Orthotropic Layers Research Articles

Analysis of Dispersion and Damping Characteristics of Love Wave Propagation in Orthotropic Visco-Elastic FGM Layer with Corrugated Boundaries

AbstractA mathematical model was designed to analyze the propagation characteristics of Love waves traversing through an orthotropic visco-elastic FGM layer under initial stress lying over a half-s...

Fuzzy estimates of resonance frequencies for three-layer composite cylindrical panels in smart aerospace and industrial structures

Is developed a numerical-analytical fuzzy-sets methodic for a analysis of the uncertain endogenous parameters of the resonant frequencies in a theoretical model of vibration of thin orthotropic composite three-layer cylindrical panels with small curvature having of symmetrical structure with two external transversally-isotropic piezoelectric layers and with internal orthotropic layer from linearly viscoelastic composite material if available of scatter errors in values of their structural exogenous physical-mechanical and geometric parameters. The technique is based on the transition to fuzzy-interval arguments in the analytical representations for the studied resonant frequencies and in the application of the heuristic principle of generalization to obtain of estimates for the values of such resonant frequencies in the form of normal fuzzy sets. An example of the implementation of the technique under consideration is presented.

Double frictional receding contact problem for an orthotropic layer loaded by normal and tangential forces

With the increasing research in the field of contact mechanics, different types of contact models have been investigated by many researchers by employing various complex material models. To ascertain the orthotropy effect and modeling parameters on a receding contact model, the double frictional receding contact problem for an orthotropic bilayer loaded by a cylindrical punch is taken into account in this study. Assuming plane strain sliding conditions, the governing equations are found analytically using Fourier integral transformation technique. Then, the resulting singular integral equations are solved numerically using an iterative method. The weight function describing the asymptotic behavior of the stresses are investigated in detail and powers of the stress singularities are provided. To control the trustworthiness and correctness of the analytical formulation and to compare the resulting stress distributions and contact boundaries, a numerically efficient finite element method was employed using augmented Lagrange contact algorithm. The aim of this paper is to investigate the orthotropy effect, modeling parameters and coefficients of friction on the surface and interface stresses, surface and interface contact boundaries, powers of stress singularities, weight function and to provide highly parametric benchmark results for tribological community in designing wear resistant systems.

Thermo-mechanical post-critical analysis of nonlocal orthotropic plates

A closed-form analytical solution for critical temperature and nonlinear post-critical temperature-deflection behaviour for nonlocal orthotropic plates subjected to thermal loading is presented. The long-range molecular interactions are represented by a nonlocal continuum framework, including orthotropy. The Von-Karman nonlinear strains are employed in deriving the governing equations. An approximate solution to the system of nonlinear partial differential equations is obtained using a perturbation type method. Series expansions up to second order of the associated field variables and the load parameter, dictating nonlinearity are employed. The behaviour in the post-critical regime is illustrated numerically by adopting an example of orthotropic Single Layer Graphene Sheet (SLGS), a widely acclaimed nano-structure, often modelled as plate. Post-critical temperature-deflection paths are presented with special emphasis on their post-critical reserve in strength and stiffness. Influence of aspect ratio and behaviour in higher modes are demonstrated. Implications of nonlocal interactions on the redistribution of in-plane forces are presented to show striking disparity with the classical plates. The obtained solution may serve as benchmark for verification of numerical solutions and may be useful in formulating simple design guidelines for plate type nanostructures liable to the thermal environment.

3D-dynamic modeling of cross-ply magneto-electro-elastic laminates based on the pseudo-Stroh formalism

This work presents methodological approaches to investigate 3D forced vibrations of simply supported multilayer magneto-electro-elastic plates. Multilayered multifunctional plates with orthotropic layers are considered. Harmonic transverse stress is used and the propagation matrix is presented at each layer. The dynamic responses of 3D displacements, stresses, and electric and magnetic displacements are analyzed with respect to the thickness direction and excitation frequency. This formalism is effective for thin and thick multilayered plates. Moreover, this study in forced vibrations makes it possible to overcome the 3D problems of computation time and storage commonly encountered in the solution of the eigenvalue problem.

