Micromechanics-based structural analysis of thick laminated composites

Pergamon 00457949(93)EOO22-G

Cumputws & Srrucrures Vol. 51. No. 2. pp. 163-179, 1994 Copyright 6 1994 Elsevier Swnce Ltd

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MICROMECHANICS-BASED STRUCTURAL ANALYSIS OF THICK LAMINATED COMPOSITES

D. A. PECKNOLD and S. RAHMAN

Department of Civil Engineering, University of Illinois at Urbana-Champaign, 205 North Mathews Ave. Urbana, IL 61801, U.S.A.

(Received 22 December 1992)

Abstract-Thick filament wound cylinders, or local regions in structural laminates around cut-outs, fasteners or stiffeners may require three-dimensional (3D) analysis and evaluation, in order to fully characterize behavior and evaluate safety margins. This paper describes a particular approach to the 3D structural-level analysis of thick laminated composites that utilizes homogenization concepts and standard displacement-based finite element modeling. Hierarchical material modeling forms the basis of the procedure. The material model consists of two modules: (1) a micro-model of a unidirectional lamina, containing the basic 3D constitutive information for fiber and matrix constituents; and (2) a sublaminate model that enforces equilibrium of tractions between laminae, and delivers 3D homogenized stresses and strains and material tangent stiffnesses. This integrated approach provides the information required for evaluating damage and failure conditions at the microstructural level, and is essential for nonlinear analysis because of possible interactions between damage and failure modes. A nonlinear elastic material model is formulated, as an example; this nonlinear model, which is suitable for epoxy matrices, has been successfully implemented in a standard finite element code and used quite extensively. However, only elastic analysis results are presented, because the important characteristics of the modeling approach are clearly revealed in this setting. Comparisons are made between material model predictions and analytical, numerical, and experimental results for a unidirectional lamina, a thick laminate, and a thick cylinder under compression and bending. These results show that the accuracy of the procedure for thick laminates is quite satisfactory for practical purposes.

INTRODUCTION

Modern engineering design of structures and structural components for high-performance applications employs numerical simulation in a central role. At the heart of this process is the realistic modeling of material behavior in the inelastic range. Comprehen- sive computational tools have evolved over the past several decades, making design evaluations feasible for traditional materials of construction. These tools make it possible to proceed from basic material properties established from standardized small- specimen tests to predictions of behavior for relatively complex components and structures. The situation is different for advanced composite materials-constitutive models for specific material systems, as well as general modeling approaches, are not yet as well developed. The variety of composite material systems in use, and the large number of possible microstructural and macrostructural forms [I], are partly responsible. In addition, various failure mechanisms occur in composites, on widely different size scales; some of these, such as fiber microbuckhng and kink-banding [24], are not yet fully understood. Fuller utilization of composites in the future-in high temperature environments, or as smart materials, for example-will require significant advances in material modeling capabilities, together

with the effective integration of these capabilities into general procedures for structural analysis. As a step towards this goal, this paper describes a particular approach to the three-dimensional (3D) structural- level analysis of thick laminated composites, that fits naturally into standard displacement-based finite element modeling.

Thick-walled filament wound cylinders, and local regions in structural laminates around cut-outs, fasteners or stiffeners may require full 3D stress analysis in order to accurately evaluate safety margins. For example, hydrostatic compression tests of small-scale model grapite/epoxy and glass/epoxy cylinders with radius-to-thickness ratios in the range of six to eight, resulted in unexpectedly low failure loads [5]. Peros [6], in his review of [5] and several other research programs carried out to evaluate the feasibility of using thick-walled composite pressure hulls for deep submergence research vehicles, con- cluded that this unexpected behavior may be due to the relatively large through-thickness tensile strains that can occur in thick anisotropic cylinders.

The stress analysis of laminated composites has generally been carried out by using either a macromechanical or a micromechanical modeling approach. Macromechanical models treat the unidirectional lamina as a homogeneous, anisotropic material with elastic or inelastic properties. The

CAS 5112-O 163

164 D. A. PECKNOLD and S. RAHMAN

elastic analysis of thin laminates has often been approached by using classical laminate theory (CLT) for synthesizing the laminate stiffness quantities needed in a plate or shell structural model [7]. For thick laminates consisting of hundreds of individual plies, the equivalent elastic properties needed for a 3D structural model have been derived by Pagan0 [8] and Sun and Li [9], by properly combining the properties of the individual laminae. Difficulties unique to the 3D problem arise due to the requirement to enforce continuity of tractions across lamina interfaces; in CLT and other plate and shell theories, these require- ments are ignored because the through-thickness normal stresses and transverse shear stresses are small relative to the in-plane stresses. More recently, 3D finite element modeling approaches which employ homogenization concepts and allow reasonable meshing, have been developed by Chang et al. [lo] and by Zywicz [ 111, and applied to the elastic analysis of thick laminates. Zywicz uses a simplified lamina constitutive relation [12] which a priori satisfies continuity of interface tractions, and he develops a finite element scheme which preserves some information related to stacking sequence. Unfortunately, both of these procedures [lo, 1 l] involve ply-by-ply sampling of material properties; therefore, extension of these procedures to nonlinear analysis will probably re-

quire modifications in order to be computationally viable. Nonlinear analysis of thin laminated composites has typically been carried out using anisotropic plasticity models at the lamina level combined with CLT [ 131, Mindlin plate theory [ 141, higher order shear deformation theories, or 3D finite element models [ 151 in which the number of laminae are small enough to allow one element-per-ply modeling.

