- Email: [email protected]
Lower and upper bounds to the macroscopic strength domain of a fiber-reinforced composite material
International Journal of Plasticity, Vol. 8, pp. 741-762, 1992
0749-6419/92 $5.00 + .00 Copyright © 1992PergamonPress Ltd.
Printed in the U.S.A.
LOWER AND UPPER BOUNDS TO THE MACROSCOPIC STRENGTH DOMAIN OF A FIBER-REINFORCED COMPOSITE MATERIAL
ALBERTO TALIERCIO Politecnico di Milano (Communicated by Cesare Davini, Universit~ degli Studi di Udine)
Abstract-The macroscopic strength domain of a composite material reinforced by long, parallel fibers is, in general, unknown but for its theoretical definition. In this note it is shown how a homogenization technique applied to yield design theory allows the derivation o f two domains (in the space o f macroscopic stresses) that are a lower and an upper b o u n d to the composite strength domain. The dependence of these domains on the fiber content and on the shape of the fiber array is pointed out. Analytical equations for the approximate uniaxial macroscopic strength o f composites with Drucker-Prager or Von Mises type matrix are derived. For more complex stress conditions, the relevant strength domains are numerically evaluated as well. The discrepancy between the two b o u n d s is in m a n y cases relatively small. In particular, the two bounds yield the same value for the uniaxial strength of the composite along the fiber direction, which, by consequence, is exactly determined.
I. INTRODUCTION
The use of fiber-reinforced composite materials, either with polymeric, metallic or ceramic matrix, is constantly expanding in a number of fields, such as aerospace, motorcraft, biomechanic, and civil engineering, especially in applications where high stiffness-to-weight and strength-to-weight ratios are required. As a consequence of their increasing use, a very large amount of theoretical and experimental work has been devoted, mainly in the last two decades, to the study of the mechanical behaviour of both composite materials and structures made of composite. In spite of the number of papers and books published on this subject, the problem of defining strength criteria, supported not simply by a good comparison with experimental results, but also by a clear mechanical interpretation, does not seem to be fully solved yet. In fact, the most widely used strength criteria for these materials are "phenomenological" criteria, such as those proposed by Azzi and TSAI [1965], HOFFMANN [1967], TSAI and Wv [1971], and they all are subsequent extensions of a criterion formulated by H~L [1950] for anisotropic materials. The strength properties of composites subjected to different kinds of stress are usually well matched by all of these criteria (see, e.g., the comparisons shown by Wtr [1974]). On the other hand, these criteria are defined by a rather large number of parameters (Tsai and Wu criterion, for instance, requires six parameters in plane stress conditions), some of which are not easily obtained by the experiments (biaxial or pure shear tests may be required). This difficulty in deriving the strength parameters is a direct consequence of the fact that phenomenologiA n abridged version of this note, in Italian, was presented at the X National Conference of the Italian Association o f Theoretical and Applied Mechanics, October 2-5, 1990, Pisa, Italy. 741
742
A. TALIERCIO
cal criteria merely aim at interpolating sets of experimental data, without considering the actual heterogeneous nature of the composite. The drawbacks related to a phenomenological approach can be avoided by directly taking into account the mechanical behaviour of fiber, matrix, and, possibly, the fibermatrix interface. This was done, for instance, by BOEHLERand RACLIN [1984], who proposed a criterion that allows for a distinction between failure modes, depending on the loading condition, and that involves parameters related to the strength properties of the components. Although their criterion allows an excellent reproduction of uni- and triaxial experimental tests, its use is limited by the number of required parameters, which amounts to 18. Similar considerations apply to the criterion proposed by HASmN [1980], which has been tested by the author under uniaxial conditions only. Another interesting criterion based on a micromechanical approach is that used by McLAUGHLIN and BATTERMAN[1970] and McLAUGHLIN [1972], who used limit analysis theory to predict the ultimate load domain of the Representative Volume Element (RVE) featuring composite members; this domain is used as strength domain for the composite. These authors do not operate within the framework of homogenization theory, which would provide a mathematical justification to their criterion, but their theoretical results are well validated by comparisons with experimental tests on unidirectional and angle-ply metal composites subjected to uniaxial tension. In order to derive criteria founded on a sound mathematical basis and defined by a reasonable number of parameters clearly related to the strength of fibers and matrix, an approach based on the application of limit analysis to homogenization theory can be used. Following McLaughlin's approach, RVEs are treated as "structures," and their ultimate loads, which are identified with the "macroscopic