An evaluation of equivalent-single-layer and layerwise theories of composite laminates
J. N. Reddy Department of Mechanical Engineering, Texas A &M University, College Station, TX 77843-3123, USA
A review of equivalent-single-layer and layerwise laminate theories is presented and their computational models are discussed. The layerwise theory advanced by the author is reviewed and a variable displacement finite element model and the mesh superposition techniques are described. The variable displacement finite elements contain several different types of assumed displacement fields. By choosing appropriate terms from the multiple displacement field, an entire array of elements with different orders of kinematic refinement can be formed. The variable kinematic finite elements can be conveniently connected together in a single domain for global-local analyses, where the local regions are modeled with refined kinematic elements. In the finite element mesh super- position technique an independent overlay mesh is superimposed on a global mesh to provide localized refinement for regions of interest regardless of the original global mesh topology. Integration of these two ideas yields a very robust and economical computational tool for global-local analysis to deter- mine three-dimensional effects (e.g. stresses) within localized regions of interest in practical laminated composite structures.
I INTRODUCTION
Numerous laminate theories have been proposed to date to describe the kinematics and stress states of composite laminates. Most of these theories are extensions of the conventional, single-layer plate theories, which are based on assumed variation of either stresses or displacements through the plate thickness:
~i(x, y, z, t )= ¢°(x, y, t)+ z~k~(x, y, t)
+(z)2¢2(x,y, t)+ ... N
= E (zy¢~(x, y, t) (1) ] = 0
where fki(x, y, z, t) denotes either a stress or a displacement component in the plate, (x, y) are the inplane coordinates, z is the thickness coor- dinate, t denotes time, and ~ , ( j = 0, 1, 2, ..., N) are functions of x, y, and t. The series in eqn (1) can be terminated at a desired degree of the thick- ness coordinate, i.e. select the value of N (N-- 1, 2, and so on). The spirit of these theories is to reduce a three-dimensional problem to a two- dimensional one; see Refs 1-13 for pioneering works in the field. A review of refined theories of plates can be found in an article by Noor and Burton 14 and the author) 5
The governing equations of motion associated with the assumed displacement or stress field can be obtained using an appropriate principle of virtual work. 16 For example, when ¢i, ( i= 1, 2, 3) is a displacement component, the dynamic ver- sion of the principle of virtual displacements is used to derive 3 ( N + l ) equations of motion, which are usually expressed in terms of stress resultants through the thickness:
where o i denotes the ith stress component (i = 1, 2, ..., 6),
21
Oxx = 0-1, 0-yy = 0"2, Ozz = (13, Oyz = 0-4,
axz = as, 0"xy = a6 (3)
The stress components are assumed to be known in terms of the displacement functions #~ through the strain-displacement relations and the stress-strain relations. When #i, (i = 1, 2, ..., 6) is a stress component, the dynamic version of the principle of virtual forces is used to derive 6 ( N + I ) equations of motion. Of course, mixed variational principles can be used to derive the governing equations associated with assumed independent expansions of displacements and
stresses. Stress-based theories 6-~ are seldom used in practice because of the difficulty in developing reliable finite element models. The displacement- based theories ~H3 have received the most atten- tion from the computational mechanics community. In the rest of this paper only the dis- placement-based theories are discussed.
The classical plate theory is also based on the displacement field in eqn (4), but with the addi- tional assumption concerning the slopes 4~ and 42:
Ou~ Ou~ 4~ - Ox ' 42 - Oy (7)
2 DISPLACEMENT-BASED SINGLE-LAYER THEORIES
The displacement-based theory of order 1 (i.e. N = 1 ) is derived using the displacement field:
ul(x, y, z, t) = u]'(x, y, t) +z4,(x, y, t)
ll2(X, y, Z, t) = blO(X, y, t) + z42(x, y, t)
u3(x, y, z, t)=u°lx3~ , Y," t )+z43(x ,y , t) (4)
where u i is the displacement component along the xi-coordinate direction (x~ = x, x 2 = y, x 3 = z). The displacement field in eqn (4) implies that straight lines normal to the x - y plane before deformation remain straight after deformation. Comparing eqns ( 1 ) and (4), we have the following:
1 41'=u~ ', 4 i=4 , , ( i = 1 , 2 , 3 ) (5)
It is clear from the displacement field in eqn (4) that (u, v, w) are the displacements of a point on the z = 0 plane in the three coordinate directions:
o ui(x, y, z, t)Iz =0 (6a) Hi =
Also, we note that
:I°u,1 16bt ~i L O z j : : °
Geometrically, 41 and - ¢ 2 denote rotations about the y and x axes, respectively, and 43 denotes the elongation of a transverse normal at the point (x, y). There will be six equations of motion in the six generalized displacements (u °, 4i). The most commonly used first-order plate theory, 16'17 is based on the displacement field of eqn (4) with 43 =0 (i.e. transverse normals are inextensible); the number of equations of motion reduce to five in terms of (Ul °, u °, u 3, 42, 42). Since the trans- verse shear strains are constant through the thick- ness, the transverse stresses would also be constant through the thickness -- a contradiction with the elasticity solution. This discrepancy is remedied in the energy sense by introducing the shear correction factors.
i.e. transverse normals before deformation remain normal to the deformed surface at z = 0. In sum- mary, the classical plate theory is based on the hypothesis, known as the L o v e - K i r c h h o f f hypo- thesis, that straight lines normal to the x - y plane before deformation, (i) remain straight, (ii) in- extensible, and (iii) normal to the z = 0 plane after deformation. The classical plate theory is governed by three equations of motion in terms of ( . I ' , . " '1 2, /33)"
The second_order ~ ~. ~9 and third-order theories 2°-3° introduce additional unknowns that are difficult to interpret in physical terms. All theories in which the normality condition in eqn (7) is not invoked account for transverse shear and normal strains. If inextensibility of transverse normals is assumed, the transverse normal strain becomes zero. For example, the third-order theory o f Reddy 27,28 is based on the displacement field, 2°-26
ul(x, y, Z, t) o =u, (x , y, t) +Z41(x, y, t)
( 4 Ir x, 1 + z ,t + ox j
u2(x, y, z, t) ~t = u2(x, y, t) + z42(x, y, t)
( 4)[ + z 3 -~-~5 42(x, Y, t)+0u~
0y J
u3(x, y, z, t )=u° (x , y, t) (8)
The displacement field accommodates quadratic variation of transverse shear strains (and hence stresses) and vanishing of transverse shear stresses on the top and bottom of a general laminate com- posed of orthotropic layers. Thus, there is no need to use shear correction factors in a third- order theory. The theory was generalized in Ref. 31.
