Mechanical Equilibrium Equations Research Articles

Martensitic microstructure evolution in austenitic steel: A thermomechanical polycrystalline phase field study

We have performed coupled thermomechanical phase field simulations of martensitic transformation in single-crystalline and polycrystalline 301-stainless steel by coupling Ginzburg–Landau equation to the mechanical equilibrium and non-stationary heat conduction equations. Simulations accounted for elastic anisotropy, under different thermal and mechanical BCs, i.e., clamped, embedded, and near-open, as well as adiabatic and isothermal with and without latent heat effects. To the best of authors’ knowledge, no such simulations are reported in literature on polycrystalline materials which provides the effects of various thermomechanical BCs incorporating latent heat. We have utilized a multivariant single embryo as an initial condition in both systems to mitigate the lack of nucleation criteria at the mesoscale; without any initial constraint, the model can select the martensitic transformation path and final microstructure. Simulations showed excellent capabilities in predicting influence of such thermomechanical BCs on martensitic microstructural formation including autocatalytic transformation process in polycrystalline steel and its evolution. Martensitic microstructure in a single-crystalline, and b polycrystalline steel, in deformed configuration (deformation scale factor is $$1\times$$ ) for near-open, embedded, and clamped boundary conditions (from left to right). Dataset edges in blue show initial undeformed configuration

Coupled mathematical modeling of cisplatin electroporation

The use of electrochemotherapy (ECT) is a well-established technique to increase the cellular uptake of cytotoxic agents within certain cancer treatment strategies. The study of the mechanisms that take part in this complex process is of high interest to gain a deeper knowledge of it, enabling the improvement of these strategies. In this work, we present a coupled multi-physics electroporation model based on a related previous one, to describe the effect of a set of electric pulses on cisplatin transport across the plasma membrane. The model applies a system of partial differential equations that includes Poisson’s equation for the electric field, Nernst-Planck’s equation for species transport, Maxwell’s tensor and mechanical equilibrium equation for membrane deformation and Smoluchowski’s equation for pore creation dynamics. Our numerical results were compared with previous numerical and experimental published data with good qualitative and quantitative agreement. These results indicate that pore aperture is favored at the cell poles by the electric field and mechanical stress forces, giving support to the dominant hypothesis of hydrophilic pore creation as the main mechanism of drug entry during an ECT treatment.

A linear poroelastic analysis of equilibrium asymptotic fields around stationary sharp V-notches in polymer gels

In this paper, an asymptotic solution is obtained for mode I stress and solvent concentration fields around stationary planar sharp V-notches in polymer gels at equilibrium state via poroelasticity theory. To this end, the mechanical equilibrium equations are first solved by employing a proper Airy stress function and then the solvent concentration field is obtained accordingly. It is shown that the stress field is similar to its corresponding linear elasticity solution with both real and complex eigenvalues. Furthermore, it is found that the solvent concentration field, around the notch tip, possesses the same singularity as the stress field and exhibits cosine variations with respect to the angular coordinate. The asymptotic solution is given for both plane stress and plane strain conditions. Next, a numerical study was performed on single edge notched (SEN) specimens with notch opening angles of 30° and 60° to explore the accuracy of the obtained asymptotic solution. It is shown that there is a very good match between the numerical results and the predictions of the asymptotic solution which confirms the validity of the present solution. The comparative study indicates that the first two or three terms of the asymptotic solution calibrated with the finite element over deterministic (FEOD) method is enough to accurately capture the stress and concentration fields at radial distances of about r/a = 0.2 from the notch tip. It is further shown that thicker SEN specimens absorb more solvent molecules per volume around their notch tips for similar far field applied strains and hence have more propensity to brittle fracture.

On a Viscoelastoplastic Porous Medium Problem with Nonlinear Interaction

A PDE system consisting of the mechanical equilibrium and mass balance equations for displacement and capillary pressure as a model for fluid diffusion in a partially saturated viscoelastoplastic p...

