Closed-form Solutions For Stresses Research Articles

Stress and displacement fields around an arbitrary shape tunnel surrounded by a multilayered elastic medium subjected to harmonic waves under plane strain conditions

This study focuses on the propagation of harmonic mechanical waves around both circular and non-circular tunnels. The excavation disturbed zone is explicitly modeled with a multilayered description of the surroundings of the tunnel. The purpose of this paper is thus to establish closed-form solutions for stress and displacement fields for circular and non-circular geometries taking into account the mechanical heterogeneity of the area adjacent to the tunnel. Particularly, the multilayered non-circular shape configuration is a great step not treated previously. The classical approach based on the wave function expansion method is used to determine the expressions of stresses and displacements around circular geometries. To handle non-circular tunnel shapes, the complex variable method based on the Muskhelishvili approach and conformal mapping functions is introduced. Results indicate that the excavation damaged zone strongly modifies the distribution of stress and displacement fields around the tunnel. Geometrical tunnel shapes also affect stress and displacement polar diagrams.

Steady Fluid–Structure Coupling Interface of Circular Membrane under Liquid Weight Loading: Closed-Form Solution for Differential-Integral Equations

In this paper, the problem of fluid–structure interaction of a circular membrane under liquid weight loading is formulated and is solved analytically. The circular membrane is initially flat and works as the bottom of a cylindrical cup or bucket. The initially flat circular membrane will undergo axisymmetric deformation and deflection after a certain amount of liquid is poured into the cylindrical cup. The amount of the liquid poured determines the deformation and deflection of the circular membrane, while in turn, the deformation and deflection of the circular membrane changes the shape and distribution of the liquid poured on the deformed and deflected circular membrane, resulting in the so-called fluid-structure interaction between liquid and membrane. For a given amount of liquid, the fluid-structure interaction will eventually reach a static equilibrium and the fluid-structure coupling interface is steady, resulting in a static problem of axisymmetric deformation and deflection of the circular membrane under the weight of given liquid. The established governing equations for the static problem contain both differential operation and integral operation and the power series method plays an irreplaceable role in solving the differential-integral equations. Finally, the closed-form solutions for stress and deflection are presented and are confirmed to be convergent by the numerical examples conducted.

Closed-form and numerical stress solution-based parameter identification for incompressible hyper-viscoelastic solids subjected to various loading modes

This paper presents closed-form and numerical stress solutions for incompressible hyper-viscoelastic solids considering the following loading modes: uniaxial and equibiaxial tension/compression, pure shear and simple shear. The analytical stress solutions are based on three widely-used hyperelastic material models (Neo-Hookean, Mooney–Rivlin, Ogden model) and assume constant engineering strain rate. On the contrary, the numerical stress solutions are independent of both the hyperelastic law and the loading history. It has been confirmed that these stress solutions can be utilised to identify the hyper-viscoelastic material model parameters. In addition, the closed-form stress solutions may allow verifying different numerical time integration schemes. The accuracy of the constitutive constants extracted for an isoprene rubber has been investigated by comparing the predicted and the measured behaviour. The very good agreement between them shows clearly the benefit of the stress solutions presented.

Assessment of the influence of hydraulic and mechanical anisotropy on the fracture initiation pressure in permeable rocks using a complex potential approach

This work aims at studying the fracture initiation pressure of a horizontal wellbore drilled in an anisotropic poroelastic medium based on an analytical model. Using the complex potential approach, the closed-form solution of stresses in the surrounding formation is deduced as a function of the orientation in respect with stratification and the pressure inside wellbore. The analytical solution is developed for the steady state flow in a permeable formation due to the difference between pressures applied at well wall and the pore pressure at the far-field. Once the stress distribution on the borehole wall is obtained, it is possible, by using a known anisotropic tensile strength criterion for the rock, to infer the explicit expression of the fracture initiation pressure as a function of different parameters, such as the anisotropic poroelastic properties, the anisotropic initial stresses, the bedding dip angle and the anisotropic tensile strength of the formation. The validation of the analytical is conducted by comparing it with the numerical results using the finite element method. Finally, through the parametric study, we highlight the influence of different factors on the fracture initiation pressure.

