Stress Continuity Conditions Research Articles

A new mixed-field theory for bending and vibration analysis of multi-layered composite plate

A novel mixed-field theory with relatively low number of unknown variables is introduced for bending and vibration analysis of multi-layered composite plates. The presented plate theory is derived from a parametrized mixed variational principle which is introduced recently by the first author. A global-local kinematic with a layer-independent number of variables is assumed for the description of the displacements of the plate. The considered kinematic stratifies the displacement and transverse stress continuity conditions at the mutual interfaces of the layers. It also fulfill the homogenous boundary conditions of the shear stresses on the upper/lower surfaces of the plates without using the shear correction factor. One-unknown variable fields which satisfy a priori the continuity conditions at the adjacent interfaces between layers and the zero boundary conditions on the bounding surfaces are considered for the approximation of the transverse shear stresses. The transverse normal stress along the total thickness of the multi-layered plate is approximated via a quadratic polynomial. The presented mixed-field plate theory has been validated through comparison of the bending and vibration analysis results with those obtained from the three-dimensional (3D) theory of elasticity and the results of the other classical and high-order plate theories.

Hybrid Numerical-Analytical Scheme for Calculating Elastic Wave Diffraction in Locally Inhomogeneous Waveguides

Numerical simulation of traveling wave excitation, propagation, and diffraction in structures with local inhomogeneities (obstacles) is computationally expensive due to the need for mesh-based approximation of extended domains with the rigorous account for the radiation conditions at infinity. Therefore, hybrid numerical-analytic approaches are being developed based on the conjugation of a numerical solution in a local vicinity of the obstacle and/or source with an explicit analytic representation in the remaining semi-infinite external domain. However, in standard finite-element software, such a coupling with the external field, moreover, in the case of multimode expansion, is generally not provided. This work proposes a hybrid computational scheme that allows realization of such a conjugation using a standard software. The latter is used to construct a set of numerical solutions used as the basis for the sought solution in the local internal domain. The unknown expansion coefficients on this basis and on normal modes in the semi-infinite external domain are then determined from the conditions of displacement and stress continuity at the boundary between the two domains. We describe the implementation of this approach in the scalar and vector cases. To evaluate the reliability of the results and the efficiency of the algorithm, we compare it with a semianalytic solution to the problem of traveling wave diffraction by a horizontal obstacle, as well as with a finite-element solution obtained for a limited domain artificially restricted using absorbing boundaries. As an example, we consider the incidence of a fundamental antisymmetric Lamb wave onto surface and partially submerged elastic obstacles. It is noted that the proposed hybrid scheme can also be used to determine the eigenfrequencies and eigenforms of resonance scattering, as well as the characteristics of traveling waves in embedded waveguides.

The scattering of steady-state SH waves in a bi-material half space with multiple cylindrical elastic inclusions

The scattering of steady-state SH waves in a bi-material half space with multiple cylindrical elastic inclusions is presented analytically. Mirror method and multi-polar coordinate systems are developed to solve the complex boundary value problem. Considering the displacement and stress continuity conditions, a series of integral equations for unknown coefficients are obtained and solved by truncation. The solution is used to calculate the dynamic stress concentration factor around the edge of the inclusion, and the analysis and numerical results are discussed. The results show that degree of dynamic stress concentration around the circular inclusion is influenced by the incident angle, the incident wave number, and the parameters of medium.

Bulk Nonlinear Elastic Strain Waves in a Bilayer Coaxial Cylindrical Rod

The problem of the propagation of long nonlinear elastic strain waves in a bilayer coaxial cylindrical rod with an ideal contact between the layers has been considered. Expressions for transverse displacements through longitudinal displacements have been derived. The former satisfies free boundary conditions and continuity conditions for displacements and stresses at the interlayer interface with the desired accuracy. It has been shown how these expressions generalize the well-known plane-section and Love hypotheses for an isotropic homogeneous rod. An equation for the propagation of a nonlinearly elastic strain longitudinal wave has been derived, and its particular solution in the form of a solitary traveling wave has been studied.

