Traction-free Boundary Conditions Research Articles

Micromechanics of stress transfer through the interphase in fiber-reinforced composites

A micromechanical model to predict the interphasial/interfacial stress transfer in a three-phase fiber-reinforced composite is presented. The axisymmetric system consists of a fiber embedded in a compliant matrix having an interphase between them. Each constituent of the composite is regarded as a linear elastic continuum. The matrix is treated as an isotropic material while the fiber and interphase are considered as a transversely isotropic material. Traction-free boundary conditions are strictly enforced. It is assumed that the interfaces are perfect and strong. A pair of uncoupled governing partial differential equations is obtained in terms of unknown displacements. Furthermore, assuming that the Eigenvalues exist for this system of equations, Eigenfunction expansion method is employed to derive an exact solution in terms of the Bessel functions. Analytical solutions are obtained for free boundary conditions at the external surface of the matrix cylinder to model a single fiber pull-out problem, and for fixed boundary conditions to approximately model a hexagonal array of fibers in the matrix material. This formulation provides an analytical framework for the analysis of interphasial and interfacial stresses as well as displacements in the entire 3D axisymmetric system. Finite element (FE) analysis was also performed to simulate stress transfer from the fiber to the matrix through the interphase. Analytically obtained stress fields are verified with FE results. Shear and radial interphasial stresses provide insight into the design of engineered interfaces/interphases.

A new higher-order shear deformation theory for static, buckling and free vibration analysis of functionally graded sandwich beams

This paper presents a new higher-order shear deformation theory for static, buckling and free vibration analysis of functionally graded sandwich beams. In this theory, the axial displacement accounts for a third-order and inverse trigonometric distribution, and the transverse shear stress satisfies the traction-free boundary conditions on the top and bottom surfaces of the beams. Governing equations of motion are derived from the Hamilton’s principle for sandwich beams with homogeneous hardcore and softcore. Navier-type solution for simply-supported beams is developed to solve the problem. Numerical results are obtained to investigate effects of the power-law index, span-to-height ratio and thickness ratio of layers on the displacements, stresses, critical buckling load and frequencies.

Lamb Waves in Plates Subjected to Uniaxial Stresses

Authors : Munawwar Mohabuth, Andrei Kotousov, Ching-Tai Ng Abstract : On the basis of the finite deformation theory, the effect of homogeneous stress on the propagation of Lamb waves in an initially isotropic hyperelastic plate is analysed. The equations governing the propagation of small amplitude waves in the prestressed plate are derived using the theory of small deformations superimposed on large deformations. By enforcing traction free boundary conditions at the upper and lower surfaces of the plate, acoustoelastic dispersion equations for Lamb wave propagation are obtained, which are solved numerically. Results are given for an aluminum plate subjected to a range of applied stresses.

An Efficient Semi-Analytical Method to Compute Displacements and Stresses in an Elastic Half-Space with a Hemispherical Pit

AbstractThis paper presents an efficient method to calculate the displacement and stress fields in an isotropic elastic half-space having a hemispherical pit and being subject to gravity. The method is semi-analytical and takes advantage of the axisymmetry of the problem. The Boussinesq potentials are used to obtain an analytical solution in series form, which satisfies the equilibrium equations of elastostatics, traction-free boundary conditions on the infinite plane surface and decaying conditions at infinity. The boundary conditions on the free surface of the pit are then imposed numerically, by minimising a quadratic functional of surface elastic energy. The minimisation yields a symmetric and positive definite linear system of equations for the coefficients of the series, whose particular block structure allows its solution in an efficient and robust way. The convergence of the series is verified and the obtained semi-analytical solution is then evaluated, providing numerical results. The method is validated by comparing the semi-analytical solution with the numerical results obtained using a commercial finite element software.

Stress triaxiality and Lode angle along surfaces of elastoplastic structures

Expressions for the stress triaxiality and the Lode angle along surfaces of elastoplastic structures are established in case of monotonic loading. The stress triaxiality is shown to be governed by the accumulated plastic strain when traction free boundary condition is considered. The exact expressions obtained are generalized to any loading thanks to the proposal of a multiaxiality rule or heuristics whose two parameters are determined from elastic computations of the structure considered: a first one with the elastic properties of the material, a second one quasi-incompressible. The multiaxiality rule proposed can then deal with both plane strain and plane stress conditions. The stress triaxiality at the surface is shown related to the Lode angle. The corresponding expressions are validated on different structures and loadings. Finally, two applications are presented: the enhancement of energetic methods for plasticity post-processing and the enhancement of homogenization localization laws.

