Bimaterial Wedge Research Articles

Interfacial fracture performance of engineered cementitious composites and fire-damaged concrete

This study investigated the interfacial fracture properties of engineered cementitious composites (ECC) and fire-damaged concrete using bi-material wedge splitting specimens. The full load versus crack mouth opening displacement (CMOD) curves of the bi-material specimens were obtained using wedge splitting tests, and the fracture toughness and fracture energy were calculated. The effects of the maximum temperature experienced by the concrete (20–700 °C) and the interfacial roughness (0–5 mm) on the interfacial damage mode, fracture toughness, and fracture energy were studied. The results indicated that when the maximum temperature experienced by the concrete was lower than 400 ℃ or the interface was smooth, the damage mode of the specimens was primarily adhesive failure. However, when the maximum temperature experienced by the concrete was equal to or greater than 400 °C and the interface was rough, cohesive failure often occurred in the specimens. Owing to the change in the damage mode, the bi-material specimens had the highest values of the fracture toughness and fracture energy at 400 °C. When the maximum temperature experienced by the concrete exceeded 400 °C, both the fracture toughness and fracture energy decreased with the increasing temperature. In this study, the fracture toughness and fracture energy of the bi-material specimens first increased and subsequently decreased with the increaseing roughness, and the maximum fracture toughness and fracture energy values were obtained when the interface roughness was 3 mm. The study results would provide a reference for the surface treatment when ECC was used in the strenthening of fire damaged concrete.

Experimental study of the effectiveness of stress reduction near V-shaped notch filled with a certain material

A variant of healing a V-shaped notch in an elastic body using an elastic material to fill the notch cavity is considered. The filling material changes the stress singularity at the apex of the V-shaped notch. The degree of changes depends on the correlation of mechanical and geometric parameters of the elastic filler with the mechanical parameters of the base material. Previously, this problem was comprehensively investigated based on the comparative analysis of the results of numerical modeling of the stress singularity exponents for a closed bimaterial wedge and a homogeneous wedge with stress-free edges at different ratios of the opening angles and elastic moduli of the homogeneous parts of the composite wedge, and at different values of Poisson's ratios. It was found that for a closed bimaterial wedge, the stress singularity in the symmetrical (with respect to the bisector of the opening angle) stress state disappears when the value of the Poisson's ratio of the material of one part (with a smaller opening angle) reaches 0.5. By means of numerical models it has been shown that the singulaity of stresses and, as a consequence, stress concentration can be eliminated even when the values of the elasticity modulus of the filling weakly compressible material is several orders of magnitude lower than the elasticity modulus of the base material. This finding made it possible to eliminate the stress singularity at the apex of the V-shaped notch when its cavity was filled with a material, whose Poisson's ratio is close to 0.5. In this work, an experimental study was carried out to substantiate and verify the effectiveness of reducing the stress concentration near V-shaped notch by filling its cavity with a material whose Poisson's ratio is close to 0.5.

Analysis of bimaterial singular regions for orthotropic and isotropic materials under thermal loading

A method to formulate an asymptotic elasticity solution for bimaterial singular regions in orthotropic materials is presented for the thermal case. In the vicinity of material and geometric singularities, approximate methods such as finite elements fail to converge and procedures to improve these solutions become necessary. The results presented herein can be used in conjunction with approximate numerical techniques for the determination of stresses and stress intensity factors in composite bonded joints, cracked bimaterial interfaces, or a junction of dissimilar materials under thermal loading. The thermal displacement and stress fields at the tip of a bimaterial wedge are developed by means of eigenfunction expansions including both singular and nonsingular terms. The thermal elasticity solution for isotropic materials as the limiting case of the orthotropic solution is also obtained. The formulations obtained are applicable to bonded joints with interfacial or interlaminar cracks, geometric and material discontinuities, and uniform thermal loading. The detailed formulation presented provides an essential component in the application of finite elements for the determination of generalized intensity factors and stresses in composites, bonded joints, or other dissimilar materials under thermal loading. Formulation results for orthotropic materials, including the limiting case of isotropic materials, are compared with published and closed form results that use different formulations, and excellent agreement is observed. Furthermore, the existence of logarithmic stress singularities of the form (k⋅ΔT⋅lnr) is investigated and discussed.

