Eulerian Strain Tensor Research Articles

A Dynamic Finite-Deformation Constitutive Model for Steels Undergoing Slip, Twinning, and Phase Changes

Depending on composition, processing, microstructure, and loading mode, steels may demonstrate various inelastic deformation mechanisms, notably dislocation glide, deformation twinning, and solid-solid phase changes. Damage in the form of voids, often leading to failure by macro-cracking, is also of current interest. A finite-deformation constitutive model is constructed in order to address such mechanisms in isotropic polycrystals under static and dynamic loading, where the latter encompasses high pressures and extreme strain rates pertinent to ballistic penetration. Slip and twinning are isochoric, and their relative contributions to kinematics are not explicitly distinguished in the present application. Phase changes among, e.g., face-centered-cubic (FCC), body-centered-cubic (BCC), body-centered-tetragonal (BCT), and hexagonal (HCP) structures are admitted, with corresponding deviatoric and volumetric strains. Porosity from voids contributes to volumetric strain. A new consistent thermodynamic framework incorporating an Eulerian strain tensor and internal state variables is developed, whereby kinetic equations for slip–twinning, phase changes, and damage evolution result in contributions to dissipation. An objective rate form of the model, derived assuming small deviatoric elastic strain, is implemented numerically. The model is applied to three different primarily austenitic, medium-high Mn steels. Specifically, representations of a slip-dominated (SLIP), a TRIP (transformation-induced plasticity) and a TWIP (twinning-induced plasticity) steel are calibrated to quasi-static tension, quasi-static compression, and dynamic compression data, at both room and high temperatures. A novel functional form of material strength distinguishes hardening profiles of the different alloys. Experimental data are reasonably well represented. Model extrapolations for dynamic strength and pressure in regimes pertinent to shock compression are analyzed. Predictions for multi-axial loading of shear with simultaneous expansion or contraction demonstrate competing physical mechanisms among the alloys that could be leveraged for optimal ballistic performance.

Family of Continuous Strain-Consistent Convective Tensor Rates and Its Application in Hooke-Like Isotropic Hypoelasticity

This paper presents a new family of objective (Lagrangian and Eulerian) continuous strain-consistent convective (corotational and non-corotational) tensor rates. Objective strain-consistent convective tensor rates are defined as having the property that there exist objective (Lagrangian and Eulerian) strain tensors from the Hill family such that the considered rates of these strain tensors are equal to the rotated/standard (Lagrangian/Eulerian) stretching (strain rate) tensors. The family of such Eulerian strain-consistent convective tensor rates was introduced by Bruhns et al. (Proc. R. Soc. Lond. A 460:909–928, 2004). On the one hand, the family of continuous strain-consistent convective tensor rates presented in this paper extends the Bruhns et al. family by including Lagrangian tensor rates and, on the other hand, it is narrower than the Bruhns et al. family due to the continuity requirement imposed on tensor rates, which is necessary for applications. The theorem that any strain tensors only from the Doyle–Ericksen family (which is a subfamily of the Hill family) provide sufficient continuity conditions for strain-consistent convective tensor rates was formulated and proved. The expressions obtained by proving this theorem for convector tensors show that each tensor from the Doyle–Ericksen family is associated with a single strain-consistent convective tensor rate (in the Bruhns et al. family, any strain tensor from the Hill family generates infinitely many convector tensors associated with infinitely many strain-consistent convective tensor rates). In addition, a new family of Hooke-like isotropic hypoelastic material models based on objective continuous strain-consistent convective rates of the rotated/standard (Lagrangian and Eulerian) Kirchhoff stress tensors was constructed. The theorem that any material model from this family is self-consistent provided that the Lame parameter $\lambda =0$ and/or the deformation of the body is isochoric was formulated and proved. By self-consistent hypoelastic material models are meant those models for which constitutive hypoelastic relations are counterparts of constitutive relations for Cauchy/Green elasticity. Some material models from the new family were tested by solving the simple shear problem. Both new and well-known solutions of this problem for material models were obtained using an approach that takes into account the self-consistency property of material models from the new family.