Stresses of orthotropic laminated beams subjected to high temperature and mechanical load

Thermo-elastic analysis of simply-supported orthotropic laminated beams subjected to high temperature and mechanical load is presented on the basis of the exact two-dimensional thermoelasticity theory. The beam is composed of several orthotropic layers, each with temperature-dependent material properties. The governing equation for each layer is analytically solved using the state space method. The displacement and stress solutions of the beam are obtained using the transfer-matrix method. A numerical example is included to study the effects of temperature on the mechanical responses of a sandwich beam. The results reveal two main effects of temperature: (i) inducing deformations and stresses by itself; (ii) affecting the deformations and stresses induced by the mechanical load.

Exact formula for the horizontal-to-vertical displacement ratio of Rayleigh waves in layered orthotropic half-spaces.

This paper is concerned with the propagation of Rayleigh waves in a compressible orthotropic half-space coated by a compressible orthotropic layer. The main purpose of the paper is to establish an explicit exact formula for the H/V ratio (the horizontal-to-vertical displacement ratio) of the Rayleigh wave. This formula is derived by using the transfer matrix for an orthotropic elastic layer and by the effective boundary condition technique. The formula recovers the H/V formula by Malischewsky and Scherbaum [(2004). Wave Motion 40, 57-67] for the compressible isotropic case and the H/V formula by Love [(1911). Some Problems of Geodynamics (Cambridge University Press, Cambridge)] for the incompressible isotropic case. The effect of anisotropy on the H/V ratio's properties is examined and it is numerically shown that the anisotropy strongly affects the peak and zero frequency of the H/V ratio and its prograde domain as well. As the H/V ratio is a dimensionless quantity and it is easily experimentally measured, the obtained formula will be a useful tool for the nondestructive evaluation of mechanical properties of thin films deposited onto half-spaces. Two numerical examples are performed to show how the obtained formula can be used to extract the material properties of deposited layers from measured values of the H/V ratio.

An orthotropic non-local approach to modeling intra-laminar damage progression in laminated composites

A non-local macroscopic constitutive model is presented to simulate the progression of intra-laminar damage in laminated composite structures. A novel feature of the model is an orthotropic, non-local averaging scheme that is specially designed to capture the preferred path of damage / fracture propagation in laminated structures consisting of strongly orthotropic layers. In addition to maintaining the objectivity of the numerical results, through rendering them mesh size and mesh orientation invariant, the orthotropic nonlocal regularization results in a more realistic prediction of damage pattern and damage growth trajectory in orthotropic laminates. The new model, CODAM2, has been implemented in the open source finite element (FE) code, OOFEM, as well as the commercial explicit FE code, LS-DYNA. Numerical examples are presented to demonstrate the capabilities of CODAM2 in predicting the overall response and extent of damage in quasi-statically loaded, notched laminated specimens with various root notch radii ranging from sharp to blunt geometries. The developed methodology serves as a useful tool for robust and efficient inelastic analysis of large-scale composite structures leading to their damage tolerant design.

A scaled boundary finite element method for bending analysis of fiber-reinforced piezoelectric laminated composite plates

The main purpose of this study is to develop a high performance numerical model for the analysis of the coupled electromechanical effects of the fiber-reinforced piezoelectric laminated composite plate under the mechanical or electric loading. Considering that the piezoelectric laminated plate is made of N orthotropic layers with the material axes oriented arbitrarily with respect to the laminate coordinate, the three-dimensional (3D) theory of elasticity for the kth orthotropic (piezoelectric) lamina is applied for the establishment of the mathematic model by using the elastic displacements and the electric potential as the nodal degree of freedom. The scaled boundary finite element method (SBFEM) coupling with the precise integration technique (PIT) are used to obtain the solution. Under the formulations of the SBFEM, only the in-plane of the plate structure is discretized into two-dimensional (2D) elements thus the spatial dimension is reduced by one. By introducing a dual vector of the internal nodal force, the SBFEM governing equation is described as an analytic formulation along the through-thickness with a matrix exponential function which can be estimated with high accuracy using the PIT. In addition, the high order spectral element is applied for the description of the basic variables so that the laminate with the complicated curved boundaries can be accurately determined. At the same time, assuming that the generalized displacement field is a quadratic function with respect to the thickness coordinate of each layer, then the generalized strain and stress fields can be determined directly based on the strain-displacement relations and the constitutive relations in the scaled boundary coordinate system. Numerical results show that a good agreement can be achieved with few discrete nodes for both the thin and moderately thick piezoelectric laminated plate under different types of loads through comparison with the 3D exact solutions and good performance has also been validated. Moreover, two numerical examples including a square piezoelectric laminated plate with the complex configuration (involving the fiber orientation and the distribution of piezoelectric layers) and a circular/annular piezoelectric laminated plate are further discussed to demonstrate the applicability of the present formulations.

