Traction-free Conditions Research Articles

Investigating the influence of stress on UV-induced degradation in cellulose acetate: A comprehensive experimental characterization

Renewable and degradable polymers have emerged in everyday applications ranging from mundane eating utensils to high-tech medical devices. However, the current literature lacks a thorough experimental characterization of the mechanical behavior change due to degradation. In this work, we characterize the microscopic chemical changes due to photo-degradation and resulting stress effects on the mechanical behavior of cellulose acetate, a renewable and degradable polymer that is used in various consumer products. Specifically, we photo-degrade this polymer under (i) traction-free conditions and (ii) under applied stress. A key finding of this work is that upon photo-degradation, this material undergoes chain scission, which affects its mechanical properties and may be further affected by the applied stress.

A novel hybrid boundary element for polygonal holes with rounded corners in two-dimensional anisotropic elastic solids

A novel hybrid boundary element is developed for polygonal holes in finite anisotropic elastic plates based on two different special fundamental solutions for holes. Since these special fundamental solutions satisfy traction-free condition along the hole's boundary, there is no mesh required on the boundary of polygonal holes. Various types of polygonal holes with rounded corners, such as triangles, rhombuses, ovals, pentagons, are considered by adding proper perturbation to an elliptical hole. The developed hybrid element is a mixture of two special boundary elements: one is based on the special fundamental solution derived through nonconformal mapping and the other is based on the solution derived through perturbation technique with conformal mapping. The special boundary element methods are combined through submodeling technique. First, the global model is solved using the perturbation solution. Then, using the displacements obtained from global model, an auxiliary submodel is set up and the results are evaluated with the nonconformal solution. The present method is compared and validated with conventional boundary element method and finite element method. The effect of hole curvature, material anisotropy, and loading condition on the stress distribution around the hole is presented.

Interaction between a crack and a circular liquid inclusion

We study the interaction between a radial matrix crack and a circular compressible liquid inclusion embedded in an elastic matrix under uniform remote in-plane normal stresses. The crack is simulated by a continuous distribution of climbing edge dislocations. A Cauchy-type singular integral equation is derived by imposing the traction-free condition on the crack surfaces. The singular integral equation is solved numerically using the Gauss-Chebyshev integration formula, resulting in the mode I stress intensity factors at the two crack tips and the internal uniform hydrostatic tension within the liquid inclusion. Our detailed numerical results indicate that the crack will propagate toward or away from the liquid inclusion depending on the inclusion-crack distance, the crack length, the compressibility of the liquid inclusion, the Poisson’s ratio of the matrix and the external loading. This fact implies that the liquid inclusion exhibits softening as well as stiffening effects.

Analytical analysis of concentration distribution and diffusion-induced stress of finite-length cylindrical electrode under galvanostatic operation

A cylindrical electrode is approximated as a long cylinder in most of existing models in which a generalized plane strain condition/plane strain is used. Based on the theory of elasticity, analytical expressions are derived for concentration distribution and stress component in a finite-length cylindrical electrode under galvanostatic operation. Using the superposition theorem, the Li-concentration is a sum of the concentration due to axial diffusion and the concentration due to lateral diffusion, and the separation of variable method is used to solve diffusion equations. By using the Boussinesq-Papkovich function, the generalized stress component distribution of a linearly combined product of the exponential-type Fourier-Bessel series is derived. The spatiotemporal distribution of concentration and diffusion-induced stresses are calculated in a cylindrical electrode with traction-free condition. The results are compared with the simulation results from a finite element software. For the concentration distribution, the numerical result and simulation result are almost the same. For the stress component, no significant difference exists between the two results, the largest relative difference for radial stress in the center is found to be about 4% and state of charge (SOC) = 17.9%. The radial stress decreases with radial position increasing, and decreases to zero at the surface, which is consistent with the results under the boundary condition. The hoop stress is tensile stress around the center of electrode, and becomes a compressive stress near the surface. Owing to the fact that the tensile hoop stress is attributed to the crack initiation, this implies that when plastic deformation is negligible, cracks first form in the center. The stress components with different length-to-radius ratios are calculated. It is found that the stress caused by lateral diffusion increases with length-to-radius ratio increasing, while the stress induced by axial diffusion decreases with length-to-radius ratio increasing. This is because the lateral diffusion has a greater influence on Li-concentration distribution in a cylinder electrode with length-to-radius ratio increasing.