Direct Eulerian Formulation of Anisotropic Hyperelasticity

Abstract Many soft materials and biological tissues comprise isotropic matrix reinforced by fibers in the characteristic directions. Hyperelastic constitutive equations for such materials are usually formulated in terms of a Lagrangian strain tensor referred to the initial configuration and Lagrangian structure tensors defining characteristic directions of anisotropy. Such equations are “pushed forward” to the current configuration. Obtained in this way, Eulerian constitutive equations are often favorable from both theoretical and computational standpoints. In the present note, we show that the described two-step procedure is not necessary, and anisotropic hyperelasticity can be introduced directly in terms of an Eulerian strain tensor and Eulerian structure tensors referring to the current configuration. The newly developed constitutive equation is applied to the particular case of the transverse isotropy for the sake of illustration.

Objective Assessment of the Quality and Accuracy of Deformable Image Registration.

Background:The increased use of deformable registration algorithms in clinical practice has also increased the need for their validation.Aims and Objectives:The purpose of the study was to investigate the quality, accuracy, and plausibility of three commercial image registration algorithms for 4-dimensional computed tomography (4DCT) datasets using various similarity measures.Materials and Methods:4DCT datasets were acquired for 10 lung cancer patients. 23 similarity measures were used to evaluate image registration quality. To ensure selected method's invertibility and assess resultant mechanical stress, the determinant of the Jacobian for the displacement field and 3-D Eulerian strain tensor were calculated. All the measures and calculations were applied on to extended deformable multi pass (EXDMP) and deformable multi pass (DMP) methods.Results:The results indicate the same trend for several of the studied measures. The Jacobian determinant values were always positive for the DMP method. The Eulerian strain tensor had smaller values for the DMP method than EXDMP in all of the studied cases. The negative values of the Jacobian determinant point to non-physical behavior of the EXDMP method. The Eulerian strain tensor values indicate less tissue strain for the DMP method. Large differences were also observed in the results between complete and cropped datasets (coefficient of determination: 0.55 vs. 0.93).Conclusion:A number of error and distance measures showed the best performance among the tested measures. The evaluated measures might detect CT dataset differences with higher precision if the analysis is restricted to a smaller volume.

Physiological Motion Reduction Using Lagrangian Tracking for Electrode Displacement Elastography

Minimally invasive treatments such as microwave ablation (MWA) have been growing in popularity for extending liver cancer survival rates in patients, when surgery is not an option. As a non-ionizing, real-time alternative to contrast-enhanced computed tomography, electrode displacement elastography (EDE) has shown promise as an imaging modality for MWA. Despite imaging efficacy, motion artifacts caused by physiological motion result in unintended speckle pattern variance, thereby inhibiting consistent and accurate ablated region visualization. To combat these unavoidable motion artifacts, a Lagrangian deformation tracking (LDT) approach based on freehand EDE was developed to track tissue movement and better define tissue properties. For validating LDT efficacy, a spherical inclusion phantom as well as seven in vivo data sets were processed, and strain tensor images were compared with identical time sampled images estimated using a traditional Eulerian approach. In vivo results revealed greater consistency among visualized LDT strain tensor images, with segmented ablated regions exhibiting standard deviation reductions of up to 98% when compared with Eulerian strain tensor images. Additionally, Lagrangian strain tensor images provided Dice coefficient improvements up to 25%, and success rates improved from approximately 50% to nearly 100% for ablated region visualization.

