Stress field near interface edge of elastic-creep bi-material

Stress fields on elastic-creep bi-material interfaces with different geometry of the interface edge are analyzed by finite element method. The results reveal that the stress highly concentrates near the interface edge at the loading instant and it gradually decreases as the creep-dominated zone expands from the small-scale creep to the large-scale creep. The stress singularity due to creep which resembles the HRR stress singularity appears near the interface edge in all cases. The stress intensity near the interface edge time-dependently decreases and becomes constant when the transition reaches the steady state. The magnitude is scarcely influenced by the edge shape of elastic material, though it depends on the edge shape of creep material. The stress intensity during the transition can be approximately predicted by the J-integral at the loading instant.     

Stress intensity factors of a surface crack near an interface end

Due to the singular behavior of the stress field near the interface edge of bonded dissimilar materials, fracture generally initiates near the interface edge, or just from the interface edge point. In this paper, an edge crack near the interface, which can be considered as being induced by the edge singularity and satisfying two conditions, is analyzed theoretically, based on the singular stress field near the interface edge and the superposition principle. It is found that the stress intensity factor can be expressed by the stress intensity coefficient of the edge singular stress field, the crack length, the distance between the interface and the crack, as well as the material combination. Boundary element method analysis is also carried out. It is found that the theoretical result coincides well with the numerical result when the crack length is small. Therefore, the theoretical representation obtained by this study can be used to simply evaluate the stress intensity factor of an edge singularity induced crack for this case. However, when the crack length becomes larger than a certain value, a significant difference appears, especially for the case with large edge singularity.     

A three-dimensional eigenfunction expansion approach for singular stress field near an adhesively-bonded scarf joint interface in a rigidly-encased plate

A three-dimensional eigenfunction expansion approach for determination of the singular stress field in the vicinity of an adhesively-bonded scarf joint interface in a plate, with its top and bottom surfaces being encased, fully or partially, between infinitely rigid blocks is presented. The plate is subjected to extension/bending (mode I) and in-plane shear/twisting (mode II) far-field loading. Both the adhesive layer and plate materials are assumed to be isotropic and elastic. The boundary conditions that are prescribed on the end-faces (free, fixed and lubricated) of the plate as well as those, prescribed at the bottom or top surface of the scarf-bonded plate on either side of the interface between the plate and adhesive layer materials (fixed-fixed, free-fixed and fixed-free), are exactly satisfied. Numerical results include the dependence of the lowest eigenvalue (or most severe stress singularity) on the wedge aperture angle of the plate material. Variation of the same with respect to the shear moduli ratio of the constituent plate and adhesive layer materials is also an important part of the present investigation. Hitherto unobserved interesting and physically meaningful conclusions in regards to the fixed edge singularity and delamination type flaw sensitivity of an adhesively bonded plate surface are also presented. Finally, hitherto unavailable results, pertaining to the through-width variations of stress intensity factors corresponding to symmetric and skew-symmetric sinusoidal loads that also satisfy the boundary conditions on the end-faces of the adhesively bonded plate, in the vicinity of the scarf joint interface, under investigation, bridge a longstanding gap in the bonded joint stress singularity/fracture mechanics literature.     

Effect of interface layer consisting of nanosprings on stress field near interface edge

The purpose of this study is to investigate the effect of an interface layer consisting of discretely arrayed nano-sized elements on stress intensified fields. A material where an interface layer consisting of Ta₂O₅ helical nanoelements (nanosprings) is inserted between dissimilar components is prepared and two types of crack initiation experiments, which possess radically different stress conditions, are carried out. The finite element analyses indicate that the stress fields in the components with and without the interface layer are completely different, and it is experimentally clarified that the fracture mechanics concept cannot be applied to the crack initiation at the dissimilar interface edge with the interface layer. The stress distributions at the crack initiation reveal that the crack initiation is governed by the apparent stress of the nanospring, σ′, at the edge. This signifies that the interface layer eliminates the stress singular field at the interface edge. The criterion of the crack initiation is evaluated as .     

