Stress-Strain Equations for Some Near-Eutectic Tin-Lead Solders

1991 ◽  
Vol 113 (4) ◽  
pp. 475-484 ◽  
Author(s):  
K. P. Jen ◽  
J. N. Majerus
Keyword(s):  
Strain Rate ◽  
Printed Circuit Boards ◽  
Volume Fraction ◽  
Total Error ◽  
Tensile Tests ◽  
Elastic Behavior ◽  
Plastic Behavior ◽  
Strain Rates ◽  
Stress Strain ◽  
Engineering Strain

This paper presents the evaluation of the stress-strain behavior, as a function of strain-rate, for three tin-lead solders at room temperature. This behavior is critically needed for reliability analysis of printed circuit boards (PCB) since handbooks list minimal mechanical properties for the eutectic solder used in PCBs. Furthermore, most handbook data are for stable eutectic microstructure whereas PCB solder has a metastable microstructure. All three materials were purchased as “eutectics.” However, chemical analysis, volume fraction determination, and microhardness tests show some major variations between the three materials. Two of the materials have a eutectic composition, and one does not. The true stress-strain equations of one eutectic and the one noneutectic material are determined from compressive tests at engineering strain-rates between 0.0002/s and 0.2/s. The second eutectic material is evaluated using tensile tests with strain-rates between 0.00017/s and 0.042/s. The materials appear to exhibit linear elastic behavior only at extremely small strains, i.e., less than 0.0005. However, this “elastic” behavior showed considerable variation, and depended upon the strain rate. In both tension and compression the eutectic alloy exhibits nonlinear plastic behavior, i.e., strain-softening followed by strain-hardening, which depends upon the strain rate. A quadratic equation σy = σy(ε˚/ε˚0) + A(ε˚/ε˚0)ε + B(ε˚/ε˚0)ε2 fit to the data gives correlation coefficients R2 > 0.91. The coefficients σy(ε˚/ε˚0), A(ε˚/ε˚0), B(ε˚/ε˚0) are fitted functions of the normalized engineering strain rate ε˚/ε˚0. Replicated experiments are used at each strain-rate so that a measure of the statistical variation could be estimated. Measures of error associated with the regression analysis are also obtained so that an estimate of the total error in the stress-strain relations can be made.

Tensile Stress-Strain Curves Modeling of Four Kinds of Solders with Temperature and Rate Dependence

2005 ◽  
Vol 297-300 ◽  
pp. 905-911 ◽  
Author(s):  
Xu Chen ◽  
Li Zhang ◽  
Masao Sakane ◽  
Haruo Nose
Keyword(s):  
Tensile Stress ◽  
Constitutive Model ◽  
Strain Rate ◽  
Constant Strain Rate ◽  
Tensile Tests ◽  
Strain Rate Range ◽  
Rate Dependence ◽  
Strain Rates ◽  
Stress Strain ◽  
Rate Range

A series of tensile tests at constant strain rate were conducted on tin-lead based solders with different Sn content under wide ranges of temperatures and strain rates. It was shown that the stress-strain relationships had strong temperature- and strain rate- dependence. The parameters of Anand model for four solders were determined. The four solders were 60Sn-40Pb, 40Sn-60Pb, 10Sn-90Pb and 5Sn-95Pb. Anand constitutive model was employed to simulate the stress-strain behaviors of the solders for the temperature range from 313K to 398K and the strain rate range from 0.001%sP -1 P to 2%sP -1 P. The results showed that Anand model can adequately predict the rate- and temperature- related constitutive behaviors at all test temperatures and strain rates.

A Statistical Constitutive Model for the Strength of Brittle Fiber at Different Strain Rates

2010 ◽  
Vol 152-153 ◽  
pp. 1213-1216
Author(s):  
Wen Huang ◽  
Zhong Wei Huang
Keyword(s):  
Constitutive Model ◽  
Strain Rate ◽  
Tensile Tests ◽  
Fiber Bundles ◽  
Strain Rates ◽  
Stress Strain ◽  
Sic Fiber ◽  
Brittle Fiber ◽  
Strain Relationship ◽  
Relationship Of

A statistical constitutive model, which takes account the effect of strain rate, was presented to describe the stress-strain relationship of brittle fiber bundles. To verify its reliability, tensile tests on two kinds of brittle fibers: glass fiber and SiC fiber, were carried out at different strain rates, and the stress-strain curves were obtained. It was found that the modulus E, the strength and the fracture strain of these fiber bundles all increase with increasing strain rate. The simulated stress-strain curves, derived from the constitutive model, fit the tested results well, which indicates that the model is valid and reliable.

