Infinite Elastic Plane Research Articles

Interacting stress of multiple random elliptical holes under remote and arbitrary surface stresses

Study of multiple elliptical hole problem is of great significance for optimization design and safety assessment in both geotechnical engineering and perforated materials concerned with elliptical holes. Currently, the existing kind of literature mainly focus on calculating the interacting stress of multiple elliptical holes under the remote stresses, less on that under the surface stresses. In this study, based on the derived fundamental solutions, the integral equations and superposition technique, a new analytical–numerical method is established to calculate the interacting stresses of an infinite elastic plane containing multiple elliptical holes with any size and orientation, under both remote stresses and arbitrarily distributed surface stresses applied on the elliptical holes. The new method is proved to be valid by existing methods (including the conformal mapping method, Laurent series method, complex function method, boundary element alternating method, and finite element method). Some computational examples of two elliptical holes (i.e., horizontal and oriented elliptical holes) under biaxial stresses at infinity (=) are given to analyze the influencing factors on the interacting tangential stresses at the elliptical boundaries. Research results show that the oriented elliptical hole usually has an amplifying effect on the horizontal one, while the horizontal elliptical hole has an amplifying or shielding effect on the oriented elliptical one, mainly depending on the orientation angle (α 2) of the oriented one. When the major axes of two elliptical holes are vertical to their center connecting-line, the plate with two elliptical holes has smaller stress concentration than that only with one elliptical hole, which is helpful for improving the strength of underground engineering and perforated materials. The proposed method has not only simple formulation (without singularity), high accuracy (due to the exact fundamental solution), and wider applicability (under both remote stresses and arbitrarily distributed surface stresses) than the common conformal mapping method, Laurent series method, complex function method and boundary element alternating method, but also the potential to be further developed for the isotropic/anisotropic problem of multiple arbitrarily shaped holes under the same complex stresses.

Stress analysis of an orthotropic elastic infinite plane with a hole

AbstractA general solution (stress functions) is derived for an orthotropic elastic plane problem using a mapping function to solve arbitrary configurations. Lekhnitskii formulism using complex variable is used. A problem of an infinite orthotropic elastic plane with a hole is solved. The boundary condition is represented by two complex variables. This makes it difficult to solve the orthotropic elastic plane problem. The problem is solved by overcoming this difficulty. Two methods of Cauchy integral and Riemann‐Hilbert problem are applied for the analysis. It is confirmed that the same exact stress functions are finally obtained by these methods and are represented by an irrational mapping function as a closed form. These stress functions are the general solution for the holed problem. The Mittag‐Leffler Theorem is used for the analysis. Therefore, there are two methods to solve an external boundary value problem. Stress components are represented by one complex variable. Therefore, it is not difficult to calculate the stress components. Arbitrarily shaped hole problem can be solved by changing the mapping function in the stress functions. The stress distributions for an infinite plane with a square hole subjected to uniform tension for Case I and III problems are shown. Symmetry of the stress distribution is lost for a Case III problem. A problem of a half plane with an edge crack is solved as Riemann‐Hilbert problem. It is confirmed that the solution coincide with that obtained by Cauchy integral method.

Analysis for an infinite orthotropic elastic plane with a hole subjected to uniform heat flux

A general solution is derived for the thermal stresses of an infinite orthotropic plane with a hole subjected to uniform heat flux. A dislocation method is applied to solve uniform heat flux problem, in which the orthotropic thermal conductivities and coefficients of thermal expansion are considered. The dislocation functions for stress analysis are also considered. The dislocations of displacements for the x and y directions due to the thermal stress and the elastic stress at infinity must be zero. Thus, the dislocation functions regarding elastic stress are determined. Then, the stress analysis is carried out by two methods, that is, methods due to the Riemann-Hilbert method and Cauchy integral method. These methods are completely different. It is confirmed that the stress functions achieved are identical. The final exact stress functions as a general solution are derived as a closed form. The stress components are represented by one complex variable. Therefore, the calculation of the stress components is simple. Arbitrarily shaped hole problem can be solved by exchanging the mapping function. As an example, an infinite plane with a square hole subjected to uniform heat flux is analyzed. The heat flux and stress distributions are shown.