Evaluation of static and dynamic responses considering thickness stretching effect for layered composite and sandwich arches using exponential shear and normal deformation theory

Present theory investigates the dimensionless vibration frequencies, deformations, and stresses for multilayered and sandwich arches using exponential shear and normal deformation theory (ESNDT). Present studies consider the effect of [1+(z/R)]radius of curvature while selecting the displacement field of arches under the action of uniform load. It includes the effects of transverse shear (γxz≠0) and transverse normal deformation(εz≠0)i.e., effect of thickness stretching. The governing relations are obtained using the Navier's method, and Hamilton's virtual work principle. The proposed layered arches are completely free from shear correction, and its top and bottom surfaces is satisfies zero traction free end conditions. Present study obtained very accurate natural frequencies for layered sandwich arches than other theories because, available literature not considered the thickness stretching effect i.e.,(εz=0). Hence, the present scientific investigation accounts the effect of thickness stretching i.e.,(εz≠0). However, the non-availability of exact elasticity results of bending and dynamic responses for layered sandwich arches. Present theory is obtained vibration frequencies, displacements, and stresses of layered composite and sandwich arches; the results are compared and validates through prior published literature.

Free-field wave motion in an inhomogeneous elastic half-plane with surface elasticity effects

Elastic wave propagation in аn inhomogeneous half-plane with surface elasticity effects is studied in this paper. All three types of travelling body waves are considered, namely pressure, vertically polarized shear and horizontally polarized shear waves propagating under time-harmonic conditions. Along the half-plane boundary, a localized constitutive law within the framework of the Gurtin-Murdoch theory is introduced, resulting in non-classical boundary conditions as opposed to the simple traction-free conditions of classical elasticity. The free-field wave motion is analytically derived here by using the wave decomposition technique, in conjunction with appropriate functional transformations for the displacement vector. Next, a series of parametric studies serves to identify the differences in the free field motion between that for the inhomogeneous half-plane with surface elasticity and the reference case of a homogeneous material with a traction-free surface. Finally, the dependence of the free-field wave motion as it develops in the material on the level of inhomogeneity and on the magnitude of the localized surface elasticity parameters, as well as on the type of travelling waves, is quantified in this work.

Free-surface elastic wave injection and wavefield reconstruction with applications to elastic reverse time migration

Accurately injecting and extrapolating multicomponent seismic data is a major step in elastic reverse time migration (RTM) and full-waveform inversion. The theoretical basis of this process is the seismic representation theorem. We propose a numerical approach based on the collocated-grid MacCormack finite-difference (FD) scheme to implement the injection and extrapolation. The MacCormack FD scheme defines wavefield parameters on the same grid. When dealing with the boundary condition in the representation theorem, it is more advantageous than the commonly used staggered-grid FD scheme which defines the stress and velocity on offset grids; thus, interpolations are required to link the boundary values to nearby nodes. In contrast, with the MacCormack scheme, the traction and particle velocity data can be directly injected into the boundary nodes as equivalent concentrated forces, force dipoles, and double couples. The accuracy of this approach is validated by comparing the regenerated wavefield with the original wavefield. Our primary purpose is to develop an injection method for elastic RTM, but this approach also can be used to numerically investigate other interesting issues in the elastic wave injection, including errors and artifacts caused by injection from unclosed boundaries, incomplete boundary values, and uncertain velocity models. In particular, seismic data are usually acquired on the ground surface where the traction-free condition is naturally maintained. We simulate a land acquisition scenario by generating a synthetic multicomponent data set on a traction-free surface. The reconstructed wavefields from velocity data are compared with the original forward-propagated waves to evaluate the accuracy of the boundary treatments. Two multicomponent elastic RTM examples are presented to demonstrate the applications of the proposed injection method. By combining the regenerated receiver-side wavefield and the vectorized imaging condition, the PP, PS, SP, and SS images are calculated.

A full field solution for an anisotropic elastic plate with a hole perturbed from an ellipse

A full-field solution for an infinite anisotropic plate containing a hole perturbed from an ellipse subjected to uniform loading at infinity is derived with Stroh formalism. With perturbation technique, the solution is expanded into a series with reference to the solution of the elliptical hole problem. Through the traction-free condition on the hole boundary, the unknown coefficients of the series are solved using the method of analytical continuation. The explicit full-field solution up to the first order and its corresponding explicit expression of the hoop stress along the hole boundary is presented. Numerical examples with different hole shapes (triangle, quadrilateral, oval, and pentagon), material types (isotropic, orthotropic, and anisotropic), and loading types (in-plane stresses and anti-plane shear) are provided. The results along the hole boundary and in the full field are verified with existing solution and commercial finite element software ANSYS. Through this verification, we conclude that although the hoop stress along the hole boundary provided by the existing analytical solutions is correct, their associated full-field solutions are incorrect because of the non-conformal mapping functions. The solutions presented in this paper are the first verified correct full-field analytical solutions published in the literature.

