Three-dimensional Anisotropic Elasticity Research Articles

Review of elasto-static models for three-dimensional analysis of thick-walled anisotropic tubes

Most shell or beam models of anisotropic tubes under bending have no validity for thick-walled structures. As a result, the need to develop three-dimensional formulations which allow a change in the stress, strain and displacement distributions across the radial component arises. Basic formulations on three-dimensional anisotropic elasticity were made either stress- or displacement-based by Lekhnitskii or Stroh on plates. Lekhnitskii also was the first to expand these analytical formulations to tubes under various loading conditions. This paper presents a review of the stress and strain analysis of tube models using three-dimensional anisotropic elasticity. The focus lies on layered structures, like fiber-reinforced plastics, under various bending loads, although the basic formulations and models regarding axisymmetric loads are briefly discussed. One section is also dedicated to the determination of an equivalent bending stiffness of tubes.

An analytical solution for dynamics of cyclic thermomechanically loaded multi-layered filament-wound composite pipes in hygrothermal environment

In this study, an analytical solution is provided to present the stress and strain analyses of the multi-layered filament-wound composite (MLF-WC) pipes subjected to the cyclic internal pressure and cyclic thermal loading in hygrothermal environment. The MLF-WC pipe is made of anisotropic, homogeneous and linear elastic materials. It is assumed that the material properties are temperature-independent. Also, applied cyclic loadings are independent of the tangential and axial coordinates. Using the three-dimensional anisotropic elasticity, time-dependent analytical expressions are provided for the stresses, strains and displacements. The accuracy of present analytical approach and numerical results is verified by comparison with those given in literature. It is remarked that the developed analytical solution can be used for the dynamic analysis of functionally graded pipes and pressure vessels under the cyclic loadings. Moreover, through the numerical results, it is found that stacking sequence has a considerable effect on the distributions of shear strain and stress. Furthermore, for the MLF-WC pipe under simultaneous cyclic internal pressure and cyclic thermal loading, the mean amplitudes of the cyclic loadings have a significant effect on the sign of axial strain.

Generalised plane strain embedded in three-dimensional anisotropic elasticity

ABSTRACT The theory of anisotropic generalised plane strain of line forces and dislocation lines is embedded in the three-dimensional elasticity of point forces and dislocation densities. Embedding in real space is achieved by slicing in reciprocal space using the projection-slice theorem.

Gradient design of tubular of the fiber-reinforced composites pressure vessel for diminishing of the edge effect

In the composite pressure vessel composed of fiber-reinforced composites tubular and metal head, the discontinuity of the structure at the joint will lead to increase of local stress and the inconsistent of internal and external deformation, which is called edge effect. An analytical procedure is developed to predict the behavior of fiber-reinforced composite circular tubes under axial unilateral loading. The stress and strain fields are solved by using the three-dimensional anisotropic elasticity theory. The results show that gradient structure with axial modulus varying along radius has obvious effect on reducing deformation difference and edge effect of pressure vessel, and the improvement of edge effect increases with the increase of gradient parameter. The numerical simulation is carried out by using finite element method, and the results are in good agreement with the theoretical calculations. The mathematical model established in this paper and the analysis results have some significance for improving the performance of fiber composite pressure vessels.

Higher order topological derivatives for three-dimensional anisotropic elasticity

This article concerns an extension of the topological derivative concept for 3D elasticity problems involving elastic inhomogeneities, whereby an objective function 𝕁 is expanded in powers of the characteristic size a of a single small inhomogeneity. The O (a 6 ) approximation of 𝕁 is derived and justified for an inhomogeneity of given location, shape and elastic properties embedded in a 3D solid of arbitrary shape and elastic properties; the background and the inhomogeneity materials may both be anisotropic. The generalization to multiple small inhomogeneities is concisely described. Computational issues, and examples of objective functions commonly used in solid mechanics, are discussed.

Pull-in experiments on electrostatically actuated microfabricated meso scale beams

We report on the results of characterization and modeling of electrostatically actuated meso scale beams. The beams with clamped ends were 5000μm long, 150μm thick and 10, 12 and 15μm wide and were operated by a parallel plate electrode located at the distance of 20μm from the beam. The devices were fabricated using deep reactive ion etching (DRIE) from a silicon on insulator (SOI) wafer with 150μm thick device layer. The beams were operated in ambient air conditions and the voltage–displacement dependence was built using image processing. In addition, pull-in behavior of the beams was modeled using several approaches, starting from geometrically nonlinear shallow Euler–Bernoulli beam model and up to fully coupled nonlinear large deflection three-dimensional anisotropic elasticity finite elements simulations. Excellent agreement between the results provided by the models and the experimental data was observed. The results of the work demonstrate an ability to achieve large displacements in simple meso scale beam structures and provide a reliable experimental reference for double clamped beams actuated by a parallel-plate electrode.

