinite series (they are given in Appendix II) C constant depending on the mechanical and optical constants of the material and the caustics experimental set-up Cn A global constant equal to C n ¼ C=ð12jλm jKR2 Þ cf the stress optic coefficient of the material. It holds that cf ¼ ν/E E Young's modulus of the ring's material Ef a fictitious value of Young's modulus F(r) a function relating the ring's and Brazilian disc's critical transverse stress components along the vertical loading axis F(ρ) The value of F(r), for n¼ 1, at the ring's critical points where the tensile stress is maximized. F(ρ)¼F(r¼ R1, n=1). It is given in Appendix IV Fo the value of F(ρ) for ρ¼0 ℑ the imaginary part of a complex number or function K constant equal to K¼ (κ1 þ1)/4μ1 þ (κ2 þ1)/4μ2 KΗ Hobbs' correction factor KH ¼6 þ38ρ2 ℓ the semi-contact length in the contact problem ℓ΄ the approximate value of ℓ on the ring, ℓ  ℓ0 ¼ R2 sinωo L1, L2 the ring's inner and outer boundaries, respectively L3 the cylindrical jaw's boundary (contact problem) ν the Poisson's ratio of the ring's material νf a fictitious value of Poisson's ratio
! the unit normal vector at a point P on the deformed np ring's front face P the arbitrary point on the front face of the ring in the caustics experimental set-up 0 P the projection of P on the reference screen Pp the new position of P after the out-of-plane deformation Pframe the magnitude of the force applied by the loading device P(τ) the distribution of the externally applied radial pressure along the contact arc in the contact problem PðϑÞ the distribution of the externally applied radial pressure along the loaded arcs of the outer boundary L2 of the ring Pc the maximum value of PðϑÞ Q the arbitrary point on the caustic curve on the reference screen ðr; ϑÞ the modulus and argument of the complex variable z ðr 1 ; ϑ1 Þ and ðr 2 ; ϑ2 Þ the moduli and arguments of the auxiliary complex variables z1 and z2, respectively, as they are defined in Fig. 10 ro the modulus of the arbitrary point on the caustic's initial curve ℜ the real part of a complex number or function R1, R2 the ring's inner and outer radii, respectively R3 the cylindrical jaw's radius, R3 ¼1.5R2 (contact problem) t the ring's (or disc's) thickness Δt the thickness change

tj

273

the ends of the loaded arcs of the ring's outer boundary L2 u, v the Cartesian (horizontal, vertical) displacement components in the ring ur ; uϑ the polar (radial, transverse) displacement components on the ring ! w the vector on the reference screen defining the deviation of light
! W the vector on the reference screen defining the caustic curve
! W x0 ; W y0 the components of the vector W in the {O0 ; x0 y0 ,z0 } Cartesian reference on the screen x0a ; y0a ; x0b ; y0b The coordinates of the end points a, b, of the caustic curve in the {O0 ; x0 ,y0 ,z0 } Cartesian reference on the screen ! ! ! ! ∇ the gradient operator ∇ ¼ ð∂=∂xÞ i þð∂=∂yÞ j iϑ z the complex variable, z ¼ x þ iy ¼ re Ζο the distance between the ring and the reference screen in the Caustics experimental set-up Ζi the distance between the focus of the incident light bundle and the ring's front face α, β the end points of the initial curve αk ; α0k Complex constants in the Laurent series representation of Φ(z), Ψ(z) ε the semi-distance between the end points of the caustic curve η the elevation corresponding to the end points α and β of the initial curve ϑcr the critical angle at which the transverse stress component of the ring changes sign on the outer ring's boundary κ1 , κ2 Muskhelishvili's constants for the ring and the jaw, respectively λm the magnification factor of the caustics experimental set-up μ1, μ2 shear moduli for the ring and jaw, respectively n the number of terms in the infinite part of the obtained formulae ρ the ratio of ring's radii, ρ ¼ R1 =R2 σ rr ; σ ϑϑ ; σ rϑ the radial, transverse and shear stress components in the ring σ ϑϑ;A ; σ ϑϑ;B ; σ ϑϑ;C and σ ϑϑ;D the transverse stress components at the points A, B, C and D of the ring the transverse stress component of the ring along the σ Ring ϑϑ loading axis, σ Ring ϑϑ ¼ σ ϑϑ ðr; ϑ ¼ 901Þ Br Br σ Br ; σ ; σ the stress components in the Brazilian disc (given rr ϑϑ rϑ in Appendix III) σt the tensile strength of the material determined by direct tension σ Br the maximum tensile stress in the Brazilian disc, t Br σ Br t ¼ σ ϑϑ ðr ¼ 0; ϑ ¼ 901Þ Ring σt the maximum tensile stress in the ring, σ Ring  σ Ring t ϑϑ ðr ¼ R1 ; ϑ ¼ 901Þ σy the yield stress σ1, σ2 the principal stresses τ the arc length within [–ℓ, þℓ] (contact problem) τ΄ the approximate value of τ on the ring, τ  τ0 ¼ R2 cosϑ Φ(z), Ψ(z) the complex potentials for the ring ωο the semi-contact angle (semi-loaded arc) ω the deviation angle of the reflected light beam in Snell's law

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shaped fractures frequently form close to the loading platens. These fractures may occur prior to tensile failure along the loaded diameter”. In order to avoid this limitation he proposed as an alternative technique the use of the “…diametrical compression of a circular disc with a small central hole” [15]. According to Hobbs, the maximum tensile stress developed in a circular ring of outer radius R2 and thickness t is given as σ Ring ¼ ðP frame K H Þ=ðπR2 tÞ where t Pframe is the overall external load and ΚH is a function of ρ. The above formula is similar to the respective approximate Hondros' formula for a solid disc simply multiplied by ΚH, which can be considered as a correction factor. For ρo0.1 Hobbs [15] approximated this correction factor as ΚH ¼ 6þ 38ρ2. Jaeger and Hoskins [16] used complex analysis for the determination of the full-field stress expressions within a circular ring loaded by uniformly distributed radial pressure along two opposite arcs of its outer (or also its inner) surface instead of concentrated line loads. The ring test (as well as the Brazilian-disc test) was critically discussed by Hudson [17] and Hudson et al. [18] who definitely concluded that “… the tensile strength demonstrated in such a test is an experimental or technological property rather than a material property” [17]. According to Hudson, even in the range of ρ-values for which the critical strength obtained from the ring test does become geometry independent, the respective value cannot be considered as representative of the true tensile strength of the ring's material. The most exhaustive study of almost all parameters related to the experimental implementation of the Brazilian-disc- and the ring-tests was carried out by Mellor and Hawkes [19]. They compared the results obtained from the two tests attempting also a matching of the respective results as the ratio ρ tends to zero. At the same moment, they compared the results of the two tests with the respective ones obtained from direct tension; however, their conclusions were not very encouraging. Moreover, they indicated that it would be beneficial to use curved jaws and presented a device which at failure gave a contact arc approximately equal to about 101. A few years later the ISRM (International Society for Rock Mechanics) standardized the procedure for implementing the Brazilian-disc test suggesting the familiar device [20] which is almost identical to that proposed by Mellor and Hawkes. Nowadays, both the Brazilian-disc- and the ring-tests are still under intensive study [21–24], since quite a few issues are still open, especially the ones related to the actual loading conditions at the ring–jaw interface (assuming that both tests are implemented using the ISRM suggested device [20]). In such a case it is known that the radial pressure is not uniformly distributed but rather it follows a distribution simulated by an elliptic arc [25]. Given that closed form solutions for the stress field developed in a ring due to this specific distribution of pressure cannot be obtained [26], an attempt is described here for the analytic determination of the stress- and displacement-fields developed assuming that the ring is under the influence of pressure that varies according to a statically equivalent parabolic distribution, closely resembling the elliptic one. Adopting Muskhelishvili's complex potentials method [27], analytic formulae are obtained in infinite series form. Assuming the ring's inner radius is small compared to the outer one, the length of the loaded arc and pressure variation along this arc are considered the same with those developed in case a solid disc of the same radius (with the ring's external radius) is compressed by the same external force in the ISRM's suggested device for the implementation of the Brazilian-disc test. After a slight modification the same formulae apply equally well for rings with bigger holes (i.e. when similarity between rings and solid discs is violated) by considering an arbitrarily predefined loaded arc. Taking advantage of these formulae some crucial aspects of the ring test are enlightened related to the stress concentration in the

immediate vicinity of the loaded arc and also to the influence of various geometric and material parameters on the overall stress field. The relation between the maximum tensile stress developed in the ring and the tensile stress developed at the centre of a solid disc is also explored. As a final step, the validity of the critical assumption of the analysis, concerning the length of the loaded arc, is critically assessed with the aid of series of experiments using the “Reflected Caustics” method [28,29].

2. Theoretical considerations 2.1. The natural problem and the respective mathematical model According to the ISRM suggestion for the standardized implementation of the Brazilian-disc test, a solid circular disc is compressed between two curved metallic jaws. To the best of our knowledge, there is not a similar suggestion for the diametral compression of ring-shaped specimens and therefore it is assumed here on that the ring test is implemented using the same as above device (Fig. 1a): a circular ring of outer radius R2, inner radius R1 and thickness t, is compressed under an overall external load Pframe between two metallic jaws the inner surface of which is cylindrical of curvature radius R3 ¼1.5R2. Assuming that the own weight of the ring and the jaws is negligible it is obvious that the ring and each jaw are initially in contact along a mathematical line, i.e. their common generatrix. As the load imposed increases contact is realised along a finite surface the projection of which on the plane normal to the axis of the ring is a finite arc of length 2ℓ. The extent of 2ℓ as well as the distribution of the contact stress is a complicated problem itself. In the present study, considering the inner radius of the ring is small enough compared to its outer one it is assumed that, in a first approximation, both the contact length and distribution of the radial pressure are accurately enough described by the relations rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6R2 Κ P f rame 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℓ¼ ð2:1Þ ℓ2 τ2 ; PðτÞ ¼ 3R2 K πt obtained in [25] in case a solid disc of radius R2 is compressed against the ISRM's metallic jaw (an assumption which is experimentally assessed in Section 7). In Eq. (2.1), K ¼ (κ1 þ1)/(4μ1) þ (κ2 þ1)/(4μ2) where κj, μj, j¼1,2 are Muskhelishvili's constants and shear moduli of the disc – here, the ring and jaw materials, respectively. Letter τ denotes the arc length within 2ℓ. Taking advantage of the above considerations one can proceed to the formulation of a First Fundamental plane problem for an isolated circular ring under the action of radial compressive stresses of magnitude P(τ) along two finite arcs of its outer boundary, each one of length 2ℓ, symmetrically lying with respect to its centre. However, such an elliptic distribution of pressure does not lead to closed-form expressions for the stress- and displacement-fields, as it is analytically outlined in Appendix I. Therefore, an alternative statically equivalent parabolic distribution is adopted [30] as   τ 2  πℓ PðτÞ ¼ 1 ð2:2Þ 8R2 Κ ℓ Thereinafter, the isolated ring's cross section, bounded by two concentric circles, L1 and L2, is considered lying in the z ¼ x þiy ¼ reiϑ complex plane with the origin of the Cartesian reference system being at its centre Fig. 1b. The loading will be of the form given by Eq. (2.2) with its axis to coincide with the y-axis. For a relatively small ℓ, the arcs τ and ℓ of L2 are approximated by the respective straight segments τ0 and ℓ0 , so that ℓ  ℓ0 ¼ R2 sinωo ;

