Isotropic Continuum Model Research Articles

Effect of cutting speed on shear band formation and chip morphology of Ti–6Al–4V alloy using nanoindentation and EBSD mapping

The hardness and orientations of the primary α grains (αp) and the transformed β (βt) grains in segmented chips obtained by turning an as-received solution treated-and-aged bar of Ti–6Al–4V (wt.%) at cutting speeds of 1, 1.5 and 2 m/s (61, 91, and 122 m/min) were analyzed using nanoindentation mapping and electron backscattered diffraction (EBSD) maps. The hardness of each αp grain in the as-received Ti–6Al–4V bar highly depends on the crystal orientation, varying from 4.5 GPa to 6.7 GPa, and the hardness values of βt were about 20% lower. In the machined chips, αp grains showed hardness values similar to the as-received material while βt became slightly harder. The homogeneous shear strain within the segments (between shear bands) was estimated to be around 0.4–0.7 at all three cutting speeds. The width of the shear bands in the 1 and 1.5 m/s chips were ∼1.3 μm and ∼2 μm in the 2 m/s chips, which are 10–40% of thicknesses predicted by Molinari's isotropic continuum model, respectively. A smaller homogeneous shear strain was correlated with larger shear cracks at each cutting speed. Grain orientations favored by prism ⟨a⟩ slip in αp when cutting at 1 m/s made the localized shear band less “adiabatic” since both the shear stress and strain were lower when these orientations were dominant. In contrast, in orientations that favored pyramidal <c+a> slip, the αp grains experienced much higher shear stress and more localized shear strain when cutting at 2 m/s, leading to higher temperatures that were much more likely to induce the α to β phase transformation. A transformation to soft β phase during shearing accelerates plastic instabilities. These observations and comparisons with finite element simulations indicate that temperatures reached the α+β phase field locally, and higher cutting speeds led to more β phase that is detrimental to the tool life.

Theoretical and experimental evaluation of thermoelectric performance of alkaline earth filled skutterudite compounds

Skutterudite compounds are one of the most promising thermoelectric materials that can be used for intermediate power generation applications because of their excellent thermoelectric and mechanical properties. In spite of extensive research efforts during the past two decades, there is still a lack of full understanding of the electronic and thermal transport in these compounds. In the present study, we carry out a combined experimental and theoretical investigation of the electronic and thermal transport properties of alkaline earth-filled (Ca,Sr,Ba)0.2Co4Sb12 skutterudites in the temperature range from 300 ​K to 800 ​K. The experimental work includes structural properties probed by PXRD and EMPA for phase purity and composition analysis. Theoretical values of the temperature dependent electronic transport parameters were determined using the nearly-free electron approximation, and various phonon thermal conductivity contributions were studied using the Debye isotropic continuum model and the theory of Price for the bipolar contribution. The theoretical and experimental transport parameters were found in good agreement with each other. They confirm that both Ba and Sr filled skutterudites are highly degenerate semiconductors, while the heat transport in the Ca filled compound is dominated by the bipolar contribution at temperatures above 300 ​K. The Sr filled skutterudite shows the highest Seebeck coefficient, likely due to its lowest donor ionisation energy. From our detailed theoretical investigations, it is seen that the key mechanism to control the phonon thermal conduction is originated from the interaction between mass defects and phonons. The maximum ZT value obtained 0.14 for Sr0.2Co4Sb12 compound at 800 ​K likely indicate that Sr is more effective filler among the alkaline earth metals.

Thermoelectric transport behaviours of n-type Mg2 (Si,Sn,Ge) quaternary solid solutions

Abstract Mg2X (X=Si, Sn, and Ge) based systems have attracted widespread attention owing to their various benefits in thermoelectric applications. In particular, to date, ternary Mg2X based solid solutions have become one of the most widely investigated thermoelectric systems. However, the investigation of temperature varied thermoelectric properties of Mg2X based quaternary systems is rather limited both theoretically and experimentally. Therefore, here, we report a rigorous theoretical work of thermoelectric properties for n-type Mg2Si 0.55 − z Sn0.4Ge0.05Biz quaternary solid solutions (z = 0.02, 0.025, 0.03, and 0.035) from 300 K to 850 K. By using nearly-free-electron model together with Fermi–Dirac statistics we define Fermi level both in extrinsic and intrinsic regimes as a function of temperature. We follow Hicks and Dresselhaus’ approach to calculate electronic transport properties. By performing the Debye’s isotropic continuum model a detailed theoretical investigation of lattice thermal conductivity is presented among with various phonon relaxation rates. From our theoretical analysis the highest ZT is attained for Mg2Si0.53Sn0.4Ge0.05Bi0.02 solid solution as 1.14 at 850 K.