Elasticity solution for the casing under linear crustal stress

The integrity of the casing is crucial for oil and gas well. Based on stress function method, a three-dimensional model of the casing-cement sheath-formation system subjected to linear crustal stress is proposed. And then an analytical solution of the model was obtained. In the process of calculation, the casing and cement sheath are simplified as the perfect cylinder. The cement sheath is closely bonded with the casing and formation. The formation is considered to be an isotropic material without the layer-block structure. And the crustal stress is assumed to be linearly increasing with the depth of the well. The analytical solution strictly meets the stress and displacement continuity condition and boundary condition, and exhibits good agreement with finite element method. The results imply that an analytical method to capture the stress and displacement field of the casing under linear crustal stress along the axis is presented. Next, a benchmark for numerical and approximate solutions is provided. In addition, a new idea about solving the casing under the non-linear loads along the axis in some special stratum (such as heterogeneity stratum, salt rock) is proposed. Finally, our understanding for the casing under complex loads will be deepened.

Bending, free vibration and buckling analyses of anisotropic layered micro-plates based on a new size-dependent model

A new size-dependent composite laminated plate model is proposed in this study based on a re-modified couple stress theory (RMCST) and a zigzag theory. In contrast to the published size-dependent models for composite laminated plates, present model a priori satisfies the continuity conditions of transverse shear stresses (TSSs) at the interfaces. In the present formulation, the discontinuous TSSs obtained from the constitutive relations are replaced by the new continuous TSSs by using the Reissner’s Mixed Variational Theorem (RMVT). In addition, the RMVT is utilized to obtain the governing equations and corresponding boundary conditions. Subsequently, the deflections, stresses, frequencies and critical loads of cross-ply simply supported plates are considered to verify the present formulation. The results prove that the scale effects of present size-dependent models are effectively investigated. Moreover, the results predicted by present size-independent models are in accordance with the exact solutions. Eventually, the differences of the mechanical performance among the following three models, i.e. present, Mindlin and Reddy for composite laminated plates are discussed in different modulus and span-to-thickness ratios.

Graphene oxide-reinforced thermo-sensitive hydrogel consistently release alendronate for enhanced cranial defect repair

A new size-dependent composite laminated plate model is proposed in this study based on a re-modified couple stress theory (RMCST) and a zigzag theory. In contrast to the published size-dependent models for composite laminated plates, present model a priori satisfies the continuity conditions of transverse shear stresses (TSSs) at the interfaces. In the present formulation, the discontinuous TSSs obtained from the constitutive relations are replaced by the new continuous TSSs by using the Reissner’s Mixed Variational Theorem (RMVT). In addition, the RMVT is utilized to obtain the governing equations and corresponding boundary conditions. Subsequently, the deflections, stresses, frequencies and critical loads of cross-ply simply supported plates are considered to verify the present formulation. The results prove that the scale effects of present size-dependent models are effectively investigated. Moreover, the results predicted by present size-independent models are in accordance with the exact solutions. Eventually, the differences of the mechanical performance among the following three models, i.e. present, Mindlin and Reddy for composite laminated plates are discussed in different modulus and span-to-thickness ratios.

Layer-element analysis of multilayered saturated soils subject to axisymmetric vertical time-harmonic excitation

The analytical layer-elements for a single poroelastic soil layer and the underlying half-space are established using an algebraic manipulation and Hankel transform. According to the boundary conditions and adjacent continuity conditions of general stresses and displacements, a global matrix equation in the transform domain for multilayered saturated soil media is assembled and solved. Solutions in the frequency domain can be further obtained with an inverse Hankel transform. Numerical examples are used to examine accuracy of the present method and demonstrate effects of soil parameters and load conditions on dynamic responses of the multilayered poroelastic saturated soils.

