Eulerian Strain Research Articles

Theoretical study of some thermodynamical properties for solid under high pressure using finite-strain EOS

This work involves the influence of high pressure on some thermodynamical properties of solid aluminum (Al) using a 3rd and 4th order Birch–Murnaghan equation of state based on the finite Eulerian strain theory.The comparison results for relative compression volume in the range of 1–0.65 versus applied high pressure for the 3rd and 4th order B–M EOS and experimental data indicated that up to 20GPa are in good agreement. But beyond this and up to 100GPa, the results for the 3rd order B–M EOS do diverge from the 4th order B–M EOS as well as from the experimental data.The isothermal bulk modulus (KT) has been worked out by using the 3rd and 4th order B–M EOS. Results were not found in good agreement above 20GPa. However, the data for first pressure derivative (KT′) for both 3rd and 4th order B–M EOS show a big divergence starting from 20GPa.Finally, the effect of high pressure on (Al) melting temperature has been studied. The Mie–Gruneison–Debye equation (based upon the 3rd and 4th order B–M EOS) and Lindemann and Kumar’s equations were investigated. Comparison results for melting temperature as a function of melting pressure by using the above mentioned equations and the experimental data have been found to be in good agreement.

Reversible high-pressure phase transition in LaN

In situ high-pressure X-ray powder diffraction experiments on LaN up to 60.1 GPa at ambient temperature in a diamond-anvil cell revealed a reversible, first-order structural phase transition starting at ∼22.8 GPa and completed at ∼26.5 GPa from the ambient cubic phase (Fm3¯m, no. 225) to a tetragonal high-pressure phase (P4/nmm, no. 19, a = 4.1060(6), c = 3.0446(6) Å, Z = 2, wRp = 0.011), which has not been claimed in theoretical predictions. HP-LaN is isotypic with a high-pressure polymorph of BaO, which crystallizes in a tetragonally distorted CsCl-type structure. The phase transition is accompanied by a volume collapse of about 11% which corresponds well with the reported data on HP-BaO. A linear extrapolation of the c/a ratio of the tetragonally distorted CsCl-type sub-cell reaches a value c/a = 1 of cubic CsCl-type HP-LaN at 91(12) GPa. In addition, the compressibility of LaN was investigated and resulted in a bulk modulus for the ambient pressure phase of B0 = 135(3) GPa and B′ = 5.0(5) after fitting a third-order Birch-Murnaghan equation of state to the experimental p–V data. The corresponding extrapolated bulk modulus of HP-LaN is found to be B0 = 278(6) GPa and its pressure derivative B′ = 1.2(2). Both as-calculated bulk moduli are compared to the respective values obtained from an Eulerian strain versus normalized stress plot to be 143(2) GPa for ambient LaN and 293(7) GPa for HP-LaN. Compared to other binary nitrides such as δ-ZrN or δ-HfN having bulk moduli of 285 GPa and 306 GPa, respectively, the extrapolated bulk moduli of HP-LaN are in the same order of magnitude, ranking HP-LaN as a highly incompressible material.