Strain-Rate Effects on Constitutive Behavior of Sn3.8-Ag0.7-Cu Lead-Free Solder

2009 ◽  
Author(s):  
Kok Ee Tan ◽  
John H. L. Pang
Keyword(s):  
Strain Rate ◽  
Yield Stress ◽  
Strain Curve ◽  
Tensile Tests ◽  
Lead Free ◽  
Indentation Hardness ◽  
Strain Rates ◽  
Test Results ◽  
Stress Strain Curve ◽  
Stress Strain

In this paper, the strain-rate dependent mechanical properties and stress-strain curve behavior of Sn3.8Ag0.7Cu (SAC387) solder is presented for a range of strain-rates at room temperature. The apparent elastic modulus, yield stress properties and stress-strain curve equation of the solder material is needed to facilitate finite element modeling work. Tensile tests on dog-bone shaped bulk solder specimens were conducted using a non-contact video extensometer system. Constant strain-rate uni-axial tensile tests were conducted over the strain-rates of 0.001, 0.01, 0.1 and 1 (s−1) at 25°C. The effects of strain-rate on the stress-strain behavior for lead-free Sn3.8Ag0.7Cu solder are presented. The tensile yield stress results were compared to equivalent yield stress values derived from nano-indentation hardness test results. Constitutive models based on the Ramberg-Osgood model and the Cowper-Symond model were fitted for the tensile test results to describe the elastic-plastic behavior of solder deformation behavior.

scholarly journals Evolution of Dynamic Recrystallization in 5CrNiMoV Steel during Hot Forming

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Zhiqiang Hu ◽  
Kaikun Wang
Keyword(s):  
Grain Size ◽  
Dynamic Recrystallization ◽  
Strain Rate ◽  
Volume Fraction ◽  
True Strain ◽  
True Stress ◽  
Hot Forming ◽  
Strain Rates ◽  
Stress Strain ◽  
Lower Strain

The dynamic recrystallization (DRX) behavior of 5CrNiMoV steel was investigated through hot compression at temperatures of 830–1230°C and strain rates of 0.001–10 s−1. From the experimental results, most true stress-strain curves showed the typical nature of DRX that a single peak was reached at low strains followed by a decrease of stress and a steady state finally at relatively high strains. The constitutive behavior of 5CrNiMoV steel was analyzed to deduce the operative deformation mechanisms, and the correlation between flow stress, temperature, and strain rate was expressed as a sine hyperbolic type constitutive equation. Based on the study of characteristic stresses and strains on the true stress-strain curves, a DRX kinetics model was constructed to characterize the influence of true strain, temperature, and strain rate on DRX evolution, which revealed that higher temperatures and lower strain rates had a favorable influence on improving the DRX volume fraction at the same true strain. Microstructure observations indicated that DRX was the main mechanism and austenite grains could be greatly refined by reducing the temperature of hot deformation or increasing the strain rate when complete recrystallization occurred. Furthermore, a DRX grain size model of 5CrNiMoV was obtained to predict the average DRX grain size during hot forming.

Multiscale Material Modeling and Simulation of the Mechanical Behavior of Dual Phase Steels Under Different Strain Rates: Parametric Study and Optimization

2018 ◽  
Vol 140 (3) ◽  
Author(s):  
Tarek M. Belgasam ◽  
Hussein M. Zbib
Keyword(s):  
Mechanical Properties ◽  
Strain Rate ◽  
Mechanical Behavior ◽  
Volume Fraction ◽  
Rate Sensitivity ◽  
Tensile Tests ◽  
Dual Phase ◽  
Strain Rates ◽  
Microstructure Parameters ◽  
Dp Steels

