Temperature Dependence Of Shear Modulus Research Articles

Finite Element Model for Investigating the Thermo-Electro-Mechanical Response of Inhomogeneously Deforming Dielectric Elastomer Actuators

Among the available soft active materials, Dielectric elastomers (DEs) possess the capability of achieving the large actuation strain under the application of high electric field. The material behavior of such elastomers is affected significantly by the change in temperature. This paper reports a 3-D finite element framework based on the coupled nonlinear theory of thermo-electro-elasticity for investigating the thermal effects on the electromechanical performance of inhomogeneously deforming dielectric elastomer actuators (DEAs). The material behavior of the actuator is modeled using the neo-Hookean model of hyperelasticity with temperature dependent shear modulus. An in-house computational code is developed to implement the coupled finite element framework. Firstly, the accuracy of the developed FE code is verified by simulating the temperature effects on the actuation response and pull-in instability of the benchmark homogeneously deforming planar DE actuator. Further, the influence of temperature on the electromechanical responses of complex bi-layered bending actuator and buckling pump actuator involving inhomogeneous deformation is investigated. The numerical framework and the associated inferences can find their potential use in addressing the effect of temperature in the design of electro-active polymer based actuators.

A modified Steinberg–Cochran–Guinan model applicable to solid–liquid mixed zone along the principle Hugoniot

A constitutive model is vital for describing the deviatoric stresses of metallic materials under intense impact loading. Although the classical Steinberg–Cochran–Guinan (SCG) model has been widely regarded as an acceptable scheme, it does not clearly describe the coupling effect of the strain hardening and temperature softening at high temperatures and pressures, especially for the solid–liquid mixed phase. In this work, a modified SCG (MSCG) model is proposed based on an idealized assumption that the shear modulus and yield strength are equal to zero at the complete melting temperature and pressure. The temperature-dependent shear modulus model is then introduced in the MSCG model. The proposed model avoids the over-temperature softening effect by discarding the linear temperature term. More importantly, the pressure–temperature coupling factor (k) is employed in the pressure term, and the k adjusts the coupling effect of the strain hardening and temperature softening. In addition, the asymmetric relation in the MSCG model indicates that the temperature softening effect is more dominant than the strain hardening effect during the whole shock loading process. Moreover, the MSCG model is compared with the SCG model, Li and Chen's SCG model, and available data; the comparison verified the ability of the model to reproduce the shear moduli and yield strengths of Al, Be, Cu, and W in the solid–liquid mixed phase.

Evolution of defect concentration in Zr<sub>50–</sub><i><sub>x</sub></i>Cu<sub>34</sub>Ag<sub>8</sub>Al<sub>8</sub>Pd<i><sub>x</sub></i> (<i>x</i> = 0, 2) amorphous alloys derived using shear modulus and calorimetric data

