Analytical Stress Research Articles

Flip-chip assembly: is the bi-material model acceptable?

The bonding layer in flip-chip assembly designs is characterized, unlike in epoxy bonded assemblies, by a relatively high effective Young’s modulus of its composite material, which is comprised of high-modulus solder and low-modulus epoxy encapsulant (underfill). Simple, easy-to-use and physically meaningful tri- and bi-material analytical stress models are developed for the evaluation of the thermally induced interfacial shearing stresses, as well as normal stresses acting in the cross-sections of the assembly components. While in a tri-material model all the three materials, the chip, the substrate and the bonding layer, are treated as “equal partners”, in a bi-material model a significant simplification is made, assuming that the bonding layer is much thinner than the bonded components, the chip and the substrate, and/or that its effective Young’s modulus is significantly lower than the moduli of the chip and the substrate materials. In the carried out numerical example based on the application of the tri-material model, the highest shearing stress occurs at the chip-bond interface and is significantly, by the factor of about 2.45, higher than the stress at the substrate-bond interface, but even the latter stress is about twice as high as the maximum shearing stress predicted on the basis of the bi-material model. As to the normal stresses acting in the cross-sections of the assembly components, the tri-material model predicts that the highest stresses occur in the chip, the lowest—in the substrate, and that the stresses in the bond are rather high, about 59% of the stresses in the chip. The bi-material model, however, simply assumes that the normal stresses in the bond are zero. The normal stresses in the chip predicted on the basis of this model are only about 78% of the stress predicted by the tri-material model. The normal stresses in the substrate evaluated on the basis of the bi-material model are almost twice as high as the tri-material model predicts, but these stresses are low anyway: it is the state of stress in the chip and in the bonding layer, and the interfacial stress at the chip-bond interface that should be of the primary concern to the device designer. It is concluded that while a simple bi-material model can be successfully used for adhesively bonded assemblies, characterized by a thin and/or low modulus bonding layer, a tri-material model should be employed for flip-chip assemblies, when high-modulus solders are used. Future work should include finite-element analyses and experimental evaluations.

Analytical Bending and Stress Analysis of Variable Thickness FGM Auxetic Conical/Cylindrical Shells with General Tractions

In the present research, a unified semi-analytical solution that incorporates influence of the auxeticity (negative Poisson ratio) of the material into elastic responses of the variable thickness functionally graded conical and cylindrical shells and circular/annular plates is developed. The top or bottom layers of the shell/plate may be subjected to general non-uniform normal and shear tractions. The mentioned material and loading complexities have not been investigated before and consequently, the presented comprehensive results are quite new. The proposed unified formulation is developed using the principle of minimum total potential energy and solved using Taylor’s transform, for some combinations of the simply supported, clamped and free edge conditions. Accuracy of results of the proposed unified solution is verified by results of the three-dimensional theory of elasticity extracted from the ABAQUS finite element analysis code. Finally, a comprehensive parametric study including evaluation of individual/simultaneous effects of the auxeticity, structure configuration, shear/normal traction, thickness variability, and boundary conditions on the resulting lateral deflection and in-plane stress distributions of the considered shell and plate structures is accomplished.

A simple analytical bending stress model of parabolic leaf spring

The evaluation of stress distribution, produced by vertical loading along a parabolic leaf spring, presents an essential aspect during the design stage. Commonly, designers utilize the finite element analysis to simulate the stress behaviour of a parabolic leaf spring. Nevertheless, the use of such method is a time-consuming process during the deterministic and the reliability-based fatigue design optimisation. In this study, we propose three analytical models describing the bending stress distribution of a simply supported single asymmetric parabolic leaf spring: (i) an initially curved single asymmetric parabolic leaf spring, subjected to a concentrated load; (ii) a straight single asymmetric parabolic leaf spring, subjected to a uniform load and (iii) an initially curved single asymmetric parabolic leaf spring, subjected to a uniform load. Bending stress distribution results of classical, finite element and proposed models are compared for several case studies. It is observed that the third model is the most precise model compared to the finite element analysis of single asymmetric parabolic leaf spring. Therefore, the suggested model can be used to generate fatigue life diagram that predicts the required mean and alternating load values for a desired fatigue life with an acceptable accuracy and a reduced computational time.

