Increase of stress intensity near interface edge of elastic-creep Bi-material under a sustained load

The transition from small-scale creep to large-scale creep ahead of a crack tip or an interface edge with strong elastic stress singularity at the loading instant causes stress relaxation and the decrease of stress intensity in general. However, this study shows that the stress near the interface edge of bi-material with no or weak elastic stress singularity increases after the loading instant and brings about the stress concentration during the transition. In addition, the creep strain distribution of this bi-material after the loading instant is different from that occurred in the transition of an interface edge with strong elastic stress singularity or a crack tip (notch root). The criterion for the increase or decrease of stress intensity near the interface edge proved by the finite element method is proposed in this study. The stress intensity near the interface edge increases when the elastic stress singularity is lower than the creep stress singularity (λ^el < λ^cr) and vice versa.     

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.     

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.     

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.     

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.     

Creep crack initiation at a free edge of an interface between submicron thick elements

To clarify the mechanics of time-dependent crack initiation at an interface edge in submicron thick elements due to creep, delamination experiments are conducted using a micro-cantilever bend specimen with a tin/silicon interface edge. After the specimen time-dependently deforms under a constant load, a delamination crack is initiated at the Sn/Si interface edge. In addition, the steady state creep property of Sn is estimated by performing an inverse analysis using a finite element method based on creep deformation experiments conducted for different specimens. Stress analysis using the obtained creep property reveals that stress and strain rate singularities exist at the Sn/Si interface edge under creep deformation. The intensity of the singular field time-dependently increases as the creep region expands, and eventually it becomes a steady state. The stress and strain rate intensities at the steady state correlate well with the crack initiation life, which indicates that the singular stress field near the interface edge governs the creep crack initiation.     

Analytical solution for bonded wedges under thermal stresses

After a change in temperature, high stresses leading to destruction may occur in bonded dissimilar materials near the point of the interface line intersection with the edge. In terms of linear elasticity, these high stresses are described by the singular terms of the stress field expansion at the corner point. In the present paper, the explicit representation of the singular terms and exact values of the stress intensity factors in the case of infinite wedge-shaped joint geometry are obtained by the Mellin transform technique. Systematic comparison with the FEM results for samples of finite size has shown that the values of stress intensity factors are in good agreement if the singularity is not too strong (the singularity orders _k<0.2). With the stronger singularity, the analytical solution is in qualitative agreement with the FEM one, such that it can be used for fast parametrical study of finite samples as well.     

Mode-I crack problem for functionally graded layered structures

This paper deals with two bonded functionally graded finite strips with two collinear cracks. Different layers may have different nonhomogeneity properties in the structure. A bi-parameter exponential function was introduced to simulate the continuous variation of material properties. The problem is solved by using the integral transform, singular integral equation methods and the theory of residues. Various internal cracks and edge crack and crack crossing the interface configurations are investigated, respectively. The asymptotic stress field near the tip of a crack crossing the interface is examined and it is shown that, unlike the corresponding stress field in piecewise homogeneous materials, in this case the “kink” in material property at the interface does not introduce any singularity. Numerical calculations are carried out, and the influences of nonhomogeneity constants, geometric parameters and crack interactions on the stress intensity factors are investigated.     

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.     

Computer Simulation of the Indentation Creep Tests on Particle-Reinforced Composites

A systematical simulation has been carried out on the indentation creep test on particle-reinforced composites.The deformation ,failure mechanisms and life are analyzed by three reasonable models.The following five factors have been considered simultaneously:creep property of the particle,creep property of the matrix,the shape of the particle, the volume faction of the particle and the size(relative size to the particle )of the indentation indenter.For all the cases,the power law respecting to the applied stress can be used to model the steady indentation creep depth rate of the indenter,and the detail expressions have been presented.The computer simulation is analyzed by the two-phase model and the three-phase model.Two places of the stress concentration are found in the composites.One is ahead of the indentation indenter, where the high stress state is deduced by the edge of theindenter and will decrease rapidly near to a steady value with the creep time The other one is at the interface,where the high stress state is deduced by the misfit of material properties between the particles and matrix.It has been found that the creep dissipation energy density other than a stress parameter can be used to be the criterion to model the debonding of the interfaces.With the criterion of the critical creep dissipation energy density, a power law to the applied stress with negative exponent can be used to model the failure life deduced by the debonding of interfaces.The influences of the shape of the particles and the matching of creep properties of particle and matrix can be discussed for the failure.With a crack model,the further growthe of interface crack is analyzed, and some important experimental phenomena can be predicted.The failure mechanism which the particle will be punched into matrix has been also discussed.The critical differences between the creep properties of the particles and matrix have been calculated, after a parameter has been defined.In the view of competition of failure mechanisms, the best matching of the creep properties of the two phases and the best shape of the particles are discussed for the composite design.     

Crack initiation strength of an interface between a submicron-thick film and a substrate

The focus of this study was to evaluate the fracture initiation criteria of the interface between a thin film and a substrate by Bogy’s, Kitamura’s and Griffth’s methods. The critical stress intensity parameter K_ijC in Bogy’s method and the concentrated stress parameter σ_ijC in Kitamura’s method were calculated based on the singular stress field near the interface edge. The work of separation per unit area Γ_ο in Griffth’s method was calculated based on the work of fracture process. The results obtained show that in comparison among interface strengths, the fracture toughness K_θθC and the concentrated stress parameter σ_θθC were respectively applied to material combinations with specific edge geometry and with weak stress singularity, while the work of separation per unit area Γ_ο was applied in all cases.     

