On Bonded Circular Inclusions in Plane Thermoelasticity

Mechanical/thermal stress intensification for mode I crack tip: Fracture initiation behavior of steel, aluminum and titanium alloys

The present paper gives a theoretical solution for a semi-infinite plate with a circular hole, which is filled with an elastic inclusion of another material, and subjected to uniform heat flow of the temperature gradient μ in the direction of the straight edge. The analysis is developed on the basis of the Airy's stress function in the generalized plane stress and by applying the bipolar coordinates. Here the plate is supposed to be insulated and the boundaries are free from the traction. The method of perturbation is adopted for the determination of unknown coefficients involved in the solution. Numerical calculations about the stress distributions along main parts of the semi-infinite plate and the inclusion are worked out in some detail, in order to clarify the effects of an inclusion. The maximum stresses on the common boundary are also calculated and compared with the results available.

This note deals with the thermal stresses in a plate containing two insulated circular holes of different sizes under uniform heat flow in an arbitrary direction. Bipolar coordinates are used, and the temperature distribution and the stress functions which are adjusted so as to satisfy the boundary conditions at the edge of one hole, are expanded into Fourier's series at the edge of another hole. The parametric coefficients involved in them are determined from the given boundary conditions at the edge of the hole and at infinity. Expression of the thermal stress along the edge of the hole is derived and the values of it are calculated. Some limiting cases are also discussed.

The problem of a stationary crack in functionally graded materials (FGM), subjected to a combination of thermal and mechanical loading is considered. An asymptotic analysis coupled with Westergaard's stress function approach is used to characterize the stress field around the crack tip. Thermal and mechanical properties (e.g., elastic modulus, coefficient of thermal expansion, and thermal conductivity) are assumed to vary exponentially. The crack is assumed to be inclined to the direction of the property gradation. The thermal loading is taken to be a uniform heat flow in a direction inclined to the crack. The principal of superposition from linear elasticity is used to solve the problem, whereby the problem is divided into a number of subproblems. The first four terms in the expansion of the stress field are derived to explicitly bring out the influence of nonhomogeneity on the structure of the stress field. It is observed that the presence of heat flow produced no additional singularity and hence the classical inverse square root singularity still prevails around the crack tip. Using these stress field contours of constant maximum shear stress are generated and the effect of thermal loading on the crack-tip stress field is discussed.

Interface crack in bi-piezothermoelastic media and the interaction with a point heat source

In the previous paper the stresses in a circular disk with an eccentric circular hole under radial force were reported. This paper, however, gives a theoretical solution to a disk containing an eccentric circular hole fixed at its centre and under a tangential force on the outer edge, and the inner edge being unstressed. In this problem the bipolar coordinates are used. The Airy stress function is used as a method of solution, and the auxiliary stress function is added to the basic function which gives the stress state in the disk fixed at its centre and under a tangential force on its edge. By the numerical calculation it appears that the edge or circular hole being near the centre of disk, has larger maximum stress. The limiting case, in which the semi-infinite plate has a circular hole in the neighbourhood of its straight edge, is also discussed.

A general solution for a three dissimilar sandwiched medium subjected to a point heat source and a three-phase composite cylinder subjected to a remote uniform heat flow is presented in this work. The solutions to the heat conduction and thermoelastic problems are derived based on the method of analytic continuation associated with the alternation technique. A rapidly convergent series solution for both the temperature and the stress field, which is expressed in terms of an explicit general term of the corresponding homogeneous potential, is obtained in an elegant form. The hereditary integral in conjunction with the Kelvin-Maxwell model is applied to simulate the thermoviscoelastic properties while a thermorheologically simple material is considered. Based on the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real-time solution can then be solved numerically by taking inverse Laplace transform. Two typical examples concerning interfacial stresses for plane-layered media and circularly cylindrical-layered media are discussed in detail.

Brittle materials like semiconductor wafers and ceramic substrates are usually cut by diamond tools, where the generation of microcracks cannot be avoided. Recently, thermal stress cleaving of a thin strip using a point heat source is studied to cut brittle materials without leaving any microcracks on the cleaved surfaces. However, this method often causes thermal damages to the workpieces because of the very high temperature reached around the heating point. It is desirable to cut the electronic substrates, on which precise circuits were already formed, with minimizing the temperature rise. In the present paper, based on the plane thermoelasticity, the time dependent thermal stress intensity factor of a line crack has been obtained for two heating ways : by two point heat sources and a circular zone heat source located symmetrically with respect to the crack line. It was found that a zone heating is better than a point heating in the case of a thin strip cleaving under the restriction of the temperature rise. The most successful cleaving is generally achieved when a radius of a heating region is expanded up to the crack tip. The present calculation was compared with the experimental results and good agreement was found.