Stress intensity factor solutions for similar and dissimilar spot welds in lap-shear specimens under clamped loading conditions

In this paper, analytical stress intensity factor solutions for similar and dissimilar spot welds in lap-shear specimens under pinned and clamped loading conditions are derived. Closed-form solutions for similar welds in lap-shear specimens of equal thickness under pinned and clamped loading conditions are derived to show explicitly the effects of the clamped edges on the stress intensity factor solutions. Finite element analyses are also employed to obtain the stress intensity factor solutions for similar and dissimilar spot welds in lap-shear specimens under pinned and clamped loading conditions. Both computational and analytical solutions indicate that the clamped loading reduces the stress intensity factor solutions for the given specimen geometries by up to 7% for similar welds and by about 20% for dissimilar magnesium/steel welds.

Effects of particle asphericity on the macro- and micro-mechanical behaviors of granular assemblies

Particle shape has been regarded as a key factor affecting mechanical behaviors of granular assemblies. The present work proposes a shape descriptor, i.e., asphericity, to isolate the effect of aspect ratio and angularity. A series of numerical triaxial compression tests are conducted on mono-disperse assemblies with different asphericities using a superball-based discrete element model. The corresponding mechanical behaviors are investigated from both macroscopic and microscopic points of view. Effects of particle asphericity are examined as well. It demonstrates that particle asphericity is able to enhance the particle–particle inter-locking with higher locked-in forces, showing an increasing proportion of sliding contacts, thereby making a granular assembly stiffer and stronger. In addition, weak contacts have a dominant proportion in the overall contact networks, positively correlated with asphericity. However, asphericity has an insignificant effect on the mean coordination number at the critical state. Furthermore, an established analytical stress–force–fabric relationship is verified using the present data. Anisotropy within an assembly is quantified in terms of several measures, i.e., contact normals, normal and tangential branch vectors, and normal and tangential contact forces. It is found that anisotropy of granular fabric is strongly determined by anisotropy sources in strong contact networks, where a larger asphericity results in a more anisotropic granular fabric.

Assessed interfacial strength and elastic moduli of the bonding material from shear-off test data

A simple and physically meaningful analytical stress model is developed in application to shear-off testing with an objective to evaluate the interfacial shearing stress in the bonding material from the measured shear off force. The model can be used also for the evaluation of the shear modulus of the bonding material, if the interfacial displacement is also measured. The general concept is illustrated by a numerical example. In the authors’ opinion, the suggested methodology, based on the concept of the interfacial compliance, suggested by the first author in his 1986 ASME J. Appl. Mech. paper, could become a basis for a new effective experimental method for assessing the interfacial shearing strength and elastic moduli of the bonding material in electronics. The methodology can be used particularly in application to the recently suggested sintered silver bonding materials to evaluate their bonding strength from the measured force-at-failure and shear modulus from the measured shearing force and displacement.

External spur gear root bending stress: A comparison of ISO 6336:2006, AGMA 2101-D04, ANSYS finite element analysis and strain gauge techniques

The International Organisation for Standardisation ISO 6336:2006 and the American Gear Manufacturers Association AGMA 2101-D04 are the two most predominant empirically based analytical gear stress analysis methods for establishing contact and root bending stress in involute spur and helical gears. Although based on the same fundamental principles, they have evolved to such an extent that, for various reasons, they are not necessarily in agreement. The use of commercial universal numerical finite element software has become an increasingly popular alternative for gear stress analysis and this research compares the root bending stresses in external spur gears established using ISO 6336:2006, AGMA 2101-D04 and numerical finite element analysis (ANSYS) with experimental strain gauge validation.

Doing and Displaying Gendered Boundary Work among Blended Families in Israel

This article seeks to introduce a more complex understanding of family change in Israel, through the case study of Israeli blended families. Going beyond the research on blended families in Israel and elsewhere, we wish to focus our analysis on how blended families are displayed in contemporary Israeli society. The analytical stress on displaying enables us to discern the fluidity and creativity in contemporary family life in Israel, as well as the boundary work through which family members present their family, to themselves and to other audiences. By analyzing data from over 40 in-depth interviews with parents who formed a blended family unit, we argue that family members embody a fuzzy mindset, which does not confine to a state of either/or, and at the same time negotiates traditional nuclear models of the “natural” family inherent in Israeli society.

