Showing papers on "Hydrostatic stress published in 2003"

A new look at the atomic level virial stress: on continuum-molecular system equivalence

Abstract: The virial stress is the most commonly used definition of stress in discrete particle systems. This quantity includes two parts. The first part depends on the mass and velocity (or, in some versions, the fluctuation part of the velocity) of atomic particles, reflecting an assertion that mass transfer causes mechanical stress to be applied on stationary spatial surfaces external to an atomic‐particle system. The second part depends on interatomic forces and atomic positions, providing a continuum measure for the internal mechanical interactions between particles. Historic derivations of the virial stress include generalization from the virial theorem of Clausius (1870) for gas pressure and solution of the spatial equation of balance of momentum. The virial stress is stress‐like a measure for momentum change in space. This paper shows that, contrary to the generally accepted view, the virial stress is not a measure for mechanical force between material points and cannot be regarded as a measure for mechanical stress in any sense. The lack of physical significance is both at the individual atom level in a time‐resolved sense and at the system level in a statistical sense. It is demonstrated that the interatomic force term alone is a valid stress measure and can be identified with the Cauchy stress. The proof in this paper consists of two parts. First, for the simple conditions of rigid translation, uniform tension and tension with thermal oscillations, the virial stress yields clearly erroneous interpretations of stress. Second, the conceptual flaw in the generalization from the virial theorem for gas pressure to stress and the confusion over spatial and material equations of balance of momentum in theoretical derivations of the virial stress that led to its erroneous acceptance as the Cauchy stress are pointed out. Interpretation of the virial stress as a measure for mechanical force violates balance of momentum and is inconsistent with the basic definition of stress. The versions of the virial‐stress formula that involve total particle velocity and the thermal fluctuation part of the velocity are demonstrated to be measures of spatial momentum flow relative to, respectively, a fixed reference frame and a moving frame with a velocity equal to the part of particle velocity not included in the virial formula. To further illustrate the irrelevance of mass transfer to the evaluation of stress, an equivalent continuum (EC) for dynamically deforming atomistic particle systems is defined. The equivalence of the continuum to discrete atomic systems includes (i) preservation of linear and angular momenta, (ii) conservation of internal, external and inertial work rates, and (iii) conservation of mass. This equivalence allows fields of work‐ and momentum‐preserving Cauchy stress, surface traction, body force and deformation to be determined. The resulting stress field depends only on interatomic forces, providing an independent proof that as a measure for internal material interaction stress is independent of kinetic energy or mass transfer.

A constitutive model for creep of lead-free solders undergoing strain-enhanced microstructural coarsening: A first report

Abstract: Lead-free solder joints in microelectronic applications frequently have microstructures comprising a dispersion of intermetallic particles in a Sn matrix. During thermomechanical cycling (TMC) of the solder joint, these particles undergo strain-enhanced coarsening, resulting in a continuously evolving, creep behavior. Because the extent of coarsening is dependent on the stress/strain state, which is dependent on the location within a joint, it is important that creep models used in joint-life prediction incorporate these effects. Here, an approach for incorporating the effect of in-situ second-phase particle coarsening in a dislocation-creep model applicable to lead-free solder alloys is proposed. The formulation, which can be expressed in a closed analytic form following some simplifications, incorporates the effects of both static- and strain-enhanced coarsening and accounts for the effects of inelastic-strain history and hydrostatic constraint. Predictions of coarsening based on the model agreed reasonably well with experimentally observed trends. Because of its simplicity, the microstructurally adaptive creep model proposed here can be easily incorporated in current finite-element codes for joint behavior simulation.

Theoretical modelling of the effect of plasticity on reverse transformation in superelastic shape memory alloys

Abstract: Stimulated by recent experimental results on superelastic NiTi shape memory alloy, a theoretical study is carried out to quantify the effect of plasticity on stress-induced martensite transformation, using a constitutive model that combines phase transformation and plasticity. A constraint equation is introduced to quantify the phenomenon of the stabilisation of plasticity on stress-induced martensite. The stabilised martensite volume fraction is determined by the equivalent plastic strain. The transformation constitutive model is adopted from a generalised plastic model with Drucker–Prager type phase transformation functions, which are pressure sensitive, while the plasticity is described by the von Mises isotropic hardening model. The martensite volume fraction is chosen as the internal variable to represent the transformation state and it is determined by the consistency transformation condition. An approach to calibrate model parameters from uniaxial tensile tests is explored, as well as the issue of elastic mismatch between austenite and martensite is discussed. Based on the proposed constitutive model, the influence of hydrostatic stress on transformation is examined. As an example of application, this new constitutive model is employed to numerically study the transformation field and the plastic deformation field near a crack tip.