A Closed-Form Solution for Tunnels with Arbitrary Cross Section Excavated in Elastic Anisotropic Ground

Closed-form solutions are of great interest for quick and accurate evaluation of stresses and displacements around underground excavations. Based on complex variable theory and on the method of conformal mapping, closed-form solutions for stresses and displacements around deep tunnels with arbitrary cross section in a homogeneous, transversely isotropic and linear elastic ground with non-isotropic far-field stress conditions are developed. The results obtained with the proposed analytical solution are compared with those obtained with the numerical code FLAC3D for some usual shapes of tunnel openings.

Closed-form solution for shear lag effects of steel-concrete composite box beams considering shear deformation and slip

Based on the consideration of longitudinal warp caused by shear lag effects on concrete slabs and bottom plates of steel beams, shear deformation of steel beams and interface slip between steel beams and concrete slabs, the governing differential equations and boundary conditions of the steel-concrete composite box beams under lateral loading were derived using energy-variational method. The closed-form solutions for stress, deflection and slip of box beams under lateral loading were obtained, and the comparison of the analytical results and the experimental results for steel-concrete composite box beams under concentrated loading or uniform loading verifies the closed-form solution. The investigation of the parameters of load effects on composite box beams shows that: 1) Slip stiffness has considerable impact on mid-span deflection and end slip when it is comparatively small; the mid-span deflection and end slip decrease significantly with the increase of slip stiffness, but when the slip stiffness reaches a certain value, its impact on mid-span deflection and end slip decreases to be negligible. 2) The shear deformation has certain influence on mid-span deflection, and the larger the load is, the greater the influence is. 3) The impact of shear deformation on end slip can be neglected. 4) The strain of bottom plate of steel beam decreases with the increase of slip stiffness, while the shear lag effect becomes more significant.

Stress analysis of functionally graded discs under mechanical and thermal loads: analytical and numerical solutions

AbstractThis study deals with stress analysis of functionally graded discs subjected to internal pressure and various temperature distributions, such as uniform T, linearly increasing To, and decreasing Ti temperatures in radial directions. For analytical study, the closed-form solutions for stresses and displacements are obtained by using the infinitesimal deformation theory of elasticity. For graded parameters, power law functions are used in analytical and numerical solutions. For numerical study, discs are modeled and analyzed by using a commercial finite element program, ANSYS®. Metal matrix composite, AlSiC, is selected as disc material. Results obtained both analytical and numerical solutions are found very well consistent with each other. The tangential stresses are found higher than the radial stresses at the inner surface for all thermal loads, and they vary from compressive to tensile and from tensile to compressive depending on the functionally graded material (FGM) properties and temperature loads. The radial stresses are found zero at the inner and outer surface and higher at one third of the disc section near the inner surface. They are also found as compressive and tensile stresses depending on the material properties and temperature loads.

Effect of the rotation on a non-homogeneous infinite cylinder of orthotropic material

The present investigation is concerned with a study effect of non-homogeneous on the elastic stresses in rotating orthotropic infinite circular cylinder subjected to certain boundary conditions. Closed form stress solutions are obtained for rotating orthotropic cylinder with constant thickness for three cases: (1) a solid cylinder; (2) cylinder mounted on a circular rigid shaft; and (3) cylinder with a circular hole at the center. Analytical expressions for the components of the displacement and the stress in different cases are obtained. The effect of the rotation and non-homogeneity on the displacement and stress are studied. Numerical results are given and illustrated graphically for each case is considered. The effects rotating and non-homogeneity are discussed. Comparisons are made with the results predicted in the presence and absence of rotation.

