Transverse Normal Stress Research Articles

Stress distributions and mechanical properties of laminates [θm/90n]S with closed and open cracks in shear loading

The classical elastic theory of laminates states that the transverse normal stresses in the local axis of the 90° layer of an undamaged laminate [θm/90n]S are positive due to in-plane shear force, the transverse normal stresses will be negative when the shear load is inthe oppositedirection. For the laminate [θm/90n]S with transverse matrix cracks, the cracks in the 90°-layer should be open for the tensile stress and closed for the compressive stress. Therefore, two cases of the transverse cracks in the unbalanced laminates subjected to shear loading are studied such that the cracks are open for one case and the cracks are closed for another case. A stress-based variational model is developed to study the effective in-plane compliance and the stress distribution in symmetrical laminates with two different types of transverse matrix cracks in 90°-ply under an in-plane shear loading. The inter-laminar shear stresses are assumed to be in the form of general functions. A partition coefficient λ is introduced into the normal stresses σx for the closed cracks. The expressions of the stress components for the closed and the open cracks are obtained from the equilibrium equations and the boundary conditions, respectively. The governing equations for the two types of cracks are also derived using a variational approach with the principle of minimum complementary energy. In addition, analysis of strain energy and analysis of deformation on surface are used to estimate the effective in-plane shear properties of the laminates. The coupling coefficients and the shear elements in the compliance matrix are investigated for the two types of cracks. The distributions of the in-planar and the inter-laminar stresses are also presented. Limits of the compliance elements and the stress distributions at the infinite crack density are compared to the results of the classical theory of laminates.

Thermal response of ceramic matrix nanocomposite cylindrical shells using Eshelby-Mori-Tanaka homogenization scheme

Abstract We find thermal stresses developed in Ceramic Matrix Composite (CMC) cylindrical shells reinforced with aggregated Carbon Nanotubes (CNTs) with heat flux prescribed on the inner surface and temperature on the outer surface. Null surface tractions are prescribed on these two surfaces and the cylinder edges are clamped. The material properties are homogenized by using a two-parameter Eshelby-Mori-Tanaka (EMT) approach. Material properties of the ceramic are assumed to depend upon the temperature, and the smooth variation of the CNT volume fraction through the shell thickness is assumed to be described either by a sigmoidal function or profile-O or profile-X often used in the literature. The one-way coupled thermo-mechanical problem is analyzed by first numerically solving the nonlinear heat equation with the Generalized Differential Quadrature Method (GDQM), and then the linear mechanical problem by using Reddy's Third-order Shear Deformation Theory (TSDT) and the GDQM. For the same thermal boundary conditions and the volume fraction of CNTs, the maximum hoop, the in-plane shear and the transverse normal stresses developed in the cylinder are highest for the profile-X of CNTs. The aggregation factor noticeably influences the maximum transverse normal and the maximum hoop stresses developed in the cylinder.

A damage mechanics model for low-velocity impact damage analysis of composite laminates

ABSTRACTA method was developed to predict numerically the damage of composite laminates with multiple plies under low-velocity impact loading. The Puck criterion for 3D stress states was adopted to model the intralaminar damage including matrix cracking and fibre breakage, and to obtain the orientation of the fracture plane due to matrix failure. According to interlaminar delamination mechanism, a new delamination criterion was proposed. The influence of transverse and through-thickness normal stress, interlaminar shear stress and damage conditions of adjacent plies on delamination was considered. In order to predict the impact-induced damage of composite laminates with more plies quickly and efficiently, an approach, which can predict the specific damage of several plies in a single solid element, was proposed by interpolation on the strains of element integration points. Moreover, the proposed model can predict specific failure modes. A good agreement between the predicted delamination shapes and sizes and the experimental results shows correctness of the developed numerical method for predicting low-velocity impact damage on composite laminates.

