Elastic Equilibrium Research Articles

Analysis of Linear Elastic Masonry-Like Solids Subjected to Settlements

A linear elastic no-tension material model is implemented in this contribution to cope with the analysis of masonry-like solids in case of either elastic or inelastic settlements. Instead of implementing an incremental non-linear approach, an energy-based method is adopted to address the elastic no-tension equilibrium. Under a prescribed set of compatible loads, and possible enforced displacements, a solution is found by distributing an equivalent orthotropic material having negligible stiffness in tension, such that the overall strain energy is minimized and the stress tensor is negative semi-definite all over the domain. A preliminary implementation of the proposed method is given by adopting a heuristic approach to turn the multi-constrained minimization problem into an unconstrained one. Numerical simulations focus on a wall with an opening subjected to either inelastic settlement or standing on elastic soil.

Решение неосесимметричной задачи эластостатики для трансверсально-изотропного тела вращения

The paper investigates the elastic equilibrium of transversely isotropic bodies of revolution under the action of stationary surface forces distributed according to the cyclic law. The proposed method for constructing the stress-strain state is a development of the method of boundary states. The method is based on the concept of spaces of internal and boundary states conjugated by an isomorphism. Bases of state spaces are formed and orthonormalized. The desired state is expanded in a series by the elements of the orthonormal basis, and the Fourier coefficients, which are quadratures, of this linear combination are calculated. The basis of the internal state space relies on the general solution of the problem of plane deformation of a transversely isotropic body and the formulas for the transition to a spatial state, the components of which depend on three coordinates. Scalar products in state spaces represent the internal energy of elastic deformation and the work of surface forces on the displacements of the boundary points. The study introduces the solution of the main mixed problem for a circular cylinder made of transversely isotropic siltstone with the axis of anisotropy coinciding with the geometric axis of symmetry. The solution is analytical and the characteristics of the stress-strain state have a polynomial form. The paper presents explicit and indirect signs of convergence of problem solutions and graphically visualizes the results

An iterative approach combined with multi-dimensional fitting of limited measured stress points to reconstruct residual stress field generated by laser shock peening

In this paper, an iterative approach combined with least squares fitting of limited measured points is introduced to reconstruct residual stress distribution after laser shock peening (LSP) treatment. The iterative approach is started by introducing prescribed stress components in the data measurement region (the mapped zone) in the finite element (FE) model, then a static equilibrium analysis is conducted, which will cause the stress distribution in the mapped zone and unmapped zone both redistribute if they don't converge to the target stresses. By repeatedly introducing the input stress in the mapped zone to converge to the prescribed stress components, the complete residual stress distribution in the whole region can be obtained. Firstly, to validate the viability of this method, based on the full stress components information of ‘measured’ sample points extracted from FE models, residual stress reconstructions of single-pass welding of a thick plate and a single shot of LSP have been carried out, respectively. After that, to examine the robustness of the method, the iterative approach combined with multi-dimensional least squares fitting is tested using a limited set of ‘measured’ sample points extracted from the FE model of single shot LSP treatment. It is found that besides the commonly studied depth direction, the in-plane stress components are also varied with its in-plane locations, which are due to “edge effects”. The residual stress fields for the whole region reconstructed by the proposed approach agree well with the actual stress fields even if some of the non-critical stress information are missing. Finally, the results are compared with those reconstructed by the eigenstrain method. It is shown that the proposed iteration approach can reconstruct the residual stress distribution in the whole region at a satisfactory level by solving a direct linear elastic static equilibrium problem, while the eigenstrain method needs to solve an inverse problem to obtain the appropriate eigenstrain distribution.

