Rigid Line Inclusion Research Articles

Theoretical Review of Weight Functions for Rigid Line Inclusions: Implications for Stress Singularities and Crack Propagation

A comprehensive theoretical analysis of weight functions for rigid line inclusions in elastic materials is presented. Classical fracture mechanics approaches were extended to accurately predict stress intensity factors (SIFs) at the tips of these inclusions, which are crucial for understanding material failure. The analysis covered both static and dynamic loading conditions, including transient Mode-III problems. Weight functions for various deformation modes were derived, and the impact of rigid line inclusions on stress singularities and crack propagation was explored. These insights are valuable for the design and analysis of composite structures and materials subjected to dynamic loading.

Transient mode-III problem of the elastic matrix with a line inclusion

The method of pull-out test has been used to identify the mechanical performance of hybrid and fiber-reinforced composite materials. This paper investigates the elastic phase preceding the pull-out of a rigid line inclusion from the polymer matrix with fixed top and bottom surfaces. The mode-III problem is investigated such that the pull-out force is applied from the out-of-plane direction and it can be either transient or static. By applying the singular integral equation technique, the semi-analytical elastic field expressions are obtained. Under the static pull-out force, the stress intensity factor (SIF) near the inclusion tip shows a monotonic increase as the length and height of the matrix increase, whereas for the transient pull-out force, the SIF displays an initial increasing and followed by a decline. The maximum SIF is obtained for (1) the matrix length is 2 to 2.5 times of the inclusion length, and (2) the matrix height is 1 to 2 times of the inclusion length. Moreover, this paper provides a solution approach that incorporates the elasticity of the inclusion, showing that there is an optimal shear stiffness that minimizes the stress singularity of system. The conclusions of this study hold significance for the design and performance evaluation of fiber-reinforced composite materials.

Are the direct and indirect BEM/BIEMs equivalent?

For a long time, direct and indirect BEM/BIEMs were always unified and seemed equivalent in real implementation for ensuring the solution. Regarding the solution space and the numerical aspects in linear algebra, it is interesting to find that they are not fully equivalent. To demonstrate this finding, antiplane shear problems containing the elliptical rigid inclusion and hole as well as rigid-line inclusion and line crack were given to demonstrate the non-equivalence between the direct and indirect BEM/BIEMs. It is interesting to find that nonunique boundary densities may yield the unique field solution in both analytical derivation and numerical implementation. The mechanism was analytically studied and numerically performed by using the degenerate kernel and the singular value decomposition technique, respectively. Besides, the Fredholm alternative theorem was employed to examine the solvability condition in both the indirect boundary integral equation method and indirect boundary element method. For continuous system and discrete system, we can explain why the boundary density does not match the exact solution but the field solution is always acceptable when setting ξ0 = 0 in the beginning of the derivation of BIE. Two approaches, direct and indirect BEM/BIEMs, show the non-equivalence of solution in case of degenerate boundary and degenerate scale.

Full Field Solution for Remotely Loaded One Side Completely Debonded Short Rigid Line Inclusion Embedded in Soft Matrix: Two-Dimensional Analytical and Experimental Insights

Abstract The problem of a completely debonded short fiber (rigid line inclusion/anticrack) embedded in a 2D isotropic elastic soft matrix subjected to the remote loading condition is of fundamental interest. The current work investigates completely debonded anticrack embedded in a soft (isotropic) matrix using Kolosov Muskhelisvili's complex potential framework. Here two configurations are studied: debonded inclusion oriented (i) parallel and (ii) perpendicular to the loading direction. In particular, the potentials take the form of a non-homogeneous Riemann—Hilbert equation for the given problem. Upon solving analytical forms of potentials, the stress fields were obtained. The stress field for the fully debonded anticrack exhibited oscillatory singular behavior between r−3/4 and r−1/4 with the dependence on the oscillatory index ε and material constants. The correctness of the analytical solution was validated using numerical simulation and experiments based on the digital photoelasticity technique. The analytical results were in good agreement with the experimental and numerically obtained stress fields confirming the accuracy of it. The magnitude of singularity is quantified by defining a complex stress intensity factor at the tip of the discontinuity and compared with the experimentally estimated value. So far in the literature, no full-field analytical solution exists for the completely debonded rigid inclusion embedded in an isotropic soft matrix. The solution obtained in the present work is of fundamental importance in developing the constitutive properties of short fiber reinforced thermoplastic (SFRT) composites.

