Mechanical Equilibrium Equations Research Articles

Behaviour of steel-fibre-reinforced geopolymer concrete-filled square steel tubular stub column under axial compression

Resource utilisation of industrial waste has become a crucial issue in construction aiming for sustainable development. Research on geopolymer concrete-filled square steel tubular (GCFST) structures is limited. Moreover, methods for calculating the bearing capacity of steel-fibre-reinforced GCFST (SFRGCFST) structures require further investigation. This study presents an experimental and analytical investigation of the axial compression performance of SFRGCFST stub columns. Twenty-four specimens were tested to assess the effect of steel tube thickness, steel fibre (SF) volume fraction, SF aspect ratio, and concrete strength grade on failure modes, load–deformation behaviour, load–strain response, ultimate resistance, and ductility of SFRGCFST columns. The results indicated that the failure modes of SFRGCFST columns are influenced by the coefficient index, which varies with the test parameters. When the coefficient index exceeded 0.88, the failure mode of the specimens transitioned from shearing failure to waist-drum-shaped failure. The average ultimate bearing capacity of the SFRGCFST columns increased by 35.13 % compared with that of the plain concrete-filled steel tubes (CFSTs). When the SF volume fraction reached 0.9 %, the ultimate bearing capacity of the SFRGCFST columns increased by 21.85 % compared with that of steel-fibre-reinforced concrete-filled steel tube (SFRCFST) columns. The incorporation of SF significantly improved the deformation capacity and ductility of the SFRGCFST columns and moderated the slope of the post-peak load–displacement and load–strain curves, slowing the descent of the curve. Utilising the multi-axial strength criterion and mechanical equilibrium equation for square CFSTs, a formula for calculating the ultimate bearing capacity of SFRGCFST stub columns was developed. This formula enables accurate prediction of the axial compressive load capacity of SFRGCFST stub columns and provides a basis for revising relevant standards.

A new spinning-extrusion forming technology for the inner-ribbed component

To solve the problem that the rib filling height is low in the traditional spinning forming (TSF) of ribbed component, a new spinning-extrusion forming technology (SEF) was proposed. A convenient device was built to conduct the SEF experiment, whose results show that the new SEF can greatly increase the rib filling height by 53% compared with the TSF. Moreover, the improvement mechanism and prediction model of rib filling height in the SEF were investigated. It is found that the rib filling in SEF is mainly dependent on two deformation stages: tension-shear deformation, which makes the material accumulate and bulge above the rib groove; compression-shear deformation, which makes the accumulated material fill into the rib groove. The final rib filling height is linear dependent on the material bulge amount in the tension-shear deformation stage. And, the improvement of rib filling by SEF lies in it can enhance the duration and deformation amount of tension-shear deformation stage, thus improve the material bulge amount in contrast to the TSF. On these bases, taking the material bulge region in SEF as object, slab method was used to analyze its deformation mechanics and establish the differential mechanical equilibrium equations; Thamasett method was adapted to calculate the spinning force; synthetically, the material bulge amount was theoretically calculated. Then, combing the linear relation between the material bulge amount and rib filling height, the model for prediction of rib filling height in the SEF was developed. The predicted rib filling heights under various processing parameters indicate that the rib filling height can be improved by increasing the extrusion force, thickness reduction rate, feed ratio and roller attack angle in the SEF.

