Equilibrium Equations In Terms Research Articles

Analytical Thermal Analysis of Radially Functionally Graded Circular Plates with Coating or Undercoating under Transverse and Radial Temperature Distributions

This study aims to provide analytical solutions for circular plates coated or undercoated with functionally graded materials (FGMs) having Young’s modulus functionality through the radius. The circular plates are subjected to thermal loads in radial and thickness directions. Because of the uncoupled stretching–bending problem of the radially FGM circular plate, the bending equilibrium equations in terms of displacements of the FGM-coated or -undercoated circular plates with Young’s modulus based on the power–law function were established individually. General solutions for the homogeneous portion or FGM ring of the radially FGM-coated or -undercoated circular plate were developed separately. Subsequently, analytical thermal solutions for the radially FGM-coated or -undercoated circular plate were evaluated by solving the simultaneous boundary and continuity conditions equations. The analytical results were validated by comparing them with finite element solutions. When degenerated, they coincided with those of the homogeneous circular plate in the literature, enhancing the obtained solutions’ reliability. These analytical solutions provide valuable insights into the plates’ responses and expand the understanding of their mechanical behaviors under thermal loads. Furthermore, the effects of the FGM thickness, the material index, and the thermal loading conditions on the mechanical behaviors were under investigation. This parameter study offers valuable perspectives into the influence of these factors on the plate’s structural response and aids in the optimization and design of FGM-coated or -undercoated circular plates.

SCHOUTEN'S FORCE STRESS TENSOR AND AFFINOR DENSITIES OF POSITIVE WEIGHT

The paper deals with the concept of the force stress pseudotensor and the derivation of equilibrium equations in terms of the Schouten's stress pseudotensor being an affinor density. The definition of Schouten's force stress pseudotensor is mainly based on the notion of a pseudoinvariant element of area. The requisute equations and notions from algebra and the analysis of pseudotensors is revisited. A fundamental orienting pseudoscalar is introduced and discussed. Conventional and non-conventional definitions of the force stress tensor are given. A unit normal vector to a level surface of a pseudoscalar field is introduced. The exceptional importance of using the theory of orientable manifolds in modeling micropolar continua in mechanics of solids is noted. The notion of M-cell and its orientation algorithm are recalled. Algorithms for constructing the tensor elements of the area of M-manifold immersed in N-dimensional space are discussed. The notions of vector, pseudovector, invariant and pseudoinvariant elements of surface area in three-dimensional space are revisited. The possibility of using pseudotensor volume elements of a given integer weight due to the formula for a pseudotensor field transformation to an absolute tensor field by a fundamental orienting pseudoscalar is discussed. Various realisations of covariant differentiation of pseudotensors are considered. Covariant derivatives are given for a pseudoscalar and a contravariant pseudotensor of the second rank of an arbitrary integer weight. The principle of virtual displacements is formulated in terms of pseudo-invariant volume and area elements. The hypothesis of the absolute invariance of the virtual work is assumed, i.e. insensitive to rotations, 3D inversion and mirror reflections. Equations of equilibrium and dynamics are derived in terms of the affinor density of Schouten's force stresses. Equilibrium equations are obtained for the case of using pseudo-invariant volume and area elements of an arbitrary integer weight.

ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ РАВНОВЕСИЯ СПЛОШНОЙ СРЕДЫ ДЛЯ ПЛОСКОЙ ДЕФОРМАЦИИ В ЦИЛИНДРИЧЕСКИХ КООРДИНАТАХ ПРИ БИЛИНЕЙНОЙ АППРОКСИМАЦИИ ЗАМЫКАЮЩИХ УРАВНЕНИЙ

