Stress Balance Equation Research Articles

A numerical model for the simulation of complex planar Newtonian interfaces

We present a numerical model for the simulation of complex planar interfaces at which moving solid objects can be immersed, reproducing a wide variety of experimental conditions. The mathematical model consists of the Navier-Stokes equations governing the incompressible viscous flow in the liquid subphase, the transport equation for the evolution of the surfactant concentration at the interface, and the interfacial stress balance equation. The equations are simplified by treating the problem as isothermal and the surfactant as insoluble. The bulk flow equations are discretized using a collocated finite volume method, while the interfacial flow equations are discretized using a finite area method. The Boussinesq-Scriven interface constitutive model and a variant form accounting for extensional viscosity are used to describe the extra surface stress tensor. The coupling between surfactant concentration, interfacial velocity, and bulk velocity is treated implicitly by solving the interfacial and bulk equations sequentially at each time step until a stopping criterion is satisfied. The motion of the solid is treated by an arbitrary Lagrangian-Eulerian method. The model has been implemented in the OpenFOAM framework and allows the incorporation of new interface models and solvers, making the developed new package a versatile and powerful tool in the field of computational rheology. Applications of the model include the numerical simulation of flow around objects, such as probes, immersed at a complex interface, reproducing given experimental conditions, and its use as a tool in the analysis and design of interfacial stress rheometers. Several test cases have been performed to validate the model by comparing the results obtained with analytical solutions and with numerical and experimental results available in the literature.

Prediction method of thermal expansion coefficient for 2D composite fabric

In this article, an analytical method for predicting the thermal expansion coefficient of 2D composite fabrics is established. First of all, different from the traditional prediction of thermal expansion coefficient by classical laminate theory, this paper extends the self-consistent method of micromechanics to 2D composite fabrics to improve the accuracy of thickness direction prediction. Analytical Prediction method of Thermal expansion coefficient of 2D Composite fabric (APTC)was established. In the x section of Representative Volume Element(RVE) model, the stress balance equation and deformation coordination equation are applied to derive the internal thermal expansion coefficient. In the z section, the coefficient of thermal expansion is derived from the Poisson effect. Secondly, the formula of thermal expansion coefficient is derived for plain fabric, twill fabric and satin fabric respectively, which is consistent with the APTC method, indicating that the method is representative. Finally, RVE finite element model is established according to the braided structure parameters of 5284/T800 composite fabric. Comparing analytical solution, numerical solution and experimental solution, the α 1 and α 2 of the numerical and experimental solutions is close enough to be considered equal, which is consistent with the transverse isotropy of the composite fabric in xOy plane predicted by theoretical analysis.When predicting α 1 and α 2 , the maximum error between numerical solution and analytical solution is 1.79%, the maximum error between numerical solution and experimental solution is 22.29%, and the error remains at about 4% during predicting α 3 . In general, the APTC method for predicting the thermal expansion coefficient of composite fabrics is accurate and effective, and the simplified formula is more convenient to popularize and apply.

Cyclic loading and unloading strain equations and damage evolution of gypsum specimens considering damping effects

This study aims to establish a strain instanton equation and damage factor evolution law for gypsum specimens by considering damping. First, damping energy is calculated based on the single-degree-of-freedom vibration model, and the instantaneous strain equation is obtained based on the stress balance equation. Second, the dissipation energy is divided into damping and damage energies, and a damage-factor correction algorithm is obtained. Third, cyclic loading and unloading tests were performed at different loading rates and stress amplitudes to verify the accuracy of the strain equation. Finally, the specimens’ magnitude curves and crack characteristics were monitored using moment–tensor acoustic emission simulations. The factors influencing the damping energy and strain equations, energy and damage evolution laws of the specimens, and damage patterns of the specimens at different loading rates were analysed. The results show that the instantaneous strain equation and the modified damage factor considering the damping effect can effectively reflect the deformation law and damage state of the specimens. In contrast, the damage to the specimens in the lower limit of the variable stress experiment was lower than that in the lower limit of the constant stress experiment. As the loading rate increases, the damage energy density of the specimen decreases, and the damage factor within a single cycle gradually decreases. As the loading rate increases, the number of crack events in the model increases significantly, size becomes more uniform, and sequentially exhibits dense and sparse distribution patterns, percentage of shear cracks decreases significantly, number of mixed cracks increases significantly, brittle behaviour of the specimen becomes obvious, and a complete damage state is attained known as the ‘crushed’ state. This study provides a theoretical reference for damage assessments of viscoelastic–plastic materials subjected to perturbing loads.