Stochastic model for migration and breakage of detrital and authigenic fines

Mobilisation of attached particles during flow in rocks occurs in geo-energy processes. Particle mobilisation, their migration through rocks and pore plugging yield significant decline in permeability and well injectivity and productivity. While much is currently known about the underlying mechanisms governing the detachment of detrital particles against attracting electrostatic forces, a critical gap exists in the theoretical understanding of detachment by breakage of widely spread authigenic particles, which naturally grow on rock grains during geological times. Previous works derived micro-scale mechanical equilibrium equations for both detrital and authigenic particles, and the upscaling procedure from particle to pore and core scales for detrital fines. In this paper, for the first time we derive a stochastic model for migration and breakage of authigenic fines and authigenic–detrital mixtures. This allows for core-scale transport modelling based on the particle-scale torque balance. We introduce a novel framework for predictive stochastic detachment modelling by particle–rock bond breakage that integrates the beam theory of elastic particle deformation, strength failure criteria and viscous flow around the attached particle. The analytical expressions for stress maxima and stress diagrams for a single particle allow determining the critical failure stresses, breakage points of the beam and breakage flow velocity. The mathematical model describing lab coreflood includes the maximum retention function for both authigenic and detrital fines. The matching laboratory coreflood data under increasing velocity at micro- and core-scales achieved high matching of the experimental data by the model. High matching validates the upscaling and downscaling procedures derived.

Coupled thermomechanical analysis using isoparametric curved shell elements

This study focuses on the coupled thermo-mechanical formulation in isoparametric shell elements and its implementation in curved laminar structures. The formulation is based on the direct isoparametric formulation, specifically designed for flat and curved shell elements, and relies on the theory of degenerated solids. The research provides a comprehensive description of the coupled thermo-mechanical analysis, including the mechanical equilibrium equations, the interrelations between stress, temperature, displacement, deformation, and the conservation of thermal energy. It adopts a simultaneous approach to addressing the impacts of temperature and mechanical loads. The paper elaborates on various numerical examples that substantiate the formulation. These examples include simulations that encompass linear scenarios, which are then compared to analytical solutions and results derived from other numerical software. The examples highlighted include one-dimensional temperature diffusion, radial diffusion, and the thermal behavior of a cylindrical cover and a pipe under a linear temperature gradient. These simulations demonstrate the formulation's capacity to accurately capture the interactions between temperature variations and structural displacements. They also confirm the alignment of the proposed model with existing analytical and numerical solutions. In conclusion, the research provides a good framework for coupled thermo-mechanical analysis. The flexibility of the formulation is evidenced across different configurations and scenarios, accommodating various types of boundary conditions such as constant and linear temperature fields, distributed loads, and constraints on displacements and rotations. It effectively manages temperature flows across one, two, and three dimensions, highlighting its extensive applicability in the realm of structural analysis. This versatility underscores its broad applicability in the field of structural analysis.

Thermodynamically consistent numerical modeling of immiscible two‐phase flow in poro‐viscoelastic media

AbstractNumerical modeling of immiscible two‐phase flow in deformable porous media has become increasingly significant due to its applications in oil reservoir engineering, geotechnical engineering and many others. The coupling between two‐phase flow and geomechanics gives rise to a major challenge to the development of physically consistent mathematical models and effective numerical methods. In this article, based on the concept of free energies and guided by the second law of thermodynamics, we derive a thermodynamically consistent mathematical model for immiscible two‐phase flow in poro‐viscoelastic media. The model uses the fluid and solid free energies to characterize the fluid capillarity and solid skeleton elasticity, so that it rigorously follows an energy dissipation law. The thermodynamically consistent formulation of the pore fluid pressure is naturally derived for the solid mechanical equilibrium equation. Additionally, the model ensures the mass conservation law for both fluids and solids. For numerical approximation of the model, we propose an energy stable and mass conservative numerical method. The method herein inherits the energy dissipation law through appropriate energy approaches and subtle treatments for the coupling between two phase saturations, the effective pore pressure and porosity. Using the locally conservative cell‐centered finite difference methods on staggered grids with the upwind strategies for saturations and porosity, we construct the fully discrete scheme, which has the ability to conserve the masses of both fluids and solids as well as preserve the energy dissipation law at the fully discrete level. In particular, the proposed method is an unbiased algorithm, that is, treating the wetting phase, the non‐wetting phase and the solid phase in the same way. Numerical results are also given to validate and verify the features of the proposed model and numerical method.

