Linear Constitutive Relations Research Articles

Acoustically Enhanced Triboelectric Stethoscope for Ultrasensitive Cardiac Sounds Sensing and Disease Diagnosis.

Electronic stethoscope used to detect cardiac sounds that contain essential clinical information is a primary tool for diagnosis of various cardiac disorders. However, the linear electromechanical constitutive relation makes conventional piezoelectric sensors rather ineffective to detect low-intensity, low-frequency heart acoustic signal without the assistance of complex filtering and amplification circuits. Herein, it is found that triboelectric sensor features superior advantages over piezoelectric one for microquantity sensing originated from the fast saturated constitutive characteristic. As a result, the triboelectric sensor shows ultrahigh sensitivity (1215mV Pa-1) than the piezoelectric counterpart (21mV Pa-1) in the sound pressure range of 50-80dB under the same testing condition. By designing a trumpet-shaped auscultatory cavity with a power function cross-section to achieve acoustic energy converging and impedance matching, triboelectric stethoscope delivers 36dB signal-to-noise ratio for human test (2.3 times of that for piezoelectric one). Further combining with machine learning, five cardiac states can be diagnosed at 97% accuracy. In general, the triboelectric sensor is distinctly unique in basic mechanism, provides a novel design concept for sensing micromechanical quantities, and presents significant potential for application in cardiac sounds sensing and disease diagnosis.

Torsional constitutive relations at finite temperature

The general form of the linear torsional constitutive relations at finite temperature of the chiral current, energy-momentum tensor, and spin energy potential are computed for a chiral fermion fluid minimally coupled to geometric torsion and with nonzero chiral chemical potential. The corresponding transport coefficients are explicitly calculated in terms of the energy and number densities evaluated at vanishing torsion. A microscopic calculation of these constitutive relations in some particular backgrounds is also presented, confirming the general structure found.

Viscoelastic response of gray matter and white matter brain tissues under creep and relaxation

Accurate measurement of the mechanical properties of brain tissue is of paramount importance for understanding its mechanics-biology relationship. Most published studies on brain viscoelasticity have been conducted using a single relaxation test, without validating the validity of linear viscoelasticity, which is insufficient to establish an accurate constitutive equation for brain tissue. We obtained the creep and relaxation profiles of fresh adult porcine white matter (N = 120) and gray matter (N = 56) under finite step-and-hold uniaxial compression, using a mechanical testing machine, with 16.67 mm/s loading rate and 80 s hold time. These curves were employed to determine viscoelastic properties and demonstrated an excellent fit with a concise power-law function. The average initial modulus for gray matter (GM) was 6.619 kPa, higher than that for white matter under transverse loading (WM-2D) at 5.579 kPa (p < 0.01), yet lower than that for white matter under axial loading (WM-1D) at 6.759 kPa (p = 0.0121). Notably, WM-2D exhibited the highest degree of fluidity (β = 0.216). Our findings reveal that gray matter behaves as a linear viscoelastic material with power-law creep compliance and relaxation modulus. Conversely, the creep and relaxation behavior of white matter deviates from the verification relationship derived from linear viscoelastic theory, indicating its nonlinearity. This fact underscores the inaccuracy of assuming a linear constitutive relationship to characterize the viscoelastic properties of white matter. By combining the power-law function with the experimentally obtained creep compliance and relaxation modulus, we offer a unique approach to determining the viscoelastic characteristics of brain tissue.

A modified approach for a scaled boundary shell formulation in structural isogeometric analysis

In this contribution, an efficient modification of the scaled boundary finite element method for shells is presented to compute accurate results for h-refinement. While previous works on the scaled boundary shell formulation showed inaccurate and unstable results for fine meshes, this modification overcomes such problems by the assumption of a linear force distribution over the thickness. Furthermore, the shell formulation is derived in the framework of isogeometric analysis by discretizing the bottom surface of the shell as a reference surface. This leads to a solution for in-plane directions in a weak sense, while the scaling direction is solved analytically by a modified Padé expansion. Therefore, the normal vector on the reference surface is constructed and the shell is scaled along that leading to an extensible shell formulation incorporating a three-dimensional linear elastic constitutive relation. Since the isogeometric analysis is utilized, an exact derivation of the normal vector and its gradient is possible. Initial curvature locking and Poisson’s thickness locking are eliminated inherently while shear locking and volumetric locking are alleviated by mesh refinement. The power of the modified approach is presented by standard benchmark problems and compared to the results of the unmodified approach and standard shell formulations.