Viscous correction to the potential flow analysis of Rayleigh–Taylor instability in a Rivlin–Ericksen viscoelastic fluid layer with heat and mass transfer

AbstractThe current investigation focuses on examining viscous corrections for viscous potential flow (VCVPF) analysis concerning the Rayleigh–Taylor instability occurring at the interface of a Rivlin–Ericksen (R–E) viscoelastic fluid and a viscous fluid during the transfer of heat and mass between phases. The R–E model is a fundamental framework in the study of viscoelastic fluids, providing insights into their complex rheological behavior. It characterizes the material's response to both deformation and flow, offering valuable predictions for various industrial and biological applications. Within the framework of viscous potential flow (VPF) theory, viscosity is exclusively accounted for in the normal stress balance equation, disregarding the influence of shearing stress entirely. This study introduces a viscous pressure term into the normal stress balance equation alongside the irrotational pressure, presuming that this addition will improve the discontinuity of tangential stresses at the fluid interface. Through derivation of a dispersion relationship and subsequent theoretical and numerical stability analyses, the stability of the interface is investigated across various physical parameters. Multiple plots are generated using the dispersion relation, and a comparative analysis between VPF and VCVPF is conducted to establish improved stability criteria. The investigation highlights that the combined impact of heat/mass transport and shearing stress serves to delay the instability of the interface.

From electron tomography of dislocations to field dislocation mechanics: application to olivine

We propose a new procedure to extract information from electron tomography and use them as an input in a field dislocation mechanics. Dislocation electron tomography is an experimental technique that provides three-dimensional (3D) information on dislocation lines and Burgers vectors within a thin foil. The characterized 3D dislocation lines are used to construct the spatial distribution of the equivalent Nye dislocation density tensor. The model dislocation lattice incompatibility equation and stress balance equation are solved with a spectral code based on fast Fourier transform algorithms. As an output of the model, one obtains the 3D distribution of mechanical fields, such as strains, rotations, stresses, resolved shear stresses (RSSs) and energy, inside the material. To assess the potential of the method, we consider two regions from a previously compressed olivine sample. Our results reveal significant local variations in local stress fields and RSSs in various slip systems, which can impact the strong plastic anisotropy of olivine and the activation of different dislocation slip systems. It also evidences the built-up of kinematic hardening down to the nanometre scale.

A numerical analysis of buckle cable force of concrete arch bridge based on stress balance method

It is difficult to calculate the buckle cable force for the cantilever casting concrete arch bridge. Relying on the 180 m-span Jiming Sansheng Bridge, this paper put forward one method to calculate the initial buckle cable force based on the stress balance method. Firstly, the stress balance equation considering only tensile stress was derived for the first time, and the feasible region of the initial cable force was calculated by the allowable tensile stress of the arch rib, which improved the original stress balance method. Then, using the influence matrix, the initial buckle cable force was optimized by reducing the allowable tensile stress of concrete in stages, and finally the optimal initial cable force was obtained. The practical engineering results show that it is feasible to calculate the initial buckle force. The maximum tensile stress of concrete arch during the cantilever casting process is 1.52 MPa, meeting the specification requirements. The deviation between the calculated and measured stress is less than 12%. The calculated cable force agrees with the measured cable force, and the deviation is less than 2%. The initial cable force is only tensioned once, improving work efficiency. The method and experience of this paper can provide a reference for the arch bridge constructed by cantilever casting.

Elastically non-linear discrete model for core of edge dislocation

To facilitate the modeling of crystal defects on their core scales, the present paper introduces a discrete stress balance equation that admits non-linear elasticity of the defected crystal cell and its arbitrary boundary conditions. Thus, the cell may be free or embedded into continuum. In the latter case, when considering a sufficiently large domain tractable by the proposed method, boundary forces can be related to long-range stresses of the defect. For a case study, an edge dislocation in the primitive cubic lattice is simulated to assess influence of the crystal dimensions and elastic properties on the microscopic discrete core.