The thermoelastic behavior of clintonite up to 10 GPa and 1,000°C

The thermoelastic behavior of a natural clintonite-1M [with composition: Ca1.01(Mg2.29Al0.59Fe0.12)Σ3.00(Si1.20Al2.80)Σ4.00O10(OH)2] has been investigated up to 10 GPa (at room temperature) and up to 960°C (at room pressure) by means of in situ synchrotron single-crystal and powder diffraction, respectively. No evidence of phase transition has been observed within the pressure and temperature range investigated. P–V data fitted with an isothermal third-order Birch–Murnaghan equation of state (BM-EoS) give V0 = 457.1(2) A3, KT0 = 76(3)GPa, and K′ = 10.6(15). The evolution of the “Eulerian finite strain” versus “normalized stress” shows a linear positive trend. The linear regression yields Fe(0) = 76(3) GPa as intercept value, and the slope of the regression line leads to a K′ value of 10.6(8). The evolution of the lattice parameters with pressure is significantly anisotropic [β(a) = 1/3KT0(a) = 0.0023(1) GPa−1; β(b) = 1/3KT0(b) = 0.0018(1) GPa−1; β(c) = 1/KT0(c) = 0.0072(3) GPa−1]. The β-angle increases in response to the applied P, with: βP = β0 + 0.033(4)P (P in GPa). The structure refinements of clintonite up to 10.1 GPa show that, under hydrostatic pressure, the structure rearranges by compressing mainly isotropically the inter-layer Ca-polyhedron. The bulk modulus of the Ca-polyhedron, described using a second-order BM-EoS, is KT0(Ca-polyhedron) = 41(2) GPa. The compression of the bond distances between calcium and the basal oxygens of the tetrahedral sheet leads, in turn, to an increase in the ditrigonal distortion of the tetrahedral ring, with ∂α/∂P ≈ 0.1°/GPa within the P-range investigated. The Mg-rich octahedra appear to compress in response to the applied pressure, whereas the tetrahedron appears to behave as a rigid unit. The evolution of axial and volume thermal expansion coefficient α with temperature was described by the polynomial α(T) = α0 + α1T−1/2. The refined parameters for clintonite are as follows: α0 = 2.78(4) 10−5°C−1 and α1 = −4.4(6) 10−5°C1/2 for the unit-cell volume; α0(a) = 1.01(2) 10−5°C−1 and α1(a) = −1.8(3) 10−5°C1/2 for the a-axis; α0(b) = 1.07(1) 10−5°C−1 and α1(b) = −2.3(2) 10−5°C1/2 for the b-axis; and α0(c) = 0.64(2) 10−5°C−1 and α1(c) = −7.3(30) 10−6°C1/2for the c-axis. The β-angle appears to be almost constant within the given T-range. No structure collapsing in response to the T-induced dehydroxylation was found up to 960°C. The HP- and HT-data of this study show that in clintonite, the most and the less expandable directions do not correspond to the most and the less compressible directions, respectively. A comparison between the thermoelastic parameters of clintonite and those of true micas was carried out.

An Eulerian approach for fluid–structure interaction problems

This paper describes a numerical method for simulating fluid–structure interaction problems on a fixed Cartesian grid. The fluid–structure interfaces are sharply represented by the aid of the level-set function and virtual flux method. The virtual flux method is one of the sharp interface Cartesian grid methods. The numerical flux across the interface is replaced with the virtual flux so that proper interface conditions must be satisfied there. For fluid and elastic structure interaction problems, the Navier–Stokes equations and unsteady evolution equations for the Eulerian strain tensor of elastic solid are simultaneously solved, so that the strain and stress tensors as well as velocities and pressures in both fluid and solid mediums can be analyzed on the fixed grid. Validity of the numerical method is demonstrated for numerical simulations about reciprocating engines, reciprocating compressors, and deformations of elastic particles in fluid flows. It is confirmed that the Eulerian approach is easily applicable to the simulation of fluid–structure interaction problems.

High-pressure structural behavior of -Fe2O3 studied by single-crystal X-ray diffraction and synchrotron radiation up to 25 GPa

In situ X‑ray diffraction experiments were carried out at pressures up to 25 GPa on a synthetic hematite (α-Fe 2 O 3 ) crystal using synchrotron radiation in an angle-dispersive setup. Experiments were performed in diamond-anvil cells using neon as a pressure-transmitting medium. Single-crystal diffraction data were collected from omega scans and structural refinements were carried out for 10 pressure points. Bulk and linear incompressibilities were obtained from least-squares fits of refined data to the Eulerian strain based Birch-Murnaghan equation of state. Finite strain analysis suggests a truncation at second order, yielding results of K 0 = 207(3), Ka 0 = 751(17), and K c0 = 492(8) for bulk and axial moduli, respectively. The a-axis is about 1.5 times stiffer than the c-axis. Compression of the main structural feature, the FeO 6 octahedra, is quite uniform, with just slight changes of distortion parameters at higher pressures.

Numerical homogenisation of an elasto-plastic model-material with large elastic strains: macroscopic yield surfaces and the Eulerian normality rule

This article presents the details of a numerical technique for computing the macroscopic response of a material with a given micro-structure to arbitrary prescribed loading histories. The method uses classical concepts of homogenisation theory in combination with the finite element method and focusses on the computation of macroscopic yield surfaces and inelastic strain rates. It places no restrictions on the magnitude of deformation and allows arbitrary combinations of stress- or strain control including the prescription of histories of the Cauchy stress. The method is illustrated by analysing a model material, consisting of a non-linearly elastic matrix with stiff elasto-plastic inclusions, which exhibits macroscopically associative elasto-plastic material behaviour with finite elastic strains. Yield surfaces and the directions of plastic flow after a prior finite simple shear deformation are computed for this material and are shown to be consistent with an additive decomposition of the Eulerian strain rate into elastic and plastic parts and a suitable formulation of the normality rule in Cauchy stress space; a novel version of the latter is derived which is valid for an arbitrary reference configuration.