Amorphous alloys exhibit unique physical and mechanical properties, which are closely connected with their microstructural heterogeneity. The correlation between structural heterogeneity and mechanical properties is one of the important issues of amorphous alloys. Micro-alloying is an effective way to tune the mechanical and physical properties of amorphous alloys. In the present study, Zr<sub>50–</sub><i><sub>x</sub></i>Cu<sub>34</sub>Ag<sub>8</sub>Al<sub>8</sub>Pd<i><sub>x</sub></i> (<i>x</i> = 0 and 2) amorphous alloys with ability to form excellent glass are chosen as model alloys. The evolutions of heat flow and shear modulus in different states (as-cast, relaxed and crystalline) with temperature of Zr<sub>50–</sub><i><sub>x</sub></i>Cu<sub>34</sub>Ag<sub>8</sub>Al<sub>8</sub>Pd<i><sub>x</sub></i> (<i>x</i> = 0 and 2) glass system are studied by differential scanning calorimetry (DSC) and electromagnetic-acoustic transformation (EMAT) technique, respectively. The experiment demonstrates that a decrease of the shear modulus is accompanied by the endothermic heat flow and vice versa. The correlation between the heat flow and shear modulus is investigated according to the interstitialcy theory. The calculations of the interstitialcy defect concentration and activation energy spectra suggest that the microstructure remains stable at relatively low temperatures. When temperature increases, the interstitialcy defect structure is activated. Compared with that in the as-cast state, the interstitialcy defect concentration in the relaxed state is reduced by structural relaxation, indicating that temperature-dependent shear modulus softening is inhibited. At temperatures above glass transition temperature, a rapid growth of interstitialcy defect concentration results in the accelerated shear softening, which is accompanied by significant endothermic heat flow. It is noted that the minor addition of palladium reduces the interstitialcy defect concentration in the Zr<sub>50–</sub><i><sub>x</sub></i>Cu<sub>34</sub>Ag<sub>8</sub>Al<sub>8</sub>Pd<i><sub>x</sub></i> (<i>x</i> = 0 and 2) metallic glass systems. It is suggested that the introduction of Pd reduces the atomic mobility and increases the characteristic relaxation time. In parallel, the change of shear modulus as a function of the aging time (below the glass transition temperature) is studied by using EMAT equipment. The results indicate that the interstitialcy defect concentration decreases in the physical aging process, which is accompanied by an increase of shear modulus. The interstitialcy defect concentration and shear modulus change towards the quasi-equilibrium state with aging time increasing. A reduction of the interstitialcy defect concentration leads to a decrease of the shear modulus change upon microalloying by Pd into Zr<sub>50–</sub><i><sub>x</sub></i> Cu<sub>34</sub>Ag<sub>8</sub>Al<sub>8</sub>Pd<i><sub>x</sub></i> (<i>x</i> = 0 and 2) metallic glass system.

Room and high temperature mechanical properties of Mg2Si, Mg2Sn and their solid solutions

Knowledge of the mechanical properties of thermoelectric materials at high temperature is required for the modelling and optimization of a thermoelectric generator under application conditions. We present room and high temperature characterization of Young’s modulus for Mg2Si, Mg2Sn and their solid solutions establishing a linear dependence, both on composition and on temperature.Through the comparison of Resonant Ultrasound Spectroscopy and free vibration techniques, we measure the elastic moduli with high precision. The comparison of our results (E = 110 GPa for Mg2Si and 78 GPa for Mg2Sn) to first principles calculations and experimentally reported data shows good agreement. Additionally, by estimating the Poisson ratio, we calculate the temperature-dependent shear modulus G and finally provide a simple bilinear function for Young’s and shear moduli as a function of temperature and composition.

The Solid Solution and Grain Boundary Hardening due to Mg in an Aluminum Alloy System at Room and Elevated Temperatures

The solid solution and grain boundary hardening due to Mg in Al-1.5wt pct Mn-0.5wt pct Cu system with 0.1 to 2.1 wt pct Mg has been studied at 25 °C to 345 °C. A transition temperature around 200 °C was found, while below it, the frictional stress, solid solution hardening from Mg and grain boundary hardening are little affected by temperature. This is due to the enhanced solute pinning on dislocations, by Cottrell clouds or other extensive segregation of solutes to dislocations. Above the transition, the solid solution hardening is controlled by the temperature-dependent shear modulus, which decreases linearly with the increasing temperature up to ~ 300 °C. Therefore, both the frictional stress and the solution hardening from Mg decrease linearly with the increasing temperature above ~ 200 °C. The grain boundary hardening obeys the Hall–Petch equation over the whole temperature range. Below the transition point the Petch slope is mainly controlled by the solute Mg, while above this temperature it is dependent on both the solute Mg level and temperature. The Petch slope increases linearly with solute Mg level, and decreases proportionally with the inverse of temperature. Semiempirical expressions for the yield strength at 25 °C to 345 °C were derived, including both solid solution and grain boundary effects.

Elastic strain response in the modified phase-field-crystal model**Project supported by the National Natural Science foundation of China (Grant Nos. 51571165 and 51371151), Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase), China, and the Fundamental Research Funds for the Central Universities, China (Grant No. 3102015BJ(II)ZS001).

To understand and develop new nanostructure materials with specific mechanical properties, a good knowledge of the elastic strain response is mandatory. Here we investigate the linear elasticity response in the modified phase-field-crystal (MPFC) model. The results show that two different propagation modes control the elastic interaction length and time, which determine whether the density waves can propagate or not. By quantitatively calculating the strain field, we find that the strain distribution is indeed extremely uniform in case of elasticity. Further, we present a detailed theoretical analysis for the orientation dependence and temperature dependence of shear modulus. The simulation results show that the shear modulus reveals strong anisotropy and the one-mode analysis provides a good guideline for determining elastic shear constants until the system temperature falls below a certain value.