Theories higher than third order are not used because the accuracy gained is so little that the effort required to solve the equations is not justi- fied. In all conventional displacement-based
An evaluation of composite laminates 23
theories, one single expansion for each displace- ment component is used through the entire thick- ness, and therefore, the transverse strains are continuous through the thickness -- a strain state appropriate for homogeneous plates.
3 EQUIVALENT-SINGLE-LAYER LAMINATE THEORIES
Extension of the single-layer theories of homo- geneous plates to laminated composite plates is straightforward.~6,17.20-42 The only difference is in accounting for the varying layer thicknesses and material properties in the evaluation of the inte- grals in eqn (2). In carrying out the integration it is tacitly assumed that the layers are perfectly bonded. For laminated composite plates, this amounts to replacing the heterogeneous laminate with a statically equivalent (in the integral sense) single layer whose stiffnesses are a weighted average of the layer stiffnesses through the thick- ness. Therefore, such laminate theories are termed equivalent single layer (ESL) theories.
For many applications, the ESL theories pro- vide a sufficiently accurate description of the global laminate response (e.g. transverse deflec- tion, fundamental vibration frequency, critical buckling load, force and moment resultants). The main advantages of the ESL models are their inherent simplicity and low computational cost due to the relatively small number of dependent variables that must be solved for. However, the ESL models are often inadequate for determining the three-dimensional stress field at the ply level. The major deficiency of the ESL models in modeling composite laminates is that the trans- verse strain components are continuous across interfaces between dissimilar materials; thus, the transverse stress components ( 6~z , 6yz, 6zz ) are dis- continuous at the layer interfaces. To see this, consider the linear strain-displacement relations for the first-order shear deformation laminated plate theory, 16,17
6i = 61°)+ ze111 (i = 1, 2, 6, 4, 5) (9a)
where
Ox ' Ox
Oy ' Oy
6~0) OUl 1_ _ - - Oy Ox ' Oy Ox
6(30) 0, e~l)= 0 e? ) ~u0 E (1)= = = + ¢ 2 , 0 Oy
e~ ~l=Ou°3 + fk~, e~')=0 (9b) 0x
We note from eqn (9a) that the strains vary linearly through the laminate thickness, and they are independent of the lamination scheme, as noted earlier. For a fixed value of z, the strains vary only with respect to the x and y coordinates. The transverse stresses according to the stress-strain relations of the kth lamina are given by
{ O'41k [ 044 Q45]klg41k (10a) O51 [Q45 Q55] [651
or
{o}k=[Q]k{6} k
where
(10b)
e3= ezz, e4= 2ey z, e5 = 2exz (11)
and [Q]k represents the matrix of material stiff- nesses of the kth layer referred to the global coordinates of the laminate. Thus, the stresses at the interface of the kth and (k + 1 )st layer, called interlaminar stresses, are not continuous because [0]k¢[(~] k+l and {6}k={e} k+l. This deficiency is most evident in relatively thick laminates, in local- ized regions of complex loading, or near geo- metric and material discontinuities.
While the inplane stresses (Oxx , Oyy, axy ) C a n be computed only through stress-strain relations, the transverse stresses ( Oxz , Cryz, Ozz ) can be computed through either the constitutive equations or the equilibrium equations of three-dimensional elas- ticity. As shown above, the constitutive equations give discontinuous interlaminar stresses. Alterna- tively, the equations of stress equilibrium
Ox 0o~ 0Oxz = O~xX+-~y + Oz 0 (12a)
0°YY+ 0-~° = 0 (12b) ~ -~+ 0y 0z
O axz +O--~+O°z---~Z=o (12c) 0x 0y 0z
24 J. N. Reddy
can be used to compute the interlaminar stresses, when the inplane stresses (axx, ayy, axy) are known. Integration of eqns (1 2a, b) with respect to z gives (Oxz, Oyz), which then can be used in eqn (12c) to determine azz. It is found that even this approach gives inaccurate, but continuous, stress fields at the layer interfaces for thick composites. This inaccuracy is attributed to inaccurate representa- tion of the inplane stress components in ESL theories.
4 LAYERWISE LAMINATE THEORIES
Unlike the ESL theories, the layerwise theories assume separate displacement field expansions within each material layer, thus providing a kine- matically correct representation of the strain field in discrete layer laminates, and allowing accurate determination of ply level stresses. Numerous displacement-based, layerwise laminate theories have appeared in the literature. 43-5° In most of these layerwise theories, displacement continuity across layer interfaces is enforced via constraint equations that allow some of the dependent vari- ables to be eliminated during the model develop- ment. However, in the layerwise theory of Reddy, 48-5° the transverse variation of the dis- placement field is defined in terms of a one- dimensional, Lagrangian, finite element representation that automatically enforces C ° continuity of the displacement components through the thickness, resulting in transverse strains that are layerwise continuous through the thickness. The variation of the displacements through the thickness can be represented to any desired level of accuracy by simply increasing the number of one-dimensional finite elements (i.e. numerical layers) or increasing the order of the transverse interpolation polynomials. Thus, the layerwise theory of Reddy provides a generaliza- tion of the layerwise displacement field concept.
The layerwise theory of Reddy is based on the following displacement expansion through the laminate thickness. The ith displacement com- ponent is expressed as (see Fig. 1 ),
N
ui(x,Y,Z,t) = Z U[(x,y,t)dPj(z) (13) J = l
where (i = 1, 2, 3), N is the number of subdivisions (e.g. finite-element discretization) through the thickness of the laminate, and ~ j are known func- tions of the thickness coordinate, z. The resulting
Fig. 1.
N - I ~
_U ~ -
_ U I ~ t
,'- uiV
" 1 It 1
Schematic of a laminate and layerwise discretization.
theory will have 3N variables and as many differ- ential equations in two dimensions. An advantage of this layerwise theory is that it requires only two-dimensional finite elements while incorpor- ating the kinematics of three-dimensional elas- ticity.
While the same interpolation functions are used in eqn (13) for all three displacements for simplicity, independent interpolation of the dis- placements (especialy u3) can be used. The func- tions • j are piecewise continuous functions, defined only on two adjacent layers, and can be viewed as the global Lagrange interpolation func- tions associated with the Jth interface of the layers through the laminate thickness, and U[ denotes the nodal values of ui at the nodes through the thickness. Because of this local nature of ~j , the displacements are continuous through the thick- ness but their derivatives with respect to z are not continuous. This implies that the transverse strains can be discontinuous at a point P on the discrete layer interfaces, {e}~#{e}~ +1, leaving the possibility that the interlaminar stresses computed from the layer constitutive equations at the point from the two layers can be continuous:
k k k + l k + l O]{e}p=[Q] {e}p
The inplane strains ei will be continuous and therefore the inplane stresses oi will be discon- tinuous at layer interfaces, as they should, because of the difference in material properties of adjacent layers.
