Deviatoric Tensor Research Articles

In situ measurement of three-dimensional intergranular stress localizations and grain yielding under elastoplastic axial-torsional loading

The three-dimensional grain-averaged response of solid bar samples under non-proportional (NP) elastoplastic axial-torsional loading was investigated using in situ high energy diffraction microscopy (HEDM) and companion crystal plasticity finite element (CPFE) modeling. Important stress metrics including applied shear (σθZ) and axial (σZZ) stress tensor components, stress and stress deviator tensor invariants (I1, J2, and J3), von Mises equivalent stress (σVMgrain), maximum resolved shear stress (mRSS), stress triaxiality (η), and lode angle parameter (θ‾) values were tracked for ∼ 300 grains under two different loading conditions: (1) Torsion-dominated loading (low NP) and (2) Tension-torsion loading (high NP) in equiatomic NiCoCr, a representative multicomponent face-centered cubic (FCC) superalloy. Overall, significant stress localizations existed within both samples as evidenced by the radial dependence of grain-resolved σθZ, σVMgrain , and J2; by comparison, I1, J3, η, and θ‾ metrics did not show discernible trends within the volume. These stress localizations reveal a complex interplay between axial and shear stress components (e.g., stress coupling) resulting in grain yielding near the sample surface largely driven by shear stress, whereas internal grain yielding was largely accommodated by axial stress. Grain-resolved stress localization trends were described well by the CPFE model, although some discrepancies in magnitude occurred, particularly for volumetric stress metrics (I1 and η) due to initial type II residual stress distributions. The superposition of initial residual stress states onto CPFE grain-resolved data significantly improved model accuracy for η. This suggests that residual stresses more strongly influence the simulation of volumetric rather than deviatoric (yield) stress metrics.

Mixed FEM for Shells of Revolution Based on Flow Theory and its Modifications

For describing elastoplastic deformation, three versions of constitutive equations are used. The first version employs the governing equations of the flow theory. In the second version, elastic strain increments are defined the same way as in the flow theory, and the plastic strain increments are expressed in terms of stress increments using the condition of their proportionality to the components of the incremental stress deviator tensor. In the third version, the constitutive equations for a load step were obtained without using the hypothesis of separating strains into the elastic and plastic parts. To obtain them, the condition of proportionality of the components of the incremental strain deviator tensor to the components of the incremental stress deviator tensor was applied. The equations are implemented using a hybrid prismatic finite element with a triangular base. A sample calculation shows the advantage of the third version of the constitutive equations.

Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields

AbstractDiscrete Helmholtz decompositions dissect piecewise polynomial vector fields on simplicial meshes into piecewise gradients and rotations of finite element functions. This paper concisely reviews established results from the literature which all restrict to the lowest-order case of piecewise constants. Its main contribution consists of the generalization of these decompositions to 3D and of novel decompositions for piecewise affine vector fields in terms of Fortin–Soulie functions. While the classical lowest-order decompositions include one conforming and one nonconforming part, the decompositions of piecewise affine vector fields require a nonconforming enrichment in both parts. The presentation covers two and three spatial dimensions as well as generalizations to deviatoric tensor fields in the context of the Stokes equations and symmetric tensor fields for the linear elasticity and fourth-order problems. While the proofs focus on contractible domains, generalizations to multiply connected domains and domains with non-connected boundary are discussed as well.

A shear history model for capturing the liquefaction resistance of sands at various cyclic stress ratios

Various constitutive formulations have been developed over the years to reproduce the cyclic resistance of sands. A common challenge for existing models is the accurate simulation of the cyclic strength of sands for a wide range of initial conditions and different cyclic stress levels when adopting a single calibration. Many liquefaction models tend to overpredict the resistance of the soil under large-amplitude loading, while underestimating the strength at low-amplitude cyclic shearing. This manifests itself in slopes of simulated cyclic resistance ratio curves (CRR-curves) which are steeper than experimental studies indicate. This paper provides a discussion on the effects of large-amplitude and low-amplitude cyclic shearing on a granular material based on micromechanical and experimental investigations presented in the literature. A constitutive model with a shear-history threshold is proposed, which accounts for a shift of the apparent angle of phase transformation under cyclic loading. In addition, a novel expression for a deviatoric fabric tensor is introduced to describe the evolution of shear-induced fabric anisotropy while a soil is dilating and contracting. Combining these two features in one formulation within the bounding surface plasticity framework enables an accurate prediction of cyclic strength of sands under a wide range of cyclic stress ratios.

