The class of problems related to the behavior of rigid disc inclusions embedded in bonded contact with an elastic medium, has been a subject of much interest in geomechanics, civil engineering, and applied mechanics. This work is motivated by both theoretical and practical interests in the problems of turbines disks, some pipes, and many industrial applications. It may give a better understanding of the behavior of foundations under external loads. In structure-medium interaction problems arising in foundation engineering, the foundation is usually modeled using a rigid or flexible inclusion having circular, strip, rectangular, or arbitrary shape. Nowadays, composites play a very important role in geomechanics engineering.
It is common knowledge that all existing structural materials contain different inter- and intra-component defects (cracks, delaminations, etc.) . The problem of the torsion of an infinite elastic medium by a rigid inclusion (deeply embedded) was considered by Selvadurai [2, 3]. His results depend on the rotational and translational stiffnesses of the embedded rigid circular disc. The problem of the torsion of an elastic half- space was considered, at first, by Reissner and Sagoci . They studied the static interaction of a rigid disc and an elastic isotropic half-space for which they obtained the solution by means of the spheroidal coordinates. The same problem was solved by Sneddon  using a different method. He used the Hankel transforms method for reducing the problem to a pair of dual integral equations. Collins  treated the torsional problem of an elastic half-space by assuming the displacement at any point in the half- space to be due to a distribution of wave sources over the part of the free surface in contact with the disc. The solution of the forced vibration problem of elastic layer of finite thickness when the lower face is either stress free or rigidly clamped was given by Gladwell . Pak and Saphores  provided an analytical formulation for the general torsional problem of a rigid disc embedded in an isotropic half-space. Besides, Bacci and Bennati  considered the torsional of circular rigid disc adhered to the upper surface of an elastic layer fixed to an undefonnable support. More recently, Singh et al.  studied the torsional of a non-homogeneous, isotropic, half- space by rotating a circular part of its boundary surface. Cai and Zue  discussed the torsional vibration of a rigid disc bonded to a poroelastic multilayered medium.. Yu  studied the forced torsional oscillations inside the multilayered solid. The elastodynamic Green’s function of the center of rotation and a point load method were used to solve the problem. Pal and Mandal  considered the forced torsional oscillations of a transversely isotropic elastic half- space under the action of an inside rigid disc. A similar problem with the rocking rotation was solved later on by Ahmadi and Eskandari . They used an appropriate Green’s function to write the mixed boundary -value problem posed as a dual integral equation. All these problems are based on the theory of Hankel integral transformation in order to bring the mixed boundary -value problem into a system of dual integral equations. Then, the corresponding solution is sought from an integral equation of Fredholm type.
The torsional of elastic layers with a penny -shaped crack was considered by some researchers. Sih and Chen  studied the problem of a penny-shaped crack in layered composite under a uniform torsional stress. The displacement and stress fields throughout the composite were obtained by solving a standard Fredholm integral equation of the second kind. Low  investigated a problem of the effects of embedded flaws in the form of an inclusion or a crack in an elastic half- space subjected to torsional deformations. The corresponding Fredholm integral equations were solved numerically by quadrature approach. The same method was used by Dhawan  for solving the problem of a rigid disc attached to an elastic half-space with an internal crack. By using Hankel and Laplace transforms and taking numerical inversion of Laplace transform, Basu and Mandal  treated the torsional load on a penny-shaped crack in an elastic layer sandwiched between two elastic half-spaces. The purpose of this article is to study an axisymmetric torsion of an embedded circular rigid disc in bonded contact with an isotropic elastic layer sandwiched between two elastic half-spaces with two penny-shaped cracks symmetrically located at each of the two interfaces between the layer and the half-spaces. A similar method was used in arecently published work by Madani and Kebli , dealing with the case of a penny-shaped crack problem in the interior of a homogeneous elastic material at the symmetry plane, under an axisymmetric torsion by two circular rigid discs symmetrically located in the elastic medium.
2 Basic Equations of the Problem
In view of the axial symmetry of the problem, it is natural to consider (r, θ, z) the cylindrical polar co-ordinates. Here we consider an axisymmetric torsion of an embedded circular rigid disc with a radius b at the symmetry plane z = 0 in an isotropic and homogeneous layer of thickness 2h and with one material sandwiched between two half-spaces of the second material. Two penny -shaped cracks with a radius a were symmetrically located at the interface between the layer and the half- spaces z = ±h. The faces of the cracks are assumed to be stress free while the discs rotate with an equal angle ω about the z- axis passing through their center as shown in Figure .1.
Owing to symmetry about the z = 0 plane, it is sufficient to consider the problem in the upper half- space where z ≥ 0. In this case of an axisymmetric torsion problem, the displacement vector assumes the form (0, uθ, 0) in the cylindrical polar coordinate system (r, θ, z). It is convenient to identify a layer region (superscript (1)) occupying the region r ∈ (0, ∞); z ∈ (0, h−) and a layer region (superscript (2)) occupying the region r ∈ (0, ∞); z ∈ (h−, ∞). The signs h+ and h−” denote the variables in the upper surface and the lower surface of plane z = h, respectively.