τ  τ0 ¼ R2 cosϑ

ð2:3Þ

S.K. Kourkoulis, Ch.F. Markides / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 272–292

275

outer periphery must be re-written as

2P frame sin2 ωo cos2 ϑ PðϑÞ ¼ 1 2 R2 tðsin2ωo  2ωo cos2ωo Þ sin ωo |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð2:6Þ

P c ¼ PðϑÞmax

Comparing Eqs. (2.5) and (2.6), it is notable that both arguments for the distribution of radial pressure and contact length apply equally well in the analysis following in Sections 2.2 and 3, leading to the same formulae differing only in the definition of the constant Pc. 2.2. The complex potentials Assuming that the problem is plane and that the material of the ring is homogeneous, isotropic and linearly elastic, Muskhelishvili's formalism is applied providing stresses and displacements throughout the ring in terms of the complex potentials Φ(z) and Ψ(z) as [27] σ rr þ σ ϑϑ ¼ 4ℜΦðzÞ

ð2:7Þ

σ rr  iσ rϑ ¼ ΦðzÞ þ ΦðzÞ  e2iϑ ½zΦ0 ðzÞ þ Ψ ðzÞ

ð2:8Þ

2μ1 ðu þ ivÞ ¼ κ 1 φðzÞ  zφ0 ðzÞ ψ ðzÞ ð2:9Þ R R where φðzÞ ¼ ΦðzÞdz, ψðzÞ ¼ Ψ ðzÞdz. In addition, ℜ denotes the real part, over-bar denotes the conjugate complex value and prime denotes first order derivative. In accordance with the configuration shown in Fig. 1b and considering the distribution of radial compressive stresses of Eq. (2.5) or (2.6), the boundary conditions are expressed as )

π=2  ωo cos2 ϑ σ rr ðϑÞ ¼  PðϑÞ ¼ P c 1  2 for 3π=2  ωo sin ωo ( π=2 þ ωo rϑr ð2:10Þ on L2 3π=2 þ ωo

Fig. 1. (a) Schematic representation of the ring – ISRM's experimental device. (b) The isolated ring and definition of symbols.

where ωo denotes the semi-contact arc. Combining Eqs. (2.1) and (2.3) yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6ΚP frame ωo ¼ Arcsin ð2:4Þ πR2 t

Introducing Eqs. (2.3) and (2.4) in Eq. (2.2) one obtains the expression for the distribution of radial pressure along the loaded arcs (t1t2) and (t3t4) of L2 in terms of ϑ as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3πP frame cos2 ϑ PðϑÞ ¼ 1 2 ð2:5Þ 32ΚR2 t sin ωo |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} P c ¼ PðϑÞmax

It must be mentioned that for rings with inner diameter approaching the outer one the assumption about the identity of the contact lengths for the ring and the solid disc ceases to be valid. In this case and in luck of a solution of the respective contact problem, the condition about the length of the contact arc (i.e. that angle ωο is provided by Eq. (2.4)) is here relaxed and a predefined value for ωο (either according to experimental observations or based on data obtained from numerical models) is considered. Accordingly (and keeping its general parabolic character), the statically equivalent distribution of pressure imposed on the ring's

σ rr ðϑÞ ¼ 0 on the unloaded part of L2

ð2:11Þ

σ rr ðϑÞ ¼ σ rϑ ðϑÞ ¼ 0 everywhere on L1 and L2

ð2:12Þ

Further, according to Muskhelishvili [27], these conditions are written in complex Fourier series form as 8 0; on L1 > < ð2:13Þ σ rr  iσ rϑ ¼ þ 1 ikϑ > : ∑ Ak e ; on L2 1

R 1 2π  ikϑ where Ak ¼ 2π dϑ are known quantities through Eqs. 0 σ rr ðϑÞe (2.10)–(2.12), while the analytic functions Φ(z) and Ψ(z) can be put in the form þ1

ΦðzÞ ¼ ∑ αk zk ; 1

þ1

Ψ ðzÞ ¼ ∑ α0k zk

ð2:14Þ

1

where αk , α0k are in general complex constants. Combining Eqs. (2.8), (2.13) and (2.14) yields on the boundaries of the ring 0;

9 r ¼ R1 > =

∑ Ak eikϑ ;

r ¼ R2 > ;

þ1 1

þ1

þ1

þ1

1

1

1

¼ ∑ ð1  kÞαk r k eikϑ þ ∑ αk r k e  ikϑ  ∑ α0k  2 r k  2 eikϑ

ð2:15Þ By comparing similar terms, Eq. (2.15) provide αk , turn Φ(z) and Ψ(z), as ΦðzÞ ¼

α0k ,

and in

  1 Pc b2 b  4n b  2ð2n þ 1Þ b0 þ b2 z2 þ 2 þ ∑ b4n z4n þ b2ð2n þ 1Þ z2ð2n þ 1Þ þ 4n þ 2ð2n þ 1Þ π z z z n¼1

ð2:16Þ

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S.K. Kourkoulis, Ch.F. Markides / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 272–292

Ψ ðzÞ ¼

0 0 Pc 0 b b 0 b0 þ b2 z2 þ 22 þ 44 π z z " 1

þ ∑

n¼1

0

0

b4n z4n þb2ð2n þ 1Þ z2ð2n þ 1Þ þ

2.4. The displacement field 0

b  4ðn þ 1Þ z4ðn þ 1Þ

0

þ

b  2ð2n þ 1Þ

#)

z2ð2n þ 1Þ ð2:17Þ

Integrating Φ(z) and Ψ(z) (assuming zero integration constants) yields  1 Pc b2 z 3 b  2 b4n z4n þ 1 b2ð2n þ 1Þ z4n þ 3 b0 z þ  þ ∑ þ φðzÞ ¼ π 3 z 4n þ 1 4n þ 3 n¼1  b  2ð2n þ 1Þ b  4n  ð2:18Þ  ð4n  1Þz4n  1 ð4n þ 1Þz4n þ 1 ( " 0 0 0 0 0 1 Pc 0 b z3 b b b4n z4n þ 1 b2ð2n þ 1Þ z4n þ 3 þ b0 z þ 2   2  34 þ ∑ π 3 z 4n þ1 4n þ 3 3z n¼1 #) 0 0 b  4ðn þ 1Þ b  2ð2n þ 1Þ  ð2:19Þ  ð4n þ 3Þz4n þ 3 ð4n þ 1Þz4n þ 1

ψðzÞ ¼

0

The constants bj , bj , in the absence of shear stresses on L2 and with the loading axis to coincide with the y-axis are all found to be real quantities. The respective expressions are very lengthy and for brevity they are given in Appendix II. 2.3. The stress field Combining Eqs. (2.7), (2.8), (2.16) and (2.17) yields the stress components in polar form at any point (r,ϑ) of the ring, as

0 0 Pc b 4b  2 b 0 0 2b0  2 2  b0  2 þ 44 cos2ϑ  b2 r 2 cos4ϑ σ rr ¼ π r r r (" # 0 1 2ð2n þ1Þb  4n b  2ð2n þ 1Þ 4n þ ∑  2ð2n þ 1Þ :cos4nϑ 2ð1 2nÞb4n r þ r 4n r n¼1

0  b2ð2n þ 1Þ r 2ð2n þ 1Þ cos4ðn þ 1Þϑ  4nb2ð2n þ 1Þ r 2ð2n þ 1Þ # 0 4ðn þ 1Þb  2ð2n þ 1Þ b  4ðn þ 1Þ 0 þ b4n r 4n  þ r 2ð2n þ 1Þ r 4ðn þ 1Þ  cos2ð2n þ 1Þϑ ð2:20Þ

0 0 Pc b b 0 0 2b0 þ 2 2 þ b0 þ 4b2 r 2 þ 44 cos2ϑ þ b2 r 2 cos4ϑ π r r (" # 0 1 2ð2n1Þb  4n b  2ð2n þ 1Þ þ 2ð2n þ 1Þb4n r 4n  :cos4nϑ þ ∑ r 4n r 2ð2n þ 1Þ n¼1

0 þ b2ð2n þ 1Þ r 2ð2n þ 1Þ cos4ðn þ 1Þϑ þ 4ðn þ 1Þb2ð2n þ 1Þ r 2ð2n þ 1Þ # 0 4nb  2ð2n þ 1Þ b  4ðn þ 1Þ 0 4n þ 4ðn þ 1Þ þ b4n r  r 2ð2n þ 1Þ r  cos2ð2n þ 1Þϑ ð2:21Þ

σ ϑϑ ¼



 0 b2 b 0 0 b0 þ 2 b2 r 2 þ 2  4 4 sin2ϑ þb2 r 2 sin4ϑ r r (" #

0 1 b  2ð2n þ 1Þ b  4n 4n b4n r 4n þ 4n  2ð2n þ 1Þ :sin4nϑ þ ∑ r r n¼1

σ rϑ ¼

Pc π

0

þ b2ð2n þ 1Þ r 2ð2n þ 1Þ  sin4ðn þ 1Þϑ    b  2ð2n þ 1Þ þ 2ð2n þ 1Þ b2ð2n þ 1Þ r 2ð2n þ 1Þ þ 2ð2n þ 1Þ r #  0 b  4ðn þ 1Þ 0 þ b4n r 4n  4ðn þ 1Þ sin2ð2n þ 1Þϑ r

ð2:22Þ

Combining Eqs. (2.9), (2.18) and (2.19) yields the Cartesian components of the displacement at any point (r,ϑ) on the ring, as   0

Pc κ1 b  2  b  2 0  u¼ cosϑ ðκ1  1Þb0  b0 r  b2 r 3  2πμ1 r   0 0 3 ðκ 1 b2  b2 Þr b2 b4  þ 3 cos3ϑ þ 3 r 3r  0 1 ðκ 1 b4n  b4n Þr 4n þ 1  b2ð2n þ 1Þ r 4n þ 3 þ ∑ 4n þ 1 n¼1 # 0 b  4n κ 1 b  2ð2n þ 1Þ  b  2ð2n þ 1Þ  4n  1  :cosð4n þ 1Þϑ r ð4n þ1Þr 4n þ 1   κ1 b  4n cosð4n  1Þϑ  b4n r 4n þ 1 þ ð4n  1Þr 4n  1 " 0 ðκ1 b2ð2n þ 1Þ  b2ð2n þ 1Þ Þr 4n þ 3 b  2ð2n þ 1Þ þ  4n þ 1 4n þ 3 r # )) 0 b  4ðn þ 1Þ þ cosð4n þ 3Þϑ ð2:23Þ ð4n þ 3Þr 4n þ 3 v¼