Dynamics of Love-Type Waves in Orthotropic Layer Under the Influence of Heterogeneity and Corrugation

The present problem deals with the propagation of Love-type surface waves in a bedded structure comprises of an inhomogeneous orthotropic layer and an elastic half-space. The upper boundary and the interface between two media are considered to be corrugated. An analytical method (separation of variables) is adapted to solve the second order PDEs, which governs the equations of motion. Equations for particle motion in the layer and half-space have been formulated and solved separately. Finally, the frequency relation has been established under suitable boundary conditions at the interface of the orthotropic layer and the elastic half-space. Obtained relation is found to be in good agreement with the classical case of Love wave propagation. Remarkable effects of heterogeneity and corrugation parameters on the phase velocity of the considered wave have been represented by the means of graphs. Moreover, the group velocity curves are also plotted to exhibit the profound effect of heterogeneity considered in the layer. Results may be useful in theoretical study of wave propagation through composite layered structure with irregular boundaries.

Two-dimensional linear models of multilayered anisotropic plates

A two-dimensional model describing the multilayered anisotropic plate deformations is proposed. The plate is assumed to consist of some orthotropic layers with arbitrary orientation of axes relative to the plate frame. The studied multilayered plate is replaced by the equivalent plate composed of a monoclinic material with piecewise elastic modules. An asymptotic solution is constructed for long-wave deformations. This problem was solved earlier in the first approximation; however, the obtained solution is not applicable for the case in which the stiffness of layers differs essentially from each other. The second asymptotic approximation is constructed in the present paper. It takes into account the effects of transversal shear and the normal fibers extension. Some special cases resulting in simple equations are studied in detail. The asymptotic solution error is estimated by comparison with the exact three-dimensional solutions for some test examples.

Simultaneous optimization of elastic constants of laminated composites using artificial bee colony algorithm

Multifunctional application of laminated composites requires multi-objective optimization of their characteristics. In this paper, the ply angles of laminated composites are determined in order to maximize the effective in-plane elastic constants simultaneously. These constants are determined by considering a representative small element of the laminated composite and imposing the conditions of uniformity of out-of-plane stresses and in-plane strains at orthotropic layer interfaces. Then, by combining the artificial bee colony algorithm and various multi-objective optimization methods, the optimal ply angles and the corresponding co-optimized constants are determined. The correctness and accuracy of the method is verified not only by providing a comparison with the existing results but also by solving a known problem. The results are presented and discussed, considering different multi-objective optimization problems. The results show that the increasing number of layers in the problem of simultaneous optimization of Young’s moduli not only does result in finding more Pareto optimal solutions in the feasible objective region but also shifts the Pareto frontier toward the utopia point. However, when the shear modulus optimization is engaged in the problem, using more than two layers only leads to obtaining more Pareto optimal answers.

Critical elastic parameters motivating divergence instability of frictional composite infinitely long media

The current study addresses the occurrence of smooth dynamic paths which rapidly diverge the mechanical system away from equilibrium configuration in an exponentially (non-oscillatory) path. The explored paths begin arbitrarily in the neighborhood of equilibrium states of an infinite orthotropic composite elastic layer. The subject treated here is handled from both the analytical and the numerical models. The divergence instability problem is reformulated as a nonlinear constrained optimization problem. Fortran implementation of sequential quadratic programming optimization algorithm for finding the minimum coefficient of friction for the onset of instability and the corresponding parameters is exploited. A finite element model is worked out to approximate the continuum. The results of several sets of numerical experiments for finding the minimum coefficient of friction for the onset of instability and the corresponding parameters are presented. The finite element method results were computed and compared with the analytic solutions: A very good matching between numerical and analytical results was obtained. It is concluded that, for normal relationships among elastic material properties and specific orthotropic directions, significantly surprising very low thresholds of the coefficient of friction were required for the onset of divergence instability.