Problems of the formulation of differential equations of equilibrium in terms of displacements for a plane strain of continuous media at bilinear approximation of closing equations are considered leaving out of account geometric nonlinearity in the cylindrical coordinate system. Based on the assumption that the curves of volumetric and shear strain are independent from each other, six main cases of physical dependencies are considered, which are the functions of the relative position of break points on the bilinear curves of the volumetric and shear strain. Obtaining of bilinear physical dependencies is based on the calculation of secant moduli of the volumetric and shear strain. On the first line of the curves, secant moduli are constant for both volumetric and shear strain, while on the second line, the secant modulus of the volumetric strain is a function of the volumetric strain, and the secant modulus of the shear strain is a function of the shear strain intensity. Putting the corresponding bilinear physical equations into differential equations of continuum equilibrium, which disregard geometrical nonlinearity, the resulting differential equations of equilibrium are obtained in terms of displacements for a one-dimensional plane strain of continuum in the cylindrical coordinate system. These equations can be used when determining stress-strain state of continuous media under one-dimensional plane strains with no regard for geometrical nonlinearity, and whose physical relations are approximated by bilinear functions.

An enhanced method of resizing support links for a planar closed-loop overconstrained deployable structure considering kinematic reliability and surface accuracy

Kinematic reliability and surface accuracy are of importance for the deployable structure in that the former directly determines whether the deployment is successful or not and the latter is greatly associated with the performance of the space antenna. These two key indexes are generally guaranteed by adjusting the length of the support link. To this end, this study proposes an enhanced method of quantitatively resizing support links for a planar closed-loop overconstrained deployable structure. First, the relationship between the successful unfolding and the combination of link adjustments is derived by taking advantage of the support vector machine to train the motion data that is collected from rigid-flexible dynamic simulations. Then, resorting to structural mechanics, the system of equilibrium equations in terms of stretching and bending is established, thereby leading to the implicit mapping from link deviations to the angular errors of the satellite panels. Thereon, a discrete optimization model with two objectives ensuring the deployable reliability and accuracy performance for link adjustment is completely developed. This optimization model is solved by an improved successive Taguchi approach, which uses the grey relational analysis coupled with principal component analysis as the multicriteria decision-making model. Finally, the implementation of the proposed method and its effectiveness are comprehensively demonstrated by a numerical example.

Mathematical solutions for the flexural analysis of Mindlin’s first order shear deformable circular plates

In this work, the problem of first order shear deformable solid circular plate under transverse load was solved mathematically. The problem considered was assumed axisymmetric. The plate and loading were considered axisymmetric. The problem was defined as a boundary value problem of a system of differential equations of equilibrium in terms of the stress resultants and the stress – resultants – displacement relations. The set of equations were considered simultaneously to express them in variable separable form. The mathematical technique of separation of variables was then used to obtain solutions for the unknown generalised displacements. Specific problems of clamped edge plates and simply supported edge plates under uniformly distributed load and point load at the centre were considered and solved using the same technique of separation of variables. The mathematical expressions obtained showed that in all cases, the deflection was expressible in terms of flexural and shear components. The maximum deflection was found to occur at the plate centre as is expected from the symmetrical nature of the problem. The shear component of the transverse deflection was found to significantly increase with significant increase in the ratio of the plate thickness to the radius (h/r0).

Hygrothermal analysis of magneto-electro-elastic plate using 3D finite element analysis

In this article, the static response of magneto-electro-elastic (MEE) plate subjected to hygrothermal loads is investigated using the finite element (FE) method. A FE formulation is derived using the principle of total potential energy and linear coupled constitutive equations of MEE materials by taking into account the thermal and hygroscopic field effects. A uniform temperature rise and moisture concentration rise has been considered. The variations of static parameters are estimated along the MEE plate length by considering the temperature and moisture dependant elastic stiffness coefficients. The coupled FE equilibrium equations in terms of displacements, electric and magnetic potentials are solved directly using condensation procedure. Numerical examples of the FE results are presented and discussed in detail to understand the significant effects of hygrothermal loading, temperature and moisture dependent material properties, boundary conditions and aspect ratio on the direct (displacements, electric potential and magnetic potential) and derived quantities (stresses, electric displacement and magnetic flux density) of MEE plate.

Form-finding of tensegrity structures based on the Levenberg–Marquardt method

This paper presents a novel and versatile form-finding method for tensegrity structures that is based on solving nonlinear equilibrium equations. Linear equilibrium equations in terms of force densities are transformed into nonlinear equilibrium equations in which the nodal coordinates vector is variable. The input parameters for the form-finding method are the topology, initial configuration of the structure, rest lengths, and axial stiffness of elements. The form-finding process is performed by solving nonlinear least-squares problems converted from nonlinear equilibrium equations, and the Levenberg–Marquardt method is used to deal with the singularity of the stiffness matrix. Several numerical examples are given to demonstrate the accuracy and efficacy of the proposed method.