Micromechanical models, which explicitly recog- nize the individual constituents of the composite, are appealing because they have the capability of pro- viding more detailed response information than macromechanical models; they are also potentially simpler in terms of material characterization because they operate at a more fundamental level. The survey article by McCullough [16] summarizes many of the approximate analytical formulae that have been derived for the equivalent elastic properties of a lamina. Detailed micromechanical studies of the linear behavior [ 171 and nonlinear behavior [ 181 of composites have been carried out by many re- searchers, using numerical finite element modeling of a representative volume element (RVE) containing a single fiber and the surrounding matrix. Although such an approach is valuable for studying fundamental local aspects of behavior including the important effects of fiber-matrix interaction, it is not practical

uivalertt Material Anbotropic & Homogeneous

‘rectional Lamina epic 1 Homogeneous

Fig

Micro-Model

I. Material modeling strategy for thick-section laminated composites

Micromechanics-based analysis of thick composites 165

for structural-level analysis of composites. Several simplified micromechanical models which use fiber and matrix constitutive information, and which have the potential to be integrated directly into a nonlinear structural analysis procedure, have been proposed. Dvorak and Bahei-El-Din [l9] and Sun and Chen [20] incorporate simple mechanics-of-materials micromodels, with elastic fiber and elastoplastic matrix, into laminate analysis using CLT. Pecknold [21] proposed a 3D mechanics-of-materials micromodel, with a nonlinear elastic matrix, and combined it with a 3D homogenization scheme for the analysis of thick laminates. Aboudi’s method of cells [22-241, and the periodic hexagonal array (PHA) model [2.5] can be viewed as coarse-mesh finite element discretizations of a RVE. These numerical micromodel procedures all trade off accuracy against simplicity of implementation and computational efficiency. The PHA model [25] appears to be by far the most computationally-intensive of the micromodels cited above; reported solution times for a simple two-element mesh of a + 0 laminate are not encouraging when one extrapolates them to structural components and finite element meshes of realistic complexity. The vanishing fiber diameter model [19] is probably the least computationally-intensive, yet it lacks accuracy. Sun and Chen [20] evaluate their 2D plane stress micromodel as too complex to be used in structural analysis and propose instead to use it to calibrate an anisotropic plasticity lamina macromodel. Aboudi’s mode1 has been successfully used since the early 1980s in a variety of applications, with the Bodner-Partom unified plasticity model [26] for metal matrix composites. It appears to be quite accurate when compared against experimental and detailed micromechanical results, and it may be viable for structural analysis even though requiring considerable computational resources.

The modeling approach which is described in this paper is illustrated in Fig. 1 [21,27]. It is a micromechanics-based procedure for structural analysis, that combines reasonable simplicity with reasonable accuracy. Because the focus of this paper is on the methodology, numerical results are presented only for the elastic response of laminae, laminates and thick cylinders. However, a nonlinear constitutive model that has been successfully integrated into a standard finite element code [27, 281 is also discussed.

MATERIAL MODELING APPROACH

Thick-section laminates may have thicknesses ranging from l/4 in. up to several inches. The laminate is typically constructed from unidirectional plies each approximately 0.005 in. thick; it may therefore consist of literally hundreds of individual plies, usually arranged in a regular stacking sequence. An example of a thick-section laminate with a repeating pattern is a [0,/90],,, laminate, which consists of a Or/90 sublaminate, repeated 32 times for a total of 96

individual laminae. The large number of plies typical of thick-section laminates produces a structure which is, in effect, much more homogenous than one with a small number of plies; this feature of thick-section laminates allows a homogenization procedure to be used effectively at the laminate level. It can reason- ably be assumed that the material in the vicinity of a generic point of the [0,/90],,, laminate consists of three plies arranged in a 0,/90 pattern. The analysis can be carried out by sampling material response at discrete locations deployed advantageously through the laminate thickness. Ply-by-ply analysis, in which the response history of each individual ply through the laminate thickness is tracked, is unnecessary and requires considerably more computational resources.

The structure is modeled using conventional displacement-based isoparametric finite elements (Fig. 1); material response is sampled at a number of representative material points, usually Gaussian integration points, within each element. The location and number of material points, as well as element type and meshing are problem-dependent structural modeling issues, quite separate from modeling of the material behavior. The material behavior at the material point is described by a 3D material model. In contrast to CLT, which has been used successfully for thin laminates for many years, this 3D modeling approach specifically recognizes and enforces stress equilibrium between laminae, and provides information on through-thickness stresses and strains.

The 3D material model is a multilevel hierarchical scheme which efficiently integrates microstructural information into structural analysis, using concepts of homogenization. Such an integrated approach provides the information required for evaluating damage and failure conditions at the microstructural level, and is essential for nonlinear analysis because of the interactions between different damage and failure modes. A complete material mode1 must provide the material tangent stiffness and perform stress updates at the material sampling points to be useful for general structural analysis purposes. These functions are performed via the two independent modules illustrated in Fig. 2:

(a) at the lower level, the micromodel of a unidirectional lamina contains the basic 3D constitutive information for the fiber and matrix materials, including damage and failure criteria; and,

(b) at the upper level, the sublaminate model syn- thesizes the responses of several unidirectional laminae that are provided by the micro-model, and delivers the 3D homogenized stresses and strains and material tangent stiffness at the material sampling points.

The sublaminate mode1 represents a 3D structural modeling strategy. In this regard, it should be noted that the micromodel can be used directly with other structural modeling approaches, such as CLT,


lca,“a - l - ffepresenrstive wilt cell Fiber Dkection

. . RepmenfaNve unrr cdl /or Micro-mode/

Fig. 2. Hierarchical material model.

Mindlin theory, or higher order shear deformation theories for plates and shells. Equally well, the sublaminate model can be used with any type of 3D constitutive model for a unidirectional lamina, ranging from a more detailed micromechanical model to an anisotropic phenomenological model.

The material tangent stiffness is built up, or com-

posed, from the tangent stiffnesses (or compliances) of the fiber and matrix materials, using the two-level material model just described. After a load step is processed at the structural level, the stress and strain increments are recovered, first in each lamina, and then in each constituent material, by the inverse process of decomposition.

HOMOGENIZATION RELATIONS

The fundamental structure of the models is determined by the definition of effective stresses and strains at the various levels of the hierarchical model. These homogenization relations are first described, followed by a description of the use of these relations to perform the tangent stiffness and stress update computations.

