General Stress Tensor Research Articles

Deep Learning Predicts Stress–Strain Relations of Granular Materials Based on Triaxial Testing Data

This study presents an AI-based constitutive modelling framework wherein the prediction model directly learns from triaxial testing data by combining discrete element modelling (DEM) and deep learning. A constitutive learning strategy is proposed based on the generally accepted frame-indifference assumption in constructing material constitutive models. The low-dimensional principal stress-strain sequence pairs, measured from discrete element modelling of triaxial testing, are used to train recurrent neural networks, and then the predicted principal stress sequence is augmented to other high-dimensional or general stress tensor via coordinate transformation. Through detailed hyperparameter investigations, it is found that long short-term memory (LSTM) and gated recurrent unit (GRU) networks have similar prediction performance in constitutive modelling problems, and both satisfactorily predict the stress responses of granular materials subjected to a given unseen strain path. Furthermore, the unique merits and ongoing challenges of data-driven constitutive models for granular materials are discussed.

Finite-strain scale-free phase-field approach to multivariant martensitic phase transformations with stress-dependent effective thresholds

A scale-free phase-field model for martensitic phase transformations (PTs) at finite strains is developed as an essential generalization of small-strain models in Levitas et al. (2004) and Idesman et al. (2005). The theory includes finite elastic and transformational strains and rotations as well as anisotropic and different elastic properties of phases. The gradient energy term is excluded, and the model is applicable for any scale greater than 100 nm. The model tracks finite-width interfaces between austenite and the mixture of martensitic variants only; volume fractions of martensitic variants are the internal variables rather than order parameters. The concept of the effective threshold for the driving force is introduced, which can be either positive (e.g., due to interface friction) or negative (e.g., due to defects and stress concentrators promoting PTs). To reproduce PT conditions obtained from experiments or atomistic simulations under general stress tensor, the effective threshold depends on the stress tensor components. Material parameters are calibrated for martensitic PT between single crystalline cubic Si I and tetragonal Si II phases, which has large transformation strains (ɛt1=0.1753;ɛt2=0.1753;ɛt3=−0.447). Finite element algorithms and numerical procedures are implemented in the code deal.II. Multiple 3D problems are solved to study the effect of mesh size, holding time during quasi-static loading, and strain rate on the multivariant microstructure evolution in Si I → Si II PT under uniaxial and hydrostatic loadings. The solution exhibits significant lattice rotations both in Si I and Si II, reproducing the appearance of diffuse grain boundaries in Si I and Si II and transforming them in polycrystals, which corresponds to known experiments. While finer mesh can produce a more detailed microstructure, the solution becomes practically mesh-independent after the mesh size is 80 times smaller than the sample size. When approaching the stationary solution, rough mesh leads to convergence to the correct microstructure faster than the fine mesh, because it neglects fine details in the microstructure. In some regions, reverse PT occurs at continuous compression, despite large transformation hysteresis. For most stationary interfaces, local thermodynamic equilibrium conditions (thermodynamic driving force for the interface motion equal to the effective threshold) are satisfied.

Effect of [formula omitted] dislocation on transformation stresses, nucleation, and growth for phase transformations between silicon I and silicon II under triaxial loading: Phase-field study

Stress-induced martensitic phase transformations (PTs) at a stationary 60∘ dislocation in single-crystalline Si are modeled by an advanced phase-field approach (PFA), which takes into account the lattice instability conditions obtained by atomistic simulations for the general stress tensor. Finite elastic, transformation, and plastic strains are considered. Finite element method (FEM) simulations elucidate two different mechanisms of nucleation and nanostructure evolution for two different stress-hysteresis cases. For a traditional finite-stress-hysteresis region, the PT starts with the barrierless nucleation of a thermodynamically-equilibrium-incomplete embryo, which loses its stability and grows forming a propagating martensitic band with distinct interfaces. However, in the unique zero-stress-hysteresis region, where PT for defect-free crystal occurs homogeneously through intermediate phases without nucleation, interfaces, and growth, the PT starts at a dislocation but spreads quasi-homogeneously, without interfaces, similar to the defect-free case; the macroscopic stress-strain curve is horizontal and without hysteresis during direct-reverse PTs. Despite large normal stresses produced by dislocation in the range of ±(6−12) GPa, a relatively small reduction in macroscopic PT stress by 1.6 GPa is quantitatively explained by mutually compensating contributions of stresses into lattice instability criterion.