Hilbert’s energy–momentum tensor extended

In classical field models, Hilbert’s energy–momentum tensor for a matter field is defined by a variational derivative of a Lagrangian with respect to the metric tensor. Constitutive relationships between fundamental field variables are a subject that is dealt with by a wide range of models in solid-state physics. In this case, a higher-order constitutive tensor replaces the second-order metric tensor. A similar premetric description with a linear constitutive relation has recently been provided for classical field models of gravity and electrodynamics. In this paper, we analyze the extension of the Hilbert definition of the energy–momentum tensor to models with general linear constitutive law. In order to treat integrals on a differential manifold covariantly, we need the differential forms description. It follows that the Lagrangian, electromagnetic current, and energy–momentum current must all be represented as twisted 4-forms, 3-forms, and vector-valued 3-forms, respectively. The constitutive relation then appears as a linear map on forms. We derive a commutative variation identity for an arbitrary linear map, enabling direct variation processes without having to deal with the individual components. We present evidence for the close relationship between the explicit form of the energy–momentum current of Hilbert’s type and the commutative-variation identity. A variety of field models with a general linear constitutive law are subject to this construction.

Dynamic fracture investigation of concrete by a rate-dependent explicit phase field model integrating viscoelasticity and micro-viscosity

To investigate dynamic fracture mechanisms of quasi-brittle materials, this work proposes a rate-dependent phase field model that integrates both macroscopic viscoelasticity and micro-viscosity to reflect the rate effects by free water and unhydrated inclusions. Based on the unified phase field theory, the model introduces a linear viscoelastic constitutive relation in effective stress space to consider the macro-viscosity of the bulk material. Additionally, the micro-force balance concept is utilized with the micro-viscosity to derive a parabolic phase field evolution law that accurately describes the dynamic micro-crack development. Explicit numerical solution schemes are established for the governing equations by developing VUEL and VUMAT subroutines in ABAQUS. This eliminates the convergence issue in implicit phase field modelling. Four typical benchmarks are investigated to validate the proposed model for macroscale and mesoscale heterogeneous problems. It is found that the proposed model can well capture the crack branching, delaying characteristic of micro-crack growth, and increase of macroscopic strength under higher strain rates. Using real meso‑structures from CT images, the complicated dynamic behaviour of concrete is investigated which yields deeper insight into stress wave propagation, crack evolution and load-carrying capacities.

Turbulent wall plumes with detrainment

This article presents a theoretical modelling framework for the previously unconsidered case of turbulent wall plumes that detrain continually with height in stably stratified environments. Built upon the classic turbulent plume model, our approach incorporates turbulent detrainment with a variable coefficient of detrainment. Based on a linear constitutive relationship between the ratio of the detrainment to entrainment coefficients and the ambient buoyancy gradient, it is found that for linear ambient stratifications, a dynamic quasi-equilibrium region, characterised by a near invariant local plume Richardson number, is achieved, downstream of which this equilibrium rapidly breaks down. With increasing ambient buoyancy gradient, while the plume becomes increasingly slender with weaker vertical motions, the level at which the plume breaks down to form a horizontal intrusion first decreases and then increases. Moreover, distinct from classic purely entraining plumes, a detraining wall plume can swell within the pre-equilibrium adjustment stage provided the local Richardson number is sufficiently low $({Ri\ll 6})$ , behaviour which is in accordance with observations made in filling-box experiments.

Prediction of urban airflow fields around isolated high-rise buildings using data-driven non-linear correction models

When it comes to predicting urban airflow, steady Reynolds-averaged Navier-Stokes (SRANS) models that rely on Reynolds stress often face a challenge called the closure problem. This problem involves unresolved structural flaws and uncertainties in the closure coefficients used in the models. Previous attempts to recalibrate coefficients for specific urban flows without breaking the linear constitutive relation have resulted in simulation results constrained by the baseline turbulence model. Therefore, this study aims to enhance the performance of SRANS models by addressing these structural flaws. To achieve this, a novel data-driven framework is proposed. It leverages the deterministic symbolic regression algorithm to discover explicit algebraic expressions for a non-linear Reynolds stress correction model. The robustness of the correction model is ensured by maintaining the linear eddy viscosity model for iterative calculations while keeping the non-linear component frozen. The proposed framework is evaluated using three isolated building cases with varying geometric configurations and inflow boundary conditions. Findings demonstrate that computational fluid dynamics (CFD) predictions incorporating the data-driven non-linear correction model consistently align closer to wind tunnel experimental results compared to both standard and non-linear versions of the k-ε turbulence model. This improvement is reflected in reduced reattachment lengths and more accurate mean velocity distributions in the wake of buildings. However, it should be noted that there is a possibility of overpredicting wind velocity in the windward area. This study introduces valuable insights and additional strategies to enhance the prediction accuracy of SRANS models in urban airflow simulations.