A generalized layered global-local beam theory for elasto-plastic analysis of thin-walled members

Based on a generalized layered global-local beam (GLGB) theory, a computationally low cost finite element model is presented for the elasto-plastic analysis of thin-walled beams. In the employed GLGB theory, the cross-section of the thin-walled beam is replaced with an equivalent layered composite one. For describing the displacement fields of the equivalent layered composite beam, the double superposition hypothesis in conjugate with an exponential shear stress function is employed. In contrast to the most of available advanced one-dimensional (1D) models proposed for thin-walled structures, the effects of the transverse normal stress and transverse flexibility are considered in the present GLGB theory. The proposed GLGB theory does not need the incorporation of any shear correction factor and it has only one general unknown parameter more than Timoshenko's beam theory. For plasticity modeling, von Mises yield criterion is employed. The non-linear 1D finite element formulations are solved using Newton-Raphson algorithm. The proposed nonlinear finite element model is validated through comparison with the three-dimensional (3D) finite element and other similar theoretical models available in the literature.

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.

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.

Analysis of pultrusion process for thick glass/polyester composites: transverse shear stress formations

In the present work process-induced residual stress development is described for a 100 mm pultruded square profile made of glass/polyester. A thermo-chemical model developed in MATLAB is coupled with a mechanical model developed in ABAQUS. The temperature and degree of cure distributions are calculated for three different preheating temperatures. In the mechanical model, a parameter study is examined to investigate the effect of total volumetric shrinkage () and the coefficient of thermal expansion (CTE) in rubbery state on the residual stresses. The non-uniform internal constraints in the part yield in an internal shear deformation during the process. The transverse shear stress and compressive normal stress levels decrease significantly as compared with the tensile normal stresses with an increase in preheating temperature. The change in and CTE in rubbery state affects the shear stresses and compressive stresses significantly, whereas the tensile stresses are less affected.

INVESTIGATION OF STRESS-STRAIN STATE OF TRANSVERSELY ISOTROPIC PLATES UNDER BENDING USING EQUATION OF STATICS {1,2} –APPROXIMATION

The study examined the construction of the fundamental solution for the equations of statics {1,2} – approximation for transversely isotropic plates under bending with the action of concentrated force. Equations {1,2} -approximation were obtained by the decomposition method in the thickness coordinate using the Legendre polynomials. These equations take into account all the components of the stress tensor, including the transverse shear and normal stresses. Since the classical theory of Kirchhoff-Love doesn’t take account of these stresses, the study on the basis of refined theories of stress-strain state of transversely isotropic plates under the action of concentrated force effects is an important scientific and technical problem. The fundamental solution of obtained equations results using a two-dimensional Fourier integral transform and inverse treatment techniques, built with the help of a special G-function. This method allows reducing the system of resolving differential equations for statics of flat plates and shells to a system of algebraic equations. After that, the inverse Fourier transform restores the fundamental solution. The work was carried out numerical studies that demonstrate patterns of behavior of components of the stress-strain state, depending on the elastic constants of transversely isotropic material. The results play a decisive role in the study of boundary value problems in the mechanics of thin-walled elements of constructions, including under the influence of concentrated and local diverse forces.

One-dimensional model of fourth order for rods with loading on lateral boundary: The case of rectangular cross section

We propose deducing from three-dimensional elasticity a one dimensional model of a beam when the lateral boundary is not free of traction. Thus the simplification induced by the order of magnitude of transverse shearing and transverse normal stress must be removed. For the sake of simplicity we consider a beam with rectangular cross section. The displacement field in rods can be approximated by using a Taylor–Young expansion in transverse dimension of the rod and we truncate the potential energy at the fourth order. By considering exact equilibrium equations, the highest-order displacement field can be removed and the Euler–Lagrange equations are simplified.

Construction of fundamental solution of {1,2}-approximation static equations of momentless stress state for transversely-isotropic plates

The problem of static for transversely-isotropic plates, which are under the action of a concentrated force, was considered. Static equations of {1,2}-approximation obtained by decomposition of the desired functions in Fourier series in Legendre polynomials in relation to the thickness of the coordinates were used. These equations take into account all the components of the stress tensor, including the transverse shear and normal stresses. Numerical studies demonstrating the effect of elastic constants on the components of the stress-strain state of a transversely- isotropic plate were conducted

One-dimensional models of fourth and sixth orders for rods derived from three-dimensional elasticity

The displacement field in rods can be approximated by using a Taylor–Young expansion in transverse dimension of the rod. These involve that the highest-order term of shear is of second order in the transverse dimension of the rod. Then we show that transverse shearing energy is removed by the fourth-order truncation of the potential energy and so we revisit the model presented by Pruchnicki. Then we consider the sixth-order truncation of the potential which includes transverse shearing and transverse normal stress energies. For these two models we show that the potential energies satisfy the stability condition of Legendre–Hadamard which is necessary for the existence of a minimizer and then we give the Euler–Lagrange equations and the natural boundary conditions associated with these potential energies. For the sake of simplicity we consider that the cross-section of the rod has double symmetry axes.