Optical-force laws for guided light in linear media

The mechanical response of transparent materials to optical forces is a topic that concerns a wide range of fields, from the manipulation of biological material by optical tweezers to the design of nano-optomechanical systems (NOMS). However, the fundamental aspects of such forces have always been surrounded by controversies, and several different formulations have been proposed. In this work, we focus on the specific case of light propagating as a superposition of guided modes in lossless dielectric waveguides as a physical example upon which to build a general stress tensor. We use this formalism to calculate optical forces for straight and curved waveguide sections and all possible excitation configurations for a given set of coupled eigenmodes, and then compare the results for each of the known proposed optical force laws as well as a novel one derived from this general stress tensor. We show that proper use of the divergence theorem is crucial to account for all force terms, many of which vanish if the procedure most commonly used is applied for situations other than eigenmodes in straight waveguides. A better understanding of how different stress tensors predict very different forces for certain waveguide geometries opens a pathway for new experimental tests of each formulation.

Effect of 60° Dislocation on Transformation Stresses, Nucleation, and Growth for Phase Transformations between Silicon I and Silicon II Under Triaxial Loading: Phase-Field Study

Stress-induced martensitic phase transformations (PTs) at a stationary 60° dislocation in single-crystalline Si are modeled by an advanced phase-field approach (PFA), which takes into account the lattice instability conditions obtained by atomistic simulations for the general stress tensor. Finite elastic, transformation, and plastic strains are considered. Finite element method (FEM) simulations elucidate two different mechanisms of nucleation and nanostructure evolution for two different stress-hysteresis cases. For a traditional finite-stress-hysteresis region, the PT starts with the barrierless nucleation of a thermodynamically-equilibrium-incomplete embryo, which loses its stability and grows forming a propagating martensitic band with distinct interfaces. However, in the unique zero-stress-hysteresis region, where PT for defect-free crystal occurs homogeneously through intermediate phases without nucleation, interfaces, and growth, the PT starts at a dislocation but spreads quasi-homogeneously, without interfaces, similar to the defect-free case; the macroscopic stress-strain curve is horizontal and without hysteresis during direct-reverse PTs. Despite large normal stresses produced by dislocation in the range of ±(6-12)GPa, a relatively small reduction in macroscopic PT stress by 1.6 GPa is quantitatively explained by mutually compensating contributions of stresses into lattice instability criterion.

Algorithmic aspects and finite element solutions for advanced phase field approach to martensitic phase transformation under large strains

A new problem formulation and numerical algorithm for an advanced phase-field approach (PFA) to martensitic phase transformation (PT) are presented. Finite elastic and transformational strains are considered using a fully geometrically-nonlinear formulation, which includes different anisotropic elastic properties of phases. The requirements for the thermodynamic potentials and transformation deformation gradient tensor are advanced to reproduce crystal lattice instability conditions under a general stress tensor obtained by molecular dynamics (MD) simulations. The PFA parameters are calibrated, in particular, based on the results of MD simulations for PTs between semiconducting Si I and metallic Si II phases under complex action of all six components of the stress tensor (Levitas et al. in Phys Rev Lett 118:025701, 2017a; Phys Rev B 96:054118, 2017b). The independence of the PFA instability conditions of the prescribed stress measure is demonstrated numerically for the initiation of the PT. However, it is observed that the PT cannot be completed unless the stress exceeds the stress peak points that depend on which stress measure is prescribed. Various 3D problems on lattice instability and following nanostructure evolution in single-crystal Si are solved. The effect of stress hysteresis on the nanostructure evolution is studied through analysis of the local driving force and stress fields. It is demonstrated that variation of internal stress fields due to differing boundary conditions may lead to completely different PT mechanisms.

Lattice Instability during Solid-Solid Structural Transformations under a General Applied Stress Tensor: Example of Si I→Si II with Metallization.