A novel approach for modeling the flow stress curves of austenite under transient deformation conditions

A novel differential form of the integrated constitutive description earlier advanced by Jonas et al., for modeling the flow stress of austenite, is developed. The temperature and strain rate dependencies are introduced in the formalism through the temperature-dependent shear modulus of the material, the yield, saturation and steady-state stresses, as well as the time to achieve 50% dynamic recrystallization. The correlation between each of the above stresses and the Zener-Hollomon parameter is carried out by means of the Sellars-Tegart-Garofalo model employing the activation energy for the self-diffusion of Fe in austenite (Q=284kJmol−1). The proposed formalism involves the determination of the flow stress of the material by means of the numerical integration of three differential equations. In this way, it is possible to compute the flow stress both during the work-hardening and dynamic recovery stage, as well as from the onset of dynamic recrystallization up to the work softening transient and final achievement of the steady-state stress. Therefore, it is possible to predict the flow stress curves of austenite under both sharp and ramped transient loading conditions leading to the occurrence of dynamic recrystallization, a novel feature that cannot be accomplished by means of the early advanced model.

Temperature dependence of the scanning performance of an electrostatic microscanner

An optical microscanner is one examples of an optical-MEMS device, which scans a laser beam across one or two dimensions by reflecting it. The microscanner has a range of applications, such as laser printers, laser displays and bio-medical imaging. For each application, the mirror is required to oscillated at a certain frequency and optical scan angle. However, its scanning performance varies with temperature. To address this issue, the temperature dependence of the natural frequency of a 1D electrostatic microscanner formed of single-crystal silicon is investigated both theorectically and experimentally in this paper. As the temperature rises from 30 °C to 80 °C, the calculated value of the natural frequency decreased from 1910.81 Hz to 1908.68 Hz, and the experimental value decreased from 2123.85 Hz to 2120.56 Hz. The percentage changes in calculated and experimental results were −0.11% and −0.15%, and thus the former was consistent with the latter. The factors of the variation of natural frequency are the deformation caused by thermal expansion and the temperature dependence of shear modulus. The results of theoretical calculations indicated that the principal factor in the change of natural frequency was the shear modulus on the temperature.

Elastic properties and activation energy profiles of viscosity in Ge–Se system across the glass-transition range

The relation between the temperature dependence of shear modulus G(T) and viscosity η(T) of GexSe1−x (0≤x≤0.30) system across the glass-transition range is investigated using several ideas: the Wachtman equation for G, the shoving model and the bond strength–coordination number fluctuation (BSCNF) model of viscosity. Firstly, the present study shows that the empirical Wachtman equation reproduces reasonably well the shear modulus data. Next, it is shown that the shoving model implemented with the analysis of G describes quite well the high viscosity data in the glass state. Besides, the viscosity in the supercooled liquid side, especially for Ge3Se7, exhibits a clear change from Arrhenius-like to VFT-like temperature dependence upon heating. From the viscosity analysis for the composition GeSe4, GeSe3 and Ge3Se7, we obtain a maximum value for the activation energy across the glass-transition range. This result implies that there is an energetic barrier that the structural units need to surmount collectively before the occurrence of the viscous flow. The temperature variation of the activation energy also suggests that in terms of the BSCNF model, the decomposition of network structure upon heating is linked with the collective excitations of the structural units, which gives a viewpoint relevant to the softening of glass.