A new equation for period vectors of crystals under external stress and temperature in statistical physics: mechanical equilibrium condition and equation of state

Starting with the rigorous derivation of the work done on the center cell by external forces, a new equation is derived for the period vectors (cell edge vectors) in crystals under external stress and temperature. Since the equation is based on the principles of statistical physics, it applies to both classical and quantum systems. The existing theory for crystals under external pressure is covered as a special case. The new equation turns out to be the mechanical equilibrium condition and the equation of state for crystals under external stress and temperature. It may be used to predict crystal structures and to study structural phase transitions and crystal expansions. For linear elastic crystals, it takes the microscopic and temperature-dependent form of the generalized Hooke’s law, and may therefore be used to calculate the corresponding elastic constants. It should be helpful in studying piezoelectric and piezomagnetic materials, as the period vectors change with external stress. It is also consistent and can be combined with the previously derived corresponding one for Newtonian dynamics.

Method of integral equations in the polytropic theory of stars with axial rotation. I. Polytropes n=0 and n=1

Calculations of characteristics of stars with axial rotation in the frame of polytropic model are based on the solution of mechanical equilibrium equation – differential equation of second order in partial derivatives. Different variants of approximate determinations of integration constants are based on traditional in the theory of stellar surface approximation, namely continuity of gravitational potential in the surface vicinity. We proposed a new approach, in which we used simultaneously differential and integral forms of equilibrium equations. This is a closed system and allows us to define in self-consistent way integration constants, the polytrope surface shape and distribution of matter over volume of a star. With the examples of polytropes n=0 and n=1, we established the existence of two rotation modes (with small and large eccentricities). It is proved that the polytrope surface is the surface of homogeneous rotational ellipsoid for the case n=0. The polytrope characteristics with n=1 in different approximations were calculated as the functions of angular velocity. For the first time it has been calculated the deviation of polytrope surface at fixed value of angular velocity from the surface of associated rotational ellipsoid.

Topology optimization and 3D printing of large deformation compliant mechanisms for straining biological tissues

This paper presents a synthesis approach in a density-based topology optimization setting to design large deformation compliant mechanisms for inducing desired strains in biological tissues. The modelling is based on geometrical nonlinearity together with a suitably chosen hypereleastic material model, wherein the mechanical equilibrium equations are solved using the total Lagrangian finite element formulation. An objective based on least-square error with respect to target strains is formulated and minimized with the given set of constraints and the appropriate surroundings of the tissues. To circumvent numerical instabilities arising due to large deformation in low stiffness design regions during topology optimization, a strain-energy based interpolation scheme is employed. The approach uses an extended robust formulation i.e. the eroded, intermediate and dilated projections for the design description as well as variation in tissue stiffness. Efficacy of the synthesis approach is demonstrated by designing various compliant mechanisms for providing different target strains in biological tissue constructs. Optimized compliant mechanisms are 3D-printed and their performances are recorded in a simplified experiment and compared with simulation results obtained by a commercial software.

Investigation of displacements and stresses in thick-walled FGM cylinder subjected to thermo-mechanical loadings

In this paper, analytical and numerical solutions are presented for computing mechanical displacements and stresses in a thick-walled cylindrical made of functionally graded materials (FGMs) under mechanical and thermal loadings. The elastic modulus and thermal conductivity are assumed to vary in the radial direction, and the Poisson’s ratio is assumed constant. Navier’s ordinary second-order differential equation derived from the mechanical equilibrium equation was solved to obtain the exact solution of the displacement and stress distributions. Furthermore and in order to validate the accuracy of the analytical solution, a numerical model was construct using the finite element method. Very good agreement has been found between the analytical formulation and the predictions of numerical simulation which confirms the accuracy of the model in the case of a thick-walled tube made of FGM under steady state temperature distribution and a mechanical loading. These results clarify the influence of the thermal field, non-homogeneity parameter and internal pressure on the thermo-elastic response of the functionally graded cylindrical vessel.

Deformation and Stresses in Solid-State Composite Battery Cathodes

Differences in thermal contraction/expansion and volume changes during discharge/charge cycles lead to internal stresses that ultimately cause degradation of solid-state composite cathodes, hindering the realization of their practical applications. The mechanical equilibrium equation is solved using the smoothed boundary method for the residual stresses induced by cooling from the sintering temperature and by (de)lithiation in polycrystalline composite microstructures that are similar to realistic solid-state composite cathodes. The composite slabs' deformations under the fabrication and operation conditions are also predicted from the simulations. The effects of cathode thickness and selections of different cathode materials on the resulting residual stresses and deformations are examined. It is found that the (de)lithiation stresses during cycling are more than twice the residual thermal stresses after sintering. Furthermore, the maximum (de)lithiation and residual thermal stresses are sensitive to the cathode thickness only when the cathode-layer thickness is comparable to that of the electrolyte separator layer. We also investigate the impact of the lithium site fraction of the cathode particles prior to sintering on the cycling stresses and deformations, which may pave the path toward an approach to mitigating mechanically induced degradation.

