Viscoplastic Finite Element Analysis Research Articles

Fracture resistance of Ceramic-Polymer hybrid materials using microscopic finite element analysis and experimental validation

The objective of this paper is to elucidate the response to contact stresses of Polymer Infiltrated Ceramic Network (PICN) using the microscopic viscoplastic finite elements, validated by clinically relevant in vitro tests. A feldspathic ceramic material, namely Vita Mark II, is an interconnected structure infiltrated with the polymer (PMMA). Axisymmetric finite element microstructure models are reconstructed from two-dimensional images of a PICN microstructure. Viscoplastic finite element analysis (FEA) with various degrees of microscopic damages occurring over contact is performed. The force-displacement responses obtained from FEA are validated with Hertzian contact tests. Finite element results for force-displacement, stresses and strains in each phase are discussed. We hypothesize that the resistance to fracture of PICN can be further improved by microstructural tailoring. The experimental evidence suggests that a composite material is both more resistant to displacement under load and more resistant to crack initiation and propagation, as hypothesized. Further parametric study on the effects of various volume fractions of two phases in PICN is done to provide some insight on increased contact damage resistance of PICN as well as potential optimization of microstructures.

Experimental and Numerical Investigation of High-Temperature Low-Cycle Fatigue and Creep-Fatigue Life of Bellows

SS316Ti is one of the widely used materials for bellows in sodium cooled fast reactor systems. Conventional design codes for bellows do not address the high-temperature failure modes. In this investigation, a methodology has been proposed for the high-temperature design of the bellows. The bellows were preliminarily designed by standards of EJMA for room temperature application. Subsequently, the detailed visco-plastic finite element analysis of the bellows has been carried out to estimate the high-temperature low-cycle fatigue (LCF) and creep-fatigue (CFI) life. The material parameters required for the visco-plastic constitutive model were generated and validated for fatigue specimens. The LCF and CFI life of the bellows based on analysis were 83 and 74 cycles, respectively. The proposed methodology has been validated by high-temperature testing of the bellows. The LCF and CFI lives of the bellows obtained experimentally were found to be 281 and 237 cycles, respectively. The life arrived based on analysis was found to be conservative compared to corresponding life by testing. The safety margin available over failure based on testing of the bellows is 3.4 under LCF and 3.2 for CFI loading.

Nonisothermal warm deep drawing of SS304: FE modeling and experiments using servo press

Experimental and viscoplastic finite element analysis (FEA) of thermo-mechanical plastic deformation in nonisothermal warm deep drawing is studied using SS304. A nonisothermal deep drawing tool is used in a servo-motor-controlled press. Drawability window of SS304 under elevated temperatures (25–225 °C) and low to high strain rates (drawing speeds of 2.5, 25, and 50 mm/s) were determined. A viscoplastic thermal material model is adopted for nonwork softening material behaviors, as seen in low-temperature forming of SS304, and found to be easily applicable and quite satisfactory. Tensile and equi-biaxial bulge tests were conducted for more accurate flow stress data to be used in FEA. Measured punch load–stroke and cup’s curvilinear thickness (rolling/transverse) curves were successfully compared with predictions from the nonisothermal FE model of the warm deep drawing.

Uncertainty analysis of solder alloy material parameters estimation based on model calibration method

Quantification of the uncertainties in the material characterization of solder joint has been one of the major concerns in the microelectronics packaging industry to predict fatigue failure accurately. Therefore, in this study, a model calibration method based on Bayesian approach is proposed to quantify these uncertainties arising in the material parameter estimation of the solder alloy. A specimen is fabricated to this end, which closely simulates solder joint behavior of the actual package under a thermal cycle. Experiment is conducted to examine the deformation by using Moiré interferometry. Viscoplastic finite element analysis procedure is constructed for the specimen based on the Anand model. The uncertainties which include inherent experimental error and insufficient data of experiments are addressed by using the likelihood estimation. Two materials, one being conventional solder of Sn36Pb2Ag and the other the lead-free solder of Sn3.0Ag0.5Cu, are considered to illustrate the approach. As a result, material parameters are identified in the form of credible interval (CI), and the displacements and strains using these parameters are given by the predictive interval (PI). The results suggest that the proposed approach can be a useful tool in the probabilistic estimation of the unknown material parameters of solder joint by accounting for the uncertainties due to the experimental data.