Application of the second mixed boundary value solution for an interaction of an edge crack and an internal crack of a half plane subjected to uniform traction

The purpose of the present paper is an application of the solution of the second mixed boundary value problem for a doubly connected region with symmetry. As a demonstration, a half plane with an edge and internal cracks in the axis of symmetry subjected to the uniform traction is analyzed. The analytical stress function can be derived; therefore, it is easy to calculate stress distribution and stress intensity factor. The stress distribution is depicted. The stress intensity factors are precisely calculated and are investigated for the crack extension.

A quantitative weld sizing criterion for fatigue design of load-carrying fillet-welded connections

Fillet welded connections are most commonly used for construction of engineering structures that are often subjected to various forms of cyclic loading in service. There exist two potential fatigue failure modes, i.e., weld root cracking through weld throat and weld toe cracking into plate thickness. The former should be prevented at design stage through a proper weld sizing. However, due to difficulties in determining stress concentration at weld throat, existing fatigue design rules, particularly on weld sizing, are largely empirical and often result in excessive conservatisms, leading to oversized welds. As structural light weighting becomes increasingly important, there is an increasing demand for more quantitative fatigue-based fillet weld sizing criterion. By taking advantage of a comprehensive set of fatigue test results on load-carrying fillet-welded cruciform joints and associated analytical developments reported by Xing et al. (2016), an analytical weld throat stress model taking into account of weld penetration and joint misalignments is presented in this paper for determining critical weld size beyond which weld throat failure become unlikely. Then, it is shown that a weld sizing criterion can be analytically developed and expressed as a function of weld penetration and joint misalignments, which are shown to agree well with the large amount of test data reported by Xing et al. (2016), including a systematic validation by means of a logistic regression method on the same test data. As a result, the present developments provide both analytical and experimental basis for achieving quantitative weld sizing so that weld throat fatigue failure mode can be effectively prevented with a clearly defined confidence level without the need of over-sizing.

An analytical multilayer source stress approach for the modelling of material modifications in machining

In the research concept of Process Signatures machining induced changes of surface and sub-surface material properties are considered as material modifications caused by the physical conditions the material is exposed to during the process. This paper presents a newly developed multilayer source stress model to analytically describe the material modifications caused by the machining process in multiple passes. The analytical model that needs measured shape deviations as input is validated via finite element simulations. The approach incorporates the effects of machining induced source stresses and the contribution of residual stresses present in the workpiece before machining. Results from milling experiments show a pronounced correlation between the identified workpiece material modifications and the width of cut.

On the tensile capacity of single-bolted connections between GFRP angles and gusset plates-testing and modelling

The polymer composite known as Glass Fibre Reinforced Polymer (GFRP) has several advantages over conventional materials. It has been slowly incorporated into civil infrastructures. Because of lack of knowledge about the behaviour and design criteria of structural connections, the wide-spread application of GFRP in this field has been rather limited. In this study, analytical and experimental behaviour of GFRP structural angle members with gusset plate connections for net-tension strength, with an emphasis on transmission line tower connections, were investigated. Thirty-five single-lap, single-bolted connections were tested. The widths of the gusset plate and edge distance were varied. A tensile load was applied on the joint and then failure modes and the corresponding loads were observed. The effect of the plate width and edge distance on the connection efficiency was also studied. Based on the studies, the optimum plate width and edge distance were determined as three and five times diameter of the bolt, respectively. A semi-empirical analytical stress concentration model proposed by Hart-Smith (1978), is adopted to determine the cross-correlation coefficient 'C' which is linearly correlated to the stress concentration factors of composite and elastic isotropic materials. Using regression analysis, the coefficient 'C' was obtained, and an ultimate net-tension strength design formula is proposed.

Nano-scratch testing of (Ti,Fe)Nx thin films on silicon

Thin films of (Ti,Fe)Nx have been produced on silicon wafers with a wide range of compositions and mechanical properties to investigate correlations between the mechanical properties measured by indentation and crack resistance in the highly loaded sliding contact in a nano-scratch test. The nano-scratch test data on the thin films using a well-worn Berkovich indenter with ~1μm end radius were supported by high resolution scanning electron microscopic (SEM) imaging and analytical stress modelling. The results show that mechanical properties of the coating, its thickness and the substrate properties all influence the deformation process. They affect the critical loads required, the type of failures observed and their location relative to the moving probe. The differences in coating mechanical properties affect how the interface is weakened (i.e. by initial substrate or coating yielding or both) and determine the deformation failure mechanism. The load dependence of the friction coefficient provides details of the sliding contact zone and the location of failure relative to the sliding probe. Improved performance was achieved at intermediate hardness and H3/E2 in the nano-scratch tests on thin films. The friction and modelling results strongly suggest that failure at low load on the hardest coatings is due to a combination of high tensile stress at the rear of the contact zone and substrate yield. Designing thin films for protective coatings with in-built dissipative structures (such as soft and low elastic modulus inclusions) and mechanisms to combat stress may be a more successful route to optimise their toughness in highly loaded sliding conditions than aiming to minimise plasticity by increasing their hardness.

