Orthotropic Composite Plates Research Articles

Stresses and Strains in Thick Perforated Orthotropic Plates

AbstractStress and strain concentrations and in-plane and out-of-plane stress constraint factors associated with a circular hole in thick, loaded orthotropic composite plates are determined by three-dimensional finite element method. The plate has essentially infinite in-plane geometry but finite thickness. Results for Sitka spruce wood are emphasized, although some for carbon-epoxy composites are included. While some results are similar to those for isotropy, there are significant consequences due to material orthotropy. Maximum stress and strain concentration factors occur at midplane for thin plates but closer to the external traction-free surfaces for thick plates. These factors decrease as the plate surface is approached and reach lower values unrepresentative of the maximum values. Differences between the midplane and/or maximum and surface stress or strain concentration factors in Sitka spruce, range from 8% if the wood grain is parallel to the vertically applied load to 15% when the grain is perpe...

Wave mode extraction from multimodal wave signals in an orthotropic composite plate

In this paper the post-processing procedure based on the mode orthogonality is applied to extract individual waveforms at a composite plate edge from multimodal signals. To obtain the amplitudes of individual modes, numerically predicted modal through-thickness stress and displacement field values are used in the orthogonality relation. The performance of the mode extraction technique is evaluated by processing signals obtained from Finite Element (FE) modeling and experimental measurements. The propagation of the overlapping wave packets of Lamb modes S0 and A0 is considered along the fiber direction and perpendicular to that direction. The required experimental two-dimensional displacement components at the plate edge are measured by 3D Scanning Laser Doppler Vibrometer (3D SLDV). It is demonstrated that S0 mode can be extracted very well from the signal but A0 mode with slightly poorer accordance with the original waveforms and numerical predictions.

A dimensionless analysis of shear modulus and stress distribution for anisotropic materials

ABSTRACTThe effect of various off­axis angles and lamina material properties on the shear modulus and the stress distribution of a circular cut-out in an orthotropic composite plate under shear loadings are presented. Based on the generalized Hooke's law, the generalized plane stress condition and the complex variable method, a dimensionless analysis is used to evaluate the influence of various elastic moduli E1, E2, G12 and ν12 on the laminate shear modulus and the stress distribution along the boundary of the circular cut-out within the laminate under shear loading. The results obtained from this dimensionless analysis provide a set of general design guidelines for structural laminates with high-precision requirements in engineering applications.

A dimensionless analysis of stress distribution for constrained pressure in an orthotropic plate

The effect of the properties of lamina material on the stress distribution of the orthotropic composite plate containing a circular cut-out under constrained pressure is investigated. Based on the generalised Hooke's law, the generalised plane stress condition and the complex variable method, a dimensionless analysis is used to evaluate the influence of various elastic moduli E1, E2, G12 and ν12 on the stress distribution along the boundary of the cut-out in the composite plate under constrained pressure. The results obtained from this dimensionless analysis provide a set of general design guidelines for structural laminates with high precision requirements in the engineering applications.

An exact spectral-dynamic stiffness method for free flexural vibration analysis of orthotropic composite plate assemblies – Part II: Applications

An exact spectral-dynamic stiffness method (S-DSM) for free vibration analysis of composite plates and plate assemblies has been proposed in Part I of this two-part paper. The main purpose of this Part II paper is twofold: (i) To validate and demonstrate the superiority of the proposed S-DSM and (ii) To establish exact benchmark solutions for free vibration of composite plate-like structures. The S-DSM is applied to a number of problems covering orthotropic composite plates and plate assemblies. It is demonstrated that the S-DSM gives exact solutions with high computational efficiency within low as well as high frequency ranges. The applications are completely general and the new development can handle complex plate shapes with any boundary conditions.

Residual Stress Determination in Orthotropic Composites by Displacement Measurements Near Through Hole

Novel method for a determination of residual stresses in orthotropic composite plates based on local displacement measurements by electronic speckle-pattern interferometry is developed and verified. The values of hole diameter increments in principal residual stress directions serve as initial experimental deformation. The technique is based on analytical solution of S.G. Lekhnitsky, which describes a stress concentration along the edge of central open hole in rectangular orthotropic plate under one-axis tension in arbitrary direction. A situation when principal directions of residual stresses coincide with principal axes of anisotropy is considered. The relations, which connect initial experimental data with residual stress components, are unequivocally solution of the properly posed inverse problem. An availability of interference fringe patterns, a quality of which is high enough for reliable recognizing of fringe orders at the hole edge immediately, is the essential experimental foundation of the approach developed. The accuracy of the proposed method has been assessed by two different ways. The results of residual stresses determination in rectangular composite plate are presented.