Nonlinear thermomechanics for analysis of weak shock profile data in ductile polycrystals

Modeling of shock compression of isotropic, polycrystalline elastic-plastic solids that deform by dislocation glide is undertaken. Internal energy density of the material depends on a thermoelastic Eulerian strain tensor, entropy, and an internal state variable representative of dislocation density. Dislocation glide is incompressible, but inelastic volume changes arising from residual local strain fields and core effects of dislocations are captured. A semi-analytical method is advanced for extracting inelastic constitutive response information from particle velocity histories of polycrystalline samples under planar shock loading. The only parameters entering the procedure are fundamental thermoelastic properties and assumed bounds on the fraction of plastic work corresponding to energy storage of generated dislocations in the lattice. Densities of statistically stored and geometrically necessary dislocations, in addition to shear stress, plastic strain, plastic strain rate, and temperature, are outcomes of the analysis. The model is implemented for polycrystalline aluminum and copper. Certain results are compared with others in the literature obtained under different kinematic and thermodynamic assumptions. Effects of choices of nonlinear thermoelastic formulations and higher-order elastic constants on extracted deviatoric stress profiles can be substantial. Stored energy of dislocations and commensurate residual volume changes together can notably affect extracted shear stress. Stored energy significantly affects predicted entropy production and its contribution to temperature rise. The calculated density of geometrically necessary dislocations required to maintain compatibility of total strain across the structured steady plastic wave front is small relative to the line density of total dislocations.

Basis-free expressions for families of objective strain tensors, their rates, and conjugate stress tensors

The objective (Lagrangian and Eulerian) strain tensors, their rates, and conjugate stress tensors used in continuum mechanics equations are considered. These tensors are represented in the eigenprojections of the right and left stretch tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . The novelty of the research lies in the simultaneous derivation of expressions for Lagrangian versions of tensors and their Eulerian counterparts with decomposition of the obtained expressions into components coaxial and orthogonal to the tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . We consider the Lagrangian and Eulerian Doyle–Ericksen families of strain tensors, which, in turn, are subfamilies of the well-known Hill family of strain tensors. Basis-free expressions are obtained for the material rates of Lagrangian tensors from the previously introduced family of generalized strain tensors, whose subfamilies are the Lagrangian and Eulerian Hill families of strain tensors, and similar expressions are obtained for the Green–Naghdi rates of Eulerian strain tensors, which are Eulerian counterparts of the material rates of Lagrangian strain tensors. Basis-free expressions for stress tensors are derived from the classical definitions of the conjugacy of stress and strain tensors. Expressions are found for the Lagrangian and Eulerian symmetric Atluri stress tensors that do not have conjugate strain tensors from the Hill family, and the obtained expressions are decomposed into components coaxial and orthogonal to the tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . The ranges of allowed ratios of different principal stretches at which the Lagrangian and Eulerian Hencky and Atluri stress tensors approximate the rotated/standard Kirchhoff stress tensors are determined. It is shown that the range of allowed ratios for the Hencky stress tensors is wider than that for the Atluri stress tensors.

Crystal thermoelasticity at extreme loading rates and pressures: Analysis of higher-order energy potentials

Several finite elastic strain measures are evaluated for use in constitutive models of crystalline elasticity and elasto-plasticity. These include the Green material strain tensor, the Eulerian material strain tensor, and the logarithmic material strain tensor, all of which are referred to locally relaxed coordinates invariant under spatial rotations. New applications of logarithmic strain-based theory towards shock compression of aluminum, copper, and magnesium single crystals and polycrystals are presented. Solutions to the planar shock problem from previous work are summarized and compared with the present results. Consideration of these new results in conjunction with previous analysis for a number of different metals, ceramics, and minerals suggests that Eulerian strain-based theory is most accurate for modeling the dynamic high-pressure response of ductile metallic crystals wherein ratios of elastic shear to bulk moduli tend to be relatively small, while logarithmic strain-based theory is recommended for modeling shocks in ceramics and minerals with larger ratios of effective elastic shear to bulk modulus.

Conjugated kinetic and kinematic measures for constitutive modeling of the thermoelastic continua

In this paper, the energy-type terms such as the stress power, the rate of the heat transferred to the system and the rate of the specific internal energy are presented in the Lagrangian, Eulerian and two-point descriptions for thermoelastic continua. In order to solve a problem based on the energy viewpoint, the mechanical, thermal and thermo-mechanical tensors conjugate to the Seth–Hill strains, and a general class of Lagrangian, Eulerian and two-point strain tensors are determined. Also, the energy pairs for thermoelastic continua are simplified for special cases of isentropic and isothermal deformation processes as well as the so-called entropic elastic materials (rubber-like materials and elastomers). At the end, a strain energy density function of the Saint Venant–Kirchhoff type in terms of different strain measures and temperature is considered for modeling the thermo-mechanical behavior of the rubber-like materials and elastomers. It is shown that this constitutive modeling can give results which are in good agreement with the experimental data.