Study on generalized BK-5th recurrent Finsler space

In this paper, we present a novel new class and investigate the connection between the K-projective curvature tensor and other tensors of Finsler space Fn, this space is characterized by the property for Cartan’s 4th curvature tensor satisfies the certain relationship with the given covariant vectors field, we define this space as a generalized BK-5th recurrent space and denote it briefly by GBK-5RFn. This paper aims to derive the fifth-order Berwald covariant derivatives of the torsion tensor and the deviation tensor . Additionally, it demonstrates that the curvature vector Kj, the curvature vector Hk, and the curvature scalar H are all non-vanishing within the considered space. We have identified tensors that exhibit self-similarity under specific conditions. Furthermore, we have established the necessary and sufficient conditions for certain tensors in this space to have equal fifth-order Berwald covariant derivatives with their lower-order counterparts.

A state-dependent multilaminate constitutive model for anisotropic sands

Experimental studies show that initial fabric and its evolution under different stress paths greatly influences soil behaviour. Even though different sample preparation methods create different inherent anisotropies and cause different material responses, the same initial fabric structure under different stress paths also results in different material behaviours. In this paper, a simple state-dependent, bounding surface-based elastoplastic constitutive model, which can simulate the anisotropic nature of sands including the effect of principal stress rotation, is described. The model is developed based on a semi-micromechanical concept within the multilaminate framework and, to include the inherent anisotropy of sand, a deviatoric fabric tensor describing the initial microstructure is introduced. In addition, a fabric evolution rule compatible with anisotropic critical state theory is employed to describe the evolving fabric structure and induced anisotropy towards the critical state. In contrast to the classical strain-driven formulation for fabric evolution, a micro-level evolution rule is proposed. This paper presents concise theoretical aspects of the multilaminate framework and the anisotropic elastoplastic constitutive formulation. The model's capability under drained and undrained monotonic loading conditions at different stress states, relative densities and principal stress orientations is demonstrated by simulating experimental data for Toyoura sand.

Determination of Yield Strength of Single-Crystals with a Hexagonal Lattice at a Given Strain Tensor and Hydrostatic Pressure

Within the framework of the Mises yield criterion generalized to hexagonal crystals proposed by the authors, the problem of determining the stress tensor depending on the strain rate tensor and the applied external pressure is solved. The cases of plane and uniaxial deformation are considered in detail for an arbitrary orientation of the crystal lattice. For the cases of plane and uniaxial deformation, diagrams of stress level lines for the onset of plastic flow are plotted. Diagrams are given for the components of the stress tensor deviator

Enstrophy production and flow topology in compressible isotropic turbulence with vibrational non-equilibrium

Enstrophy production and flow topology are numerically investigated for statistically stationary compressible isotropic turbulence in vibrational non-equilibrium with a large-scale thermal forcing. The net enstrophy production term is decomposed into solenoidal, dilatational and isotropic dilatational terms based on the Helmholtz decomposition. From the full flow field perspective, the net enstrophy production mainly stems from the solenoidal term. For the dilatational and isotropic dilatational terms, although their local magnitudes can be considerable, the positive values in the compression region and the negative values in the expansion region cancel out on average. For the solenoidal component of the deviatoric strain-rate tensor, the statistical properties of its eigenvalues and alignments between vorticity and its eigenvectors are nearly independent of the local dilatation and vibrational relaxation. The solenoidal components of enstrophy production along three eigendirections are thus mainly affected by the vorticity. For the dilatational component of deviatoric strain-rate tensor, the statistical properties of its eigenvalues and alignments between vorticity and its eigenvectors closely relate to the local dilatation and vibrational relaxation. The dilatational components of enstrophy production along three eigendirections are therefore affected by the vorticity, eigenvalues and alignments between the vorticity and eigenvectors. The topological classification proposed by Chong et al. (Phys. Fluids, vol. 2, issue 5, 1990, pp. 765–777) is employed to decompose the flow field into various flow topologies. In the strong compression and strong expansion regions, the relaxation effects on the volume fractions of flow topologies and their relative contributions to the local enstrophy production are significant.