Anti-plane pull-out of a rigid line inclusion from an elastic medium

Anti-plane pull-out of a rigid line inclusion from an elastic medium

Nonlocal fields and effective properties of piezoelectric material with a rigid line inclusion perpendicular to the poling direction

Nonlocal fields and effective properties of piezoelectric material with a rigid line inclusion perpendicular to the poling direction

A theoretical model of fracture of biological composites considering complex structural arrangement of microstructures

Although it is known that biological materials have remarkable fracture strength and toughness, the underlying mechanisms have not been fully understood. A key question is what effects of the structural arrangement of the microstructure are in biological materials. To address this problem, we developed a fracture model based on the fundamental solutions of the crack and rigid line inclusions in matrix to consider the roles of the complex arrangement of the microstructure. Firstly, we obtained the analytical solutions of the SIFs of three basic problems with two inclusions. These solutions revealed several fundamental toughening mechanisms of biological composites via controlling the geometric arrangement of microstructures. Then, we analyzed the fracture problem of multiple rigid lines and cracks with one-level of hierarchy of composite structure. We found that the large aspect ratio of rigid lines produces significant stress shielding effect on the main crack, but it simultaneously creates high possibility of producing local microcracks therein. Interestingly, these microcracks can significantly reduce the SIFs of the rigid lines, while only slightly increases the SIF of the main crack. Finally, we studied the effect of the hierarchical structure on the fracture toughness of biological materials. Compared with the one-level structure, the two-level hierarchy can effectively reduce the SIF of the main crack, and at the same time it becomes less sensitive to the change of the geometric parameters of the microstructures at higher level. This study may provide guidelines for the structural design of advanced biomimetic materials.

Static and transient thermal conductions and associated thermal stresses in an elastic strip with a rigid line inclusion

The problem of a rigid inclusion in an elastic strip under steady or transient thermal loading is studied. The inclusion is assumed to be extremely thin and thermally insulated so that it does not disturb the heat conduction in the strip. The thermal strain and stress fields are obtained by using the singular integral equation technique. The inclusion tip strain and stress are found to be singular. The near tip stresses are expressed in terms of the thermal strain intensity factor. Case studies include the influence of temperature boundary condition, inclusion location and strip width on the thermal strain intensity factor. The highest and lowest values (upper and lower bounds) of the intensity of the thermal strain are identified.

Interaction of a rigid line inclusion with various discontinuities using experimental and numerical techniques

Using boundary element method (BEM) and photoelasticity, the interaction effects of a rigid line inclusion with various discontinuities embedded in a matrix are investigated. The interaction effect of discontinuities on a rigid line inclusion has not received much attention when compared to the studies on the interaction effect of discontinuities on a crack. The following discontinuities are considered: rigid circular inclusion, rigid elliptical inclusion, circular hole, elliptical hole, and crack when a tensile load is applied along the inclusion length. A circular hole approaching normal to the line inclusion strengthens the strain fields at the rigid line inclusion tip, whereas a hole along the line inclusion alleviates the strain level. However, in the case of rigid circular inclusion, the opposite trends are observed. The influence of the aspect ratio of the elliptical hole on the elastic fields of a rigid line inclusion is also brought out. The influence of a crack near the vicinity of a rigid line inclusion is also studied. The interaction effects of a circular inclusion, a circular hole, and a crack on the rigid line inclusion are further experimentally analyzed for the first time using the photoelastic technique towards validating the boundary element model. The magnitude of singularity at the inclusion tip is quantified using the strain intensity factor. Also, the magnitude of the stress intensity factor (SIF) at the crack tip is reported. Both the strain intensity factor for rigid line inclusion and the SIF for the crack tip are estimated using the domain integral method.

Multiple elastic or rigid line inclusions in a multilayered orthotropic solid

This paper studies the problem of multiple line inclusions in a non-homogeneous medium. The formulation and solutions for rigid line inclusions and homogeneous medium are special cases when the stiffness values of the inclusions are infinite and the non-homogeneity of the material vanishes. The model built in this study is capable of dealing with any material non-homogeneity (material properties vary continuously or in step-wise form), orthotropic mechanical properties, isotropic mechanical properties, and any number of parallel or collinear inclusions. More importantly, the inclusions are not limited to rigid (they can be elastic). Dependence of the inclusion tip fields on the strain intensity factor is given in closed form. Numerical results are given for the situations of mechanical load as well as thermal load applied in the medium.

A rigid line inclusion in a nonlocal elastic medium – Mode I deformation

In this paper, the problem of a rigid line inclusion in a nonlocal elastic medium is solved. Two special inclusion orientations are considered. These are (1) inclusion is parallel to the border of the medium, and (2) inclusion is perpendicular to the border of the medium. The two-dimensional stress and strain field and the effective mechanical property of the medium with inclusion are obtained. Because of the nonlocal effect, the stresses near the tips of the inclusion are significantly smaller than those based on the classic continuum mechanics framework. Essentially, under the nonlocal theory, there is no stress singularity at the tips of the inclusion. When the nonlocal effect vanishes, the results for the classic continuum theory of elasticity are recovered. In addition to the single inclusion, results for collinear inclusions are also presented.

