Combined loading micromechanical analysis of a unidirectional composite

Combined loading micromechanical analysis of a unidirectional composite D.F.

ADAMS and D.A CRANE

A microscopic region of a unidirectional composite is modelled by a finite element micromechanical analysis using a generalized plane strain formulation, but including longitudinal shear loading. The analysis is capable of treating elastic, transversely isotropic fibre materials, as well as isotropic, elastoplastic matrix materials. Matrix material properties are considered to be temperature- and/or moisturedependent. The longitudinal shear loading capability permits the analysis of the shear response of unidirectional composites in the fibre direction. In conjunction with a laminated plate point stress analysis, the present micromechanical analysis has been used to predict the stress/strain response into the inelastic range of a graphite/epoxy [:1:4514s laminate. Available experimental data for various environmental conditions indicate excellent agreement with the analytical predictions.

Key words: composite materials; micromechanics analyses; finite element analysis; elastoplastic response

The benefits and shortcomings of micromechanical analyses of unidirectional composites have now been weighed for well over a decade. A major restriction has been the problem of relating microanalyses to the overall behaviour of a laminate. The importance of longitudinal shear stress in a composite material is evidenced by the fact that load is transferred to a fibre predominantly through longitudinal shear loading. This stress also happens to act in a weak direction of the composite, making it a critical load. Even though longitudinal shear loading is an important consideration, there have been few studies of it pertaining to micromechanics of a composite. In 1967, A d a m s and D o n e r 1 presented an elastic formulation, modelling one quadrant of a repeating fundamental region of a rectangular array of fibres. They used a finite difference representation, and an over-relaxation solution procedure. Stress concentration factors and composite shear moduli were calculated for various fibre volumes for a n u m b e r of cross-sectional shapes of fibres in an epoxy matrix. This first step was soon repeated as numerical analyses developed and computers became more advanced. T M A closed-form series solution was also developed by Sendeckyj. 5 Admittedly, the solution was tedious due to the solution technique employed and fell short of being exact due to the required truncation of the infinite series, nevertheless, determination of the effects of nonuniform fibre spacing, various filament radii, and variation of the shear modulus from fibre to fibre were some of the capabilities of the method.

The elastic solution achieved by Foye 2 in 1968 employed the finite element numerical method for the first time. This generalized plane strain study included two fibre arrangements, separate and combined loading of five of the six components of stress, contours of stresses in the matrix around a fibre, unidirectional ply composite properties and an evaluation of the accuracy of various finite element models. In addition, an incremental inelastic analysis was proposed, which was eventually employed by Baker and Foye 3 in 1969. This extended work revealed a more legitimate stress distribution because inelastic behaviour was considered. Foye continued to publish results of this analysis in 1970' and 1973.6 Although these analyses were a significant achievement, there were still some limitations to be overcome. The iterative scheme inherently accumulated error during inelastic increments which would grow to significant size as the n u m b e r of inelastic increments increased. The iterative inelastic analysis, termed the method of initial strains, had been chosen by Foye, even though it degenerates for highly inelastic behaviour, because the alternative method (the tangent modulus method) would have required an unavailable amount of computer memory for the additional longitudinal shear loading capability. Even though the method of initial strains had the advantage of requiring only one initial inversion of the stiffness matrix, the tangent modulus method was found to be slightly faster, with equal accuracy and the ability to model highly inelastic materials. 7

0010-4361/84/030181-12 $03.00 © 1984 Butterworth ~ Co (Publishers) Ltd. COMPOSITES. VOLUME 15 . NO 3. JULY 1984

181

In the past 15 years, computers have been developed considerably, and the disadvantages attributed to the tangent modulus method have been blunted by the increased size of computer memory. The advantages of the tangent modulus approach began to clearly emerge in the analysis presented by AdamsU in 1970. This was further developed in his subsequent work reported in References 10-13. This analysis method was subsequently adapted by Miller and Adams, ~4-1~ incorporating work by Crossman t~ and Branca? 8 It was a generalized plane strain formulation, including longitudinal normal loading, but not longitudinal shear. The work of Miller and Adams included the addition of a hygrothermal loading capability along with hygrothermally dependent material properties, and Branca's ~s efficient loading scheme. This generalized plane strain approach has been expanded in the present work to include a longitudinal shear loading capability.

ANALYSIS METHOD The changes in theory that arise due to the incorporation of an out-of-plane shear capability into the classical generalized plane strain formulation are detailed in References 19 and 20. Only a brief summary will be included here. The added stress and strain components require revisions in the elastoplastic constitutive equations. In addition, the special treatment of the compatibility of additional boundary conditions results in a special strain/displacement relation.

Generalizedplane strain Leknitskii 2~ defines generalized plane strain in a very general manner, which has been specialized for purposes of the present analysis. This treatment allows displacements to occur in all three coordinate directions, yet retains the advantages of the plane strain assumption. Specifically, each displacement is dependent upon the x- and y-coordinate directions, and the displacement in the z-direction has an added linear dependence on the z-coordinate, which is considered the axial coordinate of a composite in the present analysis. Including the x and y dependence of the z-displacement allows a special form of axial shear deformation corresponding to the generalized plane strain treatment. Expressions for strain can then be calculated for the continuum.

Constitutive equations

Prandtl-Reuss flow rule. Reference 14 discusses sources of more information related to these two criteria. It should be noted that these criteria do not include the possible dependency of the material response on the hydrostatic component of stress, nor the possibility that the yield or ultimate strength may be different in compression than in tension. More detailed experimental studies of polymer material response will be required before fully suitable criteria can be established. The present octahedral shear yield criterion and the Prandtl-Reuss flow rule have been selected in the interim. As more accurate criteria become available the present analysis can be modified as required to incorporate them.