Numerical simulation of stress evolution during electromigration in IC interconnect lines

Abstract: A finite element simulation of stress evolution in thin metal film during electromigration is reported in this paper. The electromigration process is modeled by a coupled diffusion- mechanical partial differential equations (PDEs). The PDEs are implemented with a plane strain formulation and numerically solved with the finite element (FE) method. The evolutions of hydrostatic stress, each component of the deviatoric stress tensor, and Von Mises' stress were simulated for several cases with different line lengths and current densities. Two types of displacement boundary conditions are considered. The simulation results are compared with Korhonen's analytical model and Black and Blech's experimentalesults.

Fluorinert as a pressure-transmitting medium for high-pressure diffraction studies

Abstract: Fluorinert is a liquid pressure-transmitting medium that is widely used in high-pressure diffraction work. A systematic study of five different fluorinerts was carried out using single-crystal x-ray diffraction in a diamond-anvil cell in order to determine the pressure range over which they provide a hydrostatic stress state to the sample. It was found that none of the fluorinerts studied can be considered hydrostatic above 1.2 GPa, a lower pressure than reported previously.

Microplane constitutive model for porous isotropic rocks

Abstract: The paper deals with constitutive modelling of contiguous rock located between rock joints. A fully explicit kinematically constrained microplane-type constitutive model for hardening and softening non-linear triaxial behaviour of isotropic porous rock is developed. The microplane framework, in which the constitutive relation is expressed in terms of stress and strain vectors rather than tensors, makes it possible to model various microstructural physical mechanisms associated with oriented internal surfaces, such as cracking, slip, friction and splitting of a particular orientation. Formulation of the constitutive relation is facilitated by the fact that it is decoupled from the tonsorial invariance restrictions, which are satisfied automatically. In its basic features, the present model is similar to the recently developed microplane model M4 for concrete, but there are significant improvements and modifications. They include a realistic simulation of (1) the effects of pore collapse on the volume changes during triaxial loading and on the reduction of frictional strength, (2) recovery of frictional strength during shearing, and (3) the shear-enhanced compaction in triaxial tests, manifested by a deviation from the hydrostatic stress- strain curve. The model is calibrated by optimal fitting of extensive triaxial test data for Salem limestone, and good fits are demonstrated. Although these data do not cover the entire range of behaviour, credence in broad capabilities of the model is lend by its similarity to model M4 for concrete-an artificial rock. The model is intended for large explicit finite-element programs.

Effect of Applied Hydrostatic Stress on the Hydration of Portland Cement and C3S

Abstract: The hydration behaviour of Portland cement and C3S with and without stress applied were studied at water to solid (w/s) ratios of 0·35 and 0·50. A greater degree of hydration and denser microstructure were obtained with hydration under stress at an early stage of the hydration process, normally within 48 h. This observation was confirmed by the determination and comparison of CH content, TGA results, surface area and pore-size distribution of samples hydrated with and without applied stress. Environmental SEM examination provided the evidence that microcracking occurred only in samples hydrated under applied stress. It is suggested that this facilitated the migration of water into the protective layers (around cement grains) that form in the early stages of hydration accelerating the reaction with the unreacted cores of the particles.

Deformation behavior of an epoxy resin subject to multiaxial loadings. Part I: Experimental investigations

Abstract: The mechanical behavior of an epoxy resin (Epon 826) was studied by performing a series of tests on thin-walled tubular specimens. These tests deal with different aspects of the mechanical behavior of this epoxy resin. The deformation behavior, such as viscoelastic behavior, hydrostatic stress effect, multiaxial behavior and loading path effect, was investigated. It was found that the Epon 826 epoxy resin is a highly nonlinear viscoelastic material. The effect of hydrostatic pressure on the deformation behavior of this epoxy is not significant. However, it shows different tensile and compressive deformation behavior. The loading path was found to have an observable effect on the deformation response of this epoxy, especially in the high stress/strain range.