Interaction Effect of Voids and Standoff Height on Thermomechanical Durability of BGA Solder Joints

This paper documents simulation studies on the interactive effect of standoff height and void volume on the thermomechanical durability of ball-grid-array solder joints using a 3-D viscoplastic finite element analysis. Surface Evolver software was used to find the optimized shape of the solder joints and standoff height by minimizing the surface energy as voids with different sizes were placed in solder balls. A global-local modeling approach was then utilized to model the thermomechanical durability of voided solder joints. Void-area-fraction ranges of 14%-40% were analyzed. A nonmonotonic behavior of durability versus void area fraction was observed. The results showed that, if the void is completely inside the solder ball and has no interface with the boundaries of the joint, it does not have a detrimental effect and even improves the durability as the void size increases. However, voids located at the interface of the solder joint and copper pads were found very detrimental to durability. Factors such as load bearing area, stress concentration factor, and overall compliance of the structure were found responsible for the nonmonotonic behavior of the joints. An analytical micromechanics approach was used to calculate the compliance of the structure, and a nonmonotonic trend in phase with the durability trend was observed. The stress concentration factor also showed the same nonmonotonic trend. The rise of these two factors for the void interfacing with copper pads in addition to the decreased-load-bearing-area effect resulted in a drastic decrease in durability.

A meso-scale damage evolution model for cyclic fatigue of viscoplastic materials

This study presents an approach to predict the degree of material degradation and the resulting changes in elastic, plastic and creep constitutive properties of viscoplastic materials, during cyclic loading in micro-scale applications. The objective of the study is to address the initiation and growth of homogeneous meso-scale damage, in the form of distributions of micro-cracks and micro-voids, due to cyclic, plastic (rate-independent inelastic) and creep (rate-dependent inelastic) deformations in viscoplastic materials and to evaluate the resulting changes in the effective meso-scale elastic, plastic and creep constitutive properties. An energy partitioning damage evolution (EPDE) model is proposed to describe the viscoplastic damage evolution. Development of the EPDE model constants is then demonstrated for a Pb-free solder, based on cyclic fatigue test data. Application of the EPDE model is demonstrated for solder joint fatigue during thermal cycling of a ball grid array (BGA) electronic assembly. A 3D viscoplastic finite element analysis is conducted, and damage evolution is modeled using a successive initiation (SI) technique reported earlier by the authors. In this approach, the local (meso-scale) material properties are progressively degraded and highly damaged sections of the macro-scale structure are ultimately eliminated, using the EPDE model. Prediction of damage initiation and propagation is presented both with and without property updating, for comparison purposes. The analysis shows that the EPDE model can realistically capture the softening observed during cyclic loading.

Packaging parameter analysis and optimization design on solder joint reliability for twin die stacked packages by variance in strain energy density (SED) of each solder joint

In accordance with vigorous development of the electronic product market as well as the consumers’ preference for smaller scales, the structure of 3D stacked die package rapidly becomes popular. Hereafter the stacked process of the silicon dies always makes the coupling effect among materials more complicated. Such an issue has been seriously paid attention to and becomes a critical problem to be solved for the product reliability. In this paper, the ANSYS software is adopted to analyze a twin die stacked package under a cyclic thermal loading condition. The viscoplastic finite element analysis and the Darveaux theory are applied to investigate the solder joint reliability (SJR) of the stacked die package. This research will verify a significant dependence between the solder joint fatigue life of the stacked die package and the distribution of the accumulated strain energy density (SED) on the solder joints by proposing a viewpoint of the variance of the strain energy density among solder joints for a 3D-Slice model.

