Non-linear constitutive model for plain-weave composites

Composites Part B 28B (1997) 627-634

PII: S1359-8368(96)00079-0

ELSEVIER

© 1997 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/97/$17.00

Non-linear constitutive model for plain-weave composites

Adnan H. Nayfeh a and G. R. Kress h

aDepartment of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH 45221-0070, USA blnstitute for Design and Construction Methods, Swiss Institute of Technology, 8092 Zurich, Switzerland (Received 1 September 1996; accepted 20 November 1996) A non-linear constitutive model for plain-weave composites is developed which is based on the micromechanical behavior of a representative unit cell. The global constitutive relationships are consistently derived from the total strain energy of the system. Compatibility of the constituent's geometric non-linearities is fully taken into account where the matrix plays the role of an elastic foundation and is thus modeled as a continuum. © 1997 Elsevier Science Limited. (Keywords: A. fabrics/textiles; B. mechanical properties; C. micro-mechanics; C. analytical modelling)

INTRODUCTION Textile composites in general and plain-weave fabrics in particular are emerging as a new generation of advanced, complex, structural materials with the potential of enhancing many of the shortcomings of conventional layered or fiber-reinforced composites. The expansion of application of composites from secondary unloaded utilization to primary load-carrying applications requires a significant improvement of their damage tolerance and reliability. In order to achieve these improvements, a high level of interlaminar strength is required. To address these emerging needs for structural composites, several design concepts have been proposed including tougher fibers and matrices, improved fiber/ matrix interfaces and the use of fiber architecture. By complementing the other methods, it is felt that the use of fiber architecture, or innovative geometric arrangements of fibers, offers the most comprehensive solution to many of the critical problems of composite materials 1-1°. With the myriad of possible fiber architectural arrangements, an almost unlimited variety of composite designs can be achieved. This leads to new classes of composites, the textile structural composites, which are already being considered--and, to a much more limited degree, applied--in order to provide additional options in the design and optimization of the performance of composites. By borrowing from the garment and textile industries, several of their general design concepts and attendant terminologies are finding their way into the synthesis textile structural composites. Plain-weave composites consist of

two sets of interlaced fibers, known as the warp and weft fibers. Figure 1 illustrates the unit cell of a plain-weave composite. The warp and weft fibers define the basic structure of such materials. Often, fabrics are coated with a different material added to the warp and weft and hence such added materials play the role of the matrix in structural composite materials. If the warp and weft are embedded in a host (matrix), then the resulting combination simulates classes of textile materials. Regardless of the architecture, all fibrous structural composites are composed of fibers embedded in a matrix of more ductile material that bonds the fibers together and acts as a load-transfer medium. However, the mechanics of plain-weave composites is set apart from that of straightfiber composites by large local deformation effects. The elastic properties of structural weaves without a supporting matrix, such as fabrics, are even more non-linear than those of reinforced plastics. Micromechanical modeling of fabrics (warp and weft) has been addressed since the late 1970s by many textile engineers (see, for example, Kawabata et al.ll-13). In 13 they used a simplified structural model for plain weaves by assuming the warp and weft axes to be straight lines which bend at the intersection points, whereas in their initial work they ignored the bending effects. A more advanced model, which includes bending effects of the yarns at these intersection points, has been addressed by them. Based upon Kawabata's model, Stubbs developed a model for predicting the loading-path-dependent response of coated fabrics 14 where the coating simulated the supporting matrix of structural composites. It is based on

627

Model for plain-weave composites: A. H. Nayfeh and G. R. Kress

m i i I q ~ l m m m m u

|

"

~

m w

m

U w

m m

Figure 1 Unit cell of the plain-weave reinforced composite

a unit cell of a plain-weave coated fabric and approximated by a number of flexible straight rods, or springs, representing different elements of the mechanical interaction of the yams and the coating. The representation of the matrix as a single rod rather than as a continuum in the model further reduces the possibilities of properly modeling the elastic interactions between the matrix and the yarns. Furthermore, the spring stiffness values of Stubbs' model are fitted to experimentally measured data by using a numerical optimization procedure and the model is then used for predicting load-path effects on the stress-strain behavior. The present model also uses rods to simulate the warp and