Phase-Field Simulation of Stress Corrosion Cracking in Stainless Steel

Stress corrosion cracking (SCC) sometimes occurs in stainless steels, which are used for steam turbines of geothermal power plants. In order to prevent SCC that causes serious accidents in the plants, understanding the SCC behavior through experiments and numerical simulations is essential. Phase-field (PF) method is one of the widely used numerical simulation methods to analyze various microscale-material phenomena including electrochemical reactions such as corrosion. In this study, we have developed a new PF model for simulating a SCC behavior in a stainless steel, which includes the stress concentration at the propagating crack tip, diffusion and electrophoresis of ions in the liquid solution. The PF model is based on the PF model for general and pitting corrosion in a pure iron [C. Tsuyuki et al., Scientific Reports, Vol. 8 (2018), 12777.], which incorporates the Butler-Volmer equation. The stress evolution during the SCC is calculated by solving the mechanical equilibrium equation by means of fast Fourier transform. The diffusion and electrophoresis of ions are analyzed by solving the Nernst-Planck equation. Through the numerical simulations of SCC in a polycrystalline stainless steel which have a notch, we revealed that the crack growth rate increase in accordance with the stress intensity factor. The results have proved the crack proceeds through the grain when the elastic strain energy at the crack tip is high, and it propagate through the grain boundary when the low energy.

Theoretical and Experimental Study on Nonlinear Flow Starting Pressure Gradient in Low Permeability Reservoir

In order to reveal the seepage characteristics of low permeability fluid in reservoir pores, the seepage mathematical equation of low permeability reservoir is derived based on the mechanical equilibrium equation. The characteristics and rules of oil and water seepage flow in low permeability reservoir are studied. The seepage process and starting mechanism are deeply analyzed. The starting pressure gradient is derived from its yield stress and surface force, and the boundary layer intensifies the nonlinear degree of seepage. In the actual reservoir development, the nonlinear characteristics of seepage should be fully grasped. The results show that the starting pressure is related to permeability and other physical properties such as viscosity and porosity.

On topology optimization of large deformation contact-aided shape morphing compliant mechanisms

A topology optimization approach for designing large deformation contact-aided shape morphing compliant mechanisms is presented. Such mechanisms can be used in varying operating conditions. Design domains are described by regular hexagonal elements. Negative circular masks are employed to perform dual task, i.e., to decide material states of each element and also, to generate rigid contact surfaces. Each mask is characterized by five design variables, which are mutated by a zero-order based hill-climbing optimizer. Geometric and material nonlinearities are considered. Continuity in normals to boundaries of the candidate designs is ensured using a boundary resolution and smoothing scheme. Nonlinear mechanical equilibrium equations are solved using the Newton–Raphson method. An updated Lagrange approach in association with segment-to-segment contact method is employed for the contact formulation. Both mutual and self contact modes are permitted. Efficacy of the approach is demonstrated by designing four contact-aided shape morphing compliant mechanisms for different desired curves. Performance of the deformed profiles is verified using a commercial software. The effect of frictional contact surface on the actual profile is also studied.

A consistent phase field model for hydraulic fracture propagation in poroelastic media

We present a novel phase field method for modeling hydraulic fracture propagation in poroelastic media. In this approach, a new phase field evolution equation is derived to account for damage dependent poro-elastic parameters (Biot’s coefficient, Biot’s modulus and porosity). The fluid flow obeys Darcy’s seepage law in the entire domain including the damage zone, where the rock permeability is assumed to be anisotropic, following the maximum principal strain. The fully coupled problem is solved by a staggered scheme in which the mechanical equilibrium and fluid flow equations are linearized and solved using a Newton–Raphson(NR) method. Several numerical results are presented to investigate the effectiveness of the proposed formulation. First, stability and convergence of the method are verified on a set of benchmark problems considering different time steps and mesh sizes. Second, it is shown that if the poroelastic parameters are kept constant and do not change with the phase field parameter, i.e. reducing to standard phase field approaches in the literature, the model will tend to underestimate the fracture length and overestimate the pore pressure. Finally, we study the interaction of a propagating hydraulic fracture in porous media with inclined natural fractures, and simulate the hydraulic fracture propagation with different perforation phase angle.