Antiplane Problem of Periodically Stacked Parallel Cracks in an Infinite Orthotropic Plate

The antiplane problem of the periodic parallel cracks in an infinite linear elastic orthotropic composite plate is studied in this paper. The antiplane problem is turned into the boundary value problem of partial differential equation. By constructing proper Westergaard stress function and using the periodicity of the hyperbolic function, the antiplane problem of the periodic parallel cracks degenerates into an algebra problem. Using the complex variable function method and the undetermined coefficients method, as well as with the help of boundary conditions, the boundary value problem of partial differential equation can be solved, and the analytic expressions for stress intensity factor, stress, and displacement near the periodical parallel cracks tip are obtained. When the cracks spacing tends to infinity, the antiplane problem of the periodic parallel cracks degenerates into the case of the antiplane problem of a single central crack.

Identification of material properties of orthotropic composite plate using hybrid non-destructive evaluation approach

Identification of material properties is one of the key issues in composite materials research. The mechanical properties of composite materials depend on diverse factors such as configuration of the laminates, constituent materials used and production method adopted. Conventional testing approach tends to be time-consuming, expensive and destructive. As an alternative, a rapid, inexpensive, hybrid and non-destructive evaluation approach which utilises experimental modal analysis and finite element analysis is proposed. Experimental modal data which consist of natural frequencies and mode shapes of an orthotropic composite plate are utilised for correlation purpose with its finite element model. This finite element model of the composite plate is continuously updated and achieves less than 5% in difference of natural frequencies and over 70% in modal assurance criterion. Material properties such as Young's moduli, in-plane shear modulus and Poisson ratio of the composite plate are then successfully determined using the well-correlated FE model.

An engineering formula for the stress concentration factor of orthotropic composite plates

An engineering formula for the theoretical stress concentration factor of orthotropic notched plates under tension is provided, as a function of the material elastic constants and the Kt of the corresponding isotropic case. The accuracy and limits of applicability of the new solution are discussed by comparison to data from the literature and results from FE analyses on notched geometries of practical interests. The proposed solution represents a very useful tool to estimate the stress concentration factor of notched orthotropic plates, composite orthotropic laminae, orthotropic unidirectional laminates and homogenised orthotropic composite laminates.

A dimensionless analysis of Poisson’s ratio and stress distribution for anisotropic materials

The effect of various offaxis angles and lamina material properties on the Poisson’s ratio and the stress distribution of a circular cutout in an orthotropic composite plate under off-axis loading are presented. Based on the generalized Hook’s law, the generalized plane stress condition and the complex variable method, a dimensionless analysis is used to evaluate the influence of various elastic moduli E1, E2, G12, and ν12on the Poisson’s ratio and the stress distribution along the boundary of the circular cutout under various off-axis loadings. The results obtained from this dimensionless analysis provide a set of general design guidelines for structural laminates with high precision requirements in the engineering applications.

Bending solution of third-order orthotropic Reddy plates with asymmetric interfacial crack

In this paper Reddy’s third-order shear deformable plate theory is applied to asymmetrically delaminated orthotropic composite plates under antiplane–inplane shear fracture mode. A double-plate system is utilized to capture the mechanical behavior of the uncracked plate portion. An assumed displacement field is used and modified in order to satisfy the traction-free conditions at the top and bottom plate boundaries. Moreover, the system of exact kinematic conditions was also implemented into the novel plate model. An important improvement of this work compared to previous papers is the continuity condition of the shear strains at the interface of the double-plate system. Applying these conditions it is shown that the nineteen parameters of the third-order displacement field can be reduced to nine. Using the simplified displacement field the governing equations are derived, as well. The solution of a simply-supported delaminated plate is presented using the state-space model and the displacement, strain and stress fields are determined, respectively. The energy release rate and mode mixity distributions are calculated using the 3D J-integral. The analytical results are compared to those by finite element computations and it is concluded that the present model is the most accurate one among the previous plate theory-based approaches.