628

weft reinforcement and uses a continuum-mechanics approach to account for the matrix. The rods are assumed to remain straight and to undergo uniform elongation under the load. Although bending effects of the reinforcement are less important when considering reinforced plastics, the outof-plane effects considered by Kawabata et al. will also be included in the present model. These effects include an average out-of-plane displacement associated with local through-the-thickness shear as well as a change of thickness. The unit cell is subjected to uniform in-plane uniaxial or biaxial loading. The resulting deformation is constrained such that different cross-sections normal to the direction of

Model for plain-weave composites: A. H. Nayfeh and G. R. Kress

,~Z |

1 ZtW

Z,W

~-

Figure 2

a

|

;I x , u I

Analytical model o f the unit cell

loading are differently but uniformly strained. This leads to non-linear geometric relationships between the strains along the rods, the in-plane strains of the unit cell, and the out-ofplane strain. The equilibrium equations are then consistently derived from the total strain-energy expression of the system. Compatibility between the matrix and the rods' non-linear deformation is reached by assuming a simplified displacement field throughout the unit cell.

displacements u and v are assumed in accordance with x u = -t~

(1)

a

YN

v = ~v

(e)

and a bilinear distribution of the out-of-plane displacement w is assumed as

w_-(1

x

y

GEOMETRY AND KINEMATICS The analytical model of the unit cell of a plain-weave composite is depicted in F i g u r e 2. Symmetry permits one to consider only one quarter of the cross-over region. The in-plane dimensions of the undeformed unit cell in the xand y-directions are a and b, respectively. The weft and warp fibers reinforcing in the x- and y-directions are modeled as extensible rods of initial lengths 11 and 12, respectively. The original crimp of the weave is modeled by distances h 1 and h2 of rods 1 and 2 from the x - y plane which also includes the cross-over point x = y = 0. In what follows the respective dimensions of the deformed unit cell will be denoted by corresponding capital letters. A biaxial state of stress, as = ay = a, will be applied to the unit cell. Throughout the unit cell, linear distributions of the in-plane

which implies that s z is a constant, fi, ~ and ~ represent the maximum values of u, v and w in the unit cell. The displacement w is a maximum at the cross-over point. Consequently, from eqns (1)-(3) we construct the remaining strain components as 1 sx = -~

(4)

a

In

% = ~v

3'yz=-

1-a

(5)

b

(6)

629

Model for plain-weave composites: A. H. Nayfeh and G. R. Kress but the following analysis uses the averages instead: ~yz --

2b

(8)

5/zx -

2a

1#

(9)

The chosen distributions of displacements and strains imply that the rods stretch uniformly under load without any bending. These assumptions allow one to take into account the geometrically non-linear effects of large displacements with relative ease. With the out-of-plane displacement w 1 of rod 1 at the cross-over point, the in-plane displacement fi and the rod extension dl, the following constraint must be satisfied for all loads: (hi-FwI)2 + ( a + fi)2 = (ll + d l ) 2

(10)

For the present problem the shear strains "Yyz and "gzxof the matrix are functions of x and y only. Eqn (18) also depends on these two coordinates. Accordingly, the average strainenergy density in the matrix is given by OOM OV

1 I

OU_

v:M

EM

2(/7;,M)

× [8x2+8y2+82+

vM e 2 . a 2+b2~2] 1--2VM kk-I- 3 - 3 - ~ J (19)

By adding the contributions due to the 'fiber' rods the total strain energy of the unit cell becomes EMVM

u_

[

VM

2(1+.M)

2 , a2 + b21~,21 8kk* 3a2b 2 1

The analogous equation holds for rod 2: (h2-w2)2+(b+f02=

(12 + d2) 2

(11)

The relationships between the out-of-plane displacements w~ and w2 of the individual rods, the common direct strain e z and the common crimp exchange # (which is the out-ofplane displacement of the cross-over point at z = 0) is given by the following two equations: W 1 = "W -'[- hl,P_,z

(12)

w2 = # - h28z

(13)

Combinations of eqns (12) and (13) yield either Wl - w2

..{-~VIE1821-{.-~V2E2

The average stresses in the unit cell are derived in the following way. First, the average strain-energy density of the unit cell follows from simply dividing the total strain energy by the volume of the unit cell V = VM+ Vt + I,'2

10U Ox -

(15)

Combination of the results, eqns (14) and (15), with those of eqns (10) and (11) yield e z = - i + ~//~I(1 + e I )2

