Exact Elasticity Solution Research Articles

On the stresses associated with heterointerfaces

ABSTRACT Considering the formation of misfit dislocations in systems with both high and low lattice mismatch, the Peierls–Nabarro mechanism is first introduced to deal with periodic dislocations. The effect of interfacial misfit dislocations on the interior of the crystal is then investigated using exact elasticity solutions. The core structure of the misfit dislocation is then given, and it is emphasised that the stress field inside the crystal can be determined under certain combinations of material constants so as to determine the atomic configuration of the material. An application of the theoretical results to interfacial dislocations of a BN/AlN heterojunction is shown, and the reliability of theory is verified by first-principles calculations.

Estimation of Transverse Shear Stresses across the Thickness of Three-Layered Orthotropic Sandwich Plates Using ABAQUS

Abstract: This paper primarily focuses on presenting the results obtained through ABAQUS Finite Element Modelling (FEM) tool. These findings have been subjected to a comparative analysis with both the 3D exact elasticity solutions and the partial FEM solutions. Notably, the ABAQUS results, as described in reference demonstrate a remarkable level of accuracy, particularly in relation to transverse shear stresses. It is crucial to underscore that the principal objective of this research revolves around the precise determination of transverse shear stresses, as these have been identified as potential contributors to delamination within the studied structures. It can be asserted that the objectives, cantered on the accurate computation of transverse shear stresses, have been successfully achieved through the implementation of ABAQUS Finite Element modelling.

An elasticity solution of laminated cylindrical panel

In this paper, an exact elasticity solution of cylindrical panel under cylindrical bending is presented. The formulation is based on the displacement approach in which 3D elasticity equations are reduced to 2-D equations using plane-strain conditions. In the present formulation, an exact elasticity solution is feasible due to the consideration of simply supported boundary conditions with applied loads expressed in harmonic forms. Single layered and multi-layered orthotropic composite panels subjected to realistic transverse normal loads are analysed. Solutions have been presented for uni-directional, bi-directional, and symmetric three-layered cross-ply laminated composite cylindrical panels. Variation of displacements and stresses through the thickness has been investigated for the panel subjected to single sinusoidal load, uniformly distributed load, hydrostatic load, and strip load. Validation of results is presented using reference results in the literature. The extensive results presented in this paper can be served as benchmark solutions for the assessment of improved panel theories.

A direct approach for three-dimensional elasto-static and elasto-dynamic solutions in curvilinear cylindrical coordinates with application to classical cylinder problems

This paper deals with introducing a unique representation of the three-dimensional Navier's equations of motion in cylindrical coordinate system in an exact simplified form without any approximation, aiming at facilitating solution procedure for different 3-D elasto-static and elasto-dynamic problems in the future. A novel form of the 3-D elasticity equations of motion including the body forces in cylindrical coordinate system is derived in an uncoupled form in terms of the longitudinal (axial) displacement component and the ‘r-θ’ in-plane anti-symmetric rotation function instead of introducing any additional auxiliary unknown potential function. The other displacement components (i.e., circumferential and radial displacement components) are shown to be obtained from two independent equations in terms of the determined axial displacement and the aforementioned rotation component. The correctness, validity and easy implementation of the introduced elasticity approach for obtaining exact elasticity solutions for various 3-D elasto-static and elasto-dynamic problems are demonstrated through solving a number of known elasticity problems. Three-dimensional static and free vibrations of finite-length solid cylinders as well as thick-walled hollow cylindrical shells are analytically solved. Numerical comparative results and discussion are conducted. Excellent agreement between the obtained results and those reported in the literature is observed in all cases, confirming the validity of the proposed new approach.

Analytical elasticity solution for accurate prediction of localized stresses in laminated composites under patch loading

Discontinuous patch-type loads cause highly localized stresses which can initiate failures like crack and delamination, and thus must be predicted accurately for efficient design of structures. This paper presents an iterative analytical elasticity solution for accurate stress analysis of laminated composite beams with arbitrary support conditions under a transverse patch loading. The beam is divided into segments with and without the patch loading. The governing equations are derived in the mixed form by employing the Reissner-type mixed variational principle and solved analytically for each segment using the multiterm extended Kantorovich approach. As both displacements and stresses are primary variables, the solution ensures the point-wise satisfaction of inter-layer and inter-segment continuity and boundary conditions. The accuracy of the solution is established by comparing the present numerical results with exact elasticity solutions available in literature for simply-supported edge conditions and compared with finite element solutions of Abaqus for other boundary conditions. An extensive numerical study is performed, and benchmark results are provided for various composite and sandwich laminates, thickness ratios, and boundary conditions under different cases of distributed patch load. The behavior of laminated metal-composite beams is also studied for patch loading, which reveals that thin aluminum layers on the composite beam are highly efficient and effective in reducing the stress concentration in the vicinity of patch load. The presented benchmark results will be beneficial to assess the accuracy and efficacy of different beam theories and elasticity-based approximate models for the analysis of multilayered anisotropic beams subjected to discontinuous loads.