Elastic interpretation of the glass transition in aluminosilicate liquids

One of the remaining puzzles of the glass transition is the origin of a glass-forming liquid's ``fragility,'' which quantifies the departure of its relaxation time from Arrhenius-activated kinetics. According to the shoving model proposed by Dyre, fragility is controlled by the instantaneous shear modulus of the liquid, since any flow event requires a local volume increase, and the related activation energy is equal to the work done in shoving aside the surrounding atoms. Here, we present an in situ high-temperature Brillouin spectroscopy test of the shoving model near the glass transition of eight aluminosilicate glass-forming systems. We find that the measured viscosity data agree qualitatively with the measured temperature dependence of shear moduli, as predicted by the shoving model. However, the model systematically underpredicts the values of fragility for our aluminosilicate liquids. This suggests that the dynamics of the glass transition are governed by additional factors beyond the evolution of the shear modulus, such as configurational entropy. We have also compared the glass transition temperature $({T}_{\mathrm{g},\mathrm{vis}})$ obtained from viscosity (temperature at 10${}^{12}$ Pa s) with the onset temperatures of the decrease in elastic moduli (${T}_{\mathrm{g},\mathrm{elas}}$) and increase in the thermal expansion coefficient (${T}_{\mathrm{g},\mathrm{CTE}}$) during heating. While we find an approximate one-to-one correlation between ${T}_{\mathrm{g},\mathrm{vis}}$ and ${T}_{\mathrm{g},\mathrm{CTE}}$, it is clear that the elastic moduli probe a different frequency response of the glass structure, since ${T}_{\mathrm{g},\mathrm{elas}}$ is systematically lower than ${T}_{\mathrm{g},\mathrm{vis}}$.

Elastic instabilities in an antiferromagnetically ordered phase of the orbitally frustrated spinel GeCo2O4

Ultrasound velocity measurements of the orbitally frustrated spinel GeCo${}_{2}$O${}_{4}$ reveal unique elastic anomalies within the antiferromagnetic phase. Temperature dependence of shear moduli exhibits a minimum within the antiferromagnetic phase, suggesting the coupling of shear acoustic phonons to molecular spin-orbit excitations. Magnetic-field dependence of elastic moduli exhibits diplike anomalies, being interpreted as magnetic-field-induced metamagnetic and structural transitions. These elastic anomalies suggest that the survival of geometrical frustration, and the interplay of spin, orbital, and lattice degrees of freedom evoke a set of phenomena in the antiferromagnetic phase.

DETERMINATION OF PARAMETERS OF STEINBERG-GUINAN CONSTITUTIVE MODEL WITH SHOCK WAVE EXPERIMENTS

Steinberg-Guinan constitutive model is widely used in impact engineering simulations. Shock wave profile measurements of Al , Cu , and W under shock pressures ranging from 4 GPa to 200 GPa were analyzed, and shear modulus G and yield strength Y of these materials were obtained by using the high-pressure sonic velocities and rate independent Lagrange analysis results. Then values of derivatives of G and Y with respect to pressure or temperature at the reference state of these three metals were determined. By analyzing the pressure and temperature dependence of shear modulus and yield strength, we conclude that the relations of Steinberg-Guinan constitutive model, Y/G=constant, are approximately correct on Hugoniot state for these materials. So this is a probable approach to solve the difficulty of measurement of Y under high-pressure and high-temperature directly.

Pressure and temperature dependence of shear modulus and yield strength for aluminum, copper, and tungsten under shock compression

Experimental data for the shear modulus and yield strength of shocked aluminum, copper, and tungsten were systematically analyzed. Comparisons between these data and calculations using the Steinberg–Cochran–Guinan (SCG) constitutive model [D. J. Steinberg, S. G. Cochran, and M. W. Guinan, J. Appl. Phys. 51, 1498 (1980)] indicate that the yield strength has the same dependence on pressure and temperature as the shear modulus for aluminum for shock pressures up to 50 GPa, for copper to 100 GPa, and for tungsten to 200 GPa. Therefore, the assumption of Yp′∕Y0=Gp′∕G0,YT′∕Y0=GT′∕G0 is basically acceptable for these materials, and the SCG model can be used to describe the shear modulus and yield strength of the shocked material at high pressure and temperature.

Pressure and temperature dependence of shear modulus and yield strength for aluminum, copper and tungsten under shock compression

Experimental data of shear modulus and yield strength of shocked aluminum, copper and tungsten were analyzed systematically. Comparison between these data and t he results calculated by using SCG(Steinberg-Cochran-Guinan) constitutive model (J.Appl.Phys.51(3):1498) shows that shear modulus and yield strength have simila r dependence on pressure and temperature under conditions for aluminum below 50 GPa, copper below 100 GPa and tungsten below 200 GPa shock pressures. So the as sumptions in the SCG model,Y′pY0=G′pG0,Y′TY0=G′TG0, are approximately correct in these pressure ranges, and the SCG model can be used to describe the shear modulus and yield strength of these metals at high shock pr essure and high temperature.