Dynamic soil impedance functions for cylindrical structures buried in elastic half-space

Spatially-distributed buried structures are highly susceptible to seismic ground motions. The macroscopic soil reactions to soil–structure relative displacements, aka. soil impedance functions (SIFs) and represented by a set of springs and dashpots, are thus very important for the assessment and design of those systems. Previous models to investigate the interaction problems between soil and horizontally buried structures (such as pipelines, tunnels) have been using spring stiffness chosen as static or frequency-independent constants, ignoring the nature of seismic loading and the energy reflected from the ground free surface. This paper presents analytical solutions for computing the frequency-domain SIFs for an infinitely-long cylindrical structure buried horizontally in homogeneous elastic half-space. The main challenge lies in mixed-boundary-value condition, where displacements are prescribed at the circular soil–structure interface and traction-free condition is satisfied along the straight-line ground surface. We used Hankel–Fourier series expansion, image technique, and Graf’s addition theorem to derive solution for axial SIFs. For a more complex in-plane SIFs problem, meanwhile, we used Hankel- and Bessel–Fourier series expansion. The in-plane solution requires numerical evaluation of contour integrals on the physical Riemann sheet, thus nested Gauss–Kronrod quadrature rule as well as Cauchy’s residue theorem are employed. We then verified our analytical solutions using results obtained from finite element simulations, in which a perfect agreement is shown between two approaches. The half-space SIFs are shown to also converge to their full-space counterparts in case of large burial depth. Additionally, parametric study was conducted to examine the variation of frequency-dependent SIFs, normalized with soil shear modulus, in response to the changes in soil Poisson’s ratio and the structure burial depth.

Zig-zag theory for concrete beams with corrugated steel webs

Owing to the large difference between the longitudinal and transverse stiffness of corrugated steel webs (CSWs), concrete beams with CSWs are idealized as sandwich beams with an orthotropic core in this study. A zig-zag displacement is then assumed to describe the deformation along the beam thickness along with a layer-wise parabolic distribution of the transverse shear stress. A new composite beam theory for a concrete beam with CSWs is finally developed, and the governing equations are derived using the variational principle of mixed energy. The transverse shear stress in the proposed theory automatically satisfies the traction-free condition at both the top and bottom surfaces of the beam. The continuity conditions of the displacements and shear stresses at the interfaces between two adjacent layers are also maintained, and thus there is no need to introduce a shear correction factor. Finally, analytical solutions for CSWs under four common boundary conditions are derived. These solutions are verified by numerical methods, and the results show that the proposed theory in this study can predict the deformation and stress of the concrete beams with CSWs more accurately.

Thermo-mechanical stress analysis of rotating fiber reinforced variable thickness disk

Circumferentially fiber reinforced composite disk, which has a variable thickness, is modeled via analytical approaches. The disk is subjected to rotation in traction free conditions and decreasing, constant, and increasing steady state radial temperature gradients along the disk radius. Limit angular velocities are calculated by operating Tsai-Wu and Norris failure indexes to the problem. Subsequently, these limit velocities are gradually decreased to examine the stress and displacement fields. Acquired results show that as the angular velocity drops, the effects of temperature gradients become more visible. At lower angular velocities, these gradients may even alter the stress field directions. Also, different failure criteria implementation may change the calculated limit velocities to a considerable degree. Therefore, the failure index should be chosen attentively to procure conservative results. In the investigation, the influence of disk geometry on the directional stresses is studied as well. Without further ado, it can be expressed that the geometry causes slight alterations in stresses and displacements.

Prediction of the spring-in of cylindrically orthotropic media and cross-ply laminates

The equation for prediction of the spring-in angle of a cylindrically orthotropic segment is shown to be independent of all material properties except for the anisotropic coefficients of thermal expansion and a stress-free state is insured for the corresponding unconstrained deformation. In contrast, the complete cylindrical geometry is shown to provide constraint to thermal deformation and thereby induce thermal residual stresses in the form of a moment. The method of superposition is demonstrated whereby traction-free conditions yield stress-free cylindrical elements with corresponding angular displacements at the element free boundaries. The first derivation of the spring-in equation is attributed to Radford, in contrast to the widely accepted view that the equation was first developed by Spencer et al. Finite-element methods, combined with the superposition approach, further validate the accuracy of the Radford equation for cylindrically orthotropic segments and explore its limitations for multiaxial composite laminates.