Finite element formulation A major consideration in the finite element formulation is the desire to have a combined loading capability, where any combination of loads can be applied simultaneously. This leads to the seemingly impossible task of applying separate and contradictory boundary conditions simultaneously. The finite element model used in this investigation is based on the assumption that the fibres are distributed in a rectangular array. While there are four axes of geometric symmetry for the typical repeating unit cell (Fig. l(a)), only two can be used due to load symmetry considerations. When considering that each load is assumed to deform the unit cell in a uniform manner, load symmetry about the x and y axes for all five stress components can be easily seen. Only one quadrant of the unit cell (Fig. l(b)) need be considered to describe the behaviour of the unit cell and of an entire continuum of unit cells. Due to the symmetry constraints of the system, normal displacements of the boundaries of the quadrant (Fig. l(b)) are restricted to

y

5y.

21I

~X

i

~

J

The key element of the relationship between load and displacement of a continuum is the constitutive equation. Each constituent material of a composite has unique properties represented in its own constitutive equation. In the present analysis, the fibre material is considered to be elastic and transversely isotropic in order to model fibres such as graphite, but can be reduced to a simple elastic, isotropic material for the purpose of characterizing fibres such as boron and glass. The matrix material is considered to be isotropic and elastoplastic, the plastic response being modelled by the Prandtl-Reuss flow rule. The constitutive equation must also represent elastoplastic material behaviour in the matrix after yielding occurs. An octahedral shear stress yield criterion is employed, and plastic strain is assumed to be proportional to the deviatoric stress tensor, using the

182

OX

~y

b Fig. 1 Symmetry simplification of the fibre arrangement: (a) unit cell; (b) quadrant to be analysed

COMPOSITES. JULY 1984

those which cause the boundary to displace only parallel to the original boundary. This is explained in great detail elsewhere '+ and attention will be focussed on the shear displacements. In the absence of longitudinal shear loading, deformation in the z-direction is simply a constant, uniform deformation of the entire frontal cross-section of the quadrant. Shear deformations constrain only the loaded boundary to displace uniformly in the z-direction, while the opposite face remains stationary in the z-direction. A Vyz shear stress, for example, causes the face at y = b (see Fig. l(b)) to move uniformly in the z-direction while the face at y = 0 is fixed in the z-direction. Meanwhile, the faces at x = 0 and x = a are free to move in the z-direction as strains are induced. Similar treatment of the other longitudinal shear stress follows. It will be noted that these restrictions are not compatible, ie while the face at x ---- 0 is required to be fixed in one case, it is required to be free in the other case. It is possible to apply the two shear boundary conditions separately by adjusting the element strain/ displacement relation. The derivation of this relation for a regular treatment of generalized plane strain is outlined in Reference 19. The resulting equation has been adapted here to manage the shear boundary conditions separately. Since the boundary conditions are concerned with displacements, the shear displacements must be considered to be separate and unique entities at each node point rather than combined into a general z-displacement. Thus, a Vxz shear stress induces a z-displacement Wxz, and a Vyz shear stress induces a z-displacement Wyz. These separate displacements can be arbitrarily specified without affecting other boundary conditions if the strain/displacement matrix equation is expanded slightly, as detailed in Reference 19. The remaining finite element formulation follows a three-dimensional formulation by Zienkiewicz. zz Two differences from Zienkiewicz's formulation are the nodal force vector and, of course, the nodal displacement vector.

Solution technique A finite element analysis computer program developed by Miller and A d a m s ~4 has been adapted for the modified form of generalized plane strain. The additional components of stress and strain result in the necessity of a thorough revision of the basic solution technique. The Branca technique '8 for applying loads and boundary conditions, and the specialized Gaussian elimination solution of the stiffness matrix, are the major burdens in the revision. Developing these new techniques also revealed a more efficient procedure for storing the stiffness matrix, as described in Reference 19.

Computer implementation The incremental procedure used by Miller and Adams ~4 remains intact in the present analysis except for the calculation of octahedral shear stress, which now includes the two added shear stress terms. This tangent modulus method enables highly inelastic materials to be managed easily.

Failure theories There are three failure theories necessary for a comprehensive prediction of failure in the present micromechanics analysis. An octahedral shear stress

COMPOSITES. JULY 1984

criterion is used to define failure of the matrix. This is a measure of the distortional energy stored in the matrix. The limitation of this criterion is that it is not suitable for a hydrostatic load, since there is no distortional c o m p o n e n t present in hydrostatic loading. Therefore, a hydrostatic criterion is also invoked? 9 In prior analyses 14 the fibre was tested for failure by a simple m a x i m u m stress criterion. Because shear stresses also exist in the present analysis, a new criterion must be invoked. The fact that the graphite fibre is assumed to be a transversely isotropic material increases the complexity of the problem. A criterion specifically designed for orthotropic materials, and which considers the entire stress tensor, is the TsaiWu 23 failure criterion. The only awkward characteristic of the Tsai-Wu criterion is the F12 term. This has been discussed widely in the literature, but Narayanswami and Adelman 24 show that neglecting this term rarely causes an error greater than 10%. This so-called 'modified Tsai-Wu' failure criterion has been incorporated into the present analysis.

CONSTITUENT MATERIAL PROPERTIES The matrix system used for experimental verification purposes in the present investigation was Hercules 3501-6 epoxy resin. Because longitudinal shear loading is a major consideration in the present study, it is appropriate to employ matrix constituent material properties derived from longitudinal shear experimental data. Solid rod torsion test shear data were available for this matrix material. 25 Data from either shear tests or tensile tests can be readily converted to octahedral shear stress/octahedral shear strain expressions for input to the analysis. Results reduced from the solid rod torsion test data are shown in Fig. 2. To use these data in the micromechanics computer program, they must be expressed in equation form, with temperature and moisture as independent variables. First, a three-parameter curve-fitting method developed by Richard and Blacklock '6 was used to fit each data curve (at each test condition). Each set of three constants from each test condition, and values for ultimate stress, are then used in another curve-fit relation which expresses each value by a second-order equation dependent upon temperature and moisture. The resulting values of these experimenal parameters are presented in Reference 19. Fibre properties were mainly obtained from Hercules '7 and Owens-Corning ~8 literature for the AS-graphite fibre and the S2-glass fibre, respectively. Shear moduli 100 21°C o 3 . 4 w t /o m o i s t u r e 6 . 8 wt°lo m o i s t u r e

i+

Dry

100°C

Ill

i +°

~ Dry

j.4

~

)~ D r y

25

160°C

I

0

0

mo,sture

6.8 wt % moisture

50

x 3 . 4 w t °/o m o i s t u r e x 6.8wt % moisture I I I I 100 150 200 250 300 O c t a h e d r e l s h e a r s t r a i n (xlO "3)

Fig. 2 Octahedral shear stress/octahedral shear strain curves for Hercules 3 5 0 1 - 6 epoxy matrix generated from solid rod torsion tests

183

Table 1.