Determination of stress and displacement state in the circular annulus steel sheet with Cartesian orthotropy

This paper deals with determination of stress, strain and displacement state of a circular annulus which is made of material with Cartesian orthotropic rheological behaviour. There is a continuous and constant load on the inner and/or outer edge. Stress and strain state in the circular annulus is in the elastic domain. Stress state is determined on the basis of an Airy stress function, where all boundary conditions have to be fulfilled. The Ritz's method was applied to determine unknown parameters of the Airy function. References F. Kosel. The stress and deflection state in the circular annulus with Cartesian orthotropy. Proceedings 14th Canadian Congress of Applied Mechanics , volume 2, pages 493--494. CANCAM, 1993. Pistonesi, C. and Laura, P. A. A., Forced vibrations of a clamped, circular plate of rectangular orthotropy, J. Sound Vibr. , 228 , 1999, 712--716. Tahan, N., PavloviE, M. N. and Kotsovos, M. D., Orthotropic rectangular plates under in-plane loading part 1: closed-form solutions for stresses, Compos. Struct. , 33 , 1995, 35--48. Wu, Z. J. and Wardenier, J., Further investigation on the exact elasticity solution for anisotropic thick rectangular plates, Int. J. Solids Struct. , 35 , 1998, 747--758. Adewale, A. O., Application of the singularity function method to semi-infinite orthotropic rectangular plates on an elastic foundation, Int. J. Mech. Sci. , 43 , 2001, 2261--2279. Bhaskar, K. and Kaushik, B., Simple and exact series solutions for flexure of orthotropic rectangular plates with any combination of clamped and simply supported edges, Compos. Struct. , 63 , 2004, 63--68. S. G. Lekhnitskii. Theory of elasticity of an anisotropic body . Mir Publishers, Moscow, 1981.

Closed-form solutions for the hollow sphere model with Coulomb and Drucker–Prager materials under isotropic loadings

Though the solution to the limit analysis problem of the hollow sphere model—with a von Mises matrix and under spherical symmetry—is well known, it is not available, to our knowledge, for both isotropic loadings (tension and compression) in the case of a Coulomb matrix and partially for a Drucker–Prager matrix. In the present Note, we establish in a unified framework, for this class of materials, closed-form solutions for stress and strain fields in a hollow sphere under external isotropic tension and compression. These analytical results not only give useful reference solutions, but can also be considered as a part of a trial velocity field in the hollow sphere submitted to an arbitrary loading. Comparisons with 3D finite element-based limit analysis approaches and with recent results in the literature are provided. In addition to the established analytical results, we present a rigorous evaluation of a recent Gurson-type macroscopic criterion corresponding to the Drucker–Prager hollow sphere under an arbitrary loading, by means of the previous 3D limit analysis codes. To cite this article: Ph. Thoré et al., C. R. Mecanique 337 (2009).

Fully-Coupled Closed-Form Solution for Stresses in Adhesively Bonded Joints in Slender Reinforced Structures Subjected to Bending and Buckling

An improved method is presented for the analysis of the stresses in adhesively bonded joints in slender reinforced structures subjected to bending and buckling. The method is based on a precise description of the kinematic equations in a reinforced beam cross-section under the assumption of isotropy and linear elasticity for both the adherends and adhesive. Second-order effects due to the member slenderness, as well as transverse deformation in the adhesive layer due to differential curvature between the two adherends, are taken into account. Governing equations for the shear and peel stresses in the adhesive layer are established in the form of two coupled differential equations. Here we take this interaction into account and propose a general solution for these two equations, which yields two fully-coupled closed-form equations for the stresses in the adhesive. These general equations can be adapted to any geometry and loading with appropriate boundary conditions. Exemplary calculations are worked out for a slender reinforced beam subjected to two different loadings. The theoretical results are found to be in good agreement with reference values obtained by means of a Finite Element Method analysis, especially at the adhesive edges where the stresses reach their critical values. It is concluded that the closed-form solution presented in this paper is suitable for a precise prediction of the adhesive stresses in bonded structural members subjected to bending and buckling.