Efficient computation of the Green's function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis

Abstract An alternative scheme to compute the Green's function and its derivatives for three dimensional generally anisotropic elastic solids is presented in this paper. These items are essential in the formulation of the boundary element method (BEM); their evaluation has remained a subject of interest because of the mathematical complexity. The Green's function considered here is the one introduced by Ting and Lee [Q. J. Mech. Appl. Math. 1997; 50: 407–26] which is of real-variable, explicit form expressed in terms of Stroh's eigenvalues. It has received attention in BEM only quite recently. By taking advantage of the periodic nature of the spherical angles when it is expressed in the spherical coordinate system, it is proposed that this Green's function be represented by a double Fourier series. The Fourier coefficients are determined numerically only once for a given anisotropic material; this is independent of the number of field points in the BEM analysis. Derivatives of the Green's function can be performed by direct spatial differentiation of the Fourier series. The resulting formulations are more concise and simpler than those derived analytically in closed form in previous studies. Numerical examples are presented to demonstrate the veracity and superior efficiency of the scheme, particularly when the number of field points is very large, as is typically the case when analyzing practical three dimensional engineering problems.

Effect of secondary orientation on notch-tip plasticity in superalloy single crystals

Numerical and experimental evolutions of slip fields in notched Ni-Base Single Crystal superalloy tensile specimens are presented as a function of secondary crystallographic orientation. The numerical predictions based on three-dimensional anisotropic elasticity and crystal plasticity are compared with experimental observations. The results illustrate the strong dependence of the slip patterns and the plastic zone size and shape on the secondary orientation of notches, which can have important consequences on crack initiation. Specific orientations or non-symmetric notch geometries lead to non-symmetric patterns on both sides of the sample. The computations show that strongly different plastic zones are expected in the core of the sample and at free surfaces. The ability of the anisotropic elastic model to anticipate the plastic domains, based on identifying dominant slip systems, is confirmed by the crystal plasticity computations, at low load levels. An important observation is that kink shear banding is a real deformation mode operating at crack tips and notches in high strength nickel-based single crystal superalloys for specific orientations.

Dynamic analysis of multi-layered filament-wound composite pipes subjected to cyclic internal pressure and cyclic temperature

Based on the three-dimensional anisotropic elasticity, the stress analysis of multi-layered filament-wound composite pipes subjected to cyclic internal pressure and temperature loading is conducted in this article. The time-dependent stress, strain and deformation distributions are numerically obtained by the use of the finite difference technique. The pressure and temperature are considered to be symmetrical about the axis of the cylinder and independent of the axial coordinate. Each layer of the pipes is made of a homogeneous, anisotropic and linearly elastic material and it is assumed that the material properties do not change with increasing the temperature. The shear extension coupling is also considered because of lay-up angles. Numerical results obtained from the present model are compared with other published results and good agreement has been achieved.

Analysis of multi-layered filament-wound composite pipes under combined internal pressure and thermomechanical loading with thermal variations

The composite pipes manufactured by filament winding technology have anisotropic behavior owing to different reinforced ply angles. Composite pipes can be exposed to the thermomechanical loading due to hot fluid that flows into them. In this paper, based on the three-dimensional anisotropic elasticity, an exact elastic solution for thermal stresses and deformations of the pipes under internal pressure and a temperature gradient has been studied. Giving heat convection conditions the variation of temperature field within the pipe is obtained by solving the conduction equation at the wall. The influence of temperature field in the governing equations of thermoelasticity has been considered via a constitutive law. The shear extension coupling is also considered because of lay-up angles. Stress, strain and deformation distributions for different angle-ply pipe designs are investigated using the present theory.

Evaluation of Galerkin Singular Integrals for Anisotropic Elasticity: Displacement Equation

Algorithms for the direct evaluation of singular Galerkin boundary integrals for three-dimensional anisotropic elasticity are presented. The integral of the traction kernel is defined as a boundary limit, and (partial) analytic evaluation is employed to compute the limit. The spherical angle components of the Green's function and its derivatives are not known in closed form, and thus the analytic integration requires a splitting of the kernel, into 'singular' and 'non-singular' terms. For the coincident singular integral, a single analytic evaluation suffices to isolate the potentially divergent term, and to show that this term self-cancels. The implementation for a linear element is considered in detail, and the extension to higher order curved interpolation is also discussed. Results from test calculations establish that the algorithms are successful.