Mathematical Formulation of Laminated Composite Thick Conical Shells

<span lang="EN-US">The </span><span lang="EN-US">mathematical formulation</span><span lang="EN-US">of thick conical shells using third order shear deformation of thick shell theory are presented. The equations of motion are obtained using Hamilton’s principle. For present analysis, we consider shell's system transverse normal stress, rotary inertia and shear deformation.</span>

Boundary layer state prediction of composite and sandwich plates via an enhanced higher-order shear deformation theory

An efficient and accurate method for the improved stress analysis using a conventional higher order shear deformation theory (HSDT) is presented. The main objective herein is to systematically establish the relationship between two independent displacement and stress fields through the mixed variational theorem (MVT). In the mixed formulation, the independent transverse shear stress field is taken from the fifth order polynomial-based zig-zag model, whereas the in-plane displacement field is assumed to be those of the conventional HSDT to amplify the benefits of numerical efficiency. Moreover, in-plane correction factor including the transverse normal stress effect is additionally considered to more accurately analyze the physical behavior for the general configuration of the laminated composite and sandwich plates. The resulting strain energy is referred to as an enhanced higher order shear deformation theory via the mixed variational theorem (EHSDTM). The EHSDTM has the same computational advantage as the conventional HSDT does, while allowing for improved local through-the-thickness variations of the displacements and stresses via the recovery procedure. Furthermore, the present theory can give a reliable solution for the shortcomings of earlier other theories, such as difficulties of predicting the edge effect in clamped boundary conditions. The accuracy and efficiency of the present theory are demonstrated by comparing results obtained herein to those of the 3-D exact solutions, 3-D FEM and other theories reported in literature.

A high-performance parametrized mixed finite element model for bending and vibration analyses of thick plates

Based on the parametrized mixed variational principle, a high-performance finite element model has been developed for bending and vibration analysis of thick plates. The proposed mixed variational formulation combines the concept of Reissner’s mixed variational theorem and Rong’s generalized mixed variational theorem. In addition to unknown parameters of the displacement fields, the transverse normal component of the stress tensor is also assumed as the independent field variable in the present variational formulation. The latter allows the fulfillment of the boundary conditions of the transverse normal stress at the top and bottom surfaces of the plate in a natural way. The variational constraint of the transverse normal strain–displacement relations is relaxed via the parametrized Lagrange multipliers method. The functional of the proposed variational formulation contains an arbitrary free parameter, called the splitting factor. Simple formulations have been proposed for selecting the splitting factor which leads to the results of higher precision. The numerical results obtained from the proposed mixed formulation have been compared with the results of three-dimensional (3D) theory of elasticity and other plate theories available in the literature. The comparison study shows that the proposed parametrized mixed plate theory is capable of achieving the accuracy of nearly exact 3D solutions.

Effects of elastically restrained edges on FG sandwich annular plates by using a novel solution procedure based on layerwise formulation

In practical applications, sandwich plates are often connected to other members, supported by damaged clamped/simply supported boundary conditions or supported by elastic restraints. Therefore, the mentioned structures may not always be simulated by the classical boundary conditions, i.e., ideal simply supported, clamped and free edges. Also, these structures may be subjected to various loads. In this study, for the first time, a novel economical analytical solution procedure is presented for axisymmetric static analysis of sandwich annular plates, by using the layerwise and 3D elasticity theories. Based on the proposed approach, functionally graded sandwich annular plates with various elastically restrained edges under arbitrary distributed loads may be analyzed and all of the displacements and stresses components may be exactly achieved. Also, imposed loads at the boundaries may be evaluated. Transverse shear and normal stresses boundary conditions on the top and bottom of the sandwich plate and the interlaminar continuity conditions of the in-plane displacement, transverse shear and normal stresses are exactly satisfied. Accuracy and efficiency of the presented solution procedure are demonstrated by comparing the obtained results for sandwich plates with the classical edge conditions as some special cases of the elastic supports with results of the three-dimensional theory of elasticity.