Homogenized Balance Equations for Nonlinear Poroelastic Composites

Within this work, we upscale the equations that describe the pore-scale behaviour of nonlinear porous elastic composites, using the asymptotic homogenization technique in order to derive the macroscale effective governing equations. A porous hyperelastic composite can be thought of as being comprised of a matrix interacting with a number of subphases and percolated by a fluid flowing in the pores (which is chosen to be Newtonian and incompressible here). A general nonlinear macroscale model is derived and is then specified for a particular choice of strain energy function, namely the de Saint-Venant function. This leads to a macroscale system of PDEs, which is of poroelastic type with additional terms and transformations to account for the nonlinear behaviour of the material. Our new porohyperelastic-type model describes the effective behaviour of nonlinear porous composites by prescribing the stress balance equations, the conservation of mass and Darcy’s law. The coefficients of these macroscale equations encode the detailed microstructure of the material and are to be found by solving pore-scale differential problems. The model reduces to the following limit cases of (a) linear poroelastic composites when the deformation gradient approaches the identity, (b) nonlinear composites when there are no pores and (c) nonlinear poroelasticity when only the matrix–fluid interaction is considered. This model is applicable when the interactions between various hyperelastic solid phases occur at the pore-scale, as in biological tissues such as artery walls, the myocardium, lungs and liver.

Application of the finite volume method for geomechanics calculation and analysis on temperature dependent poromechanical stress and displacement fields in enhanced geothermal system

The hot dry rock (HDR) development is a hot research field in recent years. Numerical simulation based on thermal-hydraulic-mechanical coupling model is a key technique for the HDR development evaluation. The calculation of geomechanics is an important part of the coupling thermal-hydraulic-mechanical simulation, and it is usually implemented by finite element method. However, meshing methods of geological modeling software are designed to serve seepage simulation, and grid quality cannot meet the requirements of finite element method. In this paper, finite volume method has been proposed to calculate the distribution of underground stress field, in order to solve the problem of discontinuous grid vertexes and calculation failure caused by finite element method. In addition, the mesh of geological modeling software is for finite difference method or finite volume method, which cannot be used directly by finite element method. In this paper, spatial discretization pattern of quasi static stress balance equation based on the finite volume method is obtained, the local grid refinement algorithm is proposed, the solution workflow of thermal-hydraulic-mechanical (THM) coupling model is presented, and the solution of displacement discretization equations and THM one way coupling model are implemented by C++ programming language. Finally, the feasibility and accuracy of the finite volume method of stress field calculation are verified through a series of numerical test examples compared with commercial finite element software. Several THM coupling simulation cases of enhanced geothermal system are presented by finite volume method. Numerical test results show that the calculation error between finite volume method and finite element method is less than 1%. Moreover, the variation of principal stress, induced by pore pressure and thermal stress, can reach 8 to 20 MPa. The finite volume method provides a unified technical framework for geomechanics and thermal-hydraulic calculation. Moreover, the method can be directly combined with geological modeling software without using two sets of grids. The results prove that the finite volume method would be a promising approach for thermal-hydraulic-mechanical simulation.

Landslide Accumulation Ice-Snow Melting for Thermo-Hydromechanical Coupling and Numerical Simulation

In this paper, we discussed the phase change coupling algorithm of accumulation bank slope under the action of ice-snow melting. We described the effect of temperature gradient on the water migration of soil. We simplified the stress balance, continuity, and energy equation in the coupled model. We discussed the variation law of temperature, seepage, and stress (deformation) field under different conditions of ice-snow melting on the bank slope of the accumulation body. Based on the three-field coupling energy balance equation of ice-snow melting with phase change, the simplified algorithm of three-field coupling is obtained. The simplified algorithm is applied to the coupling model of ice-snow thawing on indoor accumulation bank slope. We established a practical numerical model for the coupling analysis of temperature, seepage, and stress field. We established the coupled control differential equation of three fields. We investigated the three-dimensional numerical simulation of stress, displacement, plastic deformation, and other indicators. The results show that the numerical simulation results are in good agreement with the monitoring results. It is expected that the research results can more truly simulate the actual characteristics of ice and snow melting water on the bank slope of the Three Gorges Reservoir and provide reference for the prevention and prediction of extreme snow and ice disasters in the Three Gorges Reservoir area.