Stress and fracture analysis in delaminated orthotropic composite plates using third-order shear deformation theory

The third-order shear deformable plate theory is applied in this work to calculate the stresses and energy release rates in delaminated orthotropic composite plates with straight crack front. The delaminated parts are modeled by the general third-order plate theory, while a double-plate model with interface constraint is developed for the uncracked portion of the plate. The governing equations of the uncracked part are formulated by considering the equilibrium and the displacement continuity along the interface. As an example, a simply-supported delaminated orthotropic plate subjected to a point force is solved adopting Lévy plate formulation and the state-space approach. The mode-II and mode-III energy release rate distributions along the crack front were calculated by the J-integral. To verify the analytical results the 3D finite element model of the plate was constructed and the energy release rates were calculated by the virtual crack-closure technique. A previous second-order plate theory solution was also utilized in the course of the comparison. The results indicate a good agreement between analysis and numerical computation and that third-order theory is better in some cases than the second-order approximation.

A Dimensionless Analysis of Stress Distribution for Hydrostatic Tension in an Orthotropic Plate

The effect of lamina material properties on the stress distribution of the orthotropic composite plate containing a circular cutout under hydrostatic tension is investigated. Based on the generalized Hooke’s law, the generalized plane stress condition and the complex variable method, a dimensionless analysis is used to evaluate the influence of various elastic moduli E1, E2, G12 and ν12 on the stress distribution along the boundary of the cutout in the composite plate under hydrostatic tension. The results obtained from this dimensionless analysis provide a set of general design guidelines for structural laminates with high precision requirements in the engineering applications.

A Dimensionless Analysis of Young's Modulus and Stress Distribution for Orthotropic Materials

The effect of various offaxis angles and lamina material properties on the axial and transverse modulus of the orthotropic lamina under off-axis loading is studied. Also, the stress distribution of the orthotropic composite plate containing a circular cutout with normal pressure distributed uniformly along the opening edge is investigated. Through the generalized Hooke’s law and plane stress condition, a dimensionless analysis is used to evaluate the influence of various elastic moduli E1, E2, G12 and ν12 on the axial and transverse modulus of the orthotropic lamina under various off-axis loadings. Moreover, based on the generalized Hooke’s law, the generalized plane stress and the complex variable method, a dimensionless analysis is used to evaluate the influence of various elastic moduli E1, E2, G12 and ν12 on the stress distribution along the boundary of the circular cutout of the orthotropic plate with normal pressure distributed uniformly along the opening edge. The results obtained from this dimensionless analysis provide a set of general design guidelines for structural laminates with high precision requirements in the engineering applications.

Application of Reddy's third-order theory to delaminated orthotropic composite plates

In this work the third-order shear deformation theory by Reddy is applied and modified to analyze delaminated orthotropic composite plates . The delaminated plate portion is captured by Reddy's traditional theory, while a novel double-plate system is developed for the undelaminated part. It is shown that in the uncracked part four conditions are required to satisfy in symmetrically delaminated plates. The conditions involve the imposition of traction-free boundaries and the interface constraints. These four conditions enable the reduction of the parameters from nine to five in the displacement field. The governing equations show significant coupling among the stress resultants of the uncracked portion, that has to be considered in the continuity conditions between the delaminated and undelaminated parts. To demonstrate the application of the present model a simply-supported delaminated plate subjected to a concentrated force is analyzed. The distribution of the mode-II and mode-III energy release rates and their ratio are calculated using the 3-dimensional J -integral. The finite element model of the plate is also created using brick-type elements. The comparison of the analytical and finite element results shows very good agreement. It is shown that the deformations around the delamination front can be captured by the third-order plate theory with high accuracy. • We derive the governing equations of modified Reddy's theory. • The displacement field is based on the interface constraints. • The delamination causes sudden changes in the strains and stresses. • Third-order plate theory is required to capture the crack tip deformations.