The only non-zero components of stress are given by
where uθ = uθ (r, z) and Gi is the shear modulus of the material. As the torsion of the homogeneous material is static, the displacement uθ (r, z) must satisfy
By means of Hankel’s transformation, integral and its inverse given in ,
The solution of equation (2) for the regions I (0 ≤ z ≤ h) and I I (z ≥ h) is expressed as
where λ is the transform variable, J1 is the Bessel function of the first kind of order one, and Ai and Bi are unknown functions
3 Boundary and Continuity Conditions
We consider the regularity conditions at infinity, the symmetry plane condition at z = 0, and the boundary and continuity conditions at the bonded interfaces
Applying the regularity conditions at infinity given in Eq. (4a), we obtain
where A1(λ), B1(λ), and A2(λ) are arbitrary functions that need to be determined by satisfying the boundary and continuity conditions.
The above equation implies
3.1 Limiting Cases
Let’s take the limit a → 0 and the nonhomogeneity parameter γ = 1, one can obtain the closed-form solution pertinent to the torsional rotation of a rigid disc embedded in a homogeneous elastic full-space. Owing to the symmetry of the full-space case with respect to the plane of the disc, it can be deduced that τθz is zero for r > a at the disc plane. This situation corresponds exactly to the torsion of a homogeneous elastic half-space by a circular rigid disc (0 < r < a, z = 0) bonded to the surface. This is adapted to the problem concerning isotropic half-space considered by Reissner and Sagoci .
By taking the parameter γ = 1, the problem is simplified to the torsional rotation of a rigid circular inclusion attached to an elastic half-space with internal crack and the dual integral equations become
4 Reduction of the dual integral equations
The expression for p(λ) approaches to 1 for large values of λ. Equation 14 can be converted to the Abel integral equation by means of the relation
we obtain the Abel equation corresponding to Eq. (14)
Next, we invert the last equation by applying the Abel transform formula
For the left-hand side of the above equation, the integral is further simplified by using the following relationship:
we obtain the first Fredholm integral equation of the second kind
Following the similar procedure as before, Eq. (15) is reduced to the second Fredholm integral equation. Using the following formula
we obtain the following Abel-type equation:
Now, we invert the above equation by applying the Abel transform formula to get
Using the following relationships
we get the second Fredholm integral equation of the second kind
with the kernel
Next, we multiply the above two equations of the system by
we obtain the following equations:
5 Numerical Results and Discussion
As the kernels K, L, M, and N are continuous on the interval , the system of Fredholm integral equations can be solved by direct or iterative techniques . The midpoint quadrature  is used to find the numerical solution for the system given by Eqs. (25a) and (25b). By dividing the interval  into N equal subintervals, so that the midpoints are
We evaluate numerically the infinite integral K, L, M, and N using the Simpson rule. After solving the above system, the unknown coefficients can be obtained.
5.1 Stress intensity factors
The stress intensity factors at the edge of the crack and at the rim of the disc are defined, respectively, by the following equations:
On the planes z = h for r ≥ a and z = 0, the expressions of stress are given by
The second part of the integrals in Eq. (29) converge quickly as their limits r → a and r → b automatically vanishes; however, the limits of the other two integrals analyzed asymptotically as follows. Using the relation
For large values of λ, we use the following asymptotic behavior of the Bessel function of the first
Now integrating by parts, we get
We note that the infinite integrals in the preceding expressions are convergent throughout the medium except at the singular points r → a+, which occupy the crack boundary.
In this case, the integral in the above relation converges quickly and the integral is bonded as r → b−. As a result, we obtain a square root singularity at r = b and the constant ψ(b) is the measure of the strength of singularity at the vicinity of the rigid inclusion. By using the following transformations
we obtain the stress intensity factor at the edge of the crack and at the rim of the disc
Figure 2 shows the variation of the normalized stress intensity factor
Figure 4 illustrates the variation of the normalized stress intensity factor
formulation and the presented figures, conclusions may be deduced:
- Singularity at the edge of the internal crack and the internal rigid inclusion is observed (The results seem to agree with the previous works (Low , Dhawan , and Madani and Kebli ).
- The Mode III stress intensity factor
at the edge of the crack is negative, and it decreases with increasing γ.
- There is considerable interaction between the cracks and the rigid inclusion when a/b is large and the cracks are close to the rigid inclusion.
5.2 The moment required to produce rotation of the disc
The torque required to sustain the rotation of the disc can be computed using the following equation:
Using the relation
Here, as the moment is applied only to the rigid inclusion, the integrand is expressed in terms of ψ(t). Substituting the values of A1(λ) and B1(λ) from Eqs. (12) and (13) into Eq. (39) and using the asymptotic behavior of the Bessel function of the first kind
Taking into account the relation
By using the following transformations t = bu and
The moment required to effect the rotation ω, when the medium contain no crack, can be formulated as
The problem of inclusion–crack interaction has a remarkable use in the design of composite anchoring systems with flat disc inclusions, in deep foundations in a geological medium, in in situ load tests at the base of a borehole, in injection anchoring regions in soft rock masses with sealing materials, or when penetrating single-propelled anchors in steep soil masses, such as over-consolidated clays.