Behavior of epidote at high pressure and high temperature: a powder diffraction study up to 10 GPa and 1,200 K

The thermo-elastic behavior of a natural epidote [Ca1.925 Fe0.745Al2.265Ti0.004Si3.037O12(OH)] has been investigated up to 1,200 K (at 0.0001 GPa) and 10 GPa (at 298 K) by means of in situ synchrotron powder diffraction. No phase transition has been observed within the temperature and pressure range investigated. P–V data fitted with a third-order Birch–Murnaghan equation of state (BM-EoS) give V0 = 458.8(1)A3, KT0 = 111(3) GPa, and K′ = 7.6(7). The confidence ellipse from the variance–covariance matrix of KT0 and K′ from the least-square procedure is strongly elongated with negative slope. The evolution of the “Eulerian finite strain” vs “normalized stress” yields Fe(0) = 114(1) GPa as intercept values, and the slope of the regression line gives K′ = 7.0(4). The evolution of the lattice parameters with pressure is slightly anisotropic. The elastic parameters calculated with a linearized BM-EoS are: a0 = 8.8877(7) A, KT0(a) = 117(2) GPa, and K′(a) = 3.7(4) for the a-axis; b0 = 5.6271(7) A, KT0(b) = 126(3) GPa, and K′(b) = 12(1) for the b-axis; and c0 = 10.1527(7) A, KT0(c) = 90(1) GPa, and K’(c) = 8.1(4) for the c-axis [KT0(a):KT0(b):KT0(c) = 1.30:1.40:1]. The β angle decreases with pressure, βP(°) = βP0 −0.0286(9)P +0.00134(9)P2 (P in GPa). The evolution of axial and volume thermal expansion coefficient, α, with T was described by the polynomial function: α(T) = α0 + α1T−1/2. The refined parameters for epidote are: α0 = 5.1(2) × 10−5 K−1 and α1 = −5.1(6) × 10−4 K1/2 for the unit-cell volume, α0(a) = 1.21(7) × 10−5 K−1 and α1(a) = −1.2(2) × 10−4 K1/2 for the a-axis, α0(b) = 1.88(7) × 10−5 K−1 and α1(b) = −1.7(2) × 10−4 K1/2 for the b-axis, and α0(c) = 2.14(9) × 10−5 K−1 and α1(c) = −2.0(2) × 10−4 K1/2 for the c-axis. The thermo-elastic anisotropy can be described, at a first approximation, by α0(a): α0(b): α0(c) = 1 : 1.55 : 1.77. The β angle increases continuously with T, with βT(°) = βT0 + 2.5(1) × 10−4T + 1.3(7) × 10−8T2. A comparison between the thermo-elastic parameters of epidote and clinozoisite is carried out.

Compression behaviour of nitridocarbidosilicates [formula omitted][formula omitted] studied with X-ray diffraction and ab initio calculations

The compressibilities of the nitridocarbidosilicates Ln 2 [ Si 4 N 6 C ] , Ln = Ho , Er, were investigated by in situ powder synchrotron X-ray diffraction. Pressures up to 36 GPa were generated using the diamond-anvil cell technique. A fit of a third-order Birch–Murnaghan equation of state to the p – V data resulted in the bulk modulus B 0 = 162 ( 2 ) GPa and its pressure derivative B ′ = 5.1 ( 3 ) at V 0 = 605.1 ( 2 ) A ˚ 3 for Ho 2 [ Si 4 N 6 C ] . For Er 2 [ Si 4 N 6 C ] the values are B 0 = 163 ( 5 ) GPa and B ′ = 4.5 ( 6 ) at V 0 = 602.8 ( 2 ) A ˚ 3 . Complementary data of the pressure dependence of this family of compounds were obtained by density functional theory based model calculations for Y 2 [ Si 4 N 6 C ] . An anomalous compression behaviour is predicted to occur between 8 and 12 GPa for Y 2 [ Si 4 N 6 C ] . The bulk modulus was obtained from an Eulerian strain versus normalized stress plot to be about 165 GPa, i.e., a value very similar to the bulk moduli obtained experimentally for Er 2 [ Si 4 N 6 C ] and Ho 2 [ Si 4 N 6 C ] . The anomaly in the pressure dependence of the cell parameters seems to be due to an increase in the coordination number of the two symmetrically independent Y cations. This increase of coordination is mainly achieved by a rotation of the Si ( 3 ) N 3 C tetrahedron.

