Compatibility Equations Research Articles

Existence Analysis of Conformable Fractional Boundary Value Problems

This study examined the fractional boundary value issue at flexible derivatives, focusing on establishing the existence of solutions and uniqueness. We introduce conditions that advance our understanding of this complex mathematical domain by capitalizing on the innovative framework of contraction principles. Moreover, the versatility of the proposed method is emphasized by its effectiveness in dealing with a wide set for compatible fractional differential equations characterized by diverse boundary conditions.

Compatibility of strains and the three-fold differentiability of the displacement field

The problem of the necessary class of smoothness of solutions to quasi-static problems of deformable solid mechanics in terms of displacements was discussed. It is shown that in order for the equations of compatibility of deformations to become identities when displacements are substituted in them, the existence of some third mixed derivatives of displacements is required. A counterexample for a linearly elastic compressible isotropic elastic medium was given. In this counterexample, the displacement field, being a doubly differentiable solution to the boundary value problem for the system of Lame equations in the entire domain, is not a solution to the displacement problem at all points in this domain.

Novel quadrilateral strain-based finite element for static, free vibration, and buckling analysis of plates

This paper focuses on the static analysis, free vibration, and buckling validation of Reissner–Mindlin plates using a newly developed quadrilateral finite element, termed FNSBP, which is based on the strain approach. The proposed element is equipped with three essential external degrees of freedom at each of its four corner nodes. The displacement field is formulated using assumed strain functions that inherently satisfy the compatibility equations. This new element was developed to improve upon the previously introduced SBRP strain-based rectangular plate element. The effectiveness and robustness of the FNSBP element are evaluated through various numerical examples involving both thick and thin plates of different geometries. Its performance is benchmarked against displacement-based elements from the literature and analytical solutions, demonstrating superior accuracy and convergence in many cases. The results indicate that the FNSBP element provides significant improvements in computational efficiency and accuracy, especially for complex plate configurations under different loading and boundary conditions, validating its applicability in structural analysis.

Static and free vibration response of FGM plates using higher order shear deformation theory and strain-based finite element formulation

The primary objective and novelty of this work lie in the development of a new four-node quadrilateral finite element based on strain-based high-order shear deformation theory (HSDT). This is the first study to apply this innovative approach to analyze both the static and free vibration behaviors of functionally graded (FG) plates. Another key novelty is the reduction in the number of unknowns to five, unlike other high-order shear deformation theories that employ a larger number of unknowns. This reduction is achieved by applying the condition of zero transverse shear stress at the top and bottom free surfaces of the FG plate and by assuming that the transverse shear strains are sinusoidally distributed through the thickness. The material properties of FG plates are modeled to vary according to a simple power law based on the volume fractions of their constituents. The developed finite element possesses five degrees of freedom (DOFs) per node, resulting from the combination of two strain-based elements: the first is a membrane with two DOFs, and the second is a bending plate with three DOFs. The displacement fields of the proposed element are expressed using high-order terms based on the strain approach, which satisfy compatibility equations and rigid body modes. Furthermore, the concept of the neutral surface is introduced to eliminate membrane-bending coupling. The elementary stiffness and mass matrices are derived using both the total potential energy principle and Hamilton’s principle. The performance and convergence of the proposed element are validated through examples from the literature.

Use of a Refined Theory for Three-Dimensional Bending Analysis of Isotropic Rectangular Thick Plates

In this paper, a refined plate theory (Alternative II theory) is presented for the three-dimensional bending analysis of an Isotropic thick plate. The theory has similarity to the first order shear deformation theory but requires no shear correction factors. The kinematics equations were developed based on the Alternative II Refined plate theory. Thereafter, using a complete three-dimensional constitutive relation, the total potential energy was developed. A governing equation and two compatibility equations were obtained by the variation of the total potential energy with respect to displacement and rotations respectively. Solving the governing and compatibility equations, a polynomial displacement function was obtained. The stiffness coefficients were then obtained using the displacement function. Thereafter, the equations for the in-plane normal and shear stresses, transverse normal and shear stresses as well as the lateral displacement were developed using the stiffness coefficients and the displacement function. Numerical values of the lateral displacement parameters were determined for a rectangular plate of aspect ratio 2.0, 1.0 and 0.5 for span to thickness ratios of 20, 10 and 7.14286. Also, numerical values of the lateral displacement and stresses were determined for a square plate for span to thickness ratios of 4, 10, 100 and 1000. The results from this work were compared with the work of previous researchers using simple percentage difference. It was observed that refined plate theories overestimate the lateral displacement of a plate. Hence, three-dimensional analysis is recommended for thick plate analysis.