  0

Pc κ1 b  2 þ b  2 0  sinϑ ðκ1  1Þb0 þ b0 r þ b2 r 3 þ 2πμ1 r   0 0 ðκ 1 b2 þ b2 Þr 3 b  2 b  4 þ  þ 3 sin3ϑ 3 r 3r  0 1 ðκ 1 b4n þ b4n Þr 4n þ 1 b  4n þ b2ð2n þ 1Þ r 4n þ 3  4n  1 þ ∑ 4n þ 1 r n¼1 # 0 κ 1 b  2ð2n þ 1Þ þ b  ð2n þ 1Þ :sinð4n þ 1Þϑ þ ð4n þ 1Þr 4n þ 1   κ1 b  4n sinð4n 1Þϑ þ b4n r 4n þ 1 þ ð4n  1Þr 4n  1 # " 0 0 ðκ1 b2ð2n þ 1Þ þ b2ð2n þ 1Þ Þr 4n þ 3 b  2ð2n þ 1Þ b  4ðn þ 1Þ þ  4n þ 1 þ 4n þ 3 r ð4n þ 3Þr 4n þ 3  sinð4n þ 3Þϑ ð2:24Þ

3. The distribution of the stress field components along some characteristic loci The formulae obtained in the previous section for the stresses are now applied for the exploration of the stress-field characteristics in a circular ring under a parabolic pressure distribution. The ring is assumed to be made of Polymethylmethacrylate (commercially known as PMMA or Plexiglas). The modulus of elasticity and Poisson's ratio of the specific material's batch (used also in the experimental protocol of Section 7) was determined from a preliminary series of standardized tensile tests [25] and were found equal to E¼ 3.19 GPa and ν¼0.36, respectively. The choice of PMMA was based on the fact that it is a relatively brittle material and its constitutive behaviour approaches that of a linear elastic material (especially for loads not very close to the fracture load), a critical assumption of the theoretical analysis described in Section 2. From the same series of experiments the yield stress was found equal to about σy ¼28 MPa while the tensile strength was determined equal to about σt ¼ 37.8 MPa. The outer radius of the rings was assumed constant equal to R2 ¼50 mm while the ratio ρ¼R1/R2 of the inner over the outer radii varied from 0.05 to 0.3 with a step of 0.05 (the upper limit of ρ¼0.3 was set since the experimental evidence of Section 7 indicates that from this value on the assumption that the length of the contact arc in the ring is the same with that for the compact disc is no more accurate). In this case Eq. (2.5) can be used for the value of Pc.

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277

Convergence analysis indicated that the accuracy obtained using 50 terms (beyond their finite parts) of the respective series expansions for the stresses is satisfactory for all practical purposes. In particular, along L1 and for small ρ-values, the stresses are satisfactory described by keeping only one additional term (and for too small ρ-values even none of them) in the infinite part of the respective formulae. For comparison all plots are realized for an overall external load equal to Pframe ¼20 kN. For the same reason the values of the stresses in the plots following are normalized over the maximum value of the externally applied pressure Pc as it is defined in Eq. (2.5). In addition, plane strain conditions are considered. 3.1. The radial distribution of the stress field along some characteristic loci

0.5

ρ= Normalized radial stress σrr

According to previous studies [15,16,19,31] the transverse tensile stress (which is of utmost importance for the practical use of the ring test) is maximized along the radius with ϑ ¼901, or in other words along the radius coinciding with the line of symmetry of the externally applied load. Therefore, as a first step, attention is here focused to the stress distribution along the specific locus. The variation of the transverse stress components σϑϑ (normalized as previously described) along this locus for various values of the ρ ¼R1/R2 ratio is plotted in Fig. 2a. The path along which the plot is realized is denoted as AB in the sketch embedded in the figure. It is clear from Fig. 2a that the variation of the transverse stress is similar to that obtained by Jaeger and Hoskins [16] (attention should be paid to their sign notation since they considered the compressive stresses positive and also to the range of ρ-values which in their paper varies from 0.1 to 0.6) both from qualitative and quantitative point of view. As it is expected the transverse stress along the AB segment attains its maximum tensile value σϑϑ,Α at r ¼R1 and then it decreases smoothly becoming compressive and attaining its maximum compressive value σϑϑ,Β at r ¼R2. The maximum value of the transverse stress σϑϑ,Α along AB (which is the quantity somehow representing the tensile strength of the ring's material) increases non-linearly with increasing ρ-values while for ρ-0 it tends to 0.5Pc. Similarly, σϑϑ,Β (the transverse stress at point B, with ϑ ¼901 and r ¼R2) decreases again non-linearly and for ρ-0 it tends to –1.0Pc. The variation of both σϑϑ,Α and σϑϑ,Β against ρ is shown in Fig. 2b. The magnitude of the σϑϑ,Α/σϑϑ,Β ratio ranges from about 0.71 (for ρ¼0.3) to 0.52 (for ρ¼0.05). The distribution of the normalized radial stress σrr along the ϑ ¼901 radius is plotted in Fig. 2c, again for various ρ-ratios. It is seen that σrr varies smoothly from zero (at r ¼R1) to –1 (at r ¼ R2). In the immediate vicinity of the point A σrr becomes tensile; however, its sign changes quickly and becomes compressive almost all along the AB locus. This sign change of the radial stress along AB was experimentally verified by Tokovyy et al. [3] with the aid of photoelasticity. For small ρ-values (ρo0.1) the absolute value of the radial stress as r-R2 is almost equal to the respective value of the transverse stress. In other words, the state of stress in the immediate vicinity of point B in a ring of small inner radius tends to that of equibiaxial compression as it is exactly the case of the compact disc (i.e. the Brazilian-disc test). The second locus of increased importance is the radius normal to the externally applied load, i.e. the locus with either ϑ ¼01 or ϑ ¼1801. The distribution of the transverse- and radial- stress components along this locus is plotted in Fig. 3(a,b) for the same as above ρ-values. As it is perhaps expected, σϑϑ is compressive along the major portion of the ϑ ¼1801 locus (denoted as CD). Only for r-R2 it becomes tensile (Fig. 3a). From a quantitative point of view the maximum tensile stress attained along CD at point D

0.0

0.05 0.0

-0.5

0.10

0.15

0.20

0.2

ϑ=90o

0.25 0.30 0.4

0.6

0.8

1.0

B σrr A

-1.0

r/R2

Fig. 2. (a) The normalized transverse stress along the ϑ ¼ 90o locus. (b) The normalized maximum tensile and maximum compressive transverse stresses against ρ. (c) The normalized radial stress along the ϑ ¼90o locus.

(denoted as σϑϑ,D) is much lower compared to that along AB (denoted as σϑϑ,A) for all ρ-ratios. For example, for ρ¼ 0.3 it holds that σϑϑ,Α/σϑϑ,D ¼ 6.31, indicating that tensile fracture at points different than A is not to be expected in a ring test. On the contrary, the maximum compressive transverse stress along CD (denoted as σϑϑ,C) is well comparable to the respective maximum compressive stress along AB (denoted as σϑϑ,B). Concerning the radial stress distribution along the ϑ ¼1801 locus, it is seen from Fig. 3b that it is of compressive nature all

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Fig. 3. (a) The normalized transverse stress along the ϑ ¼ 1801 locus. (b) The normalized radial stress along the ϑ ¼1801 locus.

along the CD segment for all ρ-values exceeding 0.1. However, its absolute value is very small compared either to the transverse stress along the CD segment or to both the transverse and radial stresses along the AB segment. It is to be noted that as ρ decreases below 0.1 the specific stress component becomes tensile (i.e. as in the Brazilian-disc test), however, its magnitude is almost negligible. 3.2. The polar distribution of the stress field along the inner and outer circles of the ring The polar variation of the transverse stress (again normalized over Pc) along the inner circle of the ring is plotted in Fig. 4a (obviously both the radial and shear stresses are zero along this locus, denoted as AC in the sketch embedded, due to the boundary conditions imposed). As it is expected σϑϑ decreases smoothly from its maximum tensile value equal to about 0.84Pc (obtained at ϑ ¼901, i.e. along the axis of symmetry of the pressure imposed) to zero at ϑcr ¼1261 (or ϑ¼ 541 in the first quadrant). It is emphasized that ϑcr is independent from ρ. From this point on it becomes compressive and its absolute value tends smoothly to its global maximum which is equal to about –1.0Pc at ϑ ¼1801 (or ϑ ¼01). Along the r ¼R2 circle, the respective stress distributions are plotted in Fig. 4b. Again the shear stress is zero all along the specific locus (denoted as BD in the embedded figure) as dictated by the boundary conditions imposed. The radial stress is identical to the externally applied radial pressure along the loaded semi-arc (equal, for the specific conditions and numerical values adopted in

Fig. 4. The polar variation of the normalized stress components along (a) the inner circle; (b) the outer circle.

Section 3, to about 0.067π). For the remaining portion of the specific locus σrr is zero, verifying the boundary conditions. The transverse stress along the contact arc is of compressive nature attaining its absolute maximum value equal to about 1.17Pc at ϑ¼ 901. Its absolute value decreases smoothly (almost similarly to the radial stress) along the contact arc while at the end of the contact arc its slope changes abruptly. At an angle ϑ E137.51 (or ϑE 42.51 in the first quadrant) its value is zeroed and its nature changes to tensile attaining a maximum value equal to about 0.13Pc at ϑ ¼1801 (or ϑ ¼01).

4. Some characteristics of the displacement field The polar variation of the displacement field components along the outer diameter of the ring is plotted in Fig. 5a for the same as above conditions and numerical values. Two values are considered for ρ, namely ρ¼0.001 (i.e. the hole tends to disappear and the ring approaches the solid disc) and ρ¼ 0.2 (thus Pc may be defined by Eq. (2.5)). It is seen that from a qualitative point of view the curves are similar and only quantitative differences are observed, supporting the assumption that for ρ-values lower than 0.3 the outer rim is more or less insensitive to the presence of the hole.

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ρ¼0.001 and around ϑ¼ 491 for ρ ¼0.2. The variation of ur along the loci with ϑ ¼901 and ϑ ¼01 is plotted in Fig. 5b and c, respectively, for the same as previous two ρ-values. As it is perhaps expected in all cases the graphs are monotonously increasing (in terms of absolute ur-values). An exception is observed for ρ¼0.02 and ϑ ¼ 01, where a weak maximum appears at r ¼ 0.84R2. Obviously, in case of the larger inner radius, ur is systematically higher compared to the respective values of the ring with smaller inner radius, due to increased compliance. The differences for the maximum values attained are equal to about 15 % and 33 % for ϑ¼ 901 and ϑ ¼01, respectively.

5. The role of the specimen material and pressure distribution type

°

The nature of the material influences (indirectly of course) the stress field within the ring since its deformability dictates the contact arc's length which for relatively small inner radii is provided through Eq. (2.4). In order to explore this dependence, a second material of increased stiffness with respect to PMMA is considered. As such Dionysos marble is chosen since it is also extremely brittle, its constitutive law is almost linear elastic and its modulus of elasticity is equal to about 80 GPa [32], i.e. almost 25 times higher from that of PMMA. Dionysos marble is the material exclusively used in the restoration project of the Parthenon Temple on the Acropolis of Athens. An analytic description of its mechanical and physical properties can be found in the refs. [32–34]. In Fig. 6 the radial variation of the normal stress components is plotted for both PMMA and Dionysos marble along the locus with ϑ¼ 901. For both materials the overall externally applied load is now equal to 11 kN which corresponds to the external load

Fig. 5. (a) The radial and transverse displacement components on the outer ring's periphery L2, for ρ ¼0.001 and ρ ¼ 0.2. (b,c) The radial displacement components along the ϑ¼ 901 (b) and ϑ ¼ 01 (c) loci.

