Viscoplastic Stress Research Articles

Modelling with viscous and viscoplastic materials under combining and separating flow configurations

Combining and separating incompressible flow of Newtonian and inelastic Herschel–Bulkley fluids is studied numerically employing a semi-implicit Taylor–Galerkin pressure-correction algorithm, where steady solutions are obtained through a transient finite element procedure. The influence of inertia and fluid rheology is analysed on flow patterns, velocity fields and pressure drops for various flow configurations, with fixed geometric gap width that stimulates the merging and splitting in the flow. For Newtonian fluids and at larger levels of inertia, the appearance of vortices was observed, with an increase in velocity differences and pressure drops across the channel. In this case, the numerical procedure was verified with good agreement against previous numerical and experimental observations. To extend the consideration to non-Newtonian inelastic materials, the material rheological characteristics were approximated with the use of the Herschel–Bulkley fluid model, incorporating the Ostwald–de Waele power-law model and viscoplastic yield stress. Findings for unyielded power-law fluids reveal slight increase in the size of the vortices as power index (m) was decreased. Variation of the consistency index (k) shows strong influence on the streamline patterns with a rapid increase in the vortex formation as k was decreased. For Bingham model solutions, devoid of shear-thinning and increasing yield stress, a higher value of Reynolds number is required for equivalent levels of vortex formation; also one observes the appearance of yielded and unyielded regions. Under Herschel–Bulkley modelling, there was little change noted in the kinematics, but some was apparent in rheological response. Once more, observations reveal the tendency to eliminate vortices at larger yield stress levels, with the appearance of unyielded regions.

The Boundary Element Method Applied to Visco-Plastic Analysis

This paper presents a new formulation of the Dual Boundary Element Method to visco-plastic problems in a two-dimensional analysis. Visco-plastic stresses and strains around the crack tip are obtained until the visco-plastic strain rate reaches the steady state condition. A perfect visco-plastic analysis is also carried out in linear strain hardening (H’=0) materials. Part of the domain, the part that is susceptible to yield is discretized into quadratic, quadrilateral continuous cells. The loads are used to demonstrate time effects in the analysis carried out. Numerical results are compared with solution obtained from the Finite Element Method (FEM).

Simulation of Stresses during Casting of Binary Magnesium-Aluminum Alloys

A viscoplastic stress model is used to predict contraction forces measured during casting of two binary Mg-Al alloys. Force measurements from castings that did not hot tear, together with estimates from data found in the literature, are used to obtain the high-temperature mechanical properties needed in the stress model. In the absence of hot tearing, the simulation results show reasonably good agreement with the measurements. It is found that coherency of the semisolid mush starts at a solid fraction of about 0.5 and that the maximum tensile strength for the Mg-1 and 9 wt pct Al alloys at their final solidification temperatures is 1.5 and 4 MPa, respectively. In the presence of hot tearing, the measured stresses are generally overpredicted, which is attributed to the lack of a fracture model for the mush. Based on the comparison of measured and predicted stresses, it is also shown that coupling of the stress model to feeding flow and macrosegregation calculations is needed in order to accurately predict stresses in the presence of hot tearing.

Seismicity patterns and variation of the frequency-magnitude distribution of microcracks in salt

SUMMARY We analyse the statistical behaviour of about 1.6 E+6 high-frequency, acoustic emission events recorded in a rock mass in a salt mine over a 3-yr time period. The rock mass is directly located above a cavity that is backfilled during a 6 months time interval within the investigated time period. The rock mass consists mainly of rock salt with large anhydrite blocks in it. Location, event rate and strength of the events are studied in 3 months time intervals before, during, and after the backfilling process. We observe strong temporal and spatial changes of event rate and b-values that correlate with the structural features and the loading of the rock mass: two regions with a b-value maximum above the cavity start developing with the beginning of backfilling. They reach extremely high b-values of up to b= 3.18 and are located at the transition zones between the different materials. The b-value ratio in anhydrite and rock salt anti-correlates during times of equilibrium and fast loading. We conclude that b-value mapping during transient loading is useful to investigate structural heterogeneities in the rock volume, whereas b-values during equilibrium phases better indicate the level of ambient stress and visco-plastic stress relaxation in the salt rock.

An indentation system for determination of viscoplastic stress–strain behavior of small metal volumes before and after irradiation

The development of fusion materials for the first wall in future fusion reactors requires methods for the investigation of irradiation effects on the mechanical properties of materials which are only available in small volumes. Depth and force reading hardness measurement (or indentation) is one of the candidates that have the potential to extract valuable information on the stress–strain behavior of a material. A modified commercial indentation device installed in a hot cell of a fusion materials laboratory (FML) in combination with a neural network based analysis method allows identifying the material parameters of a unified viscoplasticity model with nonlinear isotropic and kinematic hardening from small metal specimens. By investigation of the same material before and after irradiation the method provides the possibility to separate irradiation effects on modulus, hardening and viscous behavior.

