Inelastic Work Research Articles

Effects of Amplitude Variations on Deformation and Damage Evolution in SnAgCu Solder in Isothermal Cycling

Although apparently simpler than in thermal cycling, the behavior of SnAgCu (SAC) solder joints in cyclic bending or vibration is not currently well understood. The rate of damage has been shown to scale with the inelastic work per cycle, and excursions to higher amplitudes lead to an apparent softening, some of which remains so that damage accumulation is faster in subsequent cycling at lower amplitudes. This frequently leads to a dramatic breakdown of current damage accumulation rules. An empirical damage accumulation rule has been proposed to account for this, but any applicability to the extrapolation of accelerated test results to life under realistic long-term service conditions remains to be validated. This will require a better understanding of the underlying mechanisms. The present work provides experimental evidence to support recent suggestions that the observed behavior is a result of cycling-induced dislocation structures providing for increased diffusion creep. It is argued that this means that the measured work is an indicator of the instantaneous dislocation density, rather than necessarily reflecting the actual work involved in the creation of the damage.

Brittle to Semibrittle Transition in Quartz Sandstone: Energetics

AbstractTriaxial compression experiments were conducted on a quartz sandstone at effective pressures up to 175 MPa and temperatures up to 900°C. Our experiments show a transition from brittle faulting to semibrittle faulting with an increase in both pressure and temperature. The yield behavior of samples deformed in the semibrittle regime follows a compactant elliptical cap at low strain, but evolves to a dilatant Mohr–Coulomb relationship with continued compaction. Optical microscopy indicates that semibrittle deformation involves cataclastic flow through shear‐enhanced compaction and grain crushing; however, transmission electron microscopy shows evidence for dislocation glide in limited portions of samples. To constrain the relative contribution of brittle and crystal plastic mechanisms, we estimate the partitioning of the inelastic work into the dissipation energy for microcracking, intergranular frictional slip, and dislocation glide. We conclude that semibrittle deformation is accommodated primarily by cataclastic mechanisms, with only a limited contribution from crystal plasticity. Mechanical data, acoustic emission records, and analysis of surface energy all indicate the activation of subcritical cracking at elevated temperature. Hence, we infer that the enhancement of subcritical cracking is responsible for the transition to semibrittle flow through promoting distributed grain‐scale fractures and millimeter‐scale shear bands. Subcritical cracking promotes the nucleation of microfractures at lower stresses, and the resulting decrease in flow stress retards the propagation of transgranular microfractures. Our study illuminates the important role of temperature on the micromechanics of the transition from brittle faulting to cataclastic flow in the Earth.

On the correlation between thermoelectricity and adhesive tool wear during blanking of aluminum sheets

Blanking is one of the most frequently used manufacturing processes. Its profitability strongly depends on the development of tool wear, since this determines the decisive factors of part quality, process stability and service intervals. Reducing the wear can only be achieved by gaining a sufficient understanding of the interactions which cause it. One important factor in this context is the temperature increase in the shear zone due to the partial dissipation of inelastic work. The resulting temperature gradient in the tool, combined with the electric contact between the workpiece and the tool, generates thermoelectric voltages and currents. The thermoelectric currents are suspected to have an especially strong influence on adhesive wear, but the mutual correlation between them during blanking operations is not clear. Therefore, several blanking examinations were carried out with the aluminum alloy 5083. The thermoelectric currents and voltages were measured instantaneously in the tool and their impact on the development of wear was shown by externally manipulating the occurring thermoelectricity.

Stability and serviceability of underground energy storage caverns in rock salt subjected to mechanical cyclic loading

Solution-mined caverns excavated in rock salt formations are recognized as the appropriate places for underground storage of energy in the form of compressed air and hydrogen. These caverns are subjected to different loading conditions during the construction and operation time. Therefore, it is essential to investigate the mechanical response of rock salt for all life stages of the cavern such as leaching phase, debrining process, first filling as well as cyclic loading operation. To achieve this goal, proper constitutive laws are required to describe the material behavior of rock salt at different time scales. In this paper, an elasto-viscoplastic-creep model is employed to predict the stress-strain relation around the cavern during the construction and cyclic operation phases. The proposed creep law is a modified version of Norton creep law which accounts for the time-dependent volumetric deformation. Additionally, a damage parameter which is dependent on the released inelastic work is included in the model to predict the material failure. To accomplish this, first, the material parameters of the employed model are determined using the relevant experimental data available in the literature. Then, the excavation process and cyclic loading operation of a typical salt cavern are numerically simulated. The stress paths around the cavern as well as the volume convergence, damage propagation and permeability changes are evaluated considering two loading scenarios, and finally, the allowable operating conditions for the simulated cavern are identified.