Effect of Wall Boundary Conditions on a Wall-Modeled Large-Eddy Simulation in a Finite-Difference Framework

We studied the effect of wall boundary conditions on the statistics in a wall-modeled large-eddy simulation (WMLES) of turbulent channel flows. Three different forms of the boundary condition based on the mean stress-balance equations were used to supply the correct mean wall shear stress for a wide range of Reynolds numbers and grid resolutions applicable to WMLES. In addition to the widely used Neumann boundary condition at the wall, we considered a case with a no-slip condition at the wall in which the wall stress was imposed by adjusting the value of the eddy viscosity at the wall. The results showed that the type of boundary condition utilized had an impact on the statistics (e.g., mean velocity profile and turbulence intensities) in the vicinity of the wall, especially at the first off-wall grid point. Augmenting the eddy viscosity at the wall resulted in improved predictions of statistics in the near-wall region, which should allow the use of information from the first off-wall grid point for wall models without additional spatial or temporal filtering. This boundary condition is easy to implement and provides a simple solution to the well-known log-layer mismatch in WMLES.

Study on One-Dimensional Large Deformation Consolidation of Soil considering Liquid Phase Inertia

Classical consolidation theory ignores the influence of soil liquid phase acceleration. This paper considers the influence of liquid phase acceleration on the stress balance equation during the consolidation of soil, obtains the one-dimensional equation governing quasi-hydrostatic consolidation under large deformation with the consideration of the inertia of the liquid phase, and solves the governing equations by finite element method. The calculation results show that the liquid phase inertia effect of the soil will cause excess pore pressure in the soil, obviously increasing in the initial stage of consolidation, and the self-weight of soil exerts an influence on the excess pore pressure at the later stages of consolidation. The liquid phase inertia effect parameter Dc determines the strength of the liquid phase inertia effect. A larger Dc value results in a larger increase in the excess pore pressure, and the later the liquid phase inertial effect occurs, the longer the duration is. In the large strain consolidation analysis, especially at the initial stage of consolidation, it is necessary to consider the liquid phase inertia effect of the soil.

Moving surface mesh-incorporated particle method for numerical simulation of a liquid droplet

In this study, a new particle method for simulating the dynamics of a liquid droplet in a two-dimensional space has been developed. The proposed method incorporates a moving surface mesh to represent a deformable free-surface boundary. The domain enclosed by the surface mesh is defined as a liquid volume (droplet), and the outer region is a gas phase with constant pressure. Fluid particles are seeded inside the liquid domain, and also, discrete nodes along the surface mesh are defined as additional computational points. The incompressible flow is solved based on the LSMPS (least squares moving particle semi-implicit) method, where all differential operators are discretized by means of consistent schemes. The stress balance equations are solved along free surfaces, where the surface tension force is directly evaluated by the geometry of the surface mesh at each surface node. As numerical tests, various problems including a hydrostatic pressure problem, circular and square patch tests, droplet oscillations and static droplets suspended by solid walls have been simulated. As a result, the proposed method shows excellent agreement with the reference solutions, even under large free-surface deformations, which verifies the validity of the current developments.

Determination of initial cable force of cantilever casting concrete arch bridge using stress balance and influence matrix methods

Cantilever casting concrete arch bridge using form traveller has a broad application prospect. However, it is difficult to obtain reasonable initial cable force in construction stage. In this study, stress balance and influence matrix methods were developed to determine the initial cable force of cantilever casting concrete arch bridge. The stress balance equation and influence matrix of arch rib critical section were established, and the buckle cable force range was determined by the allowable stress of arch rib critical section. Then a group of buckle cable forces were selected and substituted into the stress balance equation, and the reasonable initial buckle cable force was determined through iteration. Based on the principle of force balance, the initial anchor cable force was determined. In an engineering application example, it is shown that the stress balance and influence matrix methods for the determination of initial cable force are feasible and reliable. The initial cable forces of arch rib segments only need to be adjusted once in the corresponding construction process, which improves the working efficiency and reduces the construction risk. It is found that the methods have great advantages for determining initial cable force in cantilever casting construction process of concrete arch bridge.