Displacement potentials for functionally graded piezoelectric solids

• Solutions of Functionally Graded Transversely Isotropic Piezoelectric Media studied. • Displacement potential functions for the media are introduced. • The system of governing differential equations has been reduced and decoupled. • Limiting cases of the problem are discussed in detail. Two new displacement potential functions are introduced for the general solution of a three-dimensional piezoelasticity problem for functionally graded transversely isotropic piezoelectric solids. The material properties vary continuously along the axis of symmetry of the medium. The four coupled equilibrium equations in terms of displacements and electric potential are reduced to two decoupled sixth- and second-order linear partial differential equations for the potential functions. The obtained results are verified with two limiting cases: (i) a functionally graded transversely isotropic medium, and (ii) a homogeneous transversely isotropic piezoelectric solid. The simplified relations corresponding to the special case of similar variation of material properties are also given. Furthermore, the special cases of axisymmetric problems, exponentially graded piezoelectric media and transversely isotropic piezoelectric media with power law variation are discussed in detail.

Isogeometric collocation for the Reissner–Mindlin shell problem

We present an isogeometric collocation formulation for the Reissner–Mindlin shell problem. After recalling the necessary basics on differential geometry and the shell governing equations, we show that the standard approach of expressing the equilibrium equations in terms of the primal variables is not a suitable way for shells due to the complexity of the underlying equations. We then propose an alternative approach, based on a stepwise formulation, and show its numerical implementation within an isogeometric collocation framework. The formulation is tested successfully on a set of benchmark examples, which comprise important aspects like locking and boundary layers. These test show that locking effects can be conveniently avoided by using high polynomial degrees. An accompanying study on the computational time also confirms that high polynomial degrees are preferable in terms of computational efficiency.

Interpretation of cone resistance and pore-water pressure in clay with a modified spherical cavity expansion solution

A theoretical approach to interpreting the cone penetration resistance and the pore-water pressure measured on the face of the cone during piezocone tests of clays is proposed. The mechanism for the penetration of the piezocone is assumed to be spherical cavity expansion in modified Cam clay critical-state soil. Using large strain theory, the spherical cavity expansion problem is reduced to solving a system of first-order ordinary differential equations for effective stresses in the plastic zone. By taking the normal and shear stresses acting on the cone face to be the ultimate cavity expansion pressure and the effective friction at the cone–soil interface, respectively, the cone resistance was evaluated at force equilibrium. The pore-water pressures induced by cone penetration include components caused by changes in normal and shear stresses, for which the normal term can be predicted by integrating the radial equilibrium equation in terms of total stress. As it is based on an exact constitutive relationship, the present method takes more factors (such as the effects of interface shear and the traditional overconsolidation ratio) into account than other methods of tackling this problem. Comparisons between the predicted and measured values of cone resistance and pore-water pressure at different well-documented sites are in good agreement for most relatively homogeneous low- and medium-sensitivity clay sites.

Influence of Initial Stresses on the Fracture of Composite Material Weakened by a Subsurface Mode III Crack

By using the relations of linearized mechanics of deformable solids, we study the space problem of fracture of prestressed semibounded composites weakened by subsurface disk-shaped torsion (mode-III) cracks. With the help of representations of the general solutions of linearized equilibrium equations in terms of harmonic potential functions and Hankel integral transforms, we reduce the problem to a system of dual integral equations and then to the resolving Fredholm integral equation of the second kind. From the analysis of the stress distribution in the vicinity of the crack, we obtain the values of the stress intensity factors and study their dependences on the initial stresses, mechanical characteristics of the components of the composite, and geometric parameters of the problem.