Constituent material properties for AS-graphite fibre, S2-glass fibre, and 3501-6 epoxy resin

Property

Longitudinal modulus, EL (GPa) Transverse modulus, ET (GPa) Longitudinal shear modulus, GET (GPa)

Hercules AS-graphite fibre 27 220 14 34*

Owens-Corning S2-glass fibre 2s

Hercules 3501-6 epoxy matrix (room temperature, dry) 25

86 86

4.3 4.3

35

2.1

35

2.1

Transverse shear modulus GTr (GPa)

5.5

Major Poisson's ratio, VET

0.20

0.22

0.34

In-plane Poisson's ratio, Prr

0.25

0.22

0.34

Longitudinal tensile strength, GLu (MPa)

31 O0

4800

83

Transverse tensile strength, GTu (MPa)

345*

4800

83

Longitudinal shear strength, t'LTu (MPa)

1550*

2400

85

1 70*

2400

85

Transverse shear strength, l:Tru (MPa) Longitudinal coefficient of thermal expansion, eL (10-6/°C)

-0.36

5.0

40

Transverse coefficient of thermal expansion, a'r (10-6/°C)

18

5.0

40

Coefficient of moisture expansion, fl (10-3/% M)

0

0

2.0

*Estimated

and shear stress allowables necessary for stiffness calculations and the fibre failure criterion were easily found for the S2-glass (assumed isotropic). For the graphite fibre, in-plane and longitudinal shear tests required to determine these properties are not available, due to the small fibre diameter. Instead, a shear modulus value was calculated in the transverse plane of isotropy from the respective values of Young's modulus and Poisson's ratio. It was necessary to estimate values of longitudinal shear ultimate strength and longitudinal shear modulus from data for other similar graphite fibres. 29 Properties of the constituent materials are presented in Table 1 with estimated values denoted by an asterisk.

within each term of that constitutive equation. This effect is small, but significant enough to warrant its inclusion. This inelastic strain coupling contribution has been observed to be up to 6% of the total strain. To demonstrate that the longitudinal shear version of the micromechanics analysis provides accurate and useful results, two types of examples will be presented. The first uses the constituent material properties to predict results using the longitudinal shear loading analysis, which are then compared with actual solid rod torsional shear data, for two different composite materials. The second example predicts laminate behaviour using the longitudinal shear loading analysis in conjunction with a laminate point stress analysis.

NUMERICAL RESULTS

Comparison of analytical predictions with solid rod torsion test data

Test cases analysed using the present longitudinal shear loading version of the generalized plane strain micromechanics program indicate that the new analysis performs all of the same operations as the prior version, 1+ with the addition of a shear loading capability. As theory predicts, there is no coupling between shear and normal loads during elastic load increments, but during increments beyond the elastic limit the Prandtl-Reuss flow rule is in effect and coupling does occur. When the deviatoric stress tensor is non-zero (during inelastic increments), the PrandtlReuss flow rule causes each element stiffness matrix to become fully populated. This in turn forces each stress term to be dependent on each strain term. Thus, a shear stress can then induce a small normal strain, which is impossible in elasticity theory, but entirely possible in plasticity theory. In fact, all the stress components have an effect on all strain components if the deviatoric stress tensor is fully populated. The inelastic constitutive equation governs the coupling because the additional strain induced during inelastic increments is due to the two deviatoric stress terms

184

Solid rod shear data had been generated 25 for composite specimens as well as for the previously discussed solid rod torsion tests of the epoxy matrix. Fibres used in the composite specimens were the Hercules AS-graphite fibre 2~ and the Owens-Coming S-glass fibre. 28 Hercules 3501-6 epoxy resin was used as the matrix system. 25 These composites will henceforth be referred to as G R / E P and GL/EP, respectively, for the graphite and glass fibre composites. Using the properties evaluated from the previously discussed solid rod torsion tests of the matrix system, the micromechanics analysis was used to predict the composite shear stress/shear strain behaviour. Samples of these comparisons are shown in Figs 3-6 for both GR/EP and GL/EP. Tests were actually performed at room temperature, 100°C, and 160°C, at two moisture conditions, ie dry and fully saturated (6.75% moisture (M) by weight). Additional comparisons are presented in Reference 19. The present longitudinal shear micromechanics

COMPOSITES. JULY 1 984

100

analysis is only operational up to first element failure, which signifies crack initiation or some other local failure on the micro scale. The present lack of a crack propagation capability prevents the analysis from predicting the actual failure strength of a composite; but it does predict inelastic behaviour and the initiation of failure. Thus, each of Figs 3-6 show only the initial portion of the complete shear stress/shear strain curve, ie, to predicted first failure, but this is considered to be enough to reveal the accuracy of the prediction technique.

75 o_

so

"" ~

o

n s Solid rod torsion test data

b 25

0

I

0

I

10 20 30 Shear strain (xlO00) Fig. 3 Comparison of shear stress/shear strain curves obtained from micromechanics predictions and solid rod torsion tests for graphite/epoxy at room temperature, dry conditions (RTD)

-

50

-

~40 o Q..

~30 L.