Effective mobility recovery after femoral neck fractures: The necessity of gait analysis in the immediate post-operative aftercare

Mg2X (X=Si, Sn, and Ge) based systems have attracted widespread attention owing to their various benefits in thermoelectric applications. In particular, to date, ternary Mg2X based solid solutions have become one of the most widely investigated thermoelectric systems. However, the investigation of temperature varied thermoelectric properties of Mg2X based quaternary systems is rather limited both theoretically and experimentally. Therefore, here, we report a rigorous theoretical work of thermoelectric properties for n-type Mg2Si0.55−zSn0.4Ge0.05Biz quaternary solid solutions (z = 0.02, 0.025, 0.03, and 0.035) from 300 K to 850 K. By using nearly-free-electron model together with Fermi–Dirac statistics we define Fermi level both in extrinsic and intrinsic regimes as a function of temperature. We follow Hicks and Dresselhaus’ approach to calculate electronic transport properties. By performing the Debye’s isotropic continuum model a detailed theoretical investigation of lattice thermal conductivity is presented among with various phonon relaxation rates. From our theoretical analysis the highest ZT is attained for Mg2Si0.53Sn0.4Ge0.05Bi0.02 solid solution as 1.14 at 850 K.

Short note on the bending models for a membrane in capsule mechanics: Comparison between continuum and discrete models

With regard to the out-of-plane bending energy of an elastic surface membrane, capsule deformation was compared between Helfrich's isotropic continuum model and discrete models using a triangulated surface. The in-plane deformation of the membrane was modeled by the neo-Hookean or Skalak law. Two mechanical problems were numerically simulated. One was the deformation of a spherical capsule under simple shear flow and the other was the equilibrium shape of a capsule with a nonzero excess surface area, assuming the shape mechanics of a normal red blood cell. The numerical simulations demonstrated that a discrete model based on locally averaged mean curvature satisfactorily reproduces the mechanical behavior determined by Helfrich's isotropic continuum model, whereas a discrete model without the averaging exhibited different deformation behavior. Differences in the mechanical behavior between bending models may cause differences in estimated bending rigidity values of up to one order of magnitude.

Anharmonic effects in the thermoelectric properties of PbTe

In this work, we investigate the crystal anharmonic effects in the thermoelectric properties of n-type PbTe. The lattice thermal transport coefficient is computed by employing an isotropic continuum model for the dispersion relation for acoustic as well as optical phonon branches, an isotropic continuum model for crystal anharmonicity, and the single-mode relaxation time scheme. The electronic components of the transport coefficients in a wide temperature range are calculated using the isotropic-nearly-free-electron model, interaction of electrons with deformation potential of acoustic phonons, and the effect of the band non-parabolicity. It is found that the transverse optical branches play a major role in determining the phonon conductivity and the thermoelectric figure of merit of this material.

Biaxial response of ovine spinal cord dura mater

The dura mater performs a major functional role in the stability and mechanical response of the spinal cord complex. Computational techniques investigating the etiology of spinal cord injury require an accurate mechanical description of the dura mater. Previous studies investigating the mechanical response of the dura mater have reported conflicting results regarding the anisotropic stiffness of the dura in the longitudinal and circumferential direction. The aim of this study was to investigate the biaxial response of the dura mater in order to establish the tissue level mechanical behavior under physiological loading scenarios. To this end, square sections of the dura were tested in a custom biaxial setup under a comprehensive uniaxial and biaxial loading protocol. The resultant data were fit via a transversely isotropic continuum model and an anisotropic continuum constitutive model. The transversely isotropic formulation failed to accurately predict the dura mater׳s uniaxial behavior. The anisotropic formulation accurately predicted the uniaxial response in both longitudinal and circumferential directions. Significantly higher stiffness (p<0.0001) was observed in the circumferential direction as compared to the longitudinal direction. Further, the longitudinal direction displayed a significantly lower degree of nonlinearity (p<0.045) and significantly higher degree of collagen fiber dispersion (p<0.032) as compared to the circumferential direction. Results indicate that the dura mater has differential mechanical response in the longitudinal and circumferential directions and future studies should utilize an anisotropic two fiber family continuum model to accurately describe dura mater mechanics.

A Theoretical study of the Thermoelectric Transport Coefficients of n-type PbTe

ABSTRACTIn this work we present a theoretical study of the transport coefficients of n-type PbTe. The electronic transport coefficients are calculated using the isotropic-nearly-free-electron approximation, including the effect of band non-parabolicity on electron-phonon scattering. The lattice thermal transport coefficient is computed by employing the isotropic continuum model for the dispersion relation for acoustic as well as optical phonon branches, an isotropic anharmonic continuum model for crystal anharmonicity, and the single-mode relaxation time scheme. The role of transverse optical (TO) phonon modes in anharmonic interactions will be discussed in detail.