Micromodel homogenization

The 3D micromodel [21,27] is based on a simple mechanics of materials approach which nevertheless recognizes the important restraining effect of the fibers on the matrix. Aboudi’s method of cells [22] and the 2D plane stress model of Sun and Chen [20] are similar in some respects to the model described here.

The microstructure of the unidirectional composite

is idealized as a doubly periodic array as shown in Fig. 2. Also shown in Fig. 2 is the representative unit

cell which is a schematic representation of the idealized composite. The unit cell is subdivided into three subcells: one fiber subcell denoted by f and two matrix subcells, denoted by mA and B, respectively. Local stress gradients at the fiber-matrix interfaces are not recognized in this procedure. Therefore, it is explicitly assumed that constitutive relations for matrix and fiber can be defined directly in terms of spatial average stresses and strains in the subcells. Spatially-constant 3D stress and strain vectors are defined and monitored in each of the three subcells, and the effective stress and strain vectors for the lamina are synthesized from these subcell values by


an elementary homogenization process which is These relations for material element A are expressed described below. as

The relative dimensions of fiber and matrix subcells are given by weight factors W, and W,,,, which are defined in terms of fiber volume fraction V, as {-z-}A = {-:}r = { -z-}mA (3a)

w,=Jvl-> w, = 1 - w,. (1)

For computational convenience, the three subcells are grouped to form two elements which are treated sequentially: material element A, consisting of the fiber subcell f and the matrix subcell m,; and material element B, consisting of the remaining matrix subcell B. The effective stresses and strains in the lamina are determined from the three sets of subcell values in two stages: first, fiber subcell f and matrix subcell mA are used to construct material element A; then, material element A and material element B are used to construct the unidirectional lamina.

in which the subscripts denote the subcells in matrix element A. Note that the vectors shown in eqn (3a) are mixed; that is, each contains both stresses and strains.

The complementary stresses and strains in material element A are determined as simple weighted averages of the corresponding subcell quantities by the homogenization relations

{-z-}* = ?{:-},+ we {$-jm, (3b)

in which the weight factors are given in terms of fiber volume fraction by eqn (1).

In order to express compactly the necessary equilibrium and compatibility conditions relating stresses and strains in the subcells, the stress and strain vectors in the three subcells are partitioned as follows:

Then, in order to determine the effective stresses and strains in the unidirectional lamina, material elements A and B are connected in parallel, that is, their strains are the same

011 GL ---

022

{,J} = T12 E

I-III

,

033 DT

T23

713

and the lamina effective stresses are determined as weighted averages of the stresses in the A and B, by the homogenization relations

Eli CL ---

622

{t} = z:: E

r-l-H

(2)

CT.

723

YI3

in which the stress and strain vectors are (6 x l), and the subscript C denotes the unidirectional lamina. Equations (3) and (4) completely characterize the micromodel.

Sublaminate homogenization

in which I is the fiber direction, 2 lies in the plane of the lamina, and 3 is normal to the plane of the lamina. Thus, the ‘longitudinal’ stress and strain vectors a, and eL are (1 x l), and the ‘transverse’ stress and strain vectors or and cr are (5 x 1). Note the unconventional ordering of stress and strain components in eqn (2).

Pagan0 [8] and Sun and Li [9] describe procedures for determining 3D effective elastic constants for laminates. The sublaminate model described here employs the same basic assumptions as these previous works, but proceeds from a different viewpoint, and provides a complete numerical model that is suitable for both tangent stiffness and stress updating. In addition, it is not restricted by any assumptions re- garding the particular form of the lamina properties.

In the synthesis process for material element A, the longitudinal (11) components of stress and strain are combined differently than the transverse components. The longitudinal strains in the fiber and matrix subcells are assumed to be the same, and the transverse normal and shear stresses in the fiber and matrix subcells are assumed to be the same. These assumptions can be thought of on an intuitive level as connecting the subcells in parallel for the longitudinal strains, and connecting them in series for the transverse normal and shear stresses.

The sublaminate consists of the smallest typical repeating unit from which the laminate is constructed. For the [02/903r6,$ laminate described previously which consists of 96 plies, the sublaminate is [0,/90], i.e. three plies. The sublaminate model provides information on the material behavior in a neighborhood of each material point in the structural model. In this neighborhood, the actual laminate is represented as an equivalent homogeneous anisotropic material.

Some additional notation is now required to distinguish quantities referred to the lamina, or local, (1,2,3) coordinate system from those referred to the


laminate, or global, (x, y, z) coordinate system. u’ and L’ will now denote stresses and strains in a lamina coordinate system, and Q and L will denote quantities referred to the laminate system. In addition, where needed for clarity, a superscript k will indicate a particular lamina. The transformation between local and global coordinates is

{t’}” = [T]k{c}k, (u}” = IT]kJ{u’}k. (5)

The transformations for in-plane and out-of-plane stresses and strains uncouple

[Tlk =

c= s2 SC

Tf = s= cl -SC

-2sc 2X c=-s2

T; 0 _______---- 0 T; I

where ak is the angle between the fiber in lamina k and the laminate global c s cos ak, s = sin ak.

(6)

direction (1) x axis, and

The stress and strain vectors for each lamina from which the sublaminate is constructed are partitioned into in-plane and out-of-plane components

{IJ’} =

a;

I

00

(74

or, in terms of stresses and strains expressed in global laminate coordinates

(7b)

Note that the ordering of the stress and strain components is unchanged from that used for the micromodel, eqn (2) but the partitioning is different.

Interface conditions. Each lamina is assumed to be perfectly bonded to adjacent laminae. The conditions at the interface between adjacent laminae k and k + 1 are as follows:

1. Displacement continuity requires that the in- plane strains (tl, t), y,,.) be continuous across the interface, and

2. Equilibrium of tractions across the interface requires that out-of-plane stresses (a;. T,._. t,,) be continuous across the interface.

Using the partitioning defined by eqn (7b), these conditions can be stated as

In cases where compliances change across the interface, the in-plane stresses O, and out-of-plane strains co can be discontinuous across the interface.