Stress intensity solutions of thermal fatigue induced cracks in a thin plate hot spot using LEFM and finite element analysis

The chief aim of this research is to marry elegant LEFM methods, published material data, and a commercially available FEA code together to represent complex crack growth and arrest phenomena in a thin plate with residual viscoplastic stresses. Additionally, representing the variation in these stress intensities across a thermal gradient to include ultimate fatigue crack arrest is of specific interest. To the authors’ knowledge, modeling fracture mechanics due to these hot spots with viscoplastic residual stresses using LEFM methods on this type structure has not been achieved. With these stress intensities, predictions are to be made regarding not only the FCG rate but the probable radial arrest distance of a nominal crack from the hot spot center.

A finite element solution scheme for an elastic–viscoplastic soil model

A Newton–Raphson solution scheme with a stress point algorithm is presented for the implementation of an elastic–viscoplastic soil model in a finite element program. Viscoplastic strain rates are calculated using the stress and volumetric states of the soil. Sub-increments of time are defined for each iterative calculation of elastic–viscoplastic stress changes so that their sum adds up to the time increment for the load step. This carefully defined ‘iterative time’ ensures that the correct amount of viscoplastic straining is accumulated over the applied load step. The algorithms and assumptions required to implement the solution scheme are provided. Verification of the solution scheme is achieved by using it to analyze typical boundary value problems.

Viscoplastic Stress Analysis of Shrinkage and Warping in Graded Layers during Sintering

Viscoplastic Stress Analysis of Shrinkage and Warping in Graded Layers during Sintering

Thermomechanical Durability of High I/O BGA Packages

Efficient modeling strategies are developed to study thermomechanical durability of high I/O Ball Grid Array (BGA) packages, in order to facilitate virtual qualification and accelerated testing of component designs. A viscoplastic stress analysis technique is developed where the critical solder joint(s) (joint(s) where failure first occurs) are modeled in detail with a multi-domain Rayleigh-Ritz (MDRR) methodology while the load-sharing offered by noncritical joints is modeled with a simplified compact model. This hybrid technique is used to study the behavior of solder interconnects in selected Ball Grid Array (BGA) package under thermal cycling environments. Parametric studies are conducted to determine the optimal scheme for allocating a critical number of solder joints to the MDRR model, and the remaining non-critical joints to the compact models. Damage calculations are made with the Energy Partitioning Solder Durability model and cycles-to-failure predictions are compared with both finite element model predictions as well as experimental failure data provided by CALCE EPSC sponsors. Parametric studies on change in solder joint durability with interconnect volume are also discussed in this paper.

Hygro-Mechanical Durability of Underfilled Flip-Chip-on-Board (FCOB) Interconnects

The effect of constant temperature and humidity environments on the durability of interconnects in underfilled Flip-Chip-on-Board (FCOB) assemblies is investigated in this study. Polymeric underfills, used to enhance thermomechanical durability of the interconnects, are found to create new failure modes due to hygromechanical swelling. Based on experimental observations, the failure mechanism is hypothesized to be cracking of intermetallics, which have weakened due to thermal aging. Pseudo 3-D finite element analyses are conducted to quantify the moisture absorption and diffusion through the polymeric underfill, and the resulting hygromechanical viscoplastic stress history. The simulations are combined with accelerated aging tests to assess in-service durability in hot, humid environments. Model predictions are compared with results of accelerated life tests available in the literature.

Guidelines to select underfills for flip chip on board assemblies and compliant interposers for chip scale package assemblies

The effect of thermomechanical properties of underfill and compliant interposer materials, such as coefficient of thermal expansion (CTE) and stiffness (Young's modulus) on reliability of flip chip on board (FCOB) and chip scale packages (CSPs) under thermal cycling stresses is investigated in this study. Quasi-three-dimensional viscoplastic stress analysis using finite element modeling (FEM) is combined with an energy partitioning (EP) model for creep-fatigue damage accumulation to predict the fatigue durability for a given thermal cycle. Parametric FEM simulations are performed for five different CTEs and five different stiffnesses of the underfill and compliant interposer materials. The creep work dissipation due to thermal cycling is estimated with quasi 3-D model, while 3-D model is used to estimate the hydrostatic stresses. To minimize the computational effort, the 3-D analysis is conducted only for the extreme values of the two parameters (CTE and stiffness) and the results are interpolated for intermediate values. The results show that the stiffness of the underfill material as well as the CTE play important role in influencing the fatigue life of FCOB assemblies. The fatigue durability increases as underfill stiffness and CTE increase. In the case of compliant interposers, the reverse is true and durability increases as interposer stiffness decreases. Furthermore, the interposer CTE affects the fatigue durability more significantly than underfill CTE, with durability increasing as CTE decreases. The eventual goal is to define the optimum design parameters of the FCOB underfill and CSP interposer, in order to maximize the fatigue endurance of the solder joints under cyclic thermal loading environments.