The choice of N in eqn (13) provides the analyst with many options and flexibility in modeling. When N is chosen such that at least one element per layer is used, the interlaminar stress distributions can be determined accurately. The sublaminate concept can be used to model several identical layers of the laminate as one equivalent layer by choosing one element through the layers.
An evaluation of composite laminates 25
On the other hand, a layer can be modeled with more than one element to represent matrix splits or to capture local effects. In addition, the layer- wise theory of Reddy can be used to model imbedded delaminations between layers 5~ and study their growth.
The linear strain-displacement relations asso- ciated with the layerwise displacement field of eqn (13)are
OU ~ j de~ g () U ~ i~ j , eyy = C J, ~.zz = U 3 , Ox Oy dz
dtPg+ OU~ ~g, 2eyz=U~ d----z Oy
d dP___2 + 0 U___~J3 2exz = U~ dz Ox ¢J'
2exy---- k Oy I (14)
where summation on repeated indices is assumed. The governing equations for the nodal vari- J J J • . ables (U1, U2, U3) can be derived using the
principle of virtual displacements. The equations of motion of the layerwise theory are
' OMxy_Q~=Mu~)[, OM ~ + I
Ox Oy
Omlxy l + OMyy t " g Ox Oy -- Q y = MuU2,
OK~x-----z+OK~yy-o~= MuU ~ (15) 0x 0y
for I= 1, 2, .... , N, where the resultants are defined by
( hie
(ML, ' J Myy, M l y ) = (Gxx , Oyy, Oxy)lff~l(Z)dZ, -hi2
[ h/2 d~l (Qx ~, Q~, Qz~) = (O~z, Oyz, Ozz) dz
3 - t,/2 ~ Z
hi2 'J (KIx, Kyy)= (O z, dz, -h/2
r h/2 Mls = p ~ l ~ s dz (16)
~] -h/2
where h is the total thickness of the laminate and p is the material density.
Fiber-reinforced composite laminates are constructed of orthotropic layers, with the material principal axes (xl, x2) of each layer oriented at an angle with respect to the global coordinate system (x, y) while the material prin- cipal coordinate x 3 of all layers coincide with the global z axis. In the layerwise theory, the stress-strain relations of three-dimensional elas- ticity are used. For a typical lamina, we have, 17
/ axx l k aye tTzz =
% % (Txy
-Cll C12 C22
C13 0 0 C23 0 0 C33 0 0
644 C45 C55
sym.
c,: k xx] C26 gyy
C360 iEy zC'z /
0 2exz [ C6, 2exyJ (17)
The stresses can be written in terms of the dis- placements by means of eqns (14) and (17), and the resultants in (16) can be written in terms of the displacements (U j, V s, Wj) by substituting for stresses into eqn (16). Thus, the governing dif- ferential equations can be ultimately written in
• J J J terms of the displacements, (U l, U2, U3) and their derivatives with respect to x and y.
5 VARIABLE KINETIC FINITE ELEMENTS
While layerwise finite elements allow accurate determination of three-dimensional stress fields, they are computationally expensive to use due to the large number of degrees of freedom per ele- ment, comparable to stacks of three-dimensional finite elements. Thus, it is often impractical to discretize an entire laminate with layerwise finite elements. Further, for many laminate applications, the indiscriminate use of layerwise elements is a waste of computational resources since significant three-dimensional stress states are usually present only in localized regions of complex loading or geometric and material discontinuities. A logical idea is to subdivide the laminate into regions that can be adequately described by ESL models and other regions that require some type of layerwise model (i.e. a simultaneous global-local strategy). In this way, the most appropriate model is chosen for each region, thereby increasing solution
26 J. N. Reddy
economy without compromising solution accur- acy. Such global-local schemes can be developed using established finite element technology; 52 however, currently available methods make implementation extremely cumbersome. The primary source of difficulty is the enforcement of displacement continuity across boundaries that separate incompatible subdomains. Currently established methods of achieving displacement continuity between incompatible regions include (i) multi-point constraint equations via Lagrange multipliers, (ii) penalty function methods, and (iii) special transition elements. Each of these methods is too cumbersome for extensive use under a wide variety of operating conditions. Thus, there is a need for the development of a global-local analysis procedure that provides greater robust- ness, simpler computer implementation, and wider applicability to practical composite struc- tures.
To overcome the limitations of the current procedures used in the global-local finite element analysis, a variable displacement field concept was proposed by the author and his col- leagues. 5-~-56 The variable displacement field is a sum of all admissible displacement fields, single layer as well as layerwise, so that appropriate part of the displacement field can be invoked in a given region of the domain. The finite elements based on different displacement fields can be connected together in a single domain for global-local analysis. To further reduce the computational effort the mesh superposition is employed. In the mesh superposition technique, an independent overlay mesh is superimposed on a global mesh to provide localized refinement for regions of inter- est regardless of the original global mesh topo- logy. Integration of variable kinematic elements and mesh superposition technique yields a very robust, economical analysis for global-local analysis of practical laminated composite struc- tures.
The variable kinematic, finite element is devel- oped by superimposing several types of assumed displacement fields within the finite element domain. In general, the multiple assumed dis- placement field can be expressed as
ui(x, y, z) ESL, uLW(x, Y, =Ui Ix, y ,z)+ Z),
( i=1 ,2 ,3) (18)
where u 1 and u2 are the local inplane displace- ment components, and u3 is the local transverse displacement component. The coordinates (x, y)
represent the inplane coordinates and z is the transverse coordinate. The underlying foundation
ESI. of the displacement field is provided by u i which represents the assumed displacement field for any desired equivalent-single-layer theory (e.g. the first-order shear deformation theory). The second term u Lw represents the assumed displace- ment field for any desired layerwise theory (e.g. the layerwise theory of Reddy). The layerwise displacement field is included as an incremental enhancement to the basic ESL displacement field, so that the element can have full three-dimen- sional modeling capability when needed. Depend- ing on the desired level of accuracy, the element can use all, part, or none of the layerwise field to create a series of different elements having a wide range of kinematic complexity. For example, dis- crete layer transverse shear effects can be added to the element by including u~ w and u! x~', resulting in a layerwise element, which we denote as the LW1 element. Further, discrete layer transverse normal effects can be added to the element by also including u3LW, resulting in another layerwise element which we denote as the LW2 element. Displacement continuity is maintained between these different types of elements by simply enforc- ing certain homogeneous essential boundary con- ditions (see Fig. 2), thus eliminating the need for multi-point constraints, penalty function methods, or special transition elements. Such variable kine- matic plate elements have been developed by Robbins and Reddy s4-57 and show much potential for a wide variety of global-local analysis of com- posite plate problems.
