Characterization of Viscoelasticity of Molding Compounds in Time Domain

2009 ◽  
Author(s):  
Seung-Hyun Chae ◽  
Jie-Hua Zhao ◽  
Darvin R. Edwards ◽  
Paul S. Ho
Keyword(s):  
Time Domain ◽  
Viscoelastic Properties ◽  
Viscoelastic Behavior ◽  
Relaxation Modulus ◽  
Polymeric Materials ◽  
Shift Factor ◽  
Microelectronics Packaging ◽  
Molding Compound ◽  
Relaxation Experiments ◽  
The Time Domain

Although polymer-based materials are widely used in microelectronics packaging and viscoelasticity is an intrinsic characteristic of polymers, viscoelastic properties of polymeric materials are often ignored in package stress analyses due to the difficulty of measuring this property. However, it is necessary to consider the viscoelastic behavior when an accurate stress model is required. Viscoelastic properties of materials can be characterized either in the time domain or frequency domain. In this study, stress relaxation experiments were performed on a molding compound in the time domain. Prony series expansion was used to express the material’s relaxation behavior. Thermo-rheologically simple model was assumed to deduce the master curve of relaxation modulus using the time-temperature equivalence assumption. Two methods were compared to determine the Prony pairs and shift factor.

Comparison of a Novel Nonlinear Viscoelastic Finite Ramp Time Correction Method to a Heaviside Step Assumption

2011 ◽  
Author(s):  
Kevin L. Troyer ◽  
Christian M. Puttlitz
Keyword(s):  
Stress Relaxation ◽  
Soft Tissues ◽  
Viscoelastic Behavior ◽  
Relaxation Modulus ◽  
Correction Method ◽  
Nonlinear Viscoelastic ◽  
Instantaneous Strain ◽  
Relaxation Experiments ◽  
Time Correction

Connective soft tissues exhibit time-dependent, or viscoelastic, behavior. In order to characterize this behavior, stress relaxation experiments can be performed to determine the tissue’s relaxation modulus. Theoretically, the relaxation modulus describes the stress relaxation behavior of the tissue in response to an instantaneous (step) application of strain. However, a step increase in strain is experimentally impossible and a pure ramp load is intractable due to the inertial limitations of the testing device. Even small deviations from an instantaneous strain application may cause significant errors in the determination of the tissue’s relaxation modulus.

Viscoelastic Behavior of Polyisobutylene under Constant Rates of Elongation

1956 ◽  
Vol 29 (4) ◽  
pp. 1199-1208
Author(s):  
Thor L. Smith
Keyword(s):  
Time Scale ◽  
Viscoelastic Properties ◽  
Viscoelastic Behavior ◽  
Polymeric Materials ◽  
Constant Load ◽  
Distribution Functions ◽  
Large Strains ◽  
National Bureau ◽  
Stress Strain ◽  
Nonlinear Viscoelastic

Abstract A variety of methods has been used to study the viscoelastic properties of polymeric materials. These methods include the response to sinusoidal stress (dynamic measurements), stress relaxation, and creep under constant load and constant stress. The present investigation was made to determine whether or not the viscoelastic properties of rubberlike materials over an extended time scale could be obtained from stress-strain curves measured at different strain rates and temperatures. Polyisobutylene of high molecular weight was selected for study, since its viscoelastic properties have been investigated extensively in a cooperative program sponsored by the National Bureau of Standards. From the data obtained, Marvin has derived the distribution functions of relaxation and of retardation times over a time scale of 10−10 to 107 sec. These functions show quantitatively a change in properties from liquidlike to rubberlike to glasslike with decreasing time scale. The equilibrium stress-strain curves for lightly crosslinked rubber and other elastomers are closely linear for elongations up to 100 per cent. The non-equilibrium (viscoelastic) stress-strain curves for similar and noncrosslinked elastomers might be expected to be linear viscoelastic, as a first approximation, at temperatures above the glass transition, provided the strain and the strain rate are not excessively large. Nonlinear viscoelastic effects are usually pronounced for materials in their glasslike state and at large strains.

Temperature Dependence of Viscoelastic Properties of Carbon-Black-Filled Rubbers in Small Shearing Deformations

1986 ◽  
Vol 59 (4) ◽  
pp. 592-604 ◽  
Author(s):  
Koichi Arai ◽  
John D. Ferry
Keyword(s):  
Carbon Black ◽  
Temperature Dependence ◽  
Viscoelastic Properties ◽  
Viscoelastic Behavior ◽  
Relaxation Modulus ◽  
Maximum Shear Strain ◽  
Shear Moduli ◽  
Shear Relaxation ◽  
Heat Of Dissociation ◽  
Hoff Equation