Stresses in Multilayered Ceramics Subjected to Biaxial Flexure Tests

Although standard test methods for biaxial strength measurements of ceramics have been established and the corresponding formulas for relating the biaxial strength to the fracture load have been approved by American Society for Testing and Materials (ASTM) and International Organization for Standardization, respectively, they are limited to the case of monolayered discs. Despite the increasing applications of multilayered ceramics, characterization of their strengths using biaxial flexure tests has been difficult because the analytical description of the relation between the strength and the fracture load for multilayers subjected to biaxial flexure tests is unavailable until recently. Using ring-on-ring tests as an example, the closed-form solutions for stresses in (i) monolayered discs based on ASTM formulas, (ii) bilayered discs based on Roark’s formulas, and (iii) multilayered discs based on Hsueh et al.’s formulas are reviewed in the present study. Finite element results for ring-on-rings tests performed on (i) zirconia monolayered discs, (ii) dental crown materials of porcelain/zirconia bilayered discs, and (iii) solid oxide fuel cells trilayered discs are also presented to validate the closed-form solutions. With Hsueh et al.’s formulas, the biaxial strength of multilayered ceramics can be readily evaluated using biaxial flexure tests.

Updated solution for stresses and displacements in TRISO-coated fuel particles

A closed-form solution for stresses and displacements in TRISO-coated fuel particles of a high temperature reactor has been updated to enhance its application in fuel particle analysis. The modified solution is applied incrementally through irradiation, which allows the material properties and irradiation temperature to change with time. It also removes the restriction in the original solution that Poisson’s ratio in creep for the pyrocarbon layers be set to 0.5. It is presented in a manner that would enable its application to a system of any number of coating layers, not just the three layers of a TRISO-coated particle. The solution has been implemented in the PARFUME fuel performance code, where it has been demonstrated to perform efficiently in particle failure probability determinations.

Effects of intermediate principal stress on rock cavity stability

The linear Mohr-Coulomb and nonlinear Hoek-Brown failure criteria, which neglect the effects of intermediate principal stress, are widely used in soil and rock engineering. However, much experimental data shows that the failure envelope relates to the intermediate principal stress. Employing the failure criterion and the generalized plastic potential function, the stability of rock cavity driven in an isotropic and homogeneous medium was investigated under the condition of plane strain considering the effects of intermediate principal stress. The closed-form solutions for stresses and displacement around a rock cavity were given in the elastic and plastic zones. Based on the closed-form solutions, the intermediate principal stress has an important effect on cavity stability.

On the interaction between a generalized screw dislocation and circular-arc interfacial rigid lines in magnetoelectroelastic solids

The interaction of a generalized screw dislocation, which is located inside either the inclusion or the matrix, with circular-arc interfacial rigid lines under remote antiplane shear stresses, in-plane electric and magnetic loads in linear magnetoelectroelastic materials is dealt with. By using the complex variable method, the general solutions of the physical problem are presented in this paper. The closed-form solutions for stresses, electric displacement fields and magnetic induction fields are derived explicitly for the interface with a finite rigid line. The image forces acting on the generalized screw dislocation are then calculated by using the generalized Peach–Koehler formula. The influence of the length and the location of the rigid line as well as the material mismatch on the image force is evaluated. Numerical computations and discussions show that, when the length or the position of the interfacial rigid line achieves a critical value, the image force on the dislocation changes its direction due to the existence of the rigid line. In addition, the soft inclusion can repel the dislocation in magnetoelectroelastic materials owing to their intrinsic magnetoelectromechanical coupling behavior.

A General Solution for Two-Dimensional Stress Distributions in Thin Films

We present closed-form solutions for stresses in a thin film resulting from a purely dilatational stress-free strain that can vary arbitrarily within the film. The solutions are specific to a two-dimensional thin film on a thick substrate geometry and are presented for both a welded and a perfectly slipping film/substrate interface. Variation of the stress-free strain through the thickness of the film is considered to be either arbitrary or represented by a Fourier integral, and solutions are presented in the form of a Fourier series in the lateral dimension x. The Fourier coefficients can be calculated rapidly using Fast Fourier Transforms. The method is applied to determine the stresses in the film and substrate for three cases: (a) where the stress-free strain is a sinusoidal modulation in x, (b) where the stress-free strain varies only through the thickness, and (c) where a rectangular inclusion is embedded within the film, and the calculated stresses match accurately with the exact solutions for these cases.