The temperature distribution in a bi-material body with a line of cracks at the interface under uniform heat flow is solved. The boundary value problem is reduced to the solution of a singular integral equation of Cauchy type whose solution is given by Muskhelishvili. Temperature distributions for the case of a single crack, and for the case of two collinear cracks are given.

The problem of a two-dimensional piezoelectric material with an elliptic cavity under a uniform heat flow is discussed, based on the modified Stroh formalism for the piezothermoelastic problem. The exact electric boundary conditions at the rim of the hole are introduced in the analysis. Expressions for the elastic and electric variables induced within and outside the cavity are derived in closed form. Hoop stress around the hole and electric fields in the hole are obtained. The limit situation when the hole is reduced to a slit crack is discussed, and the intensity factors for the problem are obtained.

In connection with the problem of stress distribution in a deep beam with one hole(1)it is of interest to study also the more practical case where two circular holes are perforated for the purpose of the weight saving, etc. In this paper, the problem is considered as that of a deep beam having two circular holes, the depth of which is great enough compared with the size of holes. The solution is given for bipolar co-ordinates, for which the co-ordinate curves are co-axial circles.Since the problem is considered as that of plane stress, the stresses can be expressed in terms of a single functionX, say stress function, as given in (3) in this particular coordinates.(2) We assume this functionXto be (X1+X0) and suppose thatX1gives no stresses at great distance from two holes and the stresses in terms ofX0only satisfy the conditions at that point, though each function satisfies independently the linear partial differential equation of the same type as given in (4) or (5) in our particular co-ordinates.(2) Two solutions are obtained, (1) for the case of a deep beam containing two circular holes of equal radii on each side of the neutral axis and (2) for the case of a deep beam containing two circular holes on the neutral axis.The results of mathematical investigation may be summarised as follow;For the case (I). In this case, thex-axis is the neutral axis and they-axis is the vertical axis, on which the centers of two holes lie. These two circular holes lie at equal distance on either side of the neutral axis. The stress function which satisfies the conditions above mentioned is given by (8). The stresses in beam are given in (9), (10), (11) and (12). The constants in (12) which are determined by the conditions on the periphery of holes (13), namely, (14) are given in (15). The stress distribution round each hole is as shown in Fig. 3 by plotting the values of stresses on the normals to the periphery of hole. The stress round each hole has two maximum values at the symmetrical points A and B on the vertical axis through centers of holes. These two maximum values are some multiples of the stresses at the corresponding points in a non-perforated beam. These multiples are 2.85 atAand 3.22 atBfor P/D= 3.75, Pbeing the center distance and Dthe dia. cf two holes. For larger values of P/D, these multiples are nearly constant as shown in Fig. 1. The stress distribution on the neutral axis and on the vertical axis through centers of holes are given in Fig. 4 by plotting the values of stresses on the normals to the neutral and vertical axis respectively.For the case (2). In this case, the y-axis is the neutral axis of beam, on which two equal circular holes lie. The stress function for this case is given in (17) and the stresses in beam are given by (18), (19), (20) and (21). The constants in (21) are determined by the conditions round each circular hole (22), namely, (23) and are given in (24). The stress distribution round each circular hole in this case is not symmetry about the vertical axis through its center as shown in Fig. 5, since the existence of the another Hole on the same neutral axis has great influence upon the stress distribution in the neighbourhood of the other hole. But this effect due to the another hole will be decreased as the center distance of two holes is increased and, for larger value of P/D=10, the stress round of each hole may be approximately given by that for the case of a deep beam having one circular hole as shown in Fig. 2. The stress for the latter case is given byββ=MR/I (sin θ-sin 3θ) M=uniform bending moment.R=radius of circular hole.

The thermoelastic problem associated with a point heat source embedded in an anisotropic body containing an elliptic hole or a rigid inclusion is considered in this paper. By using the formalism of Stroh [1], the approach of analytic function continuation and the technique of conformal mapping, the expression for the temperature, displacements and stress functions is expressed in explicit matrix form. The present derived solutions are also valid for some special problems such as a crack or a rigid line inclusion if one lets the minor axis of the ellipse approach to zero. The stress intensity factors induced by a point heat source are also obtained.

The disturbance of a uniform heat flow by a line crack in an infinite anisotropic thermoelastic solid

An exact solution is obtained to the three-dimensional problem of thermoelectroelasticity for a piezoceramic body with a spheroidal cavity. The solutions of static thermoelectroelastic problems are represented in terms of harmonic functions. Far from the cavity, the body is in a uniform heat flow perpendicular to the axis of symmetry of the cavity

Thermal stresses due to a uniform heat flow disturbed by a pair of offset parallel cracks in an infinite plane with orthotropy