Predicting the Timing of Catastrophic Failure During Triaxial Compression: Insights from Discrete Element Method Simulations

Examining the deformation of rocks during triaxial compression may provide insights into the precursory phase that leads to large earthquakes by revealing the components of the deformation field that evolve with predictable, systematic behavior preceding catastrophic failure. Here, we build three-dimensional discrete element method simulations of the triaxial compression of rock cores that include a variety of fault geometries in order to identify the components of the deformation field, including the velocity and strain components, that enable machine learning models to predict the timing of macroscopic failure. The results suggest that the velocity field provides more valuable information about the timing of macroscopic failure than the strain field, and in particular, the velocity component parallel to the maximum compression direction. The models also strongly depend on the second invariant of the strain deviator tensor, {J}_{2}, indicative of the shear strain. The importance of {J}_{2} on the model predictions increases with confining stress, consistent with laboratory observations that show a transition from tensile- to shear-dominated deformation with increasing confining stress. In contrast to expectations from previous machine learning analyses, none of the models strongly depend on components of the strain tensor indicative of dilation, such as the first invariant of the strain tensor. This difference may arise because the simulations host less dilative strain than the experimental rocks.

Upgrading SANISAND model to address continuous stress principal axis rotation: A conservative approach

AbstractThe SANISAND models are known for their ability to simulate sands behavior with various properties under a wide range of loadings. Nevertheless, there is much room for improvement in simulating stress principal axis (PA) rotation effects. This study tried to address the continuous stress PA rotation problem following a conservative approach in preserving the SANISAND basic formulation. The inelastic response of the material under the stress rotation is regulated by a plastic intensity index as a function of the stress rotation rate with respect to the deviatoric fabric tensor and the non‐coaxiality between the stress and the plastic strain rate. The model performance is verified compared to the experimental data as well as other constitutive models. Supplementary discussions are provided to shed light on how fabric evolution results in a lower degree of anisotropy under continuous stress PA rotation. Based on continuum bounding surface plasticity, the proposed model addresses the fabric anisotropy effects on the severity of plasticity under continuous stress PA rotation.

Predicting fault reactivation and macroscopic failure in discrete element method simulations of restraining and releasing step overs

To assist the identification of precursors to ruptures in segmented fault networks, we build discrete element method simulations of two parallel faults that are underlapping or overlapping, in releasing or restraining steps. We use machine learning models to predict the timing of the reactivation of the faults, and the timing of macroscopic failure identified from the peak shear stress acting on the boundaries. The machine learning models use the evolving three-dimensional components of the strain and velocity fields to make these predictions. The models depend on the same characteristics, regardless of the preexisting geometry: the component of the velocity vector parallel to the applied loading direction, vx, the shear strain component εxy, and the second invariant of the strain deviator tensor, J2. The results suggest that crustal monitoring of strike-slip systems in which the loading is approximately parallel to the fault strike, with releasing and restraining steps and overlapping and underlapping faults, may focus on the same set of strain and velocity components. However, the results indicate that the key characteristics that control the timing of fault reactivation depend on the distance from the preexisting faults. When the models use data within one fault half-length of the preexisting fault, the predictions primarily depend on εxy and vx, whereas when they use data outside this region, they primarily depend on vx. Monitoring efforts that focus on the near-fault deformation field may benefit from tracking information that helps estimate εxy and vx, in contrast to monitoring efforts further from the main faults. Models developed with data further from the faults perform worse than models developed with all of the data, but better than models developed with near fault data, consistent with observations of distributed and subsequently localizing low-magnitude seismicity.