A computational continuum model of poroelastic beds.

Despite the ubiquity of fluid flows interacting with porous and elastic materials, we lack a validated non-empirical macroscale method for characterizing the flow over and through a poroelastic medium. We propose a computational tool to describe such configurations by deriving and validating a continuum model for the poroelastic bed and its interface with the above free fluid. We show that, using stress continuity condition and slip velocity condition at the interface, the effective model captures the effects of small changes in the microstructure anisotropy correctly and predicts the overall behaviour in a physically consistent and controllable manner. Moreover, we show that the performance of the effective model is accurate by validating with fully microscopic resolved simulations. The proposed computational tool can be used in investigations in a wide range of fields, including mechanical engineering, bio-engineering and geophysics.

Analytical solution for the dynamic response of a pile with a variable-section interface in low-strain integrity testing

Considering the displacement and stress continuity conditions at the pile-soil interface, ring-soil pile theory (RSPT) and an amended impedance function transfer method (AIFTM) are proposed as models of rigid pile-soil interaction that consider the interaction at the variable-section interface of a pile with the surrounding soil. The interactions between adjacent three-dimensional soil layers are simplified as uniformly distributed Voigt models to derive the impedance function at the pile top. An inverse Fourier transform is applied to derive the velocity response at the top of the pile under transient excitation. An engineering example is described to confirm the rationality of the proposed solution. The solution simplifies into other previously proposed solutions for specific geometric parameters of the pile shaft. The proposed solution is compared with one that neglects the interaction at the variable-section interface of the pile with the surrounding soil; additionally, the coupled effects of the related pile-soil parameters and the interaction at the variable-section interface with the surrounding soil are analyzed.

Assessment of refined high-order global–local theory for progressive failure analysis of laminated composite beams

An efficient and computationally low-cost one-dimensional (1D) finite element model is developed for the progressive failure analysis of laminated composite beams. The employed finite element formulation is based on a refined high-order global–local beam (RHGB) theory which satisfies all the kinematic and stress continuity conditions at the layer interfaces. This RHGB theory also considers effects of the transverse normal stress and transverse flexibility. By using the well-known failure theories, an iterative method is adopted to predict the first ply failure load of the laminated composite beam. After the first ply failure, the material properties of the failed elements are modified using a stiffness degradation factor. Then, the applied load is increased step-by-step. This analysis is repeated for each load increment until the ultimate strength of the laminate is reached. Hashin, Maximum stress, Hoffman, Tsai–Hill and Tsai–Wu failure criteria are used to assess the possible damage at beam elements from the initial to the final step. The present failure formulation has been validated by comparison with experimental and theoretical results available in the open literature.

Hygrothermal analysis of laminated composite plates in terms of an improved C0-type global–local model

In this paper, an efficient improved C0-type global–local model (IGLM) considering transverse normal strain is proposed to study the hygrothermal response of thick cross-ply laminated composite plates. Based on the interlaminar continuity conditions of in-plane displacement and transverse shear stresses, layer-dependent variables could be reduced. Employing shear stress free condition at the upper and the lower surfaces, derivatives of transverse displacement are eliminated from the displacement field, so that C0 interpolation functions are only required for the finite element implementation. As a result, the number of variables is independent of the number of layers of the laminate. To assess the proposed model, the classical quadratic eight-node isoparametric element is presented, in which the interelement C0 continuity conditions are satisfied. Comparing the results from available three-dimensional elasticity theory and those computed from the ninth-order model, it is found that the simple C0 finite elements could produce promising deformations and stresses for thick cross-ply laminated composite plates with different boundary conditions under hygrothermal loadings.