~ 20 /

Solidrodtorsiontestdata

r-

10 0

I

0

I

5

10

I

15

20

Shear strain (x 1000) Fig. 4 Comparison of shear stress/shear strain curves obtained from micromechanics predictions and solid rod torsion tests for graphite/epoxy at 100°C, dry conditions (ETD) )c

80

o

60

n

The local stress distribution in an individual fibre and the surrounding matrix material is also predicted by the analysis. Extensive data are presented and discussed in Reference 19. Samples of these results will be presented here. Typical local stress states predicted in the epoxy matrix surrounding an individual fibre of a dry G R / E P torsion specimen tested at room temperature are shown in Figs 7-9. Octahedral shear stresses due to curing ranged up to 75% of the elastic limit, as shown in Fig. 7(a). Octahedral shear stress is a measure of the distortional energy in the material; the highest values are seen to occur in regions of close fibre spacing, directly between the fibres, where element yield (Fig. 7(b)) and eventually first failure (Fig. 7(d)) occur. The contour value of 2.85 corresponding to first element failure represents an octahedral shear stress of 82 MPa. In Fig. 3, the data indicate yield at about 30 MPa, although the octahedral shear stress contour plot (Fig. 7(c)) shows nearly 20% of the matrix to have already yielded, revealing the fact that local matrix yielding occurs long before the composite as a whole •begins to show any sign of non-linear response. Even the predicted shear stress/strain response (shown by the triangles in Fig. 3) remains nearly linear well past the point of first element yielding.

a

\

4o

o.4~

\\

II 0.35

/

04

Solid rod torsion test data

20

05 045

'A

0

I I 10 20 30 Shear strain (×1000) Fig. 5 Comparison of shear stress/shear strain curves obtained from micromechanics predictions and solid rod torsion tests for glass/epoxy at room temperature, dry conditions (RTD)

0.55

0

60 X

0.6

0.65

oJ

~o.4 7

0 7 ~

10.---

~ 30 g

s

~) 15 /

13"

Solid rod torsion test data

/ 0

I I I I 15 20 25 30 35 Shear strain (x1000) Fig. 6 Comparison of shear stress/shear strain curves obtained from micromechanics predictions and solid red torsion tests for glass/epoxy at room temperature, saturated (6.75% moisture by weight) conditionsi(mW) 0

I 5

I 10

COMPOSITES . JULY 1984

Fig. 7 Contour plots of octahedml shear stress (normalized with respect to matdx yield stress, 28.8 MPa) within a graphite/epoxy solid rod torsion specimen tested at room temperature, dw conditions (wro): (a) after cool down from 177"C cure temperature to 21 "C; (b) at predicted first element yield, ~'xz= 14 MPa; (c) at experimentally determined composite yield stress, £xz = 31 MPa; (d) at predicted first element failure, ~t= = 65 MPa

185

\oJ

a

indicates a probable interfibre macrocrack failure, which could propagate to the interface, or merely cause a split directly between adjacent fibres. Both criteria predict failure in the same general region of close fibre spacing, but when dealing with an actual composite, the actual failure mode will also be influenced by the particular flaws in that region. For example, a local region of poor fibre/matrix bonding would be opened by the high tensile stress shown in Fig. 8(a).

,,

b

-3.5 0 3 . 5

-3.5 03.5

Fig. 8 Plots of maximum principal, minimum principal and interface normal stress within a graphite/epoxy solid rod torsion specimen tested at room temperature, dry conditions (RTO): (a) maximum principal stress (MPa) st predicted first element failure, ~xz = 65 MPa; (b) minimum principal stress (MPa) at predicted first element failure, ~'xz= 65 MPa; interface normal stress (MPa) (c) after cool down from 177°C cure temperature to 21°C; (d) at predicted first element failure,~'xz = 65 MPa

Cool down from the cure temperature induces high compressive normal stresses at the interface, due to the contraction of matrix material around the fibre (Fig. 8(c)). The contraction of the matrix material, in conjunction with the constant displacement boundary conditions, moves the boundaries together slightly, enough to compress the small volume of material near the axes of symmetry. This normal stress is unaffected by subsequently applied longitudinal shear loads, and only slightly increased by contributions due to inelastic material response at higher levels of the applied shear stress, as shown in Fig. 8(d). The reasons for this are that there is no coupling between normal and shear stresses during elastic behaviour, and that when coupling does occur (when the matrix behaviour is inelastic), the deviatoric stress tensor is so sparsely populated that the normal stress is only increased by about 3%. Due to the additional longitudinal shear stress components, the present analysis must include both inplane and out-of-plane shear stresses in the interface shear stress contour plots, in contrast to previous treatments, l~,3° Now the shear stress contour represents only the magnitude; the direction of the interface shear stress can vary from position to position along the fibre in any direction on the fibre surface.

\

\ LO

0100

0 I00 L5

2.0

0 70

\ 0 100

\, o._1

Z4 •

~

1.2

0 100

Fig. 9 Plots of interface shear stress (MPa) within a graphite/epoxy solid rod torsion specimen tested at room temperature, dry conditions (RTD): (a) after cool down from 177°C cure temperature to 21 °C; (b) at first element yield, ~xz = 17 MPa; (c) at data yield point, ~'xz= 31 MPa; (d) at first element failure, ~xz = 65 MPa

While the octahedral shear stress failure criterion predicts failure directly between fibres, a maximum stress criterion would disagree; high tensile stresses are seen to occur near the fibre in Fig. 8(a). High compressive stresses are seen to occur at the interface along the x-axis of symmetry (Fig. 8(b)) at the fibre/ matrix interface, indicating a potential failure of the interface bond. The octahedral shear stress criterion

186

Fig. 10 Plots of octahedrel shear (normalized with respect to matrix yield stress, 20 MPa) and interface normal stresses within a graphite/epoxy solid rod torsion specimen tested at room temperature, moisture-satureted conditions (6.75% moisture by weight) (R/W): (a) octahedral shear stress after cool down from 177°C cure temperature to 21 °C and moisture absorption of 6.75 %M; (b) interface normal stress (MPa) after cool down from 177°C cure temperature to 21°C and moisture absorption of 6.75 %M; octahedral shear stress at (c) experimentally determined composite yield stress, ~xz = 28 MPa; (d) predicted first element failure, "~xz= 56 MPa