Intermittent swelling and shrinkage of a highly expansive soil treated with polyacrylamide

This laboratory study examines the potential use of an anionic polyacrylamide (PAM)-based material as an environmentally sustainable additive for the stabilization of an expansive soil from South Australia. The experimental program consisted of consistency limits, sediment volume, compaction and oedometer cyclic swell–shrink tests, performed using distilled water and four different PAM-to-water solutions of P D = 0.1 g/L, 0.2 g/L, 0.4 g/L and 0.6 g/L as the mixing liquids. Overall, the relative swelling and shrinkage strains were found to decrease with increasing number of applied swell–shrink cycles, with an ‘elastic equilibrium’ condition achieved on the conclusion of four cycles. The propensity for swelling/shrinkage potential reduction (for any given cycle) was found to be in favor of increasing the PAM dosage up to P D = 0.2 g/L, beyond which the excess PAM molecules self-associate as aggregates, thereby functioning as a lubricant instead of a flocculant; this critical dosage was termed ‘maximum flocculation dosage’ (MFD). The MFD assertion was discussed and validated using the consistency limits and sediment volume properties, both exhibiting only marginal variations beyond the identified MFD of P D = 0.2 g/L. The accumulated axial strain progressively transitioned from ‘expansive’ for the unamended soil to an ideal ‘neutral’ state at the MFD, while higher dosages demonstrated undesirable ‘contractive’ states.

Origin of the isothermal B2 → B19′ transformation in NiTi-based shape memory alloys

The aim of this paper was to study the isothermal forward martensitic transformation in binary, ternary, and quaternary NiTi-based shape memory alloys to find the origin of this phenomenon. The results obtained showed that the isothermal B2 → B19′ transformation was observed only in the NiTi-based alloys with substitutional defects. The critical concentration of copper was found that completely suppressed the martensitic transformation both on cooling or isothermal holding in Ti-Hf-Ni-Cu alloys. The thermodynamics of the martensitic transformation that occurred in the presence or absence of the substitutional defects was discussed, and the reason for the influence of the substitutional defects on the martensitic transformation was clarified. It was verified that the formation of the martensite phase on isothermal holding was caused by the local fluctuation of the substitutional defect concentration which led to fulfilment of the thermodynamic condition for the martensitic transformation. It was shown that the maximum volume fraction of the isothermal martensite was limited by the maximum volume of martensite that corresponded to the thermal elastic equilibrium and appeared on cooling of the alloy without substitutional defects. The results allowed to conclude that the martensitic transformation remained to be thermal elastic in the NiTi-based alloys with a non-stoichiometric composition however, the martensite nucleation was controlled by the thermally activated migration of substitutional defects.

Magnetic field sustained by the elastic force in neutron star crusts

ABSTRACT We investigate the magneto–elastic equilibrium of a neutron star crust and magnetic energy stored by the elastic force. The solenoidal motion driven by the Lorentz force can be controlled by the magnetic elastic force, so that conditions for the magnetic field strength and geometry are less restrictive. For equilibrium models, the minor solenoidal part of the magnetic force is balanced by a weak elastic force because the irrotational part is balanced by the dominant gravity and pressure forces. Therefore, a strong magnetic field may be confined in the interior, regardless of poloidal or toroidal components. We numerically calculated axially symmetric models with the maximum shear–strain and found that a magnetic energy >1046 erg can be stored in the crust, even for a normal surface dipole-field-strength (<1013 G). The magnetic energy much exceeds the elastic energy (1044−1045 erg). The shear–stress spatial distribution revealed that the elastic structure is likely to break down near the surface. In particular, the critical position is highly localized at a depth less than 100 m from the surface.

Elasticity Solutions for Cylindrical Bending of Functionally Graded Piezoelectric Plates

Functionally graded piezoelectric materials (FGPMs), combining the properties of functionally graded materials and piezoelectric materials, provides a new idea for multi-functional and intelligent lightweight components, and it has broad application prospects in electronic devices. Based on the elasticity and electric equilibrium equations, the Mian and Spencer functionally graded plate theory is extended from elastic materials to piezoelectric materials to study the cylindrical bending of FGPM plates, where the material elastic constants, piezoelectric and dielectric parameters are assumed to vary continuously and arbitrarily along thickness direction. Accordingly, the elasticity solutions for cylindrical bending of FGPMs plates under the uniform transverse loading are obtained. Numerical examples are given to demonstrate the influence of piezoelectric effects on the static responses of the presented FGPMs plates.