New insights into dynamic disaster monitoring through asynchronous deformation induced coal-gas outburst mechanism of tectonic and raw coal seams

The coalbed methane energy resources waste, environmental pollution and miner's health remain challenging problems in worldwide mining operations accompanied by coal-gas outbursts; thus, the safe and efficient yield of these resources plays a vital role in the world's energy market. The tectonic coal development is a significant geological feature of coal and gas outburst sites, and the study of coal and gas outburst mechanism should be based on more precise geological occurrence characteristics. In this paper, the geological characteristics of coal and gas outburst sites were analyzed. The failure law of coal body under the stratified occurrence of tectonic coal and raw coal seam with different thickness was studied. The mechanical equilibrium equation of non-synchronous deformation of tectonic coal and raw coal was constructed, the initiation criterion of coal and gas outburst was established based on the support body model, and a case study was carried out. The results show that with the thickness increase of the tectonic coal seam, the peak displacement of the driving face increases. The outburst start-up and development of coal and gas outburst, coal and gas extrusion outburst and coal and gas collapse outburst were deeply analyzed, as well as the internal reasons of the occurrence of coal-gas outburst portent are explained, based on the support body model. The development process from the start to the end of coal and gas outburst is revealed, and a new perspective is proposed for the prevention and control of coal and gas outburst: improving the strength of support body is also of great significance for the prevention and control of coal and gas outburst.

Multi-phase-field approach to fracture demonstrating the role of solid-solid interface energy on crack propagation

AbstractA multi-phase-field approach for crack propagation considering the contribution of the interface energy is presented. The interface energy is either the grain boundary energy or the energy between a pair of solid phases and is directly incorporated into to the Ginzburg–Landau equation for fracture. The finite difference method is utilized to solve the crack phase-field evolution equation and fast Fourier method is used to solve the mechanical equilibrium equation in three dimensions for a polycrystalline material. The importance of the interface (grain boundary) energy is analyzed numerically for various model problems. The results show how the interface energy variations change the crack trajectory between the intergranular and transgranular fracture.

Simulation of SMA-based engineering applications considering large displacement and rotation, thermomechanical coupling and partial phase transformation

Shape memory alloys (SMAs) undergo large recoverable deformation due to their inherent diffusionless martensitic phase transformation. Two factors need to be taken into account to incisively emulate the behaviour of SMA based structures: (i) consideration of large displacement and rotation (LDR) effect, and (ii) capturing the consequence of transformation induced material level coupling. In this study, both these effects are apprehended by extending the infinitesimal strain-based constitutive model of SMA as proposed by Lagoudas et al. (2012), assuming small elastic strain but with finite inelastic strain. To preclude any spurious stress generation out of large rotation, the Jaumann stress rate is used, and incremental objectivity for finite rotation between two successive time instants is conserved by rotating the strain increment and spin tensor to the rotation-independent mid-point configuration following Hughes–Winget (Hughes and Winget, 1980) algorithm. In addition, the contribution of thermoelastic and latent heat evolved during transformation is taken into consideration in the thermal equilibrium equation. Moreover, the effect of partial phase transformation yielding minor hysteresis loop has also been captured. The equilibrium equation is expressed in the current configuration following the Updated Lagrangian formulation. The mechanical and thermal equilibrium equations are solved concurrently in block matrix form using the Newton–Raphson (NR) iterative technique, considering the coupling terms resulting from the phase transformation. The efficacy and robustness of the developed finite element model are corroborated through various practical applications of SMA-based members, e.g., SMA ring, SMA-actuated beam, morphing of corrugated airfoil, orthodontic palatal expander, compliant gripper, undergoing LDR while subjected to different thermomechanical loading conditions. The combined effects of LDR along with thermomechanical coupling yield a stiffening behaviour during loading and sluggish response at the time of thermal recovery.

Investigating the Elastic Response of Smart Cylinders Under Asymmetric Loading

This paper investigates the hygrothermal-magneto-elastic response of functionally graded piezomagnetic (FGPM) cylinders under asymmetric loading. The cylinders are supported by a Winkler-type elastic foundation, and their properties vary with the radius according to a power-law function. By solving 2D equations of Fickian diffusion and Fourier relations, the distribution of asymmetric moisture concentration and temperature field is determined. Incorporating constitutive equations into mechanical and magnetic equilibrium equations yields three second-order partial differential equations. The equations are solved using the separation of variables and complex Fourier series. Simulation results demonstrate the influence of hygrothermal loading, magnetic field, elastic foundation, and material inhomogeneity on the cylinder's response.