_ _

a2 (I + ex)2+ ~/l 2 (1 + e2) z - b2 (1 + ey) 2 hi + h2

(22)

V Oex

or = (h2Wl "q- hlW2) hi + h 2

(21)

and, secondly, the various stress components are calculated by taking the derivatives of the total strain energy with respect to the respective conjugated strain components:

(14)

8z -- hi + h2

(20)

82

10U 17y -- V Oey 10U

az -- V 08 z 10U 7gyz -- V O'yYz

(23)

=0

(24)

b OU

(25)

V 017 - - 0

(16) 10U

and

a OU

rrzx -- V O3'z:,

~=

V O#

(26)

0

hI ~/~ (I + e I)2 _ aZ (1 + ex) 2 - h I ~/l 2 (I + ez) 2 - b2 (1 + er) 2 h I d- h 2

(17)

It must be noted that the out-of-plane stresses, eqns (24)(26), must be zero. If the values of the in-plane stresses are treated as specified input data, eqn (24) yields the restriction

CONSTITUTIVE MODEL We derive the constitutive law from the unit cell's total strain energy which we develop by integrating the strain-energy densities in the matrix and the rods over the volumes of the respective matrix and reinforcement regions. The strain-energy density in any isotropic material is given by O--V---- 1 -t-v 8ij 8ij+ l ~ v

630

8kkSij '

i,j=x,y,z

(18)

-]- v1E181 ~ - ~ 08 z

v2E282 ~

08 z

=0

(27)

Similarly, eqn (27) or eqn (28) yields a 2 -F b 2 _ 081 . 082 = 0 2VMGM 3--~-~W + VlEa81 ~w -t- v2E2e2 0gv

(28)

The stresses corresponding to specified values of the in-plane

Model for plain-weave composites: A. H. Nayfeh and G. R. Kress strains follow from eqns (22) and (23) which read in detail:

and updating the rod strains 131 and /32 via the kinematics constraints (16) and (17). The objective function is

ax = 2VMGM /3x + 1 -- 2VM(/3x + % + ez

O =f? +f2

+ viE1/31 ~ . qt_IJ2E2/320/320/3x

I

.M

% = 2UMGM /3Y+ 1 --21'M (/3x+/3y+/3z 0/31 0/32 + vlElet -~v%+ vzE2/32 O/3y

(29)

)] (30)

The following changes of variables are found useful in describing the geometry of the deformed unit cell: H1 = h i +Wl

(31)

H 2 = h2 - w2

(32)

A = a ( 1 +/3x)

(33)

B = b(1 + %)

(34)

L,=l,(l+/3,)

(35)

L2 = 12( 1 +/32)

(36)

With these, the various partial derivatives in eqns (27)-(30) can be written as

0/31 aA - -O/3x llL1

0/32 bB -O/3y 12L2 0/31

0/3. --

(hl + h2)HI

IlL 1

0/32=

(hi +h2)H2

0/3z

12L2

0/31

(hi + h2)H1

O~

h2llL l

0/32 O~

(hl+hz)H2 h 112L2

where fl and f2 denote the residuals of eqns (27) and (28), respectively. We note that the minimum value of O is zero. Because of the presence of local extremes we find it necessary to search for the absolute minimum by using a scanning method rather than the gradient method. In order to economically maintain a high scanning density, we limit the domain of /3z and # by using the following kinematics arguments. Let us assume that the unit cell is in a deformed state correctly corresponding to load data ex and %. If we further assume a positive strain increment A/3x with a zero matrix stiffness value (simulating garment behavior), we obtain a lower bound of w l as

W,l=W,-Hl+¢L2-A2(l+/3x)

(39)

(40)

(41)

(42)

SOLUTION METHOD The model is used to predict the normal in-plane stresses a~ and Oy for specified strains/3x and/3y in accordance with eqns (29) and (30). However, the unknown strains,/3z,/31 and/32, and the out-of-plane displacement, v~, must be found to satisfy the equilibrium equations [eqns (27) and (28)] as well as the kinematics constraints [eqns (16) and (17)]. The non-linearity introduced by these calls for an iterative method. We minimize the residual square error of the equilibrium equations (27) and (28) by adjusting e z and ~v

2

(44)

Similarly, the upper bound of w2 for a positive strain increment A/3y can be obtained as