Phonon intrinsic mean free path in zincblende AlN

We have carried out calculations of phonon anharmonic inverse relaxation time in the cubic phase of AlN by using phonon dispersion results from a full lattice dynamical model and applying Fermi's golden rule formula. The cubic anharmonicity in crystal potential has been modelled within the anharmonic elastic continuum model. The relative roles of three-phonon Normal and Umklapp processes have been examined at different temperatures. Increase in the relaxation rate of acoustic phonons due to their interaction with optical phonons has also been quantified. The calculated results for the relaxation times of the acoustic phonon modes show appreciable differences from the results obtained from the application of the isotropic continuum model, both at low and high temperatures. The calculated results for the mean free path over a large temperature range have been compared with available estimates obtained from thermal conductivity measurements.

Dynamical shift and decay of low-energy vibrational modes: CO on Ni and Pt(111)

A surface lattice dynamics study of the effects of the coupling of vibrational modes with energies in the energy range of the bulk phonon band to a substrate lattice is presented. As illustrative examples the frustrated translational and rotational modes of a single adsorbed CO molecule on the (111) surface of Ni and Pt have been chosen. The results show that the coupling is weak for modes with energies below the zone boundary surface phonons and is well described by an elastic treatment of the substrate. Whereas modes with higher energies mix strongly with some of the zone boundary surface phonons accompanied with large broadenings and shifts from the mode energy in the rigid lattice situation. The corresponding strong satellite structures caused by this mixing could possibly be observed by vibrational spectroscopy like inelastic He scattering. Furthermore, the broadening and the shift in the elastic limit of the central nearest neighboring force constant model are compared with the results from an isotropic continuum model.

A continuum model of a solid with a free surface: I. Normal mode analysis of the model

A complete analysis is made of the normal modes of an isotropic continuum model of a solid which occupies the region (0, 0, 0) < ( x, y, z) < ( X, Y, Z), the limit ( X, Y, Z) → (∞,∞,∞) being taken. The surface z = 0 is a free surface and that z = Z is fixed; cyclic boundary conditions are used in the x and y directions. The modal displacements (vectors), modal frequencies (as a function of the wave-vectors), and wave-vector-densities are obtained for all types of normal modes; the work is a modification and extension of that of earlier workers.

Calculations of anharmonic phonon relaxation times

Anharmonic phonon relaxation times in Ge are calculated using (i) an isotropic continuum model, and (ii) a dispersive model. A complete spectrum of calculated results is presented. Frequency-averaged values for normal- and umklapp-three-phonon relaxation times are also calculated. A comparison is made between our findings and some earlier works, and disagreements are discussed. The results are applied in calculating some results in the theory of lattice thermal conductivity.

Calculation of bounds of the eigenvalue spectrum of anharmonic phonon collision operator

Extreme bounds on the eigenvalue spectrum of anharmonic phonon collision operator in Ge are calculated using (i) an isotropic continuum model, and (ii) Born-von-Karman model. The maximum occurs for short wave transverse modes. For the isotropic model the minimum occurs for longitudinal modes of the magnitude nearly 3 4 q D down to 27 K, and q D below it. For the dispersive model the minimum occurs for longitudinal modes with 3 4 q D for all temperatures.

Neutron scattering due to lattice distortion around point defects

The distortions calculated from four different models of a vacancy, one discrete lattice model, two anisotropic continuum models and one isotropic continuum model are used to calculate the differential cross section for coherent elastic scattering of long wavelength neutrons by a single vacancy in copper. The results of the local force anisotropic continuum model and the isotropic continuum model are distinctly different from those of the discrete lattice and nonlocal force anisotropic continuum models. The effect of the distortion around the vacancy on the differential scattering cross section extends to at least 600 neighbouring atoms.

On the continuum model for gas-surface interactions

A three-dimensional isotropic continuum model is used to look at the gas-surface interaction problem. Results for energy transfer to the continuum when acted upon by a transient force pulse approximating that expected during a collision process indicate a negligible amount is propagated into the solid as stress waves. This appears to support claims of researchers using discrete lattice atom models which ignore energy transfer to any but a surface array of atoms.

Variational Calculation of the Thermal Conductivity of Germanium

The thermal conductivity of germanium has been calculated using an anharmonic isotropic continuum model. This model has been extended so as to incorporate umklapp processes by introducing a pseudo-reciprocal-lattice translation vector of magnitude equal to the diameter of the Debye sphere, in such a way that the wave vectors of the three phonons involved are coplanar. Boundary and mass-difference scattering are also included, and the calculations, besides giving good agreement with experimental values of thermal conductivity, provide information on the relative contributions of the different scattering processes to the thermal resistivity and of the different polarization branches to the conductivity. In particular, it is shown that almost all the heat transport is by transverse phonons.