The radial displacement ur is positive (i.e. the ring expands radially) for ϑ-values in the [01, 501] and [01, 471] range, for ρ¼0.001 and ρ¼0.2, respectively. In the remaining portions of ϑ-values ur becomes negative, indicating that the ring is radially shrinking. For the specific configuration described here ur ranges from about 0.22 mm for ϑ ¼01 to about –0.9 mm for ϑ ¼901, for ρ¼0.001. The respective values for ρ¼0.2 range from about 0.34 mm to about –1.1 mm. The behaviour of the tangential displacement component uϑ is different. As it is expected it is zeroed at both ϑ ¼01 and ϑ ¼901 while a global maximum is observed around ϑ ¼52o in case

Fig. 6. (a) The normalized normal stresses along the ϑ ¼ 901 locus for Dionysos marble and PMMA. (b) Magnified view of the shadowed area, i.e. for r/R2-1.

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causing fracture of a Dionysos marble compact disc, according to Hondros' formula (assuming for Dionysos marble an average tensile fracture strength equal to about 7 MPa [34]). The value of ρ-ratio chosen is equal to 0.3 and data are normalized over the maximum value of the externally applied pressure Pc defined in this case by Eq. (2.5), for the specific marble-steel pair of the specimen's and jaw's materials. It is seen from Fig. 6a that for both stress components the differences between the distributions for marble and PMMA are almost negligible for r/R2 o0.7. On the contrary, as r/R2-1 the values of both stress pairs start deviating from each other. It is mentioned characteristically that for r/R2 ¼1 the radial stress in marble is almost four times higher than the respective one in PMMA and the same is true for the transverse stress. In other words, the fact that the contact arc is much shorter in the case of the marble-steel pair strongly amplifies the local stress field in the immediate vicinity of the specimen–jaw interface. This point must be carefully considered when carrying out experiments with marble rings squeezed between stiff jaws since local cracking in the immediate vicinity of the ring–jaw interface may appear. It is here emphasized that for the (r/R2, ϑ)¼(1, 901) point of the ring the radial and transverse stress components are not equal to each other as it happens at the respective point of the compact disc [11,30]. This is clearly seen in Fig. 6b which is a magnification of the shadowed area of Fig. 6a. For the specific case with ρ¼0.3 the difference in the case of marble is equal to about 12% while for PMMA it is equal to about 22%. The difference decreases for ρ-0, or, in other words, as the ring tends to the compact disc configuration. The second factor that is usually ignored when the stress field in a ring (and also in a disc) is considered is the type of the externally applied normal pressure. This is because attention is usually focused at the point where the maximum tensile stress appears (the point (r,ϑ)¼(R1,901) for the ring or the centre of the compact disc) and it is expected that as long as the inner radius of the ring is small compared to the outer one the exact profile of the pressure distribution externally applied will not seriously influence the stress at the specific point. On the other hand, it is analytically proven [30] that, at least for the Brazilian-disc test, the local stress field in the immediate vicinity of the disc–jaw interface strongly depends on the type of the pressure applied. In order to check whether this conclusion is also valid for the ring test, the results of the present analysis (where a parabolic pressure distribution is considered) are compared with the respective ones obtained by Tokovyy et al. [3] for uniform pressure. For the comparison to be feasible a semi-contact angle equal to 51 was considered in order to match with the respective assumptions by Tokovyy et al. [3]. According to the present approach this demand dictates that an overall external load equal to 10 kN must be applied for a fictitious material with Young's modulus equal to Εf ¼10 GPa and Poisson's ratio equal to νf ¼0.37. The comparison is realized for ρ¼0.15, along the ϑ ¼901 locus and the results are plotted in Fig. 7. Both the radial and transverse stresses are normalized over the maximum stress (i.e. over the maximum transverse stress) developed according to the present solution at point A. It is clear from Fig. 7 that concerning the maximum tensile stress developed, i.e. the stress at the point (r/R2, ϑ)¼(0.15, 901) (the point A in the sketch embedded in Fig. 7), the two solutions are identical. Moreover, the results of both solutions are in excellent agreement to each other for r/R values up to about 0.75. From this value on the solutions deviate from each other and this deviation is maximized at r/R ¼1. It is mentioned characteristically that according to the solution with uniform external pressure [3] the radial stress at r/R¼ 1 is lower than the respective one predicted by the solution with parabolic pressure by about 34%. This is better seen in Fig. 7b which is a magnification

Fig. 7. (a) The normalized normal stresses along the ϑ ¼ 901 locus for uniform and parabolic pressure variation. (b) Magnified view of the shadowed area, i.e. for r/ R2-1.

of the shadowed area of Fig. 7. The respective difference for the transverse stress is even higher, equal to about 47%.

6. The critical load according to the ring- and Brazilian-disc-tests As it was mentioned in the introductory section the ring test is considered as a potential substitute of the Brazilian-disc test curing the main drawback of the latter, i.e. the possible onset of fracture at the immediate vicinity of the disc–jaw interface instead of the disc's centre. In addition, at the point of maximum tensile stress in the ring test, i.e. at the point (r,ϑ)¼ (R1, 901) the stress field is uniaxial since both σrr an σrϑ are locally zeroed. The main question however, related to whether the maximum tensile stress recorded in the ring test (i.e. the stress at the point (r, ϑ) ¼(R1,901)) is representative of the tensile strength of the ring's material, is still open. As it was shown in Section 3.1 the maximum tensile stress strongly depends on ρ. According to Hobbs [15], as ρ-0 (as the ring tends to become a solid disc) the maximum tensile stress at fracture onset is only 1/6 of the respective stress recorded at the disc's centre during the Brazilian-disc test. An attempt to explain this discrepancy was provided by Mellor and Hawkes [19] in terms of the ratio of the principal stresses developed in each case and the effect of stress concentration around the hole according to Kirsh's solution [35], however, a definite and generally accepted explanation is not as yet achieved.

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Attention is now focused to the transverse stress component the maximum value of which is supposed to be somehow related with the tensile strength of the specimens' material. Concerning the compact disc the distribution is almost identical to that obtained by Hondros [11] although in his analysis the load was assumed as uniformly distributed. In fact, as it was proved recently [30] the actual load distribution does only influence the stress field in the immediate vicinity of the disc–jaw interface. At the centre of the disc a biaxial stress field is obtained and the compressive stress component (the radial stress) is equal to three times the respective tensile one (transverse stress). Theoretically speaking (and in spite quite a few criticisms [15,19]), the value of the transverse stress and the centre of the disc is equal to the tensile strength of the material. If this statement is accepted then it is seen that the respective maximum tensile stress corresponding to the two rings is equal to 6.074 and 6.453 times the tensile strength provided by the Brazilian-disc test. The respective values according to Hobbs [31] approximation are equal to 6.0038 and 6.380. In an attempt to further enlighten the relation between the two tests the transverse stress at the centre of a solid disc under parabolic distribution of radial pressure, along a contact arc 2ωο, is considered analytically, in juxtaposition to the maximum tensile stress developed in a circular ring, under the same conditions, according to the present solution. For the solid disc (of radius R2 and thickness t) the transverse stress at its centre was recently determined as [30] σ Br ϑϑ ðr ¼ 0; ϑÞ ¼ P c

4ωo cos2ωo  2sin2ωo  cos2ϑð4ωo  sin4ωo Þ 4πsin2 ωo

ð6:1Þ

For ϑ ¼ 901 Eq. (6.1) yields the maximum tensile stress component at the centre of the disc as Br σ Br t  σ ϑϑ ðr ¼ 0; ϑ ¼ 901Þ ¼

Fig. 8. (a) The normalized normal stresses along the ϑ ¼ 901 locus for a compact disc and two rings with ρ ¼ 0.01 and ρ ¼ 0.1. (b) Magnified view of the shadowed area, i.e. for r/R2-1.

The main difficulty in this direction is that as the ratio ρ¼R1/R2 increases the maximum tensile stress recorded in the ring test increases rather rapidly and in any case non-linearly (see Fig. 2b). Therefore, the correlation with the Brazilian-disc test results is not straightforward. In Fig. 8a the distribution of the normal stresses along the loaded diameter (i.e. the locus with ϑ ¼901) is plotted for a compact disc (the related formulae for its stress field is given in the Appendix III) and two rings, one with ρ¼ 0.01 and another one with ρ¼0.10, made of PMMA. All three specimens have the same external radius R2 ¼50 mm and the same external load was applied to all of them, equal to 20 kN. The results were normalized over the maximum value Pc (Eq. (2.5)) of the pressure imposed. It is seen that the results for the disc and the rings are in general in good qualitative and quantitative agreement to each other along the major portion of r/R2. Significant differences appear only as r-R1. Indeed, in the case of the rings the radial stresses are zeroed while the transverse ones increase abruptly, resulting to a uniaxial local stress field. On the contrary, in the compact disc both stresses are almost constant in the specific r/R2 region. The ratio of the magnitude of the radial over the transverse stress is equal to about three [11]. Some minor differences appear also as r-R2, since as already mentioned the radial and transverse components for the rings are not equal to each other for r ¼R2 as it happens for the compact disc. This is shown in Fig. 8b, a magnification of the elliptic shadowed area of Fig. 8a.

