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A linear mathematical model of the theory of elasticity of a three-dimensional orthotropic body is considered. The technique of integrating elastic equilibrium equations and analytical expression of elastic displacements through two functions is applied. One function satisfies a homogeneous equation in partial derivatives of the second order, and the other one of the fourth order. Modified harmonic functions, which satisfy homogeneous equations in the second-order partial derivatives are introduced to solve the fourth-order equation and describe orthotropic materials. A method of integrating elastic equilibrium equations without redundant functions and analytical expression of displacements through three modified harmonic functions was developed for the first time. The criteria for the selection of four new classes of orthotropic materials, described by three functions, are studied. Two classes contain six independent orthotropic coefficients, and the other two contain five coefficients. Other dependent orthotropic coefficients are determined from the obtained equations. The expression of deformations and stresses in an orthotropic body was recorded. It was established that there is no single representation of the general solution of the equilibrium equations for an orthotropic body.

The pile group-pile cap structure is a key foundation form for deep-water bridges. However, current effective methods for calculating the earthquake-induced hydrodynamic forces on pile caps with arbitrary cross-sections remain insufficient. In this study, the hydrodynamic force is considered as the added mass, and the dynamic equilibrium equations of the isolated pile cap structure (IC model) and the pile group-pile cap structure (PC model) under earthquakes are established, respectively, based on the structural dynamics theory. Correspondingly, the relationships between the hydrodynamic added masses and the fundamental frequencies in the IC model and the PC model are derived, respectively. The fundamental frequencies of the IC model and the PC model are obtained by numerical models built with the ABAQUS (2019) finite element software, and then the added masses on the IC and PC models are calculated accurately. The calculation method proposed in this study avoids the complex fluid–structure interaction problem, which can be applied for the seismic design of deep-water bridge substructures in real practice.

Dynamic modeling of soft pneumatic actuators are a challenging research field. In this paper, a dynamic modeling method used for a bi-directionaly soft pneumatic actuator with symmetrical chambers is proposed. In this dynamic model, the effect of uninflated rubber block on bending deformation is considered. The errors resulting from the proposed dynamic equilibrium equation are analyzed, and a compensation method for the dynamic equilibrium equation is proposed. The equation can be solved quickly after simplification. The results show that the proposed dynamic model can describe the motion process of the bi-directional pneumatic actuator effectively.

In this work, a coupled 3D thermo-elastic shell model is presented. The primary variables are the scalar sovra-temperature and the displacement vector. This model allows for the thermal stress analysis of one-layered and sandwich plates and shells embedding Functionally Graded Material (FGM) layers. The 3D equilibrium equations and the 3D Fourier heat conduction equation for spherical shells are put together into a set of four coupled equations. They automatically degenerate in those for simpler geometries thanks to proper considerations about the radii of curvature and the use of orthogonal mixed curvilinear coordinates α, β, and z. The obtained partial differential governing the equations along the thickness direction are solved using the exponential matrix method. The closed form solution is possible assuming simply supported boundary conditions and proper harmonic forms for all the unknowns. The sovra-temperature amplitudes are directly imposed at the outer surfaces for each geometry in steady-state conditions. The effects of the thermal environment are related to the sovra-temperature profiles through the thickness. The static responses are evaluated in terms of displacements and stresses. After a proper and global preliminary validation, new cases are presented for different thickness ratios, geometries, and temperature values at the external surfaces. The considered FGM is metallic at the bottom and ceramic at the top. This FGM layer can be embedded in a sandwich configuration or in a one-layered configuration. This new fully coupled thermo-elastic model provides results that are coincident with the results proposed by the uncoupled thermo-elastic model that separately solves the 3D Fourier heat conduction equation. The differences are always less than 0.5% for each investigated displacement, temperature, and stress component. The differences between the present 3D full coupled model and the the advantages of this new model are clearly shown. Both the thickness layer and material layer effects are directly included in all the conducted coupled thermal stress analyses.

Starting from the three-dimensional theory of linear elasticity, we arrive at the exact plate problem by the use of Taylor series expansions. Applying the consistent-approximation approach to this problem leads to hierarchic generic plate theories. Mathematically, these plate theories are systems of partial-differential equations (PDEs), which contain the coefficients of the series expansions of the displacements (displacement coefficients) as variables. With the pseudo-reduction method, the PDE systems can be reduced to one main PDE, which is entirely written in the main variable, and several reduction PDEs, each written in the main variable and several non-main variables. So, after solving the main PDE, the reduction PDEs can be solved by insertion of the main variable. As a great disadvantage of the generic plate theories, there are fewer reduction PDEs than non-main variables so that not all of the latter can be determined independently. Within this paper, a modular structure of the displacement coefficients is found and proved. Based on it, we define so-called complete plate theories which enable us to determine all non-main variables independently. Also, a scheme to assemble Nth-order complete plate theories with equations from the generic plate theories is found. As it turns out, the governing PDEs from the complete plate theories fulfill the local boundary conditions and the local form of the equilibrium equations a priori. Furthermore, these results are compared with those of the classical theories and recently published papers on the consistent-approximation approach.