On the well-posedness of Banach spaces-based mixed formulations for the nearly incompressible Navier-Lamé and Stokes equations

In this paper we introduce and analyze, up to our knowledge for the first time, Banach spaces-based mixed variational formulations for nearly incompressible linear elasticity and Stokes models. Our interest in this subject is motivated by the respective need that arises from the solvability studies of nonlinear coupled problems in continuum mechanics that involve these equations. We consider pseudostress-based approaches in both cases and apply a suitable integration by parts formula for ad-hoc Sobolev spaces to derive the corresponding continuous schemes. We utilize known and new preliminary results, along with the Babuška-Brezzi theory in Banach spaces, to establish the well-posedness of the formulations for a particular range of the indexes of the Lebesgue spaces involved. Among the aforementioned new results from us, we highlight the construction of a particular operator mapping a tensor Lebesgue space into itself, and the generalization of a classical estimate in L2 for deviatoric tensors, which plays a key role in the Hilbertian analysis of linear elasticity, to arbitrary Lebesgue spaces. No discrete analysis is performed in this work.

DTI Atlases Evaluations.

The cerebral atlas of diffusion tensor magnetic resonance image (DT-MRI, shorted as DTI) is one of the key issues in neuroimaging research. It is crucial for comparisons of neuronal structural integrity and connectivity across populations. Usually, the atlas is constructed by iteratively averaging the registered individual image. In tradition, the fuzzy group average image is easily generated in the initial stage, which is harmful to providing clear guidance for subsequent registration, to the performance of the final atlas. To solve this problem, an improved unbiased DTI atlas construction algorithm based on adaptive weights is proposed in this paper. The adaptive weighted strategy based on diffeomorphic deformable tensor registration is introduced. At the same time, the distance measure for tensors is used as a constraint condition, which ensures the unbiasedness of the atlas. Then, using 77 DTIs from the dataset in http://www.brain-development.org , three study-specific atlases, i.e. the constructed atlases of the proposed algorithm and two open-sourced algorithms (DTIAtlasBuilder and DTI-TK), are compared with two standardized atlases (IIT v. 4.1 and NTU-DSI-122-DTI). The performances of the atlases were evaluated in spatial normalization way with six region-based criteria (including Euclidean distances between diffusion tensors, Euclidean distances of the deviatoric tensors, standard deviation, overlaps of eigenvalue-eigenvector, cross-correlations and three sets angles of eigenvalue-eigenvector pairs between diffusion tensors) and three fiber-based criteria (including distances between fiber bundles, angles between fiber bundles and fiber property profile-based criteria). The experimental results showed that the overall performances of the study-specific atlases are better than those of the standardized atlases for specific datasets, and the comprehensive performance of the improved algorithm proposed in this paper is the best.

Effect of time–temperature, strain level and cyclic loading on the complex Poisson’s ratio of asphalt mixtures

Complex modulus and complex Poisson’s ratio are two parameters which characterise the 3D linear viscoelastic properties of asphalt mixtures. Compared to the complex modulus, the complex Poisson’s ratio is more difficult to be measured because of the great accuracy needed for the experimental device. This investigation focuses on the complex Poisson’s ratio of asphalt mixtures and the effect of loading parameters on its values. The three loading parameters considered in the paper include the time–temperature, the strain level and the number of cyclic loadings. To fulfil the objective of study, complex modulus tests, fatigue tests combined with the rest time, large amplitude cyclic loading tests and cyclic permanent deformation tests are performed. The complex modulus tests are conducted at different temperatures, frequencies and strain amplitudes. Thanks to the fatigue tests, the large amplitude cyclic loading tests and the cyclic permanent deformation tests, the change in complex Poisson’s ratio with the number of cycle, the pre-deformation strain level of specimen and the accumulated permanent deformation (up to some %) respectively is also investigated. The obtained results show that the time–temperature and the number of cyclic loading have a clear effect on the complex Poisson’s ratio of asphalt mixtures. No significant change in the complex Poisson’s ratio is observed when varying the strain cyclic loading amplitude within the small strain domain. The change in the complex Poisson’s ratio becomes more important and is observed to be proportional with the norm of the strain deviator tensor when the axial strain is accumulated or pre-loaded up to some percent.