Solving viscoelastic problems with a Laplace transform approach supplanted by ARX models, suggesting a way to upgrade Finite Element or spectral codes

Finite Element codes used for solving the mechanical equilibrium equations in transient problems associated to (time-dependent) viscoelastic media generally relies on time-discretized versions of the selected constitutive law. Recent concerns about the use of non-integer differential equations to describe viscoelasticity or well-founded ideas based upon the use of a behavior's law directly derived from Dynamic Mechanical Analysis (DMA) experiments in frequency domain, could make the Laplace domain approach particularly attractive if embedded in a time discretized scheme. Based upon the inversion of Laplace transforms, this paper shows that this aim is not only possible but also gives rise to a simple algorithm having good performances in terms of computation times and precision. Such an approach, which fully relies on the Laplace-defined Behavioral Transfer Function (LTBF) can be promoted if it uses AutoRegressive with eXogeneous input parametric models perfectly substitutable to the real LTBF. They avoid the hitherto prohibitive pitfall of having to store all past data in the computer's memory while maintaining an equal computation precision.

Effect of the diamond saw wires capillary adhesion on the thickness variation of ultra-thin photovoltaic silicon wafers during slicing

In order to improve the performance of solar cells and reduce their manufacturing costs, monocrystalline silicon wafers used for manufacturing solar cells are developing towards larger sizes and ultra-thin thickness, and the diameter of diamond saw wires used to cut silicon wafers is also continuously decreasing. These aspects cause uneven thickness and lager thickness variation (TV) of the as-cut wafer during slicing process. The reason of uneven thickness was analyzed from the point of liquid bridge with capillary force between the saw wires in this paper. The mechanical equilibrium equation between the liquid bridge and the saw wires was established based on the Young-Laplace equation, and the theoretical model of the as-cut wafer thickness was derived during the sawing process. The effect of the saw wire tensioning force, span, the diameter of the core wire and the wetting angle of coolant on the as-cut wafer thickness variation was analyzed based on the model. The results show that the increase in the saw wire tensioning force, the core wire diameter and the wetting angle of coolant improved the thickness accuracy and reduced the thickness variation of the as-cut wafer. And the increase in the saw wire span reduced the thickness accuracy and increased the thickness variation. The research results of this paper provide the mechanism analysis for the thickness variation of the ultra-thin wafer during the slicing process. It is of great significance to produce the ultra-thin photovoltaic silicon wafers with the fine saw wire in the future.

Stimuli-responsive shell theory

Soft matter mechanics generally involve finite deformations and instabilities of structures in response to a wide range of mechanical and non-mechanical stimuli. Modeling plates and shells is generally a challenge due to their geometrically nonlinear response to loads; however, non-mechanical loads further complicate matters as it is often not clear how they modify the shell’s energy functional. In this work, we demonstrate how to form a mechanical interpretation of these non-mechanical stimuli, in which the standard shell strain energy can be augmented with potentials corresponding to how a non-mechanical stimulus acts to change the shell’s area and curvature via the natural stretch and curvature. As a result, the effect of non-mechanical stimuli to deform shells is transformed into effective external loadings, and this framework allows for the application of analytical and computational tools that are standard within the mechanics community. Furthermore, we generalize the effect of mass change during biological growth to account for its effect on the stress constitution. The theory is formulated based on a standard, stress-free reference configuration which is known a priori, meaning it can be physically observed, and only requires the solution of a single-field equation, the standard mechanical momentum or equilibrium equation, despite capturing the effects of non-mechanical stimuli. We validate the performance of this model by several benchmark problems, and finally, we apply it to complex examples, including the snapping of the Venus flytrap, leaf growth, and the buckling of electrically active polymer plates. Overall, we expect that mechanicians and non-mechanicians alike can use the approach presented here to quickly modify existing computational tools with effective external loadings calculated in this novel theory to study how various types of non-mechanical stimuli impact the mechanics and physics of thin shell structures.