Recent studies on developing dual phase (DP) steels showed that the combination of strength/ductility could be significantly improved when changing the volume fraction and grain size of phases in the microstructure depending on microstructure properties. Consequently, DP steel manufacturers are interested in predicting microstructure properties as well as optimizing microstructure design at different strain rate conditions. In this work, a microstructure-based approach using a multiscale material and structure model was developed. The approach examined the mechanical behavior of DP steels using virtual tensile tests with a full micro-macro multiscale material model to identify specific mechanical properties. Microstructures with varied ferrite grain sizes, martensite volume fractions, and carbon content in DP steels were also studied. The influence of these microscopic parameters at different strain rates on the mechanical properties of DP steels was examined numerically using a full micro-macro multiscale finite element method. An elasto-viscoplastic constitutive model and a response surface methodology (RSM) were used to determine the optimum microstructure parameters for a required combination of strength/ductility at different strain rates. The results from the numerical simulations were compared with experimental results found in the literature. The developed methodology proved to be a powerful tool for studying the effect and interaction of key strain rate sensitivity and microstructure parameters on mechanical behavior and thus can be used to identify optimum microstructural conditions at different strain rates.

scholarly journals Visualization of the damage evolution for Ti–3Al–2Mo–2Zr alloy during a uniaxial tensile process using a microvoids proliferation damage model

2021 ◽  
Vol 40 (1) ◽  
pp. 310-324
Author(s):  
Ying Tong ◽  
Jiang Zhao ◽  
Guo-zheng Quan
Keyword(s):  
Plastic Deformation ◽  
Damage Evolution ◽  
Volume Fraction ◽  
Tensile Deformation ◽  
Damage Model ◽  
Tensile Tests ◽  
Uniaxial Tensile ◽  
Strain Rates ◽  
Stress Strain ◽  
Gtn Damage Model

Abstract Understanding the damage evolution of alloys during a plastic deformation process is significant to the structural design of components and accident prevention. In order to visualize the damage evolution in the plastic deformation of Ti–3Al–2Mo–2Zr alloy, a series of uniaxial tensile experiments for this alloy were carried out under the strain rates of 0.1–10 s−1 at room temperature, and the stress–strain curves were achieved. On the other hand, the finite element (FE) models of these uniaxial tensile processes were established. A microvoids proliferation model, Gurson–Tvergaard–Needleman (GTN) damage model, was implanted into the uniaxial tensile models, and the simulated stress–strain curves corresponding to different GTN parameter combinations were obtained. Based on the simulated and experimental stress–strain curves, the GTN parameters of this alloy were solved by response surface methodology (RSM). The solved GTN parameters suggest that higher strain rate can enhance the proliferation and coalescence of microvoids. Furthermore, the uniaxial tensile tests over different strain rates were simulated using the solved GTN parameters. Then, the damage processes were visualized and evaluated. The result shows that the degradation speed of this alloy is slow at the initial stage of the tensile deformation and then accelerates once the voids volume fraction reaches a critical value.

scholarly journals New Approach In The Properties Evaluation Of Ultrafine-Grained OFHC Copper

2015 ◽  
Vol 60 (2) ◽  
pp. 605-614 ◽  
Author(s):  
T. Kvačkaj ◽  
A. Kováčová ◽  
J. Bidulská ◽  
R. Bidulský ◽  
R. Kočičko
Keyword(s):  
Mechanical Properties ◽  
Strain Rate ◽  
Tensile Tests ◽  
Coarse Grained ◽  
Wear Behaviour ◽  
Ofhc Copper ◽  
Strain Rates ◽  
Ultrafine Grained ◽  
Reduction Of Area ◽  
Pin On Disk

AbstractIn this study, static, dynamic and tribological properties of ultrafine-grained (UFG) oxygen-free high thermal conductivity (OFHC) copper were investigated in detail. In order to evaluate the mechanical behaviour at different strain rates, OFHC copper was tested using two devices resulting in static and dynamic regimes. Moreover, the copper was subjected to two different processing methods, which made possible to study the influence of structure. The study of strain rate and microstructure was focused on progress in the mechanical properties after tensile tests. It was found that the strain rate is an important parameter affecting mechanical properties of copper. The ultimate tensile strength increased with the strain rate increasing and this effect was more visible at high strain rates$({\dot \varepsilon} \sim 10^2 \;{\rm{s}}^{ - 1} )$. However, the reduction of area had a different progress depending on microstructural features of materials (coarse-grained vs. ultrafine-grained structure) and introduced strain rate conditions during plastic deformation (static vs. dynamic regime). The wear behaviour of copper was investigated through pin-on-disk tests. The wear tracks examination showed that the delamination and the mild oxidational wears are the main wear mechanisms.