Interaction effects on heat conduction and thermal stress in an infinite elastic plane with two circular holes

Numerous researches have focused on the physical behavior of an elastic material in the vicinity of a single hole under the assumption that the interaction effects arising from the introduction of multiple holes remain negligible if the holes are placed sufficiently far from each other. In an effort to understand hole interaction effects on heat conduction and thermal stress, we consider the case when two circular holes are embedded in an infinite elastic material and use complex variable methods together with numerical analysis to obtain solutions describing temperature and elastic fields in the vicinity of the two circular holes. The results indicate that the interaction effects on temperature distribution and stress strongly depend on the relative size of the two holes and the distance placed between them but not on the actual size of the holes.

The symmetry and loading-independency of multiple inclusions enclosing uniform stresses in an infinite elastic plane

The identification of multiple interacting inclusions with uniform internal stresses in an infinite elastic matrix subjected to a uniform remote loading is of fundamental importance in the mechanics and design of particulate composite materials. In anti-plane shear and plane deformations, certain sufficient conditions have been established in the literature which guarantee uniform internal stresses inside multiple interacting inclusions displaying various symmetries when the matrix is subjected to specific uniform remote loading. Correspondingly, sufficient conditions which allow for the design of multiple interacting inclusions independent of any specific form of (uniform) remote loading have also been established. In this paper, we demonstrate rigorously that, in all cases, these sufficient conditions are also necessary conditions and indeed allow for the identification of all possible collections of such inclusions.

An accurate and efficient computational method for seeking two equi-tangential stress hole shapes

As an infinite elastic plane containing two holes subjected to uniform remote loading, the tangential stress concentration degree around the holes is related to the hole shapes. Conformal transformation in the complex variable method is used to achieve optimal hole shapes, in which equi-tangential stress is produced at the edge of the holes. An effective method and algorithm determining equi-stress hole shapes are proposed. The key of this method is how to map the outer domain of the two holes in physical plane into an annulus in image plane. The mapping function adopted in this article has a general explicit expression, so it is different from the previous ones. Coefficients of the mapping function are design variables in the optimization algorithm of mixed penalty function (Mult-SUMT method) to find optimal holes. The approach presented has a significant computational advantage. The numerical simulations present in detail the influence of hole sizes and external loads on the shape of the optimal holes. Some of the results are verified by ANSYS numerical method.

Torsion of elastic solids with sparse voids parallel to the twist axis

With the purpose of investigating a linear elastic solid containing a dilute distribution of cylindrical and prismatic holes parallel to the torsion axis, the full-field solution for an infinite elastic plane containing a single void and subject to torsion is derived. The obtained solution is exploited to derive the analytic expressions for the stress concentration factor related to the presence of an elliptical hole, the stress intensity factor for hypocycloidal holes and star-shaped cracks and the notch stress intensity factor for star-shaped polygons. Special sets of the void location are obtained for which peculiar mechanical behaviours are displayed, such as stress annihilation at some points along the boundary of elliptical voids and stress singularity removal at the cusps or points of hypocycloidal or isotoxal star-shaped polygonal voids. By means of finite-element simulations, it is finally shown that the presented closed-form expressions for the stress intensification provide reliable predictions, even for finite-domain realizations; in particular, the infinite-plane solution remains highly accurate when the the smooth and non-smooth external boundaries are larger than twice and five times the void dimension, respectively. Under these geometrical conditions, the derived analytical expressions represent a valid ‘guide tool’ in mechanical design.

Analysis of different damage configurations at a macro-crack tip

The solution of an infinite elastic plane containing a macro-crack and a cluster of micro-cracks is presented based on Muskhelishvili’s complex potential method. A step-by-step sub-problem procedure is used to satisfy the stress boundary conditions on each crack surface. A local damage parameter expressed by stress intensity factor is presented to evaluate the damage magnitude. Three damage configurations as parallel, radiating and arbitrarily oriented micro-cracks have been designed to simulate the damage around the macro-crack tip. The results show the micro-cracks located in front of the macro-crack tip have the amplifying effect on macro-crack growth, and the amplifying effect increases with the number of micro-cracks increase. Moreover, when the macro-crack length decreases or the distance between micro-cracks and macro-crack increases, the damage parameter of macro-crack and micro-cracks will decrease. It is found that the largest damage is the damage configuration of arbitrarily oriented micro-cracks among three damage configurations.