A refined laminated plate theory accounting for the third-order shear deformation and interlaminar transverse stress continuity

A variational consistent third-order shear laminated plate theory accounting for the transverse shear stress continuity at the interlaminar interfaces of laminated plates is developed in this paper. The transverse shear function used in this new laminated plate theory is based on the kinematics in the third-order shear deformation plate theory proposed by Shi (2007). The variational principle is employed to derive the variational consistent equilibrium equations in terms of displacements and the variational boundary conditions in terms of displacements and equivalent stress resultants. The continuity conditions of the in-plane displacements and transverse shear stresses at the interlaminar interfaces of laminated plates are enforced by the Heaviside step functions and continuity coefficients. The resulting new laminated composite plate theory accounting for interlaminar continuity has only five independent field variables. Furthermore, the number of the field variables in the present third-order shear laminated plate theory is the same as that used in the first-order shear deformation plate theory. The refined laminated plate theory is applied to solve the bending problems of four laminated composite plates with different lamination schemes and different aspect ratios to evaluate its reliability and accuracy. The resulting analytical solutions of both deflections and stresses agree well with the 3D elasticity solutions and the numerical results of finite element analysis. The result comparison with other laminated plate theories shows that new laminated plate theory accounting for the interlaminar continuity proposed in this paper yields more accurate displacements and stresses than other laminated plate theories with five global variables. Because only five field variables are used in this new laminated plate theory with interlaminar continuity, this refined laminated plate theory can be used as an accurate and efficient theoretical model for the finite element analysis of laminated composite plates.

Modeling mixed boundary conditions in a Hilbert space with the complex variable boundary element method (CVBEM)

The Laplace equation that results from specifying either the normal or tangential force equilibrium equation in terms of the warping functions or its conjugate can be modeled as a complex variable boundary element method or CVBEM mixed boundary problem. The CVBEM is a well-known numerical technique that can provide solutions to potential value problems in two or more dimensions by the use of an approximation function that is derived from the Cauchy Integral in complex analysis. This paper highlights three customizations to the technique.•A least squares approach to modeling the complex-valued approximation function will be compared and analyzed to determine if modeling error on the boundary can be reduced without the need to find and evaluated additional linearly independent complex functions.•The nodal point locations will be moved outside the problem domain.•Contour and streamline plots representing the warping function and its complementary conjugate are generated simultaneously from the complex-valued approximating function.

Slip systems and flow patterns in viscoplastic metallic sheets with dislocations

The work addresses the rate dependent crystal plasticity with hardening dependent on the dislocation density, when the local and non-local (diffusion-like) evolution equations for the dislocation densities are considered. A new procedure is proposed for solving initial and boundary value problems based on the incremental equilibrium equations in terms of variational results, coupled with a rate-type constitutive model. The problems concerning the deformation of a sheet composed of a single fcc-crystal, generated by different slip systems simultaneously activated, are solved numerically for an in-plane stress state. Necessary and sufficient conditions for obtaining the in-plane stress state are proved under the assumptions of multislip flow rule, and cubic elastic symmetry. The variational problem which defines the velocity field in the actual configuration at time t is solved by the finite element method (FEM) and the current state in the sheet is defined via an update algorithm when either local (differential type) or non-local (diffusion-like) evolution equations describe the dislocation densities. Boundary value problems resulting from the simulation of compressive and tensile tests are solved by considering all eight activated slip systems together with an activation condition. Finally, a comparison between the numerical results obtained by using the non-local (diffusion-like) evolution equations and local (differential type) equations which describe the dislocation densities is carried out.

Stress singularities in an anisotropic body of revolution

To fill the gap in the literature on the application of three-dimensional elasticity theory to geometrically induced stress singularities, this work develops asymptotic solutions for Williams-type stress singularities in bodies of revolution that are made of rectilinearly anisotropic materials. The Cartesian coordinate system used to describe the material properties differs from the coordinate system used to describe the geometry of a body of revolution, so the problems under consideration are very complicated. The eigenfunction expansion approach is combined with a power series solution technique to find the asymptotic solutions by directly solving the three-dimensional equilibrium equations in terms of the displacement components. The correctness of the proposed solution is verified by convergence studies and by comparisons with results obtained using closed-form characteristic equations for an isotropic body of revolution and using the commercial finite element program ABAQUS for orthotropic bodies of revolution. Thereafter, the solution is employed to comprehensively examine the singularities of bodies of revolution with different geometries, made of a single material or bi-materials, under different boundary conditions.