High-order finite difference modeling of tsunami generation in a compressible ocean from offshore earthquakes

To study the full seismic, ocean acoustic, and tsunami wavefields generated by subduction zone earthquakes, we have developed a provably stable and accurate finite difference method that couples an elastic solid to a compressible fluid subject to gravitational restoring forces. We introduce a linearized dynamic traction-free boundary condition for the moving sea surface that is valid for small amplitude perturbations about an ocean initially in hydrostatic balance. We derive an energy balance for the continuous problem and then use high-order summation-by-parts finite difference operators and weak enforcement of boundary conditions to derive an equivalent discrete energy balance. The discrete energy balance is used to prove stability of the numerical scheme, and stability and accuracy are verified through convergence tests. The method is applied to study tsunami generation by time-dependent rupture on a thrust fault in an elastic solid beneath a compressible ocean. We compare the sea surface evolution in our model to that predicted by the standard tsunami modeling procedure, which assumes seafloor uplift occurs instantaneously and neglects compressibility of the ocean. We find that the leading shoreward-traveling tsunami wave in our model has a noticeably smaller amplitude than that predicted by the standard approach.

Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory

This paper proposes a new higher-order shear deformation theory for buckling and free vibration analysis of isotropic and functionally graded (FG) sandwich beams. The present theory accounts a new hyperbolic distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion are derived from Lagrange's equations. Analytical solutions are presented for the isotropic and FG sandwich beams with various boundary conditions. Numerical results for natural frequencies and critical buckling loads obtained using the present theory are compared with those obtained using the higher and first-order shear deformation beam theories. Effects of the boundary conditions, power-law index, span-to-depth ratio and skin-core-skin thickness ratios on the critical buckling loads and natural frequencies of the FG beams are discussed.

Theoretical modeling of low-attenuation lamb wave modes generation in three-layer adhesive joints using angle beam transducer

IN THIS PAPER, AT FIRST THE ATTENUATION OF LAMB WAVES IN THREE-LAYER ADHESIVE JOINTS, INCLUDING TWO ELASTIC PLATES BONDED TOGETHER BY A VISCOELASTIC ADHESIVE LAYER, IS INVESTIGATED USING GLOBAL MATRIX METHOD AND THEN SUITABLE INCIDENCE ANGLE IS THEORETICALLY CALCULATED TO GENERATE LOW-ATTENUATION LAMB WAVES USING ANGLE BEAM TRANSDUCER. THEORETICAL BOUNDARY VALUE PROBLEM IN THREE-LAYER ADHESIVE JOINTS WITH PERFECT BOND AND TRACTION-FREE BOUNDARY CONDITIONS ON THEIR OUTER SURFACES IS SOLVED TO FIND A COMBINATION OF FREQUENCIES AND MODES WITH LOWEST ATTENUATION. CHARACTERISTIC EQUATION IS DERIVED BY APPLYING CONTINUITY AND BOUNDARY CONDITIONS IN THREE-LAYER JOINTS USING GLOBAL MATRIX METHOD. PHASE VELOCITY DISPERSION CURVES AND ATTENUATION INTENSITY PLOT IN HIGH AND LOW FREQUENCIES ARE OBTAINED WITH NUMERICAL SOLUTION OF THIS EQUATION BY A COMPUTER CODE FOR A THREE-LAYER JOINT, INCLUDING AN ALUMINUM REPAIR PATCH BONDED TO THE AIRCRAFT ALUMINUM SKIN BY A LAYER OF VISCOELASTIC EPOXY ADHESIVE. TO VALIDATE THE NUMERICAL SOLUTION RESULTS OF CHARACTERISTIC EQUATION, WAVE STRUCTURE CURVES ARE PLOTTED FOR A SPECIAL MODE IN TWO DIFFERENT FREQUENCIES IN THE ADHESIVE JOINT. ALSO, TRANSDUCER INCIDENCE ANGLE IS CALCULATED IN TERMS OF FREQUENCY FOR DIFFERENT MODES USING THEORETICAL METHOD TO GENERATE LAMB WAVE MODES WITH LOW ATTENUATION LEVEL BY ANGLE BEAM TRANSDUCER. THESE MODES ARE RECOGNIZABLE BY TRANSDUCERS IN INSPECTIONS WITH LAMB WAVES BECAUSE OF LOW ATTENUATION LEVEL.