Multi-scale finite element analyses of sheet metals by using SEM-EBSD measured crystallographic RVE models

In this study, two multi-scale analyses codes are newly developed by combining a homogenization algorithm and an elastic/crystalline viscoplastic finite element (FE) method (Nakamachi, E., 1988. A finite element simulation of the sheet metal forming process. Int. J. Numer. Meth. Eng. 25, 283–292; Nakamachi, E., Dong, X., 1996. Elastic/crystalline viscoplastic finite element analysis of dynamic deformation of sheet metal. Int. J. Computer-Aided Eng. Software 13, 308–326; Nakamachi, E., Dong, X., 1997. Study of texture effect on sheet failure in a limit dome height test by using elastic/crystalline viscoplastic finite element analysis. J. Appl. Mech. Trans. ASME(E) 64, 519–524; Nakamachi, E., 1998. Elastic/crystalline viscoplastic finite element modeling based on hardening–softening evaluation equation. In: Proc. of the 6th NUMIFORM, pp. 315–321; Nakamachi, E., Hiraiwa, K., Morimoto, H., Harimoto, M., 2000a. Elastic/crystalline viscoplastic finite element analyses of single- and poly-crystal sheet deformations and their experimental verification. Int. J. Plasticity 16, 1419–1441; Nakamachi, E., Xie, C.L., Harimoto, M., 2000b. Drawability assessment of BCC steel sheet by using elastic/crystalline viscoplastic finite element analyses. Int. J. Mech. Sci. 43, 631–652); (1) a “semi-implicit” finite element (FE) code and (2) a “dynamic explicit” FE code. These were applied to predict the plastic strain induced yield loci and the formability of sheet metal in the macro scale, and simultaneously the crystal texture and hardening evolutions in the micro scale. The isotropic and kinematical hardening laws are employed in the crystalline plasticity constitutive equation. For the multi-scale structure, two-scales are considered. One is a microscopic polycrystal structure and the other a macroscopic elastic plastic continuum. We measure crystal morphologies by using the SEM-EBSD apparatus with a unit of about 3.8 μm voxel, and define a three dimensional (3D) representative volume element (RVE) for the micro polycrystal structure, which satisfy the periodicity condition of crystal orientation distribution. A “micro” finite element modeling technique is newly established to minimize the total number of finite elements in the micro scale. Next, the “semi-implicit” crystallographic homogenization FE code, which employs the SEM-EBSD measured RVE, is applied to the 99.9% pure-iron uni-axial tensile problem to predict the texture evolution and the subsequent yield loci in the various strain paths. These “semi implicit” results reveal that the plastic strain induced anisotropy in the micro and macro levels can be predicted by our FE analyses. The kinematical hardening law leads a distinct plastic strain induced anisotropy. Our “dynamic-explicit” FE code is applied to simulate the limit dome height (LDH) test problem of the mild steel DQSK, the high strength steel HSLA and the aluminum alloy AL6022 sheet metals, which were adopted as the NUMISHEET2005 Benchmark sheet metals (Smith, L.M., Pourboghrat, F., Yoon, J.-W., Stoughton, T.B., 2005. NUMISHEET2005. In: Proc. of 6th Int. Conf. Numerical Simulation of 3D Sheet Metal Forming Processes, PART A and B(Benchmark), pp. 409–451) to estimate formability. The “dynamic explicit” results reveal that the initial crystal orientation distribution has a large affects to a plastic strain induced texture and anisotropic hardening evolutions and sheet formability.

Effect of Voids on Thermomechanical Durability of Pb-Free BGA Solder Joints: Modeling and Simulation

The effect of process-induced voids on the durability of Sn–Pb and Pb-free solder interconnects in electronic products is not clearly understood. Experimental studies have provided conflicting ambiguous conclusions, showing that voids may sometimes be detrimental to reliability, but they may sometimes even increase the reliability of joints, depending on the size and location. Because of the higher level of process-induced voids in Pb-free solders, this debate is more intensified in Pb-free joints. This study presents finite element analysis (FEA) of the influence of void size, location, and spacing on the durability of Pb-free solders. A three-dimensional, global-local, viscoplastic FEA is conducted for a CTBAG132 assembly under thermal cycling. The displacement result of the global FEA at the top and bottom of the critical ball is used as the boundary condition in a local model, which focuses on the details of a single ball of the CTBGA package under temperature cycling. Parametric study is conducted to model a solder ball with voids of different sizes and locations. The maximum void area fraction modeled is from 1% to 49% of the ball area. An energy-partitioning model for cyclic creep-fatigue damage is used to estimate the damage and to monitor the trends as the size and location of voids are varied. Potential sites for maximum damage and crack initiation are identified. FEA results show that as void size increases up to about 15% of the area fraction of the ball, durability increases. For voids bigger than that, the durability starts to decrease. This study also confirms that as voids are located closer to the damage initiation site and the propagation path, their lifespan decreases.