Effects of Strain Rate and Amplitude Variations on Solder Joint Fatigue Life in Isothermal Cycling

The behavior of lead-free solder alloys under realistic service conditions is still not well understood. Life prediction of solder joints relies on conducting accelerated tests and extrapolating results to service conditions. This can be very misleading without proper constitutive relations and without understanding the effects of cycling parameter variations common under realistic service conditions. It has been shown that the fatigue life depends on the inelastic work accumulation, independently of cycling-induced material property variations, which explains the breakdown of damage accumulation rules and allows the development of a modified Miner's rule. This paper discusses the interacting effects of strain rate and amplitude variations on solder joint fatigue life. Individual SnAgCu solder joints with two different Ag contents (SAC305 and SAC105) were tested in low cycling shear fatigue under single and varying amplitudes with different strain rates. Such a shear fatigue experiment allows the measurement of work accumulation and the evolution of solder deformation properties during cycling. The results showed that cycling with a lower strain rate at fixed amplitude causes more damage per cycle. Alternating between mild amplitude at a high strain rate and harsh amplitude at a low strain rate leads to ongoing increases in the rate of damage at the mild amplitude and thus relatively rapid failure. In comparing SAC305 with SAC105, the effect of strain rate on both alloys is almost the same, and SAC305 is still more fatigue resistant than SAC105 in varying amplitude cycling with any strain rate.

Neighborhood Effects, Peer Classification, and the Decision of Women to Work

We examine the influence of neighborhood peer effects on the decision of women to work using panel data that follows clusters of adjacent homes between 1985-1993. Modeling assumptions imply rank order restrictions that enable us to classify individuals into peer groups while identifying peer effects and underlying mechanisms. For women, peer effects influence labor supply in part because women appear to emulate the work behavior of nearby women with similar age children. For men, peer effects are mostly absent, consistent with inelastic work decisions. Geographically concentrated panel data are crucial for these estimates. Our approach could also be applied to other instances in which neighborhood peer effects are important.

МАТЕМАТИЧЕСКАЯ МОДЕЛЬ ПРОЦЕССА ДЕФОРМИРОВАНИЯ КРЕПИ ВЕРТИКАЛЬНОЙ ГОРНОЙ ВЫРАБОТКИ С УЧЕТОМ НАЧАЛЬНОЙ ПОРИСТОСТИ МАТЕРИАЛА И УПРУГОВЯЗКОПЛАСТИЧЕСКИХ СВОЙСТВ СЖАТОГО СКЕЛЕТА

The mathematical model describing process of deformation monolithic is constructed fix a vertical excavation for materials with porous structure which squeezed skeleton possesses at the same time elastic, viscous and plastic properties. Deformation of the porous environment under the influence of the set evenly distributed squeezing loadings is divided into two interconnected stages: elastic deformation of the porous environment and inelastic deformation of the squeezed matrix. A problem of finding of the intense deformed state fix vertical development with a circular form of cross section at each stage of deformation decides within the flat deformed state. Thus the effects connected by that development has final depth aren't considered. The ratios defining fields of tension and movements at the first stage of deformation are received. Dependence of the squeezing loadings at which initial porosity of material in all area fix is defined reaches zero value. At the second stage of process of deformation analytical expressions for finding of fields of tension and movements in elastic and plastic zones of deformation of the squeezed skeleton are removed, and the equation for determination of radius of elasto-plastic border is also received. As conditions of compatibility continuity conditions a component of tension and movements on elasto-plastic border, and also equality to zero plastic deformations on it got out. The assessment of influence on the size of limit of the section of environments of elastic and plastic deformation of initial porosity, hardening and a limit of fluidity of material is given. The asymptotic behavior of elasto-plastic border is shown over time. Graphic dependences a component of tension from coordinate are constructed at various values of size of initial solution of a time and other physicomechanical and geometrical parameters of material and a design. Keywords: porous materials, during the inelastic work of the squeezed skeleton, monolithic strengthening design, vertical development, the intense deformed state.