Porous Media Model of Reservoir considering Seepage Stress Coupling and Seepage Field Analysis

In the process of reservoir exploitation, the range of rock stress field changes is much larger than the reservoir area where the seepage occurs; so the amount of calculation of the existing seepage stress full coupling method is often very huge. Based on the theory of seepage mechanics and rock mechanics, a coupling analysis model of reservoir multiphase seepage and stress is established, and a numerical solution is established by using finite element and finite difference methods. The evolution law of the seepage field and stress field and the change of rock mechanics parameters can be studied, with emphasis on the readjustment of rock stress distribution and its deformation characteristics under the influence of the seepage field. At the same time, to improve the calculation efficiency of numerical simulation, considering the difference between the seepage calculation area and stress calculation area, the finite element method is improved. The percolation area of the reservoir is calculated with a fine grid to obtain a more accurate distribution of underground fluid. The coarse grid is used to calculate the stress calculation area and to reduce the calculation time. The mechanical equilibrium equation in the fully coupled theory is discretized on the coarse grid by the finite element method. The mass conservation equation is discretized on the fine grid by the finite volume method. The numerical simulation of Terzaghi’s one-dimensional consolidation problem and Mandel’s two-dimensional consolidation problem shows that the calculation results of this method are in good agreement with the analytical solution. Through the numerical calculation of a two-dimensional single-phase flow single well production problem and a three-dimensional two-phase flow five-point well pattern production problem, the influence of the seepage stress coupling effect in reservoir numerical simulation is analyzed.

Development of a Thermal Model of Welding by the Finite Element Method in Software "Bazis"

The paper presents a model and an algorithm for calculating the operational properties of a metal and a structure after its assembly-welding. The model is based on the equations of non-stationary thermal conductivity and mechanical equilibrium, as well as the analysis of structural-phase transformations using thermokinetic diagrams. It is implemented in the original BAZIS software package. Validation of software was carried out when calculating thermal fields and deformations as applied to arc welding, arc deposition, and laser welding of standard joints.

Model matematyczny silnika skokowego z wirnikiem aktywnym

The use of step-motor drives is required where it is necessary to precisely determine the position of the device component performing the movement. The modeling of any stepping motor consists of the equilibrium equations of voltages and the equilibrium equation of moments. Based on the proposed mathematical model of a stepping motor, a simulation model can be built, which can provide a lot of information about the operation of the drive before it is built. The following work shows a mathematical model of a stepping motor taking into account the equilibrium equations of mechanical and electrical equilibrium. Thermal phenomena were not considered in this model

A deep learning approach for solving diffusion-induced stress in large-deformed thin film electrodes

Understanding the effects of large deformation on electrochemical performance of electrodes, such as silicon and tin, in lithium-ion battery has stimulated great interest in analyzing stress evolution during electrochemical cycling and in developing mechanochemical models for the stress analysis. As the complexity of mechanochemical models continues to increase, there is an urgent need to develop computational methods to reduce computational costs and increase computational efficiency. In this work, we propose a physics-inspired neural network (PINN) based on the DeepXDE deep learning library in analyzing diffusion-induced stresses (DIS) in a thin film electrode with large deformation, in which a loss function, which is associated with the loss functions of partial differential equations (PDEs) in the domain and the initial/boundary conditions for the mechanochemical problem, is introduced. Using the proposed physics-inspired neural network, the mechanical equations, including geometric, constitutive and equilibrium equations, in the framework of finite deformation theory as well as the mass transport equation are solved to obtain the stress evolution in the large-deformed thin-film electrode. The numerical results are in accord with the results obtained from finite element method. This work provides a unique approach for deep learning to solve the coupling mechanochemical problems in energy storage.