Elasticity of stishovite and acoustic mode softening under high pressure by Brillouin scattering

Brillouin scattering measurements on single-crystal stishovite, a high-pressure polymorph of SiO 2, were carried out to 22 GPa. Acoustic velocities in three 30-μm thick crystal platelets of synthetic stishovite were measured in a forward symmetric scattering geometry, and the full set of elastic constants were retrieved to 12 GPa. The measured velocity data were fit to Christoffel’s equation, yielding ambient-pressure elastic constants of C 11 = 455(1) GPa, C 33 = 762(2) GPa, C 12 = 199(2) GPa, C 13 = 192(2) GPa, C 44 = 258(1) GPa, and C 66 = 321(1) GPa. The elastic modulus ( C 11 − C 12)/2 was observed to decrease with pressure, indicating acoustic mode softening, consistent with theoretical predications for the behavior of stishovite as it approaches the transition to the CaCl 2-type phase. The bounds on the aggregate adiabatic bulk and shear moduli are K S0 = 315(1) GPa, G 0 = 240(1) GPa for the Voigt bound, K S0 = 301(1) GPa, G 0 = 216(1) GPa for the Reuss bound. Pressure derivatives of aggregate bulk and shear moduli were constrained to be ( ∂K S/ ∂P) T0 = 4.34(16) and ( ∂G/ ∂P) 0 = 0.7(1) for the Reuss bound, and ( ∂K S/ ∂P) T0 = 4.0(1) and ( ∂G/ ∂P) 0 = 1.1(1) for the Voigt–Reuss–Hill (VRH) average, respectively, by fitting the data to Eulerian finite strain equations. The volume compression curve obtained from our Brillouin measurement is in very good agreement with previous compression studies up to 50 GPa.

Leucite at high pressure: Elastic behavior, phase stability, and petrological implications

Elastic and structural behavior of a natural tetragonal leucite from the volcanic Lazium district (Italy) were investigated at high pressure by in situ single-crystal X-ray diffraction with a diamond anvil cell under hydrostatic conditions. A first-order phase transition, never reported in the literature, was observed at P = 2.4 ± 0.2 GPa from tetragonal ( I 4 1 / a ) to triclinic symmetry (analysis of diffraction intensities suggests the space group P 1), accompanied by a drastic increase in density of about 4.7%. The transition pressure was bracketed by several measurements in compression and decompression. No further phase-transition has been observed up to 7 GPa. Fitting a second-order Birch-Murnaghan equation of state (BM-EoS) to the pressure-volume data of the tetragonal polymorph, we obtain K 0 = 41.9(6) GPa and K ′ = 4 (fixed). In the case of the triclinic polymorph, a second-order BM-EoS gives K 0 = 33.2(5) GPa. The eulerian finite strain ( f e ) vs. normalized stress ( F e ) curves were calculated for the low- and high- P polymorphs, providing F e (0) = 42(1) and F e (0) = 33.2(4) GPa, respectively. The axial bulk modulus values of the tetragonal polymorph, calculated with a linearized BM-EoS, are K 0 ( a ) = 34.5(5) and K 0 ( c ) = 78(1) GPa. For the triclinic polymorph, we obtain K 0 ( a ) = 35.9(5), K 0 ( b ) = 34.9(7), and K 0 ( c ) = 35.5(7) GPa. The elastic behavior of the low- P polymorph appears to be more anisotropic than that of the high- P polymorph. The HP-crystal structure evolution of the tetragonal polymorph of leucite was studied on the basis of six structural refinements at different pressures between 0.0001 and 1.8 GPa. The main deformation mechanisms at high-pressure are due to tetrahedral tilting, giving rise to an increase of the ellipticity of the four- and six-membered rings of the tetrahedral framework. The T-O bond distances are practically invariant within the stability field of the tetragonal polymorph. The complex P -induced twinning, due to the tetragonal → triclinic phase-transition, and the low quality of the diffraction data at pressure above the phase-transition, did not allow the refinement of the crystal structure of the triclinic polymorph.