Wedge and crack solutions for Mindlin–Reissner plates

Displacement potentials are used to define the complete set of displacement and stress-resultant fields in moderately thick plate bending problems. The asymptotic singular terms are used to define and solve the eigenproblems associated with all alternative homogeneous wedge boundary conditions, and extended to model bimaterial wedges. This solution is used to model edge cracks with a filler and thus simulate the effect of crack repair. Definitions for the stress intensity factors are given, as well as the constitutive relations used to model a repaired crack with a contact element.

A complex potential method for the asymptotic solution of wedge problems using first-order shear deformation plate theory

Abstract In the present work a complex potential approach is derived in order to investigate stress singularities in wedges or sharp notches using first-order shear deformation plate theory. The focus is on the calculation of the singularity exponent as a fundamental quantity in fracture mechanics. Isotropic homogeneous and bi-material wedges are considered. The effect of different boundary conditions along the notch faces and the influence of the elastic contrast on the singularity exponent are discussed in detail. Within an asymptotic analysis, the governing PDE-system is solved introducing three holomorphic potentials. The requirement that the boundary and continuity conditions must be fulfilled leads to an eigenvalue problem determining the singularity exponent and the angular distribution of the field variables. The underlying eigenvalue equation is a highly non-linear transcendental equation which in general is solved numerically. For specific bi-material configurations closed-form analytical solutions are obtained. The findings are compared with results from literature and with finite element calculations. It is shown, that the proposed complex potential method is a very efficient approach to study the asymptotic solution behaviour. In contrast to methods based on real-valued eigenfunction expansions the physical requirement that all field variables can take only real values is fulfilled automatically due to the nature of the complex potential formalism. The obtained near fields allow for further applications such as an embedding in numerical methods.

Finite element analysis of the Rayleigh wave scattering in isotropic bi-material wedge structures

The numerical study is performed of the harmonic Rayleigh wave scattering in a composite structure constructed from two elastically isotropic 90°-wedges. These wedges are in contact along one pair of their faces. It is assumed that either the perfectly sliding contact or the perfectly rigid one is realized. The other pair of faces forms a plane border between the resulting bi-material wedge and the exterior half-infinite space occupied by vacuum. The finite element method is used. The perfectly matched layer spatially confines the computational domain. The dependences of the reflection and transmission coefficients of the Rayleigh wave on the angle of incidence, the Poisson ratio and the type of contact are obtained and analyzed for different combinations of materials. The behavior of the coefficient of the Rayleigh wave conversion into the interfacial wave which may exist on the internal boundary of the structure is also investigated. A number of relations between the coefficients of conversion are derived from symmetry considerations for structures with sliding contact and composed of identical isotropic materials.

A semi-infinite crack of mode III in the bimaterial wedge

An exact solution of the antiplane problem for a semi-infinite interface crack in a piecewise-homogeneous wedge under a self-balanced load on its sides has been obtained. Three types of boundary conditions on the wedge sides were examined: the both sides being stress-free; both sides being clamped, and one side being stress-free with the second one clamped. As a result of using the Wiener–Hopf method, the solution was represented in quadratures. Green's functions were obtained for stress intensity factors; in the case of a geometrically symmetrical wedge structure simple formulae were found for these functions. The stress singularity in the apex of the wedge was studied. In contrast to the homogeneous wedge structure the asymptotic of the stresses near the apex was established to have sometimes two singular terms for some values of the composite parameters.

The anti-plane shear elastostatic fields near the wedge vertex of an incompressible hyperelastic bimaterial

The anti-plane shear transformation corresponding to a bimaterial wedge problem is studied within the framework of finite elasticity. Two wedges of arbitrary angles are assumed to be incompressible hyperelastic Neo–Hookean materials, and have a common perfect interface and a traction free surface. The resolution of the boundary value problem near the vertex of a bimaterial wedge by means of asymptotic procedure leads to an eigenequation connecting wedges angles and materials properties. Results proved that, contrary to linear elasticity predictions, it is needed to extend development to higher orders because of their contribution in stress singularity.

2D fracture analysis of magnetoelectroelastic composites by the SBFEM

This paper extends the scaled boundary finite element method (SBFEM) to 2D fracture analysis of magnetoelectroelastic composites. The generalized extended stress intensity factors, i.e., the stress intensity factors, the electric displacement intensity factor and the magnetic induction intensity factor, for different crack configurations and material combinations are evaluated. The governing equations of the SBFEM for magnetoelectroelastic materials are derived based on the virtual work principle. The generalized extended stress intensity factors are determined from the solution of the singular stress, electric displacement and magnetic induction fields by extending the definition of the generalized stress intensity factors. The electrically and magnetically impermeable and permeable crack face boundary conditions are implemented and discussed. Numerical examples including the cracks in magnetoelectroelastic materials, the interface cracks between magnetoelectroelastic bimaterials and magnetoelectroelastic bimaterial wedge are presented to validate the proposed technique. The accuracy of the proposed technique is verified by comparison with the reference results.