COMPOSITES. JULY 1984

Interface shear stresses developed during cool down (Fig. 9(a)) are only 25% of the shearing stress at failure (Fig. 9(d)). It will be noted that the curing stresses become purely hydrostatic on the horizontal and vertical axes of symmetry and at 45 °, due to geometric symmetry. Viewing Figs 9(a)-(d) consecutively shows that the initial curing residual shear stress is small relative to the load-induced shear stress, which eventually initiates failure. The G R / E P in a room temperature test, when moisture-saturated (6.75% M by weight), reveals very high octahedral shear stresses and interface normal stresses due to moisture dilation, as seen in Figs 10(a) and 10(b). Although thermal contraction during cool down counteracts the subsequent moisture dilation, the extent of moisture dilation far overshadows the thermal contraction. The majority of matrix material is already inelastic before loads are applied (Fig. 10(a)), which reduces the subsequent load-carrying capability, while the moisture also softens the matrix, thus reducing its ultimate strength. Figs 10(c) and 10(d) show the increased inelastic behaviour that entirely envelopes the matrix at failure. The moisture loading governs the contours, ie they remain more nearly symmetric than in the loading condition without moisture (Fig. 7(d)). Again a high interface normal stress is present (Fig. 1 l(a)), while a high interface shear stress (Fig. 1 l(b)) contributes to the high octahedral shear stress state shown in Fig. 10(d). The maximum and minimum principal stress contours (Figs 1 l(c) and 1l(d)) have opposite results than in the room temperature, dry specimens, due to the opposite effects of moisture and curing stresses. The minimum principal stress shows a highest absolute value at 45 ° from the x-axis of symmetry at the interface. Because the absolute value of the maximum principal stress is higher than the minimum principal stress, failure would be predicted to occur at the fibre/matrix

-70

0 70

0100

-83 55-83~

C

,2 ~

Fig. 11 Plots of interface normal, interface shear, maximum principal and minimum principal stresses within a graphite/epoxy solid red torsion specimen tested at room temperature, moisture-seturated conditions (6.7596 moisture by weight) (RWV) at predicted first element failure Fxz == 56 MPa: (a) interface normal stress (MPa); (b) interface shear stress (MPa); (c) maximum principal stress (MPs); (d) minimum principal stress (MPa)

COMPOSITES. JULY 1984

interface nearest the x-axis if the maximum stress criterion were used. A different result is again shown by the octahedral shear stress criterion (Fig. 10(d)), which predicts an interfibre failure, although it is near the same area. The elevated test temperature (100°C) G R / E P specimen under dry conditions indicated a relatively less intense octahedral shear stress state due to curing than the room temperature, dry specimen. This occurs not only because of the smaller temperature excursion, but also because of the lower stiffness properties at the elevated temperature. This results in lower stress concentrations when flaws are present in real specimens. Thus the assumption of no flaws and perfect bonds becomes less severe, and the analysis predicts more accurately. Interface normal stresses are negligible compared with the magnitude of the octahedral shear stresses occurring at failure, while maximum and minimum principal stress contours predict maximum absolute values at the same point as the octahedral shear stress contours; between the fibres on the x-axis of symmetry. Therefore, an elevated temperature G R / E P specimen would tend to fail within the matrix, where a crack would initiate and subsequently begin to propagate. Because the G L / E P tested at room temperature and 6.75% M had a lower fibre volume (50.5%) than the G k / E P (55%) tested at similar conditions, and because the fibre properties are different from graphite, slightly different results can be expected from the microanalysis. But after the cool down from the curing temperature and subsequent moisture conditioning, the G R / E P and G L / E P indicated very similar octahedral shear stress contours. Coefficients of thermal expansion vary markedly between the glass fibre and the graphite fibre (see Table 1); the graphite fibre expands axially when temperature is lowered, and contracts in the transverse direction. The glass fibre, being isotropic, contracts in all material directions when temperature is lowered. However, the matrix thermal contraction is much higher than that of either fibre. These large differences between coefficients of thermal expansion from fibre to matrix cause stresses to be induced. These differences are highest in the axial direction of the GR/EP, and lowest in the transverse direction of GR/EP, while the G L / E P is inbetween, although quite high. Considering this and the fact that more matrix is present in the G L / E P because of its lower fibre volume, the overall thermal and moisture effects tended to be equal for both composites. Therefore, similar contours were obtained for both G R / E P and GL/EP. As the applied shear stress is increased until the first element failure occurs, the contours also propagate in a similar manner. But the average stress at failure for the G L / E P is higher (63 MPa) than that for the G R / E P (56 MPa). The matrix in the G L / E P has yielded to a greater extent than in GR/EP, and more energy has been absorbed. Thus, the lower fibre volume makes a more uniform stress distribution possible in the matrix, ie the stress concentration due to the fibre decreases. The brief results presented here show good correlation with experimental data, suggesting that the micromechanics analysis is valid. These results cannot be considered conclusive, of course. A much more comprehensive study should be undertaken in the future, to reinforce the present results.

187

Laminate analysis The use of micromechanics analyses was restricted to unidirectional laminates in prior works, since no longitudinal shear considerations were taken into account. Longitudinal shear stresses are a necessary consideration in any laminate analysis, since even a simple normal loading will induce shear stresses in offaxis plies. The laminate analysis used must be compatible with the restrictions of the micromechanics analysis, however, and certain assumptions must be made. Because direct application of the interlaminar shear stress cxy (in the micromechanics coordinate system, see Fig. 1) is not a capability of the present micromechanics analysis, a two-dimensional classical laminated plate point stress analysis was chosen over a three-dimensional finite element laminate analysis. The AC-3 laminate point stress analysis computer program 3' is operational at the University of Wyoming. Basically, it describes elastic stresses and strains induced in each ply due to loads applied to the laminate. These can be temperature and moisture loadings, as well as mechanical loads. Compatibility of the two analyses and the scheme used to arrive at an inelastic stress/strain curve for a particular laminate is described by the following four steps: 1.