P c ð1 þ cos2ωo Þð2ωo  sin2ωo Þ π 2sin2 ωo

ð6:2Þ

For the case of the ring the transverse stress component along the ϑ¼ 90o radius is obtained from Eq. (2.21) as (  Pc 0 0  2b0  b0  4b2  b2 r 2 ðr; ϑ ¼ 901Þ ¼ σ Ring ϑϑ π 0

0

1  b2 b4 0   4 þ ∑ 2ð2n þ 1Þb4n  b4n r 4n 2 r r n¼1 h i 0  4ðn þ1Þb2ð2n þ 1Þ  b2ð2n þ 1Þ r 2ð2n þ 1Þ

þ

0

þ

4nb  2ð2n þ 1Þ þ b  2ð2n þ 1Þ

r 2ð2n þ 1Þ )) 0 2ð2n  1Þb  4n b  4ðn þ 1Þ   4ðn þ 1Þ r 4n r

ð6:3Þ

Solving Eq. (6.2) for Pc/π and substituting in Eq. (6.3), it is found that σ Ring ϑϑ ðr; ϑ ¼

901Þ ¼ σ Br t FðrÞ

ð6:4Þ

where F(r) reads as

0 0 2sin2 ωo b b 0 0 2b0  b0  ð4b2  b2 Þr 2 þ 2 2  4 4 ð1 þ cos2ωo Þð2ωo  sin2ωo Þ r r ( 1

0  2ð2n þ 1Þb4n  b4n :r 4n þ ∑

FðrÞ ¼

n¼1

h i 0  4ðn þ 1Þb2ð2n þ 1Þ  b2ð2n þ 1Þ r 2ð2n þ 1Þ 0

þ

4nb  2ð2n þ 1Þ þ b  2ð2n þ 1Þ r 2ð2n þ 1Þ 0

2ð2n  1Þb  4n b  4ðn þ 1Þ   4ðn þ 1Þ r 4n r

)) ð6:5Þ

As far as attention is focused to the critical point of the ring (r ¼R1, ϑ ¼901), and small ρ-values are considered, then keeping (according to Section 3) only one term of its infinite part the

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maximum transverse stress component in the ring is provided through Eqs. (6.4) and (6.5) as Br σ Ring ϑϑ ðr ¼ R1 ; ϑ ¼ 901Þ ¼ σ t

 2sin2 ωo 0 0 2b0  b0  ð4b2  b2 ÞR21 : ð1 þ cos2ωo Þð2ωo  sin2ωo Þ 0

0

þ ð6b4  b4 ÞR41  ð8b6  b6 ÞR61 þ 0

þ

4b  6 þ b  6 R61

0



b8 R81

#

0

b2 R21

0



2b  4 þ b  4 R41

ð6:6Þ

 σ Ring Introducing the notation σ Ring t ϑϑ ðr ¼ R1 ; ϑ ¼ 901Þ and F(ρ)¼ F(r ¼R1,n ¼1) expressed in terms of ρ¼R1/R2, and setting R2 ¼50 mm, Εq. (6.6) is written as σ Ring ¼ σ Br t FðρÞ t

ð6:7Þ

The exact expression of F(ρ) due to its lengthiness is given in Appendix IV. Concerning the convergence of the solution presented and the dependence of the results obtained on the number n of the additional terms (beyond the finite parts of the formulas involved) it is mentioned that there is not a general rule dictating the appropriate number that should be used (apart perhaps from the

fact that the bigger the inner radius the smaller the number of terms required and that in cases of small to very small holes only one term or even none terms are quite enough as far as restriction is made on these holes). In general, the number of additional terms that will be used depends on the accuracy desired and also to the area of ring of interest. In any case as it is seen, for example from Fig. 9(a1, a2) (where the polar distribution of the stress components is plotted along a path with r¼ 5R2/6), a number of additional terms exceeding n ¼3 (apart from their finite parts), is enough to sufficiently describe stresses. Obviously, as one approaches the outer boundary of the ring the number of terms should be increased. Concerning the displacements it is easily verified that four (n ¼4) additional terms are enough to fully describe deformation all over the ring. Moreover, for the critical points of the ring, i.e. the points where the tensile stress is maximized the situation is even better. In this case even one (n ¼1) additional term is enough as it can be seen in Fig. 9(b1, b2) where the maximum tensile stress developed is plotted against the number of additional terms for two ρ-values, equal to 0.1 and 0.5, for the as above ring's properties. In order now for a direct comparison with the results obtained by Hobbs [31] to be feasible, the analytic expression for F(ρ) given

Fig. 9. (a) The polar distribution of the stress components for n¼ 3 (a1) and n ¼4 (a2) additional terms of the series expansion. Both plots are realized along a path with r¼ 5RO/6, for a PMMA (E¼ 3.19 GPa, ν ¼ 0.36) disc (R2 ¼ 0.05 m, R1 ¼ R2/2, t ¼0.01 m) squeezed between the ISRM's steel (E¼ 210 GPa, ν ¼ 0.3) curved (radius of curvature R3 ¼1.5R2) jaws under an overall load Pframe ¼20 kN. (b) The maximum tensile stress developed in a ring for two different ρ-values equal to 0.1 (b1) and 0.5 (b2); it is seen that in this case convergence is satisfactory even for n=1 additional term.

S.K. Kourkoulis, Ch.F. Markides / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 272–292

in the Appendix IV, is further transformed, neglecting insignificant terms (for ρr0.1) to the much simpler one 4sin2 ωo Fðρr 0:1Þ ¼ F o þ ð1 þ cos2ωo Þð2ωo  sin2ωo Þ  20sin2ωo þ sin4ωo þ 7ðωo csc2 ωo  cot ωo Þ  3  þ 8cos3 ωo sin ωo  2ωo ρ2 where F o  Fðρ ¼ 0Þ ¼

 4sin2 ωo 2sin2ωo  3cotωo þ ωo ð3csc2 ωo  2Þ ð1 þcos2ωo Þð2ωo  sin2ωo Þ

ð6:8Þ

ð6:9Þ

For a semi-contact angle equal to ωo ¼2.51 (which is in fact the value considered by Hobbs) Eqs.(6.8) and (6.9) yield Fðρr 0:1; ωo ¼ 2:51Þ  6 þ 38ρ2

283

Pframe (Fig. 10a). The reflected light rays are received on a screen parallel to the ring at a distance Zo forming an illuminated locus of points Q, the caustic [28]. A Cartesian system {O; x,y,z} is considered at the apex O on the front face of the ring in the unloaded state and a second one {O0 ; x0 ,y0 ,z0 } (obtained from {O; x, y,z} by a translation equal to Zo) is considered on the screen. If P΄ is the normal projection of P on the screen the caustic is described by   

! 
!
!

! ! ! the vector W ¼ O0 P 0 þ w , where O0 P 0  ¼  OP  and w defines the deviation of light at P. According to Snell's law this deviation corresponds to an angle 2ω where ω is the angle between the
! incident light ray and the normal np to the distorted surface at P. For small ω angles it holds   ! ! νt ! w ¼ Z o ∇ ½Δtðx; yÞ ¼ Z o ∇ ðσ 1 þ σ 2 Þ ð7:2Þ E

ð6:10Þ

which is essentially the same with the respective formula for KH provided by Hobbs [31], in spite of the fact that he considered a uniform pressure rather than a parabolic one (This should be expected since as it has been shown previously the loading type does not influence the ring's critical point for very small ρ-values). Concluding it can be said that as long as the Brazilian-disc test results represent the tensile strength then in order for the same quantity to be obtained by a ring test one should divide the value of the maximum tensile stress by the parameter F(ρ) as it is indicated by Eq. (6.7), using the appropriate expression for F(ρ) (according to the accuracy desired).

7. The length of the contact arc by the method of Reflected Caustics 7.1. Theoretical formulation for the ring–jaw contact problem As a final step the assumption concerning the identity of the contact length of a ring with small inner radius and a compact disc is here assessed experimentally by the experimental method of Caustics. The method, in the form of Transmitted Caustics, was introduced by Manogg [36,37] in 1964. A few years later Theocaris [28,29,38] introduced the method of Reflected Caustics. The latter is a unique tool for non-transparent materials and is based on simple principles of Geometric Optics: if a light beam impinges on a specimen at the vicinity of an intense stress field the reflected rays (received on a reference plane parallel to the specimen) will concentrate along a strongly illuminated curve, the caustic, due to the strong thickness variations of the specimen. The shape and size of the caustic permit quantitative investigation of critical features of the stress field and the geometry of the deformed area. It is exactly this property that will be used here for the determination of the contact length developed during the compression of discs or rings between the jaws of the ISRM device. According to the general approach [28], the application of the method necessitates first the determination of the complex potential characterizing the elastic equilibrium. Considering the ring–jaw system, the complex potential for the equilibrium of a ring with a relatively small ρ-value may be given in terms of the respective solid disc, as [25] ffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΦðzÞ ¼ ð ℓ2  z2 þ izÞ ð7:1Þ 6R2 Κ It is here emphasized that Φ(z) of Eq. (7.1) refers to the ring–jaw elastic system and should not be confused with the formula of Eq. (2.16) which is valid for the isolated ring. Consider now a parallel light beam impinging normally on a ring with a small hole in equilibrium under the action of a force

Fig. 10. (a) The basic principles of the Reflected Caustics method. (b) The initial curve and the formation of the caustic. (c) The definition of symbols for the determination of the contact length using the caustic curve.

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In Eq. (7.2) Δt(x,y) is the thickness change due to Poisson's effects and σ 1 ; σ 2 are the principal stresses. Assume that the undeformed ring's front face lies in the z ¼ x þ iy ¼ re  iϑ ; ϑ A ½0; π complex plane (Fig. 10b). The origin of the Cartesian system is again (as in Fig. 10a) the apex O (the complex variable z should not be confused with the z-coordinate axis). Points P providing the caustic curve correspond to points z ¼ r o e  iϑ (the subscript at ro distinguishes the specific points from any other arbitrary point z on the ring). In Fig. 10b the translation of P (or z ¼ r o e  iϑ ) to Pp due to the Δt(x,y)/2 change of thickness is, also, shown. According to the classic terminology [28], the locus of points P is called the initial curve (denoted in Fig. 10b by A). Combining Eqs. (2.7), (7.1) and (7.2) and introducing the magnification factor of the set-up λm ¼(Zo 7Zi)/Zi (þ/– stand for divergent/convergent light beams, respectively and Zi is the distance of the focus of the respective light bundle from the ring's front face) it is obtained that z}|{cf ν W ¼ λm z þ4Z o t Φ0 ðzÞ ¼ λm z þ CΦ0 ðzÞ ð7:3Þ E |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}C Separating real ℜ from imaginary parts ℑ the parametric form of the caustic becomes     W x0 ¼ λm x þCℜ Φ0 ðzÞ ; W y0 ¼ λm yþ Cℑ Φ0 ðzÞ ð7:4Þ Zeroing the Jacobian determinant of the transformation of Eq. (7.4) leads to   Φ″ðzÞ ¼1 ð7:5Þ C  λm  which represents the initial curve, i.e. the locus of points which if properly illuminated provide the caustic curve (the double prime denotes the second derivative). Substituting Φ(z) from Eq. (7.1) in Eq. (7.5) and introducing the complex variables z1 ¼ z ℓ ¼ r 1 e  iϑ1 ; z2 ¼ z þ ℓ ¼ r 2 e  iϑ2 (ϑ1 ; ϑ2 A ½0; π ) defined in Fig. 10c, one obtains 0 12=3 B C C B C r 1 r 2 ¼ B2 C @ 12jλm jKR2 A |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

ℓ4=3 ¼ ð2jC n jÞ2=3 ℓ4=3

ð7:6Þ

Cn

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Moreover, noticing that r 1;2 ¼ r 2o þℓ2 8 2r o ℓcosϑ (Fig. 10c), Eq. (7.6) yields ð7:7Þ r 4o  2r 2o ℓ2 cos2ϑ þ ℓ4  ð2jC n jℓ2 Þ4=3 ¼ 0  n whence for 2C  Z ℓ the radius of the initial curve is obtained as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7:8Þ r o ¼ ℓ cos2ϑ þ cos2 2ϑ  1 þ ð2jC n j=ℓÞ4=3 In addition, Eq. (7.4) provides the classic parametric equations of the caustic as 2 3  !2=3

2C n  ϑ1 þϑ2 5 4 W x0 ¼ λm r o cosϑ  r o sin ϑ  ℓ 2 2 3  n !2=3

2C  ϑ1 þ ϑ2 5 ð7:9Þ W y0 ¼ λm 4  r o sinϑ  2C n þ r o cos ϑ  ℓ 2 The initial curve and the respective caustic according to Eqs. (7.8) and (7.9) respectively, are plotted in Fig. 11a for plane stress conditions and a PMMA disc with the characteristics described in Section 3. An external load Pframe ¼20 kN was assumed, creating at the disc's centre a stress equal to about 12.7 MPa, well below the respective yield stress. λm was set equal to 2. It is seen from Fig. 11a that the radius of the initial curve is more than two times the contact semi-length while for the specific load the caustic is of almost cyclic shape.