6 MESH SUPERPOSITION TECHNIQUE
The use of hierarchical, variable kinematic, finite element concept provides a convenient means of
: =°
I 2
1 _ Trans la t ion =
A u 0 U~ = 0
Fig. 2. Superposition of the first-order shear deformation theory displacement field and layerwise displacement field.
Deformation of the transverse normal AB in the xz plane.
An evaluation of composite laminates 27
simultaneously discretizing subregions of a single computational domain with different theories. Thus, the analyst can use the most appropriate theory for each subregion of the domain. How- ever, the efficiency of a global-local analysis based solely on variable kinematic elements is still strongly dependent on a priori knowledge of the locations of subregions that require the more powerful layerwise elements. This limitation is due to the requirement that the inplane discretiza- tions of adjacent subregions must be compatible across subregion boundaries. The removal of this restriction would result in a very general, robust global-local computational model. In keeping with the hierarchical modeling philosophy of the previous section, the finite element mesh super- position technique (see, for example, the recent paper by Fish 58) is chosen as the means to remove the strong dependence of initial mesh topology upon global-local solution efficiency.
The mesh superposition method is an adaptive refinement technique that creates refined areas within a chosen crude finite element mesh by superimposing independent, refined meshes (overlay meshes) on the original mesh. No changes are made to the original mesh during the superposition process. The mesh superposition method is adaptive in the sense that the size, shape and ultimate location of the overlay mesh is based on the solution provided by the original mesh alone. The overlay mesh and the original mesh need not have a compatible discretization; the overlay mesh can be used to provide enhanced interpolation capability precisely where it is most needed, regardless of the original mesh.
To illustrate the mesh superposition idea, 57 consider the two-dimensional, displacement-based, finite element model of an elastic solid shown in Fig. 3. The original coarse mesh occupies a region
B C
G
,~ ~: D
Fig. 3. Finite element mesh superposition showing a coarse 4 x 3 global mesh and an independent, refined, 5 x 5
overlay mesh (shaded).
R 0 with boundary So (i.e. the rectangle ABCD). Assume that a solution obtained with the original 4 x 3 mesh indicates high displacement gradients in a subregion R 1 enclosed by boundary $1 (i.e. the quadrilateral region EFGD). An independent overlay mesh is then constructed corresponding to the subregion R1 and this overlay mesh is superimposed on the original mesh to form a composite mesh. The total displacement field u for the resulting composite mesh is then defined as follows:
u = u ° + u 1 i n R l (19a)
and
u = u ° in R o - R 1 (19b)
where u is the total displacement field, u ° is the displacement field interpolated on the original 4 x 3 mesh, and u 1 is the additional displacement field interpolated on the 5 × 5 overlay mesh. Note that u I serves as an incremental enhancement to the global solution within region R1. A new solution is then computed based on the multiple displacement field, which permits an enhanced representation of the solution within subregion R 1. Note that the process of forming a composite mesh does not require altering the original mesh. An important consequence of the multiple dis- placement fields is that the original mesh and the overlay mesh need not have compatible discreti- zations along the boundary S1, since Co continuity of the displacements across S~ is imposed by pre- scribing homogeneous essential boundary condi- tions on the additional displacement field u l:
u l = 0 on S I~S 0 (20)
(i.e. along EFG). Homogeneous essential bound- ary conditions on u I are not required along the boundary $1~ So (i.e. along EDG) unless EDG happens to be part of a Dirichlet boundary (i.e. boundary where the solution u is specified) for the problem as a whole. By circumventing the mesh compatibility requirement along $1, a tremendous amount of flexibility is provided for the construc- tion of the overlay mesh, and the adaptive refine- ment process can proceed in an optimal manner regardless of the original mesh topology.
The assembled finite element equations of the global-local analysis with mesh superposition technique are of the form,
IK,11 IK,211 IId,/l I , l [K2,] [K22]J [{d2}J [{FE}J
(21)
28 J. N. Reddy
where [Ku] and [K22 ] represent the independent stiffness matrices for the global mesh and local overlay mesh respectively, the submatrices [K~2] and [K21] represent the stiffness matrices associ- ated with the coupling between the global and overlay meshes, and {d l} and {d2} denote the nodal degrees of freedom associated with the global and local meshes respectively. Thus, the process of creating a composite mesh by adding an overlay mesh to an original global mesh destroys the banded nature of the composite system of equa- tions. This process is illustrated in Figs 4 and 5. In Fig. 4, one quadrant of a 2 × 2 global mesh is superposed by an independent 2 × 2 local mesh to form a composite mesh. The local elements can be different from the global elements, both in terms of the degree of interpolation as well as the theory (or mathematical model) on which they are based. In the composite stiffness matrix, Fig. 5 shows the composite stiffness matrix in terms of zero and nonzero entries for the composite mesh of Fig. 4. While both [Kn] and [K22 ] retain the usual banded form, the presence of the coupling
Global Mesh Oveday Mesh Composite Mesh
O - - - ~ ,
Fig. 4.
Composite System of Equations
I K,,] tK,2]] [{d,}] = [{ F1
Finite element mesh superposition showing the structure of the composite system of equations.
I K-i-i] [K,2]-] [K22]_J
Fig. 5. Form of the stiffness matrix for the composite mesh. Nonzero entries are indicated by dark areas. Banded nature is lost due to extensive coupling between global and overlay
meshes.
stiffnesses ([Kt2 ] and [K21]) destroys the banded nature of the composite stiffness matrix, thus a conventional direct banded equation solver is inefficient. Due to the incremental additive nature of the variables interpolated on the overlay mesh, {d2} = {0} provides a reasonable starting estimate for {d2}. Thus, the following iterative method is used to solve the composite system of equations:
Step 1. Set {d2} = {0}
Step 2. Solve[K~t]{d~}={Ft}-[Ktz]{d2} for {dl}
Step 3. Solve[Kz2]{d2}={Fz}-[Kzt]{d,} for {d2}
Step 4. Repeat steps 2 and 3 until convergence is achieved.
7 NUMERICAL RESULTS
7.1 Accuracy of the layerwise theory
Numerical results are presented here to illustrate the accuracy of the layerwise theory. The numeri- cal results were obtained using a displacement finite element model of the layerwise theory described above. The reader is referred to the paper by Robbins and Reddy 5° for a description of the finite element model and additional numerical results.