Calculate ply properties from the micromechanics analysis.

2.

Apply the laminate point stress analysis to the prescribed laminate, loaded in the prescribed manner, to obtain the stress state induced in each ply.

3.

Use the micromechanics analysis to analyse each individual ply of the laminate, holding loads in the same ratios as the laminate analysis predicted. This will result in inelastic stresses and strains in each ply when loaded to high levels.

4.

Transform the stress and strain states back to laminate coordinates to obtain a laminate stress/ strain curve. This curve can be used for comparisons with laminate experimental data.

Experimental data were available for a 57.5% fibre volume, [±4514s G R / E P laminate at four combinations of temperature and moisture conditions, a° viz room temperature, dry (RTD); room temperature, 1% M by weight (RTW); elevated temperature (103°C), dry (ETD); and elevated temperature (103°C), 1% M (ETW). Both constituent materials were the same graphite and epoxy previously defined in Table 1. The micromechanics analysis is assumed to model a single unidirectional ply of the laminate in ply coordinates, by modelling one quadrant of the repeating unit cell, as depicted in Fig. 12. The micromechanics assumption that boundaries displace uniformly was described earlier. The laminate analysis intrinsically makes this assumption, since classical point stress analyses do not include interlaminar shear stresses. In an actual composite, certain interlaminar shear stresses cause shear deformation in the plane transverse to the fibre direction. This is in-plane deformation for the micromechanics model, which is inadmissible in this analysis due to the constant displacement boundary conditions. Therefore, interlaminar shear stresses are assumed to be negligible in the present example. Another assumption is that the load ratios will change only negligibly due to inelastic material behaviour. With the above assumptions in effect, the entire curing 200

150

~

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COMPOSITES . J U L Y 1 9 8 4

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and conditioning history of each ply is approximated, as well as the effect of the applied load increments. The resulting inelastic strains at each incremental applied load are in ply coordinates and must be transformed back to laminate coordinates. This done, longitudinal stress and strain at various increments for each condition are the final results needed to compare with experimental data. Figs 13-16 show typical predicted results compared with the experimental data curves. Reasons for the good correlations of the analysis with the experimental data can be investigated by observing the loading history recorded in the local matrix stress contour plots. Normalized octahedral shear stress contours for the RTD test case are shown in Fig. 17, ranging from curing stresses to the failure stress state. Referring back to the original step-by-step description of the laminate analysis presented earlier, it will be noted that hygrothermal effects are considered in separate steps. First there are stresses induced by hygrothermally-influenced constituent material properties (see Fig. 17(a)). But the other plies are also influenced by hygrothermal effects, which in turn influence the stresses in the representative ply being

COMPOSITES.

JULY 1984

considered (see Fig. 17(b)). These will be termed 'ply curing effects' and 'laminate curing effects', respectively. Because Fig. 17(a) is also representative of stresses in a unidirectional composite, contrasting Figs 17(a) and 17(b) shows curing stress differences between a unidirectional laminate and a [___45] laminate. But, for the present purposes, it is important to note the additional curing effects of the ply configuration in Fig. 17(b), and their magnitude. Any material defect which might exist in a local region of high stress will cause a severe stress concentration, which in turn can cause premature matrix yielding and failure. This is a possible reason for the lower stiffness exhibited by the experimental data at room temperature conditions (Figs 13 and 14). Fig. 17(c) shows the amount of matrix that has already yielded locally when the overall composite response first indicates a yield point experimentally. While the tangent modulus of the yielded matrix is still close to the initial modulus, the non-linearity is not apparent in the composite response (Fig. 13). Only after the yielded region has propagated upwards a distance equal to about half the width of the quadrant do the experimental data begin to suggest an elastic limit. When the micromechanics analysis predicts first element failure, this implies the initiation of a crack. To prevent this damage initiation, the composite should thus not be loaded past the stress represented by the final predicted point in Fig. 13. A more comprehensive study would reveal just how much load could safely be added beyond yield, while preventing major microcracking. After curing, the composite experiences large normal stresses along the interface, as shown in Fig. 18(a), suggesting a high probability of debonding behaviour. There are also fairly high shear stresses along the interface in this as-cured condition, which grow enormously as load is applied and increased to predicted first failure (Figs 18(b) and 18(d)). This shear stress is the combined effect of the

189

reduced to 36 MPa in the ETW case. This is a significant change, considering that it is due to a relatively modest temperature change.

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Fig. 18 Plots of interface normal and interface shear stresses within a graphite/epoxy [4-45]4 s laminate ply at room temperature, dry conditions (RTD) under an applied axial load, (xx = - 1 4 5 MPa; (a) interface normal stress (MPa) after laminate cure and cool down; (b) interface shear stress (MPa) after laminate cure. cool down and moisture-absorption effects; (c) interface normal stress (MPa) at predicted first element failure; (d) interface shear stress (MPa) at predicted first element failure

applied longitudinal shear stress and the transverse normal stress, which are acting in weak directions of the material, and which eventually lead to failure. It will also be noted in Fig. 18(c) that the interface normal stress has been reduced by the applied loads. This is further evidence of how applied loads can relieve some of the stresses induced during cool down; it is an inelastic coupling effect. The ETW test case involves two separate forms of curing and conditioning stress. First, stresses are induced independently in each ply by thermal and moisture changes due to the difference in properties between the fibre and matrix. Secondly, since the individual plies exhibiting these hygrothermal strains are bonded to each other, they mutually affect each other because of their differing ply orientations. The combination of thermal and moisture loads has farreaching consequences because of the opposing effects in cool down and moisture absorption. The temperature drop during cool down causes the matrix to contract by a relatively large amount, the fibre to contract transversely by a lesser amount, and the fibre to actually expand slightly in its axial direction (see Table 1). The composite axial strain is predicted to be negative. The octahedral shear stresses induced by curing are almost fully negated by the absorption of 1% M by weight by the composite. The small hygrothermal stresses cause only small average laminate hygrothermal stresses. However, shifting of the octahedral shear stress contours due to the application of the induced loads was noted. The detrimental effect of unfavourable temperature and moisture combinations can be clearly seen by comparing the octahedral shear stress contours at first failure of the ETW test case with the RTD case. An octahedral first element failure stress of approximately 80 MPa (shown normalized in. Fig. 17(d) for the RTD case as the 2.8 contour) is