Correlation Between Solder Joint Fatigue Life and Accumulated Work in Isothermal Cycling

The fatigue behavior of solder joints in realistic service applications is still not well understood. Service life prediction based on conducting accelerated tests and extrapolating test results therefore involves a high potential for error. Understanding both the evolution of solder joint properties and the damage accumulation has proved to be critical to reliability modeling. Damage accumulation in isothermal cycling is shown to scale with the accumulated inelastic work even in complex cycling scenarios, so that the life of a solder joint ends upon accumulation of a given amount of work. Individual ball grid array solder joints were cycled in shear fatigue experiments with different load amplitudes and strain rates. The effects of load amplitudes and strain rates on the work accumulation and fatigue life were systematically addressed. The correlation between different loading scenarios and the accumulated work to failure was also discussed. The results showed that the accumulated work until the development of a major crack is constant regardless of the load amplitude. After that the accumulated work to failure is lower for larger load amplitudes. For some reason, a larger fraction of the work appears to be dissipating as heat at lower load amplitude, but only during crack growth. On the other hand, the strain rate affects the fraction of the work going to heat even before the development of a major crack.

Statistical Variations of Solder Joint Fatigue Life Under Realistic Service Conditions

Common damage accumulation rules fail to predict the fatigue life of solder joints under realistic service conditions where cycling amplitudes vary over time. A modification of Miner’s rule of linear damage accumulation has been proposed that accounts for effects of amplitude variations in, for example, the vibration of microelectronic assemblies with lead-free solder joints on the average or characteristic fatigue life. We are, however, obviously much more concerned with the first failure across a very large sample set. Prediction of, say, the first failure out of 10000 or a million would require the extrapolation of experimental failure distributions and the assumption of a shape of this distribution. Even qualitative comparisons of accelerated test results and their scatter should account for effects of amplitude variations. We have argued that the long-term life of solder joints in vibration or cyclic bending is limited by the accumulation of inelastic work, and that much can be learned from the low cycle fatigue behavior in shear. Individual ball grid array scale SAC305 and SAC105 solder joints were cycled in shear at room temperature with combinations of two different stress amplitudes. Relying on our modified Miner’s rule and the associated understanding of the effects of amplitude variations, we show that the statistical uncertainty in the fatigue life of solder joints under a specific set of realistic service conditions must be significantly greater than measured in fixed amplitude cycling tests. The predicted failure distribution was best fit by a Weibull distribution over a limited range, but we argue that the assumption of such a distribution is likely to be increasingly conservative when it comes to the prediction of earlier failure. Estimates are provided for the potential errors.

Damage Evolution in Lead Free Solder Joints in Isothermal Fatigue

The extrapolation and generalization of accelerated test results for lead free solder joints require the identification of a damage function that can be counted on to apply beyond the region of the test. Individual ball grid array (BGA) scale Sn3Ag0.5Cu (SAC305) solder joints were subjected to isothermal shear fatigue testing at room temperature and 65 °C. The resulting mechanical response degradation and crack behavior, including strain hardening, crack initiation, and propagation, were correlated with the inelastic work and effective stiffness derived from load–displacement hysteresis loops. Crack initiation was found to scale with the accumulated work, independently of cycling amplitude and strain rate. The subsequent damage rate varied slightly with amplitude.

Probability of observable flaw initiation at high cycle fatigue

Fatigue evolution during the flaw initiation phase and under high cycle fatigue conditions is described at the level of the representative volume element by the probability that this element fails in a given period of time under given stimulus. This probability is calculated as a function of the accumulated inelastic work density. For the calculation of small amounts of inelastic work in the quasi-elastic domain of materials, a microplasticity model is developed using the hypothesis of strain energy density equivalence and a statistical distribution of work concentration within the RVE. The model is successfully fitted to experimental data.