Bottom nozzle failure mechanism of water reactor pressure vessel under severe accident conditions

Most of past studies devoted to the creep rupture of a nuclear reactor pressure vessel (RPV) lower head under severe accident conditions, have focused on global deformation and rupture modes. Limited efforts were made on local failure modes associated with penetration nozzles as a part of TMI-2 vessel investigation project (TMI-2 VIP) in 1990s. However, it was based on an excessively simplified shear deformation model. In the present study, the mode of nozzle failure has been investigated using data and nozzle materials from Sandia National Laboratory's lower head failure experiment (SNL-LHF). Crack-like separations were revealed at the nozzle weld metal to RPV interfaces indicating the importance of normal stress component rather than the shear stress in the creep rupture. Creep rupture tests were conducted for nozzle and weld metal materials, respectively, at various temperature and stress levels. Stress distribution in the nozzle region is calculated using elastic–viscoplastic finite element analysis (FEA) using the measured properties. Calculation results are compared with earlier results based on the pure shear model of TMI-2 VIP. It is concluded from both LHF-4 nozzle examination and FEA that normal stress at the nozzle/lower head interface is the dominant driving force for the local failure. From the FEA for the nozzle weld attached in RPV, it is shown that nozzle welds failure occur by displacement controlled fracture of nozzle hole not by load controlled fracture of internal pressure. Considering these characteristics of nozzle weld failure, new concept of nozzle failure time prediction is proposed.

Viscoplastic finite element analysis of matrix crack propagation in model continuous-carbon fibre/epoxy composites

Abstract Non-linear finite element (FE) analysis was used to investigate the effect of matrix viscoplasticity in a continuous carbon fibre/epoxy composite subject to tensile loading. During fibre fracture, the sudden increase in the strain rate near the fracture location reduces matrix ductility which results in catastrophic failure involving rapid matrix crack-growth. By applying the minimum strain energy density criterion to the FE results, the semi-cone angle of a matrix crack was predicted to be in the range 32°–36°, twice the value for the corresponding short fibre system. The FE results were compared with experimental results obtained from laser Raman spectroscopy (LRS), and it was found that the fibre stress profiles are characteristic of conical matrix crack geometries. Good correlation was observed with the experimental results confirming matrix cracking as the primary mode of failure for this composite system. The energy balance model was used to determine the available strain energy to propagate the crack.

A Crystallographic Homogenization Elastic/Crystal Viscoplastic Finite Element Analysis of Pure Iron Polycrystal Material by Employing the Kinematic Hardening Rule

In this study, multi scale analyses, by using the crystallographic homogenization based semi-implicit finite element code, have been carried out to indicate a plastic strain induced texture evolution and macro-continuum yield loci evolution. This crystallographic homogenization finite element code has employed the double scale hierarchical structure, which consists of a polycrystal microstructure and a macro-continuum. We focus to discuss “How to define a proper microstructure for the double scale finite element analysis, by employing the real polycrystal aggregation, which has been measured by SEM-EBSD apparatus”. For scaling up from the polycrystal microstructure to the macro-continuum, we looked at the periodicity of crystal orientation distribution, which is one of morphological factors to feature the polycrystal structure itself, which is named as “texture”. Through a statistical stydy of these measured polycrystal morphologies, finally we obtaine a realistic polycrystal unit cell of the microstructure, which can be adopted for multi scale finite element analyses. The availability of this code to predict the plastic deformation induced anisotropy and kinematical hardening is confirmed.