5.3 Displacement and stress fields
The results for the variation of the normalized displacement u(i)(ρ, ξ)/ωa and stress τ(i)(ρ, ξ)/Giωa with ρ = r/b are shown graphically in Figures 5–8 for the different values of the dimensionless axial distances ξ = z/b. For each region, five different axial distances are selected as I (ξ =0; H/5; 2H/5; 3H/5; H) and I I (ξ = H ; 6H/5; 7H/5; 8H/5; 2H), with the particular values of the height H = 1, the dimensionless
crack size c = 1 and the shear parameter γ = 1 and γ = 2. The variation of the normalized displacements is shown in Figures 5 and 6. We notice that the displacements in the two regions increase at first, reach maximum values at ρ = c, and then decrease out of the disc band with increasing ρ. The distribution of the shear stresses in the elastic medium are also discussed and shown in Figures 7 and 8. The stresses initially increase, attain its maximum values, and then with the increase in the value of ρ, the stresses go on decreasing.
In this article, an axisymmetric torsion of a rigid disc embedded in the interior of a homogeneous elastic layer sandwiched between two half-spaces containing two interface cracks is analytically addressed. By using the Hankel integral transformation and its inverse, the mixed boundary value problem is reduced to a system of dual integral equations, which are reduced to a Fredholm integral equation system of the second kind. The numerically computed results of the displacements, the stresses, and the stress intensity factors are presented graphically for some dimensionless parameters. The numerical results show that the discontinuities around the crack and the inclusion cause a large increase in the stresses that decay with distance from the disc loaded. Furthermore, the dependence of the stress intensity factor on the disc size, the distance between the crack and the disc and the nonhomogeneity parameter is observed.
Menshykov O.V. Menshykov V.A. and Guz I.A.: The contact problem for an open penny-shaped crack under normally incident tension-compression wave. Eng. Fract. Mech. 75(5) 1114-1126 (2008).
Selvadurai A.P.S.: Asymmetric displacements of a rigid disc inclusion embedded in a transversely isotropic elastic medium of infinite extent . Int. J. Sci. 18 979-686 (1980)
Selvadurai A.P.S.: Rotary oscillations of a rigid disc inclusion embedded in an isotropic elastic infinite space. lnt. J. Solids. Struct. 17 493-498 (1981)
Collins W.D.: The forced torsional oscillations of an elastic halfspace and an elastic stratum Pro. London. Math. Society 12 (1962) 226–244
Pak R.Y.S. Saphores J.D.M.: Torsion of a rigid disc in a half-space Int. J. Eng Sci. 29 (1991)
Bacci A. Bennati S.: An approximate explicit solution for the local torsion of an elastic layer Mech. Struct. Mach. 24 (1996) 21–38
Singh B.M. Danyluk H.T. Vrbik J. Rokne J. Dhaliwal R.S.: The Reissner-Sagoci Problem for a Non-homogeneous Half-space with a Surface Constraint Meccanica 38 (2003) 453–465
Guo-cai W. Long-zhu C.J.: Torsional oscillations of a rigid disc bonded to multilayered poroelastic medium Int. Zheijang. Univ. Sci. 6(3) (2005) 213–221
Pal P.C. Mandal D. Sen B.: Torsional Oscillations of a Rigid Disc Embedded in a Transversely Isotropic Elastic Half-Space Adv. Theor. Appl. Mech. 4 (2011) 177–188
Ahmadi S.F.; Eskandari M.: Rocking rotation of a rigid disk embedded in a transversely isotropic half-space Civil Eng. Infra. J. 47 (2014) 125–138
Sih G.C. Chen E.P.: Torsion of a laminar composite debonded over a penny-shaped area J. Franklin Inst. 293 (1972) 251–261
Dhawan G. K.: On the torsion of elastic half-space with penny-shaped crack Defense. Sci. J. 24 (1974) 15–22
Basu S.; Mandal S.C.: Impact of Torsional Load on a Penny-Shaped Crack in an Elastic Layer Sandwiched Between Two Elastic Half-Space Int. J. Appl. Comput. Math 2 (2016) 533–543
Debnath L. Bhatta D.: Integral transforms and their applications Chapman Hall CRC (2007) Madani F. Kebli B.: Axisymmetric Torsion of an Internally Cracked Elastic Medium by Two Embedded Rigid Discs Mechanics and Mechanical Engineering 21 (2017) 363–377
Kythe P. K.; Puri P.: Computational methods for linear integral equations. Birkhäuser Boston (2002)
Atkinson K.: The numerical solution of integral equations of the second kind. Cambridge University Press. New York (1997)