Load introduction to composite columns revisited—Significance of force allocation and shear connection stiffness

The AISC 360-16 Specification recommends that the design shear force between parts of a composite column in the load introduction area shall be calculated based on the force allocation at ultimate limit state. Applicability of this straightforward method to the load levels that usually arise in slender composite columns is questionable, as this capacity-based force allocation is only true when the axial force is equal to the plastic resistance of the composite cross-section. Next, the number of required shear connectors is calculated as a quotient of the design shear force and the strength of a single shear connector. We demonstrate that: first, for the lower load levels, the stiffness-based force allocation gives a more accurate estimate of the shear force; second, the number of shear connectors satisfying the strength requirement can lead to insufficient force transfer between parts of the composite cross-section. To investigate the shear transfer mechanism in composite columns, we derive an analytical model with linear elastic constitutive relations both for steel and concrete and three types of shear force slip laws: elastic, elastic plastic, and rigid plastic. The case studies carried out for different shear transfer scenarios demonstrate the importance of the shear connection stiffness on the effectiveness of the load introduction. The remaining portion of the shear force is transferred outside the load introduction area, which hampers the column’s ability to withstand shearing from varying bending moments or incipient buckling. To control the shear force transfer efficiency by enhancing the shear connection stiffness, we propose an original Stiffness Method and provide design charts as an aid in the design process.

Machine-learned constitutive relations for multi-scale simulations of well-entangled polymer melts

We investigate the use of a machine-learning-based simulation scheme to predict flows of complex fluids with a non-linear constitutive relation. This scheme, originally proposed by Seryo et al. for general complex fluids [N. Seryo et al., “Learning the constitutive relation of polymeric flows with memory,” Phys. Rev. Res. 2, 033107 (2020)], is used to learn the time derivative of the stress as a function of the stress and velocity gradient. However, previous assessments were limited to fluids with linear constitutive relations. In this study, we employ the dual sliplink model to express the dynamics of well-entangled polymers as a fluid exhibiting a non-linear stress response. We test the simulation scheme by using it to predict the flow of a viscoelastic fluid between two parallel plates with only shear deformations and compare the results with those of a multi-scale simulation using microscopic simulators. Overall, our machine-learning method possesses good predictive capabilities, for both the transient response and the non-linear behavior at steady-state, i.e., the shear-thinning. We are able to accurately track the evolution of the stress, for both the weak and strong elastic cases, although the velocity predictions for the latter show a decreased accuracy in the transient regime. Furthermore, our simulation scheme is more computationally efficient than the conventional multi-scale simulation approach, which uses microscopic simulators, containing a system of coarse-grained polymers, to evaluate the macroscopic stress. We discuss possible extensions and improvements for enhancing the predictive capabilities and generality of the method.

Non-Fickian macroscopic model of axial diffusion of granular materials in a long cylindrical tumbler

We provide new concepts for understanding transport phenomena in flows of granular materials by using a non-Fickian macroscopic model of axial diffusion of a granular material in a finite cylindrical tumbler. The model accounts for diffusion induced by particle collisions only in a thin surface flowing layer due to localization of shear within the cross-section of the drum. All model parameters are related to measurable quantities in a granular flow. It is shown that the proposed model is a member of the general class of linear constitutive relations with memory. An exact solution for the spreading of a finite-width pulse initial condition under the proposed non-Fickian model is derived and compared to the solution of a Fickian model (i.e., the “classical” diffusion equation).