Classical laminate theory (CLT) for thin-sections ignores the second interface condition, on the as- sumption that the out-of-plane stresses are negligible, and that the in-plane stresses and strains can be accurately determined without considering equilibrium of tractions at the interface.

Calibrating patterns. The philosophy of homogenization methods is to replace the actual material by a well-defined equivalent material; the properties of this equivalent material are determined by requiring that the actual and equivalent materials respond in the same way when subjected to certain calibrating patterns of stress or strain. These fundamental patterns should obviously include appropriately chosen spatially homogeneous stress and/or strain patterns. but may also include higher-order spatial variations as well, depending on the character of the postulated equivalent material. The similarity to the well-known patch test [29] for nonconforming finite elements may be noted here. In the proposed sublaminate model, the equivalent material is homogeneous_ and as a result, only spatially homogeneous calibrating patterns are needed.

In the actual structure which is analyzed, the stress and strain conditions will generally be more complex than these fundamental patterns. Thus, the structural modeling should be done in such a way that the

Micromechanics-based analysis of thick composites

Deformed shape

169

a. Typical in-plane strain calibrating pattern $) = d2’ = Cx

Z

L X

b. Typical out-of-plane stress calibrating pattern o!” = a!” = 4

Fig. 3. Examples of calibrating patterns for sublaminate model.

variation of stress and strain over a material neighborhood is not ‘too large’, so that the conditions experienced by the equivalent material do not differ greatly from the fundamental patterns with which it was calibrated, i.e. a sublaminate should not experi- ence significant stress or strain gradients.

In view of the interface conditions eqn (8), the cali- bration patterns that are selected are homogeneous in -plane strains and homogeneous out -of-plane

stresses. Figure 3 illustrates the idea, showing for clarity only a 2D view, the x-z plane of the lamina. In one of the six calibrating patterns, the sublaminate, consisting in this example of two plies, is subjected to an elongation cX = CX with the other membrane strains (c,,, ya) constrained to zero, and with the out-of-plane stresses (uZ, TV:, T.~;) equal to zero. These conditions, applied at the boundaries of the laminate, are experienced throughout both laminae. The equivalent homogeneous material (not shown) is subjected to the same conditions. The homogenized stress 6, in the equivalent material is determined from t6, = t%v) + t(‘b$), and the homogenized strain C, is determined from tC2 = t”)cl.” + tc2)ct2). A second calibrating pattern, typical of the stress patterns that are used, is also shown in Fig. 3-in this case, the sublaminate is subjected to a through-thickness stress tr, = 5: with the other out-of-plane stresses (T,v_, TV:) equal to zero, and with all the in-plane strains (.cr, L,., yl, ) con-

strained to zero. The homogenized stress CY and the homogenized strains E, in the equivalent material

are determined in the same way as for the first case above.

The six independent calibrating patterns are collected together, using the partitioning scheme of eqn (7), as

{-z-r={-:-}, fork=1 ,..., N, (9)

where the overbar indicates homogenized sublaminate quantities.

The complementary in-plane stresses and out-of- plane strains in the equivalent material are determined from the homogenization relations

{A} = j, (;){-;-y, (IO)

where tk is the thickness of lamina k, t is the sublaminate thickness, and N is the number of laminae in the sublaminate.

Equations (9) and (lo), which are analogous to eqns (3) and (4) for the micromodel, completely characterize the sublaminate model.

Stacking sequence effects are not recognized in this procedure; however these effects are expected to be small in the thick laminates which are of interest here. Roy and Tsai [30], using comparisons with exact elasticity solutions, conclude that stacking sequence effects in symmetric laminates are negligible when

170 D. A. F’ECKNOLD and S. RAHMAN

they are composed of approximately 20 or more sublaminates.

MATERIAL TANGENT STIFFNESS

The material tangent stiffness at the material sampling point is assembled from the tangent constitutive information for the fiber and matrix materials. The homogenization relations [eqns (3), (4), (9) and (lo)], expressed in incremental form, are used in this process. The resulting computational procedure consists of a series of simple matrix operations, as described below.

Micromodel tangent stiffness

The tangent stiffness of the unidirectional lamina is built up from tangent stiffness or compliance relations for the three subcell components. These may be linear relationships or they may reflect various types of nonlinearities.

The relations between incremental stresses and strains within each of the subcells are expressed most con- veniently in terms of tangent compliance matrices S

X (11)

in which AT is a uniform temperature increase, and (t(, , a,, q) are thermal conductivities. Equation (11) is partitioned in accordance with eqn (2), as follows:

The tangent compliance matrix is now partially inverted to provide relations of the form

in which P,, is (I x I), P,, is (I x S), and P,, is (5 x 5) and are given by

p,,= (l/S,,)

p,, = - p,, 5,

p,, = s,, + s,,p,, * (14)

where S,, is (I x I), and the thermal effects are defined

by

P,, = P,,% > P,, = ur + Pior,. (15)

Material element A. With the tangent relations for

each subcell expressed in the form of eqn (13), the homogenization relations eqns (3) are applied to give the tangent relations for material element A in partially inverted form as

do, i-l =[ p,, p,; dc, A - p; PI,.

where

I p,, p,, + w, ---_---- -K p,,

The partial inversion is now completed to give the tangent stiffness for material element A as

{~aiZ=[_~~_~~~]Ai~~}A+AT{_~}~

(18)

in which

c,, = P,; ’

c,, = p,, c,,

c,, = p,, + Cl, K

Lb= --cup,,

81. = p,, + P,,Br (19)

or, in nonpartitioned form

Unidirectional lamina. The tangent compliance relations, eqns (I I), for material element B, which consists of a single matrix constituent. is inverted to give the tangent stiffness relation as

Micromechanics-based analysis of thick composites

where Finally, the homogenization relations, eqns (4) are applied to give the 3D tangent stiffness relation for the unidirectional lamina as

where

(22)

[Cl, = W,[Cl, + w,“nl

{8Ic = Wf{81, + w,“{BLl. (23)

Sublaminate tangent stiffness

The incremental tangent stiffness relations for a lamina given in eqn (22) are now denoted as

{da’}k = [C’lk{dt’Jk + AT{p’}k (24)

to indicate that they are expressed in lamina coordinates. They are transformed to global laminate coordinates to give

The inversion is then completed to give

where

- - G = P,tl Coo

e,, = C;

{do}k = [C]k{dc}k + AT{/?}k (25)

in which

- - C, = P,, + c, Pi

F” = - C, P&

[Cl” = [Tlk.qC’lk[Tlk, {/I}” = [T]kr{/3’}k. (26)

The partitioned form of eqn (25) is

{-d!!;)” = [-;C$!?~ {-;c!;r + AT{;-i” .