Stress analysis of rotating thick-walled nonhomogeneous sphere under thermomechanical loadings

In this work, a rotating thick-walled spherical vessel made of nonhomogeneous materials subjected to internal and/or external pressure under thermal loading was analyzed within the context of three-dimensional elasticity theory. An analytical formulation was established for computing the displacement and stress fields. It has been assumed that the mechanical and thermal properties are varying through thickness of the functionally graded material (FGM) according to a power-law nonlinear expression, while Poisson's ratio is considered as constant. Based on equilibrium equation, Hooke's law, stress-strain relationship in the spheres and other theories from mechanics a second-order ordinary differential equation well-known as Navier equation is obtained that represents the thermoelastic field in hollow FGM sphere. Navier equation derived from the mechanical equilibrium equation was solved to obtain the exact solution of the displacement and stress distributions. Different results of thermoelastic field are presented across the thickness of the sphere. The analysis of the different results reveals that stress and strain in the FGM sphere are significantly depend upon variation made in temperature profile, rotation and inhomogeneity parameter on the thermoelastic field. Thus, the inhomogeneity in material properties can be exploited to optimize the distribution of displacement and stress fields.

Phase field simulation of martensitic transformation in Ti–24Nb–4Zr–8Sn alloy

A phase field model for cubic to orthorhombic martensitic transformation (MT) at the nanoscale in a β titanium (Ti) alloy Ti–24Nb–4Zr–8Sn (in wt.%) is investigated by finite element simulation. The approach is based on phase field theory, time-dependent Ginzburg-Landau theory, and mechanical equilibrium equations. Partial differential equations (PDEs) were solved using the commercial software COMSOL Multiphysics. The morphology of the product phase exhibits plate-like or needle-like shapes that reduce the elastic strain energy of the system. The simulation result for random initial order parameters is in agreement with previous experimental observations. The final volume fractions of two different orthorhombic martensitic variants are not dependent on the initial conditions.

Coupled Field Analysis of Phenomena in Hybrid Excited Magnetorheological Fluid Brake

The paper presents a field model of coupled phenomena occurring in an axisymmetric magnetorheological brake. The coupling between transient fluid dynamics and electromagnetic and thermal fields as well as mechanical equilibrium equations is taken into account. The magnetic field in the studied brake is of an excited hybrid manner, i.e., by the permanent magnets (PMs) and current Is in the excitation winding. The finite element method and a step-by-step algorithm have been implemented in the proposed field model of coupled phenomena in the considered brake. The nonlinearity of the magnetic circuit and rheological properties of a magnetorheological fluid (MR fluid) as well as the influence of temperature on the properties of materials have been taken into account. To solve equations of the obtained field model, the Newton-Raphson method and the coupled block over-relaxation method have been implemented. The elaborated algorithm has been successfully used in the analysis of the phenomena in the considered magnetorheological brake. The accuracy of the developed model and its usefulness have been verified by a comparative analysis of the results of simulation and laboratory tests carried out for the developed prototype of the studied brake.

Analytical and numerical analysis of functionally graded (FGM) axisymmetric cylinders under thermo-mechanical loadings

This study examines the thermo-mechanical behavior of FGM hollow cylinders subjected to mechanical loading and steady thermal stresses. The distribution of temperature, displacement, and stresses on the cylinder's inside and outside surfaces is considered a function of radius. The analytic solution of the displacement and stress distributions was obtained by solving Navier's second-order differential equation using the integral and differential transform method, derived from the mechanical equilibrium and heat equations. Moreover, numerical models were built using the finite element method with USDFLD and the finite difference method to test the accuracy of the analytical solution. The analytical formulation and numerical simulation predictions have a high level of agreement, indicating that the model is accurate. The effect of inhomogeneity in the FGM thick cylinder was addressed in this study by selecting a non-dimensional parameter β=[−2,2]that may be assigned any value influencing the stresses in the cylinder. It is concluded that by altering the valueβ, the properties of FGM can be altered to achieve the lowest stress levels. It is also shown that the effect of various Boundary Conditions (B.Cs) is significant in stress fields. This present work also observed that the significance of thermal B.C. is more dominant than fixed pressure B.Cs.