Nonlinear Eulerian thermoelasticity for anisotropic crystals

A complete continuum thermoelastic theory for large deformation of crystals of arbitrary symmetry is developed. The theory incorporates as a fundamental state variable in the thermodynamic potentials what is termed an Eulerian strain tensor (in material coordinates) constructed from the inverse of the deformation gradient. Thermodynamic identities and relationships among Eulerian and the usual Lagrangian material coefficients are derived, significantly extending previous literature that focused on materials with cubic or hexagonal symmetry and hydrostatic loading conditions. Analytical solutions for homogeneous deformations of ideal cubic crystals are studied over a prescribed range of elastic coefficients; stress states and intrinsic stability measures are compared. For realistic coefficients, Eulerian theory is shown to predict more physically realistic behavior than Lagrangian theory under large compression and shear. Analytical solutions for shock compression of anisotropic single crystals are derived for internal energy functions quartic in Lagrangian or Eulerian strain and linear in entropy; results are analyzed for quartz, sapphire, and diamond. When elastic constants of up to order four are included, both Lagrangian and Eulerian theories are capable of matching Hugoniot data. When only the second-order elastic constant is known, an alternative theory incorporating a mixed Eulerian–Lagrangian strain tensor provides a reasonable approximation of experimental data.

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.

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.

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.

In-Vivo Dynamic Deformation of the Mitral Valve Anterior Leaflet

Surgical techniques have been developed for mitral valve repair for a wide range of pathologies. However, excessive tissue stress and damage have been identified as etiologic factors limiting long-term durability. Before computational models to optimize valve repair can be realistically developed, in-vivo dynamic mitral valve leaflet strain data are required. However, these data do not presently exist. In the present study, a sheep model and sonomicrometry were used to compute the in-surface Eulerian strain tensor of the anterior leaflet over the cardiac cycle at varying afterloads. The anterior leaflet of nine Dorsett sheep (35 kg to 45 kg) was instrumented with nine 1-mm hemispherical piezoelectric transducers in a 15-mm square array. Three-dimensional crystal spatial positions were recorded at 250 Hz over several cardiac cycles, with peak left ventricular pressures varying from 90 mm Hg to 200 mm Hg. The in-surface Eulerian strain tensor was computed from the crystal displacements. The mitral valve anterior leaflet experiences large anisotropic strains and peak strain rates of 400%/s, followed by an absolute cessation of any deformation during systole. Increasing left ventricular pressure also increased the effective leaflet stiffness but not the peak strains. We report the first data on the dynamic in-vivo strain tensor of a functioning mitral valve anterior leaflet, which indicated large anisotropic strains and very high strain rates. Our observations also suggest that changes in left ventricular pressure and annular geometry result in altered effective leaflet stiffness, and may be an important factor in reducing leaflet stress and as such potentially affect mitral valve repair longevity.

Nonlinear fluid dynamics description of non-Newtonian fluids

Nonlinear hydrodynamic equations for visco-elastic media are discussed. We start from the recently derived fully hydrodynamic nonlinear description of permanent elasticity that utilizes the (Eulerian) strain tensor. The reversible quadratic nonlinearities in the strain tensor dynamics are of the “lower convected” type, unambiguously. Replacing the (often neglected) strain diffusion by a relaxation of the strain as a minimal ingredient, a generalized hydrodynamic description of viscoelasticity is obtained. This can be used to get a nonlinear dynamic equation for the stress tensor (sometimes called constitutive equation) in terms of a power series in the variables. The form of this equation and in particular the form of the nonlinear convective term is not universal but depends on various material parameters. A comparison with existing phenomenological models is given. In particular we discuss how these ad-hoc models fit into the hydrodynamic description and where the various non-Newtonian contributions are coming from.