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Interface normal stresses induced during cool down and 1% M absorption are relatively small tensile stresses, but subsequent mechanical loading of the laminate counteracts the induced tensile stresses, so that near the x-axis of symmetry a small negative value exists at first element failure. Small tensile stresses are induced near the y-axis of symmetry at first element failure, but are small enough that they have no noticeable effect on subsequent laminate behaviour. Although the interface shear stress after curing and moisture conditioning is small (4% of the ultimate stress), the subsequently applied mechanical loads result in shear stresses along the interface which are very high at firs( element failure. Although the interface shear stress is high, the octahedral shear stress is shown to be highest at the outer edge of the model, along the x-axis of symmetry. Failure in this elevated temperature, 1% M condition would probably occur between the fibres in the region of highest octahedral shear stress. But at room temperature, dry conditions, a higher stress would be induced along the interface (see Fig. 18(d)), increasing the the probability of failure of the fibre/ matrix bond. A resulting hypothesis is that because this failure is likely to occur more often in the RTD laminate, an actual laminate at these conditions has a higher probability of failure initiation due to inherent flaws in the material, especially along the fibre/matrix interface. Thus, experimental data show a lower initial modulus and a lower overall stiffness than theory predicts, as in Fig. 13.

CONCLUSIONS The establishment of a generalized plane strain finite element formulation that includes a longitudinal shear loading capability has further advanced composites technology, by making detailed information concerning the micromechanical shear loading response of composite materials fully available. Such a quasi-threedimensional formulation approaches the capability of a true three-dimensional analysis, while retaining the conciseness of a two-dimensional analysis. Although the present generalized plane strain analysis only approximates a true three-dimensional analysis, the present results show that this is quite accurate, and therefore valuable in many potential applications. A full three-dimensional study of the micromechanical response should be performed, to establish the amounts of error incurred by the special assumptions of generalized plane strain. Such a three-dimensional program is now operational? 2 With the longitudinal shear loading capability, interfacing with a laminate analysis, either the point stress analysis 31 used here or a full three-dimensional analysis, 32 is made possible, enabling the investigation of complicated laminates. The results of the comparisons between theory and experiment for the [+4514s laminate presented here are encouraging, although limited to one simple laminate. However, because this particular laminate is dominated by a longitudinal shear loading mechanism, it is assumed that further investigation of various other laminates will again show good agreement with experiment. One such application to a much more complex laminate 33 has in fact indicated equally good agreement.

COMPOSITES . J U L Y 1 9 8 4

The combined micromechanics/laminate analysis predicts o n l y a slightly n o n - l i n e a r stress/strain curve u p to the p o i n t w h e r e first e l e m e n t failure is p r e d i c t e d to occur. In the p r e s e n t f o r m u l a t i o n , this is the t e r m i n a tion p o i n t o f the analysis. I f a c r a c k i n i t i a t e d at the first e l e m e n t failure is a l l o w e d to grow, p e r h a p s a n entire stress/strain curve c o u l d be m o d e l l e d , a n d u l t i m a t e stresses predicted. This c r a c k p r o p a g a t i o n c a p a b i l i t y has b e e n d e v e l o p e d in a n o t h e r microm e c h a n i c a l analysis, 34 w h i c h c o u l d be closely followed in i n c o r p o r a t i n g a s i m i l a r c a p a b i l i t y in the p r e s e n t p r o g r a m . This w o u l d p r o v i d e a c a p a b i l i t y to m o d e l m o r e c o m p l e t e l y the b e h a v i o u r o f l a m i n a t e s over their entire l o a d i n g range. Such further d e v e l o p m e n t s o f the a n a l y s i s p r e s e n t e d here s h o u l d l e a d to the c o n t i n u e d a d v a n c e m e n t o f the design o f high p e r f o r m a n c e c o m p o s i t e structures. A s such a n a l y s e s develop, design p a r a m e t e r s for c o m p o s i t e s will b e c o m e m o r e useful a n d well-known. C o m p o s i t e s will in t u r n b e c o m e the logical m a t e r i a l choice for m a n y m o r e a p p l i c a t i o n s , p r e s e n t i n g a n i m p o r t a n t alternative to the use o f h o m o g e n e o u s metals.

ACKNOWLEDGEMENTS T h e p r e s e n t w o r k was s p o n s o r e d b y the U S A r m y M a t e r i a l s a n d M e c h a n i c s R e s e a r c h Center, u n d e r the direction o f D r R. Lewis, Chief, C o m p o s i t e s Division. T h e a s s i s t a n c e o f g r a d u a t e students M.M. M o n i b a n d D.P. M u r p h y is also gratefully a c k n o w l e d g e d .