Consider a cross-ply laminate (0/90/0) sub- jected to sinusoidal transverse load at the top surface of the plate. This problem has the three- dimensional elasticity solution. 36 The plies are of equal thickness (h/3), and the material properties of each ply are
Ej = 25 msi, E2 = 1 msi, E 3 = E2, Gl2 = 0"5 msi,
Gl3 = G23 = 0"2 msi, v12 = v13 = v23 = 0-25 (22)
The intensity of the sinusoidally distributed load is denoted q0. Two different finite element meshes are used. The two meshes differ from each other only in the mesh refinements through the thick- ness. A 2 × 2 mesh of eight-node quadratic ele- ments is used in a quadrant of the laminate. The mesh used through the thickness are as follows (see Fig. 6):
Mesh 1 -- three quadratic elements through the thickness (441 dof).
Mesh 2 -- six quadratic elements through the thickness (969 dof).
An evaluation of composite laminates 29
z
1-D × 7
2 - D ~
Fig. 6. Finite element meshes of layerwise elements used for a three-layer (0/90/0), simply-supported square laminate under sinusoidal transverse load. Mesh 1 is shown in the figure; mesh 2 differs only in doubling the number of layer-
Fig. 8. Distribution of transverse shear stress Oxz through the thickness of a three-layer (0/90/0), simply-supported
square laminate under sinusoidal transverse load.
Figures 7 and 8 contain plots of nondimensional stresses (axx, trxz) through the thickness of the square, thick, laminate (b/h = 4). The stresses are nondimensionalized as follows:
6xx = Oxx( ac, ac, z)h2 /( b2qo)
6xz = axz( bc, a~, z)h/( bqo) (23)
where ac=O.lO5662(b/2 ), bc=O'894338(b/2) are the (reduced order) Gauss points closest to the points of maximum stresses. The coordinate system is taken in the midplane of the laminate, with the origin of the coordinate system being at the center of the laminate. In Figs 6 and 7, the solid line represents the three-dimensional elas- ticity solution of Pagano, 36 the solid circles repre- sent the finite element stresses at the Gauss points for mesh 1, the open circles represent the finite element solution at the Gauss points for mesh 2 (refined), and broken lines correspond to the classical and first-order theories. Excellent agree- ment is found between the three-dimensional elas- ticity results and the finite element results based on the layerwise laminate theory. The deflection u°3(x, y) coincides with the exact three-dimen- sional elasticity solution and is not shown here.
All stresses in the layerwise theory were com- puted in the post-computation using the displace- ment field, linear strain-displacement relations, and linear constitutive relations. The inplane normal stress (Oxx) in the classical laminate theory (CLT) and the first-order shear deformation (FSD) theory were post-computed at the Gauss points using the constitutive equations. The trans- verse shear stress Oxz in the CLT was post- computed from the first two equilibrium equations of the three-dimensional elasticity, whereas they were post-computed in the FSD both from constitutive and three-dimensional elasticity equations.
From the plot of the inplane normal stress Oxx, it is seen that both CLT and FSD predict wrong sign of the stress at the layer interfaces. This is due to the fact that the stress is approximated in the classical and first-order theories by a linear expansion. In trying to best approximate the non- linear stress distribution by a linear variation, both CLT and FSDT yield wrong interface stress values. This can lead to inaccurate prediction of failure load and failure mode. The equilibrium- based stress 05 from the single-layer theories is in considerable error compared to the three-dimen- sional elasticity solution; in fact, they predict maximum value of Oxz in the middle layer while
30 J. N. Reddy
the three-dimensional elasticity gives in the outer layers. Note that the error introduced in the com- putation of the inplane stresses (Oxx, ayy, ox;.) through constitutive equations will influence the accuracy of the transverse stresses computed using the equilibrium equations. The transverse shear stress computed in the FSD by constitutive equations is in qualitative agreement with the three-dimensional elasticity results. For all stresses, the layerwise theory yields accurate results.
7.2 Global-local analysis with variable kinematic elements
To demonstrate the accuracy and efficiency of the variable kinematic finite elements, a global-local analysis is performed to determine the nature of the free edge stress field of the free edge effect in a thick, symmetric angle-ply laminate under imposed axial extension. 55 Consider a thick, sym- metric, angle-ply laminate (45/-45)~ subjected to axial displacements on the ends. The laminate has a length of 2L, width 2W, and thickness 4h, with L=10W and W=8h (see Figs 9 and 10). Each of the four material layers is of equal thickness h, and is idealized as a homogeneous, orthotropic material with the following properties expressed in the material coordinate system:
EL= 20 ×106 psi, ET=Ez=2.1x l0~ps i ,
GLT=GLz = 0"85 X 106 psi,
~ULT = ktcz = ~tTz = 0"21 (24)
where subscript L denotes the direction parallel to the fibers, subscript T denotes the inplane direction perpendicular to the fibers, and sub- script z denotes the out-of-plane direction. The origin of the global coordinate system coincides with the centroid of the three-dimensional com- posite laminate. The x-coordinate is taken along the length of the laminate; the y-coordinate is taken along the width of laminate; and the z- coordinate is taken through the thickness of the laminate. Since the laminate is symmetric about the xy-plane, only the upper half of the laminate is modeled. Thus the computational domain is defined by ( -L<x<_L, -W<_y<_W, 0<z-<2h). The displacement boundary conditions for this problem are:
Ul(L, y, z) = Uo,
u2( -L, 0, 0) =0,
u3(x, y, 0) =0
ul( - L , y, z) =0,
u2(L , 0, 0) =0,
(25)
The variable kinematic finite elements are used in a global-local analysis to determine interlaminar free edge stresses near the middle of one of the two free edges (see Figs 9 and 10). The global region is modeled using first order shear deform- able elements; the local region is modeled with LW2 elements in order to capture the three- dimensional stress state near the free edge. Five different finite element meshes are used. The inplane discretization for all five meshes is exactly the same, consisting of a 5 x 11 mesh of eight- node quadratic two-dimensional finite elements. All elements have the same length (2L/5). How- ever, the width of the elements decreases as the free edge at (x, W,, z) is approached. The widths of the eleven rows of elements, as one moves away from the refined free edge are h/16, h/16, h/8, hi4, hi2, h, h, 2h, 3h, 3h, 5h, where h is the ply thickness. The five meshes differ only in the size of the local region that is discretized with LW2 elements. The LW2 elements used in the local region employ eight quadratic layers through the
2W
L/W=~O ~ Loc~Reglon (LW2) W/h --- 8 Y ~ !. ! Global Region (FSD)
/
X
Fig. 9.
2L
Inplane discretization of a ( + 4 5 / - 4 5 ) , laminate under axial extension.