Abstract Measurements of dynamic storage and loss shear moduli G′ and G″ (0.12 to 2 Hz) and shear relaxation modulus G(t) (up to 104 s) have been made on six vulcanized and one unvulcanized carbon-black-filled rubber compounds over a temperature range from −22.5° to 63°C. The maximum shear strain in the oscillatory deformations was less than 0.005 and in the stress relaxation measurements, 0.015. The temperature dependence of viscoelastic properties could not be fully described in terms of horizontal shifts (αT) of logarithmic time or frequency scales. It could, however, be largely described by vertical shifts (ST) corresponding to uniform temperature dependence of the magnitudes of contributions to modulus from a spectrum of relaxation mechanisms. There were some departures from this behavior, especially in a blend containing two rubber species and in the unvulcanized compound at long times. The temperature dependence of the ST shift factors followed the van't Hoff equation with values of ΔH from 5.9 to 14.7 kJ/mole, attributable to a heat of dissociation of contacts between particle aggregates. The slow relaxation over many logarithmic decades of time scale in the rubbery zone of viscoelastic behavior is attributed to adjustments of such contacts by Brownian motion, which leave the density of the structure unchanged as shown by constancy of the differential dynamic modulus measured by superposed small oscillating deformations.

Characterization of Cure-Dependent Viscoelastic Behavior for Molding Compound and Application to Package Warpage Simulation

2009 ◽  
Author(s):  
Tz-Cheng Chiu ◽  
Je-Li Kung ◽  
Yi-Shao Lai
Keyword(s):  
Finite Element ◽  
Viscoelastic Model ◽  
Viscoelastic Behavior ◽  
Relaxation Modulus ◽  
Cure Kinetics ◽  
Constitutive Relationship ◽  
Element Analysis ◽  
Molding Compound ◽  
Modeling Methodology ◽  
Bga Package

In this study a process-dependent viscoelastic model is developed for considering the constitutive relationship of an epoxy molding compound. The process dependence is realized by incorporating the phenomenological models for the cure kinetics, the cure-dependent volume shrinkage, and the cure-dependent viscoelastic stress relaxation modulus into the constitutive model for the molding compound. The cure-dependent viscoelastic model is incorporated into numerical finite element analysis to simulate warpage of an overmolded chip scale ball grid array (BGA) package under uniform cooling from reflow to room temperature. The simulation results are compared to Shadow Moire´ experimental data for validating the modeling methodology. Additional finite element analyses are performed to investigate the influence of molding compound constitutive behavior (temperature-dependent elastic or viscoelastic) on the package warpage prediction, and to consider the package warpage evolution during the post-mold curing (PMC) process.

scholarly journals Numerical and Experimental Investigations of Polymer Viscoelastic Materials Obtained by 3D Printing

Polymers ◽  
2021 ◽  
Vol 13 (19) ◽  
pp. 3276
Author(s):  
Jusuf Ibrulj ◽  
Ejub Dzaferovic ◽  
Murco Obucina ◽  
Manja Kitek Kuzman
Keyword(s):  
Numerical Analysis ◽  
3D Printing ◽  
Analytical Solutions ◽  
Relaxation Modulus ◽  
Polymeric Materials ◽  
Shift Factor ◽  
Basic Material ◽  
Experimental Investigations ◽  
Relaxation Functions

The aim of this research is to determine the relaxation and creep modulus of 3D printed materials, and the numerical research is based on the finite volume method. The basic material for determining these characteristics is ABS (acrylonitrile butadiene styrene) plastic as one of the most widely used polymeric materials in 3D printing. The experimental method for determining the relaxation functions involved the use of a creep test, in which a constant increase of the stress of the material was performed over time to a certain predetermined value. In addition to this test, DMA (dynamic mechanical analysis) analysis was used. Determination of unknown parameters of relaxation functions in analytical form was performed on the basis of the expression for the storage modulus in the frequency domain. The influence of temperature on the values of the relaxation modulus is considered through the determination of the shift factor. Shift factor is determined on the basis of a series of tests of the relaxation function at different constant temperatures. The shift factor is presented in the form of the WLF (Williams-Landel-Ferry) equation. After obtaining such experimentally determined viscoelastic characteristics with analytical expressions for relaxation modulus and shift factors, numerical analysis can be performed. For this numerical analysis, a mathematical model with an incremental approach was used, as developed in earlier works although with a certain modification. In the experimental analysis, the analytical expression for relaxation modulus in the form of the Prony series is used, and since it is the sum of exponential functions, this enables the derivation of a recursive algorithm for stress calculation. Numerical analysis was performed on several test cases and the results were compared with the results of the experiment and available analytical solutions. A good agreement was obtained between the results of the numerical simulation and the results of the experiment and analytical solutions.

scholarly journals A Combined Exponential-Power-Law Method for Interconversion between Viscoelastic Functions of Polymers and Polymer-Based Materials

Polymers ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 3001
Author(s):  
Vitor Dacol ◽  
Elsa Caetano ◽  
João R. Correia
Keyword(s):  
Mechanical Response ◽  
Creep Compliance ◽  
Viscoelastic Behavior ◽  
Relaxation Modulus ◽  
Creep Tests ◽  
General Applicability ◽  
Wide Range ◽  
Long Time ◽  
Linear Viscoelastic ◽  
The Time Domain