Stress softening of elastomers in hydrostatic tension

Abstract: This study is concerned with inelastic effects of non-reinforcing carbon-black filled elastomers when subjected to periodic hydrostatic loading-unloading cycles in tension. During cyclic testing of sufficient magnitude, a critical state may be reached where microcavities suddenly grow inside the rubber, possibly initiated at sites of internal imperfections. As a result of cavitation damage the tensile bulk modulus in the natural configuration is reduced. A series of hydrostatic tension tests are performed at room temperature to provide new insight into the progressive deterioration of the bulk stiffness. We define dilatational stress softening as a phenomenon where the hydrostatic stress on unloading and subsequent submaximal reloading is significantly less than that on primary loading for the same volumetric strain. Dilatational stress softening during initial loading cycles and the permanent volumetric change upon unloading are not accounted for when the mechanical properties are represented in terms of a strain-energy function, i.e. if the material is modelled as hyperelastic. In this paper a constitutive model is derived to include the progressive reduction of the bulk stiffness and the permanent volumetric change of carbon-black filled elastomers subjected to quasi–static loading. The basis of the model is the theory of pseudo-elasticity, which including a softening variable modifies the dilatational strain energy function. An acceptable correspondence between the theory and the data is obtained.

Reduction in the Number of Independent Parameters for the Tsai—Wu Tensor Polynomial Theory of Strength for Composite Materials:

Abstract: It is shown that the number of required parameters for the Tsai-Wu tensor polynomial strength criterion for fiber composites can be reduced from seven to five for composite materials that do not fail under practical levels of either hydrostatic or transverse pressure. For these materials, the interactive strength parameters can be defined in terms of their commonly measured uniaxial or noninteracting strength parameters, thereby eliminating the need to conduct combined stress tests. The derived parameters are given by F12 = − F11/4 F23 = − F22 These parameters fall within the stability limits of the theory, yet they lead to open failure surfaces in the compressive stress quadrant. The assumptions used to derive the interactive parameters were supported by measurements that showed typical carbon fiber composites did not fail under significant levels of hydrostatic pressure or unequal transverse compression. Comparison with previous work where the interactive parameters had been determined from combined str...

In situ measurement of reinforcement stress in an aluminum-alumina metal matrix composite under compressive loading

Abstract: The phenomena of stress partitioning between the matrix and the reinforcements in a loaded metal matrix composite dominate the mechanical behavior of these materials. Numerical models for estimating the stress in the matrix and the reinforcement under load are well developed. However, direct experimental measurements (e.g. measurement of reinforcement stress) are more difficult and have not been widely undertaken at present. The objective of the present work was to measure in situ the hydrostatic stress in the ceramic reinforcements in a continuously reinforced metal matrix composite loaded under transverse compression (i.e. loading perpendicular to the fiber axis). A single crystal sapphire reinforced AA6061 matrix model composite (reinforcement volume fraction ∼10%) was used for the measurements, which were undertaken at applied strains of 5, 10 and 20%. The stress measurements utilized the piezo-spectroscopic property of the Cr 3+ ions which were present as impurities in the sapphire reinforcements. The compressive deformation of the composite was simulated using an isotropic, plane strain finite element model. The reinforcement hydrostatic stress estimates from the isotropic FEM model were suitably modified to incorporate the effects of anisotropy in properties of the sapphire single crystal. The mean values of the experimental measurements of reinforcement hydrostatic stress matched well with the numerical estimates.

Binding energy and density of shallow impurity states in GaAs–(Ga, Al)As quantum wells: effects of an applied hydrostatic stress

Abstract: The effects of hydrostatic stress on the binding energy and the density of shallow-donor and shallow-acceptor impurity states in a GaAs–(Ga, Al)As quantum well are calculated using a variational procedure within the effective-mass approximation. Results are for different well widths and hydrostatic stresses, as a function of the impurity position along the growth direction of the structure. We have found that in the low-pressure regime the binding energy changes linearly for both donor and acceptor impurities, independently of the sizes of the well. However, for high pressures (greater than 13.5 kbar) this is valid for acceptors but not for donors due to the Γ-X crossover. We have shown that there are two special structures in the density of impurity states, one associated with on-centre and the other with on-edge impurities. Also, we have observed that the density of impurity states depends strongly on the applied hydrostatic stress.