REFERENCES 1 Adams, D.F. and Doner, D.R. 'Longitudinal shear loading of a unidirectional composite' J Composite Mater 1 (January

2

3

4 5 6 7 8 9

10

11 12

1967) pp 4-17 Foye, R.L. 'Advanced design concepts for advanced composite airframes' Technical Report AFML-TR-68-91 (North American-Rockwell Corporation, Columbus, OH, USA, July 1968) 1 Baker, D.J. and Foye, R.L. 'Advanced design concepts for advanced composite airframes' Technical Management Report No 2 (North American-Rockwell Corporation, Columbus, OH, USA, October 1969) Foye, R.L. 'Stress concentrations and stiffness estimates for rectangular reinforcing arrays' J Composite Mater 4 (October 1970) pp 562-566 Sendeckyj, G.P. 'Longitudinal shear deformation of composites II, stress distribution' J Composite Mater 5 (January 1971) pp 82-93 Foye, R.L. 'Theoretical post yielding behavior of composite laminates, part I - - inelastic micromechanics" J Composite Mater 7 (April 1973) pp 178-193 Marcal, P.V. 'A comparative study of numerical methods of elastic-plastic analysis'AL4A Journal 6 No 1 (January 1968) pp 157-158 Adams, D.F. "Inelastic analysis of a unidirectional composite subjected to transverse normal loading' J Composite Mater 4 (July 1970) pp 310-328 Adams, D.F. 'Inelastic analysis of a unidirectional composite subjected to transverse normal loading' Report RM-6245-PR (The Rand Corporation, Santa Monica, CA, USA, May 1970) Adams, D.F. 'High performance composite materials for vehicle construction: an elastoplastic analysis of crack propagation' Report R-IO70-PR (The Rand Corporation, Santa Monica, CA, USA, March 1973) Adams, D.F. 'Elastoplastic crack propagation in a transversely loaded unidirectional composite' J Composite Mater 8 (January 1974) pp 38-54 Adams, D.F. 'Practical problems associated with the application of the finite element method to composite material micromechanical analyses' Fibre Sci and Tech 7 No 2 (April 1974) pp 111-122

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13 Adams, D.F. 'A micromechanicai analysis of crack propagation in an elastoplastic composite material' Fibre Sci and Tech 7 No 4 (October 1974) pp 237-256 14 Miller, A.K. and Adams, D.F. 'Micromechanical aspects of the environmental behavior of composite materials" Department Report UWME-DR-701-111-1 (Mechanical Engineering Department, University of Wyoming, USA, January 1977) 15 Adams, D.F. and Miller, A.K. 'Hygrothermal microstresses in a unidirectional composite exhibiting inelastic material behavior' J Composite Mater 11 No 3 (July 1977) pp 285-299 16 Miller, A.K. and Adams, D.F. 'Inelastic finite element analysis of a heterogeneous medium exhibiting temperature and moisture dependent material properties' Fibre Sci and Tech 13 No 2 (March-April 1980) pp 135-153 17 Crossman, F.W. "Computer simulation of the deformation of composite materials by finite element analysis' Ph D thesis (Stanford University. Stanford, CA, USA, May 1972) 18 Branca, T.R. 'Creep of a unidirectional metal matrix composite subjected to axial and normal lateral loads' TAM Report 341 (Theoretical and Applied Mechanics Department, University of Illinois, Urbana, IL, USA, June 1971) 19 Crane, D.A. and Adams, D.F. 'Finite element analysis of a unidirectional composite including longitudinal shear loading' Department Report UWME-DR-O01-104-1 (Mechanical Engineering Department, University of Wyoming, USA, December 1980) 20 Adams, D.F. and Crane, D.A. 'Finite element micromechanical analysis of a unidirectional composite including longitudinal shear loading' submitted to Computers and Structures 21 Leknitskii, S.G. 'Theory of Elasticity of an Anisotropic Elastic Body' (Holden-Day, Inc, San Francisco, CA, USA, 1963) 22 Zienkiewiez, O.C. "The Finite Element Method' (McGraw-Hill Book Co, New York, NY, USA, 1977) 23 Tsai, S.W. and Wu, E.M. 'A general theory of strength for anisotropic materials' J Composite Mater 5 (January 1971) pp 58-80 24 Narayanswami, R. and Adelman, H.M. 'Evaluation of the tensor polynomial and Hoffman strength theories of composite materials' J Composite Mater 11 (October 1977) pp 366-377 25 Unpublished experimental data for Hercules 3501-6 epoxy matrix (Composite Materials Research Group, Department of Mechanical Engineering, University of Wyoming, USA, 1978) 26 Richard, R.M. and Blacklock, J.M. 'Finite element analysis of inelastic structures'A/AA Journal 7 No 3 (March 1969) pp 432-438 27 'Hercules Magnamite graphite fibers' (Hercules, Inc. Wilmington, DE, USA, 1978) 28 'Textile fibres for industry' (Owens-Coming Fiberglass Corporation, Toledo, OH, USA, 1971) 29 Kriz, R.D., Stinchcomb, W.W. and Tenney, D.R. "Effects of moisture, residual curing stresses and mechanical load on the damage development in quasi-isotropic laminates" Report No 16 (NASA-Virginia Tech Composites Program, February 1980) 30 Grimes, G.C. and Adams, D.F. "Investigation of compression fatigue properties of advanced composites' Northrop Technical Report NOR 79-17 (Northrop Corporation, Hawthorne, California, USA and University of Wyoming, USA, October 1979) 31 'AdvancedCompositesDesign Guide Vol II-Analysis~ Appendix B (Air Force flight Dynamics Laboratory, Dayton, OH, USA, January 1973) 32 Monib, M.M. and Adams, D.F. 'Nonlinear three-dimensional finite element analysis of composite laminates' Department Report UWME-DR-O01-102-1 (Department of Mechanical Engineering, University of Wyoming, USA, November 1980) 33 Grimes, G.C., Adams, D.F. and Dusablon, E.G. 'The effects of discontinuities on compression fatigue properties of advanced composites' Northrop Technical Report NOR 80--158 (Northrop Corporation, Hawthorne, California, USA and University of Wyoming, USA, October 1980) 34 Adams, D.F. and Murphy, D.P. 'Analysis of crack propagation as an energy absorption mechanism in metal matrix composites' Report UWME-DR-IOI-102-1 (University of Wyoming, Department of Mechanical Engineering, USA, February 1981)

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AUTHORS

A graduate student at the University of Wyoming, D.A. Crane is presently with General Dynamics Corporation,

192

Fort Worth, TX, USA. D.F. Adams, to whom inquiries should be directed, is Professor of Mechanical Engineering in the Composite Materials Research Group, Department of Mechanical Engineering, University of Wyoming, Laramie, WY 82071, USA.

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