stress" in the composite structure, are determined. By applying the static and kinematic theorems of limit analysis, lower and upper bounds, respectively, to the carrying capacity of RVEs can be obtained. DE BUHAN and TALIERCIO[1988,1991] applied this approach to composites reinforced by one or more arrays of parallel fibers, embedded in a matrix conforming with Von Mises strength criterion. By defining suitable stress fields over the RVE, these authors obtained lower bounds to the macroscopic strength properties of the composite, which, in many cases, were in excellent agreement with available experimental results. However, these lower bounds had to be proved not to be overconservative when compared with upper bounds. Also, the use of a Von Mises matrix does not permit allowance for the effects of isotropic pressure on the composite strength, which, as shown by BOEHLER and RACLIN [1984], can be far from being negligible. This work is mainly devoted to the formulation of upper bounds to the macroscopic strength properties of fiber composites, by means of a kinematic limit analysis approach requiring the definition of suitable failure mechanisms for the RVE. This is done by extending the results of DE BUHAN et al. [1991], which actually were of interest only for composites in plane strain conditions. After having briefly reviewed the fundamental aspects of the theoretical approach used and recalled the definition of the lower bound (Section II), in Section I I I a simplified version of this bound particularly useful in applications is presented, and the assumptions under which the approximated lower bound is valid are discussed. Section IV is devoted to the definition of upper bounds for composites with any type of components. These theoretical results are specialized in Section V to composites with Drucker-Prager type matrix. The use of Drucker-Prager criterion permits us to obtain macroscopic strength criteria sensitive to the effects of hydrostatic pressure and, consequently, more
Macroscopic strength domain of a fiber-reinforced composite material
743
realistic than those derived with the Von Mises model. The lower bound of de Buhan and Taliercio is reformulated accounting for Drucker-Prager criterion, and upper bounds are computed as well. Both bounds are defined by only four strength parameters, for any stress state, which are related to the uniaxial tensile and compressive strengths of the components and can be rather easily obtained by means of experiments. In spite o f the simplicity o f the failure mechanisms used in the kinematic approach, lower and upper bounds tend to be rather close for composites with different fiber arrangement and volume fraction, subjected to uni- and biaxial stress.
11. PROBLEM FORMULATION
Consider a composite material consisting of an array of parallel, cylindrical fibers with circular cross-section, embedded in a matrix of bonding material. Suppose that the fibers are "long," in the sense that their length is comparable to the size of the structural member considered. If the fibers form a regular array, the composite can be regarded as heterogeneous material with periodic internal structure. Periodic materials are featured by a "representative volume element" ( R V E - s e e , e.g. I-IAsHIN [1983]). The volume of the RVE will be indicated by (~; by a suitable scale change, its measure can always be made equal to 1, so that in subsequent analysis It~l = 1 will be assumed for simplicity. Furthermore, (~f and (~m will denote the parts of the RVE relevant to the fiber and the matrix material, respectively. Let the volume fraction of fibers be ~ = [t~f[/[ (~1(= [(~f[), so that the volume fraction of matrix is 1 - ~/= [(~,, [. The points _xof the RVE will be referred to an orthogonal reference frame Oxyz, with the x-axis collinear with the fibers. Let e_x,ey, e~ be the base vectors. The length of the RVE along the x-axis is apparently arbitrary. As a consequence, subsequent analysis will be developed in any cross-section of the composite perpendicular to the x-axis (thus staying in the (y, z) plane) and, for the sake of brevity, "RVE" will actually denote the intersection of the representative element with this plane. Regular arrays o f fibers can be fundamentally o f two kinds: . Rectangular arrays, see Figure la, such that, in the (y, z) plane, the centers of the fibers coincide with the centers of rectangles with sides parallel to the axes y and z. In this case, dy and dz will denote the length o f the sides o f the RVE. . Hexagonal arrays, see Figure lb, such that the centers of the fibers, in any section perpendicular to the x-axis, are located at the centers of more or less regular hexagons. In this case, the reference frame Oxyz is chosen in such a way that z is the direction defined by the centers o f a row of fibers in any cross-section and y is perpendicular to z in this section, dy and dz will denote the distance between adjacent fibers along these axes. Reinforcing arrays will be featured by an angle/3, defined as 13 = arctg dJdz. The reference frame will be chosen in such a way that dy _> d z (so that 45 ° _
744
A. TALIERCIO
1
dy
4-
a)
Z
Fig. 1. Unidirectional fiber reinforced composite with (a) rectangular and (b) hexagonal reinforcing array. The relevant RVEs are also shown.