A framework based on physics-informed neural networks and extreme learning for the analysis of composite structures

This paper presents a novel approach for solving direct problems in linear elasticity involving plate and shell structures. The method relies upon a combination of Physics-Informed Neural Networks and Extreme Learning Machine. A subdomain decomposition method is proposed as a viable mean for studying structures composed by multiple plate/shell elements, as well as improving the solution in domains composed by one single element. Sensitivity studies are presented to gather insight into the effects of different network configurations and sets of hyperparameters. Within the framework presented here, direct problems can be solved with or without available sampled data. In addition, the approach can be extended to the solution of inverse problems. The results are compared with exact elasticity solutions and finite element calculations, illustrating the potential of the approach as an effective mean for addressing a wide class of problems in structural mechanics.

Improved Electromechanical Response in Laminated Piezoelectric Plates Using a Zigzag Theory

The zigzag theories are a class of the shear deformation theories that combine the accuracy of the layerwise theories and the efficiency of the equivalent single-layer theories. A nine variable zigzag theory is presented in this work for the laminated piezoelectric plates. Unlike any existing zigzag theory, the transverse shear boundary conditions on the lateral surfaces are not imposed. The theory satisfies interlaminar continuity of displacements and transverse shears stresses. A variational principle is used in deriving governing equations. Bending responses have been obtained under sinusoidal and uniformly applied mechanical and electromechanical loads. Three-dimensional (3D) exact elasticity and piezoelasticity solutions have been used to assess the accuracy of the present theory for a number of simply supported rectangular laminated composite and piezoelectric plates. The transverse shear stress relaxation condition at the boundary surfaces has led to improved predictions in the displacements, stresses, and electric field entities. Present theory is also compared with a number of shear deformation theories to illustrate how significantly the responses get improved due to the transverse shear relaxation.

Thin wall approximation revisited

A thin wall mechanical framework for calculation of stresses, strains and displacements, e.g. in fuel rod cladding deformed plastically by contact with expanding fuel pellets, is proposed. The standard thin wall approximation to exact elastic solution of a long closed cylindrical tube loaded by inner and outer pressure is extended by considering displacement boundary conditions at inner or outer radius instead of inner pressure. A suitable projection of strains enables application of displacement boundary conditions, as well as treatment of tube dimensional changes in this thin wall framework. Applications to several problems relevant for fuel cladding deformations are discussed. Analytical solution of the problem of partially plastic tube and FEM simulations of mechanical tests are utilized to assess the quality of the proposed approach.

Forced vibration analysis of composite beams with piezoelectric layers based on the variable separation method

This paper proposes a new strategy based on the proper generalized decomposition method to solve the forced vibration problem for bi-dimensional piezoelectric composite beams. The approach considers a classical harmonic space-frequency description of the dynamic problem and redefines it by approximating the displacement and electric potential fields as a sum of separated functions of x (beam axis coordinate), z (thickness coordinate) and ω (load frequency). The methodology consists in an iterative algorithm that solves three one-dimensional linear problems at each iteration. The result is a 2D solution in frequency domain with 1D computational complexity. Several numerical tests with wide range of slenderness ratios are considered in order to assess the validity of the method. Moreover, a study of different combinations of boundary conditions is carried out. The results in terms of modal parameters and frequency response functions are analysed and compared with exact elasticity solution and finite element simulations.

Limit behaviour of Eringen’s two-phase elastic beams

In this paper, the bending behaviour of small-scale Bernoulli–Euler beams is investigated by Eringen’s two-phase local/nonlocal theory of elasticity. Bending moments are expressed in terms of elastic curvatures by a convex combination of local and nonlocal contributions, that is a combination with non-negative scalar coefficients summing to unity. The nonlocal contribution is the convolution integral of the elastic curvature field with a suitable averaging kernel characterized by a scale parameter. The relevant structural problem, well-posed for non-vanishing local phases, is preliminarily formulated and exact elastic solutions of some simple beam problems are recalled. Limit behaviours of the obtained elastic solutions, analytically evaluated, studied and diagrammed, do not fulfil equilibrium requirements and kinematic boundary conditions. Accordingly, unlike alleged claims in literature, such asymptotic fields cannot be assumed as solutions of the purely nonlocal theory of beam elasticity. This conclusion agrees with the known result which the elastic equilibrium problem of beams of engineering interest formulated by Eringen’s purely nonlocal theory admits no solution.