Reliability physics behind the QFN state of stress

A physically meaningful analytical stress model is developed for the explanation of the reliability physics underlying the relief in the thermally induced interfacial shearing stresses in a typical Quad Flat No-lead (QFN) assembly. The stress relief in a QFN design is, in effect, an implementation of the general concept developed and quantified by the first author about a decade ago of using inhomogeneous bonds for lower thermal stresses in adhesively bonded or soldered assemblies. It is shown also that the maximum stress in the QFN design can be minimized, if the lengths of the peripheral zones are established based on the requirement that the stresses at the ends of the assembly are made equal to the stresses at the ends of its mid-portion. In the carried out numerical example the gain in the relief in the maximum interfacial shearing stress owing to the application of a bond with an elevated compliance is 9.94 %. When the optimized inhomogeneous bond is applied, the gain is 16.87 %. The gain in the stress relief owing to the optimized inhomogeneous bond (with respect to the homogeneous compliant bond) is 10.66 %. The computed interfacial stresses are below the yield stress of the solder material.

Improving the fatigue performance of porous metallic biomaterials produced by Selective Laser Melting

Additive Manufacturing (AM) techniques such as Selective Laser Melting (SLM) are increasingly being used for producing customized porous metallic biomaterials. These biomaterials are regularly used for biomedical implants and hence a long lifetime is required. In this paper, a set of post-built surface and heat treatments is presented that can be used to significantly improve the fatigue life of porous SLM-Ti6Al4V samples. In addition, a novel and efficient analytical local stress method was developed to accurately quantify the influence of the post-built treatments on the fatigue life. Also numerical simulation techniques were used for validation. The developed methods and techniques can be applied to other types of porous biomaterials and hence provide new and useful tools for improving and predicting the fatigue life of porous biomaterials.

Closed-form solution of the Ogden–Hill’s compressible hyperelastic model for ramp loading

This article deals with the visco-hyperelastic modelling approach for compressible polymer foam materials. Polymer foams can exhibit large elastic strains and displacements in case of volumetric compression. In addition, they often show significant rate-dependent properties. This material behaviour can be accurately modelled using the visco-hyperelastic approach, in which the large strain viscoelastic description is combined with the rate-independent hyperelastic material model. In case of polymer foams, the most widely used compressible hyperelastic material model, the so-called Ogden–Hill’s model, was applied, which is implemented in the commercial finite element (FE) software Abaqus. The visco-hyperelastic model is defined in hereditary integral form, therefore, obtaining a closed-form solution for the stress is not a trivial task. However, the parameter-fitting procedure could be much faster and accurate if closed-form solution exists. In this contribution, exact stress solutions are derived in case of uniaxial, biaxial and volumetric compression loading cases using ramp-loading history. The analytical stress solutions are compared with the stress results in Abaqus using FE analysis. In order to highlight the benefits of the analytical closed-form solution during the parameter-fitting process experimental work has been carried out on a particular open-cell memory foam material. The results of the material identification process shows significant accuracy improvement in the fitting procedure by applying the derived analytical solutions compared to the so-called separated approach applied in the engineering practice.

Lattice spring methods for arbitrary meshes in two and three dimensions

SummaryThe computation of elastic continua is widely used in today's engineering practice, and finite element models yield a reasonable approximation of the physically observed behaviour. In contrast, the failure of materials due to overloading can be seen as a sequence of discontinuous effects finally leading to a system failure. Until now, it has not been possible to sufficiently predict this process with numerical simulations. It has recently been shown that discrete models like lattice spring models are a promising alternative to finite element models for computing the breakdown of materials because of static overstress and fatigue. In this paper, we will address one of the downsides of current lattice spring models, the need for a periodic mesh leading to a mesh‐induced anisotropy of material failure in simulations. We will show how to derive irregular cells that still behave as part of a homogeneous continuum irrespectively of their shape and which should be able to eliminate mesh‐induced anisotropy. In addition, no restraints concerning the material stiffness tensor are introduced, which allows the simulation of non‐isotropic materials. Finally, we compare the elastic response of the presented model with present lattice spring models using mechanical systems with a known analytical stress field. Copyright © 2016 John Wiley & Sons, Ltd.