The Effects of Physical Ageing and of Prior Immersion on the Esc Behaviour of Polycarbonate in Ethanol

1993 ◽  
Vol 305 ◽  
Author(s):  
J. C. Arnold ◽  
A. R. Eccott
Keyword(s):  
Strain Rate ◽  
Immersion Time ◽  
Constant Strain Rate ◽  
Tensile Tests ◽  
Strain Rates ◽  
Physical Ageing ◽  
Diffusion Effects ◽  
Craze Initiation ◽  
Comparison Of Tests

AbstractThe effects of physical ageing and prior immersion time on the ESC behaviour of polycarbonate in ethanol were studied. Constant strain rate tensile tests were performed at a range of strain rates for samples with ageing times varying from 100 hours to 3000 hours and for prior immersion times of between 1 hour and 500 hours. Comparison of tests performed in ethanol and in air gave a good indication of the point of craze initiation. The results showed that there was a reduction in strain to crazing as the strain rate decreased, apart from with the lowest strain rate used. A longer prior immersion time also promoted craze formation. Both of these results are attributable to diffusion effects. Physical ageing had little effect on the ESC behaviour, due to the large amounts of deformation encountered in this system.

Understanding Dynamic Recrystallization Behavior through a Delay Differential Equation Approach

2007 ◽  
Vol 558-559 ◽  
pp. 441-448 ◽  
Author(s):  
Jong K. Lee
Keyword(s):  
Differential Equation ◽  
Strain Rate ◽  
Critical Strain ◽  
Strain Curve ◽  
Single Peak ◽  
Strain Rates ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Strain Behavior ◽  
Delay Differential

During hot working, deformation of metals such as copper or austenitic steels involves features of both diffusional flow and dislocation motion. As such, the true stress-true strain relationship depends on the strain rate. At low strain rates (or high temperatures), the stress-strain curve displays an oscillatory behavior with multiple peaks. As the strain rate increases (or as the temperature is reduced), the number of peaks on the stress-strain curve decreases, and at high strain rates, the stress rises to a single peak before settling at a steady-state value. It is understood that dynamic recovery is responsible for the stress-strain behavior with zero or a single peak, whereas dynamic recrystallization causes the oscillatory nature. In the past, most predictive models are based on either modified Johnson-Mehl-Avrami kinetic equations or probabilistic approaches. In this work, a delay differential equation is utilized for modeling such a stress-strain behavior. The approach takes into account for a delay time due to diffusion, which is expressed as the critical strain for nucleation for recrystallization. The solution shows that the oscillatory nature depends on the ratio of the critical strain for nucleation to the critical strain for completion for recrystallization. As the strain ratio increases, the stress-strain curve changes from a monotonic rise to a single peak, then to a multiple peak behavior. The model also predicts transient flow curves resulting from strain rate changes.

scholarly journals A Strain Rate-Dependent Damage Evolution Model for Concrete Based on Experimental Results

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Bin Xu ◽  
Xiaoyan Lei ◽  
P. Wang ◽  
Hui Song
Keyword(s):  
Strain Rate ◽  
Uniaxial Compression ◽  
Damage Evolution ◽  
Damage Model ◽  
Evolution Model ◽  
Experimental Results ◽  
Strain Rates ◽  
Compression Tests ◽  
Stress Strain ◽  
Actual Damage

There are various definitions of damage variables from the existing damage models. The calculated damage value by the current methods still could not well correspond to the actual damage value. Therefore, it is necessary to establish a damage evolution model corresponding to the actual damage evolution. In this paper, a strain rate-sensitive isotropic damage model for plain concrete is proposed to describe its nonlinear behavior. Cyclic uniaxial compression tests were conducted on concrete samples at three strain rates of 10−3s−1, 10−4s−1, and 10−5s−1, respectively, and ultrasonic wave measurements were made at specified strain values during the loading progress. A damage variable was defined using the secant and initial moduli, and concrete damage evolution was then studied using the experimental results of the cyclic uniaxial compression tests conducted at the different strain rates. A viscoelastic stress-strain relationship, which considered the proposed damage evolution model, was presented according to the principles of irreversible thermodynamics. The model results agreed well with the experiment and indicated that the proposed damage evolution model can accurately characterize the development of macroscopic mechanical weakening of concrete. A damage-coupled viscoelastic constitutive relationship of concrete was recommended. It was concluded that the model could not only characterize the stress-strain response of materials under one-dimensional compressive load but also truly reflect the degradation law of the macromechanical properties of materials. The proposed damage model will advance the understanding of the failure process of concrete materials.

Sign in / Sign up

Export Citation Format

Share Document