The analytical curvature distribution model of columns and mathematical solution for pushover analysis

AbstractThe acquisition of a flexural backbone curve for columns primarily relies on finite element analysis and experiments, of which the complexity and high cost are significant challenges. Hence, this study proposes an analytical curvature distribution model (CDM) along the full height of the column, which is the key point of formula derivation and theoretical solution of the column backbone curve. The proposed CDM is a piecewise function model composed of linear and quadratic functions, with the characteristics of continuity and differentiability at the intersection point. The deformation compatibility equations, equilibrium equations, and material constitutive equations based on the variables of CDM are constructed to form an equation system, the solution of which can be theoretically determined by an iterative algorithm. Furthermore, 155 test specimen columns of different materials, for example, reinforced concrete, high‐strength reinforced concrete, and shape memory alloy are used to validate the proposed CDM and mathematical solutions of pushover analysis through three crucial indicators, that is, goodness of fit of backbone curve, ultimate displacement, and peak force, indicating strong practicability, high accuracy, and wider applicability. This theoretical method can be applied to deal with the key issues of interest during pushover analysis, that is, predicting the flexural backbone curves with different materials, determining the curvature distribution throughout the entire process of pushover analysis, and characterizing the evolution process of the plastic region.

A new membrane finite element based on the strain approach

Membrane elements with in-plane rotation are applicable for modeling thin planar structures that are only stressed by volume and surface loads acting in their plane. In this paper a new finite element designed for static and free vibration analysis, featuring a four-node configuration is presented. Each node incorporates two engineering displacements and a fictive rotation, aligning with strain approach principles. The displacements field of this element is based on the assumed functions for the various strains satisfying the compatibility equations. This developed element passed all benchmark tests in the case of bending and shear problems. In static analysis, the displacements, strains, and stresses are calculated, while the free vibration analysis aims to determine natural frequencies, including both axial and fundamental frequencies. The proposed membrane element is validate through a series of selected examples, demonstrating its effectiveness and accuracy. The results consistently indicate that the new element exhibits high performance and exactitude in both types of analysis.

Finite element analysis of plates by an accurate strain-based element

Plates analysis has been a very important research area for a long time. In this perspective, in this paper, finite element analysis of thin plates in bending is carried out using a new four node strain-based plate element. The strain-based approach and Kirchhoff theory for thin plates have been used to create the strain field of this element. This approach consists in having a displacement field from a proposed strain field. This developed model contains essential degrees of freedom at each corner node which are three, one displacement and two rotations. The displacement field is based on the assumed functions for various deformations which satisfy the compatibility equations, and the displacement functions are obtained by integrating these deformation functions. The numerical validation of this developed element was held through several examples from the literature. The obtained results from these numerical tests prove the good accuracy and performance of this element.

Equilibrium Solution of Two – Dimensional Non-Homogeneous Equations in the Theory of Elastic Mixtures

Abstract: The problem of plane elasticity for a doubly connected body with inner and outer boundaries in a regular polygonal form with common centre and parallel sides has been studied. The sides of the polygon were exposed to external forces. The nature of the force term was determined by application of complex variable theory. Kolosov’s method of solution was applied to obtain the biharmonic equation of the forcing term. The forces on the particle were studied under 2-dimensions from which the compatibility and equilibrium equations were derived. The compatibility and equilibrium equations were used to derive the force – stress relations. The results shows that there is a significant relationship between the angle of the force term on the plane of the particle and the stress state of the particle, which is in conformity with existing experimental results.