Three-dimensional stress analyses around cutouts in laminated composites by special hybrid finite elements

The distribution of stresses around some cutouts in laminated composites is analyzed by using the combination of two kinds of special hybrid stress multilayer finite elements. One kind of the three-dimensional special finite element contains a traction-free cylindrical surface and the other contains a traction-free planar surface. Both of these special finite elements are derived based on a modified complementary energy principle. The expressions of six stress components in each kind of element are derived, such that the homogeneous equilibrium equations in each layer, the continuity of the transverse stresses on the interlayer surface, and the traction-free boundary conditions over the designated surfaces are satisfied a prior, while the interelement reciprocity traction is satisfied a posterior in a variational sense. All components of displacement are included since bending–stretching coupling may occur. Examples have indicated that the combination of these special multilayer finite elements is far superior in predicting the distribution of circumferential stresses and transverse stresses for laminates with holes and notches when only a very coarse element mesh is used near the free edge of the cutouts.

A refined higher-order shear deformation theory for bending, vibration and buckling analysis of functionally graded sandwich plates

A refined higher-order shear deformation theory for bending, vibration and buckling analysis of functionally graded sandwich plates is presented in this paper. It contains only four unknowns, accounts for a hyperbolic distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion are derived from Hamilton\'s principle. The Navier-type and finite element solutions are derived for plate with simply-supported and various boundary conditions, respectively. Numerical examples are presented for functionally graded sandwich plates with homogeneous hardcore and softcore to verify the validity of the developed theory. It is observed that the present theory with four unknowns predicts the response accurately and efficiently.

Interlaminar stress analysis of piezo-bonded composite laminates using the extended Kantorovich method

An iterative method has been applied to analyze the free edge interlaminar stresses of piezo-bonded composite laminates. Electric fields are applied to two symmetrically bonded piezo actuators that can generate induced strain, resulting in pure extension of the whole structure. The stresses, which satisfy the traction-free and free edge boundary conditions, are obtained by taking the principle of complementary virtual work, and conducting the extended Kantorovich method. In order to obtain accurate interlaminar stress distributions, static and kinematic continuity conditions are applied at the interfaces between plies through iterations. The stress components were obtained under the plane strain assumption as well as antiplane shear assumption. The results were compared with those obtained by the finite element method, to demonstrate the validity of the proposed method. Through the iteration processes, the interlaminar stresses converged well, and predicted maximum peak stress at the interface of piezo and composite layers. The present method provides accurate stresses distribution near the free edges, of piezo-bonded composite laminates.

Free-edge stress analysis of functionally graded material layered biocomposite laminates.

A stress function based theory is proposed to obtain free-edge stress distributions for three-dimensional, orthotropic, linearly elastic rectangular biocomposite laminates with surface-bonded functionally graded materials (FGM). The assumed stress fields automatically satisfy the pointwise equilibrium equation, as well as traction-free and free edge boundary conditions. The complementary virtual work principle, followed by the general eigenvalue solution procedure, is used to obtain 3-D free edge stress states. A typical stacking sequence of composite laminate is used as numerical investigation with surface bonded FGMs. It is shown that with proper exponential factor of FGMs, the interlaminar stresses at the FGM layer interface can be reduced significantly, in return to prevent debonding of FGM layers. This approach can be useful in the design of functionally graded material layered biocomposite structures.

Rayleigh waves with impedance boundary conditions in incompressible anisotropic half-spaces

In this paper, the propagation of Rayleigh waves in an incompressible elastic half-space with impedance boundary conditions is investigated. The half-space is assumed to be orthotropic and monoclinic with the symmetry plane x3=0. The main aim of the paper is to derive explicit secular equations of the wave. For the orthotropic case, the secular equation is obtained by employing the traditional approach. It is an irrational equation. For the monoclinic case, the method of polarization vector is used for deriving the secular equation. This is an algebraic equation of eighth-order. When the impedance parameters vanish, the equations obtained coincide with the corresponding secular equations of Rayleigh waves with traction-free boundary conditions.

Investigation of Lamb waves attenuation in elastic–viscoelastic three-layer adhesive joints in low and high frequencies: Theoretical modeling

In this paper the attenuation of Lamb waves in three-layer joints, including two elastic plates bonded together by a viscoelastic adhesive layer, is investigated in high and low frequencies using global matrix method. Theoretical boundary value problem in three-layer adhesive joints with perfect bond and traction-free boundary conditions on their outer surfaces is solved to find a combination of frequencies and modes with lowest attenuation. Characteristic equation is derived by applying continuity and boundary conditions in three-layer joints using global matrix method. Attenuation and phase velocity dispersion curves in high and low frequencies are obtained with numerical solution of this equation by a computer code for a three-layer joint, including an aluminum repair patch bonded to the aircraft aluminum skin by a layer of viscoelastic epoxy adhesive. To validate the numerical solution results of characteristic equation, wave structure curves are plotted for a special mode in two different frequencies in the adhesive joint. Also, attenuated displacement amplitude curves in a round-trip of Lamb wave along the adhesive joint are plotted to investigate the reduction of the signal amplitude generated by a single transducer. The purpose of present paper is to find a combination of frequencies and modes with minimum attenuation in high and low frequencies. These frequencies and modes are recognizable by transducers in inspections with Lamb waves because of low attenuation level.