Creep Stress Redistribution of Single-Lap Weldbonded Joint

Weldbonding is a new joining technique that combines the spot welding with the adhesive bonding. It has found more and more applications in recent years because of its unique advantages. However, researches on the new technique are few especially on the long-term stress and strength of weldbonded structures. In the present paper, three-dimensional viscoplastic finite element analyses (FEA) of a single-lap weldbonded joint are conducted to investigate the stress redistribution. The results show that stresses increase obviously near the interface between the adhesive and the weld spot with the creep of adhesive. The adhesive tends to distribute its stresses more uniformly and transfer some loads to the weld spot. The stress concentration factor of weldbonded joint is significantly lower than that of the direct spot-weld in connection. The simulations illustrate that increasing of creep parameter will enhance the maximum stress level in the weld spot.

Solder Joint Shape Prediction Using a Modified Perzyna Viscoplastic Model

Ball grid array solder joint reliability is known to be dependent on the shape of solder joints after reflow. To ensure good solder joint formation and prevent solder bridging, it is critical to understand the amount of paste volume needed during assembly and reflow. The final shape of the solder joint is a function of surface tension, wetting area, gravity, and applied forces. In this paper, a new methodology to simulate solder joint shape is presented. Large deformation viscoplastic finite element analysis is used to simulate incompressible fluid flow. A numerical model for surface tension is outlined and validated with closed-form solutions. The results of the numerical model are compared to other known solder joint shape prediction methods. The effects of package weight, coplanarity, warpage, paste volume, pad misregistration, and joint construction on solder joint shape are then analyzed. Recommendations are provided on ways to maximize standoff height and avoid bridging. Finally, the formation of leadless solder joints is studied and compared to experimental data.

Three-dimensional modeling and simulation of superplastic forming

The computational details of a viscoplastic finite element analysis of three-dimensional superplastic forming (SPF) are discussed in this paper. Two typical superplastic forming shapes, conical bulging and rectangle box bulging, were analyzed and the experiment of conical bulging was performed to verify the numerical simulation. The inverse identification of superplastic material parameters incorporating the finite element method and an optimization technique was developed and an activity rule was proposed for effective selection of measurement locations in the experiment. Based on the numerical modeling, the influence of many factors such as the frictional coefficient, strain rate sensitivity and strain rate on the thickness distribution was studied and some useful conclusions were drawn.

Formability assessment of FCC aluminum alloy sheet by using elastic/crystalline viscoplastic finite element analysis

This study looks at the influence of crystallographic texture on the formability of FCC aluminum sheet metal using our elastic/crystalline viscoplastic finite element (FE) analysis code “ROBUST-CRYSTAL”. First, the crystallographic textures of Al–2.5% Mg aluminum alloy generated by employing different annealing temperatures were obtained by X-ray diffraction and orientation distribution function (ODF) analyses. The measured ODF results revealed clearly different textures of sheets, featured by orientation fibers, skeleton lines and selected orientations in Euler angle coordinate space, which can be related to plastic anisotropy. The orientation probability assignment method was used in FE modeling. The orientations determined from the measured ODF results were assigned to FE integration points. A large number of integration points, which represent crystallites and can rotate individually, are employed to represent textures of the sheet metals, which take into account the initial and evolutional plastic anisotropy without introducing Taylor or Sachs homogenization assumption. An actual sheet metal forming process, VDI benchmark problem, was adopted to assess the texture effects on strain localization and failure. It was confirmed by comparison with experimental results that our crystalline plasticity FE code could predict strain localization and assess formability with good accuracy.

The effect of crystallographic textures on the formability of high-strength steel sheets

The effect of crystallographic textures on the formability of high-strength steel sheets was investigated by using the elastic/crystalline viscoplastic finite element (FE) analysis. First, the preferred orientations, which correlate with the plastic anisotropy, were revealed by orientation fibers, skeleton lines and selected orientations in Euler angle space. Secondly, the mechanical properties such as R-values, yield loci and stress–strain relationships, and drawability were evaluated by crystallographic numerical analyses and experiments. It was confirmed that one of the reasons why high-strength steel sheets show low formability was due to lack of γ texture components. Then, a textural design method was proposed for improving the formability of high-strength steel sheets. An artificial γ fiber texture was specified in terms of a rotationally symmetric Gaussian distribution of misorientations with scatter widths along the given skeleton line. Traces of the artificial γ texture orientations in terms of volume fractions were randomly selected and assigned to the original high-strength steel model by using the orientation probability assignment method. Finally, from the standard limit dome height (LDH) test, it was found that there was distinct improvement on formability for the new high-strength steel sheets even with a little proportion of γ texture components.

