• View

  • Download

Embed Size (px)


  • Compurers & S~luclures Vol. 38, No. 3, pp. 361-376, 1991 Printed in Great Britain.

    o!Ms-7949/91 s3.00 + 0.00 0 1991 Perpnon Press pk


    S. TOLS~N and N. ZASARAS Department of Mechanical Engineering, University of Minnesota, 111 Church Street SE.,

    Minneapolis, MN 55455, U.S.A.

    (Received 8 February 1990)

    Abstract-Acceptance and utilization of composite materials require confidence in their load carrying capacity. Therefore, it is desirable to develop a computational model capable of determining the ultimate strength of laminated composite plates under conditions of complex loading. A new seven degree of freedom finite element model for laminated composite plates is developed. The model utilizes three ~sp~~ments, two rotations of normals about the plate midplane, and two warps of the normals, to accurately and efficiently determine the laminate- stresses, Based on these stresses a failure model for determining first ply failure (FPF) and last ply failure (LPF) by a progressive stiffness reduction technique has been developed. The progressive failure model produces results in good agreement with experimental data. The calculated FPF and LPF results form lower and upper bounds within which the true load carrying capacity lies.


    The objective of this paper is to construct a method- ology for determining when failure of a composite plate occurs and to develop the mathematical and computations tools to carry out the analysis. Research into the analysis of plates has been extensive. Kirchhoff was one of the first to develop a comprehensive theory for plates. His classical plate theory yields inaccurate results for thick plates (S = a/h < 20) and for thin plates with holes. Reissner was one of the first to develop a plate theory that considered transverse shear deformations in static analysis [ 1,2]. Mindlin later expanded the shear deformation theory to accommodate rotary inertia terms [3]. These first order shear deformation theories fail to produce accurate results for very thick plates (S c 10) and they do not satisfy the condition of vanishing transverse shear stress on the top and bottom plate surfaces. They also require the use of a shear correction factor to obtain accurate stress results.

    Higher order plate theories seem to begin with Levinson [4]. The displacement field for his theory includes up to cubic variation in the through- thickness direction. The theory predicts zero trans- verse shear stresses on the top and bottom plate surfaces, does not require a shear correction factor, and is accurate for both thin and thick piates.

    The preceding theories were all developed for isotropic materials, but each one has been generalized for anisotropic materials. Reissner and Stavsky devel- oped an orthotropic laminated plate theory based on classical plate assumptions [S]. A first order shear deformation theory for laminated anisotropic plates was presented by Yang ef al. [6]. Finally, higher order

    shear deformation theories for laminated composite plates were developed concurrently by Reddy [A and Murthy [8].

    Reddys penalty plate bending fmite element tech- nique is one of the first finite element models for first order shear deformation theory [9]. The theory uses eight-noded quadrilateral elements with five degrees of freedom at each node. The degrees of freedom include three displacements and two rotations. Higher order theories contain second order derivatives of the transverse displacement, Therefore, a finite element model requires C conti- nuity. Construction of such an element requires many degrees of freedom at each node and, therefore, much computer processing time. In an attempt to efficiently solve this problem, Reddy developed a mixed formu- lation for his simple higher order theory [IO]. The resulting finite element model consists of 11 degrees of freedom per node, three displacements, two ro- tations, and six moment resultants. While this may have been an improvement, it remains computation- ally intensive and a more efficient alternative is required.

    Failure of a composite material can be described in a variety of ways. First ply failure (FPF) occurs when initial failure of a single layer in a laminate fails in either the fiber direction or in the direction perpen- dicular to the fibers. Last ply failure (LPF) occurs after the structure has degraded to the point where it is no longer capable of carrying additional load. Most authors beheve failure is caused by (1) lon~tudinal tensile loads in the fiber direction, (2) longitudinal compressive loads in the fiber direction, (3) tensile loads transverse to the fibers, (4) compressive loads transverse to the fibers, or (5) shear loads [l 1,121. Utilizing these five loading cases, the failure modes of


  • 362 S. TOLSON and N. ZABARAS

    a composite material can be described as (1) breaking of fibers, (2) cracking of the matrix, (3) separation of the fiber and the matrix (debonding), and (4) separ- ation of one iamina from another (delamination) (131. The consequence of individual failure modes is not of great interest for FPF, but will be of great importance in determining LPF.