A parallel domain decomposition method for large eddy simulation of blood flow in human artery with resistive boundary condition

In this paper, we present a parallel domain decomposition algorithm for the simulation of blood flows in patient-specific artery. The flow may be turbulent in certain situations such as when there is stenosis or aneurysm. An accurate simulation of the turbulent effect is important for the understanding of the hemodynamics. Direct numerical simulation is computationally expensive in practical applications. As a result, most researchers choose to focus on a portion of the artery or use a low-dimensional approximation of the artery. In this paper, we focus on the large eddy simulation (LES) of blood flows in the abdominal aorta. To make the model more physiologically accurate, we consider a resistive outflow boundary condition which is more accurate than the traditional traction free condition. Different from the common decoupled approach where the resistive boundary condition is pre-calculated and then imposed as a Neumann condition, we prescribe it implicitly as an integral term on the LES equations and solve the coupled system monolithically. The governing system of equations is discretized by a stabilized finite element method in space and an implicit second-order backward differentiation scheme in time. A parallel Newton–Krylov–Schwarz algorithm is applied for solving the resulting nonlinear system with analytic Jacobian. Due to the integral nature of the resistive boundary condition, the Jacobian matrix has a dense block corresponding to all the variables on the outlet boundaries. Impacts of the resistive boundary condition with different parameters on the simulation results and the performance of the algorithm are investigated in detail. Numerical experiments show that the algorithm is stable with large time step size and is robust with respect to other parameters of the solution algorithm. We also report the parallel scalability of the algorithm on a supercomputer with over one thousand processor cores.

Mixed boundary value problems in power-law functionally graded circular annulus

The present work proposes a semi-analytical technique for solution of mixed boundary value problem in functionally graded circular annulus wherein shear modulus varies radially in power-law form while Poisson’s ratio is constant. The technique relies on two main steps. In the first step, corresponding to terms in periodic Fourier series applied individually as traction along the annulus surface, stress and displacement field in the annulus is computed harnessing Airy stress functions approach. In the second step, leveraging the strain–displacement relations in polar co-ordinates, mixed boundary conditions are rendered in terms of displacement all along the annulus surface. Assuming the unknown traction along the annulus surface in terms of periodic Fourier series with finite terms, the modified displacement boundary condition, family of solution from the first step and orthogonality of sine and cosine functions is used to generate a system of simultaneous linear equations for the series coefficients. Knowing the coefficients of periodic Fourier series, stress and displacement field can be computed everywhere in the annulus. The first step is digitized in terms of MAPLE functions, exhaustively validated through traction distribution comprising of normal, shear traction on part of the boundary and pair of equal and diametrically opposite point load along the boundary. The second step is corroborated through a problem where the inner surface is subjected to a specified traction distribution and mixed boundary conditions exist along the outer surface in the form of complete constraint along a part and traction-free condition elsewhere.

Large amplitude free vibration of porous skew and elliptical nanoplates based on nonlocal elasticity by isogeometric analysis

The current work presents an investigation on size-dependent nonlinear free vibration of porous skew and elliptical nanoplates. Formulations are based on a four-variable higher-order shear deformation plate theory to satisfy automatically the traction-free condition, von Kármán geometric nonlinearity to account for the mid-plane stretching, and Eringen's nonlocal elasticity theory to capture the scale effect. The isogeometric analysis and a displacement control strategy are employed synthetically to obtain large amplitude responses of the nanoplates. New results show the influences of porosity distribution pattern, porosity coefficient, and nonlocal parameter on the behaviors of vibration behaviors of the skew and elliptical nanoplates.

Meshfree analysis of non-rectangular sandwich plates based on refined C0 higher order shear deformation theories

In the present work, bending and free vibration characteristics of non-rectangular laminated composite and sandwich plates are investigated using C0 meshfree formulation based on element free Galerkin (EFG) method with moving kriging (MK) shape function and newly proposed higher-order shear deformation theories. The five new refined higher-order theories with non-polynomial transverse shear stress functions are proposed, which automatically satisfy traction free conditions on top and the bottom surfaces of the plate. The governing differential equations of motion for the continuum system are derived through the minimization of Lagrange functional and are discretized into the algebraic form using MK based meshfree method. The accuracy and applicability of the proposed models are examined first for benchmark problems on the bending and vibration analysis of thin and thick laminated composite and sandwich square plates. Thereafter, several new results on the flexural and free vibration behaviour of sandwich skew, trapezoidal and L-shaped plates, hitherto not found in the literature, are presented for various geometrical parameters and boundary conditions. The presented results for the sandwich plates may serve as the benchmark solutions for the other numerical methods employed for structural analysis of complicated geometry.