Multiple rigid line inclusions (anti-cracks) in a multilayered orthotropic medium under anti-plane loading

Multiple rigid line inclusions (anti-cracks) in a multilayered orthotropic medium under anti-plane loading

A photoelastic investigation of partially debonded rigid line inclusion

An inverse three-quarters singularity with an oscillatory behavior exists at the tip of a partially debonded rigid line inclusion. The studies on partially debonded rigid line inclusion are hitherto confined to analytical solutions. There exists no work until now that studies partially debonded rigid line inclusions experimentally. In this paper, we experimentally investigate the mechanics of a rigid line inclusion that is bonded to one of its surface. Debonded rigid line inclusion specimens oriented along the direction of loading and perpendicular to the loading direction are considered. Digital photoelastic technique is employed to extract the whole field stress distribution near the debonded inclusion tip. Here, the emphasis is laid on the experimental aspects, starting from the specimen realization to data extraction. The nature of singularity at the tip is investigated, and a multi-parameter stress field equation for a partially debonded rigid line inclusion problem is derived. A complex stress intensity factor is defined at the inclusion tip to quantify the magnitude of singularity and is estimated experimentally using a linear over-deterministic least-squares approach involving the digital photoelasticity.

Circular inclusion problem of two‐dimensional decagonal quasicrystals with interfacial rigid lines under concentrated force

AbstractThe plane problem of two‐dimensional decagonal quasicrystals with a rigid circular arc inclusion was investigated under infinite tension and concentrated force. Based on complex representations of stresses and displacements of two‐dimensional decagonal quasicrystals, the above problem is transformed into Riemann boundary problem by using the analytic continuation principle of complex functions. The general solutions of two‐dimensional decagonal quasicrystals under the action of plane concentrated force and infinite uniform tension are derived. The closed solutions of complex potential functions in several typical cases are obtained, and the formula of singular stress field at the tip of rigid line inclusions is given. The results show that the stress field at the tip of circular arc rigid line inclusions has singularity of oscillation under plane load. Numerical examples are given to analyze the effects of inclusion radius, different inclusions, the coupling coefficient and phason field parameter on stress singularity coefficients.

A comparison of domain integral and multi-parameter methods for the strain intensity factor estimation of rigid line inclusion using digital image correlation

A comparative study is carried out between two methods to estimate the strain intensity factor of a rigid line inclusion using the displacement and strain fields from digital image correlation (DIC). The two methods are (i) a proposed hybrid domain integral method and (ii) the linear over-deterministic least-squares technique involving multi-parameter displacement field equations (multi-parameter method). The multi-parameter method uses the displacement data obtained from the DIC experiments to find the unknowns in the multi-parameter equation. The proposed hybrid methodology uses a domain integral method to calculate the strain intensity factor using the full-field displacement and strain data obtained from DIC. The strain intensity factor estimated using the proposed domain integral method, and the multi-parameter method agrees with the analytical estimate. The influence of the size of the annular region and the mesh sensitivity is also reported. The sensitivity to the variance and bias errors in measurement for both the methods is investigated. To investigate the effect of variance error, a Gaussian noise is introduced over the reference and deformed images before post-processing, and its influence on the strain intensity factor estimate is discussed. The effect of random perturbations of displacements and strains on the strain intensity factor estimates is also investigated. The domain integral method is less sensitive to the variance error when compared to the multi-parameter solution approach. Further, the bias errors due to uncorrected lens distortions, over-smoothing of data, and DIC algorithm adopted are introduced into the measurements, and its effect on both the methods is investigated. The multi-parameter method is less sensitive to bias errors when compared to the domain integral method. Therefore, it is recommended that the domain integral method may be used for the experimental determination of strain intensity factor in measurements with variance errors and the multi-parameter method for measurements with bias errors.