To the derivation of relativistic hydrodynamic equations for a rarefied non-ideal gas system of high-energy particles in the framework of Tsallis statistics

The paper discusses the construction of non-extensive relativistic dissipative hydrodynamics of an anomalous hadronic fluid on the basis of relativistic kinetic equation, obtained earlier in the context of the Tsallis statistics, characterized by the nonextensivity parameter q, and taking into account correlation effects (by rejecting the standard hypothesis of molecular chaos) in the collision term for heavy ions. It is shown that some specific form of local thermal equilibrium quark-gluon matter is described by a generalized version of the relativistic Yüttner distribution. With the help of this distribution all thermodynamic parameters of state are defined in explicit form. Linear constitutive relations and transport coefficients such as shear viscosity, bulk viscosity and heat conductivity are derived from the linearized collision integral written in the Anderson-Witting form and evaluated using a relaxation time approximation. The designed non-extensive relativistic fluid dynamics is designed to simulate a wide range of phenomena in astrophysics, cosmology and high-energy physics.

К выводу в рамках статистики Тсаллиса релятивистских гидродинамических уравнений для разреженной неидеальной газовой системы высокоэнергетических частиц

In the paper the construction of non-extensive relativistic dissipative hydrodynamics of identical particles on the basis of the relativistic kinetic equation obtained in a q-non-extensive context of Tsallis statistics and taking into account inclusion of correlation effects (by rejecting the molecular chaos hypothesis ) into the collision term. It is shown that the local collision equilibrium is described by a generalized version of the Yuttner relativistic distribution. With the help of this distribution all thermodynamic parameters of state are defined in explicit form. Linear constitutive relations and transport coefficients such as shear viscosity, bulk viscosity and heat conductivity are derived from the linearized collision integral written in the Anderson-Witting form and evaluated using a relaxation time approximation. The designed non-extensive relativistic fluid dynamics is designed to simulate a wide range of phenomena in astrophysics, cosmology and high-energy physics

Nonlinear coupled constitutive relation model for monatomic gas in the OpenFOAM framework

In high-speed rarefied gas flow simulations, the linear constitutive relations of the classical Navier-Stokes-Fourier (NSF) equations are no longer successfully applied. In the present work, the two-dimensional nonlinear coupled constitutive relations (NCCR) model of the monatomic gas in an algebraic-type form is successfully implemented in the rhoCentralFoam in OpenFOAM to simulate the high-speed rarefied gas flows. Methods: This solver initially resolves the NSF equations using high-resolution central schemes in the finite volume method. The implicit NCCR model is solved by the iterative process with the initial values from the linear constitutive relations of the NSF model. After the convergence, these nonlinear constitutive relations are embedded in the NSF equations to continue the solution. Results: This new modified solver would be validated for the argon gas flows past a circular cylinder in cross-flow at Mach of 5 and Kn = 0.05 and the sharp leading edge flat plate at Mach of 4 and Kn = 0.0042. Conclusion: The simulation results show that the NCCR model agrees well with the DSMC data for the temperature contours for both cases and the temperature and velocity along the shock stand-off distance and is better than the NSF model in the high-speed rarefied gas simulations for the circular cylinder case.

Global weak attractors in the dynamics of bodies with vector-type microstructure

We investigate the dynamics of bodies with vector-type microstructure. We consider linear constitutive relations and a nonlinear coupling between macroscopic and microscopic motions, determined by gyroscopic-type inertia. Based on an existence result obtained in the presence of viscous-type stress components, we determine the existence of a global attractor; its weak nature derives from the lack of uniqueness determined by the nonlinear coupling.

Dynamic simulation of aircraft electro-impulse de-icing using bond-based peridynamics

Electro-impulse de-icing is a lightweight type of mechanical de-icing system that consumes little energy, provides high reliability, and offers wide application prospects. In this study, the bond-based peridynamics theory was used to simulate the dynamic damage process and evolution law of aircraft electro-impulse de-icing. The ice layer was simplified as a brittle material that satisfies the linear elastic constitutive relation. A numerical analysis model of the ice layer, aircraft skin substrate, and their interface was established to simulate the dynamic responses under a high strain rate. Considering the anisotropy of ice adhesion on the substrate, the interfacial bonds were described as shear bonds and tensile bonds, and the critical elongations of the two were derived. The tensile and shear capacities of the ice on the substrate were then simulated using these critical elongations, and the peeling rates of single ice particles and interfacial ice layers were taken as indexes describing the efficacy of electro-impulse de-icing. The process of electro-impulse de-icing was then analyzed for an aluminum substrate with two adjacent clamped edges. Finally, the results of the peridynamic simulation were compared with those of existing experiments and finite element models to verify the effectiveness of the peridynamic approach.