Modeling and Analysis of Single Point Incremental Forming Force with Static Pressure Support and Ultrasonic Vibration.

In order to solve the problem of low accuracy caused by instability and springback during the single point incremental forming (SPIF) process, static pressure support (SPS) and ultrasonic vibration (UV) are introduced into the technology for auxiliary forming. In order to qualitatively and quantitatively study the mechanism of static pressure support–ultrasonic vibration-single point incremental forming (SPS-UV-SPIF) force, a typical truncated cone is used as the research object. The working principle and motion rules of the technology are analyzed. The sheet micro-element of the sidewall area is taken as an analysis object. The spatial stress balance equation of the sheet is constructed. The various stresses are integrated and calculated. The forces in each area of the sheet are analyzed and modeled. Finally, an analytical model for SPS-UV-SPIF force is established. The influence law of the static pressure parameter and the vibration parameter on the forming force is obtained. The corresponding SPS system and UV system are designed. The Kistler forming force test system is built. The experimental results are consistent with the theoretical analysis results, which verifies the correctness of the analytical model.

A Seepage-Stress Coupling Model in Fractured Porous Media Based on XFEM

Large cracks are important seepage channels inside fractured-vuggy reservoirs. Therefore, in this thesis, the calculation method of fully coupled modeling of the fractured saturated porous medium based on the extended finite element method (XFEM) is established to study the expanding regularity of cracks in fractured-vuggy reservoirs. Fully coupled governing equations are developed for hydro-mechanical analysis of deforming porous medium with fractures based on the stress balance equation, the seepage continuity equation and the effective stress principle. The final nonlinear fully coupled equations reflect not only the coupling effect of the physical quantity within the porous medium but also the coupling between the medium and the fracture. During the spatial dispersion of coupled equations based on XFEM, two kinds of additional displacement functions are introduced in the displacement model of the fracture area to reflect the strong discontinuity of the fracture surface. The pore pressure enhancement function is also applied to represent the weak discontinuous features of the normal pore pressure. The validity and efficiency of this model and calculation are verified through three calculating examples. The following crack propagation laws are obtained: (1) The larger the water flow rate is, the longer the crack propagation length is, and the larger the propagation width is (2) The greater the crack angle and the crack length, the easier it is to expand the crack. Besides, compared with dip angle, the crack length has a more sensitive influence to the crack propagation. (3) When multiple cracks exist, the larger the fracture spacing is, the easier the crack will expand.

Heated transcritical and unheated non-transcritical turbulent boundary layers at supercritical pressures