Dominant stress region for crack initiation at interface edge of microdot on a substrate

In order to examine the mechanics of crack initiation at the free interface edge of a microcomponent on a substrate, delamination tests are carried out for two specimen shapes of Cr microdots on a SiO₂ substrate. The microdots of the first specimen are shaped like the frustum of a round cone. The Cr microdots are successfully delaminated from the SiO₂ substrate in a brittle manner and the critical load is measured by atomic force microscopy (AFM) with a lateral loading apparatus. Stress analysis reveals that a singular stress field exists near the interface edge and the strength for the crack initiation is governed by the intensified normal stress field. The critical stress intensity parameter is evaluated as K_σC ≈ 0.24 MPa m^0.39. Similar delamination tests are conducted for microdots shaped like the frustum of an oval cone. The stress distributions at the crack initiation of this specimen shape show a higher normal stress than the first specimen shape in the region near the interface edge of about x < 40 nm, while it is lower in the region of about x > 50 nm (x: distance from the edge). This suggests a limitation of conventional fracture mechanics: namely, the crack initiation in these specimens is not uniquely governed by the intensity of the singular field. It is found that the delamination crack is initiated when the averaged stress σ_ya in the region of 90-130 nm reaches 190-270 MPa, regardless of the specimen shape. This indicates that the dominant stress region of crack initiation is roughly estimated as 90-130 nm and the criterion is given in terms of the averaged stress in the region.     

Effect of residual stress on delamination from interface edge between nano-films

The focus in this study is on the effect of residual stress on the delamination crack initiation from the interface edge between thin films, Cu/TiN, where the stress is intensified by the free edge effect. The delamination tests, where the mechanical stress is applied on the interface, show that the specimen with the thinner Cu film has an apparently higher strength at the interface edge. The residual stress in the films is then evaluated by curvature measurement of film/substrate coupon and the influence on the delamination is analyzed. The residual stress increases with the increase of film thickness and remarkably intensifies the stress near the edge. By superimposing the contributions of the applied load and the residual stress, a good agreement is obtained in the normal stress intensity near the interface edge at the delamination independent of the Cu thickness. This signifies that the combination of intensified stresses due to the applied load and the residual stress governs the crack initiation at the interface edge, and the toughness at the interface edge is evaluated by the stress intensity factor on the basis of the fracture mechanics concept.     

Disappearance of stress singularity at interface edge due to nanostructured thin film

The purpose of this study is to examine the stress distribution near the interface between a nanostructured thin film and a solid body. We focus on a nanostructured thin film that consists of Ta₂O₅ helical nanosprings fabricated on a Si substrate by dynamic oblique deposition. The mechanical properties of the thin film are obtained by vertical and lateral loading tests using a diamond tip built into an atomic force microscope. The apparent shear and Young’s moduli, G′ and E′, of the thin film are 2-3 orders of magnitude lower than those of a conventional solid Ta₂O₅ film. Moreover, the thin film shows strong anisotropy. A finite element analysis for two types of components with different interface edges between the thin film and an elastic solid body is conducted under uniform displacement. One has a free edge where the surface-interface angle is 90°-90°, and the other has a short interface crack. These analyses indicate the absence of not only stress singularity but also high stress concentration near the free edge and the interface crack tip. The characteristic stress distributions near the interface are due to the nanoscopically discrete structure of the thin film.     

A method for eliminating the effect of 3-D bi-material interface corner geometries on stress singularity

The stress singularities at three-dimensional (3-D) interface corners and edges have been investigated numerically using common finite element methods. The effects of variation of edge angle and vertex angle between the two free side surfaces on the order of stress singularity at bi-material interface corners are determined. It is found that the order of stress singularity at interface corner depends not only on the combination of materials and edge angles but also on the vertex angle between the two free side surfaces. The effect of the vertex angle on the order of stress singularity at interface corner can be eliminated by smoothing the intersection of interface edges, which can be achieved simply by generating a circular-arc fillet at the intersection of the two free side surfaces. The numerical results show that, after smoothing the intersection of interface edges, the order of stress singularity along the interface edges become continuous, i.e. the order of the corner singularity can be reduced to the level of the edge singularity.     

Complex variable method of computing Jk for bi-material interface cracks

A method using functions of a complex variable is developed for evaluation of J₁ and a modified J₂ integrals for bi-material interface cracks. This method, used in conjunction with the finite element method, would be useful in the prediction of stress intensity factors for cracks lying between the interface of two dissimilar materials. Since the direct evaluation of J₂ poses difficulties in modeling the singular behavior in the near vicinity around the crack tip for bi-material crack problems, it is modified by evaluating it around a contour path of small radius from the crack tip within the singularity dominated zone. It is shown that the stress intensity factors for a bi-material interface crack can be accurately evaluated using these j_k integrals.     

Stress intensity factors of an inclined elliptical crack near a bimaterial interface

In this paper the stress intensity factors are discussed for an inclined elliptical crack near a bimaterial interface. The solution utilizes the body force method and requires Green’s functions for perfectly bonded semi-infinite bodies. The formulation leads to a system of hypersingular integral equation whose unknowns are three modes of crack opening displacements. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of stress intensity factors along the crack front accurately. Distributions of stress intensity factors are presented in tables and figures with varying the shape of crack, distance from the interface, and elastic modulus ratio. It is found that the inclined crack can be evaluated by the models of vertical and parallel cracks within the error of 24% even for the cracks very close to the interface.     