Pressure and temperature dependence of the elasticity of pyrope–majorite [Py 60Mj 40 and Py 50Mj 50] garnets solid solution measured by ultrasonic interferometry technique

Compressional (P) and shear (S) wave velocities have been measured for two synthetic polycrystalline specimens of pyrope–majorite garnets [Py 60Mj 40 and Py 50Mj 50] by ultrasonic interferometry to 8 GPa and 1000 K, in a DIA-type cubic anvil high pressure apparatus (SAM-85) interfaced with synchrotron X-radiation and X-ray imaging. Elastic bulk ( K S) and shear ( G) moduli data obtained at the end of the cooling cycles were fitted to functions of Eulerian strain to third order yielding pressure derivatives of the elastic moduli (∂ K S/∂ P) T = 4.3 (3); (∂ G/∂ P) T = 1.5 (1) for Py 60Mj 40 garnet and (∂ K S/∂ P) T = 4.4 (1); (∂ G/∂ P) T = 1.3 (1) for Py 50Mj 40 garnet. Both (∂ K S/∂ P) T and (∂ G/∂ P) T are identical for the two garnet compositions and are also consistent with Brillouin scattering data for polycrystalline Py 50Mj 50. Moreover, the new pressure derivatives of the elastic moduli are equal within experimental uncertainties to those of end-member pyrope garnet from ultrasonic studies [Gwanmesia, G.D., Zhang. J, Darling, K., Kung, J., Li, B., Wang, L., Neuville, D., Liebermann, R.C., 2006. Elasticity of polycrystalline pyrope (Mg 3Al 2Si 3O 12) to 9 GPa and 1000 °C. Phys. Earth Planet. Inter. 155, 179–190] and from Brillouin spectroscopic studies [Sinogeikin, S.V., Bass, J.D., 2002a. Elasticity of majorite and majorite–pyrope solid solution to high pressure: implications for the transition zone. Geophys. Res. 9(2), 1017], thereby demonstrating that the pressure derivatives of the elastic moduli are independent of the physical acoustics technique employed and unaffected by substitution of Si for Mg and Al within the Py–Mj solid solution in the range (Py 100–Py 50) of the present measurements. Temperature dependence of the elastic obtained from linear regression of entire P– T– K and P– T– G data are (∂ K S/∂ T) P = −14.6 (4) MPa/K; (∂ G/∂ T) P = −9.4 (4) MPa/K for Py 60Mj 40 garnet, and (∂ K S/∂ T) P = −14.6 (4) MPa/K; (∂ G/∂ T) P = −9.33 (2) MPa/K for Py 50Mj 50 garnet. These values are essentially identical for the two compositions and are also in excellent agreement with the Brillouin scattering data of [Sinogeikin, S.V., Bass, J.D., 2002b. Elasticity of pyrope and majorite–pyrope solid solutions to high temperatures. Phys. Earth Planet. Inter. 203, 549–555] for Py 50Mj 50 garnet; however, they are higher than those for pyrope by about 23% for (∂ K S/∂ T) P and 11% for (∂ G/∂ T) P [Gwanmesia, G.D., Zhang. J, Darling, K., Kung, J., Li, B., Wang, L., Neuville, D., Liebermann, R.C., 2006. Elasticity of polycrystalline pyrope (Mg 3Al 2Si 3O 12) to 9 GPa and 1000 °C. Phys. Earth Planet. Inter. 155, 179–190].

Objective Tensor Rates and Applications in Formulation of Hyperelastic Relations

Following Ogden, a class of objective (Lagrangian and Eulerian) tensors is identified among the second-rank tensors characterizing continuum deformation, but a more general definition of objectivity than that used by Ogden is introduced. Time rates of tensors are determined using convective rates. Sufficient conditions of objectivity are obtained for convective rates of objective tensors. Objective convective rates of strain tensors are used to introduce pairs of symmetric stress and strain tensors conjugate in a generalized sense. The classical definitions of conjugate Lagrangian (after Hill) and Eulerian (after Xiao et al.) stress and strain tensors are particular cases of the definition of conjugacy of stress and strain tensors in the generalized sense used in the present paper. Pairs of objective stress and strain tensors conjugate in the generalized sense are used to formulate constitutive relations for a hyperelastic medium. A family of objective generalized strain tensors is introduced, which is broader than Hill’s family of strain tensors. The basic forms of the hyperelastic constitutive relations are obtained with the aid of pairs of Lagrangian stress and strain tensors conjugate after Hill (the strain tensors in these pairs belong to the family of generalized strain tensors). A method is presented for generating reduced forms of the constitutive relations with the aid of pairs of Lagrangian and Eulerian stress and strain tensors conjugate in the generalized sense which are obtained from pairs of Lagrangian tensors conjugate after Hill by mapping tensor fields on one configuration of a deformable body to tensor fields on another configuration.