As pointed out in Section I, several authors have studied the problem of defining macroscopic strength domains for periodic fiber-reinforced materials, that is domains suitable for the description o f the overall strength properties of fiber composites subjected to any stress. To this end, it is customary to define suitable measures for the average stress at any point of the composite structure; these are called macroscopic stresses and are the volume average of the microscopic stresses over the RVE (see, e.g. HASmN 8, ROSEr~ [1964]). Thus, denoting by =a(x) the microscopic stress field in the RVE and by ( . ) volume average over RVE, the macroscopic stress =27is
____2:= (o) = fe e(x) d6t. Now, suppose that the strength domains of fibers and matrix are given convex domains in the stress space, denoted by Gf and Gin, respectively. SUQUET [1983a,b] proved that a convenient macroscopic strength domain can be defined as
[G h°m =
~[~
= (__o); div g = 0 ¥x E if; [[~] .ns = 0 v x E S;
o periodic over 6t; __a._nanti-periodic over 06~;
O)
g e GmVx_e ~m ; ~ 6 Gsvx e as I, and that this domain also is convex. Here S is any possible discontinuity surface for the microscopic stress field ~ in (t and _ns is the normal unit vector at any point of S. Remark. In the definition of G h°m, eqn (1), only the strength domains of matrix and fibers are involved. I f no further assumption is made about the constitutive law of the component materials, G h°" is called "domain of the potentially admissible macroscopic stresses" (SALEN?ON [1983]). This means that, if _27~ G h°'n, S__cannot be associated with any microscopic stress field being statically admissible and, at the same time, compatible with the strength properties of fibers and matrix. On the other hand, if the components are elastic-perfectly plastic, with associated flow rule, and if Gin, Gf are their
Macroscopic strength domain of a fiber-reinforcedcomposite material
745
elastic domains, Gh°m actually contains macroscopic stresses compatible with the strength properties o f the composite as a whole. For further details, see SAtENqON [1983] and SUQtrET [1983a]. Convexity allows the definition of G h°m in a form, dual to that of eqn (1), which will be used later. To this end, it is worth recalling the definition of support function 7r(d) o f a convex domain G C R N (see, e.g. TYRRELL ROCKArELLAR [1970]):
~r(_x) -- sup[_x._yl_y ~ G} where _x._ydenotes inner product o f two vectors _x,_yin R N. Convex domains coincide with the intersection in R N of the subspaces defined by means of their own support functions: O = [_yl_y._x__ < ~(_x)v_x E RN]. Refer now to the macroscopic stress domain G h°m, eqn (1), and denote by 7rh°m its support function. By identifying the macroscopic stress space with R 6, G h°m can be alternatively defined as O h°m =
I I :P
h°m( o)V O ~-" R 6 ] •
(2)
The definition of 7r h ° m w a s given by SUQUET [1983a], to which readers are referred for further details. Let v be a piecewise differentiable velocity field over the RVE, ~ d the relevant microscopic strain rate field, S any possible discontinuity surface for p, _ns the normal to this surface and Ilp]l the jump in p across this surface. If v can be expressed as
p=D._x+_u, where D is any symmetric second order tensor and u has the same kind o f periodicity as the considered medium, 2 it is possible to show that the support function of G h°m is
7rh°m(D) = inf I ( ~ r ( d ) ) I D = (_d), _D E R6}.
(3)
Because of the assumed periodicity, in eqn (3) ( d ) can be computed as
( d) = faa v_ ~) n_dS, where ~ denotes "symmetric part of dyadic product" (i.e. v ~ n = ½(_v ® n + _n ® p)) and n is the outward normal to the boundary o f 6L Finally
= ] - ~
;
]
r(d=)dV + ,is 7r(n_s;[v])dS ,
(Y)
1The definition given by Suquet actually involves velocity fields more general than those used here. 21.e. U is Q-periodic.