Derivation of an effective plate theory for parallelogram origami from bar and hinge elasticity

Periodic origami patterns made with repeating unit cells of creases and panels bend and twist in complex ways. In principle, such soft modes of deformation admit a simplified asymptotic description in the limit of a large number of cells. Starting from a bar and hinge model for the elastic energy of a generic four parallelogram panel origami pattern, we derive a complete set of geometric compatibility conditions identifying the pattern’s soft modes in this limit. The compatibility equations form a system of partial differential equations constraining the actuation of the origami’s creases (a scalar angle field) and the relative rotations of its unit cells (a pair of skew tensor fields). We show that every solution of the compatibility equations is the limit of a sequence of soft modes — origami deformations with finite bending energy and negligible stretching. Using these sequences, we derive a plate-like theory for parallelogram origami patterns with an explicit coarse-grained quadratic energy depending on the gradient of the crease-actuation and the relative rotations of the cells. Finally, we illustrate our theory in the context of two well-known origami designs: the Miura and Eggbox patterns. Though these patterns are distinguished in their anticlastic and synclastic bending responses, they show a universal twisting response. General soft modes captured by our theory involve a rich nonlinear interplay between actuation, bending and twisting, determined by the underlying crease geometry.

Limit analysis of planar steel frames, in-element plastic-hinge for uniformly distributed loads

This work calculates the collapse load and collapse mechanism of 2D frames with slender structural members and uniformly distributed loads. The search for the collapse mechanism and the collapse load is carried out using step by step method: the load factor is increased and at each step the balance and compatibility equations must be satisfied that the value of the plastic moment is not exceeded in any section. It is verified that the results are different in the cases of point loads and uniform distributed loads, both from a qualitative and quantitative point of view.

Alternative II Theory Solution for a Thick Rectangular Anisotropic Plate Under in-Plane and Lateral Loads

This work investigated the application of the Alternative II refined plate theory in the analysis of an anisotropic plate subjected to in-plane and lateral loads. The kinematic equations developed from the Alternative II Refined plate theory were used together with a complete three-dimensional constitutive relation to obtain the total potential energy of an anisotropic plate under lateral and in-plane loads. General variation of the total potential energy was done, a governing equation and two compatibility equations were obtained. A polynomial displacement function was obtained by solving the governing and compatibility equations. This was used to obtain peculiar displacement functions by satisfying the boundary conditions of any plate. The stiffness coefficients were obtained using the displacement function. With the displacement functions and the stiffness coefficients, the equations for the in-plane normal and shear stresses as well as the transverse normal and shear stresses were determined for any applied lateral load when the applied in-plane load is a fraction of the buckling load. Also, the equations for the displacements of the plate were determined. Numerical values of the stresses and displacement parameters were determined for span to thickness ratios of 5, 10, 20 and 100 at angle of fiber orientations of 0 and aspect ratios of 1, 1.5 and 2.0 when the ratio of applied in-plane load to buckling load are 0, 0.25 and 0.5. Using simple percentage difference, the results from this work were compared with the works of previous researchers.

Investigating the modal behaviors of a deep beam with a transverse open crack

This paper investigates the modal behaviors of a deep beam with a transverse open crack. Unlike cracked shallow beams with aspect ratio (L/h) > 10, cracked deep beams with aspect ratio < 5 simultaneously exhibit near the crack both axial-bending coupling mode caused by mode I crack and shear (or sliding) mode caused by mode II crack, referred to as axial-bending-shear modal coupling. Therefore, for accurate modal behaviors prediction, appropriate compatibility equations accounting for both modes I and II crack should be employed for deriving a frequency equation. Because the derived frequency equation is nonlinear with respect to modal frequency, the modal frequencies are calculated through the Newton–Raphson method. The corresponding mode shapes are determined by substituting the calculated modal frequencies into the spectral solutions for either side of the crack. The validity of the proposed method was demonstrated through comparison to the finite element analysis and experimental results.

Two parallel expansions for improving supersonic axisymmetric nozzle performance

The aim of this work is to develop a numerical computation program allowing designing new contours of a supersonic axisymmetric nozzle having two expansions at the throat, named by DEN (Dual Expansion Nozzle). This new nozzle gives a uniform and parallel flow at the exit section, to improve considerably the performances compared to the conventional Minimum Length Nozzle (MLN), and the Best Performances Nozzle (BPN). The present nozzle has a two unknowns external and central body curved walls. Each of them is started by an initial expansion angle to give a uniform and horizontal flow at the exit section. Two others transition regions are calculated in parallel with the contours points to give the desired exit Mach number. The walls are determined point by point by the High Temperature Method of Characteristics (HT MOC) model. The resolution of the four compatibility and characteristics equations is done numerically by the finite difference predictor corrector algorithm. The validation of the results is controlled by the convergence of the calculated critical sections ratio to that given by the theory. The design depends on four parameters, where MLN and BPN become special cases of DEN. A comparison is made with MLN, since it is currently used in the aerospace propulsion and with BPN aiming to improve their performances. The comparison is made for the same critical mass flow rate. The results demonstrate a remarkable reduction up of 45 %, and 52 % in the mass of DEN when the exit Mach number ME = 3.00 and the stagnation temperature T0 = 2000 K. The application is made for air and for future aerospace missiles in order to improve their trajectory parameters. The chosen example demonstrates an improvement of 13 % and 16 % on the missile range compared, respectively to MLN, and BPN.