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.

A novel economical analytical method for bending and stress analysis of functionally graded sandwich circular plates with general elastic edge conditions, subjected to various loads

Abstract In this study a novel economical analytical approach is presented for static analysis of sandwich circular plates based on a layerwise theory linked with three-dimensional theory of elasticity. Each layer of the sandwich plate may be made of functionally graded materials, the boundary condition may be elastically restrained, and the sandwich plate may be subjected to non-uniform normal and/or shear tractions. As first time, elastically restrained edge sandwich plates are investigated and two types of translational and rotational restraints are used for modeling of general boundary conditions: I) by using in-plane translational springs at the mid plane of the core, rotational springs at the mid plane of each layer and transverse translational springs II) by using a distributed in-plane translational springs along the thickness direction of the edge and transverse translational springs. The proposed analytical method enables to obtain the in-plane and transverse displacements and stresses components. Transverse shear and normal stresses boundary conditions on the top and bottom of the sandwich plate and continuity conditions at the interfaces between layers are exactly satisfied. The efficiency of the proposed approach is demonstrated by comparison of the obtained results for classical edge conditions with those extracted from the ABAQUS software based on the 3D-elasticity.

High-order plate finite elements for smart structure analysis

This paper presents new finite elements for plate analysis of smart composite structures. Based on the sinus model, Murakami’s Zig–Zag functions are introduced in the three directions, improving the accuracy for multilayered modeling. The transverse normal stress is included allowing use of the three-dimensional constitutive law. Three different eight-node finite elements are developed using C0 approximations, each with a different number of unknown functions: 9, 11 or 12. For the piezoelectric approximation, a layer wise description is used with a cubic variation in the thickness of each layer while the potential is assumed to be constant on each elementary domain for the in-plane variation. These finite elements aims at modeling both thin and thick plates without any pathologies of the classical plate finite elements (shear and Poisson or thickness locking, spurious modes, etc.). This family is evaluated on classical piezoelectric problems of the literature and special emphasis is pointed towards the introduction of equipotential conditions.

Coupled three-dimensional piezoelasticity solution for edge effects in Levy-type rectangular piezolaminated plates using mixed field extended Kantorovich method

A coupled three-dimensional piezoelasticity solution for rectangular piezoelectric laminated plates with Levy-type boundary conditions is presented using the mixed-field multi-term extended Kantorovich method (EKM) and Fourier series expansion. The solution considers two-way electromechanical coupling, and satisfies all the essential and natural boundary conditions as well as the continuity conditions at the interface exactly at all points. The objective is to accurately characterize the edge effects in finite dimensional piezolaminated plates under mechanical and actuation potential loading, and also to provide a benchmark for assessing the accuracy of the two-dimensional laminate theories. The interlaminar shear stress is also accompanied by a singular stress field of the transverse normal (peel) stress near the edge. The numerical study is conducted for both single layer PZT plates for sensory applications and hybrid sandwich plates for smart structural applications. It is shown that the EKM solution predicts the displacement, stress, electric potential and electric displacement fields very accurately including the sharp variations of stresses near the edge, under the potential loading with just two/three terms in the trial function.

Interacting wakes of a narrow and a wide flat plate in tandem arrangement

Flow structures behind two different sized flat plates in tandem arrangement normal to flow at high Reynolds number have been investigated experimentally. A narrow flat plate, as a control plate, has been placed upstream of a wide plate to investigate the interacting wakes behind the wide flat plate. The near wake downstream of the wide plate has been measured by employing constant-temperature hot wire anemometer, quantitatively. The effects of different width ratio (h/D) range from 0.1 to 1.0 together with gap ratio (g/D) ranging from 0.5 to 2.0 have been probed. It was found that Strouhal number variation is directly proportional to gap ratio between the plates. Moreover, it was observed that turbulent kinetic energy production is mostly contributed by transverse normal turbulent stress and therefore follows the transverse stress pattern rather than the stream wise stress.