The Modelling of Freezing Process in Saturated Soil Based on the Thermal-Hydro-Mechanical Multi-Physics Field Coupling Theory

The freezing process of saturated soil is studied under the condition of water replenishment. The process of soil freezing was simulated based on the theory of the energy and mass conservation equations and the equation of mechanical equilibrium. The accuracy of the model was verified by comparison with the experimental results of soil freezing. One-side freezing of a saturated 10-cm-high soil column in an open system with different parameters was simulated, and the effects of the initial void ratio, hydraulic conductivity, and thermal conductivity of soil particles on soil frost heave, freezing depth, and ice lenses distribution during soil freezing were explored. During the freezing process, water migrates from the warm end to the frozen fringe under the actions of the temperature gradient and pore pressure. During the initial period of freezing, the frozen front quickly moves downward, the freezing depth is about 5 cm after freezing for 30 h, and the final freezing depth remains about 6 cm. The freezing depth of the soil column is affected by soil porosity and thermal conductivity, but the final freezing depth mainly depends on the temperatures of the top and lower surfaces. The frost heave is mainly related to the amount of water migration. The relationship between the amount of frost heave and the hydraulic conductivity is positively correlated, and the thickness of the stable ice lens is greatly affected by the hydraulic conductivity. With the increase of the hydraulic conductivity and initial void ratio, the formation of ice lenses in the soil become easier. With the increase of the initial void ratio and thermal conductivity of soil particles, the frost heave of the soil column also increases. With high-thermal-conductivity soil, the formation of ice lenses become difficult.

Phase-field modeling of crack propagation in polycrystalline materials

A phase-field model based on a modified form of the regularized formulation of Griffith’s fracture theory is presented to investigate intergranular and transgranular crack propagations in polycrystalline brittle materials. Grains and grain boundaries are incorporated in the crack initiation and propagation model based on a phase-field model for grain growth, in which the elastic anisotropy varies based on the grain orientation angle, and the grain boundary energy is related to the misorientation angle of the adjacent grains. Correction parameters are utilized in the total free energy functional and mechanical equilibrium equations to consider the effect of material strength on crack nucleation and propagation independent of the regularization parameter. This allows controlling the strength and crack surface energy along the grain boundaries as a function of the misorientation angle in order to mediate intergranular and/or transgranular crack propagation. To demonstrate the capability of the proposed model, intergranular and transgranular crack propagation in ZrB2 bicrystal systems under tensile loading are studied in detail. The effects of grain boundary misorientation angle, grain boundary inclination with respect to initial crack direction, and grain boundary strength (and/or crack surface energy) on the crack propagation path are investigated. Intergranular crack propagation can be promoted by specific combinations of grain boundary strength and crack surface energy, which can contribute to the fracture toughness of polycrystalline materials.

Damping characteristic study of ship bionic anti-rolling device

ABSTRACT According to the shape and structure characteristics of ray in the ocean, a passive damping anti-rolling device was designed for fishing boats, platforms on the ocean, pontoon, heavy lift vessel and small ships that were used in maritime navigation or offshore operations. The dynamics model of the systems were established using three-dimensional (3D) computational fluid dynamics (CFD) software, the damping characteristics of the anti-rolling device was analyzed. The mathematical equation of mechanical equilibrium that satisfied the damping characteristics were deduced for the first time, and it is verified through numerical simulation analysis. The engineering feasibility and satisfactory anti-rolling effect of the device were demonstrated, and it can be seen from the experimental results that the average anti-rolling effect can reach more than 45% and more than 70% at zero speed.

Buckling behavior and axial load transfer assessment of coiled tubing with initial curvature in constant-curvature wellbores

This paper presents a mechanical equilibrium equation for coiled tubing (CT) with an initial curvature for use in a constant-curvature wellbore, based on the beam–column method. The critical sinusoidal and helical buckling loads, the rules for the contact between the CT and wellbore, and the transmission of the axial force along the length are determined. The proposed model shows that the initial curvature of the CT has a greater influence on the critical sinusoidal buckling than on the helical buckling. In addition, both critical buckling loads increase with the initial angular radian value, and the buckling loads in constant-curvature wellbores are greater than those in vertical wells. The critical buckling loads decrease exponentially with an increase in the borehole curvature radius, which changes slightly when the curvature radius of the well increases to a certain extent. When the CT is undergoing sinusoidal buckling, the contact force between the CT and well is sensitive to the initial angular radian value, axial force, and curvature radius of the well. However, when the CT experiences helical buckling, the contact force is primarily affected by the axial force. When the CT has a small amount of residual buckling, its buckled shape is similar to that of a straight pipe. Thus, the distribution and magnitude of the contact force are similar. Meanwhile, the transmission of an axial force along the length is affected more significantly by the friction coefficient and outer diameter of the CT than by the initial angular radian value and curvature radius of the well.