Spatial thermoviscoplastic problems

Methods and results of studies of the three-dimensional viscoplastic stress–strain state of engineering structures under thermomechanical loading are presented. The following classes of thermoviscoplastic problems are considered: axisymmetric problems, nonaxisymmetric problems for bodies of revolution, three-dimensional problems for bodies of arbitrary shapes, and three-dimensional problems for anisotropic bodies of revolution

Thermomechanical Durability Analysis of Flip Chip Solder Interconnects: Part 1—Without Underfill

A generalized multi-domain Rayleigh-Ritz (MDRR) approach developed by Ling and Dasgupta (1995), is extended in this paper, to obtain the stress field in flip chip solder interconnects, under cyclic thermal loading. Elastic, Plastic and time-dependent visco-plastic analysis is demonstrated on flip chip solder interconnects. The method has been applied to other surface-mount interconnects in the past such as J-lead (Ling and Dasgupta, 1996a) and ball-grid joints (Ling and Dasgupta, 1997). The analysis results for the J-lead and ball grid joints have confirmed that the MDRR technique is capable of providing stress-strain hysteresis with adequate accuracy, at a fraction of the modeling effort required for finite element model generation and analyses. Nonlinear viscoplastic stress analysis results for flip chip interconnects without underfill are presented in this paper. The fatigue endurance of the solder joints is assessed by combining results from this stress analysis model with an energy-partitioning damage model (Dasgupta et al., 1992). The life predicted by the analytical damage model is compared with experimental results.

A Stress Model for Multiobjective Design Optimization of a Power Electronic Module*

One of the most common failure mechanisms in power electronic modules is fatigue of the die attach due to shear stresses created by a mismatch in the thermal expansion of the die and the substrate. This paper presents a shear stress model in the die attach that incorporates viscoplastic stress components that become more important at the elevated temperatures in power electronic modules. The paper also describes a multiobjective design optimization approach utilizing a genetic algorithm for minimizing the chance of module failure, not only by die attach fatigue, but also by other competing failure mechanisms such as overheating of the chip, die fatigue, and substrate fatigue. It is shown that a successful power electronic module (PEM) design must balance the tendencies to fail by these competing failure mechanisms. Several tradeoff solutions in PEM design are presented for different size and material choices for a PEM design.

Development of non‐iterative self correcting solution (NONSS) method for the viscoplastic analysis with the Chaboche model

In a finite element program for the viscoplastic analysis with Chaboche model, the non-iterative and self-correcting solution (NONSS) method, proposed by Tanaka and Miller has been implemented. Towards this, the necessary Jocobian coefficients have been derived analytically for the sopisticated 23-parameter Chaboche viscoplastic model and finally the tangent modulus matrix D* and the residual viscoplastic stress vector ξ* have been obtained in order to determine the global tangent stiffness matrix as well as pseudo-load vector. The computational efficiency of this method is demonstrated by solving a variety of bench mark problems over a wide range of rate sensitivity domain under complex monotonic and cyclic loading histories relevant to fast breeder reactor applications. An improved Self-Adaptive Eulers Forward (SAEF) method is taken as the reference for comparision. © 1998 John Wiley & Sons, Ltd.