Initiation of interface crack at free edge between thin films with weak stress singularity

Delamination tests using sandwich type specimens are conducted for eight combinations of materials: thin films formed on silicon substrates which are relatively popular in micro-electronic industry, to develop a method for quantitative evaluation and comparison of crack initiation strength at the free edge. The difficulty stems from the difference of stress singularity, K_ij/r^λ (K_ij: stress intensity, r: distance from free edge and λ: order of stress singularity), where λ is depending on the combination of materials. Thus, the critical K_ij has different dimensions, MPa m^λ, in each interface. Using the experimentally observed delamination load, the stress distribution along the interface is analyzed by boundary element method. Since the orders of stress singularity, λ, in the materials are less than 0.07 (weak singularity), the stress field near the interface edge is almost constant in atomic (nanometer) level. Then, the critical strength for the interface cracking is quantitatively represented by the concentrated stress near the edge. The effects of the several factors such as species of thin films, oxidized interlayers and deposition processes of thin films on the interface strength are evaluated on the basis of this critical stress as well.     

The effect of material combinations and relative crack size to the stress intensity factors at the crack tip of a bi-material bonded strip

Several types of singular stress fields may appear at the corner where an interface between two bonded materials intersects a traction-free edge depending on the material combinations. Since the failure of the multi-layer systems often originates at the free-edge corner, the analysis of the edge interface crack is the most fundamental to simulate crack extension. In this study, the stress intensity factors for an edge interfacial crack in a bi-material bonded strip subjected to longitudinal tensile stress are evaluated for various combinations of materials using the finite element method. Then, the stress intensity factors are calculated systematically with varying the relative crack sizes from shallow to very deep cracks. Finally, the variations of stress intensity factors of a bi-material bonded strip are discussed with varying the relative crack size and material combinations. This investigation may contribute to a better understanding of the initiation and propagation of the interfacial cracks.     

Stress intensity factors of a central interface crack in a bonded finite plate and periodic interface cracks under arbitrary material combinations

Although a lot of interface crack problems were previously treated, few solutions are available under arbitrary material combinations. This paper deals with a central interface crack in a bonded finite plate and periodic interface cracks. Then, the effects of material combination and relative crack length on the stress intensity factors are discussed. A useful method to calculate the stress intensity factor of interface crack is presented with focusing on the stress at the crack tip calculated by the finite element method.     

Stress intensity factor analysis of an interface crack between dissimilar anisotropic materials under thermal stress using the finite element analysis

New numerical methods were presented for stress intensity factor analyses of two-dimensional interfacial crack between dissimilar anisotropic materials subjected to thermal stress. The virtual crack extension method and the thermal M-integral method for a crack along the interface between two different materials were applied to the thermoelastic interfacial crack in anisotropic bimaterials. The moving least-squares approximation was used to calculate the value of the thermal M-integral. The thermal M-integral in conjunction with the moving least-squares approximation can calculate the stress intensity factors from only nodal displacements obtained by the finite element analysis. The stress intensity factors analyses of double edge cracks in jointed dissimilar isotropic semi-infinite plates subjected to thermal load were demonstrated. Excellent agreement was achieved between the numerical results obtained by the present methods and the exact solution. In addition, the stress intensity factors of double edge cracks in jointed dissimilar anisotropic semi-infinite plates subjected to thermal loads were analyzed. Their results appear reasonable.     

Numerical methods for determination of stress intensity factors of singular stress field

The asymptotic solution of the singular stress field near a singular point is generally comprised of one or more singular terms in the form of Kr^λ-1f_ij(θ). Based on the asymptotic solution of the singular stress field and the common numerical solution (stresses or displacements) obtained by an ordinary tool such as the finite element method or boundary element method, a simple and effective numerical method is developed to calculate stress intensity factors for one and two singularities. Three examples show that the stress intensity factors evaluated using the method proposed in this paper are very accurate.     