Fig. 11. (a) The initial curve and the respective caustic for a PMMA ring under low load levels. The size of the contact length is also shown. (b) The optic experimental arrangement.

For ϑ ¼ϑ1 ¼ϑ2 ¼0,π the coordinates of the end-points, (α,β) and (a,b) of the initial curve and caustic, respectively (Figs. 10b and 11a), are given as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) u  !2=3 u xα 2C n  t ð7:10Þ ; yα ¼ yβ ¼ 0 ¼ 7 r o ðαÞ ¼ 7 ℓ 1 þ xβ ℓ x0a ¼ W x0 ðαÞ ¼  x0b ¼ W x0 ðβÞ ¼ λm r o ðαÞ 2 y0a

¼ W ðαÞ ¼ y0

y0b

¼ W ðβÞ ¼ λm 4r o ðαÞ y0

3  !2=3 2C n  n5 2C ℓ

ð7:11Þ

ð712Þ

Combination of Eqs. (7.10)–(7.12) yields the contact length ℓ, in terms of the distance 2ε ¼ ðabÞ ¼ 2W x0 ðαÞ or of the elevation η ¼ W y0 ðαÞ ¼ W y0 ðβÞ of points α and β, respectively, as  2=3 ½2W x0 ðαÞ4 ℓ2 þ 2C n  ℓ4=3  ¼0 4λ2m 2 ℓ ¼ 2C

n4

W y0 ðαÞ   þ1 2C n λm

!2

ð7:13Þ

33=2  15

ð7:14Þ

In other words, employing the Reflected Caustics method the contact length is directly determined in terms of experimentally measurable quantities. Eq. (7.14) although simpler than Eq. (7.13) is

S.K. Kourkoulis, Ch.F. Markides / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 272–292

rarely used since it is difficult to experimentally define the y0 ¼0 line. 7.2. Experimental arrangement Taking advantage of the above conclusions series of tests were carried out employing the experimental arrangement schematically shown in Fig. 11b. It consists of a He-Ne laser and two collimating lenses: lens 1 which transforms the diverging beam emitted from the laser to a collimated one and lens 2 which transforms the collimated beam into a converging one with its focal point in front of the specimen (in order to control the magnification factor λm of the setup). The beam impinges on the loaded specimen and the reflected rays are driven with the aid of a semi-reflector, placed at an angle equal to 451 with respect to the specimen's plane, towards a screen forming the caustic curve. The semi-reflector improves the quality of the experimental results [39] since optimum placement of the screen and normal incidence of the rays on the ring is achieved (recall however that using the semi-reflector the distance Zo becomes the sum of the specimen – semi-reflector and the semi-reflector – screen distances). The standardized ISRM apparatus for the Brazilian-disc test was mounted to a 250 kN servo-hydraulic INSTRON loading frame. A semi-spherical head was interposed between the traverse of the frame and the upper jaw to ensure load normality. All tests were carried out under displacement-control mode at a rate of 0.02 mm/min. The resulting force was measured by a 10 kN load cell calibrated with a verified Wykeham Farrance compression ring. The displacement rate was calibrated using a verified High Mag micrometric calibrator. The specimens were made of PMMA. Both intact (solid) discs and rings were tested. The thickness of the specimens was equal to t¼ 10 mm and the radius of the compact discs and the outer radius of the rings were equal to R2 ¼ 50 mm. The values of the ρ-ratio for the rings were equal to ρ¼ 0.01, ρ¼ 0.02 and also from ρ¼ 0.05 to ρ¼0.40 with a step of 0.05. Five specimens were tested for each configuration. In the state considered as “undeformed” the specimen was placed between the jaws and therefore it was loaded by the own weight of the upper jaw which is equal to about 60 N. The caustic curve formed on the screen was photographed at predefined load intervals. Its critical quantities were then measured from these photos using of a standard optical arrangement and suitable commercial software. 7.3. Experimental results Every effort was paid for the specimens to be identical and the loading conditions to be the same for all experiments. Typical fractured specimens can be seen in Fig. 12a. It is very interesting to note that for ρ-values lower than 0.1 the fracture did not propagate through the whole length of the loaded diameter: a small ligament in the immediate vicinity of the contact arc, equal to about 2–3 mm, was unbroken (see the specimens of the upper row of Fig. 12a). As it can be seen from the lower row of Fig. 12a successive crack arrests are visible as the crack approaches the jaw. This could be explained since in this region the stress field tends to that of equal bi-axial compression which is not in favour of crack propagation. The fracture surface for ρr0.1 was almost a perfect mathematical plane while on the contrary for increasing ρ-values it becomes wavy (see the two specimens of the lower row of Fig. 12a). A series of caustics for a specimen with ρ ¼0.02 is shown in Fig. 12b, for various load levels. The first photo corresponds to the unloaded specimen (own weight of the jaw) while the remaining ones to loads equal to 5%, 20% and 40% of the respective fracture load. Higher load levels were not considered to avoid deviations

285

from linear elasticity. Caustics are formed in three specimen's regions characterized by strong strain gradients: around the hole and in the vicinity of the two contact arcs. In each region two caustics are formed, one from reflection on the rear and one from reflection on the front face of the specimen. Attention is focused to the caustic from reflection on the front face since according to Eq. (7.13) its “opening”, Wx0 , is directly related to the contact arc's length. The evolution of Wx0 with increasing load is clearly seen. In Fig. 13 the caustics from the upper contact region are shown for two load levels corresponding to 20% (Fig. 13a) and 40% (Fig. 13b) of the fracture load of the specimens with ρ¼0.35. Fig. 13(a1,b1) correspond to a specimen with ρ¼ 0.35, Fig. 13(a2,b2) to ρ¼ 0.15, Fig. 13(a3,b3) to ρ¼ 0.10, Figs. 13(a4,b4) to ρ¼ 0.01 and finally Figs.13(a5,b5) correspond to an intact specimen (i.e. ρ ¼0). It is clear from this sequence of photos that, independently from the load level, no differences for Wx0 are detectable as long as ρo 0.15. From this value on differences start appearing gradually which for ρ¼0.35 (Fig. 13(a1, b1)) range in the region from about 8% to about 13%. The differences increase with increasing ρ-values. Taking advantage of the data gathered from the whole experimental protocol it was finally concluded that the assumption of Section 2.1 concerning the identity of the contact length in the ring–jaw and the disc–jaw complexes is valid or at least well acceptable for ρ-values smaller than 0.3. In this region the differences of Wx0 for the disc and the rings does not exceed 5%.

8. Discussion and conclusion The ring test, i.e. the compression of a circular ring between the jaws of the device suggested by ISRM for the Brazilian-disc test, was revisited analytically. A crucial assumption made was that when the inner radius of the ring is very small compared to the outer one, the contact length developed at the ring–jaw interface will be considered identical to the respective one developed at the compact disc–jaw interface, thus being provided by the related contact problem rather than being arbitrarily described. Using complex potentials method the stress- and displacement-fields on the ring were obtained in the form of infinite series. The number of terms required for satisfactory accuracy varies according to the specific needs. For practical purposes however (determination of the tensile strength) as in case of very small holes, the accuracy offered by the present solution is quite satisfactory even for a single additional term beyond its finite part. The main advantage of the present study is that the externally imposed boundary conditions approach in a more realistic manner the actual conditions developed at the ring–jaw interface: the load imposed is a parabolically varying pressure (rather than a point(line) force or a uniform pressure) while the contact arc, at least for small holes, was a function of the load level (instead of being arbitrarily predefined and constant throughout the loading procedure). It could be anticipated at this point that the parabolic pressure distribution considered differs from the one provided by the contact problem even by 20%. Indeed, although the two distributions are statically equivalent they are not identical. However, it is to be taken into account that even the “actual” (elliptic) pressure distribution obtained by both Muskhelishvili [27] and Timoshenko [1] in their milestone books (similar results were obtained also by the authors of this paper [25,26]) in case of contact of two elastic bodies, was a result of several simplifying assumptions. For example, Muskhelishvili's approach, adopted here for the analytic continuation and further derivation of Φ(z) (Eq. (7.1)), is only valid when the disc and jaw occupy the lower and upper half planes (or vice-versa), respectively. In other words it is assumed that considering any point z within each one of the two semi-infinite regions its complex conjugate point should

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Reflected from the rear face

Reflected from the front face

Wx΄ Fig. 12. (a) Typical fractured ring-specimens. In the specimens of the upper row (ρ o0.10) the fracture is perfectly plane and does not run all along the loaded diameter: Small ligaments remain unbroken. (b) The fracture surface of a ring with ρ ¼ 0.10. The arrest of the propagating fracture as the crack enters the biaxial compression field close to the loaded rim is clear. (c) Typical caustics for a ring with ρ ¼ 0.02 at various load steps. The evolution of the reflected caustic's critical quantity Wx0 , characterizing the contact length is evident.

S.K. Kourkoulis, Ch.F. Markides / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 272–292

287

Fig. 13. (a) Typical caustics for rings with ρ ¼0.35 (a1,b1), ρ¼ 0.15 (a2,b2), ρ ¼ 0.10 (a3,b3), ρ ¼0.01 (a4,b4) and a compact disc (a5,b5). Row (a) corresponds to 20% and row (b) to 40% of the fracture load of the ring with ρ ¼ 0.35.

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belong to the other one and vice versa (so that on the x-axis it should hold that z  z  x, which is clearly not the case for curved boundaries even for small contact lengths). On the other hand, indications exist, both from experiments [40,41] and also from numerical analyses by the Finite Element Method (from an ongoing research project of the authors), that the actual pressure distribution is neither exactly elliptic nor parabolic. It is described by a distribution between them and it seems that it is somehow influenced also by the friction stresses developed along the contact arc. In any case, it can be safely stated that the parabolic distribution is much closer to reality compared to the uniform distribution. The results of the present analysis are in very good qualitative agreement with previous studies [3,15,19,31] in spite of the above mentioned differences. Moreover, for very small ρ-values the stressand displacement-fields in the ring are very close to the respective distributions in the compact disc [26,30]. The differences are localized in the immediate vicinity of the specimen–jaw interface and also in the vicinity of the central area of the specimens. Taking into account that the ring test is considered as a potential substitute of the Brazilian-disc test, the critical quantities provided by the two tests (i.e. the maximum tensile stress developed), were studied in juxtaposition to each other. It was concluded that even for rings with infinitesimally small holes (i.e. ρ-0) the respective critical quantity is some six times higher from the respective quantity of the Braziliandisc test. In general, the relation between the critical quantities of the two tests is a complicated function of ρ. However, for ρ-values tending to zero the simple approximate relation between these quantities proposed by Hobbs [31] approaches the one here derived in an excellent manner. Taking advantage of the present analysis it is possible to obtain a direct (yet theoretical) relationship between the parameter ρ and the load required for a specific level of tensile stress to be developed in a ring. As a practical application, the load, Pframe, required to be exerted on a ring (of outer radius 50 mm and thickness 10 mm) made of PMMA, in order for the maximum stress developed to be equal to the respective tensile strength (equal to σt ¼ 37.5 MPa), was obtained. The results indicate that while for a ring with ρ¼0.3 the force required to cause failure is equal to only about 6.024 kN for the ring with ρ-0 (in fact with ρ¼0.01) the load becomes equal to about 9.902 kN. Recalling that the respective force that should be applied on a compact disc (Brazilian-disc test) as it is calculated by Hondros [11] formula P frame ¼ πR2 tσ Br is equal to about 58.905 kN it is concluded that t frames of much lower capacity could be used for the ring tests, highlighting an additional advantage of the latter (besides the uniaxiality of the stress field at the point where the strength is calculated and the fact that fracture at the ring–jaw interface is avoided).