Geometrical Nonlinear Analysis of Plane Problems by Corotational Formulation

AbstractThis study deals with the geometric nonlinear analysis of the plane problem based on the corotational formulation. Both analytical solutions and hybrid stress functional will be utilized in the proposed technique. A quadrilateral four-node element with drilling degrees of freedom is proposed for the finite-element analysis. The corotational method is applied for the nonlinear behavior. In this way, small strains and rigid body motions can be separated. Based on analytical solution, the hybrid stress scheme is used in the local coordinates for small strains. By using Allman’s quadratic displacement, the boundary condition for this element is introduced. In this approach, added drilling degrees of freedom increase the accuracy and robustness of the element. Furthermore, the corotational formulas are written in the local and global coordinates system to derive the nonlinear relations. These equations were solved by using the arc-length algorithm. To investigate the accuracy and capability of the sugg...

Dynamic simulation of CO2-injection-induced fault rupture with slip-rate dependent friction coefficient

Poro-elastic stress and effective stress reduction associated with deep underground fluid injection can potentially trigger shear rupture along pre-existing faults. We modeled an idealized CO2 injection scenario, to assess the effects on faults in the first phase of a generic CO2 aquifer storage operation. We used coupled multiphase fluid flow and geomechanical numerical modeling to evaluate the stress and pressure perturbations induced by fluid injection and the response of a nearby normal fault. Slip-rate dependent friction and inertial effects have been taken into account during rupture. Contact elements have been used to take into account the frictional behavior of the rupture plane. We investigated different scenarios of injection rate to induce rupture on the fault, employing various fault rheologies. Published laboratory data on CO2-saturated intact and crushed rock samples, representative of a potential target aquifer, sealing formation and fault gouge, have been used to define a scenario where different fault rheologies apply at different depths. Nucleation of fault rupture takes place at the bottom of the reservoir, in agreement with analytical poro-elastic stress calculations, depending on injection-induced reservoir inflation and the tectonic stress scenario. For the stress state considered here, the first triggered rupture always produces the largest rupture length and slip magnitude, both of which correlate with the fault rheology. Velocity weakening produces larger ruptures and generates larger magnitude seismic events. Heterogeneous faults have been considered including velocity-weakening or velocity strengthening sections inside and below the aquifer, with the upper sections being velocity-neutral. Nucleation of rupture in a velocity-strengthening section results in a limited rupture extension, both in terms of maximum slip and rupture length. For a heterogeneous fault with nucleation in a velocity-weakening section, the rupture may propagate into the overlying velocity-neutral section, if the extent of velocity-weakening and associated friction drop are large enough.

Analytical modeling of residual stress formation in workpiece material due to cutting

Residual stress on surface layer of workpiece has significant effects on service life of parts, including fatigue strength and corrosion resistance. Metal cutting process is widely used on manufacturing parts, which will cause severe mechanical and thermal stress on the material and then result in the formation of residual stress on workpiece surface layer. A thorough analysis of residual stress formation due to cutting is essential to create parts with desired residual stress distribution. After analyzing stress field in workpiece based on previous work, the authors in this paper observe that stress of a point during cutting is time dependent, and then proposed that initial condition is mandatory in order to determine initial stress state in stress relaxation procedure. Then the authors propose a criterion to determine initial condition based on the calibration of surface residual stress. Using this criterion and calibration method, the residual stress in different depth of machined surface can be compute using the proposed analytical residual stress model, and the computed results are consistent with experiment results. Moreover, the computed results show that the residual stress component in cutting speed direction is higher than that in axial direction, which is consistent with the general conclusion in the existing literatures. The proposed concept of initial condition and its calibration method in stress relaxation promote the understanding of physical mechanism of residual stress formation in cutting, and can be extended to the situations where the evolution of flow stress in workpiece during cutting is similar with that of in orthogonal cutting.