Investigating the viscous damping effects on the propagation of Rayleigh waves in a three-layered inhomogeneous plate

The present paper investigates the propagation of anti-plane Rayleigh waves in an elastic three-layered inhomogeneous plate in the presence of viscous damping. The core layer is sandwiched between skin layers with appropriate perfect interfacial boundary conditions, while traction-free conditions are prescribed on the outer faces. The eigenfunctions expansion method is employed for the study. Various cases of damped and undamped situations have been examined in connection to the obtained displacements and shear stresses. Finally, the effects of viscous damping on the propagation of Rayleigh waves are reported graphically using some physical available data.

Hybrid stress analysis of a near-surface circular hole in finite structures

The ability to stress-analyze complicated structures from recorded load-induced temperatures is demonstrated. The considered structures have a near-surface hole and subjected to a concentrated load. The complexity of the structure is simplified by conformal mapping, the traction-free condition on the boundary of the hole is analytically satisfied by analytic continuation, and the equilibrium and compatibility conditions are satisfied by means of Airy stress function in complex-variable formulation. For isotropic member that is cyclically loaded within its elastic range, the produced in-phase temperature variations are linearly proportional to the local changes in the normal stresses. Even though no recorded thermal data were used at or near to the edges, the present hybrid method simultaneously separates the load-induced temperatures into the individual stress components, determines reliably the boundary stress and hence the stress concentration, and smooths the measured input data. Unlike prior capabilities of using geometrical symmetry to simply the stress function representation, the present analysis retains all the terms in the stress functions. Therefore, the considered hybrid stress analysis approach of such complex structures extends significantly the applicability of thermoelastic stress analysis compared to prior capabilities and is considered to be the most complicated formulation of the hybrid complex-variable method to date. To support the reliability of the present hybrid method, the results were compared with finite element predictions and previous results based on Mitchell solution.

Free-edge stress evaluation of general laminated composites using a novel multifield variational beam formulation

Free-edge interlaminar stresses in general laminated composites are evaluated using a multifield (stress and displacement) finite element (FE) beam sectional formulation based on the Hellinger-Reissner principle considering three-dimensional (3D) material constitutive model. Both 3D stress and warping displacement fields are modeled as the primary variables while the interlaminar stress continuity and free-edge traction-free conditions of the laminates are ensured without introducing any peculiar assumptions. Nodal stresses are therefore computed directly at each constituent lamina using a domainwise approach while preserving the desired continuity (or discontinuity) at the interfaces through a global equilibrium constraint. The predicted free-edge stresses are validated for laminated composites with various layups subjected to extension and bending loads. Good correlation is shown against other state-of-the-art analytical methods and detailed 3D FE analysis. The proposed approach leads to a quick convergence of interlaminar stresses using fewer FEs with the benefit of assured continuity at the lamina interfaces. The boundary layer regions due to the free-edge are also identified for angle-ply and cross-ply laminates under different mechanical loading cases. The free-edge interlaminar stresses vanish slowly for the cross-ply layups, indicating dominant effect over a maximum of 60% of the laminate width, than the angle-ply ones.

Application of distributed dislocation method to curved crack moving near a press-fitted inclusion in a two-dimensional infinite plate

This paper describes the application of the distributed dislocation method to the problem of a curved crack moving near a press-fitted inclusion in a two-dimensional infinite plate which is subjected to a tensile loading. The formulation is based upon the superposition of the following three fundamental problems: a continuous distribution of edge dislocations along the crack line in an infinite plate with an inclusion, residual stress generated by the press-fitted inclusion without the crack, and stress field in the infinite elastic plate containing the circular inclusion, but without the crack, which is subjected to a uniform tensile loading. The superposition leads to simultaneous singular integral equations, which are introduced by relating the traction-free condition along the curved crack line to the dislocation density functions as unknown functions in the integral equations. The stress intensity factors can be derived directly from those dislocation density functions. In this study, the crack-propagation program, which is considered that the crack tip is automatically moved in a direction satisfying the restriction that the stress intensity factor KII is zero, was developed. The developed program is applied to the elastic problem including the press-fitted inclusion and crack, and the influence of the initial crack locations and residual stress generated by inserting the inclusion into the hole on the geometry of the crack path is discussed.