Electric field intensity factors for conducting paths in anisotropic dielectric bimaterials

Conducting paths in an anisotropic dielectric bimaterial as well as in a homogeneous anisotropic material subjected to electric loading are investigated. The conducting path problems are formulated by using a linear transformation method. Electric field intensity factors are obtained for conducting paths emanating from a surface electrode in an orthotropic material. The asymptotic problem of a kinked conducting path in dissimilar anisotropic dielectric materials is considered. The electric field intensity factor for the asymptotic problem is obtained in the infinite product form. To ascertain validity of the solution obtained from the linear transform method, numerical computations are carried out by using finite element method. The electric field intensity factor for a conducting path emanating from the vertex of a bimaterial wedge with a tilt boundary is also obtained in the closed form.

Thermo-Mechanical Stress near Apex of a Bi-Material Wedge by a Novel Finite Element Analysis

A super wedge tip element for application to a bi-material wedge is develop utilizing the thermo-mechanical stress and displacement field solutions in which the singular parts are numerical solutions. Singular stresses near apex of an arbitrary bi-material wedge under mechanical and thermal loading can be obtained from the coupling between the super wedge tip element and conventional finite elements. The validity of this novel finite element method is established through existing asymptotic solutions and conventional detailed finite element analysis.

Interfacial wedge cracks in dissimilar anisotropic materials under antiplane shear

An interfacial crack in an anisotropic bimaterial wedge under antiplane shear is investigated. An interfacial wedge crack in an isotropic bimaterial with concentrated forces is first solved by using the Mellin transform and the Wiener–Hopf technique. The solution of an interfacial crack in an anisotropic composite wedge is then derived from the isotropic solution based on a linear transformation method. An expression for the stress intensity factor for the interfacial crack is obtained. The asymptotic problem of an interfacial wedge crack is also analyzed. The stress intensity factor is obtained in terms of the stress intensity factor of a wedge without cracks. Numerical calculations of the stress intensity factor are carried out for various wedge types such as symmetric and asymmetric wedges as well as wedges with edge cracks and kinked cracks.

Analysis of a wedge in an anisotropic bimaterial under antiplane loads

An anisotropic bimaterial wedge under antiplane shear is investigated. The problem is formulated by using a linear transformation method. Employing complex function representations of elastic fields, a complete form of elastic fields near the apex of an anisotropic composite wedge is derived. An anisotropic composite wedge subjected to concentrated loads is also analyzed. The transformed isotropic bimaterial wedge is first solved, and the solution for the anisotropic wedge is then obtained based on the linear transformation method. Expressions of the stress intensity factor for the composite wedge are obtained. Numerical computations are carried out to obtain the order of stress singularity and the stress intensity factors for three types of wedge shapes.

The use of the J-integral vector to analyse adhesive bonds with and without a crack

The singularity problem in adhesive joints is mentioned in finite element analyses but it is often ignored. Adhesive joints may have cracks and certainly many sources of singularities due to the adherend corners. Cracks are easily dealt with using fracture mechanics concepts. However, angular wedged notches or bi-material wedges are more difficult to treat because the strength of the singularity must be known. Another point that must be born in mind is the fact that crack propagation in adhesive joints occurs in a mixed mode, especially for ductile materials. The present study shows that the use of the J-integral can treat most of the above problems. A very useful fracture mechanics method using the J-integral vector approach is discussed in connection with adhesive bonds but can be generalized to other singular problems. It is shown that the J vector can be used in adhesive bonds where there is always discontinuity in material properties regardless of whether there is a crack or not. One of the advantages of using the J vector in numerical analyses for homogeneous materials is that the accurate stresses (better known) and displacements in the far field can be used to calculate J instead of the not-so-accurate values near the singular point. A method for dealing with the singularity in adhesive bonds, which can be used in engineering design and analysis, has been established in this investigation.

Effect of a thin plastic adhesive layer on the stress singularities in a bi-material wedge

Geometric singularities for perfect bond constitute a fairly well explored area in linear elasticity. In this paper, an effect of a thin plastic adhesive layer on the singular stress field at the interface corner of a glued bi-material wedge is studied within a simple model. Following the conventional one-dimensional analysis of bonded joints, the adhesive layer is treated as a layer of displacement discontinuity between two materials. A new characteristic equation for the determination of the characteristics of the singular stress field is derived based on the classical works of Williams and Bogy. The new characteristic equation is studied for certain particular composites and angles, which commonly occur due to joining dissimilar materials. The focus of this paper is the problem associated with elimination of the singular stress field, with the goal of providing an approach for designing singular-free adhesive joints.