An Enhancement of the Warping Shear Functions of Refined Zigzag Theory

Abstract The paper presents an enhancement in Refined Zigzag Theory (RZT) for the analysis of multilayered composite plates. In standard RZT, the zigzag functions cannot predict the coupling effect of in-plane displacements for anisotropic multilayered plates, such as angle-ply laminates. From a computational point of view, this undesirable effect leads to a singular stiffness matrix. In this work, the local kinematic field of RZT is enhanced with the other two zigzag functions that allow the coupling effect. In order to assess the accuracy of these new zigzag functions for RZT, results obtained from bending of angle-ply laminated plates are compared with the three-dimensional exact elasticity solutions and other plate models used in the open literature. The numerical results highlight that the enhanced zigzag functions extend the range of applicability of RZT to the study of general angle-ply multilayered structures, maintaining the same seven kinematic unknowns of standard RZT.

A new hybrid homogenization theory for periodic composites with random fiber distributions

A new homogenization theory is constructed for unidirectional composites with periodic domains containing random fiber distributions. Periodic domains are partitioned into subdomains comprised of single fibers embedded in matrix phase. The subdomain displacement field solutions combine finite-volume and locally-exact elasticity approaches. Inclusions are regarded as meshfree components whose displacement fields are represented by discrete Fourier transforms that satisfy exactly Navier’s equations. In contrast, the matrix is discretized into subvolumes and handled within the finite-volume micromechanics framework. Novel use of traction and displacement continuity conditions at the inclusion/matrix interface seamlessly connect inclusion and matrix phases. Subdomains’ assembly enforces traction, displacement continuity and periodicity conditions in a surface-average sense. Quadratic convergence of stress fields with decreasing matrix discretization relative to exact elasticity solution is obtained, yielding accurate homogenized moduli with relatively coarse matrix meshes. Fourfold and greater reductions in execution times relative to the finite-volume micromechanics justify the new theory’s construction. The proposed theory facilitates accurate and efficient studies of the rarely reported effect of random fiber distributions on homogenized moduli and local stress fields as a function of inclusion content and number of microstructural realizations. We illustrate this by computing probability distributions of homogenized moduli for up to 60,000 microstructural realizations of multi-inclusion periodic domains that may be used in machine-learning algorithms.

Three-dimensional exact elasticity solutions for antisymmetric angle-ply laminated composite plates

This paper is concerned with the derivation of the benchmark three-dimensional exact solutions for the static analysis of rectangular antisymmetric angle ply plates. The expressions for the exact solutions of displacements and stresses can be used in a straight forward manner to find the numerical results for any fiber orientation angle. It has been explored that the expressions for the exact solutions of displacements and stresses in the square antisymmetric angle-ply plates with 45° fiber orientation angle cannot be used for other fiber orientation angles and vice versa. Comparison of the exact solutions with the analytical solutions employing the first order shear deformation theory (FSDT) reveals that the FSDT can be efficiently used for thin and two layered thick antisymmetric angle-ply plates.

An Exact Elasticity Solution for Monoclinic Functionally Graded Beams

In this study, an elasticity solution is presented for monoclinic functionally graded beams subject to a transverse pressure distributed sinusoidally. Monoclinic material properties are assumed to vary exponentially throughout the thickness of the beam’s layers. An analytical formulation based on the classical Euler–Bernoulli beam theory is also derived for comparison purposes of simply supported monoclinic functionally graded beams. In benchmark examples, the numerical results of normal stresses, transverse shear stress, as well as axial and vertical displacements are presented. The effect of material grading, fiber angle, and beam length to thickness ratio on the stress and displacement distributions is comprehensively investigated. The proposed elasticity-based analytical solution and presented numerical results can be used for verification or comparison purposes of numerical procedures.

A novel finite element formulation based on extended higher-order theory for sandwich plates of arbitrary aspect ratio

The extended higher-order sandwich plate theory for plates with arbitrary aspect ratio was formulated for two-dimensional orthotropic sandwich plates. The novelty of the theory is that it considers five generalized co-ordinates in the core (two axial and one transverse displacements at centroid of the core, one rotation at the centroid of the core about x-axis and one rotation at the centroid of the core about y-axis). Theory is very accurate when compared with the exact elasticity solution in terms of stresses and displacement both. In the current paper, a novel two dimensional rectangular element is developed based on the extended higher-order sandwich plate theory. Elemental equations along with the procedure to derive these is given in the paper. Developed finite element model is validated by comparing the results with elasticity solution and the theory itself for two sandwich plate configurations. The comparison shows that results obtained from the proposed finite element are in very good agreement with elasticity in terms of displacements and stresses both. Thus the proposed element is a powerful analysis tool which can be used for accurately analyzing the real world structures involving sandwich plates at a low computational cost.

Antiplane point load solution for a linear slot with rounded ends

Abstract An exact linear elastic solution is derived for a pair of opposing point loads, which act in the middle of a straight slot having rounded ends. This problem is analogous to the mode III crack problem where two concentrated forces act in opposite directions to open the surfaces of a finite-length crack. The corresponding path independent J integral for this slot problem is also determined.