Critical study of existing solutions for a penny-shaped interface crack, comparing with a new boundary element solution allowing for frictionless contact

A numerical study of the fundamental problem of a pressurized penny-shaped crack at the interface of two dissimilar half spaces is carried out allowing for the possibility of frictionless contact between crack faces. A new, highly accurate axi-symmetric formulation of the boundary element method (BEM) for the solution of elastic contact problems is employed. The correctness and accuracy of available predictions of different kinds for several key characteristics of the solution of this problem are checked. First, comparison of the BEM results for the near-tip contact length shows a very good agreement with some existing predictions. Second, the global solution obtained by BEM is compared with existing asymptotic solutions, obtained with both the open and the frictionless contact models. BEM results show that at the closest neighborhood to the crack tip the global solution of the problem is governed by the first term of the asymptotic solution of the frictionless contact model (up to a distance of the order of a fraction of the near-tip contact length). After a small transition region, in an adjacent surrounding zone whose extent is almost independent of the near-tip contact length, the global solution of the problem is governed by the first term of the asymptotic solution of the open model. As a result of the comparison presented, the regions in which the classical fracture parameters, stress intensity factor (SIF) and energy release rate, can be accurately obtained from the global numerical solution of a crack of this kind have been determined. Third, BEM results and previous estimations show certain discrepancies with a recently published closed form solution of the near-tip contact length and the mode II SIF of the frictionless contact model. A new closed form expression of this mode II SIF, derived from the asymptotic solution of the open model, is proposed in this paper.     

Computation of stress intensity factors of interface cracks based on interaction energy release rates and BEM sensitivity analysis

Stress intensity factors of bimaterial interface cracks are evaluated based on the interaction energy release rates. The interaction energy release rate is defined based on the energy release rates of a cracked body, corresponding to two independent loading conditions, actual field and an auxiliary field, and is related to the sensitivities of the potential energies for crack extensions. The potential energy of a cracked body is expressed with a domain integral, which is converted to a boundary integral expression by applying the divergence theorem. By differentiating this expression with the crack length, a boundary integral expression for the interaction energy release rate is obtained. The boundary integral representation for the interaction energy release rate involves the displacement, the traction, and their sensitivity coefficients with respect to the crack length. The boundary element sensitivity analyses are used to calculate these quantities accurately. A regularized boundary integral equation relating the boundary displacement and traction is differentiated with respect to an arbitrary shape parameter to derive the regularized boundary integral equation for the sensitivity coefficients of the boundary displacement and traction. The proposed approach is applied to several cracks in dissimilar media and the results are compared with those obtained by the conventional approach based on the extrapolation method. The analytical displacement and stress solutions for an interface crack between two infinite dissimilar media subjected to uniform stresses at infinity are used to give the auxiliary field, in which the values of the stress intensity factors are known. It is demonstrated that the present method can give accurate results for the stress intensity factors of various bimaterial interface cracks under coarse mesh discretizations.     

An interaction integral method for computing mixed mode stress intensity factors for curved bimaterial interface cracks in non-uniform temperature fields

In finite element analysis the interaction integral has been a useful tool for computing the stress intensity factors for fracture analysis. This work extends the interaction integral to account for non-uniform temperatures in the calculation of stress intensity factors for three dimensional curvilinear cracks either in a homogeneous body or on a bimaterial interface. First, the derivation of the computational algorithm, which includes the additional terms developed by the non-zero gradient of the temperature field, is presented in detail. The algorithm is then implemented in conjunction with commercial finite element software to calculate the stress intensity factors of a crack undergoing non-uniform temperatures on both a homogeneous and a bimaterial interface. The numerical results displayed path independence and showed excellent agreement with available analytical solutions.     

Stress intensity factor analysis for an interface crack between dissimilar isotropic materials under thermal stress

Thermal stresses, one of the main causes of interfacial failure between dissimilar materials, arise from different coefficients of linear thermal expansion. Two efficient numerical procedures in conjunction with the finite element method (FEM) for the stress intensity factor (SIF) analysis of interface cracks under thermal stresses are presented. The virtual crack extension method and the crack closure integral method are modified using the superposition method. The SIF analyses of some interface crack problems under mechanical and thermal loads are demonstrated. Very accurate mode separated SIFs are obtained using these methods.     

Debonding of concrete-epoxy interface under the coupled effect of moisture and sustained load

Fiber reinforced polymer (FRP) is a prevalent material for strengthening or retrofitting concrete structures. It has been found that the effectiveness of entire strengthening or retrofitting scheme highly depends on the bond performance of concrete-epoxy interface. During service life, structures are exposed to complicated and unavoidable mechanical and environmental situations that can cause bond deterioration. In order to ensure the structural safety in a long run, a comprehensive experiment focusing on the coupled effect of sustained load and moisture on the bond property of concrete-epoxy interface is conducted. A drastic deterioration of concrete-epoxy interfacial fracture toughness, up to 77%, is observed under coupled sustained load and moisture exposure. Based on our experimental investigation, a predictive model is developed to describe the bond property variations of concrete-epoxy interface against long-term sustained load and moisture. The proposed predictive model can be used to obtain a more reliable and accurate service life prediction and maintenance planning for FRP-bonded concrete structures.