Interaction between a Macrocrack and a Cluster of Microcracks by Muskhelishvili’s Complex Potential Method

The interaction between a macrocrack and a cluster of microcracks has been investigated based on Muskhelishvili’s complex potential method. A step-by-step subproblem procedure is used to satisfy the stress boundary conditions on each crack surface. The interactions between a cluster of microcracks and a macrocrack and the interaction among microcracks are analyzed. Three damage configurations as chained, reverse-chained, and randomly distributed microcracks have been designed to simulate the damage around the macrocrack tip. The solution of an infinite elastic plane containing a macrocrack and a cluster of microcracks is presented for the plane subjected to a uniform tensile load. The stress intensity factor (SIF) at the macrocrack tip and the microcrack tips is obtained. The results show that the inclination angle of the microcrack and the distance between the macrocrack and microcracks have a great influence on SIF. When the inclination angle is small, the SIF at microcrack tips may be larger than other inclination angles. These results are helpful to analyze the fracture or damage behaviors of materials.

Analytical solution with validity analysis for an elliptical void and a rigid inclusion under uniform or nonuniform anti-plane loading

An elliptical inclusion (covering both void and rigid inclusions) embedded in an infinite and finite elastic plane subject to uniform or nonuniform (m-th order polynomial) anti-plane loading conditions is analyzed. An analytical solution in terms of the stress field for an infinite plane is developed through the method of analytic function and conformal mapping. Closed-form complex potentials and analytical expressions for Stress Concentration Factors (SCFs) are obtained. The results show that (i.) the SCF value decreases with an increasing loading order, so that the influence of the non-uniformity of the anti-plane loads on the SCF is revealed to be beneficial from the failure point of view; (ii.) decrease in the SCF value for an infinite plane is monotonic, which does not hold true for a finite plane. The results for an infinite plane are confirmed and extended for finite planes by exploiting the well-known heat–stress analogy and the finite element method. It is worth mentioning that the comparison between the analytical solution for an infinite plane and the numerical solution for finite plane is provided, showing that the analytical solution of an infinite plane can be used as an accurate approximation to the case of a finite plane. Moreover, the proposed heat–stress analogy can be exploited to study the crack–inclusion interaction or multiply connected bodies. The computational efficiency of the proposed methodology makes it an attractive analysis tool for anti-plane problems with respect to the full scale three-dimensional analysis.

An elliptical liquid inclusion in an infinite elastic plane

Beyond recent related literature, which focused on spherical incompressible liquid inclusions, the present work studies an elliptical compressible liquid inclusion in an infinite elastic plane under static remote mechanical loading. Here, it is assumed that the change of pressure inside the liquid inclusion is linearly related to the change of inclusion volume with the bulk modulus of the liquid as the proportionality coefficient. Also, the role of the liquid surface tension on the solid–liquid interface is examined especially when the size of the liquid inclusion is comparable to or smaller than the elastocapillary length. Our results show that both the surface tension and the change of liquid pressure have a significant effect on reducing the stress concentration factor at the endpoints of an elliptical liquid inclusion. In addition, the pressure change inside the liquid inclusion is studied when a uniaxial remote stress is applied perpendicular or parallel to the major axis of the elliptical liquid inclusion. In particular, the effective plane-strain Young's modulus of a solid–liquid composite containing circular liquid inclusions predicted by the present model is linearly related to the volume fraction of the liquid inclusions, in reasonable agreement with existing experimental data.

Analytic solution for a circular nano-inhomogeneity with interface stretching and bending resistance in plane strain deformations