Single-crystal elasticity of diaspore, AlOOH, to 12 GPa by Brillouin scattering

The high-pressure elasticity of diaspore (AlOOH) has been determined by Brillouin spectroscopy to 12 GPa in diamond anvil cells. Experiments were carried out using a 16:3:1 methanol–ethanol–water mixture as pressure medium, and ruby as pressure standard. Acoustic velocities were measured in three roughly orthogonal planes at ambient and eight elevated pressures. The nine individual elastic stiffness constants of the orthorhombic crystal were obtained by fitting the velocity data to Christoffel's equation. Aggregate elastic moduli and pressure derivatives were calculated from the C ij s by fits to Eulerian finite strain equations, yielding: K S 0 = 152 ( 1 ) GPa , G 0 = 117.2(5) GPa, ( ∂ K S / ∂ P ) T 0 = 3.7 ( 1 ) , ( ∂ G / ∂ P ) 0 = 1.5 ( 1 ) for the Voigt–Reuss–Hill average. All individual C ij s increase with pressure but C 23 and C 55 exhibit anomalously low pressure derivatives. From calculated linear compressibilities, the a-axis is the most compressible. The b-axis becomes the least compressible axis at high pressures. Over the examined pressure range, the azimuthal P-wave anisotropy decreased from 22% to 16%, while the azimuthal S-wave anisotropy increased from 15% to 21%. Both volume and axial compression curves calculated using our Brillouin results are in good agreement with the results from static compression studies. High-pressure sound velocities in diaspore exceed those of other hydrous minerals as well as many anhydrous phases relevant to Earth's upper mantle.

Compression behavior of WC and WC-6%Co up to 50 GPa determined by synchrotron x-ray diffraction and ultrasonic techniques

The equations of state (pressure-volume relations) for WC and WC-6%Co have been determined by synchrotron x-ray diffraction measurements on polycrystalline powder samples loaded in a diamond anvil cell as well as by ultrasonic measurements on hot-pressed polycrystalline, cylindrical samples loaded in a multianvil high-pressure apparatus. The third-order Birch–Murnaghan equation of state fitted to the x-ray diffraction pressure-density sets of data, collected up to 50 GPa, yields ambient pressure isothermal bulk moduli of KoT=411.8±12.1 GPa and KoT=402.4±14.1 GPa, with pressure derivatives of KoT′=5.45±0.73 and KoT′=7.50±0.86 for WC and WC-6%Co, respectively. The ultrasonic measurements, conducted up to 14 GPa, enabled the determination of the pressure dependences of both bulk and shear moduli. Using Eulerian finite strain equations to fit the ultrasonic data, we obtain for WC an ambient pressure adiabatic bulk modulus of Kos=383.8±0.8 GPa, and Kos′=2.61±0.07 for its pressure derivative, while values of Gos=304.0±0.3 GPa and Gos′=1.50±0.09 were determined for the shear modulus and its pressure derivative, respectively. Meanwhile, for WC-6%Co, we obtain Kos=357.5±1.0 GPa, Kos′=5.18±0.14, Gos=253.5±0.3 GPa, and Gos′=1.09±0.09. The equations of state derived from the ultrasonic data are in good agreement with extrapolated results reported previously by Day and Ruoff [J. Appl. Phys. 44, 2447 (1973)] and Gerlich and Kennedy [J. Appl. Phys. 50, 3331 (1978)] who carried out measurements up to 0.2 and 1.0 GPa, respectively.

Eulerian conjugate stress and strain

New results are presented for the stress conjugate to arbitrary Eulerian strain measures. The conjugate stress depends on two arbitrary quantities: the strain measure f(V) and the corotational rate defined by the spin \Omega. It is shown that for every choice of f there is a unique spin, called the f-spin, which makes the conjugate stress as close as possible to the Cauchy stress. The f-spin reduces to the logarithmic spin when the strain measure is the Hencky strain log(V). The formulation and the results emphasize the similarities in form of the Eulerian and Lagrangian stresses conjugate to the strains f(V) and f(U), respectively. Many of the results involve the solution to the equation AX-XA=Y, which is presented in a succinct format.