Calculation on Formation of Longitudinal Cracks in Reinforced Concrete Structures Taking into Account Nonlinearity of Concrete Straining

Actuality of study of the problem of longitudinal crack formation along reinforcement taking into account concrete nonlinear strain is pointed out in this article. It is known that at collaboration of steel and concrete plastic strains occur in concrete surrounding bar. Previously, in the work [6], solution on determination of splitting cracks at elastic work of concrete in the area of active bond to steel was proposed. In this article, theoretical solution on determination of longitudinal cracks both in supporting area and in the area of collaboration of steel and concrete between cracks along element is suggested. For establishing equations taking into account inelastic work of concrete, the theory of plastic properties suggested by G.A. Geniev was used. Component stress-strain relation is recorded by analogy with studies performed by G.A. Geniev and N.N. Malinin. At that, design dependencies are linked with normative factors of concrete, such as concrete ultimate tensile strength, Poisson ratio and concrete elastic strain ratio at stretching. All this allowed to significantly simplify the use of expressions offered.

Assessment of Solder Joint Fatigue Life Under Realistic Service Conditions

The behavior of lead-free solder alloys under complex loading scenarios is still not well understood. Common damage accumulation rules fail to account for strong effects of variations in cycling amplitude, and random vibration test results cannot be interpreted in terms of performance under realistic service conditions. This is a result of the effects of cycling parameters on materials properties. These effects are not yet fully understood or quantitatively predictable, preventing modeling based on parameters such as strain, work, or entropy. Depending on the actual spectrum of amplitudes, Miner’s rule of linear damage accumulation has been shown to overestimate life by more than an order of magnitude, and greater errors are predicted for other combinations. Consequences may be particularly critical for so-called environmental stress screening. Damage accumulation has, however, been shown to scale with the inelastic work done, even if amplitudes vary. This and the observation of effects of loading history on subsequent work per cycle provide for a modified damage accumulation rule which allows for the prediction of life. Individual joints of four different Sn-Ag-Cu-based solder alloys (SAC305, SAC105, SAC-Ni, and SACXplus) were cycled in shear at room temperature, alternating between two different amplitudes while monitoring the evolution of the effective stiffness and work per cycle. This helped elucidate general trends and behaviors that are expected to occur in vibrations of microelectronics assemblies. Deviations from Miner’s rule varied systematically with the combination of amplitudes, the sequences of cycles, and the strain rates in each. The severity of deviations also varied systematically with Ag content in the solder, but major effects were observed for all the alloys. A systematic analysis was conducted to assess whether scenarios might exist in which the more fatigue-resistant high-Ag alloys would fail sooner than the lower-Ag ones.

A unified model for metal failure capturing shear banding and fracture

Dynamic fracture of metals may be brittle or ductile depending on factors such as material properties, loading rate and specimen geometry. At high strain rates, a thermo plastic instability known as shear banding may occur, which typically precedes fracture.Experiments on notched plate impact show a ductile–brittle failure transition, where lower impact velocities lead to brittle behavior, while higher impact velocities lead to shear banding. For more complex problems such as armor penetration, both brittle fracture and shear banding have been observed in the same specimen, however, current failure models can either account for fracture or shear banding. For predictive numerical simulations of dynamic failure, it is thus crucial to account for both failure modes, since exclusion of either mode neglects important physics observed in experiments.In this work a thermodynamically consistent model which accounts for both shear banding and dynamic fracture and can thus capture both failure bodes at intermediate strain rates, is presented. The model consists of an elastic–viscoplastic material with strain hardening, strain rate hardening, and thermal softening. Fracture is modeled with the phase field method, for which a novel modification is presented here to account for the creation of fracture surfaces by inelastic work. Numerical examples are presented to illustrate the basic behavior of the model, and to compare it to three special cases: a damage free case, an isothermal case, and an isothermal case where the contribution of inelastic work to fracture is excluded.

Damage evolution and thermal coupled effects in inelastic solids

Mechanical degradation and ductile failure in metal forming operations can be successfully modelled using Continuum Damage Mechanics. In addition to elastic–plastic deformations, heat transfer affects material behaviour, imposing further effects upon damage evolution. This paper addresses modelling aspects and presents a numerical discussion of the coupled effects between ductile damage and temperature evolution. The heat transfer problem is formulated based on a transient heat conduction approximation, in which heat transfer at the free surfaces and heating due to dissipation of the inelastic work are accounted for. Damage is modelled using an elastic–plastic fully coupled approximation, in which differences in tensile and compressive stress states are included.