Materials used in a naval collision calculation with finite element software

Collisions and groundings have a significant contribution to structural damages occurring to ships. For collision calculation are used non-linear relations between strains and deformations, considering a significant difference between the initial shape and final deformed shape of the structure, and the mechanical equilibrium equations are written on the deformed shape of the structure. Using a mechanical test to a material sample will result the connection between the engineering strain and engineering deformation. During numerical simulation, we are using the real mechanical characteristic curve, which represents the reliance between real strains and real deformations, those being calculated from the distorted material sample. For calculation, we are using Johnson – Cook plasticity model, frequently used by other authors as well during collision calculation. There is included new information, also tabular calculations using specific formulas, resulted during carried out studies, to establish the model parameters considering the measured data or using information from another material model (exponential law). Considering the materials used within ANSYS software and calculus processor LS-DYNA, are presented types of materials useful for collision calculation, describing the elasticity, plasticity and yielding. Following, it is used a procedure for dynamic explicit calculus because it is suitable for solving of nonlinear systems. During simulation with finite element of the collision, there are no strains and deformations concentrations and it is necessary to generate an artificial curve, which leads to same results, as in reality and as well finite element analysis.

A finite element based simulation framework for robotic grasp analysis

The commonly used grasp simulators such as GraspIt! and OpenRAVE use wrench space formulations and grasp quality metrics such as ϵ and v to identify stable grasps in dynamic conditions. However, wrench space formulations are derived based on static mechanical equilibrium equations, and the physical attributes of the object such as stiffness and mass are also not considered for grasp analysis. Above all, these grasp analysis frameworks cannot be experimentally validated, thereby resulting in grasps that are not reliable. In this paper, an experimentally validatable Finite Element (FE) based grasp analysis framework is proposed for evaluation of robotic grasps in dynamic conditions. By applying standard solutions of Hertzian contact theory to a few robotic grasps, Finite Element Method (FEM) is validated. A real-world grasp situation is then simulated using FEM, and the FE model is validated based on the contact area measured in real-time using a pressure sensor. By applying dynamic perturbations to the validated FE model, the stability of the grasp is evaluated, and the most stable grasp is identified using the contact area based metric, π. It is observed that FE simulations agree with the analytical solutions and experimental results, with a relative error of not more than 7%.

Investigation of Mechanical Numerical Simulation and Expansion Experiment of Expandable Liner Hanger in Oil and Gas Completion

The expansion experiment of the expansion liner hanger is a one-time failure process, so in order to save cost, the finite element technology needs to be used to simulate the expansion experiment. Obtaining the mechanical parameters of the expansion liner hanger can effectively optimize the size of the expansion liner hanger structure and guide the expansion completion. Firstly, main structure and principle of expandable liner hanger were introduced. Secondly, mechanical equilibrium equations of the expandable process were established to obtain pressure of the expandable fluid, and pressure of the expandable fluid is obtained. Thirdly, finite element (FE) simulation mechanical model of the expansion of the Ø244.5 mm × Ø177.8 mm expandable liner hanger was established to analyze the hang mechanism and the change rule of mechanical parameters during the expansion. The FE results have shown that radial displacement and residual stress of the inner wall of hanger varied in 5 cycles, and the expansion ratio of the expandable tube was 7.4% during the expansion. The expansion force did not change stably but gradually increased in stages. And the hydraulic pressure required for the expandable cone to continuously move down was 18 MPa. According to the contact stress generated on five rubber cylinders and the contact stress generated on five metal collars, the total hang force has been calculated, which exceeds 1000 kN and meets the design requirements. Lastly, the expansion test results have shown that expansion pressure was 19 MPa, and the expansion rate was 7.1%. The mechanical analysis results of the expandable liner hanger were in good agreement with the experiment results in this study, which provide important mechanical parameters for well completion with expandable liner hanger.