碳纳米管复合材料的界面剪切应力研究

本论文采用剪切滞后模型，建立了碳纳米管复合材料的代表性体元模型，并且将碳纳米管看做一个有效纤维，模拟分析碳纳米管复合材料的主要形态特性。研究重点主要集中在碳纳米管与基体接触面的剪切应力，假设有效纤维与碳纳米管具有相同的直径和长度，运用应力连续条件和位移连续条件以及相关平衡方程等，推导得出计算所需要的复合材料界面处的切应力表达式，本论文推论得到的结果可以校核碳纳米管复合材料因滑动位移而产生的破坏。 In this paper, the shear lag model is used to establish the representative element model of carbon nanotube composite and the carbon nanotube is considered as an effective fiber to simulate the main morphological characteristics of composites. The investigation focuses on the shear stress of interface, and it is assumed that the effective fibers and carbon nanotubes have the same diameter and length. The interface shear stress expression is derived using the continuity conditions of stress and displacement, and equilibrium equations. Based on the results obtained, the damage caused by the sliding displacement of carbon nanotube composites could be checked.

Higher order flutter analysis of doubly curved sandwich panels with variable thickness under aerothermoelastic loading

In this study, the supersonic panel flutter of doubly curved composite sandwich panels with variable thickness is considered under aerothermoelastic loading. Considering different radii of curvatures of the face sheets in this paper, the thickness of the core is a function of plane coordinates (x,y), which is unique. For the first time in the current model, the continuity conditions of the transverse shear stress, transverse normal stress and transverse normal stress gradient at the layer interfaces, as well as the conditions of zero transverse shear stresses on the upper and lower surfaces of the sandwich panel are satisfied. The formulation is based on an enhanced higher order sandwich panel theory and the vertical displacement component of the face sheets is assumed as a quadratic one, while a cubic pattern is used for the in-plane displacement components of the face sheets and the all displacement components of the core. The formulation is based on the von Karman nonlinear approximation, the one-dimensional Fourier equation of the heat conduction along the thickness direction, and the first-order piston theory. The equations of motion and boundary conditions are derived using the Hamilton principle and the results are validated by the latest results published in the literature.

A three-dimensional analytical solution for sandwich pipe systems under linearly varying external pressures

A three-dimensional analytical model is presented to calculate the stress and deformation of a sandwich pipe (SP) system subjected to linearly varying external pressures. Based on a stress function method, an analytical solution for a thick-walled pipe was researched, and then an analytical solution for a SP system was obtained according to the boundary and continuity conditions of stress and displacement. Furthermore, the analytical solution of a SP system was verified by comparing with finite element method (FEM). The results indicate that the stresses from present method (PM) are in a good agreement with that from FEM. Finally, our researches provide a benchmark for approximate or numerical solutions, and are beneficial to evaluate the safety and integrality of SP systems.

Analysis of dynamic stress intensity factors for interfacial crack near shallow circular inclusion in bi-material half-space to SH wave

An analytical approach is used to investigate the SH-wave scattering problems of the bi-material half-space including a circular inclusion embedded and an interfacial crack. Considering the multiple scattering phenomena between the boundaries and the circular inclusion, the “image” principle is used to establish the expressions of the wave field. With the aid of interfacial conjunction, the interfacial crack is constructed. According to the displacement and stress continuity conditions, the Fredholm integral equations of the first kind are obtained to determine the unknown forces attached to the linking sections. Finally, analytical expressions for dynamic stress intensity factors are obtained. The numerical results for dynamic stress intensity factors are presented graphically considering the factors of the crack length, the incident wave number, the inclusion defect and the different media parameters.

Free Edge Effects in Sandwich Laminates under Tension, Bending and Twisting Loads

An accurate analytical solution for predicting the free edge effects in sandwich laminates under tension, bending and twisting loading is presented. The recently developed mixed-field multiterm extended Kantorovich method (MMEKM) has been used to obtain the solution of the governing equations, which are developed using the Reissner-type variational principle. The present mixed-field approach enables the exact and point-wise satisfaction of traction-free edge and interlaminar continuity conditions for displacements and stresses. The numerical results presented for different loadings and lay-up show rapid convergence of the iterative series solution. The comparison of the present results with the detailed FE solution shows good agreement. The present solution captures the singularity of stresses in the free edge region by showing the rise in its peak magnitude with the number of terms in the solution. The presented accurate 3D elasticity based solution can act as a useful benchmark for assessing the accuracy of solutions obtained from other approximate methods.