Energy release rate, the crack closure integral and admissible singular fields in fracture mechanics

One of the assumptions of Linear Elastic Fracture Mechanics is that the crack faces are traction-free or, at most, loaded by bounded tractions. The standard Irwin’s crack closure integral, widely used for the computation of the Energy Release Rate, also relies upon this assumption. However, there are practical situations where the load acting on the crack boundaries is singular. This is the case, for instance, in hydraulic fracturing, where the fluid inside the crack exerts singular tangential tractions at its front. Another example of unbounded tractions is the case of a rigid line inclusion (anticrack) embedded into an elastic body. In such situations, the classical Irwin’s crack closure integral fails to provide the correct value of the Energy Release Rate. In this paper, we address the effects occurring when square-root singular tractions act at the boundary of a line defect in an elastic solid and provide a generalisation of Irwin’s crack closure integral. The latter yields the correct Energy Release Rate and allows broad applications, including, among others, hydraulic fracturing, soft materials containing stiff inclusions, rigid inclusions, shear bands and cracks characterized by the Gurtin-Murdoch surface stress elasticity. We present the results in the most general form, where six Stress Intensity Factors are present: three of them are classical SIFs corresponding to the Modes I-II-III and computed under the assumption of homogeneous boundary conditions at the defect surfaces, while the other three SIFs are associated with singular admissible tractions (those that lead to a finite ERR value). It is demonstrated that this approach resolves an ambiguity in using the same SIF’s terminology in the cases of open cracks and rigid inclusions, among other benefits.

A crack on a cycloid rough bimaterial interface

We conduct an analytical study of a finite mode III Griffith or Zener-Stroh crack lying on a cycloid rough interface between an elastic body and a rigid substrate. Closed-form expressions of the analytic function characterizing the stress and displacement fields are derived via the solution of a Riemann-Hilbert problem with discontinuous coefficients. Stress distributions are obtained in both Cartesian and curvilinear coordinate systems. In general, the stresses exhibit a square root singularity at the two interface crack tips and at all cusp tips. Cusps located on the interface crack surface behave as cracks while those outside the interface crack act as anticracks or rigid line inclusions. The stress intensity factors scaling the square root singularity in the stresses are extracted from the full-field solutions. When one crack tip is lodged only at a cusp tip, the stronger r−3/4 and the weaker r−1/4 power-type singularities at this crack-cusp tip are observed and the stress intensity factors scaling these power-type singularities are derived.

On the path independence and invariant of the J-integral for a slant crack and rigid-line inclusion using degenerate kernels and the dual BEM

The J-integral and stress intensity factor (SIF) are two major parameters in linear elastic fracture mechanics (LEFM) for the fracture criterion. In this paper, we focus on the J-integral of the slant crack and the slant rigid-line inclusion under the remote anti-plane shear. By employing the degenerate kernel, the path independence of J-integral is analytically demonstrated by using the elliptic coordinates. The positive and negative J-integrals are also analytically derived and numerically implemented by using the dual BEM for the crack and the rigid-line inclusion, respectively. It is interesting to find that the J-integral is not an invariant by using different observer systems but is one component of the vector of the first order tensor. Transformation law of the J-integral with respect to different observers is analytically proved and numerically demonstrated. Finally, the tensor property of order one is examined.

Singular elastic field induced by a rigid line inclusion in a thin nanoplate with surface elasticity

This paper analyzes the mechanical behavior of a Kirchhoff thin nanoplate with surface stresses containing a thickness-through rigid line inclusion. Based on the bulk and surface elasticity, a rigid line inclusion problem of a prescribed displacement is reduced to an associated mixed boundary value problem. By using the Fourier transform, the problem is converted to dual integral equations, then to a weakly singular integral equation with the logarithmic kernel. An exact solution for a rigid line inclusion with an off-nanoplate rigid rotation is obtained and full elastic fields including the deflection, bending moments, and effective shear forces at any position of the nanoplate are determined explicitly in terms of the elementary functions. The singularity coefficients of the bending moments and effective shear forces near the tip of the rigid line inclusion are evaluated. The obtained results show that the bending moments and the stress components exhibit an r−3/2 singularity near the rigid line inclusion tip, where r is the distance from the rigid line inclusion tip. The effective shear forces have an r−5/2 singularity. The influence of surface elasticity on the singularity coefficients is examined.

Dispersion of rigid line inclusions as stiffeners and shear band instability triggers

A dispersion of stiff and thin (‘rigid line’) inclusions (RLIs) in a matrix material may result beneficial for stiffening in the elastic range, but might be detrimental to strength, as material instabilities may be triggered by inclusions when the matrix is brought to a viscoplastic-damaging state. This dual role of RLIs is investigated by means of the embedded reinforcement model. Validated against available analytical predictions, this numerical model is employed to assess the roles of RLIs’ orientation, interaction, volume fraction, and distribution, considering up to 1500 inclusions. When the matrix material deforms inelastically, RLIs produce stress concentrations that promote the nucleation of shear bands. These are characterized at collapse for many distributions of RLIs, showing that their effects range from almost negligible to a disrupting alteration of the dominant failure mechanism. In the latter case, it is shown that the dominant shear bands can be fragmented by RLIs into a mosaic of tiny localization bands. These results offer new insights into energy dissipation mechanisms of reinforced materials, as they are promoted or inhibited by the interactions of rigid line inclusions.