In the existing literature, there are only two in-plane equilibrium equations for membrane problems; one does not take into account the contribution of deflection to in-plane equilibrium at all, and the other only partly takes it into account. In this paper, a new and exact in-plane equilibrium equation is established by fully taking into account the contribution of deflection to in-plane equilibrium, and it is used for the analytical solution to the well-known Föppl-Hencky membrane problem. The power series solutions of the problem are given, but in the form of the Taylor series, so as to overcome the difficulty in convergence. The superiority of using Taylor series expansion over using Maclaurin series expansion is numerically demonstrated. Under the same conditions, the newly established in-plane equilibrium equation is compared numerically with the existing two in-plane equilibrium equations, showing that the new in-plane equilibrium equation has obvious superiority over the existing two. A new finding is obtained from this study, namely, that the power series method of using Taylor series expansion is essentially different from that of using Maclaurin series expansion; therefore, the recurrence formulas for power series coefficients of using Maclaurin series expansion cannot be derived directly from that of using Taylor series expansion.

In this paper, the well-known Hencky problem—that is, the problem of axisymmetric deformation of a peripherally fixed and initially flat circular membrane subjected to transverse uniformly distributed loads—is re-solved by simultaneously considering the improvement of the out-of-plane and in-plane equilibrium equations. In which, the so-called small rotation angle assumption of the membrane is given up when establishing the out-of-plane equilibrium equation, and the in-plane equilibrium equation is, for the first time, improved by considering the effect of the deflection on the equilibrium between the radial and circumferential stress. Furthermore, the resulting nonlinear differential equation is successfully solved by using the power series method, and a new closed-form solution of the problem is finally presented. The conducted numerical example indicates that the closed-form solution presented here has a higher computational accuracy in comparison with the existing solutions of the well-known Hencky problem, especially when the deflection of the membrane is relatively large.

Structural analysis and construction control of staged construction process is a major subject for modern long-span bridges. This paper introduces the concept of stress-free-state variable of structural elements and deduces the mechanical equilibrium equations and geometric shape governing equations for staged construction structures utilizing the minimum potential energy theorem. As the core of stress-free-state theory, the two aforementioned equations demonstrate following principles, 1) when the stress-free-state variable of a structural element is set, the internal force and deformation of the element are unique at the completion state of the structure regardless of its construction process; 2) the stress-free length of a cable is independent of its external loads, change in stress-free length of the cable corresponds to a unique variation of the cable force when load is constant; and 3) the internal force of a structural element can be independent from its geometric shape within the completion state of a staged construction structure through an active manipulation of stress-free-state variables of the element. Stress-free-state theory establishes the stage-to-stage and stage-to-completion relationships for staged construction bridges, provides a direct and efficient method for theoretical calculations and a flexible and convenient approach for the control of staged construction, and makes parallel construction and auto-filtering of thermal and temporary loading effect possible.

Droplet dynamics on a solid substrate is significantly influenced by surfactants. It remains a challenging task to model and simulate the moving contact line dynamics with soluble surfactants. In this work, we present a derivation of the phase-field moving contact line model with soluble surfactants through the first law of thermodynamics, associated thermodynamic relations and the Onsager variational principle. The derived thermodynamically consistent model consists of two Cahn–Hilliard type of equations governing the evolution of interface and surfactant concentration, the incompressible Navier–Stokes equations and the generalized Navier boundary condition for the moving contact line. With chemical potentials derived from the free energy functional, we analytically obtain certain equilibrium properties of surfactant adsorption, including equilibrium profiles for phase-field variables, the Langmuir isotherm and the equilibrium equation of state. A classical droplet spread case is used to numerically validate the moving contact line model and equilibrium properties of surfactant adsorption. The influence of surfactants on the contact line dynamics observed in our simulations is consistent with the results obtained using sharp interface models. Using the proposed model, we investigate the droplet dynamics with soluble surfactants on a chemically patterned surface. It is observed that droplets will form three typical flow states as a result of different surfactant bulk concentrations and defect strengths, specifically the coalescence mode, the non-coalescence mode and the detachment mode. In addition, a phase diagram for the three flow states is presented. Finally, we study the unbalanced Young stress acting on triple-phase contact points. The unbalanced Young stress could be a driving or resistance force, which is determined by the critical defect strength.

Large values and gradients of stress and strain, triggering concentrated stress and strain, arise in angular areas of a structure. The strain action, leading to the finite loss of contact between structural elements, also triggers concentrated stress. The loss of contact reaches an irregular point and a line on the boundary. The theoretical analysis of the stress–strain state (SSS) of areas with angular cutouts in the boundary under the action of discontinuous strain is reduced to the study of singular solutions to the homogeneous problem of elasticity theory with power-related features. The calculation of stress concentration coefficients in the domain of a singular solution to the elastic problem makes no sense. It is experimentally proven that the area located near the vertex of an angular cutout in the boundary features substantial strain and rotations, and it corresponds to higher values of the first and second derivatives of displacements along the radius in cases of sufficiently small radii in the neighborhood of an irregular boundary point. As far as these areas are concerned, it is necessary to consider the plane problem of the elasticity theory, taking into account the geometric nonlinearity under the action of strain, to analyze the effect of relationships between strain orders, rotations, and strain on the form of the equation of equilibrium. The purpose of this work is to analyze the effect of relationships between strain orders, rotations, and strain on the form of the equilibrium equation in the polar system of coordinates for a V-shaped area under the action of temperature-induced strain, taking into account geometric non-linearity and physical linearity.