Hypoplastic modeling of anisotropic sand behavior accounting for fabric evolution under monotonic and cyclic loading

A unified hypoplastic model is formulated by incorporating the anisotropic critical state theory to describe the fabric effect in sand under both monotonic and cyclic loading conditions. An evolving deviatoric fabric tensor that characterizes the internal microstructure of sand is introduced into the hypoplastic model in conjunction with the intergranular strain concept. A scalar-valued fabric anisotropic variable indicating the interplay between the fabric and the loading direction is employed to account for the impact of fabric anisotropy on both the dilatancy and shear strength of sand. The model is demonstrated to be capable of simulating the anisotropic behavior of sand, using a single set of parameters under both monotonic and cyclic loading conditions, as evidenced by the satisfactory match with experimental results from various sources. In particular, by considering the influence of fabric evolution on the dilatancy of sand, the model adequately accounts for the fabric change effect and accurately captures the deviatoric strain accumulation, cyclic mobility, and the flow liquefaction phenomenon under cyclic loading condition.

Generalised neo-pseudo projective recurrent Finsler space

Objectives: The purpose of this paper is to obtain several results in the field of generalised neo-pseudo projective recurrent Finsler space. Methods: A generalization technique is employed to solve the resulting problem. We provide its application in the study of space-time. Findings: In section 1, we have defined and studied some of the basic and useful results for later work. Section 2 deals for the neo-pseudo projective recurrent curvature tensor. The notion of neo-pseudo projective recurrent space of second order has been delineated in the section 3. In the section 4 we have studied the generalised neo-pseudo projective recurrent space and established several new results. Novelty/Conclusion: In this paper we have studied some recurrent properties of neo-pseudo projective curvature tensor in a Finsler space. We have obtained several new results which are as follows:• If the space Fn admits a neo-pseudo projective curvature tensor Qa bgd then Qa bgd is skew-symmetric with regard to last two indices.• If the neo-pseudo projective deviation tensor Qab and pseudo deviation tensor field Tab coincides to each other for q = 1 then space is W-flat.• If Fn admits the projectively flat Q-recurrent space then the relationÑeQabg +ÑbQage +ÑgQaeb = 0 holds good.• If a Finsler space Fn admits projectively flat Q-birecurrent space then the relation KerQabg +KbrQage +KgrQaeb = 0 holds good.• If the space is Q-birecurrent then the generalised Q-recurrent space is Qsymmetric.• For the projective flat generalised Q-recurrent space the relationÑrÑeQagd +ÑrÑgQade +ÑrÑdQaeg = 0 holds good.AMS Subject Classification: 58B20, 53C20, 53C60.

Influence of different loading paths on the multiaxial fatigue behavior of 2024 aluminum alloy under the same amplitude values of the second invariant of the stress deviator tensor

2024 aluminum alloy is a common aeronautic material. During operations, construction elements made of aluminum alloys undertake complex cyclic loadings. Therefore, it is important to estimate the influence of these loadings on the durability of the material. Hereby, multiaxial fatigue tests with the same amplitude values of the second invariant of the stress deviator tensor are conducted, and test data are analyzed. The modified Sines method is utilized to predict fatigue experimental data. Results show that the model is accurate enough to fatigue behavior prediction of 2024 aluminum alloy.