Creep deformation behavior of Sn–3.5Ag solder/Cu couple at small length scales

In order to adequately characterize the behavior of solder balls in electronic devices, the mechanical behavior of solder joints needs to be studied at small length scales. The creep behavior of single solder ball Sn–Ag/Cu solder joints was studied in shear, at 25, 60, 95, and 130 °C, using a microforce testing system. A change in the creep stress exponent with increasing stress was observed and explained in terms of a threshold stress for bypass of Ag 3Sn particles by dislocations. The stress exponent was also temperature dependent, exhibiting an increase in exponent of two from lower to higher temperature. The activation energy for creep was found to be temperature dependant, correlating with self-diffusion of pure Sn at high temperatures, and dislocation core diffusion of pure Sn at lower temperatures. Normalizing the creep rate for activation energy and the temperature-dependence of shear modulus allowed for unification of the creep data. Microstructure characterization, including preliminary TEM analysis, and fractographic analysis were conducted in order to fully describe the creep behavior of the material.

Simple constitutive relationship for C–Mn steels deformed at elevated temperatures

The strain, strain rate and temperature dependence of a C-Mn steel (BS grade 430) deformed under plane strain compression conditions have been investigated in the temperature range 1173-1373 K, at strain rates of 0.25-25 s-1. It has been determined that the stress-strain behaviour of this material can be modelled satisfactorily in a rational manner by combining the exponential saturation law advanced by Sah et al., for description of the evolution of the flow stress, and modified appropriately to account for dynamic recrystallisation effects, with the kinetic models advanced by Kocks and Sellars-Tegart-Garofalo. The Kocks model has been introduced for description of the temperature and strain rate dependence of both the initial flow stress, at the beginning of plastic flow, and the apparent saturation stress, achieved at elevated strains. The Sellars-Tegart-Garofalo model has been employed for the description of the actual saturation stress, achieved after the work softening transient brought about by the operation of dynamic recrystallisation, also in terms of temperature and strain rate. The relaxation strain, in the Sah et al. law has been found to be dependent on the effective strain rate and it has been determined that such a dependency can be described by means of a simple parametric power law relationship. Also, it has been found that the optimised value for activation energy for deformation at elevated temperatures is very close to the activation energy of self-diffusion of iron in austenite, which emphasises the need to introduce into this kind of analysis scaling parameters as the temperature dependent shear modulus of the material. The stress-strain/strain rate/temperature relationship that has been developed in the present work is able to characterise satisfactorily both the flow stress and work hardening rate of the material under the deformation conditions explored, allowing a valid interpolation of the flow stress at any strain, temperature, and strain rate.

Degradation of niobium in molten aluminium: A. Robin, H.R.Z. Sandim. (Faculty of Chemical Engineering (Materials), Lorena, Brazil.) Int. J. Refract. Metals/Hard Mater., Vol 20, No 3, 2002, 221–225

Knowledge of the mechanical properties of thermoelectric materials at high temperature is required for the modelling and optimization of a thermoelectric generator under application conditions. We present room and high temperature characterization of Young’s modulus for Mg2Si, Mg2Sn and their solid solutions establishing a linear dependence, both on composition and on temperature.Through the comparison of Resonant Ultrasound Spectroscopy and free vibration techniques, we measure the elastic moduli with high precision. The comparison of our results (E = 110 GPa for Mg2Si and 78 GPa for Mg2Sn) to first principles calculations and experimentally reported data shows good agreement. Additionally, by estimating the Poisson ratio, we calculate the temperature-dependent shear modulus G and finally provide a simple bilinear function for Young’s and shear moduli as a function of temperature and composition.