    Standard laminate strength analysis is the most common and oldest of the composite analysis methods [II, 121. The method neglects local effects such as fiber misalignment, material discontinuities, and free edge effects, and assumes that the stiffness of the laminate receives no contribution from failed layers.

    The first finite element based failure analysis of composite materials was performed by Lee [14]. Lee used his own direct mode dete~ning failure criterion and standard laminate strength analysis methods to determine the ultimate strength of plates with circular holes. The major drawback of a three- dimensional failure analysis such as Lees is the tremendous amount of memory space and calculation time required. Hwang and Sun attempted to improve the computational aspects of the three-dimensional formulation by incorporating a Newton-Raphson type iteration process [ 151.

    The search for more efficient finite element analysis of composite plates, therefore, leads to two- dimensional plate fo~ulations. Reddy and Pandey developed a first ply failure analysis of composite laminates based on first order shear deformation plate theory [16]. The limiting factor of this analysis is the inadequacy of first order shear deformation theory for thick composite plates.

    Engblom and Ochoa developed a two-dimensional plate analysis similar to the above, but with increased interpolation in the through thickness direc- tion [17, 181. Their analysis is carried out to LPF. The stiffness reduction and progressive damage accumulation are treated in a manner similar to standard laminate analysis. The generated stresses are less accurate than those obtained from higher order shear deformation plate theory. Also, since the interpolation is of a different order through the thickness than in the plane, the analysis cannot be easily adapted to elements other than the eight- noded variety discussed in their paper.

    The objective of this paper is to develop a two- dimensional finite element failure analysis for com- posite plates that is more accurate and more flexible than previously developed plate analyses, while at the same time more efficient than current higher order shear deformation or three-dimensional formu- lations. The plan of this paper is as follows. First, a new seven degree of freedom finite element composite plate formulation for laminate stress calculation will be presented. Then a brief review of some failure theories will be given. An algorithm for analyzing progressive failure in laminated plates will be given and tested on several problems with various stacking

    sequences and load conditions. experiments will be reported.

    Comparison with


    Kinematic assumptions

    The following displacement field is assumed


    + Z3t;,kY)


    + Z35&*Y)

    W& Y) = we, YX (1)

    where u, v, and w denote the displacements at the midplane (z = 0), and Y, r, and [ are appropriately selected functions of x and y.

    The above displacement field can be simplified by utilizing the condition that the transverse shear stresses a,, and cry2 vanish on the top and bottom surfaces of the plate. The final expressions for the displacement field in an orthotropic plate are as follows:



    w= WCG Y), (2)

    where YX and Y,, are rotations of the normals to the midplane about the x and y axes respectively, and 6, and {, describe the warping of the normal in the x and y directions, respectively. It can be seen that for this displacement field, normals to the midplane of the plate before deformation do not necessarily remain normal or straight after deformation.

    The strains can now be derived using the final form of the ~splacement field, in eqn (2).

    au ay, 3 ai, C,=z$-Z-g+Z z

    au ay if& 4&i-z ay ay

    -G =o

    y,,= -YX+; +3.r21,

    yxy ay ax _a,+_z(!T$+!s)+zf!!+~). (3)

  • Progressive failure in laminated composite plates 363

    Note that the assumption 6, = 0 is retained from first The reduced stiffness matrix, Q, from the material order shear deformation theory. coordinates to the global coordinates, can be calcu-

    lated as Constitutive model

    If 6, = 0 and a transversely isotropic material is assumed, the stress-strain relationship can be stated as[ll]

    [Ql= PIIQW-I. (6)

    where TV, = o,, 02 = (rm, a,