Singular full-field stresses in composite laminates with open holes
Introduction
Development of the methods, for an efficient stress analysis of the composite structures containing curvilinear edges such as cutouts, etc., is of significant practical interest. Ply level models of the laminated composites, according to which a lamina is modeled as a homogeneous orthotropic material, result in a singular stress behavior in the vicinity of the ply interface and laminate edge. The present article deals with a three-dimensional elasticity analysis in the presence of the singular stresses and focuses on representing the stress field by using a superposition of the asymptotic solution and polynomial spline approximation.
A valuable experience was gained due to considerable effort devoted to the solution of the straight free edge problems. A hybrid approximation, based on an assumed equilibrium stress field was proposed by Pagano, 1978a, Pagano, 1978b using Reissner’s variational principle. Without including the precise singular stress terms, highly accurate stress predictions for various laminates were demonstrated. The singular term of the asymptotic solution for the composite wedge near the ply interface and the wedge edge was obtained by Mikhailov (1979). The power of singularity was determined as a function of ply anisotropy and wedge angle. Independently, Wang and Choi (1982) constructed an infinite series elasticity solution for the same problem based on Lekhnitskii’s complex variable stress function. The singular stress term was precisely determined. A polynomial particular solution was added to satisfy the axial loading condition. Determination of the unknown multiplicative factors in the homogeneous solution, including the coefficient of the singular term, was accomplished by the boundary collocation method. The results for a [±45]s laminate were shown to converge using 30 eigenfunctions of the homogeneous solution. A hybrid finite element formulation (Tong et al., 1973) based on this solution was developed by Wang and Yuan (1983).
Folias (1992) and Wang and Lu (1993) considered stresses in laminated composites at the interface and the open hole edge. They showed that the zeroth order term of the asymptotic expansion of the three-dimensional elasticity equations in terms of the parameter λ=h/D (ply thickness/hole diameter) yields a two-dimensional elasticity problem. Thus, the singular stress term at the ply interface and the curvilinear edge is the same as that for the straight edge, provided the ply orientations are the same, relative to the tangent to the curved edge. However, extending these results to obtain a full-field solution is not trivial. The critical difference is that, the analytically obtained eigenfunctions of the asymptotic two-dimensional field do not satisfy the three-dimensional equations in any finite volume. Thus, no exact homogeneous solution is constructed in a finite volume surrounding the intersection of the hole edge and the orthotropic ply interface. It should be noted that an impressive convergence of the hybrid singular finite element formulation was demonstrated (Tong et al., 1973; Wang, 1983) for problems, where the assumed stress functions in relatively large singular elements actually provided elasticity solutions over the entire element. An assumed, displacement based finite element formulation including the singular asymptotic term was developed by Wang and Lu (1993), and the stress intensity factor for a ±45 laminate was obtained as a function of the circumferential coordinate. However, insufficient information was given to allow one to comment on the rate of convergence. It was noted that the asymptotic solution was included in the formulation only over a small region near the free edge of the cutout.
Iarve (1996) developed a B-spline based approximate three-dimensional solution for the multilayered composite laminates containing open holes. It was also shown that a two-dimensional problem (with θ as a parameter) identical to the one obtained asymptotically in Folias (1992), and Wang and Lu (1993) follows from the three-dimensional formulation assuming vanishing of the spatial derivatives in the circumferential (θ) direction. The numerically obtained stress distributions near the hole edge were compared to the stresses given by the singular term of the asymptotic solution. At the singularity, the polynomial spline approximation does not capture the directional non-uniqueness of the singular stress functions, Pagano and Kaw (1995), and resulted in an interfacial traction discontinuity. However, it was observed that the singular term of the asymptotic solution with appropriate coefficient and constant additive terms matched the full-field spline solution at approximately one half-ply thickness from the singular point. The surprisingly large area of agreement suggests that superposition of the singular term and the polynomial approximation, may be utilized for the determination of the stress intensity factor.
Morley, 1969, Morley, 1970 pioneered the idea of superposition of the analytical and finite element solution in the problems with local field singularities. His approach is based on the Rayleigh–Ritz method where the polynomial displacement approximation is enriched through the entire domain, by the local analytical (asymptotic) solution minus its finite element approximation. The analytical solution must satisfy the field equations and the homogeneous boundary conditions of the problem. The finite element approximation is obtained under the boundary conditions generated by the analytical solution with unit stress intensity factor. For sufficiently fine meshes, the analytical solution and its approximation will differ only in the vicinity of the singular point. The scaling factors, which are the coefficients of the additional terms used for enrichment of the finite element basis, are obtained through a variational procedure. Yamamoto and Tokuda (1973) applied this method to crack stress intensity factor determination, using a multiple term asymptotic expansion for the analytical solution. They used a boundary collocation method to obtain the coefficients of the terms containing analytical solutions.
For the curvilinear edge singularities considered in the present article, the analytical solution in the finite domain near the singularity is unknown. The asymptotic solution obtained in Folias, 1992, Wang, 1993, Iarve, 1996 is a two-dimensional solution in nature and cannot be used directly in the approach described in Morley, 1969, Morley, 1970 and Yamamoto and Tokuda (1973). It should be mentioned that Yamamoto and Sumi (1978) considered an axisymmetric problem of a twisted round isotropic bar with an annular crack. The asymptotic solution, which was used as the basis for the analytical solution near the crack tip, was equivalent to a local plane strain solution which did not satisfy the axisymmetric equilibrium equations throughout the domain. The asymptotic solution for the round isotropic bar problem, which was reduced to a single unknown function – the circumferential displacement component – was augmented by a higher-order term added to the asymptotic solution to satisfy the equilibrium equations. However, in a general orthotropic case, these complementary terms are not obvious and have not been reported in the literature.