Creep behavior of eutectic Sn–Ag lead-free solder alloy

Precipitation-strengthened tin-based eutectic Sn–3.5 Ag alloy was investigated for its creep behavior at three temperatures ranging from 303 to 393 K, under the tensile stress range of σ/E = 10−4 to 10−3. The steady-state creep rates cover seven orders of magnitude (10−3 to 10−9 s−1). The initial microstructure was found to have Ag3Sn intermetallic compound finely dispersed in the matrix of β–Sn. By incorporation of a threshold stress, σth, into the analysis, the creep data of eutectic Sn–Ag at all temperatures can be fitted by a single straight line with a slope of seven after normalizing the steady-state creep rate and the effective stress, indicating that the creep rates are controlled by the dislocation-pipe diffusion in the Sn matrix. The steady-state creep rate, , can then be expressed as , where QC is the creep activation energy, G is the temperature-dependent shear modulus, b is Burger's vector, R is the universal gas constant, T is the absolute temperature, σ is the applied stress, A is a material-dependent constant, and , in which σOB is the Orowan bowing stress and kR is the relaxation factor.

The isotropic shear modulus of multicomponent Fe-base solid solutions

A critical analysis of the available experimental data for the effect of alloying elements on the isotropic shear modulus of bcc (body-centered cube) Fe– X ( X=Al, Be, C, Co, Cr, Ge, Ir, Mn, Ni, Pt, Re, Rh, Ru, Si and V) solid solutions is carried out. The total effect of a solute on the shear modulus is decomposed into two contributions: the electronic (or chemical) and the volumetric. A systematic trend of the electronic contribution is demonstrated as a function of electron-to-atom ( e/a) ratio and the ground-state electronic configuration of the solute atom. Based on the demonstrated trend, we predict the chemical contribution of the shear modulus of Cu, Mo, N, Nb, Ti and W in ferromagnetic α-Fe (bcc), and that of Ti and V in paramagnetic γ-Fe [face-centered cube (fcc)]. These along with the corresponding volumetric contributions enable us to predict the total effect of a solute on the shear modulus in α-Fe and γ-Fe. In the case of γ-Fe, we derive the chemical and volumetric contributions of Ni and Pt from the experimental shear modulus data of paramagnetic Fe–Ni and Fe–Pt alloys while those of C, Co, Cr, Mn, Mo, N and Si are derived from the shear modulus of paramagnetic Fe–Ni– X alloys. In the case of Al, Be, Cu, Ge, Ir, Nb, Re, Rh , Ru and W, the total effect on the shear modulus is calculated by assuming that the electronic contribution to the shear modulus in γ-Fe is the same as in α-Fe. To calculate the isotropic shear modulus of multicomponent bcc and fcc solid solutions, we propose linear superposition laws. The proposed relationships are validated using the experimental data of a large number of multicomponent alloys having austenitic, ferritic, and lath martensitic microstructures. It is demonstrated that for all three microstructures, in most cases the shear modulus can be predicted with an accuracy of ±3% in multicomponent solid solutions. It is also found that the high dislocation density in lath martensite accounts for a decrease in shear modulus by about 5% compared to the ferritic counterpart. We also demonstrate that the temperature dependence of shear modulus in multicomponent bcc and fcc solid solutions is similar to that of pure α- and γ-Fe, respectively, for up to about 800 K.

Creep behavior of eutectic Sn-Cu lead-free solder alloy

Tensile creep behavior of precipitation-strengthened, tin-based eutectic Sn-0.7Cu alloy was investigated at three temperatures ranging from 303-393 K. The steady-state creep rates cover six orders of magnitude (10-3 - 10-8 s-1) under the stress range of σ/E = 10-4 - 10-3. The initial microstructure reveals that the intermetallic compound Cu6Sn5 is finely dispersed in the matrix of β-Sn. By incorporating a threshold stress, σth, into the analysis, the creep data of eutectic Sn-Cu at all temperatures can be fitted by a single straight line with a slope of 7 after normalizing the steady-state creep rate and the effective stress, indicating that the creep rates are controlled by the dislocation-pipe diffusion in the tin matrix. So the steady-state creep rate, e, can be expressed as e = A Gb/RT(σth/G) exp (-Qc/RT), where Qc is the activation energy for creep, G is the temperature-dependent shear modulus, b is the Burgers vector, R is the universal gas constant, T is the temperature, σ is the applied stress, A is a material-dependent constant, and σth = (σOB√1-kR2, in which σOB is the Orowan bowing stress, and kR is the relaxation factor.