Thermo-mechanically coupled model of diffusionless phase transformation in austenitic steel

A constitutive model of thermo-mechanically coupled finite strain plasticity considering martensitic phase transformation is presented. The model is formulated within a thermodynamic framework, giving a physically sound format where the thermodynamic mechanical and chemical forces that drive the phase transformation are conveniently identifiable. The phase fraction is treated through an internal variable approach and the first law of thermodynamics allows a consistent treatment of the internal heat generation due to dissipation of inelastic work. The model is calibrated against experimental data on a Ni–Cr steel of AISI304-type, allowing illustrative simulations to be performed. It becomes clear that the thermal effects considered in the present formulation have a significant impact on the material behavior. This is seen, not least, in the effects found on forming limit diagrams, also considered in the present paper.

High-speed Impact of Two Identical Spheres Made of SUJ2

Impact of spheres is one of the basic problems in mechanical dynamics. The severity of impact is generally characterized by the Coefficient of Restitution. The authors have already dealt with direct central impacts of two identical spheres in low speed impact and revealed that strain rate sensitivity of material properties could not be ignored even in low speed impacts. Therefore, this paper deals with the direct central impact of two identical spheres in high speed range. The experiments of direct central impact of two spheres were conducted by using steel balls for use in journal ball bearing (JIS SUJ2). Two steel balls were impacted by air gun. The impact analyses considering the material properties as well as the strain rate effect have been carried out using Finite Element Method (F. E. M). Then the experimental and analytical results were compared and they matched well. The effect of strain rate takes an important role in high speed impact as well as in low speed impact. If the strain rate effect of the material properties was ignored, it results in overestimation of inelastic work during impact in the numerical simulation. Further, it was found that the experimental equation for the coefficient of restitution derived from the experiments in low speed impact was still effective in high speed range.

Dynamically driven phase transformations in heterogeneous materials. II. Applications including damage

A model, developed for heterogeneous materials undergoing dynamically driven phase transformations in its constituents, has been extended to include the evolution of damage. Damage is described by two mechanisms: interfacial debonding between the constituents and brittle failure micro-crack growth within the constituents. The analysis is applied to silicon carbide-titanium (SiC-Ti) unidirectional metal matrix composites that undergo the following phenomena: Ti has a yield stress of approximately 0.5 GPa and above a pressure of about 2 GPa undergoes a solid-solid phase transformation. The inelastic work from plastic dissipation contributes to the temperature and pressure rise in the Ti. SiC behaves elastically below a critical stress, above which it is damaged by microcrack growth. Finally, under tensile loading, the interface between Ti and SiC debonds according to an interfacial decohesion law. Each process is first examined independently in order to understand how its characteristic behavior is manifested in the stress-strain response of the composite. The complex interplay between loading states, viscoplasticity, damage, and solid-solid phase transformations is then studied at both the micromechanics and macromechanics levels.

Crack deflection in tungsten carbide based laminates

Laminate materials were manufactured by alternating WC–6%Co layers with equal thickness layers of one of the following compositions: WC–16%Co, WC–16%Ni, Co, a W–Ni–Fe alloy, and Ni. One composition used WC–6%Ni in place of WC–6%Co, layered with the W–Ni–Fe alloy. Flexure specimens, consisting of nine alternating layers, were fabricated with either WC–6%Co or WC–6%Ni on the tensile surface of the specimen. The ability of these composites to deflect or arrest cracks was investigated in four point bending. Two of the compositions, WC–6%Co/W–Ni–Fe and WC–6%Co/Ni, exhibited the ability to fail non-catastrophically, showing inelastic work of fracture as high as 34,400 J/m 2.

A method for the evaluation of design limits for structural materials in a cyclic state of creep

The purpose of the present paper is to demonstrate how the minimum theorems proposed in an accompanying paper (Ponter and Boulbibane, 2002) can be utilised in the prediction of the deformation and life assessment of structures subjected to cyclic mechanical and thermal loadings. The developed method, which is based upon bounding theorems and an associate programming method, the Linear Matching method, takes into account the changes in residual stress field occurring within a cycle. Although the solution provided a bound on the inelastic work, it also appears that generally the displacements predicted by this solution are smaller than those that would be predicted by the rapid cycle solution. By way of illustration a simple non-linear viscous model is adopted and a number of solutions are presented involving a Bree plate problem subjected to cyclic histories of load and temperature. An elastic follow-up factor is identified as a key design parameter for high temperature dwell periods.