On The Objective Corotational Rates of Eulerian Strain Measures

In the present paper, some new basis-free expressions for an arbitrary objective corotational rate of the general Eulerian strain measures are provided which are in compact form. Moreover, a complete discussion on the requirements for the continuity of the objective corotational rates are presented.

Three-dimensional phenomenological thermodynamic model of pseudoelasticity of shape memory alloys at finite strains

The familiar small strain thermodynamic 3D theory of isotropic pseudoelasticity proposed by Raniecki and Lexcellent is generalized to account for geometrical effects. The Mandel concept of mobile isoclinic, natural reference configurations is used in order to accomplish multiplicative decomposition of total deformation gradient into elastic and phase transformation (p.t.) parts, and resulting from it the additive decomposition of Eulerian strain rate tensor. The hypoelastic rate relations of elasticity involving elastic strain rate \(\underline{{{\mathbf{d}}}}^{\rm e}\) are derived consistent with hyperelastic relations resulting from free energy potential. It is shown that use of Jaumann corotational rate of stress tensor in rate constitutive equations formulation proves to be convenient. The formal equation for p.t. strain rate \(\underline{{{\mathbf{d}}}}^{{\rm in}}\) , describing p.t. deformation effects is proposed, based on experimental evidence. Phase transformation kinetics relations are presented in objective form. The field, coupled problem of thermomechanics is specified in rate weak form (rate principle of virtual work, and rate principle of heat transport). It is shown how information on the material behavior and motion inseparably enters the rate virtual work principle through the familiar bridging equation involving Eulerian rate of nominal stress tensor.

Analysis of deformation coupled surface remodeling in porous biomaterials

Surface remodeling of biological tissues through tissue growth or dissolution is deemed critical to their proper functioning, and is influenced by the deformation of the tissues during physiological activities. The present work attempts to develop a constitutive framework for deformation modulated surface remodeling of biological tissues. The framework is developed assuming finite deformation of the tissue, and the effect of deformation on the driving force for surface remodeling is determined from thermodynamic principles. The microscopic trends are upscaled to yield the remodeling-induced change in a macroscopic porous tissue. By way of application, the effect of deformation on the remodeling kinetics is determined for an incompressible elastic tissue. Depending on the ratio of the specific elastic stiffness and the specific Gibbs energy variation induced by the cell, the effect of deformation on the remodeling kinetics can be significant. It is found that both tensile and compressive deformation aid tissue dissolution (and dissuade growth). However, the magnitude of the effect is found to be different under tensile and compressive loadings, and critically depends on the reference frame used for the strain measurements. For Lagrangian strain measures (e.g., stretch, engineering strain), the increase in the dissolution kinetics per unit strain is higher under compressive loadings. On the other hand, for Eulerian strain measures (e.g., logarithmic or true strain), the effect of tensile loading on the dissolution kinetics is higher. This reinforces the need for proper reference frame definition for experimental strain measurements.

Spin tensors associated with corotational rates and corotational integrals in continua

In many cases of constitutive modeling of continua undergoing large deformations, use of corotational rates and integrals is inevitable to avoid the effects of rigid body rotations. Making corotational rates associated with specific spin tensors is a matter of interest, which can help for a better physical interpretation of the deformation. In this paper, for a given kinematic tensor function, say G , a tensor valued function F as well as a spin tensor Ω 0 is obtained in such a way that the corotational rate of F associated with the spin tensor Ω 0, becomes equal to G . In other words, F is the corotational integral of G associated with the spin tensor Ω 0. Here, G is decomposed additively into the principally diagonal and the principally off-diagonal parts. The symmetric tensors with the same principally diagonal parts have the same tensor valued corotational integrals, but associated with different spin tensors, which depends on the principally off-diagonal part. For the validity of the results, the Eulerian logarithmic strain tensor is shown to be the tensor valued corotational integral of the strain rate tensor, associated with the logarithmic spin. This result has separately been derived by Reinhardt & Dubey and Xiao et al. In addition, a specific corotational rate called Γ-rate and its associated spin tensor are introduced.

Stresses conjugate to the Jaumann rate of Eulerian strain measures

In this paper basis-free expressions are derived for the stresses conjugate to the Jaumann rate of the general Eulerian strain measures. The component-form of the expressions is also presented. The basis-free results are compact and also valid for all different cases of distinct and coalescent principal stretches. As an example, the compact basis-free expressions for the stress conjugate to the Jaumann rate of logarithmic strain lnV are determined.