Elastic–brittle–plastic analysis of circular openings in Hoek–Brown media

Abstract A closed-form solution is presented in this paper for the prediction of displacements around circular openings in a brittle rock mass subject to a hydrostatic stress field. The rock mass is assumed to be governed by Hoek–Brown yield criterion and a non-associated flow rule is used. For the elastic–brittle–plastic analysis of circular openings in an infinite Hoek–Brown medium, the existing analytical solutions were found to be incorrect. The present closed-form solution is based on a theoretically consistent method and the solution does not require the use of any numerical method. The present closed-form solution was validated by using the finite element method. In the finite element analysis, the infinite boundary was simulated “exactly” by using the newly developed elastic support method. Several cases were analyzed and the present closed-form solutions for stresses and displacements were found to be in an excellent agreement with those obtained by using the finite element method.

Exact solutions for stresses in functionally graded pressure vessels

Closed-form solutions for stresses and displacements in functionally graded cylindrical and spherical vessels subjected to internal pressure alone are obtained using the infinitesimal theory of elasticity. The material stiffness obeying a simple power law is assumed to vary through the wall thickness and Poisson's ratio is assumed constant. Stress distributions depending on an inhomogeneity constant are compared with those of the homogeneous case and presented in the form of graphs. The inhomogeneity constant, which includes continuously varying volume fraction of the constituents, is empirically determined. The values used in this study are arbitrarily chosen to demonstrate the effect of inhomogeneity on stress distribution.

Stress and shape evolution of irregularities in oxide films on elastic–plastic substrates due to thermal cycling and film growth

Initial geometric imperfections in oxide films on elastic–plastic substrates often develop local shape irregularities upon thermal cycling, even though the same system shows no such shape change under similar constant temperature exposure. This paper outlines a mechanics-based explanation of the evolution of such irregularities during thermal cycling, based on reversed plasticity in the material surrounding the irregularity and stress-dependent oxide film growth. Idealized models consisting of spherical or cylindrical elastic shells embedded in an elastic–plastic matrix are used to identify regimes where thermal strains, cyclic reversed plasticity, geometry and oxide growth combine to promote rapid evolution of irregularities. The entirely closed-form solutions for stress and displacements during temperature cycling allow for quick evaluation of the role of various parameters, including: the temperature amplitude, size of the irregularity, oxide thickness, yield stress of the substrate and stress-dependent oxide growth strains. The usefulness of the solutions is illustrated by comparing two oxide film-growth scenarios with preliminary experimental results on a representative thermal barrier system that exhibits significant feature growth. The comparison demonstrates that oxide growth enhanced by tensile stresses created during reversed plastic deformation may be a significant factor in accounting for the large geometry changes observed in experiments.

Elastodynamic Analysis of Antiplane Anisotropic Interface Cracks

The transient elastodynamic full-field response and the dynamic stress intensity factor of a semi-infinite interface crack lying between dissimilar anisotropic media subjected to a dynamic body force are investigated. At t = 0, a concentrated antiplane dynamic point loading is suddenly applied at Medium 1. The total wave field is due to the effect of this point loading and the scattering of the incident wave by the interface crack. We introduce a linear coordinate transformation that can transform the anisotropic interface crack problem to the isotropic case. The relationship between the problem of anisotropic material and the corresponding isotropic problem is established for shear stresses and displacement in a Cartesian coordinate system. Exact transient closed-form solutions for stresses and stress intensity factors are obtained. Numerical results for the time history of stresses and stress intensity factors during the transient process are discussed in detail.