The density functional theory was employed to study the stress-strain behavior and elastic instabilities during the solid-solid phase transformation (PT) when subjected to a general stress tensor, as exemplified for semiconducting Si I and metallic Si II, where metallization precedes the PT, so stressed Si I can be a metal. The hydrostatic PT occurs at 76GPa, while under uniaxial loading it is 11GPa (3.7GPa mean pressure), 21 times lower. The Si I→Si II PT is described by a critical value of the phase-field's modified transformation work, and the PT criterion has only two parameters given six independent stress elements. Our findings reveal novel, more practical synthesis routes for new or known high-pressure phases under predictable nonhydrostatic loading, where competition of instabilities can serve for phase selection rather than free energy minima used for equilibrium processing.

Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor

We prove global existence of appropriate weak solutions for the compressible Navier–Stokes equations for more general stress tensor than those covered by P.–L. Lions and E. Feireisl’s theory. More precisely we focus on more general pressure laws which are not thermodynamically stable; we are also able to handle some anisotropy in the viscous stress tensor. To give answers to these two longstanding problems, we revisit the classical compactness theory on the density by obtaining precise quantitative regularity estimates: This requires a more precise analysis of the structure of the equations combined to a novel approach to the compactness of the continuity equation. These two cases open the theory to important physical applications, for instance to describe solar events (virial pressure law), geophysical flows (eddy viscosity) or biological situations (anisotropy).

Phase-field approach for stress- and temperature-induced phase transformations that satisfies lattice instability conditions. Part 2. simulations of phase transformations Si I[formula omitted] Si II

A complete system of equations of the advanced phase-field theory for martensitic phase transformations (PTs) under a general stress tensor is presented. Theory includes a fully geometrically nonlinear formulation for the general case of finite elastic and transformational strains as well as anisotropic and different elastic properties of phases. Material parameters are calibrated, in particular, based on the crystal lattice instability conditions from atomistic simulations for martensitic PTs between cubic Si I and tetragonal Si II phases under complex triaxial compression-tension loading. A finite element algorithm and numerical procedure is developed and implemented in the code deal.II. Various 3D problems on lattice instabilities and following nanostructure evolution in single-crystal silicon are solved for compression in one direction under lateral stresses and analyzed. Strong effects of the stress states and local stress hysteresis on the interface width and nanostructure evolution are presented. In particular, the interface width diverges when lateral stress tends to the region in which instability stresses for direct and reverse PTs coincide. Direct and reverse transformations both occur in the unique homogeneous way without hysteresis, energy dissipation, and damage due to internal elastic stresses. Stress fields within a sample and especially within interfaces are determined and their effect on the nanostructure evolution is analyzed. Problems with definition of the elastic interfacial tension (stress) are analyzed. It is demonstrated that the instability stresses for initiation of the PTs are independent of the prescribed stress measure; however, this does not mean that PT will be completed at such stresses.

Braginskii magnetohydrodynamics for arbitrary magnetic topologies: coronal applications

We investigate single-fluid magnetohydrodynamics (MHD) with anisotropic viscosity, often referred to as Braginskii MHD, with a particular eye to solar coronal applications. First, we examine the full Braginskii viscous tensor in the single-fluid limit. We pay particular attention to how the Braginskii tensor behaves as the magnetic field strength vanishes. The solar corona contains a magnetic field with a complex and evolving topology, so the viscosity must revert to its isotropic form when the field strength is zero, e.g. at null points. We highlight that the standard form in which the Braginskii tensor is written is not suitable for inclusion in simulations as singularities in the individual terms can develop. Instead, an altered form, where the parallel and perpendicular tensors are combined, provides the required asymptotic behaviour in the weak-field limit. We implement this combined form of the tensor into the Lare3D code, which is widely used for coronal simulations. Since our main focus is the viscous heating of the solar corona, we drop the drift terms of the Braginskii tensor. In a stressed null point simulation, we discover that small-scale structures, which develop very close to the null, lead to anisotropic viscous heating at the null itself (that is, heating due to the anisotropic terms in the viscosity tensor). The null point simulation we present has a much higher resolution than many other simulations containing null points, so this excess heating is a practical concern in coronal simulations. To remedy this unwanted heating at the null point, we develop a model for the viscosity tensor that captures the most important physics of viscosity in the corona: parallel viscosity for strong fields and isotropic viscosity at null points. We derive a continuum model of viscosity where momentum transport, described by this viscosity model, has the magnetic field as its preferred orientation. When the field strength is zero, there is no preferred direction for momentum transport and viscosity reverts to the standard isotropic form. The most general viscous stress tensor of a (single-fluid) plasma satisfying these conditions is found. It is shown that the Braginskii model, without the drift terms, is a specialization of the general model. Performing the stressed null point simulation with this simplified model of viscosity reveals very similar heating profiles to those of the full Braginskii model. The new model, however, does not produce anisotropic heating at the null point, as required. Since the vast majority of coronal simulations use only isotropic viscosity, we perform the stressed null point simulation with isotropic viscosity and compare the heating profiles to those of the anisotropic models. It is shown that the fully isotropic viscosity can overestimate the viscous heating by an order of magnitude.