2h 48
LW'2 Bomonm
2 w y
Fig. 10. Layerwise discretization of the local region (through the thickness) near the free edge of a (+45 / -45)5
laminate under axial extension.
1 / I h -f- h
An evaluation of composite laminates 31
laminate thickness (four per material layer). The thickness of the numerical layers decreases as the ( + 4 5 / - 4 5 ) interface is approached. From bot- tom to top, the layer thicknesses are 0.533h, 0.267h, 0.133h, 0-083h, 0.083h, 0.133h, 0.267h, 0-533h (see Fig. 9).
The five meshes used in this problem are sum- marized below (see Table 1).
• Mesh 1 - - 3 x 4 local mesh of LW2 ele- ments, centered about the point (0, W, 0). The LW2 elements extend a distance of hi2 away from the free edge (2354 active global dof).
• Mesh 2 - - 3 x 5 local mesh of LW2 ele- ments, centered about the point (0, W,, 0). The LW2 elements extend a distance of h away from the free edge (2740 active global dof).
• Mesh 3 - - 3 x 6 local mesh of LW2 ele- ments, centered about the point (0, W,, 0). The LW2 elements extend a distance of 2h away from the free edge (3226 active global dof).
• Mesh 4 - - 3 x7 local mesh of LW2 ele- ments, centered about the point (0, W,, 0). The LW2 elements extend a distance of 3h away from the free edge (3512 active global dof).
• Mesh 5 - - 5 x 11 mesh of LW2 elements in the entire domain. This mesh is used as a control mesh for comparison (9116 active global dof).
Figures 11 and 12 show the distribution of the interlaminar stress Oxz and azz, respectively, through the laminate thickness. All stresses are nondimensionalized by multiplying them by the factor (20eo/Ec), where e0 is the nominal applied axial strain of(uo/2L). The stresses are computed
at the reduced Gauss points nearest the middle of the refined free edge, i.e. along the line ( - 0.115 L, 0.998W, z). In Fig. 11, all four global-local meshes compare very well with the control mesh. In Fig. 12, meshes 1 and 2 show some error, indi- cating that Ozz is more sensitive to the mesh and the boundary layer thickness is larger than that of Uxz. Meshes 3 and 4 are practically indistinguish- able from the control mesh.
Figures 13 and 14 show the distribution of the interlaminar stresses axz and Ozz, respectively, across the width of the laminate near the (+45/ -45) interface. The stresses are computed at the reduced Gauss points closest to the line (0, y, h), i.e. along the line (-0.115L, y, 1.014h). In both Figs 13 and 14, the interlaminar stresses com- puted with meshes 3 and 4 are very close to the stresses obtained with the control mesh. Once again, the stresses computed with meshes 1 and 2 show a slight error; however, the distributions are qualitatively similar to the other meshes.
2 . . . . i . . . . i , , , , i . . . .
Mesh 1 7 " /
0 Mesh 2 1.5 • Mesh 3, Mesh 4 ~ 7 -
0.5 ~
0 , , , , I , , , , I ,
- 2 - 1 . 5 - 1 - 0 . 5 0
Transverse Shear Stress ( ~
Fig. 11. lnterlaminar shear stress oxz distribution near the free edge of a symmetric (+45/-45)s laminate under axial
extension.
Table 1. Global-local finite element meshes used in the study of the free edge problem a
Number of Width of Total elements in local active local region region dof
Mesh 1 3 x 4 hi2 2354 Mesh 2 3 x 5 h 2740 Mesh 3 3 x 6 2h 3226 Mesh4 3 x 7 3h 3512 Mesh 5 5 x 11 16h 9116
aThe total inplane discretization for all five global-local meshes is 5 x 11.
Fig. 12. Interlaminar normal stess azz distribution near the free edge of a symmetric (+45/-45)s laminate under axial
extension.
32 J. N. Reddy
l Mesh3. Mesh 4
0.4 0.5 0.6 0.7 0.8 0.9 1
YM
Fig. 13. Interlaminar shear stress a,, distribution across the width of a ( + 45/- 4.5), laminate under axial extension.
Mesh
Mesh Mesh
0.7 0.75 0.6 0.85 0.9 0.95 1
YMf
Fig. 14. Interlaminar normal stress a,,( - 0.115 L , y, 1.014/z) distribution across the width of a (+45/-45),
laminate under axial extension.
7.3 Variable kinematic elements and mesh superposition
To illustrate the utility of combining the variable kinematic element concept with the mesh super- position technique, consider the cylindrical bend- ing of a cantilever plate by a surface bonded piezoelectric actuator shown in Fig. 15. The objective of the analysis is to determine the trans- verse stresses in the thin adhesive layer near the end of the actuator. The local region of interest, where significant three-dimensional stresses are expected, is shown in Fig. 15. If this problem is to be solved by variable kinematic finite elements, in order to capture these local three-dimensional stresses while maintaining an overall economical solution, the majority of the plate should be dis- cretized using FSD elements while the local region of interest be discretized using the LW2 elements (see Fig. 16).
Figure 17 shows the computed transverse stresses in the center of the adhesive layer verses
Fig. 15. Cylindrical bending of a cantilever plate with a surface bonded piezoelectric actuator (strain induced in the
actuator = - 10 - 3).
j Global Region, 8 FSD elements
Br_ : Local Region, 13 LW2 elements
4 layers through piezoelectric
3 layers through adhesive
4 layers through aluminum
Fig. 16. Finite element mesh used to model the cylindrical bending of a cantilever plate with a surface bonded piezo- electric actuator. Variable kinematic elements are used to permit the simultaneous use of two different mathematical
models (or theories).
SOW
-5000 -
-1oOW 3 3.2 3.4 3.6 3.8 4
Y W-0 Fig. 17. Distribution of transverse stresses along the center of the adhesive layer near the free edge of the actuator, as determined using a mesh of variable kinematic finite
elements.
y. Several points of interest should be noted. First, both of the transverse stresses approach zero as the global-local boundary AA is approached (i.e. as y decreases toward 3.0); this suggests that the size of the local region is sufficient to capture all
An evaluation of composite laminates 33
of the free edge stresses near the end of the actu- ator. Secondly, the transverse shear stress appears to satisfy the traction free boundary conditions at the free edge of the actuator and adhesive. Third, a significant transverse normal stress exists in the adhesive layer near the free edge of the actuator. This transverse normal stress is of particular concern since many adhesives are relatively weak in tension. It is also interesting to note that if this problem is solved using FSDT elements only, then both the transverse shear stress and transverse normal stress are predicted to be zero over the entire domain.