Drawability assessment of BCC steel sheet by using elastic/crystalline viscoplastic finite element analyses

Elastic/crystalline viscoplastic finite element (FE) analyses were carried out to asses the drawability of three kinds of BCC steel sheets, such as mild steel, dual-phase steel and high-strength steel, in the cylindrical cup deep drawing processes. In this study, the crystal orientations were obtained by X-ray diffraction and orientation distribution function (ODF) analyses. The measured ODF results have revealed clearly different textures of sheets, featured by orientation fibers, skeleton lines and selected orientations in Euler angle coordinate space, which can be related to the plastic anisotropy. An orientation probability assignment method, which can be categorized as an inhomogenized material modeling, was used in this FE modeling. The orientations were determined from the measured ODF and assigned to FE integration points one by one. Numbers of integration points, which represent crystallites and can rotate individually, are employed to represent textures of the sheet metals for taking account of the initial and evolutional plastic anisotropy without introducing Taylor or Sachs homogenization assumption. The FE analyses showed how the fiber textures affect the strain localization and earing in the deep drawing operation. It was confirmed by comparison with experimental results that this FE code could predict the extreme strain localization and earing with good accuracy and assess the sheet drawability.

Rigid–viscoplastic finite element analysis of the piercing process in the automatic simulation of multi-stage forging processes

In this paper, an application-oriented piercing simulation technique in automatic forging simulation by the rigid–viscoplastic finite element method is presented. In this approach, it is assumed that fracture in piercing takes place at the instant when the maximum accumulated damage around the shearing region reaches the critical damage value of the material and that separation is made along the line connecting the two die edges. A method of obtaining the critical damage value by comparing the tensile test result with that of the analysis is given, with emphasis on the effect of the strain hardening exponent on the critical damage value. The approach presented is verified by a test piercing process with a medium-carbon steel and is applied successfully to the automatic simulation of a sequence of six-stage compound forging processes.

Elastic/crystalline viscoplastic finite element analyses of single- and poly-crystal sheet deformations and their experimental verification

Abstract The elastic/crystalline viscoplastic constitutive equation, based on a newly proposed hardening-softening evolution equation, is introduced into the dynamic-explicit finite element code “Itas-Dynamic.” In the softening evolution equation, the effective distance and the angle between each slip system of a crystal are introduced to elucidate the interaction between the slip systems, which causes a decrease of dislocation density. The polycrystal sheet is modeled by Voronoi polygons, which correspond to the crystal grains; and by the selected orientations, which can relate to the texture, they are assigned to the integration points of the finite elements. We propose a direct crystal orientation assignment method, which means that each integration point of finite element has an assigned orientation, and its orientation can be rotated independently. Therefore, this inhomogeneous polycrystal model can consider the plastic induced texture development and subsequent anisotropy evolution. The parameters of the constitutive equation are identified by uni-axial tension tests carried out on single crystal sheets. Numerical results obtained for sheet tensions are compared with experimental ones to confirm the validity of our finite element code. Further, we investigate the following subjects: (1) how the initial orientation of single crystal affects slip band formation and strain localization; (2) how the grain size and particular orientations of the grain affect the strain localization in case of a polycrystal sheet. It is confirmed that the orientation of a single crystal can be related to the primary slip system and the deformation induced activation of that system, which in turn can be related to the slip band formation of the single crystal sheet. Further, in case of a polycrystal sheet, the larger the grain size, the more the strain localizes at a specific crystal, which has the particular orientation. It is confirmed through comparisons with experiments that our finite element code can predict the localization of strain in sheets and consequently can estimate the formability of sheet metals.