A Nonlinear Multi-Domain Stress Analysis Method for Surface-Mount Solder Joints

Solder joint fatigue failures are a potential reliability hazard in surface-mount electronic packages under cyclic thermal loading environment. Proper design and reliability assessment are thus crucial to ensure the fatigue endurance of the electronic packages. Accurate modeling of the stress and strain fields within the solder joint under cyclic thermal loading condition is of extreme importance since ultimately, a reasonable fatigue life estimation depends not only on a appropriate fatigue model, but more fundamentally, on accurately predicted stress and strain fields. Modeling stress and strain fields in solder joint in surface-mount electronic packages have never been an easy task since solder undergoes elastic, plastic and time dependent creep during each loading and unloading cycle. Some of the existing closed-form stress analysis models tend to oversimplify this complicated viscoplastic stress state, thus failing to give a reasonable prediction of the solder joint fatigue endurance. Extensive finite element analyses require prohibitive investment in terms of the analysis time and analyst expertise, especially when full scale elastic, plastic and creep analyses are performed. A generalized multi-domain approach proposed earlier by the authors is further developed in this paper to obtain the stress and strain fields in J-leaded surface-mount solder joint undergoing elastic-plastic deformation, under cyclic thermal environment (Ling et al., 1995). The Rayleigh-Ritz energy method based on a multi-field displacement assumption is used. In a previous paper (Ling et al., 1995), the results for analysis within elastic region had been demonstrated and were proved to be in agreement with finite element analysis. In this paper we further develop the methodology into plastic deformation region. Hysteresis loops for both the global and the local CTE mismatch problem can finally be generated. Results for two-dimensional elastic-plastic analysis are presented in the current paper. Creep deformation can be further modeled with this scheme by using time-stepping incremental techniques, and will be presented in a future paper. The final goal of this research is to predict the stress, strain and energy density distributions in the solder joint with reasonable accuracy. The fatigue assessment of the solder joint can then be performed by combining results from this stress analysis model with an appropriate damage model, for example, the energy-partitioning fatigue model (Dasgupta et al., 1992).

The viscoplastic mixed boundary element method for an axisymmetric body due to arbitrary distributed load

The viscoplastic mixed boundary element method (VM-BEM) for an axisymmetric body due to arbitrary distributed load is proposed in terms of displacement integral equations. The displacements, tractions and viscoplastic stresses, including both symmetric and antisymmetric terms in boundary and internal displacement integral equations, are expanded into harmonic functions along the tangential direction of an axisymmetric body. The displacement integral equations and viscoplastic iteration schemes (explicit scheme, implicit scheme and implicit trapezoidal scheme) are introduced. The advantage of a mixed BEM is that the singularity of the integral equation at internal points is one order lower than that of pure BEM. Numerical examples are given in a later section.

Simulation of Structural Change in HAZ and Viscoplastic Stresses during Welding Process of SUS304 Steel.

In this paper, the structural change in the heat-affected zone during welding of an SUS-304 plate is investigated in relation to the reverse transformation from work-induced martensite to austenite. The transformation occurs in HAZ and affects the mechanical properties of the welded component. The kinetic equations for the transformation are evaluated by comparing the calculated structural change with experimental results. The stress/strain analysis of the welding process is conducted based on the viscoplastic constitutive equations associated with the structural change. Emphasis is placed on the structural change in HAZ and its analysis in the simulation of the welding process, and then the properties of the welded component are also discussed using the calculated results of distributions of stresses and structures.

A unified viscoplastic boundary element approach for residual stress analysis of axisymmetric bodies under axisymmetric and non-axisymmetric loads

This paper presents a unified viscoplastic boundary element method (BEM) for the residual stress analysis of axisymmetric structures subjected to axisymmetric and non-axisymmetric loads. By using the Fourier expansions of displacements, tractions and viscoplastic stresses including both the symmetric and anti-symmetric terms, the BEM formulations for an axisymmetric body subjected to axisymmetric and non-axisymmetric loads are developed. A unified viscoplastic model including the actual loading and unloading stress-strain material behaviour can be used to generate plasticity solutions in a simple manner when stationary conditions are reached, and at the other extreme, can reproduce standard creep phenomena. Hence, the viscoplastic BEM program developed in this paper for computing residual stresses of axisymmetric structures allows the treatment of a wide range of materially non-linear problems. The solution algorithm employed is relatively simple and possesses many advantages over the conventional computational procedures utilized in elastoplastic material analysis. Several examples of interest to residual stresses of axisymmetric structures due to axisymmetric and non-axisymmetric loads are presented.

Simulation of solidification and viscoplastic stresses during vertical semicontinuous direct chill casting of aluminum alloy

Abstract A method to simulate temperature and stress fields in solidifying cylindrical ingots during the vertical semicontinuous direct chill casting process is presented in this paper. As the first approximation of the whole process, a steady state in the casting ingot with enough length is examined, and the steady heat conduction equation with heat generation due to solidification incorporated with material flow is solved by the finite element method. Based on the obtained temperature distribution, elastic-viscoplastic stresses were also calculated to obtain residual stresses. Here, material parameters appearing in the constitutive equation were determined by the data of tensile stress-strain curves under some strain rates at several temperature levels close to the melting temperature. Calculated temperature and residual stress fields were compared with the experimental data to verify the analytical procedure, and discussions are also given on the effect of operating conditions on the fields.