Dynamic Instability of Laminated Composite and Sandwich Plates Using a New Inverse Hyperbolic Zigzag Theory

AbstractThe present study deals with the dynamic stability behavior of laminated composite and sandwich plates subjected to in-plane static and periodic compressive loads based on a recently developed inverse hyperbolic zigzag theory by the authors. The present model satisfies the traction-free boundary conditions on the surfaces of the plate and interlaminar continuity conditions at the layer interfaces, thus obviating the need for a shear correction factor. An efficient C0 continuous isoparametric serendipity element with seven field variables is employed for the usual discretization of the plate structure. The effect of span-thickness ratio, modular ratio, boundary conditions, static load factor, and thickness ratios are examined by solving a variety of numerical problems on laminated composite and sandwich structures. The principal instability regions of plate structures are obtained using Bolotin’s approach and are represented either in the nondimensional load amplitude-excitation frequency plane or ...

Isogeometric analysis of laminated composite plates based on a four-variable refined plate theory

In this paper, a simple and effective formulation based on isogeometric approach (IGA) and a four variable refined plate theory (RPT) is proposed to investigate the behavior of laminated composite plates. RPT model satisfies the traction-free boundary conditions at plate surfaces and describes the non-linear distribution of shear stresses without requiring shear correction factor (SCF). IGA utilizes basis functions, namely B-splines or non-uniform rational B-splines (NURBS), which reveals easily the smoothness of any arbitrary order. It hence handles easily the C1 requirement of the RPT model. Approximating the displacement field with four degrees of freedom per each node, the present method retains the computational efficiency while ensuring the reasonable accuracy in solution.

A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates

A new inverse trigonometric shear deformation theory is proposed for the static, buckling and free vibration analyses of isotropic and functionally graded (FG) sandwich plates. It accounts for a inverse trigonometric distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion obtained here are solved for three types of FG plates: FG plates, sandwich plates with FG core and sandwich plates with FG faces. Closed-form solutions are obtained to predict the deflections, stresses, critical buckling loads and natural frequencies of simply supported plates. A good agreement between the obtained predictions and the available solutions of existing shear deformation theories is found to demonstrate the accuracy of the proposed theory.

Rayleigh waves with impedance boundary conditions in anisotropic solids

The paper is concerned with the propagation of Rayleigh waves in an elastic half-space with impedance boundary conditions. The half-space is assumed to be orthotropic and monoclinic with the symmetry plane x3=0. The main aim of the paper is to derive explicit secular equations of the wave. For the orthotropic case, the secular equation is obtained by employing the traditional approach. It is an irrational equation. From this equation, a new version of the secular equation for isotropic materials is derived. For the monoclinic case, the method of polarization vector is used for deriving the secular equation and it is an algebraic equation of eighth-order. When the impedance parameters vanish, this equation coincides with the secular equation of Rayleigh waves with traction-free boundary conditions.

Accurate prediction of free-edge and electromechanical coupling effects in cross-ply piezoelectric laminates

3D hybrid analyses on the interlaminar stresses near the free edges of piezoelectric laminated plates are presented on the basis of three-dimensional piezoelasticity. The state space equations for cross-ply piezoelectric laminates subjected to a uniform axial extension are obtained by considering all the independent elastic and piezoelectric constants. With the application of the transfer matrix and recursive solution approach, the equations satisfy the traction-free and open-circuit boundary conditions at free edges and the continuity conditions across the interfaces between material layers. Three-dimensional exact solution is sought and validated by comparing the present analytical results with those from existing approximate and finite element models. The singularities of interlaminar stresses near free edges are observed and the significant electromechanical influence on the free-edge effect is found.

Elastic lattice modelling of seismic waves including a free surface

Elastic lattice methods (ELMs) have been shown to accurately model seismic wave propagation in a heterogeneous medium. These methods represent an elastic solid as a series of interconnected springs arranged on a lattice and recover a continuum wave equation in the long wavelength limit. However, in the case of a regular lattice, the recovery of the continuum equation depends on the symmetry of the lattice. By removing particles above a free surface this symmetry is broken. Therefore, this free surface implementation leads to errors when compared with a traction free boundary condition. The error between a traction free boundary condition and the ELMs grows as the Poisson׳s ratio deviates from 0.25. By modifying the interaction constants with a scalar, the error can be reduced while keeping the flexibility of the nearest neighbour interaction rule. We present results of simulations where modified spring constants reduce the misfit with a traction free boundary solution and hence increase the accuracy of the elastic lattice method solution on the free surface.