746
A. TALIERCIO
a-(_d) being the support function of the convex strength domain of the material at each point n of the RVE. In eqn (3') the term computed on S is given by 7r(ns;l[v]) = supl_g.es. II_v]l,_oE G(_x)}. o_
At this stage, it is worth emphasizing that generally G h°m is known only through its definition, eqn (1) or eqn (2), but not explicitly. By defining suitable statically admissible microscopic stress fields over the RVE, DE BUICAN and TALIERCIO [1988,1991] proved that the domain Gs = [S= = o=,,,+ nofe_x ® e_,:; a,, E Gm; ~rn + ofe_x @ e_r E G f l
(4)
is contained in Gh°m; this domain is entirely known if G,,,, G f (and r/) are. The support function of G~, which will be used later, reads (see DE BUHAN & TALmRCIO [1991]) 7rs(__D) =
inf [(1 - ~ ) r m ( d m )
dm,df
+ ~Trf(__df)},
(5)
where 7r,~, Try are the support functions of Gm, Gf respectively, and tensors din, _df fulfill the following conditions: D = (1 - ,/)__d., + ~/__df;
D xx = d xx = d f x.
(6)
In particular, two tensors d~,,=d7 fulfilling eqns (6) exist, at which the infimum in eqn (5) is reached: 71-s(D ) :
( | - ~/)71m(__d~) + ~71"f(__d;).
(5')
In the next two sections, two domains constituting a lower and an upper bound to G h°m, denoted Go and G1, respectively, will be defined. Go will constitute a lower
bound stricter than Gs, defined by eqn (4), so that the set of inequalities Go C Gs C G h°m C G1
will be obtained. This will allow the bilateral bounding of the unknown domain G h°m and the use as macroscopic strength domain of either one of the domains defined, provided that the maximum gap between the two bounds is sufficiently small. Finally, in Section V the theoretical results obtained will be applied to composites with DruckerPrager matrix and the difference between the strength properties predicted by the two bounds will be quantified for some stress conditions. i!1. A LOWER BOUND TO G h°m Consider now a particular composite material with fibers reducing to filaments, that is to unidimensional reinforcing elements (7 -~ 0). Let $ + (resp. b - ) be the tensile (resp. compressive) strength of the filaments. As shown by DE BUHAN and TALmRCIO [1991], for this special case the definition of the lower bound Gs, eqn (4), here denoted Go, reads: GO :
[ ~ = ff=m -]- (Ye_x (~ C_x; ~=m ~- Gin, - 6 -
(7)
Macroscopic strength domain of a fiber-reinforced composite material
747
and its support function is 71"0(O ) -~- 7~m(D ) -~- m a x l # + D X X , - # - D ~ X l .
(8)
It is rather evident that the case considered now is mostly o f academic interest, since the assumption made about the fibers is unlikely to be met in real composites. The importance o f the domain now defined is that, under conditions to be specified later, Go can be proved to be contained in the macroscopic strength domain of a composite with fibers of any strength and volume fraction, provided that the parameters ~+ and # - be suitably defined. This feature makes Go more interesting to be used in comparison with G~; in fact, in the definition of Gs, eqn (4), the entire strength domain of the fiber is involved, whereas only the uniaxial strength values o f the fibers are required to define Go. Let us now prove what is stated above, namely that Go is contained in G h°m for any 7- To this end, for a composite with any 7/define the parameters #-, #+ as follows: ~-
n(a7
am);
~+
'7(al+ -
+
where am, + am- are the uniaxial strength values of the matrix along the fiber direction. By setting r + Uf+/am,+ r - = of~am (>- 1), assume that the strength domains of the components fulfill the following conditions: =
7 r i ( d ) _> r % r m ( _ d ) v _ d l d ~ _ 0,
(9)
7rf(d=) >_ r-Trm(d_)vd_ ldXX <_ O. In particular, eqns (9) are fulfilled if the strength domains o f the components are homothetic, for instance if fiber and matrix are both formed by Von Mises or Tresca materials. Under these assumptions, it is easy to show that Go C Gs(C Gh°m). The p r o o f is based on the inequality 7r0(D) --- ~rs(D) VD E R 6, which can be obtained by decomposing D into the two particular tensors __d~and __d] at which the infimum in eqn (5) is attained. By making use of Jenkin's inequality for convex functions, i.e. 91"m(D = (1 -- T/)d* + 7/__d7)) _< (1 - 1'/)71"re(d*) -~- l~71"m(_d;)