界面对硅酸铝短纤维增强AZ91D镁基复合材料蠕变性能的影响

界面对复合材料蠕变性能的影响很大。在试验分析的基础上建立了硅酸铝短纤维增强AZ91D镁基复合材料理论分析模型,利用三维有限元分析方法,系统研究了界面特性、界面上应力应变分布和短纤维位向变化对硅酸铝短纤维增强AZ91D镁基复合材料蠕变性能的影响。研究表明：界面特性,如厚度、模量,均对纤维最大轴应力和稳态蠕变速率有影响,当界面厚度增加,纤维最大轴应力减小而稳态蠕变速率增大;当界面模量增大,纤维最大轴应力增大而稳态蠕变速率减小,但当界面模量高于基体模量时,纤维最大轴应力和稳态蠕变速率均保持不变;纤维位向也影响轴应力分布和稳态蠕变速率,纤维在其末端界面上存在较大的应力和应变,此处容易产生微裂纹而使材料抗蠕变能力下降;界面对硅酸铝短纤维增强AZ91D镁基复合材料的蠕变曲线和蠕变断裂机制也有影响,其影响程度还与纤维位向有关。     

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.     

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.     

Interface strength of structured nanocolumns grown by glancing angle deposition

In order to investigate the mechanics of interfacial fracture in structured nano-elements grown by glancing angle deposition (GLAD), fracture experiments were conducted on oblique Ti nanocolumns grown on a Si substrate using a micro-brick specimen. Two types of specimens, a Forward specimen (loading with the column tilt direction) and a Reverse specimen (loading against the column tilt), were prepared to clarify the effect of an asymmetric nanostructure on the interface strength. The specimens fractured at the interface or near the interface between the Ti nanocolumns and the Si substrate for both specimens. The critical force and displacement at fracture in the Forward specimens were about 2.0 times and 1.6 times as large as those in the Reverse specimens, respectively, showing clear anisotropy in the interface strength. The local stress distribution along the interface in the single nanocolumn at fracture was analyzed by finite element analysis. While the stress singularity in the Reverse specimen was greater than that in the Forward specimen, the normal stresses in a region 1–3 nm near the interface edge were almost identical regardless of the loading direction, suggesting that intensified stress in the nanoscale region dominated fracture.     

Effects of inclusion on the creep rate of piezoelectric films

This paper presents an analytical method to study the effect of inclusions in piezoelectric materials on the creep rate. The driven force for the creep rate of piezoelectric materials with inclusion is from diffusional mass transport along the inclusion interface. The results show that the creep rate of piezoelectric materials containing the rigid inclusion with the shape parameter m = 0.8 appears at the minimum, and the effect of inclusion volume fraction on the creep rate of piezoelectric materials with soft inclusion becomes the smaller and smaller, as the stiffness of soft inclusion decreases. Thus, the effect of inclusion on creep characteristics can be improved by controlling the material property, the sizes, shapes, and volume ratio of inclusions.     

Stresses near the edge of bonded dissimilar materials described by two stress intensity factors

Stresses near the free edges of the interface of bonded dissimilar materials can be described by the sum of a regular stress term and one or two stress singularity terms. A method for the calculation of the corresponding stress intensity factors from finite element results is presented which is useful to determine two stress intensity factors together. Results for some geometries show that all three terms may contribute significantly to the stress distribution near the free edge of the interface for thermal stress.     

Application of virtual crack closure integral method for interface cracks in low-k integrated circuit devices under thermal load

The virtual crack closure integral (VCCI) method is used to evaluate the stress intensity factor (SIF) and energy release rate (ERR) of an interface crack under thermal load. The VCCIs used in this work include the traditionally known “Mode I” and “Mode II” VCCIs and an additional coupling VCCI. The singularity element is used in the finite element method (FEM) implementation. The SIF and ERR calculated by the FEM are compared to the exact solution in the case of a joint dissimilar semi-infinite plates with double edge crack under thermal loading. The FEM result agrees well with the exact solution for relatively coarse meshes. The contribution of the mesh density and material mismatch to the FEM error is also explored. The VCCI method is used with the multi-scale FEM in a delamination risk assessment of a low-k integrated circuits device in flip-chip plastic ball grid array packages. The ERR is calculated for different package configurations and the prediction of the delamination risk is confirmed by reliability tests.     

Impression creep behavior of Al–1.9%Ni–1.6%Mn–1%Mg alloy

In the present study, microstructure and creep behavior of an Al–1.9%Ni–1.6%Mn–1%Mg alloy were studied at temperature ranging from 493 to 513 K and under stresses between 420 and 530 MPa. The creep test was carried out by impression creep technique in which a flat ended cylindrical indenter was impressed on the specimens. The results showed that microstructure of the alloy is composed of primary α(Al) phase covered by a mantle of α(Al)+Ni₃Al intermetallic compound. Mn segregated into Al_xMn_yNi_z or Al₆Mn phases distributed inside the matrix phase. It was found that the stress exponent, n, decreases from 5.2 to 3.6 with increasing temperature. Creep activation energies between 115 kJ/mol and 151 kJ/mol were estimated for the alloy and it decreases with rising stress. According to the stress exponent and creep activation energies, the lattice and pipe diffusion- climb controlled dislocation creep were the dominant creep mechanism.