Assessment of recent zig-zag theories for laminated and sandwich structures

Abstract The flexural response of laminated composite and sandwich beams/plates under static distributed loading, classical and non-classical boundary conditions (simply-supported, cantilever and propped-cantilever) and geometric/constitutive heterogeneity of layers is analysed. As structural models, a recently developed equivalent single-layer zig-zag model by the authors, a discrete-layer model and a sublaminate model developed from it in this paper are used. Their contribution is to consider the continuity of the transverse normal stress and its through-thickness gradient at layer interfaces, as prescribed by the elasticity theory in addition to kinematic and transverse shear stress interlayer continuity customary considered in the literature. To this purpose, a piecewise variation of the three displacement components is adopted. The zig-zag amplitude expressions are obtained in closed-form from the enforcement of stress continuity conditions. To be refined without affecting costs, the equivalent single-layer model has variable kinematics and just five unknowns. Instead, sublaminate and discrete-layer models are refined by increasing the number of computational layers and variables. The aim of this paper is to assess whether the equivalent single-layer model having a computational cost comparable to that of classical models can be as accurate ad discrete-layer and sublaminate models. Benchmarks are presented, for which exact elasticity and approximate solutions are available for comparisons in literature. It is illustrated the utility of considering a variable kinematics for obtaining accurate stress predictions from constitutive equations and the transverse normal deformability for keeping equilibrium. The equivalent single-layer model is shown as accurate as discrete-layer and sublaminate models in all cases examined.

Free edge stresses in composite laminates with imperfect interfaces under extension, bending and twisting loading

An accurate analytical three-dimensional elasticity solution is presented for the free edge stresses in composite laminates under extension, bending and twisting loading, accounting for possible bonding imperfections at the layer interfaces. The bonding imperfection is modeled using the linear spring-layer model. A mixed formulation with both displacements and stresses as primary variables is developed using the Reissner-type variational principle and the solution is obtained using the recently developed mixed-field multiterm extended Kantorovich method (MMEKM). The method ensures exact and point-wise satisfaction of the traction-free edge boundary conditions and interlaminar stress continuity conditions, which is key to the accurate solution of free edge stresses. A detailed numerical study established excellent convergence characteristics of iterative series solution. The presence of bonding imperfection at an interface leads to nonsingular free edge interlaminar stresses at that interface, in contrast with the singular stress field occurring at a classical perfect interface. The effect of loading type on the relative reduction of free edge stresses with the increase in imperfection compliance is investigated for cross-ply and angle-ply laminates. The effect of ply-angle of angle-ply laminates on the same is also ascertained.

An improved dynamic model for hub and laminated composite plate system considering warping effect

In previous investigations of rigid-flexible coupling dynamics of the laminated plate, warping effect caused by the deformation of the normal vector of the mid-plane was neglected, which may lead to a significant error of the inter-laminar transverse shear stress. In this paper, an improved dynamic formulation for the rigid-flexible coupling dynamics of hub and laminated composite plate system is proposed based on the global–local higher-order shear deformation theory. By enforcing free shear traction conditions for both surfaces and continuity conditions of displacements and transverse shear stresses on the laminate interfaces, the dynamic equations for multi-body system are established based on the principle of virtual work. A numerical example for dynamic analysis of a hub and laminated composite plate system is presented to show different results obtained by different displacement theories. It is demonstrated that the results obtained by the present formulation satisfy both the continuity conditions of inter-laminar shear stresses and the free shear traction conditions of upper and lower surfaces.