Evaluation of Galerkin Singular Integrals for Anisotropic Elasticity: Displacement Equation

Algorithms for the direct evaluation of singular Galerkin boundary integrals for three-dimensional anisotropic elasticity are presented. The integral of the traction kernel is defined as a boundary limit, and (partial) analytic evaluation is employed to compute the limit. The spherical angle components of the Green's function and its derivatives are not known in closed form, and thus the analytic integration requires a splitting of the kernel, into 'singular' and 'non-singular' terms. For the coincident singular integral, a single analytic evaluation suffices to isolate the potentially divergent term, and to show that this term self-cancels. The implementation for a linear element is considered in detail, and the extension to higher order curved interpolation is also discussed. Results from test calculations establish that the algorithms are successful.

厚肉フィラメントワインディングパイプの構造解析

The stress distribution of anisotropic multilayer filament-wound composite pipes was analyzed. Based on the three-dimensional anisotropic elasticity and the classical laminated-plate theory, exact elastic solutions of stress and deformation were presented for the filament-wound pipes subjected either to internal pressure, axial force or torsion. The shear coupling effect is considered in the analysis. The influence of winding angle and winding sequence on stress distribution in pipes was investigated. The ratio of hoop-to-axial stress is 2:1 in orthotropic pipes; however, the ratio proved to be not 2:1 in anisotropic pipes due to the coupling effect. The stress and deformation of filament-wound pipes strongly depend on the winding angle and the winding sequence.

Analysis of axisymmetrically-loaded filament wound composite cylindrical shells

Exact elasticity solutions for laminated composite cylindrical shells under axisymmetric loading are presented. The material of the shell is assumed to be cylindrically anisotropic. Based on the three-dimensional anisotropic elasticity, three coupled governing partial differential equations in terms of displacements are developed. The general expressions for the stresses and displacements by using the Frobenius method in the radial coordinate are discussed. The closed form solutions based on Flügge thin shell theory are also derived for comparison purposes. Detailed stress and displacement distributions for long laminated shells with layups [90/0/45/−45] s and [45/−45] s under periodic band loads are investigated.

On an elliptic crack embedded in an anisotropic material

A generalized Stroh’s formalism for three-dimensional anisotropic elasticity is applied to study the elliptic crack problem. The traction on the crack plane is expressed in a simple one-dimensional integral. The integrand contains one of the Barnett–Lothe tensors which can be calculated directly from the elastic constants. It is shown that with respect to a local coordinate system, the traction on the crack plane and relative crack face displacement in the vicinity of the crack edge have the same form as their two-dimensional counterparts. A systematic method to derive the stress intensity factors for polynomial loadings is discussed. Explicit results are given for constant, linear and quadratic loadings.

Generalization of the Stroh Formalism to 3-Dimensional Anisotropic Elasticity

A method of analysis for three-dimensional anisotropic elasticity is developed, based on Stroh's two-dimensional formalism. By use of the present method a three-dimensional solution can be constructed by the corresponding two-dimensional solution. The method is applied to solve several problems concerning a point force or a dislocation loop. Existing results for these problems are recovered in a unified manner and additional new results are obtained.

Stress functions and internal stresses in linear three-dimensional anisotropic elasticity

A stress function method is presented in order to give a general solution of the incompatibility problem of three-dimensional linear anisotropic elasticity theory. A relation between the internal stress tensor in terms of derivatives of a sixth-order stress function tensor will be derived. Using this formulation the so far open problem of the representation of the second-order stress function tensor by the fourth-order stress function tensor in anisotropic elasticity theory is solved in general.

The Contact Behavior Between Laminated Composite Plates and Rigid Spheres

The contact behavior between a composite laminate and a rigid sphere is discussed in this paper using a method derived from the three-dimensional anisotropic elasticity theory. Response of the plate and the force-indentation ( F − α ) relationship were obtained by constructing an exact Green’s function of the laminate. Agreement between the prediction and the experimental data reported by others is excellent before laminates are damaged. The F - α relationship is approximately proportional to the out-of-plane Young’s modulus and is influenced by the plate thickness. The effects of changing the in-place Young’s moduli, all the shear moduli, stacking sequences and span of the laminates are insignificant unless identation is large. Further, the modified Hertz law underestimates the contact force for the same indentation.