Phase-field simulation of void evolution in UO<sub>2</sub> under applied stress

Owing to the migration and aggregation of point defects produced by cascade collision, a large number of cavities form easily during irradiation of the uranium dioxide (UO<sub>2</sub>) that is an important nuclear fuel. In addition, cavities are also inevitably introduced into the ceramic fuel during sintering. Moreover, the creep strain and thermal strain, caused by the extreme environment of high temperature and strong irradiation, significantly increase the applied stress of nuclear fuel. Therefore, it is crucial to investigate the microstructure evolution of the cavities in UO<sub>2</sub> fuel under applied stress. In this work, a phase-field model of void evolution in UO<sub>2</sub> under applied stress is established. Firstly, the elastic equilibrium equation is solved by the perturbation-iterative method, and the stress distribution around a single void under applied stress is calculated. The results show that the stress concentration is observed at the edge of the void, and the simulated stress distribution is consistent with the theoretically analytical results. Then, the evolution processes of a single void under different applied stresses are simulated by the phase-field model. The results show that the growth rate of void increases with the augment of applied stress. Finally, the effect of applied stress on grain growth and void evolution in polycrystalline are also studied. The results show that the applied stress will accelerate the void growth. With the increase of the applied stress, the effect of the applied stress on accelerating the void evolution increases.

Positions of Elastic Equilibrium of a Pipeline with Vibrating Supports

Positions of Elastic Equilibrium of a Pipeline with Vibrating Supports

Implementation of Tawhidi String Relations (TSR) in Demand, Supply, Elasticity, and Market Equilibrium

This study intends to explore the implementation of Tawhidi String Relations (TSR) in demand, supply, elasticity, and market equilibrium and the impact of implementing the TSR approach in these functions. Tawhidi String Relations (TSR), a methodology derived from Tawhid that uses the Al-Quran and Sunnah as an ontology in each solution variable with the ultimate goal of achieving social wellbeing. This TSR method has the advantage of a process of interaction, integration, and evolution which is understood as the evolution of the formation of human knowledge that is subject to Divine Laws through a shuratic process and ijma’, thus realizing a universal social wellbeing function. This study uses qualitative research methods by exploring through literature studies. Basically, demand, supply, elasticity, and market equilibrium in conventional economics have several differences with those found in Islamic economics where what distinguishes them is demand, supply, elasticity and market equilibrium in Islamic economics has limitations and principles that must be met by individuals to behave accordance with Islamic law which is based on the Al-Quran, Sunnah, Ijma', and Qiyas as the agreed source of Islamic law.

A diffuse-interface compact difference method for compressible multimaterial elastic–plastic flows

In this paper, a novel high-order numerical method is proposed for the simulation of compressible multimaterial solid–fluid problems involving elastic–plastic solid behaviors. The solid, liquid, and gas behaviors are described by an Eulerian interface-capturing model derived from hyperelasticity theory. The deformations in each component of the solids are tracked using a single deviatoric strain tensor. The numerical algorithm employs a tenth-order compact finite difference scheme and a fourth-order Runge–Kutta time-stepping scheme. To prevent numerical oscillations near the material interface, a mechanical equilibrium assumption is introduced for the elastic–plastic problems, and a localized artificial diffusivity method satisfying this assumption is designed and applied in the numerical algorithm. Numerical tests in one and two dimensions are shown to demonstrate the accuracy and feasibility of the proposed approach. In particular, the accuracy and numerical resolution of the method are verified using problems with analytical solutions. Numerical tests of interface advection problems demonstrate that our method preserves the velocity, pressure, and elastic stress equilibria at the material interface and prevents overshoots in the rest of the domain. The two-dimensional Richtmyer–Meshkov instabilities between two elastic or two elastic–plastic solids are simulated to verify the suitability of the numerical method for solving problems involving large deformations. Impact of a 2D Taylor bar to a rigid wall is also simulated to test the robustness of the numerical method.

Contact problems for an inhomogeneous elastic wedge with variable Poisson’s ratio