A mixed Legendre-Galerkin spectral method for the buckling problem of simply supported Kirchhoff plates

In this paper, we develop a mixed Legendre-Galerkin spectral method to approximate the buckling problem of simply supported Kirchhoff plates subjected to general plane stress tensor. By the spectral theory of compact operators, the rigorous error estimates for the approximate eigenvalues and eigenfunctions are provided. Finally, we present some numerical experiments which support our theoretical results.

Dynamics of the Ericksen–Leslie equations with general Leslie stress I: the incompressible isotropic case

The Ericksen–Leslie model for nematic liquid crystals in a bounded domain with general Leslie and isotropic Ericksen stress tensor is studied in the case of a non-isothermal and incompressible fluid. This system is shown to be locally strongly well-posed in the $$L_p$$ -setting, and a dynamical theory is developed. The equilibria are identified and shown to be normally stable. In particular, a local solution extends to a unique, global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold. In this case, the solution converges exponentially to an equilibrium, in the topology of the state manifold. The above results are proven without any structural assumptions on the Leslie coefficients and in particular without assuming Parodi’s relation.

A finite element method for the buckling problem of simply supported Kirchhoff plates

The aim of this paper is to develop a finite element method to approximate the buckling problem of simply supported Kirchhoff plates subjected to general plane stress tensor. We introduce an auxiliary variable w:=Δu (with u representing the displacement of the plate) to write a variational formulation of the spectral problem. We propose a conforming discretization of the problem, where the unknowns are approximated by piecewise linear and continuous finite elements. We show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments supporting our theoretical results.

Unified Stress Tensor of the Hydration Water Layer

We present the general stress tensor of the ubiquitous hydration water layer (HWL), based on the empirical hydration force, by combining the elasticity and hydrodynamics theories. The tapping and shear component of the tensor describe the elastic and damping properties of the HWL, respectively, in good agreement with experiments. In particular, a unified understanding of HWL dynamics provides the otherwise unavailable intrinsic parameters of the HWL, which offer additional but unexplored aspects to the supercooled liquidity of the confined HWL. Our results may allow deeper insight on systems where the HWL is critical.

Non-dissipative hydrodynamics: effective actions versus entropy current

While conventional hydrodynamics incorporating dissipative effects is hard to derive from an action principle, it is nevertheless possible to construct classical actions when the dissipative terms are switched off. In this note we undertake a systematic exploration of such constructions from an effective field theory approach and argue for the existence of non-trivial second order non-dissipative hydrodynamics involving pure energy-momentum transport. We find these fluids to be characterized by five second-order transport coefficients based on the effective action (a three parameter family is Weyl invariant). On the other hand since all flows of such fluids are non-dissipative, they entail zero entropy production; one can therefore understand them using the entropy current formalism which has provided much insight into hydrodynamic transport. An analysis of the most general stress tensor with zero entropy production however turns out to give a seven parameter family of non-dissipative hydrodynamics (a four parameter sub-family being Weyl invariant). The non-dissipative fluids derived from the effective action approach are a special case of the fluid dynamics constrained by conservation of the entropy current. We speculate on the reasons for the mismatch and potential limitations of the effective action approach.

Comparison of different failure criteria for fiber-reinforced plastics in terms of fracture curves for arbitrary stress combinations

Abstract For fiber-reinforced plastics exists a big number of different criteria for the failure prediction. The intention of this paper is to compare the TSAI-HIL-, the LaRC04- and PUCK’s criterion in terms of their fracture curves for a unidirectional glass-fiber reinforced composite layer. Therefore after the implementation of these three criteria, the two-dimensional fracture curves for all possible stress combinations, which can be derived from a general spatial stress tensor, are computed. In this way, the characteristics of the criteria, similarities and differences and possible weak points become obvious.