Construction of a measurement system and analytical solution for docking pose of large aircraft components based on draw-wire displacement sensors

Abstract Relative pose detection is a key technology in large component automatic docking systems. In this paper, a measurement system has been developed to detect the relative pose of large aircraft components with draw-wire displacement sensors, while an analytical calculation method has been proposed to determine the docking pose of large aircraft components. This method establishes a measurement model for the docking pose of large components and separates the position and orientation solutions based on the distances between seven sets of measurement points measured by the draw-wire displacement sensors. First, the principle of three-dimensional rendezvous positioning is refined to ascertain the relative position of the components. Then, by analyzing the properties of the rotation matrix and the coupling relationship between the position and orientation variables of the movable component, 15 compatible equations containing only two variables with a maximum degree of four are obtained. Based on this set of equations, a unique solution for the rotation matrix was obtained through the variable substitution method, gradually eliminating higher-order terms. Finally, the correctness of the method is verified through numerical examples. Unlike typical parallel mechanism models, this model can directly obtain 15 compatible equations without the need for methods such as Gröbner bases. Moreover, compared to the measurement method using six draw-wire displacement sensors, a unique solution can be directly determined, solving the problem of multiple solution selection. This measurement method only requires measuring the distance between measurement points to solve the relative pose, with low equipment costs and less susceptibility to external environmental factors.

Form-finding of thermal-adaptive pin-bar assemblies based on eigenvalue modification

Lattice structures with tunable expansion properties have been investigated in multidisciplinary fields to control the temperature effects of structures or materials. The expected thermal adaptivity can be achieved by optimizing the structural geometry. A novel method for the form-finding of thermal-adaptive pin-bar assemblies is developed in this paper by considering the control of structural temperature effects as the minimization of the potential energy of the system. Based on the stationarity condition of the potential energy with respect to the nodal coordinates, the compatibility relationship between the thermal elongations of members and the target nodal displacements is proven to be the sufficient and necessary condition for structural thermal adaptivity. The solvability of the compatibility equation is determined by the rank equality between the compatibility matrix and its augmented form, which can be measured by the number of nonzero eigenvalues of its Gramian matrix. The analytical relationship between the eigenvalues of the Gramian matrix and the nodal coordinates is established using the matrix perturbation theory. A numerical strategy based on Newton’s method is proposed in which the eigenvalues are gradually modified by adjusting the nodal coordinates until the rank equality is satisfied. To address the existence of multiple solutions with structural thermal adaptivity, structural symmetry and periodicity constraints are introduced to narrow the solution space. The thermal-adaptive configurations of three illustrative pin-bar assemblies are analyzed using the proposed form-finding method, and the expected thermal deformations are verified for the obtained configurations using the finite element software ABAQUS. Comparing the results obtained by the proposed method with those obtained by nonlinear programming and the genetic algorithm validates the advantages of the proposed method in terms of computational time, optimality of the obtained configuration and applicability to complex structural geometries.

An approach to describe the twinning and detwinning processes of the martensitic structure in ferromagnetic alloy with shape memory in magnetic and force fields