In the mechanical analysis of composites containing nano-inhomogeneities, it is customary to consider only the stretching resistance of the inhomogeneity-matrix interface but neglect the bending resistance of the interface. In this paper, we consider a circular nano-inhomogeneity in an infinite elastic plane subjected to an arbitrary uniform remote in-plane loading with both stretching and bending resistance incorporated on the interface. Analytic solutions are obtained for the stress field both inside and outside the inhomogeneity by using an integral-type boundary condition representing the jump in traction across the interface. We show that the presence of interface bending resistance has no influence on the average of the mean stress in the inhomogeneity, and for certain interface stretching and bending rigidities the stress field inside the inhomogeneity can remain uniform regardless of the specific uniform remote loading. Numerical examples are presented to examine the influence of the interface bending resistance on the interfacial tractions imposed on the inhomogeneity and matrix for a uniform remote uniaxial loading. It is found that the introduction of interface bending resistance perturbs the (interfacial) tractions imposed on the inhomogeneity only slightly whether the inhomogeneity is softer or harder than the matrix, while it may influence the (interfacial) tractions imposed on the matrix significantly when the inhomogeneity is much softer than the matrix. Moreover, it is shown that the peak of the interface bending resistance-induced jump in traction across the interface initially increases and then decreases as the inhomogeneity becomes harder (from an initial state in which the inhomogeneity is softer than the matrix).

Surface tension-induced interfacial stresses around a nanoscale inclusion of arbitrary shape

Investigated in this paper is the surface tension-induced stress field around a nanoscale inclusion of arbitrary shape embedded in an infinite elastic plane, with an emphasis on the combined effects of surface tension and arbitrary inclusion shape. Muskhelishvili’s complex variable method is employed to formulate the basic equations that are solved with the aid of conformal mapping and series expansion methods. Accuracy of the present solution is verified by comparing its predictions with known results of an arbitrarily shaped hole. The surface tension-induced stress field is demonstrated for four shapes of inclusions (ellipse, approximately regular triangle, square and regular pentagon with round corners). The numerical results show that as the inclusion changes from a “soft” to a “hard” one (compared to the matrix), hoop and normal stresses along the inclusion–matrix interface on the inclusion side will increase, while those on the matrix side will decrease. However, shear stress along the inclusion–matrix interface does not considerably change as the inclusion changes from a “soft” to a “hard” one. It is also found that the maximum hoop stress on the matrix side of a “soft” inclusion and the maximum shear stress always occur nearby, but not exactly at the round corners where all other stresses for all four discussed inclusion shapes attain their maximums. Besides, for the four inclusion shapes discussed, shear stress along the inclusion–matrix interface vanishes at all corners as a consequence of the symmetry.

Prediction of Fragmentation Zone Induced by Blasting in Rock

This paper presents a simple method to evaluate the two-dimensional fragmentation zone induced by gas pressure during blasting in rock. The fragmentation zone is characterized by analyzing crack propagation from the blasthole. To do this, a model of the blasthole with a number of radial cracks of equal length in an infinite elastic plane is considered. In this model, the crack propagation is simulated by using two conditions only, the crack propagation criterion and the mass conservation of the gas. As a result, the stress intensity factor of the crack decreases as crack propagates from the blasthole so that the crack length is determined. In addition, gas pressure inside blasthole also continues to decrease during crack propagation. To validate suggested analytical solution, discrete element method is used by comparing length of propagated crack due to blasting.

Influence of a micro-crack on the finite macro-crack

The numerical solution of an infinite elastic plane containing a macro-crack and an arbitrary oriented micro-crack is presented based on the distributed dislocation technique and Gauss-Chebyshev quadrature method, which is verified by finite element method. The stress intensity factor (SIF), stresses field and strain energy density near the macro-crack tip are obtained. The effect of the micro-crack on SIF of the macro-crack is analyzed and the macro-crack propagation direction is predicted based on the minimum strain energy density criterion (SED). The results show that as the micro-crack length decreases or the distance between micro-crack and macro-crack increases, the effect of the micro-crack on SIF of the macro-crack will be getting weaker. The micro-crack increases SIF of the macro-crack at some orientation, while it decreases SIF at other orientation. The micro-crack acts an attraction effect on the macro-crack propagation at some orientation, while it acts a repulsion effect at other orientation.

Anti-plane analysis of an infinite plane with multiple cracks based on strain gradient theory

The anti-plane stress analysis in an infinite elastic isotropic plane is carried out by using the distributed dislocation techniques in the framework of strain gradient theory. The technique involves the solution of a screw Volterra dislocation in the region. Employing a distributed dislocation solution indicated that the stress components reveal the familiar hypersingularity at the site of the dislocation. The solution for stress fields in integral form is obtained for a plane containing a Volterra-type screw dislocation via Fourier transform of the biharmonic equation. This method is extended to allow the solution of integral equations in the regions weakened by multiple smoothed curved cracks under anti-plane deformation. The integral equations are solved numerically for the density of dislocation on the crack surfaces. The numerical method in Chebyshev series forms is used to solve the hypersingular integral equations. The effects of interaction among cracks with different arrangements and those of the size effect on the stress intensity factors are discussed. To confirm the validity of the formulations, numerical values of the results are compared. The predicted results via the aforementioned approach are in excellent agreement with those in the literature. Some new examples are solved to demonstrate the applicability of the procedure.