W2u=W2-~-n2--¢t2--B2(1-~-/3y) 2

(45)

The highest absolute values for the upper bound of Wl and the lower bound of w2 given below are consistent with the assumption that no change in thickness occurs:

(37)

(38)

(43)

Wlu = W2u

(46)

w21 = wu

(47)

We recognize that the true displacement values for any given combination of reinforcement and matrix stiffness values will lie between these bounds. The scanning routine defines a mesh covering the two-dimensional space spanned by the bounds, calculates the residuum O at each point, and identifies the point with the lowest value of O. High accuracy is obtained by repeating the process in an appropriate neighborhood of this point.

RESULTS We consider a square unit cell with side lengths a = b = 10 mm and the crimp is defined from hi = h2 = 2.68 mm, which corresponds to a 15° crimp angle and a total thickness of t -----5.36 mm. The volume fractions of the reinforcements are Vl = v2 = 0.4. The Young's modulus of the rods is 135 GPa, which is a typical value for unidirectional T300carbon-fiber composites with 60% fiber volume content. The influence of the supporting matrix is simulated by plotting the stress-strain curves for various values of Young's modulus for the matrix chosen arbitrarily in

Table 1. We present the results of the model by plotting the response of the in-plane strain/3x, the out-of-plane strain/3z, and the effective in-plane Young's modulus for the biaxial (Ox = Cry= o) load case. All curves labeled '1' simulate the case of non-existing matrix. Matrix stiffness values typical for structural parts made from carbon-fiber-reinforced

631

Mode/for plain-weave composites: A. H. Nayfeh and G. R. Kress Table 1

to curves 1 in Figures 3-6, Figure 3 shows the limiting case of a bilinear strain-stress curve with instantaneous strain at very low stress. This effect is due to the total straightening of the rods, which is plotted in Figure 4. Only after the completion of straightening can the rods start to experience extension and thus try to resist further increase in in-plane strains. This is due to the fact that the yams do not resist bending and can only resist elongation when they are fully straight. In considering the tangent stiffness behavior, Figure 5, this is shown in a constant structural stiffness modulus. However, one has to consider that the secant modulus (Figure 6) is zero for very small loads (reflecting the instantaneous strain at very low loads) and then increases

Matrix modulus values

Curve no.

E (MPa)

1 2 3 4 5 6

0.0 10.0 100.0 1000.0 10 000.0 100 000.0

plastics (CFRP) range between the values attributed to curve nos 4 and 5. The symmetry of the model with respect to the x- and y-directions lead to a zero-crimp exchange in the biaxial load case. For zero matrix modulus, EM = 0, corresponding

0.05

0.04 1 0.03

£x 0.02

0.01 6

0.00

I

0

IO0

200

300

400

500

600

_

700

,J

800

(MPa) Figure 3

Variation o f in-plane strain ex as a function o f

o x = a), = a

0.0

6

-0.1

5

-0.2 -0.3 -0.4

Ez -o.5 -0.6 -0.7 -0.8

-0.9

0

100

200

300

400

500

o (MPa) Figure 4

632

Variation o f out-of-plane strain e z as a function o f Ox = O-y = a

600

700

800

Mode/for plain-weave composites: A. H. Nayfeh and G. R. Kress 70000

60000 50000 ILl v

e~

2

40000 0

o e..

I--

3

30000 20000 10000 0

0

100

200

300

400

500

600

700

BOO

o (MPa) Figure 5

Variation of the tangent modulus

Ey as a function

70000 F

of ~x = o, = a

6

60000 Q. ~E

50000

~"

40000

= lo o

30000

:E

20000

u

10000

4

0

0

1 O0

200

300

400

500

600

700

800

a (MPa) Figure 6

Variation of the secant modulus E~ as a function of o~ = Ov -----a

with increasing load. With the highest value of the matrix modulus, i,e. curve no. 6, we approach the other limiting stiffer case. Figure 3 shows an almost linear increase of the in-plane strain e~ with applied stress and the through-thethickness strain ~z, see Figure 4, remains very small indicating that the geometry of the unit cell does not considerably change. The model approaches linear behavior and, naturally, the tangent and the secant structural moduli coincide. For intermediate values of the matrix modulus, the structural tangent modulus, Figure 5, exhibits the most interesting behavior. For small but non-zero values of E~ (curves 1 and 2) the structural modulus is very small at low loads but increases quickly with increasing load, asymptotically approaching the stiffness value for zero matrix