Equilibrium equations for transversely accreted shells

AbstractA particular case of the theory of growing thin‐walled structures, namely the theory of transversely accreted shells is considered. A correspondence between equilibrium equations in terms of rates for transversally accreted shells and equilibrium equations in terms of displacements for permanent shells is established. The equilibrium equations for accreted shells that obey the kinematics hypothesis of Mindlin type are derived in terms of displacements as well as in terms of rates.

On the interaction of a periodic system of parallel coaxial radial-shear cracks in a prestressed composite

Based on approaches of the linearized theory of elasticity, we consider an axisymmetric problem of stress-strain state of a composite with initial (residual) stresses containing a periodic system of parallel coaxial radial-shear cracks. Using representations of general solutions of the linearized equilibrium equations in terms of harmonic potential functions and the apparatus of integral Hankel transformations, we reduce the problem to a system of dual integral equations and then to a Fredholm integral equation of the second kind. We obtain expressions for the stress intensity factors near the crack edges and, for a laminated composite with isotropic layers, analyze dependences of these factors on the initial stresses, mechanical characteristics of components, and geometric parameters of the problem. On the basis of analysis of the sharp “resonance-like” behavior of the stress intensity factors, for certain values of the initial compressive stresses, we determine critical parameters of compression along the periodic system of parallel coaxial cracks.

Three-Dimensional Elastic Solution of a Transversely Isotropic Functionally Graded Rectangular Plate

Three-dimensional elastic solution of a simply supported, transversely isotropic functionally graded rectangular plate is presented in this paper. Suppose that all elastic coefficients of the material have the same power-law dependence on the thickness coordinate. By introducing two new displacement functions, three equations of equilibrium in terms of displacements are reduced to two uncoupled partial differential equations. Exact solution for a second-order partial differential equation expressed by one of displacement functions is obtained and analytical solution for another fourth-order partial differential equation expressed by another displacement function is found by employing the Frobenius method. The validity of the present solution is first investigated. And the effect of the gradation of material properties on the mechanical behavior of the plate is studied through numerical examples.

Exact undrained elasto-plastic solution for cylindrical cavity expansion in modified Cam Clay soil

This paper presents an exact semi-analytical analysis for cylindrical cavity expansion in modified Cam Clay critical state soil under the undrained condition. By assuming small-strain deformation in the elastic region and large-strain deformation in the plastic region, the formulation of the problem is reduced to solving a system of first-order ordinary differential equations for the radial, tangential and vertical effective stresses in the plastic zone. The pore pressure distribution can subsequently be determined from the radial equilibrium equation in terms of the total stresses. The semi-analytical solution derived in this paper follows the rigorous definitions for the two effective stress invariants instead of approximate ones, as usually assumed in the existing literature, and applies to the more general case where the out-of-plane (vertical) in situ stress may be different from the in-plane (horizontal) one. The results show clearly that the stress history (overconsolidation ratio) has a significant influence on the stress and pore pressure distributions around the cavity. The present solution provides an exact and realistic theoretical framework for predicting the soil behaviour around the cylindrical cavity, with applications to the interpretation of pressuremeter tests and to pile installation problems. It also serves as a valuable benchmark for verifying various cavity expansion numerical methods involving the critical state plasticity model.

Three-dimensional thermo-elastic general solutions of one-dimensional hexagonal quasi-crystal and fundamental solutions

This Letter presents three-dimensional general solutions for static problems in thermo-elasticity of one-dimensional hexagonal quasi-crystals. Two displacement potentials are introduced to simplify the equilibrium equations in terms of displacement and temperature. Rigorous operator theory and generalized Almansiʼs theorem are applied to derive the general solutions in terms of five quasi-harmonic functions. To show the significance of the general solutions, a semi-infinite space and an infinite space, both of which are subjected to a heat source, are considered. In these two cases, closed form fundamental phonon–phason-elastic fields are expressed by elementary functions, which play an important role in numerical simulations.