Integral Equations of a Plane Problem of the Elasticity Theory for a Multiply Connected Quasiorthotropic Body

We construct a system of singular integral equations for the first fundamental problem of the plane elasticity theory for a quasiorthotropic body containing holes and cracks. For this purpose, we use the wellknown integral equations obtained for a system of curvilinear cracks (cuts) in a quasiorthotropic plane. The integral equations for a multiply connected region with holes are constructed with the help of the limit transition from open cuts in an infinite elastic plane to closed cuts. These singular integral equations of the first kind on closed contours (boundary of the body) are supplemented with the corresponding regularizing functionals guaranteeing the unique solvability of the integral equations for arbitrary right-hand sides.

Hypocycloidal Inclusions in Nonuniform Out-of-Plane Elasticity: Stress Singularity vs. Stress Reduction

Stress field solutions and Stress Intensity Factors (SIFs) are found for $n$-cusped hypocycloidal shaped voids and rigid inclusions in an infinite linear elastic plane subject to nonuniform remote antiplane loading, using complex potential and conformal mapping. It is shown that a void with hypocycloidal shape can lead to a higher SIF than that induced by a corresponding star-shaped crack; this is counter intuitive as the latter usually produces a more severe stress field in the material. Moreover, it is observed that when the order $m$ of the polynomial governing the remote loading grows, the stress fields generated by the hypocycloidal-shaped void and the star-shaped crack tend to coincide, so that they become equivalent from the point of view of a failure analysis. Finally, special geometries and loading conditions are discovered for which there is no stress singularity at the inclusion cusps and where the stress is even reduced with respect to the case of the absence of the inclusion. The concept of Stress Reduction Factor (SRF) in the presence of a sharp wedge is therefore introduced, contrasting with the well-known definition of Stress Concentration Factor (SCF) in the presence of inclusions with smooth boundary. The results presented in this paper provide criteria that will help in the design of ultra strong composite materials, where stress singularities always promote failure. Furthermore, they will facilitate finding the special conditions where resistance can be optimized in the presence of inclusions with non-smooth boundary.

Geometrical shape of in-plane inclusion characterized by polynomial internal stress field under uniform eigenstrains

The internal stress field of an inhomogeneous or homogeneous inclusion in an infinite elastic plane under uniform stress-free eigenstrains is studied. The study is restricted to the inclusion shapes defined by the polynomial mapping functions mapping the exterior of the inclusion onto the exterior of a unit circle. The inclusion shapes, giving a polynomial internal stress field, are determined for three types of inclusions, i.e., an inhomogeneous inclusion with an elastic modulus different from the surrounding matrix, an inhomogeneous inclusion with the same shear modulus but a different Poisson’s ratio from the surrounding matrix, and a homogeneous inclusion with the same elastic modulus as the surrounding matrix. Examples are presented, and several specific conclusions are achieved for the relation between the degree of the polynomial internal stress field and the degree of the mapping function defining the inclusion shape.

Periodic inclusions with uniform internal hydrostatic stress in an infinite elastic plane

We develop a new method to construct periodic inclusions with uniform internal hydrostatic stress in an elastic plane subjected to uniform remote in‐plane loading. The method is based on two particular conformal mappings which lead to a system of nonlinear equations from which the inclusion shapes are determined. We illustrate our results with several examples. In particular, we show that the ratio of the inclusion size to the period of the inclusion‐matrix system changes with the inclusion aspect ratio and, in the specific case of uniform remote shear loading, the orientation of the inclusions is also altered. Finally, we show that if the period of the inclusion‐matrix system exceeds roughly six times the inclusion size, such periodic inclusions can be treated approximately as periodic elliptical inclusions with specific aspect ratio and orientation determined by the corresponding elastic constants and uniform remote loading.