A Bi-Material Wedge – a Model for the Prediction of Failure Initiation at Shape and Material Discontinuities

Geometrical and material discontinuities in constructions lead to singular stress concentrations and consequently to a crack initiation. The model of a bi-material wedge makes it possible to analyse such construction points to assess their stability. The presented approach is based on the knowledge of the strain energy density factor distribution in the concentrator vicinity.

A Novel Hybrid Element Analysis for Piezoelectric-parent Material Wedges

A new super wedge-tip hybrid element together with standard hybrid finite elements is developed to determine singular electro-elastic fields at the vertex of piezoelectric-parent material wedges. With the technique, stress and electric displacement intensity factors and energy release rates in a PZT5H panel containing a central crack are computed and compared with the available theoretical solutions. It is shown that the numerical results converge to exact solutions rapidly with fewer elements and proper number of high order terms. Then, the fracture parameters named as general stress intensity parameters and general electric displacement intensity parameters for three kinds of bimaterial wedges such as bi-piezoelectric wedge, piezoelectric-conductor wedge and piezoelectric-composite wedge are computed and depicted in graphical forms.

Antiplane electro‐mechanical field of a bimaterial piezoelectric wedge under a pair of concentrated forces and surface charges

This paper presents the general solutions of the antiplane electro‐mechanical field for a bimaterial piezoelectric wedge subjected to a pair of concentrated forces and surface charges. Identical wedge angles are assumed here. Using the Mellin transform method, the singularity orders and the generalized stress, strain, electric displacement and electric field intensity factors are derived analytically. The discussion of a semi‐infinite crack at the interface of two bonded piezoelectric half‐planes is also included. Some degenerated cases are investigated to check the accuracy of the obtained solutions.

Analysis of Stress Singularities at Bi-Material Corners in Reddy's Theory of Plate Bending

AbstractThe order of stress singularity at a sharp corner of a plate needs to be known before a numerical approach can be taken to determine accurately the stress distribution of a plate with irregular geometry (such as a V-notch) under loading. This work analyzes the order of the stress singularity at a bi-material corner of a thick plate under bending, based on Reddy's third-order shear deformation plate theory. An eigenfunction expansion technique is used to derive the asymptotic displacement field in the vicinity of the sharp corner by solving the equilibrium equations in terms of displacement functions. This paper explicitly shows the first known characteristic equations for determining the order of the stress singularity at the interface corner of a bonded dissimilar isotropic plate. Moreover, the numerical results are given in graphic form for the order of stress singularity at the interface corner in bonded dissimilar isotropic plates and at the vertex of a bi-material wedge with free radial edges. The results presented herein fill some of the gaps in the literature

Three-dimensional asymptotic stress field at the front of an unsymmetric bimaterial wedge associated with matrix cracking or fiber break

A combined approach for prediction of the singular stress fields at the interfacial fronts of fiber breaks and matrix cracks is presented. A recently developed eigenfunction expansion method is employed for obtaining three-dimensional asymptotic displacement and stress fields in the vicinity of a point located at the front of a bimaterial wedge of general (unsymmetric) geometrical configuration (with respect to bimaterial interface), and subjected to extension/bending (mode I) and in-plane shear/twisting (mode II) far field loading and free–free wedge-side boundary condition. Each material is isotropic and elastic, but with different material properties. The material 2 (fiber in the case of matrix cracking or matrix in the case of fiber break) is always taken to be a half-space, while the wedge aperture angle of the material 1 is varied to represent varying matrix cracking or fiber break incidence angles at the interfaces. Numerical results pertaining to the variation of the lowest eigenvalues (or stress singularities) for various wedge aperture angles of the material 1, subjected to the afore-mentioned wedge-side boundary condition, are also presented. Variation of the same with the shear moduli ratio of the component material phases is also an important part of the present investigation. The conclusion drawn from the present asymptotic analysis of the matrix cracking problem is in agreement with that of an earlier investigation based on the energy based criterion. Similar conclusion drawn from the present asymptotic analysis of fiber break problem is in agreement with that of an energy based criterion derived here in analogy to the corresponding matrix crack problem by another investigator.