(27)

-- ai = P, + PJ, (33)

The material tangent stiffness and thermal load

vectors at the material sampling point are composed by means of the systematic process just described. Element-level numerical integration and structural-

level assembly processes then follow as usual.

MATERIAL NONLINEARITIES

The lamina stiffness is partially inverted to give

{;;$ = [--;;--;;~ {;!?-y + AT{-;;}

(28)

in which

P, = C,’

P,=C,P,

It is possible to model certain types of nonlinear

material behavior, such as delaminations, by modify- ing the assumed continuity conditions between fiber and matrix elements in the lamina micromodel, or between laminae in the sublaminate model. However, the primary vehicle for modeling nonlinear material behavior is through the specification of appropriate nonlinear constitutive models for the fiber and matrix constituents in the subcells of the micromodel.

P,, = C,, - P, C,,

P,,{, = - PO0 Bo

P/$=/J-Pn,Bo. (29)

Using eqn (9) in (28), and substituting the result in the homogenization relation (10) gives

It is well known that epoxy matrices exhibit rather

pronounced nonlinear softening response in shear, which appears to be reversible on unloading. Com- pression tests by Camponeschi [31] indicate that the nonlinear response of thick laminates is essentially reversible over a substantial loading range. There- fore, only a very simple model of nonlinear material behavior, that of nonlineur ehstic behavior of the matrix constituent, is discussed here.

Nonlinear elastic model for nzatri.rc

The observed nonlinear softening in pure shear can be modeled by a Ramberg-Osgood power-law relation of the form

171


where G is the initial shear modulus, and /I, Q, and n are material constants to be determined for the in

situ matrix material. These relations are stated in a general form valid

for multi-axial stress states, i.e.

(35)

in which 6; are linear elastic strains and

s,, = cTi, - f rrkk 6, ) q=&. (36)

The incremental form of eqn (35) is

bn-1 7rn-’

+2G2 70 0 &,G d+ (37)

in which the normalized deviator stresses are

- % SUE-. (38)

r,Z

Equations (35) and (37) are expressed in matrix form as

{de1 = k,P,l+ czP,l+ cJS,lIW) (40)

in which the stress and strain vectors are ordered as in eqn (7a). The matrices S,, SZ, S, are

[%I=

M =

0 0 0

1 1 1 0

0 0 0 0 0

000000

1

1 1 sym.

0 1 sym.

0 0 2

0 0 0 1

0 0 0 0 2

000002

(41)

and the stress-dependent coefficients c,, c2, cj are given by

c,= &&,(l+b,)] I.

c,=&(l+b,)

1 n-l b c3=c2 I> (42)

where

(43)

and K, G are the initial bulk and shear moduli, respectively.

Equation (40) gives the nonlinear tangent compli-

ance matrix for the matrix subcells in the lamina micromodel.

STRESS UPDATE FOR NONLINEAR ANALYSIS

After a load step is processed at the structural level,

the decomposition process is performed using the tangent material properties. Ultimately, the stress and strain increments in each of the three subcells in the micromodel are extracted, for every lamina in the sublaminate, and at every material sampling point.

New stress and strain totals calculated from these

tangential approximations satisfy all the equilibrium and compatibility relations which define the microstructure of the material model. However, if nonlinear material behavior has occurred during the load step, violations of the nonlinear constitutive relations may have occurred, because of the finite size of the load step. Thus, unless one is willing to use very small load steps and to accept the errors associated with a series of tangential approximations, a correction procedure, the stress update, is required.

Because of the hierarchical structure of the

material model, this stress correlation procedure is inherently more complex, and involves more computation, than the equivalent process for a non- hierarchical material model, such as the von Mises metal plasticity model. Fortunately, the particular architecture of the material model proposed here allows an efficient relaxation scheme in which cycles of tangential approximations, followed by constitutive corrections. are repeated until satisfaction of specified convergence criteria. This complete procedure will be described in a subsequent publication.

EXAMPLES OF LINEAR ELASTIC RESPONSE

Comparisons are made between material model predictions and analytical, numerical, and experimental results for the elastic responses of a unidirectional lamina, a thick laminate, and a thick cylinder.


Table 1. Elastic properties of graphite fibers and epoxy matrix given by Chen and Cheng[l7]

E, E2 (Mpsi) (Mpsi) (h$i) “12 “23

Graphite 24 2 4 0.3 0.15 Epoxy 0.6 0.6 0.231 0.3 0.3

Unidirectional lamina

Equivalent elastic properties of a unidirectional graphite/epoxy lamina with a hexagonally-packed fiber array were determined by detailed numerical micromodeling of a unit-cell by Chen and Cheng [ 171, and by Aboudi [31] using the Method of Cells.

Equivalent elastic properties of the lamina are calculated with the micromodel, over the full range of fiber volume fraction, using the graphite and epoxy properties from Chen and Cheng [17] shown in Table 1. In order to obtain the equivalent elastic properties for comparison purposes, the tangent stiffness, eqn (23), is inverted, and the engineering moduh are extracted from the compliance matrix. Note that this extra computational step is not needed for general applications of the model in structural analysis.