Natural Frequency Analysis of Functionally Graded Orthotropic Cross-Ply Plates Based on the Finite Element Method

This paper aims to present a finite element (FE) formulation for the study of the natural frequencies of functionally graded orthotropic laminated plates characterized by cross-ply layups. A nine-node Lagrange element is considered for this purpose. The main novelty of the research is the modelling of the reinforcing fibers of the orthotropic layers assuming a non-uniform distribution in the thickness direction. The Halpin–Tsai approach is employed to define the overall mechanical properties of the composite layers starting from the features of the two constituents (fiber and epoxy resin). Several functions are introduced to describe the dependency on the thickness coordinate of their volume fraction. The analyses are carried out in the theoretical framework provided by the first-order shear deformation theory (FSDT) for laminated thick plates. Nevertheless, the same approach is used to deal with the vibration analysis of thin plates, neglecting the shear stiffness of the structure. This objective is achieved by properly choosing the value of the shear correction factor, without any modification in the formulation. The results prove that the dynamic response of thin and thick plates, in terms of natural frequencies and mode shapes, is affected by the non-uniform placement of the fibers along the thickness direction.

Modal and aeroelastic analysis of trapezoidal corrugated-core sandwich panels in supersonic flow

Corrugated-core sandwich panels have a great potential in the application to acreage thermal protection system of aerospace vehicles. However, due to their structural complexity, it takes high computational cost to conduct modal analysis with detailed 3D finite element models and the aeroelastic behaviors of these sandwich structures have not been fully investigated. In this paper, an analytical model of a trapezoidal corrugated-core sandwich panel using homogenization technique and layerwise theory is presented for modal and aeroelastic analysis. The trapezoidal corrugated core is homogenized as an equivalent orthotropic layer based on an energy approach and the overall sandwich panel is treated as a three-layer continuum. The proposed layerwise theory which is developed for a sandwich plate, assumes higher-order displacement field for the core layer and first-order displacement field for the top and bottom layers. The unsteady aerodynamic pressure is evaluated by the supersonic Piston theory. An eight-noded C0 isoparametric element with 13 degrees of freedom per node is used for finite element method. The accuracy and reliability of the present method are verified by comparing natural frequencies and mode shapes as well as flutter speeds with those obtained from commercial software. Parametric studies concerning different boundary conditions and parametric variables are also conducted. It is shown that the proposed method has sufficient accuracy and requires less computational effort, providing a theoretical basis for the utilization of the trapezoidal corrugated-core sandwich panel in aircraft designing.

Solution to the Task of Elastic Axial Compression–Tension of the Composite Multilayered Cylindrical Beam

The authors present an accurate solution to the task of elastic axial compression (tension) of the multilayered cylindrical beam with axial orthotropic layers surrounding a central core. The description of the geometry and structural framework of the beam requires the employment of the circular cylindrical system of coordinates where the mechanical characteristics of its inhomogeneous materials serve as the functions of the only variable. The task is solved via direct integration of the entire system of equations of the theory of elasticity within the selected system of coordinates upon the condition of rigid contact at the interfaces of the layers. The analytical relations for all the components of the features of the stress-strain state are obtained, their application is illustrated by the results of the solution to the test task of compression of the four-layered beam with the isotropic core.