Understanding and modeling the viscoelastic behavior of polymers and polymer-based materials for a wide range of quasistatic and high strain rates is of great interest for applications in which they are subjected to mechanical loads over a long time of operation, such as the self-weight or other static loads. The creep compliance and relaxation functions used in the characterization of the mechanical response of linear viscoelastic solids are traditionally determined by conducting two separate experiments—creep tests and relaxation tests. This paper first reviews the steps involved in conducting the interconversion between creep compliance and relaxation modulus in the time domain, illustrating that the relaxation modulus can be obtained from the creep compliance. This enables the determination of the relaxation modulus from the results of creep tests, which can be easily performed in pneumatic equipment or simple compression devices and are less costly than direct relaxation tests. Some existing methods of interconversion between the creep compliance and the relaxation modulus for linear viscoelastic materials are also presented. Then, a new approximate interconversion scheme is introduced using a convenient Laplace transform and an approximated Gamma function to convert the measured creep compliance to the relaxation modulus. To demonstrate the accuracy of the fittings obtained with the method proposed, as well as its ease of implementation and general applicability, different experimental data from the literature are used.

Assessment of respiratory pressure-volume nonlinearity in rabbits during mechanical ventilation

1996 ◽  
Vol 80 (5) ◽  
pp. 1637-1648 ◽  
Author(s):  
R. Peslin ◽  
M. Rotger ◽  
R. Farre ◽  
D. Navajas
Keyword(s):  
Mechanical Ventilation ◽  
Nonlinear Model ◽  
Time Domain ◽  
Viscoelastic Properties ◽  
Static Pressure ◽  
Viscoelastic Model ◽  
Positive Pressure ◽  
Flow Data ◽  
Volume Dependence ◽  
The Time Domain

The volume dependence of respiratory elastance makes it difficult to recognize actual changes in lung and chest wall elastic properties in artificially ventilated subjects. We have assessed in six anesthetized, tracheotomized, and paralyzed rabbits whether reliable information on the static pressure-volume (PV) curve could be obtained from recordings performed during step variations of the end-expiratory pressure without interrupting mechanical ventilation. Pressure and flow data recorded during 5- and 10-hPa positive-pressure steps were analyzed in the time domain with a nonlinear model featuring a sigmoid PV curve and with a model that, in addition, accounted for tissue viscoelastic properties. The latter fitted the data substantially better. Both models provided reasonably reproducible coefficients, but the PV curves obtained from the 5- and 10-hPa steps were systematically different. When the PV curves were used to predict respiratory effective elastance, the best predictor was the curve derived from the 10-hPa step with the viscoelastic model: unsigned differences averaged 8.6 +/- 11.1, 26.9 +/- 36.4, and 5.5 +/- 5.8% at end-expiratory pressures of 0, 5, and 10 hPa, respectively. This approach provides potentially useful, although not highly accurate, estimates of respiratory effective elastance-volume dependence.

Estimation of Master Curves of Relaxation Modulus and Tensile Properties for Solid Propellant

2013 ◽  
Vol 871 ◽  
pp. 247-252 ◽  
Author(s):  
Boh Wi Seo ◽  
Jae Hoon Kim
Keyword(s):  
Stress Relaxation ◽  
Tensile Properties ◽  
Solid Propellant ◽  
Master Curve ◽  
Classical Method ◽  
Viscoelastic Behavior ◽  
Temperature Shift ◽  
Relaxation Modulus ◽  
Tensile Tests ◽  
Shift Factor

The stress relaxation modulus E(t) is one of the most important properties of viscoelastic materials such as solid propellant, and it is used to define the viscoelastic behavior based on the influence of time and temperature. In this paper, stress relaxation tests are conducted under constant strain 2% for 600 seconds in the range of temperature 60°C to-60°C and tensile tests are performed for solid propellants under constant cross head rate 5 mm/min in the same temperatures as stress relaxation tests. Based on the results, time-temperature shift factors are obtainedby shifting the relaxation modulus curves horizontally and the master curve of relaxation modulus is generated. The master curve of relaxation modulus according to classical method and Williams-Landel-Ferry (WLF) method are discussed. Also, the master curve of tensile properties are drawn using predetermined shift factor and the results are discussed.

Characterization on the Viscoelastic Property of PDMS in the Frequency Domain

2011 ◽  
Vol 1301 ◽  
Author(s):  
Ping Du ◽  
I-Kuan Lin ◽  
Hongbing Lu ◽  
Xi lin ◽  
Xin Zhang
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Practical Interest ◽  
Relaxation Modulus ◽  
Robust Fitting ◽  
Storage And Loss Moduli ◽  
The Fourier Transform ◽  
The Time Domain ◽  
Force Calculation

ABSTRACTA key issue in using Polydimethylsiloxane (PDMS) based micropillars as cellular force transducers is obtaining an accurate characterization of mechanical properties. The Young’s modulus of PDMS has been extended from a constant in the ideal elastic case to a time-dependent function in the viscoelastic case. However, the frequency domain information is of more practical interest in interpreting the complex cell contraction behavior. In this paper, we reevaluated the Young’s relaxation modulus in the time domain by using more robust fitting algorithms than previous reports, and investigated the storage and loss moduli in the frequency domain using the Fourier transform technique. With the use of the frequency domain modulus and the deflection of micropillars in the Fourier series, the force calculation can be much simplified by converting a convolution in the time domain to a multiplication in the frequency domain.

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