The micromodel values for the lamina transverse moduli E, and Gn are compared with the results from Chen and Cheng, and from Aboudi in Fig. 4. The good quality of these results is typical of numerous comparisons which have been made with published analytical and numerical results [27]. Based on these

Table 2. Elastic properties of graphite fibers and epoxy matrix

E, E* G12 (Mpsi) (Mpsi) (Mpsi) v, “21

AS-4 Graohite 27.0 2.5 5.0 0.30 0.25 3501-6 Epoxy 0.728 0.728 0.26 0.40 0.40

comparisons, it is evident that the micromodel provides accurate predictions of equivalent elastic moduli for a unidirectional lamina, as compared to other available or numerical approaches. This is a minimum requirement that should be met by any nonlinear modeling approach.

Thick laminate

Camponeschi [3 l] experimentally determined unidirectional properties for AS4/3501-6 prepreg tapes by testing [0], laminates in compression. These observed prepreg tape properties are first used to backcalculate constituent fiber and matrix properties by trial and error, involving a few trials; no attempt is made to determine best-fit values. The resulting constituent properties are shown in Table 2. Table 3 shows the degree of fit that is obtained with the experimental values. With these constituent properties as input, the material model is then used to determine the equivalent elastic properties of a OS-in thick [0,/90]i6, laminate. Initial elastic properties determined from compression tests by

Table 3. Properties of AS4/3501-6 nrepreg tape. Fiber volume fraction = 0.60

Source (M:si) EZ

(Mpsi)

Experiment,? Camponeschi 16.48 1.40 0.871 0.55§ 0.33 0.54

Micromodel 16.50 1.56 0.87 0.53 0.34 0.47

t Compression tests of [0], laminates. 1 From [ *45], tension test. 0 From literature.

Graphite/Epoxy

2.0 ----- Chen and Cheng (1970)

----- Aboudi (1991)

^ - Micro-model

a z s 1.5

B

p 1.0

m

0.0 0.0 0.2 0.4 0.6 0.8 1 .o

Fiber Volume Fraction Vt

Fig. 4. Calculated transverse elastic moduli of a graphite/epoxy lamina.

174 D. A. F%CKNOLD and S. RAHMAN

Table 4. Equivalent elastic properties of [0,/90],, thick laminate

EL Source (I&i) (Mpsi) (M:si) (MG;;si) (G&i) (MG;;i) “ry Y,: “g:

Experiment,t Camponeschi 11.63 0.07 0.47 -

Material model 11.60 6.58 1.85 0.82 0.69 0.60 0.08 0.46 0.47

t Compression tests of [0,/90],,, laminates

10,ooopi 1oopi

Fig. 5. Geometry of the thick cylinder.

Camponeschi [31] are shown in Table 4, along with the model results. Unfortunately, only the longitudinal elastic modulus E, and the transverse Poisson’s ratios vXr, v,; are reported by Camponeschi. Although good agreement with the longitudinal elastic modulus does not prove much, the transverse Poisson’s ratios are predicted with reasonable accuracy.

(a) Compress/on (b) Bending

Fig. 6. Cylinder in compression and circumferential bending.

Thick cylinder

Comparisons are made with exact elasticity solutions and finite element numerical solutions for thick cylinders under compression and bending. In these solutions, a condition of plane strain is assumed in the cylinder axis direction. A model thick-walled

’ Finite Element Homogenized ----- Exact Homogenized

- Exact Ply-by-Ply

-0.5 0.0 0.5 1.0 Hoop Strainco (%)

One Sublamina

Radial Stresso, (ksi)

Fig. 7. (c,,, a,) responses of thick-section laminated cylinder in compression.


cylinder (Fig. S), with approximately the same dimensions as those tested at DTRC Carderock[S] is considered. The cylinder OD is 8.25 in; the wall thickness is 0.48 in with stacking sequence [0,/90],, . The wall is built up from 96 plies of AS4/3501-6 prepreg tape, each 0.005 in thick, with two unidirectional plies in the circumferential direction for each one in the axial direction. This is the type of thick-section construction which is of primary interest; that is, many thin plies interleaved to produce an effectively homogeneous wall structure.

A general circumferentially varying pressure load on a cylinder can be decomposed into its harmonic components, i.e.

p(B)=p,+~p,cod. n

The axisymmetric pressure p,, produces primarily a compressive mode of response in the cylinder wall; the ovaling pressure p2 cos 20 produces primarily circumferential flexure. These two pressure distributions are selected as representative loading cases, with p,, = 10,000 psi and pZ = 100 psi (Fig. 6). Note that the ovaling pressure distribution on the geomet- rically perfect cylinder can provide an indication of the effects of uniform pressure on a slightly out-of- round cylinder.

Figures 7 and 8 show the results for compression loading, and Figs 9 and 10 show the results for circumferential bending. Three sets of results are shown in these figures. The exact ply-by-ply results are obtained from exact 3D elasticity solutions [32, 331 for a layered cylinder having the lamina properties predicted by the micromodel (Table 3); the exact homogenized results are obtained from the exact elasticity solution for a homogeneous anisotropic cylinder having the equivalent elastic properties predicted by the material model (Table 4); and the finite element homogenized results are obtained from a finite element numerical solution which incorporates the material model, and uses fiber and matrix properties given in Table 2. The finite element solution uses 20-node isoparametric solid elements with 2 x 2 x 2 material sampling and integration. An 18 x 2 x 1 mesh was used on one quadrant of the cylinder (18 elements circumferentially x 2 elements through the thickness x 1 element axially).

Comparisons of exact ply-by-ply with exact homogenized results test the accuracy of the sublaminate homogenization approach, without intro- ducing additional questions of the accuracy of the structural analysis. Comparisons of the exact ply-by- ply with finite element homogenized results test the accuracy of the complete integrated structural analysis approach.

o Finite Element Homogenized -____

Exact Homcu@nized

- Exacl fly-by-fly

-1m -100 -so 0 54 im is0

Hoop Stress a8 (ksl)

One Sublamina

4.3 -0.5 0.0 0.5

Radial Slrainc, (%)

Fig. 8. (u,, tO) responses of thick-section laminated cylinder in compression.