Effects of hydrostatic stress on the density of impurity states and donor-related optical absorption spectra in GaAs–(Ga,Al)As quantum wells

Abstract: The effects of hydrostatic stress on the density of donor impurity states and donor-related optical absorption spectra in a GaAs–(Ga,Al)As quantum well are investigated. The shallow-donor binding energy for different well widths and different values of the hydrostatic stress has been calculated. It has been found that for wider well widths the binding energy increases slowly with hydrostatic stress contrary to the behavior of the binding energy for wells with smaller widths. In particular, it has been found that the binding energy does not change appreciably with the impurity position when the width of the well is small and for large values of hydrostatic stress. Two structures in both the density of states and the optical absorption spectra, associated with impurities located close to the center and to the edges of the structure, are obtained. Also, it has been observed that the density of states and the optical absorption spectra depend strongly on the applied hydrostatic stress.

Investigation on flip chip solder joint fatigue with cure-dependent underfill properties

Abstract: A cure-dependent viscoelastic constitutive relation is applied to describe the curing process of epoxy underfill in flip chip on board (FCOB). The chemical shrinkage of the epoxy underfill during the curing process is applied via incremental initial strains. Thus, the stress and strain build-up, caused by the simultaneous increase in stiffness and shrinkage during the curing process, are simulated. Accelerated fatigue experiments with thermal cycles from -55/spl deg/C to 80/spl deg/C are carried out for a specially designed flip chip configuration. Based on the obtained curing induced initial stress and strain fields, thermo-mechanical predictions are presented for the test carriers. The solder bumps are modeled with temperature dependent visco-plastic properties. A combination of a Coffin-Manson based fatigue relation and a creep fatigue model is used as fatigue failure criterion. The results show that the finite element method (FEM)-based fatigue life predictions match better with the experimental results, if the curing induced initial stress state is taken into account. The effect of cure-induced hydrostatic stress is qualitatively investigated by using a modified energy partitioning damage model with a correction factor in the creep damage formulation to take into account the effect of the hydrostatic stress.

Observations of barrelling in aluminium solid cylinders during cold upsetting using different lubricants

Abstract: This work deals with experimental derivation of friction factor values m for different lubricants and their combinations using ring compression tests and standard ring geometry 6:3:2. The values thus obtained were used to generate data on cold upsetting of annealed aluminium solid cylinders. Two aspect ratios and three strain levels were considered for the upsetting experiment. Calculations were made with the assumption that the curvature of the bulge followed a circular arc. The measured radius of curvature of the bulge was shown to agree with the value calculated using experimental data. Relationships were established between various bulge parameters such as new hoop strain, hydrostatic stress, geometrical shape factor, and stress ratio factor, and the friction factor m of different lubricants.

Criteria for cavitation of rubber particles: Influence of plastic yielding in the matrix

Abstract: The principles of the cavitation criteria for rubber particles in polymeric matrices are briefly reviewed. Although these criteria are based on a linear elastic analysis, it is shown that it is possible to extend them to take into account the elastic-plastic behaviour of the matrix. In this objective, the representative volume element of a periodic material was meshed and computations were performed using a finite element method. The results reported in this paper focus mainly on cavitation under uniaxial tension and examine the influence on the hydrostatic stress in the rubber particles of different parameters such as the volume fraction of rubber, the plastic behaviour of the matrix or the ratio of the elastic moduli. In all cases, plastic yielding in the matrix leads to saturation of the hydrostatic stress in the rubber phase. It is also shown that the history of cavitation barely influences the progression of plasticity in the matrix.

Stress distributions and energetics in the laterally ordered systems of buried pyramidal Ge/Si(001) islands: An atomistic simulation study

Abstract: Stress distributions in laterally ordered arrays of coherent Ge islands of shallow pyramidal shape buried in a Si(001) matrix are studied via large-scale atomistic simulations, using Stillinger-Weber Ge/Si systems as a vehicle. The existence of tensile hydrostatic stress regions is observed on the spacer surface, above the buried islands. Our previously reported finding [M. A. Makeev and A. Madhukar, Phys. Rev. Lett. 86, 5542 (2001)] that the hydrostatic stress at the spacer layer surface above the island apex is nearly inversely proportional to the spacer layer thickness is validated by a comparison with experimental data. The lateral variations of the hydrostatic stress on the spacer layer surface show ``bell-shape'' profiles, with the effective size of the tensile regions above the island apex varying as a power law with the spacer layer thickness, with the power exponent being greater than 1. Studies of the energetics of twofold stacks of island systems show that the elastic interaction energy between the islands is minimized for the vertically aligned geometry. The spacer layer thickness dependence of the hydrostatic and biaxial stress field distributions in the interior of the Si(001) matrix are presented as these define the behavior of the electron and hole three-dimensional confinement potentials that determine the electronic properties of the pyramidal island quantum dots.