Hodge duality between stress space and strain space in anisotropic media

AbstractDuality in a linear elasticity theory is studied based on a theory of differential form. A generalized expression of a Hodge star operator with an index is introduced. The index in the Hodge star operator means a superposition of ordinary Hodge star operators. By using the superposed Hodge star operator, a linear constitutive relation in anisotropic media can be expressed as a duality between a stress and strain spaces. Then, a basic elastodynamic equation is derived from the conservation law and the linear constitutive relation in the stress and strain spaces. Moreover, a geometric expression of equation of stress function and displacement function is derived by using the differential form and the dual relation of linear constitutive relation. These geometric results imply that the approach of differential forms is applicable to an analysis of deformation in anisotropic media. The superposed Hodge star operator discussed in this study has similar properties of a discrete Hodge operator, which represents the summation of differential forms on the decomposed regions. This means that the superposed Hodge star operator and the discrete Hodge operator express the constitutive relation in which physical quantities act on each other.

Investigating the Dynamic Compressive and Tensile Properties of Polymer Binder Explosive Based on the Split-Hopkinson Bar Technique

Split Hopkinson bar technique was often used to measure the dynamic mechanical properties of engineering materials. In this paper, the dynamic tensile and compression mechanical properties of polymer explosive bonded (PBX) under different strain rates were obtained by using split Hopkinson pressure/tension bar. The thickness of the specimen and the shape of the incident wave are designed to ensure the rationality of the experimental results. By comparing the experimental results, it was found that the PBX had different dynamic tensile and compression properties. The PBXs have been tested and shown tensile and compressive strengths ratios that range between 5 and 7. A constitutive relation is developed for modeling the dynamic mechanical response of PBX-I by using the Boltzmann superposition principle with a Prony series representation. The parameters of the PBX-I were fitted by using least square method. A finite element model was used to simulate the dynamic compressive and tensile behavior of PBX-I, and the numerical simulation results were in good agreement with the experimental results, which proved that the linear viscoelastic constitutive relation can be applied to the PBX-I.

On the torsional buckling moment of cylindrical shells consisting of functionally graded materials resting on the Pasternak-type soil

In this study, the buckling analysis of cylindrical shells made of functionally graded materials (FGMs) under the torsional moment resting on the Pasternak-type soil is performed. After establishing the linear constitutive relations of FGM cylindrical shells within the framework of the modified Donnell type shell theory, the governing equations of FGM cylindrical shells under the torsional moment are derived considering the influence of Pasternak-type soil. Analytical formula for the torsional moment is obtained by choosing the approximation functions that satisfies the boundary conditions in an integral sense. From the obtained formula, the formulas for the critical torsional moment in the presence of Winkler soil and absence of soils are obtained as a special case. Variations of critical torsional moment for different soil coefficients, volume fraction ratio and shell characteristics are investigated in detail. Keywords: functionally graded materials; cylindrical shell; buckling; critical torsional moment; Pasternak-type soil.

Using gene expression programming to discover macroscopic governing equations hidden in the data of molecular simulations

The unprecedented amount of data and the advancement of machine learning methods are driving the rapid development of data-driven modeling in the community of fluid mechanics. In this work, a data-driven strategy is developed by the combination of the direct simulation Monte Carlo (DSMC) method and the gene expression programming (GEP) method. DSMC is a molecular simulation method without any assumed macroscopic governing equations a priori and is employed to generate data of flow fields, while the enhanced GEP method is leveraged to discover governing equations. We first validate our idea using two benchmarks, such as the Burgers equation and Sine–Gordon equation. Then, we apply the strategy to discover governing equations hidden in the complex fluid dynamics. Our results demonstrate that in the continuum regime, the discovered equations are consistent with the traditional ones with linear constitutive relations, while in the non-continuum regime such as shock wave, the discovered equation comprises of high-order constitutive relations, which are similar to those in the Burnett equation but with modified coefficients. Compared to the Navier–Stokes–Fourier equations and the Burnett equation, the prediction of the viscous stress and heat flux in the shock wave via the presented data-driven model has the best match to the DSMC data. It is promising to extend the proposed data-driven strategy to more complex problems and discover hidden governing equations which may be unknown so far.