Line Inclusion Research Articles

Numerical estimation of strain intensity factors at the tip of a rigid line inclusion embedded in a finite matrix

In this paper, the strain intensity factor at the tip of a rigid line inclusion, embedded in an isotropic matrix, is studied using analytical and numerical techniques. We first revisit the elasticity solution of a rigid inclusion embedded in an infinite elastic matrix using Stroh formulation as a basic framework. This study reveals that the strain intensity factor is appropriate for quantifying the magnitude of singularities at the inclusion tip. Next, we propose a numerical methodology, based on the reciprocal theorem, to calculate the strain intensity factor of the inclusion problem embedded in a matrix of finite geometry. The input to this method is the actual elasticity solution, which is obtained using finite element analysis (FEA). The FEA model is qualitatively validated by comparing FEA results with that of photoelasticity experiments. Finally, strain intensity factor in the case of finite geometry specimen is also estimated.

English

Hydraulic fracturing technology is widely used in most oil-water wells to improve production. However, the mechanism of fracturing in a reservoir with inclusion fissures is still unclear. In this study, a theoretical model was developed to determine the stress distribution during hydraulic fracturing. The line inclusion fissure was regarded as a thin bar and the stress around the artificial fracture, which is affected by a single line inclusion, was determined using the Eshelby equivalent inclusion theory. Stress intensity factors at the tip of both the artificial fracture and the inclusion were achieved, and initiation of the fracture was predicted. Furthermore, to validate the theoretical model, re-fracturing experiments were performed on a large-scale tri-axial system. The results showed that the defects reduce the intensity of the rock, which introduces the possibility that more complex fractures emerge in the reservoir. The results also showed that the fracture direction is governed by far-field stress. The obtained conclusions are helpful to better understand the mechanism of hydraulic fracturing in reservoirs.

Antiplane analysis for an elliptical inclusion in 1D hexagonal piezoelectric quasicrystal composites

An elliptical inclusion embedded in an infinite 1D hexagonal piezoelectric quasicrystal matrix is analysed in the framework of linear piezoelasticity of quasicrystals. Using the technique of conformal mapping, the closed-form solutions of the complex potentials, all the field quantities in the matrix and the inclusion are obtained under far-field antiplane mechanical loads of the phonon and phason fields and an inplane electrical load. Several special cases, such as a homogeneous material, a soft and permeable inclusion, an impermeable inclusion, a line inclusion, a rigid inclusion and a crack are reduced by the present results. Some numerical examples are provided to shows the variations of the stresses of the phonon and phason fields and the electric field with the shape of hole/inclusion, the dielectric permittivity and the distance from the hole/inclusion.

A computational model for large deformations of composites with a 2D soft matrix and 1D anticracks

• Anticracks are rigid line inclusions in a two-dimensional deformable matrix. • For the first time, we present a meshfree method without remeshing for anticracks. • The matrix undergoes finite deformation, with anticracks moving rigidly. • The method matches available analytical solutions for stress intensity factors. • The method can handle nonlinear constitutive models and multiple anticracks. Anticracks (also known as rigid line inclusions) occur frequently in a variety of natural and engineered composites as very stiff and extremely sharp (almost zero-thickness) fibers or lamellae embedded in a softer matrix. In the linear elastic regime, similarly to cracks, anticracks generate a singularity in the stress distribution around the tip. Because of this similarity, existing analytical techniques and solutions (for simple cases) can be easily translated to anticracks. However, despite their importance in many biological and engineering composites, there has been surprisingly little development of numerical methods that would account simultaneously for the presence of multiple fibers or lamellae, arbitrary loadings and nonlinear behavior of the matrix. This paper presents the first numerical approach for rigid line inclusions, based on a meshfree scheme recently developed for multiple crack growth in elastic media. The inclusion of zero thickness is created as a crack, and a rigid motion (rotation and translation) is enforced at the anticrack faces. The equations of motion are solved according to a Total Lagrangian framework, and the matrix supposed hyperelastic. Contrarily to available analytical solutions, the degrees of freedom of the rigid motion are determined a posteriori as a consequence of the (discretized) elastic equilibrium, expressed in a variational approach. Results show that the proposed approach match well the analytical solutions and provides accurate stress intensity factors (SIFs) for relatively little computational cost. Moreover, the method can reproduce some peculiar features of the anticracks: unlike cracks, singularities also appear under compressive and parallel loads; moreover, for a certain combination of biaxial load, stress concentrations disappear. Finally, the paper presents examples drawn from biological and engineering composites: the reorientation of one or more fibers under large strains, resulting in a smart stiffening and strengthening mechanism. Reorienting towards the direction of applied load has structural importance since reinforcements can have the most effectiveness in withstanding loads. If the matrix is compliant, the reorientation is eased.