176 D. A. FWKNOLD and S. RAHMAN

Compression loading. Figure I shows the distribution through the thickness of the hoop strain E,, (the in-plane strain) and the radial stress o, (the out-of- plane stress). Fine details of the distributions are shown in insets on the figures; a single 0,190 sublaminate is shown on the inset for scale. Overall, it is difficult to distinguish these distributions from those in a homogeneous cylinder, aside from the local, unimportant irregularities which appear at the ply level. The homogenized properties produce quite accurate results, when used in both the exact and finite element stress analyses. The hoop strain and radial stress (Fig. 7) are continuous across interfaces between plies. In the homogenization procedure, these values are the same in all plies at the sampling point [eqn (9)]; note that the actual gradients of these quantities across a sublaminate thickness (three plies) are quite small. Figure 8 shows the complementary quantities: crII, the in-plane stress and c,, the out-of- plane strain. In the homogenization scheme, these

latter quantities represent weighted averages of the individual ply values [see eqn (IO)]. The individual ply values, which are recovered from eqn (28) using the computed values of c,, and c,, are not shown on the figures for clarity, but they are obviously as accurate as the homogenized values of E,, and cr, from which they are determined. Even though the radial strains are indeed discontinuous between plies, (see inset, Fig. 8) this produces only minor local irregularities in the overall distribution of radial strain. It is interest- ing to note that the tensile radial strain at the inside surface of the cylinder (Fig. 8) is significant, about half as large as the compressive hoop strain [6].

Circumferential bending. Figure 9 shows the distribution through the thickness of the hoop strain t,, (the in-plane strain) and the radial stress 0, and interlaminar shear stress T,~ (the out-of-plane stresses). These distributions are plotted at the circumferential locations where each is a maximum: at 0 = 0” for Q, and a,; and at 0 = 45” for T,“. Again,

-o.tII., -0.15 -0.10 -0.05 0.00 0.03 0.10 CL15

Hoop Strain IQ (%)

-uD 0 500

Radial Stress u, (psi) One Sublaminate

-0.5 0 SW 1OW 1500

Shear Stress r,+ (psi)

Fig. 9. (c,, o,, TV) responses of thick-section laminated cylinder in circumferential bending.


-30 40 -10 0 10 20 30

Hoop Stress oe (ksi)

o Finite Element HOmOQenized _____

Exact Homogenized

- Exact Ply-by-Ply

-0.1 0.0 0.1

Radial Strain 6, (%)

--0.00 0.0s 0.10 0.15 020

Shear Strain Yd (%)

Fig. 10. (a,,, t,, y,,,) responses of thick-section laminated cylinder in circumferential bending.

these quantities follow macroscopically smooth distributions that are difficult to distinguish from those in a homogeneous cylinder. The in-plane strain cI) and out-of-plane stresses Q, and 7,” shown in Fig. 9 are the same in all plies at a sampling point according to the homogenization procedure. The through-thickness distributions of the complementary quantities (the in-plane stress eO, and out-of-plane strains 6, and yrO) are shown in Fig. 10. The individual ply values are extracted according to the homogenization procedure by using eqn (28). These are not shown for clarity; instead the weighted-average responses, eqn (IO), are shown. Large radial strains, of the same magnitude as the hoop strains, are evident as in the compression loading case. These radial strains are tensile at the inner surface at 0 = 90” and 270”, and tensile at the outer surface at 0 = 0” and 180”. There are also significant radial tensile stresses near the midsurface of the shell. This is a well-known effect, and it

becomes more severe for thinner cylinders (larger R/t ratios).

These comparisons demonstrate that the sublaminate model is accurate enough for practical purposes when the number of plies is large, i.e. the section is a typical thick section. It is, of course, expected that accuracy will deteriorate as the number of sublaminates decreases [21]. Although the minimum number of sublaminates required for a specified accuracy has not been established, guidelines can obviously be developed. From a practical standpoint, it is expected that about IO-20 sublaminates will usually be suffi- cient for reasonable accuracy [30].

DISCUSSION AND CONCLUSIONS

The purpose of this paper is to explain and illumi- nate a particular approach to the 3D nonlinear structural-level analysis of laminated composites.


Hierarchical material modeling, utilizing modular

components (the lamina micromodel and the sub-

laminate model). forms the basis of the procedure. This lcads to a flexible, self-contained numerical

method for modeling nonlinear material behavior. The micromodel utilizes familar mechanics-of-

materials concepts; yet, to the authors’ knowledge,

it has not previously been proposed in precisely this form. The 3D character of the micromodel appears to be an important factor in its demonstrated accuracy [27] because fiber matrix restraint effects are better represented. The sublaminate model utilizes the same basic assumptions that are embodied in the previous work of Pagan0 [8] and Sun and Li [9]. but it is formulated in a way that is intended to provide additional insight, and that is particularly convenient for numerical evaluation of tangent stiffness matrices and for stress updating. The important feature of this material modeling approach is that it provides information for evaluating damage and failure conditions at the microstructural level&yet. it is computationally efficient enough to be used in finite element analysis of structural models of realistic size and geometric complexity. Because the material model fits into standard displacement-based finite element modeling, geometric nonlinearitics in the structural response are handled directly using existing formulations and algorithms.

Only elastic analysis results are presented. because the important characteristics of the modeling approach are clearly revealed in this setting. These results. and others presented in [27], show that the accuracy of the procedure is quite satisfactory for practical purposes. A nonlinear elastic power-law material model is formulated, as an example; this nonlinear model, which is suitable for epoxy matrices, has been successfully implemented in a standard finite element code and used quite extensively [27,28]. In nonlinear analysis, one expects a computational premium to be paid for the hierarchical material modeling approach proposed here, as compared to, say, the much simpler von Mises model for metal plasticity. The authors felt at the outset that if material model computation time could be limited to about ten times that required for von Mises plasticity, that would be acceptable from a computational standpoint. Even though special efforts have not been directed towards optimizing the efficiency of the material model algorithms, that goal appears to be reasonable.