Exact elasticity solutions to rotating pressurized axisymmetric vessels made of functionally graded materials (FGM)

AbstractIn this study the plane elasticity equations of pressurized and rotating structures, generalized analytical closed‐form solutions are presented for hollow cylinders, spheres and thin disks. Three well known material pairs are graded in radial direction by using the power rule for various boundary and load conditions. Stress and displacement solutions are obtained. The effect of a constant rotational velocity is also investigated separately in absence of the internal and external pressures. Crucial results are also presented in tabular form.

Parametric finite-volume method for Saint Venant’s torsion of arbitrarily shaped cross sections

We extend our recent finite-volume based approach to the solution of Saint Venant’s torsion problems of bars and beams comprised of rectangular sections to enable analysis of arbitrary cross sections characterized by curved boundaries. This is accomplished by incorporating parametric mapping based on transfinite grid generation to enable discretization of the bar cross section by quadrilateral rather than rectangular subvolumes employed in the original version. The construction of the local stiffness matrix that relates the surface-averaged subvolume warping functions to the corresponding tractions is carried out in the reference plane such that the subvolume equilibrium in the physical plane is satisfied in a surface-averaged sense. This produces explicit expressions for the stiffness matrix elements that may be readily coded. Orthotropic subvolumes are intrinsic in the method’s construction so that bars with heterogeneous and composite microstructures may be analyzed. The convergence and accuracy of the parametric finite-volume method are assessed and verified upon comparison with exact elasticity solutions for cross sections with convex and concave boundaries. Examples involving structural applications of prismatic bars with curved boundaries illustrate the utility of the developed methodology. These include cross sections that resemble biological constructs with homogeneous and graded regions aimed at enhancing torsional rigidities, as well as homogeneous and graded elliptical cross sections with orthotropic shear moduli aimed at reducing and eliminating warping. We demonstrate for the first time that by laminating an elliptical cross section with alternating stiff and soft isotropic layers in a manner that mimics the required orthotropic moduli at the homogenized level, warping can be practically eliminated with sufficient microstructural refinement.

Singular Nature of the Elastocapillary Ridge

The functionality of soft interfaces is crucial to many applications in biology and surface science. Recent studies have used liquid drops to probe the surface mechanics of elastomeric networks. Experiments suggest an intricate surface elasticity, also known as the Shuttleworth effect, where surface tension is not constant but depends on substrate deformation. However, interpretations have remained controversial due to singular elastic deformations, induced exactly at the point where the droplet pulls the network. Here we reveal the nature of the elastocapillary singularity on a hyperelastic substrate with various constitutive relations for the interfacial energy. First, we finely resolve the vicinity of the singularity using goal-adaptive finite element simulations. This confirms the universal validity, also at large elastic deformations, of the previously disputed Neumann's law for the contact angles. Subsequently, we derive exact solutions of nonlinear elasticity that describe the singularity analytically. These solutions are in perfect agreement with numerics, and show that the stretch at the contact line, as previously measured experimentally, consistently points to a strong Shuttleworth effect. Finally, using Noether's theorem we provide a quantitative link between wetting hysteresis and Eshelby-like forces, and thereby offer a complete framework for soft wetting in the presence of the Shuttleworth effect.

Saint Venant’s torsion of homogeneous and composite bars by the finite volume method

We propose a new finite-volume based approach to the solution of Saint Venant’s torsion problems of homogeneous and composite prismatic bars subjected to pure torsion. The semi-inverse approach is employed to construct an approximate displacement field characteristic of pure torsion within the subvolumes of the beam’s discretized cross section such that the governing differential equation for the warping function is locally satisfied in each subvolume in an integral sense. Orthotropic subvolumes are admitted in the formulation to enable analysis of bars comprised of heterogeneous microstructures, including composite materials with orthotropic shear moduli. Continuity of both displacements and tractions across subvolumes’ interfaces is satisfied in a surface-average sense, together with traction-free lateral boundary conditions. Closed-form expressions for the stiffness matrix elements that relate surface-averaged displacements to tractions are provided for ease of programming. The convergence and accuracy of the proposed method are assessed and verified upon comparison with the exact elasticity solutions for prismatic homogeneous and composite bars of rectangular cross section and the corresponding finite-difference solution for the homogeneous bar. The utility of the developed solution methodology is first illustrated by analyzing torsion of homogeneous T and box beams, focusing on the singular behavior at the re-entrant corners and the concomitant membrane analogy’s applicability. Then, composite cross sections reinforced by angle braces as well as continuous fiber-reinforced wraps with graded shear moduli are investigated with the aim of mitigating the re-entrant corner shear stress singularities, not easily analyzed using the finite-element or finite-difference approaches.