Interfacial line inclusion between two dissimilar thermo-electro-magneto-elastic solids

In this study, a rigid line inclusion on the interface between two dissimilar thermo-electro-magneto-elastic solids is considered. Based on the Muskhelishvili’s theory and extended Stroh formalism, it is reduced to the non-homogeneous Hilbert equation. According to the holomorphic continuation technology and Muskhelishvili’s theory, the exact solutions for both thermal and electro-magneto-elastic fields are derived. The explicit expressions of the multiple coupling fields are presented, and the generalized stress intensity factors are illustrated with the closed form. From the results, it can be observed the coupling of the thermo-electro-magneto-elastic effects. Moreover, the generalized stress has singularities and oscillatory properties near the inclusion tip.

Wave propagation through periodic lattice with defects

The discrete periodic lattice of masses and springs with line and point defects is considered. The matrix integral equations of a special form are solved explicitly to obtain the Floquet---Bloch dispersion spectra for propagative, guided and localised waves. Explicit form of the dispersion equations makes possible detailed analysis of the position and other characteristics of the spectra. For example in the case of the uniform lattice with one line inclusion along with one single defect we obtain the sharp explicit upper bound $$\frac{3}{4}-\frac{1}{2\pi }$$34-12? for the mass of single defect for which there exist localised waves in the spectral gaps. The developed method can be applied to various problems in optics, solid-state physics, or electronics in which lattice defects play a major role.

Mixed boundary value problem for quasi-orthotropic elastic plane

The quasi-orthotropic elastic plane in which the characteristic roots of the fundamental differential equation for the orthotropic elastic plane are doubled is investigated for a mixed boundary value problem. One of the main purposes of the present paper is to derive a general solution of a mixed boundary value problem. For the associated stress analysis, a rational mapping function is used and a closed-form stress function is obtained. The stress analysis is hence rigorous for the rational mapping function, which gives a comparative arbitrary configuration. The quasi-orthotropic elastic plane problem includes that of an isotropic elastic plane. As a demonstration of the stress analysis, a half plane with an oblique edge rigid line inclusion in which the rim of the half plane and the principal axis for the orthotropic elasticity coincides is analyzed. A uniform tension is applied over the half plane, and the rigid line inclusion is expressed as a fixed boundary. The homogenous solution of the stress function can be obtained without any integration. Stress distributions and stress intensity factors at the tip of the rigid line inclusion are investigated for Poisson’s ratio. The relationship between the quasi-orthotropic elastic plane and the isotropic elastic plane with respect to stress intensity factors is derived. Stress functions in which the rim of the half plane and the principal axis of the orthotropic elasticity do not coincide are also described.

Stress concentration near stiff inclusions: Validation of rigid inclusion model and boundary layers by means of photoelasticity

Photoelasticity is employed to investigate the stress state near stiff rectangular and rhombohedral inclusions embedded in a ‘soft’ elastic plate. Results show that the singular stress field predicted by the linear elastic solution for the rigid inclusion model can be generated in reality, with great accuracy, within a material. In particular, experiments: (i.) agree with the fact that the singularity is lower for obtuse than for acute inclusion angles; (ii.) show that the singularity is stronger in Mode II than in Mode I (differently from a notch); (iii.) validate the model of rigid quadrilateral inclusion; and (iv.) for thin inclusions, show the presence of boundary layers deeply influencing the stress field, so that the limit case of rigid line inclusion is obtained in strong dependence on the inclusion’s shape. The introduced experimental methodology opens the possibility of enhancing the design of thin reinforcements and of analyzing complex situations involving interaction between inclusions and defects.

Large-scale thermal analysis of fiber composites using a line-inclusion model by the fast boundary element method

Abstract The fast boundary element method is applied for the three-dimensional large-scale thermal analysis of fiber-reinforced composites based on a line inclusion model. In this approach, fibers are treated as inclusions with temperature assumed constant over the circular cross-section and varying along the length direction. Therefore, fibers can be meshed with line elements, making both the modeling complexity and the number of unknowns significantly reduced. An interface integral boundary element method introduced by Gao in 2009 (Engineering Analysis with Boundary Elements 2009; 33: 539–546) is extended to generate a single-domain boundary integral equation for governing this line-inclusion problem. Thus in principle, fibers with arbitrary length can be modeled. The fast multipole method is employed for the fast analysis of such problems with large-scales. The largest composite model in a personal desktop computer has the number of fibers reaching 20,000. Numerical results clearly demonstrate validity of the proposed model and its potential for large-scale analysis of fiber-reinforced composites.