In the article, based on the theory of micromagnetism, a special case of the moment theory of elasticity for an anisotropic material and the Hadamard compatibility condition for deformations, a microstructural model of the behavior of the Heusler alloy with shape memory in magnetic and force fields is constructed. The dynamics of the magnetic process is described by the Landau-Lifshitz-Gilbert equation. Using the Galerkin procedure, variational equations corresponding to the differential relations of the magnetic problem are written out and the finite element method is used for numerical realization of these equations. Considering the problems of the moment theory of elasticity we come to a special case of this theory in which, under the action of force and magnetic fields, the tensor of moment stresses initiated only by the magnetic field is zero. Using this result, the general expression for the state equation is constructed that describes, within the framework of small deformations, the behavior of an anisotropic material under force and magnetic influences. The stress tensor in this equation consists of a symmetric part and a skew-symmetric part, depending only on the magnetic field. For the problem of deformable solid mechanics with such a state equation, the Lagrange variational equation was constructed, which made it possible to reduce the requirements for the smoothness of the desired solution by an order of magnitude. The general expression for the state equation has been concretized for an anisotropic material having one axis of symmetry of the fourth order and two axes of symmetry of the second order. The solution of the Hadamard compatibility equation for deformations made it possible to determine the sliding surfaces and sliding directions in a tetragonal crystal cell of a ferromagnetic material in the martensitic state, along which a shift occurs, leading to the appearance and dissipation of a twinned structure. The microstructural model, constructed using all these positions, is used to describe experimental results on the combined action of magnetic and force fields on the Heusler alloy.

Effect of steel fibers and the fiber-concrete adhesion stress on the nonlinear shear behavior of beams

In this article, a theoretical model in nonlinear elasticity is presented to analyze the influence of the percentage of steel fibers and the effect of the fiber-concrete bond stress on the shear force at the rupture of beams subjected to the combined effect of bending moment, normal force, and shear force. For a given beam section, it is defined by a succession of layers of concrete and longitudinal steel elements. Each layer is defined by its height hi, width bi, and position relative to one end of the section YGi. Each longitudinal steel element is also defined by its cross-sectional area and position relative to one end of the section. The steel fibers are defined by volume percentages of 0.5%, 1%, 1.5%, and 2%, taking into account the mechanical nonlinearity of the materials. This model is based on the multilayer analysis of sections and an iterative solution procedure for each layer and each section, considering a given longitudinal deformation state and shear stress. The global equilibrium of the sections is analyzed under the assumption of flat longitudinal deformations but with, in principle, an interdependence of longitudinal normal stresses and shear stresses. In this study, using the principle of virtual work, equilibrium equations for deformations and stresses, as well as partial compatibility equations between concrete deformations and mean deformations, are derived. Comparative examples between ordinary reinforced concrete beams with variable shapes and reinforcement details, and those reinforced with steel fibers, are presented to demonstrate the accuracy of the proposed model for simulating the nonlinear shear response of beams.

Motion state factor driven for doubly-curved shallow shell deformation reconstruction

The conventional shape sensing method, based on C0 continuity, is recognized for its computational efficiency. However, it is prone to shear locking due to low-order polynomial interpolation in shell analysis, introducing two primary challenges. Firstly, it lacks consideration for C1 continuity. Secondly, existing high-order polynomial interpolation methods significantly increase the number of degrees of freedom (DOF), reducing computational efficiency. Therefore, a brand-new real-time structural health monitoring (SHM) method driven by motion state factor instead of nodal factors is proposed. This method accommodates all deformation modes of shallow shells with C1 continuity and improves the shape function's order without additional DOF. It begins with deriving motion states from compatibility equations, leading to a polynomial shape function expression. Subsequently, conversion equations for real constants and DOF are developed, establishing a least-squares model for theoretical strains and measured values. Following that, the higher-order element formula, optimized for minimal DOF, is then inferred from varied displacement data. Finally, simulations and experiments demonstrate this method's effectiveness, accuracy, and reliability in analyzing shallow shell elements.

A stiffness model for EHL contact on smooth/rough surfaces and its application in mesh stiffness calculation of the planetary gear set

A method for calculating time-varying mesh stiffness (TVMS) is proposed by developing a stiffness model for elastohydrodynamic lubrication (EHL) contact and integrating it into three-dimensional loaded tooth contact analysis (3D-LTCA). To enhance the accuracy and efficiency of the calculation, three projects have been implemented: Firstly, the contact deformation is modified, and the average slope is employed for calculating contact stiffness. Secondly, the lubricant film and EHL contact deformation are introduced into the deformation compatibility equations. Finally, the polynomial chaos expansion (PCE) surrogate model is employed to enhance computational efficiency. The proposed method is used to calculate TVMS of the planetary gear set, and the influence of speed, torque, and rough surface are analyzed. The results show that the previous method underestimating the TVMS of planet-ring gear pair in double-teeth meshing area under fluid lubrication, while the proposed method effectively addresses the issue. The different effects of the torque and speed on TVMS under mixed and fluid lubrication also have been revealed, emphasizing the significance of surface topography in gear meshing characteristics.