Recurrence Decompositions in Finsler Space

Finsler geometry is a kind of differential geometry originated by P. Finsler. Indeed, Finsler geometry has several uses in a wide variety and it is playing an important role in differential geometry and applied mathematics of problems in physics relative, manual footprint. It is usually considered as a generalization of Riemannian geometry. In the present paper, we introduced some types of generalized $W^{h}$ -birecurrent Finsler space, generalized $W^{h}$ -birecurrent affinely connected space and we defined a Finsler space $F_{n}$ for Weyl's projective curvature tensor $W_{jkh}^{i}$ satisfies the generalized-birecurrence condition with respect to Cartan's connection parameters $\Gamma ^{\ast i}_{kh}$, such that given by the condition (\ref{2.1}), where $\left\vert m\right. \left\vert n\right. $ is\ h-covariant derivative of second order (Cartan's second kind covariant differential operator) with respect to $x^{m}$ \ and $x^{n}$ ,\ successively, $\lambda _{mn}$ and $\mu _{mn~}$ are\ non-null covariant vectors field and such space is called as a generalized $W^{h}$ -birecurrent\ space and denoted briefly by $GW^{h}$ - $BRF_{n}$ . We have obtained some theorems of generalized $W^{h}$ -birecurrent affinely connected space for the h-covariant derivative of the second order for Wely's projective torsion tensor $~W_{kh}^{i}$ , Wely's projective deviation tensor $~W_{h}^{i}$ in our space. We have obtained the necessary and sufficient condition forsome tensors in our space.

Density ratio effects on the topology of coherent turbulent structures in two-way coupled particle-laden channel flows

This investigation considers the effect of density ratio on the modification of coherent turbulent structures in particle-laden channel flows at a shear Reynolds number of Reτ = 180. Direct numerical simulation and Lagrangian particle tracking are used to accurately predict the motion of particles dispersed within the flow at three Stokes numbers, St+ = 0.1, 50, and 92. Particle–fluid coupling is achieved through a local element-based force feedback field in the Navier–Stokes equations, which are solved using a seventh-order accurate spectral element method. After an initial transitory period wherein the effects of particle–fluid interaction are emphasized, the low density ratio particles are found to enhance the turbulence field, increasing the frequency of Q-criterion satisfying regions, while the inertial particles suppress the turbulence, reducing the number of quasistreamwise vortices. Results indicate that the topology of the quasistreamwise vortices is altered by the presence of the particles in the viscous sublayer, the buffer layer, and the log-law region such that the distribution of the third invariant of the deviatoric tensor, R, is widened by the presence of tracer-like particles and made thinner by the inertial particles. This effect reduces the amount of unstable focus/compressing regions and stable focus/stretching regions, which account for the streamwise vortical structures observed in these types of flow. Investigating the instantaneous coupling force field surrounding turbulent structures and low speed streaks shows that particles exert their greatest influence on the fluid in the regions noted, and mechanisms by which the particulate phase interacts with turbulent vortices are analyzed.

The extrema point deviatoric moment component

The concern plate’s behavior was the growth of the extrema positive tensors of both mean (hydrostatic) and deviatoric moments. Systematic consideration of the limits and growth responses of the mean and deviatoric tensors at the extrema positive sections were calculated. It was found that the use of the shorter dimension for the moment normalization in both directions preserves the directional influence of the dimensions and isolates clearly the hydrostatic phenomenon. Moreover, the presence of non-identical boundary conditions in the plate on two perpendicular directions was found to result in the deviation of the two-way action of behavior. Consequently, a novel understanding of the two-way plate action has been achieved. In such understanding, the pure two-way action accompanies the occurrence of a hydrostatic curvature (or moment) at a plate’s extrema positive point. At the later state, the hydrostatic and deviatoric tensor components would be equal. Based on linear and multi-Stage regression analysis, the growth of the normalized hydrostatic moment component (CHMc) with the geometrical aspect has been deemed to follow a linear regression model, as well as the growth of the deviatoric moment component with the geometrical aspect ratio, has been deemed to follow the sigmoidal, Morgan-Mercer-Flodin (MMF) nonlinear regression model.