To illustrate the utility of combining the vari- able kinematic element concept with the mesh superposition technique, consider once again the example problem shown in Fig. 15. Figure 18 shows the global mesh, the local overlay mesh and the resulting composite mesh used to solve this problem. The global mesh is a coarse uniform mesh of ten FSD elements. To capture the local three-dimensional stress field in the adhesive layer near the end of the actuator, an independent local overlay mesh of 13 LW 2 layerwise elements are used. The LW2 elements used to form the overlay mesh contain four layers through the aluminum substrate, three layers through the adhesive, and four layers through the piezoelec- tric actuator. The local overlay mesh is super- imposed over the fourth and fifth global elements to achieve the composite mesh. The composite mesh shown in Fig. 18 has the exact same inter- polating capability as the mesh shown in Fig. 16. Thus, it is not surprising that the solution obtained from the composite mesh of Fig. 18 is exactly the same as the previous solution obtained using variable kinematic elements only; therefore the results are not repeated here. The advantage of the present model over the previous model is that the same global mesh can be used in the investigation of many different local regions of interest. For each new local region of interest, an independent local overlay.mesh of variable kine- matic elements is formed and superimposed on the global mesh. Since the global and overlay meshes need not be compatible, the effectiveness of the global-local analysis is not strongly tied to the global mesh topology.
8 CONCLUSIONS
A review of the single-layer theories of composite laminates is presented, and the layerwise theory
Global Mesh lO fir= order ~,mr a~ormab~ Comma 63 degrees of fresdom
Fig. 18. Finite element mesh superposition showing the coarse global mesh of FSD elements, the independent overlay mesh of LW2 elements, and resulting composite
mesh.
proposed by the author is described. A finite element modeling methodology is presented for the hierarchical, global-local analysis of lami- nated composite plates. The method incorporates a new variable kinematic, displacement-based, finite element that is based on a multiple assumed displacement field approach. The variable kine- matic elements provide a great degree of flexi- bility in defining the transverse (through thickness) variation of the assumed displacement field. The resulting finite element model permits different subregions of the computational domain to be described by different mathematical models. Enforcing displacement continuity along sub- region boundaries requires only the specification of certain homogeneous essential boundary con- ditions, thus avoiding multi-point constraints, penalty function methods, or special transition elements.
To increase the utility of the variable finite elements, an integration of the variable kinematic element concept with the finite element mesh superposition technique is developed. The result- ing global-local model permits selected local subregions in a composite plate to be conveni- ently discretized with an independent refined overlay mesh composed of FSDT, LW 1, and/or LW2 elements. Due to the independent nature of the local overlay mesh, the present integrated method allows several different local regions of interest to be accurately and conveniently investi- gated regardless of the original global mesh topol- ogy. The computational procedure described
34 J. N. Reddy
herein offers greater flexibility in modeling and it can be used in a variety of applications.
ACKNOWLEDGEMENT
It is a pleasure to acknowledge the help of Mr Donald Robbins, a research assistant of the author, in computing the numerical results reported herein.
REFERENCES
1. Cauchy, A. L., Sur requilibre et le mouvement d'une plaque solide. Exercises de Mathematique, 3 (1828) 328-55.
2. Poisson, S. D., Memoire sur l'equilibre et le mouvement des corps elastique. Mem. Acad. Sci., 8 (1829) 357.
3. Kirchhoff, G., Uber das Gleichgwich und die Bewegung einer Elastischen Scheibe, J. Angew. Math., 40 (1850) 51-88.
4. Basset, A. B., On the extension and flexure of cylindrical and spherical thin elastic shells. Phil, Trans. Royal Soc. (London) Ser. A, 181 (6) (1890) 433-80.
5. Goodier, J. N., On the problem of the beam and the plate in the theory of elasticity. Tram. Royal Soc. Canada, 32 (1938)65.
6. Reissner, E., On the theory of bending of elastic plates. J. Math. Physics, 23 (1944) 184-91.
7. Reissner, E., The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech., 12 (1945) 69-77.
8. Reissner, E., Reflections on the theory of elastic plates, Appl. Mech. Rev., 38 ( 11 )(1985) 1453-64.
9. Boll6, E., Contribution au probleme lineare de flexion d'une plaque elastique. Bull. Tech. Suisse. Romande., 73 (1947) 281-5 and 293-8.
10. Hencky, H., Uber die Berucksichtigung der Schubver- zerrung in ebenen Platten. Ing.-Arch., 16 (1947) 72-766.
11. Hildebrand, E B., Reissner, E. & Thomas, G. B., Notes on the Foundations of the Theory of Small Displacements of Orthotropic Shells. NASA TN-1833, Washington, DC, USA, 1949.
12. Mindlin, R. D., Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl, Mech., Trans. ASME, 18 (1951) 31-8.
13. Viasov, B. E, Ob uravneniyakh teovii isgiba plastinok (on the equations of the theory of bending of plates), Izv. Akd. NaukSSR, OTN, 4 (1958) 102-9.
14. Noor, A. K. & Burton, W. S., Assessment of shear defor- mation theories for multilayered composite plates. Appl. Mech. Rev., 42 (1)(1989) 1-13.
15. Reddy, J. N., A review of refined theories of laminated composite plates. Shock & Vibration Digest, 22 (7) (1990) 3-17.
16. Reddy, J. N., Energy and Variational Methods in Applied Mechanics. John Wiley and Sons, New York, USA, 1984.
17. Ochoa, O. O. & Reddy, J. N., Finite Element Analysis of Composite Laminates. Kluwer, Dordrecht, The Nether- lands, 1992.
18. Lo, K. H., Christensen, R. M. & Wu, E. M., A higher- order theory plate deformation. Part 1: Homogeneous
plates. J. Appl. Mech., Tram. ASME, 44 (4) (1977) 663-8.
19. Lo, K. H., Christensen, R. M. & Wu, E. M., A higher- order theory of plate deformation. Part 2: Laminated plates. J. Appl. Mech., 44 (4)(1977) 669-76.
20. Librescu, L., Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures. Noordhoff, Leyden, Netherlands, 1975.
21. Jemeilita, G., Techniczna teoria plyt S'redniej Grubosci (Technical theory of plates with moderate thickness), Rozprawy Inzynierskie (Engrg. Trans.), Polska Akad. Nauk, 23 (3)(1975) 483-99.
22. Schmidt, R., A refined nonlinear theory of plates with transverse shear deformation. J. Ind. Math. Soc., 27 (1) (1977) 23-38.
23. Krishna Murty, A. V., Higher order theory for vibration of thick plates. AIAA J., 15 (12)(1977) 1823-4.
24. Levinson, M., An accurate, simple theory of the statics and dynamics of elastic plates. Mech. Res. Comm., 7 (6) (1980) 343-50.