PLATE BUCKLING ANALYSIS USING A GENERAL HIGHER-ORDER SHEAR DEFORMATION THEORY

The buckling of higher-order shear plates is studied in this paper with a unified formalism. It is shown that usual higher-order shear plate models can be classified as gradient elasticity Mindlin plate models, by augmenting the constitutive law with the shear strain gradient. These equivalences are useful for a hierarchical classification of usual plate theories comprising Kirchhoff plate theory, Mindlin plate theory and third-order shear plate theories. The same conclusions were derived by Challamel [Mech. Res. Commun.38 (2011) 388] for higher-order shear beam models. A consistent variational presentation is derived for all generic plate theories, leading to meaningful buckling solutions. In particular, the variationally-based boundary conditions are obtained for general loading configurations. The buckling of the isotropic or orthotropic composite plates is then investigated analytically for simply supported plates under uniaxial or hydrostatic in-plane loading. An analytical buckling formula is derived that is common to all higher-order shear plate models. It is shown that cubic-based interpolation models for the displacement field are kinematically equivalent, and lead to the same buckling load results. This conclusion concerns for instance the plate models of Reddy [J. Appl. Mech.51 (1984) 745] or the one of Shi [Int. J. Solids Struct.44 (2007) 4299] even though these models are statically distinct (leading to different stress calculations along the cross-section). Finally, a numerical sensitivity study is made.

Analysis of classical and first-order shear deformable cracked orthotropic plates

The Kirchhoff and Mindlin plate theories are applied in this study to calculate the stresses and the energy release rates in delaminated orthotropic composite plates. A novel double-plate system is developed with the imposition of the kinematic continuity constraints in the interface plane. The governing equations of the system were derived in both cases. As a demonstrative example a simply-supported plate subjected to a point force was analyzed using Lévy plate formulation and the problem was solved by a state-space model. The distribution of the stress resultants and the interlaminar stresses in the uncracked part were also determined. Moreover, the distributions of the mode-II and mode-III energy release rates along the crack front were calculated by the J-integral. The 3D finite element model of the plate was created providing reference data for the analytical model. The results show that the displacement and stress fields obtained from the Kirchhoff and Mindlin theories are quite similar, but in the case of the energy release rates, transverse shear effect is necessary to consider to obtain reasonably good agreement between the analytical and numerical results.

Comparison of finite element models simulating the interaction of ultrasonic guided waves with sites of disbonding in composites

Disbonding in composite structures is a serious defect which can dramatically reduce the structures’ life, and can lead to catastrophic failure. To avoid this, non-destructive testing or structural health monitoring techniques are needed. One such technique involves ultrasonic guided waves, which has recently found use in this field thanks to its ability to inspect in non-accessible areas and over long distances. Numerical models are often used because they can help explain experimental results, and offer the ability to simulate different damage scenarios, predicting results with less cost than experiments. In this work, sites of disbonding between an orthotropic composite plate and an isotropic polyamide plate were modeled via the finite element method. Several methods for modeling the damage site are employed, and results are compared with experimental ones. Model types range from the introduction of a geometrical void at the interface boundary to addressing boundary conditions between adjacent layers. The amount of mode conversion after interaction with the damage site is used to evaluate the validity of each model type. Results are discussed in terms of computational effort and accuracy in predicting true physical behavior.

An approximate model of cylindrical and spherical contact on composite laminates of finite thickness

An empirical force–contact area relationship for the two-dimensional (2D) plane strain line contact between a stiff cylindrical indenter and a composite laminate of finite thickness supported on a rigid substrate is presented. The validity of a 2D Hertzian line contact force–contact length relation originally meant for isotropic contact problems on laminates of finite thickness is first investigated. The same relationship can be applied to orthotropic composite plates provided that an equivalent indentation modulus is known. The indentation modulus can then be corrected for the finite thickness of the plates. The results correlate well with experimental observation. Similar procedures can be performed to find the corresponding correction for a 3D axisymmetric contact problem.

Interface fracture in orthotropic composite plates using second-order shear deformation theory

The second-order shear deformation theory is used in this study to calculate the stresses and the energy release rates in orthotropic composite plates. A novel double-plate system is utilized with the imposition of the proper kinematic constraints in the interface plane of a double-plate system. The governing equations of the system were derived and as a demonstrative example a simply-supported plate subjected to a point force was analyzed. Using Lévy plate formulation, the plate problem was solved by a state-space model, incorporating four different regions. The distribution of the stress resultants and the interlaminar stresses in the uncracked part were also determined. Moreover, the distributions of the mode-II and mode-III energy release rates along the crack front were calculated by the J-integral. The 3D finite element model of the plate was created providing reference data for the analytical model. The results show reasonably good agreement between the analytical and numerical results. Also, the present model eliminates the physical inconsistency of previous models and reveals that under mixed-mode II/III condition, the energy release rate is not contributed by the bending, twisting moments and shear forces at all.