At this point it is mentioned that the fracture loads measured during the experimental protocol described in Section 7 are not in very good agreement with the above predictions especially for ρ0. This discrepancy could be attributed to the linearity assumption adopted in the analytic approach. Moreover, it seems that the 6:1 relation for the fracture stress between the impact disc and the ring with a negligibly small inner radius does not reflect experimental evidence accurately enough and in any case it should be explored further. Before concluding it should be emphasised that the mathematical procedure described in the present paper for the solution of the problem is not itself limited to the case the ring's inner radius is very small. The procedure is easily extended over the whole domain of ρ-values, from ρ-0 to ρ-1. Clearly, for the latter case (ρ-1) the assumption concerning the length of the contact arc (and the distribution of pressure along this arc) is not longer valid. In such a case one should consider a predefined contact length (either experimentally or numerically determined or even arbitrarily chosen) assuming of course that now Eq. (2.6) is used in the analysis following Section 2.1 instead of Eq. (2.5). In this direction, keeping the parabolic shape of the external loading, the stress variation along the AB locus (see Fig. 2a) for a ring with R2 ¼0.05 m, R1 ¼0.8R2, t¼0.01, made of PMMA (E¼ 3.19 GPa, ν¼0.36) and loaded by a Pframe ¼20 kN being applied along a prescribed loaded arc equal to 2ωo ¼501 is shown in Fig. 14. It is seen that the magnitude of the stresses at the ring's critical point are well comparable to the respective ones at the immediate vicinity of the loaded arc. It is evident that for such geometric configurations adoption of simplified pressure distributions (i.e. uniform pressure) or point loading would provide erroneous results not only in the vicinity of the contact arc but also at the ring's critical points. Obviously, the problem of rings with inner radius exceeding the 30% of their outer one must be also studied further as a contact problem. Although the limit of ρ¼ 0.3 is more or less arbitrary the experimental investigation of the problem with the method of Reflected Caustics pointed out that at least for the material here considered (PMMA) the conclusions drawn according to the present analysis are safe. It is quite possible that for rings made of stiffer materials (like marble) the range of validity of the present analysis is wider. In any case, it is clarified also at this point that the stress distribution in a ring (even in case the inner diameter tends to zero) does not much the respective distribution in a compact disc [17]. Therefore, some geometrical correction factor is required for the derivation of the tensile strength of the material and clearly this is a weak point of the ring test. Both the Brazilian-disc and ring test are structural and the relation of their outcome with the actual tensile strength obtained from a direct tension test is to be further studied. Moreover, quite a few additional problems related to the ring test are still open for study. Among them the initiation, crack propagation path and crack arrest and the application of a triaxial rock failure model for the stress path imposed by the test geometry and load conditions are perhaps the hottest ones. For the first one prior experience [42–44] indicates that a dynamic analysis taking into account the actual boundaries of the finite disc's domain and the role of the stress waves reflected on them should be carried out, while the second one should be studied by properly validated numerical models (due to their complex nature). In this direction, (validating the models) the present analysis could be proved a valuable tool.

Acknowledgements Fig. 14. The stress distribution along the AB critical locus (see Fig. 2) for a ring with R2 ¼0.05 m, R1 ¼0.8R2, t¼ 0.01, made of PMMA for an overall external force Pframe ¼ 20 kN over a predefined loaded semi-arc equal to ωo ¼ 251.

This research has been co-financed by the European Social Fund (ESF) and Greek National Funds through the “Education and Lifelong Learning” Program of the National Strategic Reference

S.K. Kourkoulis, Ch.F. Markides / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 272–292

Framework (NSRF) - Research Funding Program: THALES: Reinforcement of the interdisciplinary and/or inter-institutional research and innovation (Grant: THALES NTUA MIS380147 / 2012). The authors kindly acknowledge this support. The authors express their gratitude to Professor Dimitrios N. Pazis of the Mechanics Department, National Technical University of Athens. His deep knowledge on the founding principles of the

289

Caustics method and his unique experience on its laboratory application was a decisive factor for the successful implementation of the experimental protocol. Finally, the authors express their gratitude to the anonymous reviewers of the manuscript. Their constructive comments helped the authors in the direction of significantly improving the initial version of the manuscript.

Appendix I. The expressions obtained in case Muskhelishvili's approach is adopted for the pressure distribution along the contact arc Consider the elliptic distribution of radial pressure along the contact arc (Eq. (2.1)) 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℓ2  τ 2 3R2 Κ With the aid of the obvious simplifications ℓ  ℓ0 ¼ R2 sinωo , τ  τ0 ¼ R2 cosϑ Eq. (AI.1) becomes

PðτÞ ¼

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2 ωo  cos2 ϑ 3Κ Using Eq. (AI.2), Eq. (2.10) would become

ðAI:2Þ

PðϑÞ ¼

σ rr ðϑÞ ¼ PðϑÞ ¼

ðAI:1Þ

π=2  ωo  1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2 ωo  cos2 ϑ for 3π=2  ωo 3Κ

)

( rϑr

π=2 þ ωo 3π=2 þωo

ðAI:3Þ

on L2

Then introducing Eq. (AI.3) in Eq. (2.13) would give

σ rr  iσ rϑ ¼

8 > <

þ1

> : ∑1

Z

0;

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin ωo  cos2 ϑe  ikϑ dϑ eikϑ ; 6πΚ

on L1 ðAI:4Þ

on L2

The elliptic form of the integrals involved in Eq. (AI.4) does not permit derivation of closed-form expressions for the boundary conditions (for a certain k) fact that would be added up to the complexity of the solution.

0

Appendix II. The expressions for the constants bj and bj b0 ¼

2ωo  sin2ωo R2  ωo 2 2 2 2 4sin ωo R2  R1



ωo  sin2ωo þ ðsin2ωo cos2ωo =2Þ b2 ¼ þ sin2ωo 2sin2 ωo



    R42 R2 2  R1 2  3 R22 R21  2    3 R22  R21 þ R62  R61 R2 2  R1 2

   

R42 R22  R21 þ R62  R61 ωo  sin2ωo þðsin2ωo cos2ωo =2Þ þsin2ωo  b2 ¼ 2    2sin2 ωo 3 R22  R21 þ R62 R61 R2 2  R1 2 b4n ¼

ðsin4nωo =nÞ þ 2ððsin2ωo cos4nωo  2ncos2ωo sin4nωo Þ=ð4n2  1ÞÞ sin4nωo  2n 4sin2 ωo   h i ð1 þ 4nÞ R22  R21 R2 2ð2n  1Þ  R2 2ð4n  1Þ  R1 2ð4n  1Þ R22ð2n þ 1Þ  2 h ih i  1  16n2 R22  R21  R22ð4n þ 1Þ R21ð4n þ 1Þ R2 2ð4n  1Þ  R1 2ð4n  1Þ



ðsin4nωo =nÞ þ 2ððsin2ωo cos4nωo  2ncos2ωo sin4nωo Þ=ð4n2  1ÞÞ sin4nωo  2n 4sin2 ωo   h i 2ð2n þ 1Þ 2ð4n þ 1Þ 2ð4n þ 1Þ  2ð2n  1Þ 2 2 R2 ð1  4nÞ R2  R1 R2  R2  R1  2 h ih i  2 1  16n2 R2  R21  R22ð4n þ 1Þ R21ð4n þ 1Þ R2 2ð4n  1Þ  R1 2ð4n  1Þ



b  4n ¼

b2ð2n þ 1Þ ¼

  sin2ð2n þ 1Þωo ðð2sin2ð2n þ 1Þωo Þ=ð2n þ1ÞÞ þ ððsin2ωo cos2ð2n þ1Þωo  ð2n þ 1Þcos2ωo sin2ð2n þ 1Þωo Þ=ð2nðn þ1ÞÞÞ  2n þ 1 4sin2 ωo   h i  2 ð 4n þ 1 Þ  2 ð 4n þ 1 Þ 4 ð n þ R 2 1Þ ð4n þ3Þ R22  R21 R2 4n  R2  R1 h i 2 h ih i 1  4ð2n þ 1Þ2 R22  R21  R22ð4n þ 3Þ  R21ð4n þ 3Þ R2 2ð4n þ 1Þ  R1 2ð4n þ 1Þ

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  sin2ð2n þ 1Þωo ðð2sin2ð2n þ 1Þωo Þ=ð2n þ1ÞÞ þ ððsin2ωo cos2ð2n þ 1Þωo  ð2n þ1Þcos2ωo sin2ð2n þ 1Þωo Þ=ð2nðn þ 1ÞÞÞ  2n þ 1 4sin2 ωo   h i 4ðn þ 1Þ 2ð4n þ 3Þ 2ð4n þ 3Þ 2 2  4n  ð4n þ 1Þ R2  R1 R2  R2  R1 R2 h i 2 h ih i 1  4ð2n þ 1Þ2 R22  R21  R22ð4n þ 3Þ R21ð4n þ 3Þ R2 2ð4n þ 1Þ  R1 2ð4n þ 1Þ

b  2ð2n þ 1Þ ¼

0 b0

0 b2



ωo sin2ωo þ ððsin2ωo cos2ωo Þ=2Þ ¼ þ sin2ωo 2sin2 ωo

  2R21 R22  2R21 þ R1 4 R62  2    3 R22  R21 þ R62  R61 R2 2  R1 2





sin4ωo þ2ððsin2ωo cos4ωo 2cos2ωo sin4ωo Þ=3Þ sin4ωo ¼  2 4sin2 ωo

0

b2 ¼

0 b4

0





  2R21 R22 3R42  2R21 R22 R41  2    3 R22 R21 þ R62  R61 R2 2 R1 2

  sin2ð2n þ 1Þωo ð2sin2ð2n þ 1Þωo =2n þ 1Þ þ ðsin2ωo cos2ð2n þ1Þωo  ð2n þ 1Þcos2ωo sin2ð2n þ 1Þωo =2nðn þ 1ÞÞ  2 2n þ 1 4sin ωo n h   h i i   ð4n þ 1ÞR21 ð4n þ 3Þ R22  R21 R2 4n  R2 2ð4n þ 1Þ  R1 2ð4n þ 1Þ R42ðn þ 1Þ h   h i io h i 2  R1 2ð4n þ 1Þ ð4n þ1Þ R22  R21 R42ðn þ 1Þ þ R22ð4n þ 3Þ  R21ð4n þ 3Þ R2 4n = 1 4ð2n þ 1Þ2 R22  R21 h ih io  R22ð4n þ 3Þ  R12ð4n þ 3Þ R2 2ð4n þ 1Þ  R1 2ð4n þ 1Þ