Fourth‐Order Adiabatic Linear Compressibilities for Non‐Cubic Solids

AbstractExpressions for the linear adiabatic compressibilities of an at least orthorhombic crystal and their pressure derivatives are derived in terms of the second‐, third‐, and fourth‐order elastic constants within a finite‐strain theory which includes thermal effects according to the quasi‐harmonic approximation of lattice dynamics. The expressions are derived in terms of the Lagrangian strain tensor and the frame‐indifferent analogue of the Eulerian strain tensor. Numerical applications are given for three hexagonal metals: cadmium, magnesium, and zinc, namely the temperature variations of the linear adiabatic compressibilities and their pressure derivatives, including recent estimations of the combinations of fourth‐order elastic constants appearing in the present work. When possible, the comparison with experimental data shows a rather good agreement with the compressibilities calculated within the present theory using the Green‐Lagrange strain measure of the deformation. About the variations of the first pressure derivatives of the adiabatic compressibilities with temperature, the present work provides us with theoretical values in the lack of experimental data.

Time rates of generalized strain tensors Part I: Component formulas

Hill derived a simple component formula for the material time derivative of a generalized Lagrangian strain tensor. We examine Hill's derivation in detail and explain why it is generally valid only when the principal stretches are distinct. We then give a proof of Hill's formula which is valid for any C 2 motion and any C l strain measure. Our proof is based on a component form of the chain rule for a tensor-valued function of a time-dependent symmetric tensor. This result is also used to derive component formulas for the Jaumann rate of a generalized Eulerian strain tensor. Finally, we apply the general formulas to the logarithmic strain tensors.

Time rates of generalized strain tensors Part II: Approximate basis-free formulas

Utilizing the component formulas derived in Part 1, we obtain approximate basis-free formulas for the material time derivative of a generalized Lagrangian strain tensor and for the Jaumann rate of a generalized Eulerian strain tensor. These formulas provide good approximations when the shear strains are small, regardless of the volumetric strain and the strain rate. We also derive explicit bounds for the error in the approximate formulas. By applying our general results to the logarithmic strain tensors, we obtain improved versions of approximate basis-free formulas due to Hill and Gurtin and Spear.

X‐ray studies of thermal expansion and phase transitions in (NH4)2SbF5 crystals

AbstractX‐ray diffraction studies of phase transitions in (NH4)2SbF3 single crystals have been made by using the Bond‐type diffractometer for high‐precision measurements of the unit‐cell dimensions in a function of temperature. The principal linear thermal expansion coefficients and the linear Eulerian strain tensor have been calculated in the temperature interval 110 – 298 K. The phase transition diagram was proposed and the sequence of the phases has been compared with the results of other experimental techniques (NMR, NQR, DTA, electrophysical methods).

A high-pressure single-crystal X-ray diffraction study of V3O5 including the phase transition at 6.2 GPa

The unit-cell parameters of the mixed-valence oxide V3O5 have been studied as functions of pressure from 0.3 up to 7.5 GPa in a single-crystal experiment involving measurements at 21 different pressure values. A diamond-anvil high-pressure (h.p.) cell of new construction [Malinowski (1987). Submitted to J. Appl. Cryst.] mounted on a four-circle X-ray diffractometer was used. At Pt = 6.25(9) GPa a phase transition was observed at room temperature. This is analogous to the semiconductor-metal transition observed at Tt = 428 K and normal pressure, which implies a change in the ordering of the metal-atom valencies. The linear Eulerian strain tensor for the transition has been calculated. The bulk modulus K has been determined for the low-pressure (l.p.) as well as for the h.p. phase: K l.p.(0.3–5.5 GPa) = 269(3), K h.p.(6.3–7.5 GPa) = 175(11) GPa. Certain anomalies in the cell-parameter values in a small pressure interval around the phase transition indicate a preparation in the crystal structure for the transition in both the l.p. and the h.p. phase.