Three-dimensional exact hygro-thermo-elastic solutions for multilayered plates: Composite laminates, fibre metal laminates and sandwich plates

Three-dimensional (3D) exact hygrothermal elasticity solutions are developed to study the behaviour of simply supported rectangular multilayered composite plates, considering any combination of orthotropic and/or isotropic layers, under hygro-thermo-mechanical loadings. In agreement with the 3D exact elasticity solutions derived by Pagano in the early 1970s, the form of the through-thickness exact solutions of any given layer depends, in fact, on whether the layer material is isotropic or (at most) orthotropic. The 3D exact thermoelastic solutions found, already available, consider either only isotropic layers or only (at most) orthotropic composite layers. This work aims to be all-inclusive, in addition to a further development by considering hygrothermal elasticity. The results here presented consider composite laminates, fibre metal laminates and sandwich plates with different side-to-thickness ratios under a series of hygro-thermo-mechanical loadings. A transverse mechanical load and/or thermal load and/or hygroscopic load are imposed on the multilayered plate top and bottom surfaces to study the effects of hygrothermal environments in the multilayered plate behaviour. These effects are demonstrated by the through-thickness distributions of displacements, stresses, temperature and weight percent moisture content for the selected multilayered plates under different hygro-thermo-mechanical loadings, which may serve as 3D benchmark exact solutions.

Frictional moving contact problem of an orthotropic layer indented by a rigid cylindrical punch

In this study frictional contact problem of an orthotropic layer indented by a rigid moving cylindrical punch is considered. The rigid punch moves over the orthotropic layer with a constant subsonic velocity. The layer is fully bonded to the rigid foundation and the external and tangential forces are applied via rigid punch. Utilizing the Fourier integral transform, the plane contact problem is reduced to a second kind integral equation in which the unknowns are the contact stress and contact widths. To solve the singular integral equation numerically Gauss-Jacobi integration formula is employed. Two material type which are Glass-Epoxy (GI/Ep) and Boron-Aluminum (B/Al) are chosen two analyze the effect of the moving velocity V2ρ/C66, the friction coefficient η and the punch radius R/h on the contact width, contact stress and in-plane stress.

On the plane frictional contact problem of a homogeneous orthotropic layer loaded by a rigid cylindrical stamp

In this study, contact problem for a homogeneous orthotropic layer loaded by a rigid cylindrical stamp is considered. The rigid cylindrical stamp slides over the contacting medium whose bottom surface is fixed to the ground in all directions. Using the integral transformation technique, the contact problem is formulated analytically into a singular integral equation. The resulting integral equation is converted to algebraic equations by using Gauss–Jacobi integration formulas and solved numerically. In addition to the analytical formulation, a finite element method (FEM) study is also conducted. The results that are obtained using FEM are compared with the results found using analytical formulation. It is found that the results obtained from analytical formulation and FEM study are in good agreement with each other. The primary intention of this paper is to demonstrate the effects of orthotropic material properties, geometrical properties and the coefficient of friction on the stresses generated due to the sliding motion of the rigid cylindrical stamp. The results of this study may provide benchmark results for engineers to be used in tribology applications involving friction and wear mechanisms.

Development of Large-Scale Finite Element Solver for Wind Turbine Blade Structure Using Balancing Domain Decomposition Methods

In a wind turbine blade, laminated plates consisting of fiber reinforced plastic materials are generally used due to its high specific strength. We have been developing a large-scale finite element solver to analyze the wind turbine blade structure. For such a structure, the laminated finite element is frequently used in modeling. Each laminated finite element has multiple layers, each of which is an orthotropic body in order to model the layered fiber reinforced materials with different fiber directions. Also, since a realistic wind turbine blade structure generally requires a large number of finite elements for discretization, we adopted a domain decomposition method, which is one of the numerical methods to parallelize the finite element method for large-scale analyses. Domain decomposition methods have been studied mainly in the mathematics field and applied to large-scale finite element analyses of realistic structures. However, the application studies do not sufficiently cover practical engineering models such as the model with laminated finite elements with orthotropic layers. Some researchers in the field of domain decomposition methods stated that the change or the jump of material parameters may degrade convergence performance of the iterative solver. Therefore, we investigated the convergence performance in the present study. The convergence performance of the balancing domain decomposition preconditioner and its variations implemented in our solver were compared. The numerical tests show that thin laminated finite elements with high aspect ratios significantly increase the number of iteration counts. Moreover, the consideration of material parameter changes in the balancing domain decomposition preconditioner (the Scaled-BDD method) attained a smaller number of iteration counts than the straightforward implementation.