0

b  4ðn þ 1Þ ¼

0 b2ð2n þ 1Þ

0

  4R21 3 4R21 R2 2 þ R1 8 R82  2    10 15 R22  R21 þ R10 R2 6  R1 6 2  R1

2 2 2ωo sin2ωo R1 R2  2ω o 2sin2 ωo R22  R21

ωo  sin2ωo þ ððsin2ωo cos2ωo Þ=2Þ ¼ þ sin2ωo 2sin2 ωo

b4n ¼



  sin2ð2n þ 1Þωo ð2sin2ð2n þ 1Þωo =ð2n þ1ÞÞ þ ððsin2ωo cos2ð2n þ1Þωo  ð2n þ 1Þcos2ωo sin2ð2n þ 1Þωo Þ=ð2nðn þ1ÞÞÞ  2 2n þ 1 4sin ωo n h   h i i h    ð4n þ 3ÞR21  ð4n þ 1Þ R22  R21 R42ðn þ 1Þ  R22ð4n þ 3Þ  R21ð4n þ 3Þ R2 4n þ R21ð4n þ 3Þ ð4n þ 3Þ R22  R21 R2 4n h i i h i 2 h ih i  R2 2ð4n þ 1Þ  R1 2ð4n þ 1Þ R42ðn þ 1Þ g= 1  4ð2n þ 1Þ2 R22  R21  R22ð4n þ 3Þ  R21ð4n þ 3Þ R2 2ð4n þ 1Þ  R1 2ð4n þ 1Þ

  2  ðsin4ðn þ 1Þωo Þ=ðn þ 1Þ þ 2 ðsin2ωo cos4ðn þ 1Þωo  2ðn þ 1Þcos2ωo sin4ðn þ1Þωo Þ=4ðn þ 1Þ2  1 sin4ðn þ 1Þωo  ¼4 2 2ðn þ 1Þ 4sin ωo n h h i i  2ð2n þ 1Þ  2ð4n þ 3Þ  2ð4n þ 3Þ 2ð2n þ 3Þ 2 2 2 R2   ð4n þ3ÞR1 ð4n þ 5ÞðR2  R1 ÞR2  R2  R1 h h i io nh i h i R1 2ð4n þ 3Þ ð4n þ 3ÞðR22 R21 ÞR22ð2n þ 3Þ þ R22ð4n þ 5Þ  R12ð4n þ 5Þ R2 2ð2n þ 1Þ = 1 16ðn þ 1Þ2 ðR22  R21 Þ2  R22ð4n þ 5Þ  R12ð4n þ 5Þ h io  R2 2ð4n þ 3Þ R1 2ð4n þ 3Þ   h ððsin4nωo Þ=nÞ þ 2ððsin2ωo cos4nωo  2ncos2ωo sin4nωo Þ=4n2  1Þ sin4nωo n ð1 þ 4nÞR21 ð1  4nÞðR22  R21 ÞR22ð2n þ 1Þ  2 2n 4sin ωo h i i h io  R22ð4n þ 1Þ  R12ð4n þ 1Þ R2 2ð2n  1Þ : þ R12ð4n þ 1Þ ð1 þ4nÞðR22  R21 ÞR2 2ð2n  1Þ  ½R2 24n  1  R1 2ð4n  1Þ R22ð2n þ 1Þ   2 = ð1  16n2 Þ R22  R21  ½R22ð4n þ 1Þ  R12ð4n þ 1Þ ½R2 2ð4n  1Þ  R1 2ð4n  1Þ 

b  2ð2n þ 1Þ ¼

Appendix III. The stress field on the isolated solid disc of radius R2 due to a P(ϑ) parabolically varying radial pressure along the actual loaded rim [30]. 8 > <

ðR22  r2 Þ2 2r 4

Pc σ rr ¼ 2r6  R62  r 4 R22 4πsin2 ωo > : 2r 4 R22 ϑϑ Br

+

ðR2 þ r 2 Þ2  ð2rR2 sinðωo  ϑÞÞ2 þ U sin2ϑ U ℓn 22 ðR2 þ r 2 Þ2  ð2rR2 sinðωo þ ϑÞÞ2

* þ 4ωo R22 ðR22  2r2 Þcos2ϑ þ r4



4ωo R42 cos2ϑ þ r4





r 4  R42 þ 2r2 R22 cos2ϑ þ2cos2ωo r4

2r 6 þ R62  r 4 R22 cos2ϑ þ 2cos2ωo r 4 R22



+



9  1 R2 cosωo  rsinϑ  1 R2 cosωo þ rsinϑ ! * 2π tan R2 sinωo þ rcosϑ  tan R2 sinωo  rcosϑ  = " 0 r x r R2 sinωo r2  R22 cos2ωo  r 2 cos2ϑ R cosω  rsinϑ R cosω þ rsinϑ ; sin4ϑ þ 2cos2ω sin2ϑ 7ðR22 r 2 Þ  tan  1 R22 sinωoo  rcosϑ  tan  1 R22 sinωoo þ rcosϑ o 2 2 R2 ðR2 þ r 2 Þ2 ð2rR2 sinðωo þ ϑÞÞ2 the same expression without 2π R2 sinωo o x r R2

S.K. Kourkoulis, Ch.F. Markides / International Journal of Rock Mechanics & Mining Sciences 71 (2014) 272–292

! R22 cos4ϑ þ 2cos2ω cos2ϑ þ o r2 R22 ðR22 þ r 2 Þ2  ð2rR2 sinðωo  ϑÞÞ2 !# ) R22 sin2ωo  r 2 sin2ϑ R22 sin2ωo þ r 2 sin2ϑ þ cos2ω  4ω  o o ðR22 þ r 2 Þ2  ð2rR2 sinðωo þ ϑÞÞ2 ðR22 þ r 2 Þ2  ð2rR2 sinðωo  ϑÞÞ2 þ:

σ Br rϑ ¼

R22 cos2ωo þ r 2 cos2ϑ

!



291

r2

ðAIII:1Þ

" P c ðR22  r 2 Þ r 4  R42 ðR2 þr 2 Þ2  ð2rR2 sinðωo  ϑÞÞ2 cos2ϑ ℓn 22 4πsin2 ωo 2r 4 R22 ðR2 þr 2 Þ2  ð2rR2 sinðωo þ ϑÞ2 Þ þ

4ωo R22 r 4 þ R42 sin2ϑ  sin2ϑ 4 r r 4 R22

9  1 R2 cosωo  rsinϑ  1 R2 cosωo þ rsinϑ * 2π  tan R2 sinωo þ rcosϑ  tan R2 sinωo  rcosϑ  = 0 r x r R2 sinωo R cosω  rsinϑ R cosω þ rsinϑ ;   tan  1 R2 sinω o  rcosϑ  tan  1 R2 sinω o þ rcosϑ 2

o

2

o

the same expression without 2π R2 sinωo o x r R2 ! r2 R22 sin2ωo  r 2 sin2ϑ sin4ϑ þ 2cos2ω sin2ϑ þ o R22 ðR22 þ r 2 Þ2 ð2rR2 sinðωo þ ϑÞÞ2 ! ! 2 R2 sin2ωo þ r 2 sin2ϑ r2 R22 cos4ϑ þ 2cos2ωo cos2ϑ þ 2 þ þ 2 r R22 ðR2 þ r 2 Þ2  ð2rR2 sinðωo  ϑÞÞ2 !# 2 2 2 2  R2 cos2ωo  r cos2ϑ R2 cos2ωo þ r cos2ϑ þ  ðR22 þ r 2 Þ2  ð2rR2 sinðωo þ ϑÞÞ2 ðR22 þ r 2 Þ2  ð2rR2 sinðωo ϑÞÞ2

ðAIII:2Þ

Appendix IV. The analytic expression for the factor F(ρ) ( 2sin2 ωo 2 FðρÞ ¼ ð1 þ cos2ωo Þð2ωo  sin2ωo Þ 3ðρ2  1Þ3 ð1 þ 4ρ2 þ 10ρ4 þ 4ρ6 þ ρ8 Þ

  3ð  3 10ρ2  21ρ4 þ 12ρ6 þ 15ρ8 þ 6ρ10 þ ρ12 cotωo þ2ð3 þ 13ρ2 þ 28ρ4 þ ρ6  26ρ8  8ρ10  3ρ12 Þsin2ωo

þ ρ2 ð1 þ ρ2 þ ρ4 þ ρ6 þ 4ρ8 Þsin4ωo  3ωo ð 1  3ρ2 6ρ4 þ 6ρ6 þ 3ρ8 þ ρ10 Þ 2ðρ2 1Þ 

 þ ð3 þ ρ2 Þcsc2 ωo þ 16ρ2 cos3 ωo  15  100ρ2  379ρ4  1071ρ6  2710ρ8  5706ρ10  9924ρ12  13155ρ14  10665ρ16 þ 3909ρ18 þ35613ρ20 þ 100205ρ22 þ 173029ρ24 þ 230977ρ26 þ 249815ρ28 þ203077ρ30 þ 141080ρ32 þ 89303ρ34 þ 50790ρ36 þ 25562ρ38 þ 11257ρ40 þ 4225ρ42

 þ 1333ρ44 þ 310ρ46 þ 40ρ48 Þsinωo þ 3ρ2 ð1 þ 4ρ2 þ 10ρ4 þ 4ρ6 þρ8 Þ  5  19ρ2  45ρ4  85ρ6  140ρ8 210ρ10 325ρ12  160ρ14  21ρ16 þ112ρ18 þ 257ρ20 þ 430ρ22 þ 645ρ24 þ 595ρ26   þ 350ρ28 þ 185ρ30 þ 83ρ32 þ 28ρ34 þ 5ρ36 sin3ωo ρ2 1 þ ρ2 þ ρ4 þ ρ6 þ ρ8 þ ρ10 þ ρ12 þ ρ14    ½1 þ ρ½2 þ ρ½4 þ ρðρþ 1Þð6 þ ρðρþ 1Þ þ 8ρ16 ½1 þρ½  2 þ ρ½4 þ ρðρ 1Þ 6 þ ρðρ  1Þð3 þ ρ2 Þ h     3 2 2 2 4  3þρ sin5ωo = 15ðρ  1Þ 1 þ 12ρ þ 78ρ þ 348ρ6 þ 1203ρ8 þ 3428ρ10 þ 8338ρ12 þ 17588ρ14 þ 32679ρ16 þ 53856ρ18 þ78768ρ20 þ 101024ρ22 þ 110954ρ24 þ101024ρ26 þ 78768ρ28 þ 53856ρ30 þ 32679ρ32 þ17588ρ34 þ 8338ρ36 þ 3428ρ38 þ 1203ρ40 þ 348ρ42 þ 78ρ44 þ 12ρ46 þ ρ48

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