Asymptotic behaviour at a tip of a rigid line inclusion in linearized elasticity

AbstractWe consider an asymptotic behaviour of a solution near a tip of a rigid line inclusion in two dimensional homogeneous isotropic linearized elasticity. By means of Goursat‐Kolosov‐Muskhelishvili stress functions we derive convergent expansions of the solution around there. Furthermore, we give expressions of the invariant integral and the Irwin's formula.

Mode III crack propagation in a bimaterial plane driven by a channel of small line defects

We consider the quasi-static propagation of a Mode III crack along the interface in a bimaterial plane containing a finite array of small line defects (microcracks and rigid line inclusions). The microdefects are arranged to form a channel around the interface that can facilitate (or prevent) the crack propagation. The two dissimilar elastic materials are assumed to be weakly bonded, so that there is no kinking of the main crack from the straight path. On the basis of asymptotic formulae obtained by the authors, the propagation is analysed as a perturbation problem and the incremental crack advance is analytically derived at each position of the crack tip along the interface relative to the position of the defects. Numerical examples are provided showing potential applications of the proposed approach in the analysis of failure of composite materials. Extension to the case of infinite number of defects is discussed.

On a partially debonded rigid line inclusion penetrating a circular inhomogeneity

We derive closed-form solutions to the mixed boundary value problem of a partially debonded rigid line inclusion penetrating a circular elastic inhomogeneity under antiplane shear deformation. The two tips of the rigid line inclusion are just mutual mirror images with respect to the inhomogeneity/matrix interface, and the upper part of the rigid line inclusion is debonded from the surrounding materials. By using conformal mapping and the method of image, closed-form solutions are derived for three loading cases: (i) the matrix is subjected to remote uniform stresses; (ii) the matrix is subjected to a line force and a screw dislocation; and (iii) the inhomogeneity is subjected to a line force and a screw dislocation. In the mapped ξ-plane, the solutions for all the three loading cases are interpreted in terms of image singularities. For the remote loading case, explicit full-field expressions of all the field variables such as displacement, stress function and stresses are obtained. Also derived is the near tip asymptotic elastic field governed by two generalized stress intensity factors. The generalized stress intensity factors for all the three loading cases are derived.

Perturbation analysis of Mode III interfacial cracks advancing in a dilute heterogeneous material

The paper addresses the problem of a Mode III interfacial crack advancing quasi-statically in a heterogeneous composite material, that is a two-phase material containing elastic inclusions, both soft and stiff, and defects, such as microcracks, rigid line inclusions and voids. It is assumed that the bonding between dissimilar elastic materials is weak so that the interface is a preferential path for the crack. The perturbation analysis is made possible by means of the fundamental solutions (symmetric and skew-symmetric weight functions) derived in Piccolroaz et al. (2009). We derive the dipole matrices of the defects in question and use the corresponding dipole fields to evaluate “effective” tractions along the crack faces and interface to describe the interaction between the main interfacial crack and the defects. For a stable propagation of the crack, the perturbation of the stress intensity factor induced by the defects is then balanced by the elongation of the crack along the interface, thus giving an explicit asymptotic formula for the calculation of the crack advance. The method is general and applicable to interfacial cracks with general distributed loading on the crack faces, taking into account possible asymmetry in the boundary conditions. The analytical results are used to analyse the shielding and amplification effects of various types of defects in different configurations. Numerical computations based on the explicit analytical formulae allows for the analysis of crack propagation and arrest.

Exact solution to the antiplane problem of doubly periodic conducting rigid line inclusions of unequal size in piezoelectric materials

AbstractThe problem of a doubly periodic array of conducting rigid line inclusions in piezoelectric materials under far‐field antiplane mechanical load and inplane electric load is investigated, where the fundamental cell contains four rigid line inclusions of unequal size. An exact solution to the problem is presented by employing the conformal mapping technique and the elliptical function theory. The closed form formulae for the stress and electrical displacement intensity factors at the rigid line inclusion tip and the effective electroelastic moduli of such composites are derived. Many new and existing solutions can be regarded as the special or degenerated cases. Numerical examples are provided to show the interesting electroelastic interaction phenomenon induced by multiple rigid line inclusions. The present work is helpful in understanding the optimization mechanism of some naturally occurring composites and in designing novel materials by direct engineering of their microstructure.