25 Seide, P., An improved approximate theory for the bend- ing of laminated plates. Mech. Today, 5 (1980) 451-66.
26. Murthy, M. V. V., An improved transverse shear defor- mation theory for laminated anisotropic plates. NASA Tech. Paper 1903, 1981, pp. 1-37.
27. Reddy, J. N., A simple higher-order theory for laminated composite plates. J. Appl. Mech., 51 (1984) 745-52.
28. Reddy, J. N., A refined nonlinear theory of plates with transverse shear deformation. Int. J. Solids and Struc- tures, 20 (9/10) (1984) 881-96.
29. Bhimaraddi, A. & Stevens, L. K., A higher order theory for free vibration of orthotropic, homogeneous, and laminated rectangular plates. J. Appl. Mech., 51 (1984) 195-8.
30. Di Sciuva, M., A refined transverse shear deformation theory for multilayered anisotropic plates. Atti Accad Sci. Torino, 118 (1984) 279-95.
31. Reddy, J. N., A general non-linear third-order theory of plates with transverse shear deformation. J. Non-Linear Mech., 25 (6)(1990) 677-86.
32. Stavsky, Y.. On the theory of symmetrically hetero- geneous plates having the same thickness variation of the elastic moduli. In Topics in Applied Mechanics, E. Schwerin Memorial Volume. ed. D. Abir, F. Ollendorff & M. Reiner. Elsevier, New York, USA, 1965, pp. 105-66.
33. Yang, P. C., Norris, C. H. & Stavsky, Y., Elastic wave propagation in heterogeneous plates. Int. J. Solids and Structures, 2 (1966)665-84.
34. Whitney, J. M., The effect of transverse shear defor- maton in the bending of laminated plates. J. Comp. Mater., 3 (1969) 534-47.
35. Whitney, J. M. & Pagano, N. J., Shear deformation in heterogeneous anisotropic plates. J. Appl. Mech., 37 (4) (1970) 1031-6.
36. Pagano, N. J., Exact solutions for rectangular bidirec- tional composites and sandwich plates. J. Comp. Mater., 4 (1970)20-34.
37. Whitney, J. M., Shear correction factors for orthotropic laminates under static load. J. Appl. Mech., 40 (1) (1973) 302-4.
38. Reissner, E., A consistent treatment of transverse shear deformations in laminated anisotropic plates. AIAA J., 10 (5)(1972) 716-18.
39. Reissner, E., Note on the effect of transverse shear deformation in laminated anisotropic plates. Comp. Meth. Appl. Mech. Engng, 20 (3)(1979) 203-9.
40. Librescu, L. & Reddy, J. N., A critical review and generalization of transverse shear deformable aniso-
An evaluation of composite laminates 35
tropic plates theories. In Euromech Colloquium 219, Refined Dynamical Theories 'of Beams, Plates and Shells and their Applications. ed. I. Ehshakoff & H. Irretier. Springer-Verlag, Berlin, Germany, 1987, pp. 32-43.
41. Di Sciuva, M., Bending, vibration and buckling of simply supported thick multilayered orthotropic plates: an evaluation of a new displacement model. J. Sound & Vibration, 105 (3)(1986) 425-42.
42. Di Sciuva, M., An improved shear deformation theory for moderately thick multilayered anisotropic shells and plates. J. Appl. Mech., 54 (3)(1987) 589-96.
43. Sreenivas, S., A refined analysis of composite laminates. J. Sound & Vibration, 30 (4) (1973) 495-507.
44. Mau, S. T., A refined laminated plate theory. J. Appl. Mech., 40 (2)(1973) 606-7.
45. Epstein, M. & Glockner, P. G., Nonlinear analysis of multilayered shells, Int. J. Solids and Structures, 13 (1977) 1081-9.
46. Murakami, H., Laminated composite plate theory with improved in-plane responses. J. Appl. Mech., 53 (3) (1986) 661-6.
47. Hinrichsen, R. L. & Palazotto, A. N., Nonlinear finite element analysis of thick composite plates using cubic spline functions. AIAA J., 24 (11 ) (1986) 1836-42.
48. Reddy, J. N., A generalization of two-dimensional theories of laminated composite plates. Comm. Appl. NumericalMethods, 3 (1987) 173-80.
49. Reddy, J. N., On the generalization of displacement- based laminate theories. Appl. Mech. Rev., 42 (11)(2) (1989) $213-22.
50. Robbins, D. H. & Reddy, J. N., On the modeling of free- edge stress fields and delaminations in thick composite laminates. In Composites Structures for Aerospace Appli- cations, ed. J. N. Reddy & A. V. Krishna Murty. Narosa Publishing House, New Delhi, India, 1992, pp. 33-74.
51. Barbero, E. J. & Reddy, J. N., Modeling of delamination in composite laminates using a layer-wise plate theory. Int. J. Solids and Structures, 28 (3) ( 1991) 373-88.
52. Reddy, J. N., On computational schemes for global-local stress analysis. Workshop on Computa- tional Methods for Structural Mechanics and Dynamics. NASA Langley Research Center, Hampton, VA, USA, 1985, pp. 29-31.
53. Robbins, D. H. & Reddy, J. N., Modeling of thick com- posites using a layerwise laminate theory. Int. J. Numeri- calMethods Engng, 36 (1993) 655-77.
54. Robbins, D. H. & Reddy, J. N., Global/local analysis of laminated composite plates using variable kinematic finite elements. Paper presented at AIAA/ASME/ ASCE/AHS/ASC 33rd Structures, Structural Dynam- ics, and Materials (SDM) Conference, Dallas, TX, USA, 13-15 April 1992.
55. Reddy, J. N., Robbins, D. H. & Reddy, Y. S. N., Analysis of interlaminar stresses and failures using a layer-wise laminate theory. Local Mechanics Concepts for Compo- site Material Systems, ed. J. N. Reddy & K. L. Reifsnider. Springer-Verlag, New York, USA, 1992, pp. 307-39.
56. Reddy, J. N. & Robbins, D. H., Analysis of composite laminates using variable kinematic finite elements. In Proceedings of the 7th Brazilian Symposium on Piping and Pressure Vessels (SIBRAT), ed. C. A. C. Selke & C. S. BaeceUos. University of Santa Catarina, Floriano- polis, Brazil, 1992, pp. 47-68.
57. Robbins, D. H. & Reddy, J. N., Modeling of actuators in laminated composite structures. Paper presented at the North American Conference on Smart Structures and Materials, Albuquerque, NM, USA, 31 Jan.-3 Feb. 1993.
58. Fish, J., The s-version of the finite element method. Computers and Structures, 43 (3) (1992) 539-47.