Plane contact problems of the elasticity theory are investigated for a wedge when Poisson’s ratio is an arbitrary smooth function with respect to the angular coordinate while shear modulus is constant. For this case Young’s modulus is also variable with respect to the angular coordinate. A finite contact domain is given on one wedge face, it does not include the wedge apex, while the other wedge face is rigidly fixed (problem A) or stress-free (problem B). To reduce the problems to integral equations with respect to the contact pressure, we use the general Freiberger type representation for the solution of elastic equilibrium equations written in polar coordinates with variable Poisson’s ratio. Exact solutions of auxiliary problems are constructed with the help of Mellin integral transforms. The regular asymptotic method employed is effective for contact domains relatively distant from the wedge apex. It is shown that logarithmic terms appear in the asymptotic solutions for the inhomogeneous material which are missing in the well-known asymptotics for the homogeneous one. In contact problem C which is corresponding to problem A, the friction and roughness are taken into account in the contact region. The roughness of the wedge surface is simulated by a Winkler type coating. The collocation method is used for solving integral equations of the second kind. Unlike problem A, in problem C the contact pressure does not have square root singularities at end-points where it takes finite values. Calculations are made for the cases when Poisson’s ratio and Young’s modulus increase or decrease from the surface of the wedge.

Investigation of the formation of a machined surface profile during end milling with a tool with a long overhang

Purpose. Investigate the mechanism of forming the profile of the machined surface when milling with a tool with a long overhang.
 Methodology. Experimental method was used with a special stand, which makes it possible to simulate an elastic system of a part with different dynamic properties, record oscillograms of oscillations of a part and a tool during the cutting process. Milling was performed with a milling cutter with cutting modes in which self-oscillations occur. To determine the patterns of movement of the part and the tool, the method of aligning the fragments of oscillograms was used.
 Results. The scheme of the formation of the profile of the machined surface under the conditions of milling with self-oscillations by a tool with a long overhang has been investigated. It was found that the free vibrations of the tool after the tooth leaves the cutting zone have a higher intensity than that of the part. This affects the position of the infeed point on the cutting surface. To determine the period and range of self-oscillations of the cutter and the part, the section of the oscillogram with self-oscillations was straightened using the Savitsky-Galei filter. Comparison with the cutting surface of the sections of the oscillogram of the part and the tool during milling showed an identical vibration mode of the cutter. Based on the results of experimental studies, the deviation from the line of elastic equilibrium of the first wave of self-oscillation of the cutter was determined and a graph of its change was plotted. The studies performed have confirmed the previously established effect of deviation from the elastic equilibrium position of the first self-oscillation wave during counter milling on the shaping of the treated surface.
 Originality. It has been established that the profile of the machined surface when milling with long overhang end mills is directly related to the deviation of the tool from the elastic equilibrium position during self-oscillation.
 Practical value. The obtained results show the effect of a tool with a long overhang on the formation of a machined surface during milling with self-oscillation.

Elastic equilibrium of a hollow cylinder of finite length subjected to axisymmetric force loading

Elastic equilibrium of a hollow cylinder of finite length subjected to axisymmetric force loading

Elastic equilibrium of anisotropic bimaterial bodies with thin elastic anisotropic inclusions under longitudinal shear

Elastic equilibrium of anisotropic bimaterial bodies with thin elastic anisotropic inclusions under longitudinal shear

Calculation of the tense-deformed state of two ortogonal attended plates at the regional terms of symmetry through the matrices of Grina type

Purpose. Calculate the tense-deformed state of two ortogonal attended plates through special the built matrices of Grina type.
 Research methods. Bases of theory of laminas, apparatus of trigonometric rows of Fourier, methods: border-component tasks, variation of arbitrary permanent, matrices of Grina type.
 Results. A task of elastic elastic equilibrium of plate pairs connected at right angle was considered. On parallel edges of component body to connection rib special edge conditions – conditions of symmetry – were chosen. From the physical point of view the probed body can be the model of lateral walls of parallelepiped. It is provided the special terms of symmetry on both edges of component body, which are parallel to the rib of connection of plates. The method of calculation allows to calculate the tense-deformed state of spatial construction consisting of two plates in edge conditions of arbitrary surface loading. The results of calculation (as lines of level) of basic characteristics of static deformation of considered rectangular connection of two plates are given.
 Scientific novelty. The method of calculation of pair of plates, united at right angles was improved at the regional terms of symmetry, with subsequent graphic illustration of achived results.
 Practical value. The task probed in-process designs the phenomena which take place, at deformation of elements of vulcanization equipment. Achived results allow to find pequliarities of elements work of construction of complext structure and to promote its efficiency by optimization of component parameters parts.