Closed form solution for a line inclusion in magnetoelectroelastic media

This paper studies the problem of a line inclusion in a medium possessing coupled piezoelectric, piezomagnetic and magnetoelectric effects. Based on the extended Stroh formalism, the general two-dimensional solutions to the magnetoelectroelastic problem are obtained, involving five analytic functions of different variables. The magnetic, electric and elastic fields around the inclusion are given. Expressions of the stresses, electric field and magnetic field in the vicinity of the inclusion tip are derived and the field intensity factors are provided. The energy release rate is written in terms of those field intensity factors. Solutions for some special cases, including the electrically and/or magnetically insulated/conductive inclusions/cracks, are also established. Explicit algebraic results are given for a special case of an anti-plane crack in a magnetoelectroelastic medium.

The stress intensity near a stiffener disclosed by photoelasticity

Can the thickness of a thin inclusion (in a matrix material) be made so small (though retaining sufficient stiffness and matrix adhesion) to generate ‘in practice’ a stress state in agreement with the analytical (square-root singular) solution for a rigid line inclusion (so-called ‘stiffener’) embedded in a linear elastic plate? Can this inhomogeneous stress state be generated for tensile loading parallel to the stiffener? We provide a direct and positive answer to these questions, by showing how to produce elastic materials containing thin inclusions and by providing photoelastic investigation of these structures. The experiments fully validate the stress state calculated for an elastic plate containing a rigid (finite-length) line inclusion, until a distance from the inclusion tip on the order of its thickness, corresponding to a stress concentration up to seven.

Growth of slip surfaces and line inclusions along shear bands in a softening material

A perturbative approach applied to prestressed elastic solids exhibiting strain softening (of the true-stress/true-strain response) provides an explanation to the common observation that shear bands formed within ductile and quasi-brittle materials tend to grow rectilinearly under Mode II and that slip surfaces tend to nucleate and grow inside them. The effects of an ‘obstacle’ (a rigid line inclusion) within a shear band are also analyzed and quantified in terms of inhibition of shear band growth.

A Modified Technique of Anastomosis Between Dissected Aortic Stump and a Dacron Tube Graft: The Suture Line Inclusion Technique

We describe a modified technique of anastomosis between dissected aortic stump and a Dacron tube graft (Sulzer Vascutek, Renfrewshire, Scotland) with the aim of minimizing suture line complications. After a stented graft was implanted into the stump or a Teflon felt strip (C.R. Bard, Inc, Tempe, AZ) was circumferentially placed on the intima of the stump, a series of nonpledgeted 2-0 horizontal mattress sutures were placed from inside to outside. Certain tension on those sutures should be created to keep the intraluminal Teflon felt strip in place. The end of the Dacron tube graft was everted outward, and its double-folded end was anastomosed to the aortic stump with incorporation of the proximal end of the stented graft or the intraluminal Teflon felt strip. The everted Dacron portion was then returned to its original position, and the previously placed horizontal mattress sutures were appropriately passed through the corresponding site, and all sutures were securely tied in place. Our initial application showed that this simple technique provides a more hemostatic anastomosis.

Exact solution for elliptical inclusion in magnetoelectroelastic materials

This paper considers the magneto-electro-mechanical coupling between an inclusion and matrix, which are both of magnetoelectroelastic materials. The general cases including the mode I, mode II and mode III are studied. Analytical solutions for an elliptical cylinder inclusion inside an infinite magnetoelectroelastic medium under combined mechanical–electrical–magnetic loadings are formulated via the Stroh formalism. Crack problem is also investigated and the stress, electric and magnetic fields in the vicinity of the crack tip are determined by a complex vector of intensity factors. Various special cases, including an impermeable inclusion, a permeable crack, a rigid and permeable inclusion, a rigid and permeable line inclusion, a permeable cavity, and an impermeable cavity are discussed.

Collinear periodic cracks and/or rigid line inclusions ofantiplane sliding mode in one-dimensional hexagonal quasicrystal

For a one-dimensional (1D) hexagonal quasicrystal (QC), there is the periodic ( x 1 , x 2 ) -plane of atomic structures with the quasiperiodic direction x 3 -axis along which there exists a phason displacement. The macroscopically collinear periodic cracks and/or rigid line inclusions are placed on the periodic ( x 1 , x 2 ) -plane for finding out the influence of phason displacement on the related physical quantities. These two models are reduced to the Riemann–Hilbert problem of periodic analytic functions to obtain the closed-form solutions for the antiplane sliding mode. It is found that the phonon and phason stress intensity factors of cracks as well as the phonon and phason stress field intensity factors of rigid line inclusions are not related to the coupling of phonon and phason fields. These mean that there is not the influence of phason displacement on both the phonon stress intensity factor (usual stress intensity factor) of cracks and the phonon stress field intensity factor of rigid line inclusions. However, the energy release rates of periodic cracks and/or rigid line inclusions are obtained and affected not only by the periodicity of cracks and/or rigid line inclusions but also by the phason displacement.