Dielectric pressure in continuum electrostatic solvation of biomolecules

Continuum solvation representations based on the Poisson-Boltzmann equation have become widely accepted in biomolecular applications after years of basic research and development. Since analytical solution of the differential equation can be achieved only in a few specific cases with simple solute geometry, only numerical solution is possible for biomolecular applications. However, it is conceptually difficult to assign solvation forces in the numerical methods, limiting their applications into direct simulations of energy minimization and molecular dynamics. In this study a dielectric pressure formulation was derived from the general Maxwell stress tensor for continuum solvation of biomolecules modeled with the widely used abrupt-transitioned dielectrics. A charge-central strategy was then proposed to improve the numerical behavior of the computed pressure. An interesting observation is the highly similar charge-central formulations between the smooth-transition dielectric and the abrupt-transition dielectric models utilized in the biomolecular solvation treatments. The connections of the new formulation with both the Davis-McCammon and Gilson et al. approaches were further presented after applying the normal field approximation. The consistency was verified with the numerical tests on a realistic biomolecule. The numerical experiments on the tested biomolecule further indicate that the charge-central strategy combined with the normal field approximation not only improves the accuracy of the dielectric boundary force but also reduces its grid dependence for biomolecular applications.

A discontinuous Galerkin formulation of non‐linear Kirchhoff–Love shells

AbstractDiscontinuous Galerkin (DG) methods provide a means of weakly enforcing the continuity of the unknown‐field derivatives and have particular appeal in problems involving high‐order derivatives. This feature has previously been successfully exploited (Comput. Methods Appl. Mech. Eng. 2008; 197:2901–2929) to develop a formulation of linear Kirchhoff–Love shells considering only the membrane and bending responses. In this proposed one‐field method—the displacements are the only unknowns, while the displacement field is continuous, the continuity in the displacement derivative between two elements is weakly enforced by recourse to a DG formulation. It is the purpose of the present paper to extend this formulation to finite deformations and non‐linear elastic behaviors. While the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple‐stress from the displacement field at the mid‐surface of the shell, the non‐linear formulation considered implies the evaluation of the general stress tensor across the shell thickness, leading to a reformulation of the internal forces of the shell. Nevertheless, since the interface terms resulting from the discontinuous Galerkin method involve only the resultant couple‐stress at the edges of the shells, the extension to non‐linear deformations is straightforward. Copyright © 2008 John Wiley & Sons, Ltd.

Influence of viscosity on the scattering of an air pressure wave by a rigid body: a regular boundary integral formulation

This paper gives a regular vector boundary integral equation for solving the problem of viscous scattering of a pressure wave by a rigid body. Firstly, single-layer viscous potentials and a generalized stress tensor are introduced. Correspondingly, generalized viscous double-layer potentials are defined. By representing the scattered field as a combination of a single-layer viscous potential and a generalized viscous double-layer potential, the problem is reduced to the solution of a vectorial Fredholm integral equation of the second kind. Generally, the vector integral equation is singular. However, there is a particular stress tensor, called pseudostress, which yields a regular integral equation. In this case, the Fredholm alternative applies and permits a direct proof of the existence and uniqueness of the solution. The results presented here provide the foundation for a numerical solution procedure.

General expression of the gyroviscous force

Assuming only small gyromotion periods and Larmor radii compared to any other time and length scales, and retaining the lowest significant order in δ=ρi∕L⪡1, the general expression of the ion gyroviscous stress tensor is presented. This expression covers both the “fast dynamics” (or “magnetohydrodynamic”) ordering, where the time derivative and ion gyroviscous stress are first order in δ relative to the ion gyrofrequency and scalar pressure, respectively, and the “slow dynamics” (or “drift”) ordering, where the time derivative and ion gyroviscous stress are, respectively, second order in δ. This general stress tensor applies to arbitrary collisionality and does not require the distribution function to be close to a Maxwellian. Its exact divergence (gyroviscous force) is written in a closed vector form, allowing for arbitrary magnetic geometry, parallel gradients, and flow velocities. Considering, in particular, the contribution from the velocity gradient (rate of strain) term, the final form of the momentum conservation equation after the “gyroviscous cancellation” and the “effective renormalization of the perpendicular pressure by the parallel vorticity” is precisely established.