Elastic equilibrium of spherical particle composites with transversely isotropic interphase and incoherent material interface

The rigorous analytical solution to the unit cell model of spherical particle composite with transversely isotropic interphase and incoherent material interface has been obtained by the multipole expansion method. To this end, a general expression of the displacement vector in spherical transversely isotropic solid in terms of the vector spherical harmonics has been derived. By accurate fulfillment of the contact conditions, the model boundary value problem is reduced to the linear algebraic system. The anisotropic interface model has been developed as the first order approximation of the transversely isotropic interphase. Relevance of this model to the theory of material interfaces and its applicability in the nanomechanics context is discussed. The developed model expands the theory of curved deformable interfaces of Gurtin et al. [Phil. Mag. A 78 (1998) 1093] on the incoherent interfaces between the dissimilar elastic materials and provides a certain insight into the interface elastic moduli. Taking incoherency of the nano level interfaces into account enables more realistic models and thus increases the predictive power the homogenization theory of nanostructured solids. The obtained accurate numerical data discover a significant effect of the interphase/interface anisotropy on the local stress and effective stiffness of spherical particle composite.

The Solution of Boundary Value Problems of Various Types with Consideration of Volume Forces for Anisotropic Bodies of Revolution

In this work, we studied the axisymmetric elastic equilibrium of transversely isotropic bodies of revolution, which are simultaneously under the influence of surface and volume forces. The construction of the stress-strain state is carried out by means of the boundary state method. The method is based on the concepts of internal and boundary states conjugated by an isomorphism. The bases of state spaces are formed, orthonormalized, and the desired state is expanded in a series of elements of the orthonormal basis. The Fourier coefficients, which are quadratures, are calculated. In this work, we propose a method for forming bases of spaces of internal and boundary states, assigning a scalar product and forming a system of equations that allows one to determine the elastic state of anisotropic bodies. The peculiarity of the solution is that the obtained stresses simultaneously satisfy the conditions both on the boundary of the body and inside the region (volume forces), and they are not a simple superposition of elastic fields. Methods are presented for solving the first and second main problems of mechanics, the contact problem without friction and the main mixed problem of the elasticity theory for transversely isotropic finite solids of revolution that are simultaneously under the influence of volume forces. The given forces are distributed axisymmetrically with respect to the geometric axis of rotation. The solution of the first main problem for a non-canonical body of revolution is given, an analysis of accuracy is carried out and a graphic illustration of the result is given

SANM

Solving nonlinear systems is an important problem. Numerical continuation methods efficiently solve certain nonlinear systems. The Asymptotic Numerical Method (ANM) is a powerful continuation method that usually converges faster than Newtonian methods. ANM explores the landscape of the function by following a parameterized solution curve approximated with a high-order power series. Although ANM has successfully solved a few graphics and engineering problems, prior to our work, applying ANM to new problems required significant effort because the standard ANM assumes quadratic functions, while manually deriving the power series expansion for nonquadratic systems is a tedious and challenging task. This paper presents a novel solver, SANM, that applies ANM to solve symbolically represented nonlinear systems. SANM solves such systems in a fully automated manner. SANM also extends ANM to support many nonquadratic operators, including intricate ones such as singular value decomposition. Furthermore, SANM generalizes ANM to support the implicit homotopy form. Moreover, SANM achieves high computing performance via optimized system design and implementation. We deploy SANM to solve forward and inverse elastic force equilibrium problems and controlled mesh deformation problems with a few constitutive models. Our results show that SANM converges faster than Newtonian solvers, requires little programming effort for new problems, and delivers comparable or better performance than a hand-coded, specialized ANM solver. While we demonstrate on mesh deformation problems, SANM is generic and potentially applicable to many tasks.

Convergence of fixed-point algorithms for elastic demand dynamic user equilibrium

In this paper we present sufficient conditions for convergence of projection and fixed-point algorithms used to compute dynamic user equilibrium with elastic travel demand (E-DUE). The assumption of strongly monotone increasing path delay operators is not needed. In its place, we assume path delay operators are merely weakly monotone increasing, a property assured by Lipschitz continuity, while inverse demand functions are strongly monotone decreasing. Lipschitz continuity of path delay is a very mild regularity condition. As such, nonmonotone delay operators may be weakly monotone increasing and satisfy our convergence criteria, provided inverse demand functions are strongly monotone decreasing. We illustrate convergence for nonmonotone path delays via a numerical example.