Acknowledgements-The research reported here was begun while the first author was a Summer Faculty Fellow in the Navy/ASEE Summer Faculty Research Program at the David Taylor Research Center. Annapolis (now Carderock Division, Naval Surface Warfare Center) in the summer of 1990. Support was provided by the DTRC Independent Research program. The important contributions of Karin Gipple and Gene Camponeschi at DTRC are gratefully . . . . . acknowledged. Contrnumg support has also neen provmed

by the Center for Composite Materials Research at the University of Illinois at Urbana-Champaign, under an ONR-URI grant (ONR N00014-86-K-0799) and their Industrial Affiliates Program.

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REFERENCES

T.-W. Chou, R. L. McCullough and R. B. Pipes, Composites. Scienr$c American 10, 1933203 (1986). B. Budianskv. Micromechanics. Comrmr. Sfrucr. 16, 3--12 (1983). . M. J. Shuart, Short wave-length buckling and shear failures for compression-loaded composite laminates. NASA TM87640 (1985). K. H. Lo and E. S-M. Chim, Compressive strength of unidirectional composites. J. R&forced Plastics Corn - posites 11, 838-896 (1992). H. J. Garala, Experimental evaluation of graphites epoxy composite cylinders subjected to external hydrostatic compressive loading. Proc. of‘ 1987 Spring Conference on Experimentul Mechanics, pp. 948 -95 I. Society for Experimental Mechanics (1987). V. Peros. Thick-walled composite material pressure hulls: three-dimensional laminate analysis consider- ations. J. Composife Mafer. 24, 1213-1225 (1990). R. M. Jones, Mechanics of Composite Materiul.~. McGraw-Hill, New York (1975). N. J. Pagano, Exact moduli of anisotropic laminates. In Mechanics of Composite Materials (Edited by G. P. Sendeckyj), pp. 23344. Academic Press (1974). C. T. Sun and S. Li, Three-dimensional effective elastic constants for thick laminates. J. Composite Mater. 22, 629-639 (1988). F.-K. Chang, J. F. Perez and K.-Y. Chang, Analysis of thick laminated composites. /. Composite Mater. 24, 801.-822 (1990). E. Zywicz, Implementation of a lamina-based constitutive relationship for analysing thick composites. ht. J. Numer. Meth. Engng 35, 1031~1043 (1992). R. M. Christensen and E. Zywicz, A three-dimensional constitutive theory for fiber composite laminated media. J. Appl. Mech. 57, 948-955 (1990). R. Vaziri, M. D. Olson and D. L. Anderson, A plasticity-based constitutive model for fibre-reinforced composite laminates. J. Composite Mater. 25, 512-535 (1991). C. T. Sun and G. Chen, Elastic-plastic finite element analysis of thermoplastic composite plates and shells. AIAA Jnl 30, 513-518 (1992). 0. H. Griffin, M. P. Kamat and C. T. Herakovich, Three-dimensional finite element analysis of laminated composites. J. Composite Mater. 15, 543-560 (1981).

16. R. L. McCullough, Micro-models for composite materials--continuous fiber composites. In The Delaware Composites Design Encyclopedia 2 (Edited by L. A. Carlsson and J. W. Gihesnie). Technomic (1990).

17. C. H. Chen and S. Cheng, Mechanical properties of anisotropic fiber-reinforced composites. J. Appl. Mech. 37, 186-189 (1970).

18. D. F. Adams and D. A. Crane, Finite element micromechanical analysis of a unidirectional composite including longitudinal shear loading. Comput. Struct. 18, 1153-1165 (1984).

19. G. J. Dvorak and Y. A. Bahei-El-Din, Plasticity analysis of fibrous composites. J. Appl. Mech. 49, 327-335 (1982).

20. C. T. Sun and J. L. Chen, A micromechanical model for plastic behavior of fibrous composites. Compo.sites Sri. Techno/. 40, 115-129 (1991).

21. D. A. Pecknold, A framework for 3-D nonlinear modeling of thick-section composites. DTRC-SME-90/92, David Taylor Research Center, Bethesda, MD (1990).


23

24.

25

22. J. Aboudi, Micromechanical analysis of composites by the method of cells. Appt. Mech. Rev. 42, 193-221 (1989). J. Aboudi, Mechanics of Composite Materials-d Unified Micromechanical Approach. Elsevier ( I99 I). M.-J. Pindera, C. T. Herakovich, W. Becker and J. Aboudi, Nonlinear response of unidirectional boron/aluminum. J. Composite Mater. 24, 2-21 (1990). J. F. Wu, M. S. Shepard, G. J. Dvorak and Y. A. Bahei-El-Din, A material model for the finite element analysis of metal matrix composites. Composires Sci. Technol. 35, 347-366 (1988). S. R. Bodner and Y. Partom, Constitutive equations for elasticviscoplastic strain hardening materials. J. Appl. Mech. 42, 385-389 (1975). S. Rahman and D. A. Pecknold, Micromechanics-based analysis of fiber-reinforced laminated composites. Structural Research Series No. 572. UILU-ENG-92- 2012, Department of Civil Engineering, University of Illinois at Urbana-Champaign (I 992).

26.

27.

28. D. A. Pecknold and S. Rahman, Nonlinear compression response of thick-section laminates using micromechan- its and homogenization. ASTM Symposium on Com- pression Response of Composite Structures, Miami (1992).

29. B. Irons and S. Ahmad, Techniques of Finite Elements. Halstead Press, New York (1980).

30. A. K. Rov and S. W. Tsai. Three-dimensional effective moduli oforthotropic and symmetric laminates. J. Appl. Mech. 114, 3947 (1992).

31. E. T. Camponeschi, Jr, Compression response of thick-section composite materials. DTRC-SME-90/60, David Taylor Research Center, Bethesda, MD (1990).

32. S. G. Lekhnitskii, Anisotropic plafes (Translated from the second Russian edition by S. W. Tsai and T. Cheron). Gordon & Breach, New York (1968).

33. J. G. Ren, Exact solutions for laminated cylindrical shells in cylindrical bending. Composites Sci. Technol. 29, 169-187 (1987).

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Micromechanics-based structural analysis of thick laminated composites