The present article extends the superposition approach to problems where no analytical solutions in the finite domain are known. The model developed is based on Reissner’s variational principle and is intended to reflect the singularities, which arise at each interface at the boundary of the hole. The hybrid approximation functions to be developed, have the following characteristics:
(1) They include the asymptotic solution thus representing the directional non-uniqueness of the solution. It is only in this manner that one can embed the proper singular field. The fact that the asymptotic solution results from the three-dimensional problem by truncating the spatial derivatives in the circumferential direction (Iarve, 1996) will be used to construct hybrid stress functions.
(2) Two independent (B-spline) displacement functions are considered: One is related to the regular and the other to the singular portion of the stress field. It is undesirable to use the asymptotic displacement functions in the displacement approximation because, the calculation of their derivatives in the circumferential direction, required in the variational formulation, is only possible numerically. It is assumed that the displacements related to the singular stresses will also be approximated with splines. Thus, the approximations of stresses and displacements are made independently.
Examples to demonstrate the convergence with respect to meshing parameters will be considered.
Section snippets
Problem statement
Consider a rectangular N-layer laminate built of orthotropic layers with length L in the x-direction, width A in the y-direction, and thickness H. Individual ply thicknesses are hp=z(p)−z(p−1), where z=z(p) and z=z(p−1) are upper and lower surfaces of the p-th ply, respectively. The origin of the x, y, z coordinate system is in the lower left corner of the plate, as shown in Fig. 1. A circular opening of diameter D with the center at x=xc and y=yc is considered. Uniaxial loading is applied via
Asymptotic solution
Consider a region around the hole edge and the interface between plies p and p+1. A local coordinate system η, ψ is introduced in the radial cross section θ=const according to Eq. (2). In this coordinate system 0⩽ψ⩽π/2 in the upper ply and −π/2⩽ψ⩽0 in the lower one.
For an arbitrary function F,whereIn Cartesian coordinates, the derivatives can be calculated as
If η
Variational formulation
Displacement components are represented aswhere the displacements ui satisfy boundary conditions (1). The term uis is associated with singular stress components and the second term uir with bounded stress components. The stresses will be assumed asThe stresses σijhyb and displacements uis are independently assumed. The stresses σijr and displacements uir are related as follows:where Cijklq and αklq are elastic modulii and thermal expansion
Hybrid approximation
Consider the exact stresses associated with the displacement uis in the qth ply:The thermal expansion term is not included with the singular displacement portion, since it was accounted for in Eq. (9). The stresses resulting from displacement field uis are modified to include the singular asymptotic stress field (6). We shall calculate the strain field generated by the truncated derivatives of the displacements, as follows:where are
Governing equations
Taking into account Eqs. (15) and the change to independent displacement functions ui and uis, Eqs. (13a) and (13b) will attain the following form:where are based on truncated derivatives (4). Eq. (17) follows from Eq. (13a) after substituting Eq. (7) and allows one to calculate the total displacement in an independent problem under the given traction and displacement boundary conditions (1).
Determination of kp(θ)
The error in boundary conditions near the singularity from Eq. (17) is a result of approximating the directionally non-unique singular stress field by polynomials. The directional non-uniqueness means that for η→0, the stresses aijp may tend to ±∞ depending upon ψ. The polynomial approximation provides unique and finite stress values at every point of a given ply. The values of the stress intensity factors are obtained to enforce singlevaluedness of the non-singular portion
Spline approximation of displacement components
The x, y and z displacement components are approximated by using cubic spline functions in curvilinear coordinates. The total displacement is approximated aswhere X is a vector of three-dimensional spline approximation basis functions, and Ui are the unknown spline approximation coefficients. The non-square matrices Ci and constant vector E are defined so that approximation (24) is kinematically admissible, i.e., it satisfies boundary conditions (1) for arbitrary coefficients
[45/−45]s laminate
A square [45/−45]s plate similar to that in Wang and Lu (1993) is considered. The geometric properties are M, and ply thickness h=0.00254 M. Orthotropic ply properties were E1=138 GPa, E2=E3=14.5 GPa, G12=G13=G23=5.86 GPa, and ν12= ν13= ν23= 0.21, where index 1 corresponds to the fiber direction, and νij the poisson ratio meaning strain εi under uniaxial stress σj in contracted notations. The displacement boundary conditions (1) were applied so that u0/L=0.001. The
Conclusions
(1) A method of superposition of hybrid and displacement approximations was developed to provide accurate stress fields in the vicinity of the ply interface and the hole edge in a multilayered composite. The asymptotic analysis was used to derive the hybrid stress functions. The displacement approximation was based on polynomial B-spline functions.
(2) The coefficients of the singular terms in stress solution near the ply interfaces and the open-hole edge were determined in [45/−45]s and
Acknowledgements
The first author acknowledges the support of the Materials Directorate, Air Force Research Laboratory, Wright-Patterson AFB OH under Contract No. F33615-95-D-5029.
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