modulus. This is also true for higher matrix moduli (curves 3 to 5) but, due to incomplete straightening of the rods, the limit value for the structural stiffness corresponding to zero matrix modulus is not reached within the considered load range. CONCLUSION (1) A non-linear constitutive model for plain-weave composites is developed. (2) The model accounts both for crimp exchange and thickness change of the composite. (3) The influence of matrix stiffness on the non-linear composite behavior is simulated. (4) The degree of non-linearity is most obvious for the tangent modulus.

633

Model for plain-weave composites: A. H. Nayfeh and G. R. Kress ACKNOWLEDGEMENT This work has been supported by AFOSR grant no. F4962095-1-0269.

REFERENCES 1. 2.

3.

NOMENCLATURE 4.

a,b dl, d2 El, E2 EM hi, h2 H1, Hz 11, 12 L],L2 U, V, W

U VI, V2 VM

Vi, V2 VM x, y, Z

length and width of the undeformed unit cell elongations of rod 1 and rod 2, respectively Young's moduli of rod 1 and rod 2, respectively Young's modulus of the matrix rod distances from midplane at cross-over points crimp of rod 1 and rod 2, respectively, after deformation lengths of rod 1 and rod 2, respectively lengths of rod 1 and rod 2, respectively, after deformation displacements in the x-, y- and z-direction, respectively total strain energy in the unit cell volume fractions of rod 1 and rod 2 in the unit cell, respectively matrix volume fraction in the unit cell absolute volumes of rod 1 and rod 2, respectively volume of the matrix reference coordinates of the unit cell

5.

6. 7.

8. 9. 10.

11. 12.

Greek symbols 13. "~ yz, "~ xz /3x, /3y, /3z ~'1, /32 O'x, O-y, O"z II M

634

out-of-plane shear strains direct strains in the x-, y- and z-direction, respectively extensional strains in rod 1 and rod 2, respectively out-of-plane shear stresses Poisson's ratio of the matrix material

14.

Ishikawa, T. and Chou, T.W.J. Mater. Sci. 1982, 17, 3211-3220. Ko, F., Pastore, C., Yang, J. and Chou, T.W. in 'Composites 86, Recent Advances in Japan and the United States'. Japan Society for Composite Materials, Tokyo, pp. 21-28. Isly, F. in 'Proceedings of 42nd Annual Conference', Reinforced Plastics/Composite Institute, Society of the Plastics Industry, Inc., New York, 1987. Ko, F. and Pastore, C. in 'Recent Advances in Composites in the United States and Japan', ASTM STP 864 (Eds. J.R. Vinson and M. Taya), American Society for Testing and Materials, Philadelphia, PA, 1985. Ko, F. in 'Proceedings of ICCM IV (Fourth International Conference on Composite Materials)', Japan Society for Composite Materials, Tokyo, 1982, pp. 1609-1616. Tsai, S. 'Composite Design', Think Composites, Dayton, OH, 1986. Riley, C. in 'Proceedings of 41 st Annual Conference', Reinforced Plastics/Composite Institute, Society of the Plastics Industry, Inc., New York, 1986. Ko, F. Am. Ceram. Soc. BulL 1989, 68(2), 401-414. Nayfeh, A.H. and Chimenti, D.E.J. Acoust. Soc. Am. 1995, 97(4), 2056-2062. Nayfeh, A.H. 'Wave Propagation in Layered Anisotropic Media with Application to Composites', Elsevier Publishing Co., Amsterdam, 1995. Kawabata, S. and Niwa, M., J. Text. Inst., 1979, 70, 417. Kawabata, S. and Niwa, M. in 'Proceedings of 13th Textile Research Symposium (Mt. Fuji, 1984)', The Textile Machinery Society of Japan, Osaka, 1984, p. 1. Kawabata, S. in 'Textile Structural Composites' (Eds. T.-W. Chou and F.K. Ko), Composite Materials Series, 1989. Elsevier, Amsterdam, p. 67. Stubbs, N. in 'Textile Structural Composites', (Eds. T.-W. Chou and F.K. Ko), Composite Materials Series, 1989. Elsevier, Amsterdam, p. 331