Nominally zero-pressure-gradient fully developed flat-plate turbulent boundary layers with heated and unheated isothermal walls at supercritical pressures are studied by solving the full compressible Navier–Stokes equations using direct numerical simulation. With a heated isothermal wall, the wall temperature sets such that the flow temperature varies through the pseudo-critical temperature, and thus pseudo-boiling phenomena occur within the boundary layers. The pseudo-boiling process induces strongly nonlinear real-fluid effects in the flow and interacts with near-wall turbulence. The peculiar abrupt density variations through the pseudo-boiling process induce significant near-wall density fluctuations $\sqrt{\overline{\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D70C}^{\prime }}}/\overline{\unicode[STIX]{x1D70C}}\approx 0.4{-}1.0$ within the heated transcritical turbulent boundary layers. The large near-wall density fluctuations induce a turbulent mass flux $\unicode[STIX]{x1D70C}^{\prime }u_{i}^{\prime }$, and the turbulent mass flux amplifies the Favre-averaged velocity fluctuations $u_{i}^{\prime \prime }$ in the near-wall predominant structures of streamwise low-speed streaks that are associated with the ejection (where $u^{\prime \prime }<0$ and $v^{\prime \prime }>0$), while reducing the velocity fluctuations in the high-speed streaks associated with the sweep ($u^{\prime \prime }>0$ and $v^{\prime \prime }<0$). Although the near-wall low-speed and high-speed streak structures dominate the Reynolds-shear-stress generation, the energized Favre-averaged velocity fluctuations in the low-speed streaks enhance both the mean-density- and density-fluctuation-related Reynolds shear stresses ($-\overline{\unicode[STIX]{x1D70C}}\overline{u^{\prime \prime }v^{\prime \prime }}$ and $-\overline{\unicode[STIX]{x1D70C}^{\prime }u^{\prime \prime }v^{\prime \prime }}$) in the ejection event and, as a result, alter the Reynolds-shear-stress profile. The large density fluctuations also alter the near-wall viscous-stress profile and induce a near-wall convective flux $-\overline{\unicode[STIX]{x1D70C}}\widetilde{u}\widetilde{v}$ (due to non-zero $\widetilde{v}$). The changes in the contributions in the stress-balance equation result in a failure of existing velocity transformations to collapse to the universal law of the wall. The large density fluctuations also greatly contribute to the turbulent kinetic energy budget, and especially the mass flux contribution term becomes noticeable as one of the main positive terms. The unheated non-transcritical turbulent boundary layers show a negligible contribution of the real-fluid effects, and the turbulence statistics agree well with the statistics of an incompressible constant-property turbulent boundary layer with a perfect-gas law.

11011 回転分裂法を用いた静電浮遊液滴の粘性係数測定に関する研究

In order to establish a viscosity measurement technique using the rotational breakup of an electrostatically levitated droplet, new stress balance equation was derived and the viscosity was estimated in consideration of deformation of the droplet center. The glycerol droplet with approximately 2.5 mm in diameters was suspended in the air by an electrostatic levitator and rotated by a torque which was acoustically generated by the two loud speakers. The test section was established in the chamber, and the dry air was filled to prevent the moisture absorption. In order to prevent an acoustic effect, output of the two speakers stopped before the breakup. The elongational deformation behavior of the droplet was observed by high-speed video camera. Inclining deformation of rotational direction of the droplet center was observed just before the breakup. New stress balance equation was constructed from the momentum conservation equation of the axially symmetric droplet. The viscosity was evaluated by using the stress balance and image analysis. As a result, viscosity measurement accuracy was improved during small inclining deformation.

The Thin Ice Layer Brittle Fracture Stress Model and de-icing Angle Optimization

Through the establishment of stress balance equation and analysis of stress status, de-icing model and brittle fracture model under the conditions of vertical scraping and tilt scraping are presented respectively. According to material strength theory, the thesis analyses necessary external force to eliminate ice layer and its influencing factors, and then gains energy consumption forecast formula of mechanical scraping deicing, which provides the basis for motor selection of ice-scraping mechanism. Conclusion shows that tilt scraping can greatly reduce deicing energy consumption, that the smallest scraping force and energy consumption are obtained with tilt angle of 40°.

Hydrostatic grounding line parameterization in ice sheet models

Abstract. Modeling of grounding line migration is essential to accurately simulate the behavior of marine ice sheets and investigate their stability. Here, we assess the sensitivity of numerical models to the parameterization of the grounding line position. We run the MISMIP3D benchmark experiments using the Ice Sheet System Model (ISSM) and a two-dimensional shelfy-stream approximation (SSA) model with different mesh resolutions and different sub-element parameterizations of grounding line position. Results show that different grounding line parameterizations lead to different steady state grounding line positions as well as different retreat/advance rates. Our simulations explain why some vertically depth-averaged model simulations deviate significantly from the vast majority of simulations based on SSA in the MISMIP3D benchmark. The results reveal that differences between simulations performed with and without sub-element parameterization are as large as those performed with different approximations of the stress balance equations in this configuration. They also demonstrate that the reversibility test is passed at relatively coarse resolution while much finer resolutions are needed to accurately capture the steady-state grounding line position. We conclude that fixed grid SSA models that do not employ such a parameterization should be avoided, as they do not provide accurate estimates of grounding line dynamics, even at high spatial resolution. For models that include sub-element grounding line parameterization, in the MISMIP3D configuration, a mesh resolution finer than 2 km should be employed.