and by virtue o f the definition itself o f support function, which yields r m ( D ) --- maxl amD + ~", - a m-D xx }, from eqn (8) one gets: r 0 ( D ) < (1 - ~)a'm(d,~) + ~/Tr,,(__d~) + m a x l O + D ~ , - 6 - D ~1 ~/(r + - 1)Trm(=j), ifd~XX>0; d* _ 1)Trm(__tiT), if d]=___ 0,
~-~ ( l -- ~)71"m(=d~) "~" 7]Tl'm(=d;) + l r / ( r _
where eqns (6) have been accounted for. Finally, substitution o f the hypothesis concerning the strength domains o f fibers and matrix, expressed by eqns (9), into the latter inequality, yields:
748
A. TALIERCIO
• "o(O) -< (1
-
yl)"/rm(dr%)
Jr
~f(d;)
= "xs(D)VD E R 6
(see eqn (5')), q.e.d. Remarks: i. The inclusion Go C Gs has already been proved by DE BUHAN and TALIERCIO [1991], though under conditions more restrictive than eqns (9). ii. The definition of Go coincides with that given by MCLAUGHLIN [1972] for the strength domain of a fiber-reinforced composite. However, this author directly formulated his strength criterion with reference to composites with infinitely thin fibers and did not operate within the framework of homogenization theory, which actually shows that this is only a lower bound to the actual macroscopic strength domain of a composite with any fiber volume fraction. iii. The definition of Go can be rather easily extended to composites reinforced by N arrays of parallel fibers, each one having a certain orientation defined by the unit vector _e,t")(n -- 1 . . . N ) . Let G) m be the strength domain of the material forming the fibers of the n-th array, tr~ ~") the uniaxial tensile/compressive strength of the matrix along _elm (o ±(") = a ± for any n if the matrix is isotropic), a f (") the uniaxial tensile/compressive strength of the fibers of the n-th array and y, their volume fraction. Let r ±(") = o~(")/~,(")(>
1);
a +-(") = T / . ( o ~ (") -
a~.(")),
n = 1...N.
If the strength domains of the components fulfill the following conditions: r~
( d ) ->- r +<")
e_n
"u='gx
> O,
(9') 7rs(") ( = d ) >_ r - ( " ) T r m ( _ d ) v _ d
I ~- (. " ) .__u._~ . _(-) -0,
the domain Go
: a
=gm +
, ~m E Gin;
n=l
<--
-< a +(")
I
is a lower bound to the macroscopic strength domain of the composite. The proof is a trivial extension o f that given for unidirectional composites.
IV. AN U P P E R BOUND TO G h°m
The dual definition given for 7rh ° ' , eqn (3), shows that, if _/: is interpreted as microscopic velocity field, upper bounds to G h°m can be obtained by defining failure mechanisms for the RVE featured by velocity fields obtained by combination of a term linearly dependent on the position of any point x and another term (u) possessing the same kind of periodicity as the medium considered. Regardless of whether the reinforcing array is rectangular or hexagonal, the simplest
Macroscopic strength domain of a fiber-reinforced composite material
749
failure mechanism of this kind is featured by constant strain rate throughout the RVE (__d(_x) = =D v_x E (~). As a consequence, one immediately gets (see eqn (3))
7rh°m(D)
~--- ( T r ( d ) ) ~--
(1 - '0')'/rm(D ) "l- ' q T r f ( D ) .
Further, provided that the strength domains of matrix and fibers fulfill eqns (9), by setting p+ = 7/+ (1 - ~)/r +, p - = ~ + (1 - ~ ) / r - (p+- <_ 1), the inequality
•"lrh°m(D) <_ <71"(d)) _< p+'lrf(D) i f D ~ >
O; (10)
_< p - x f ( D ) if D x~ _< 0 holds. In order to obtain a relationship being equivalent to eqn (10), but involving domains and not scalar functions, define two domains G~, G f as follows:
o / = [__rl_r:_D__ o; E=:_D_<_p+xy(_D_ - DXXe_~® e_.)v_O_lDX" <_ 01,
I~I_~:_D___p-~-,(__D)vDID~_<0; ~=:_D_ <_ p-Tr1( ~ - O=e_x ® e_x)VDlD =
>_
01.
(lla)
(1 lb)
The geometrical meaning of these domains in the macroscopic stress space is as follows. Gy+ (resp. G f ) is obtained by first homothetically contracting GI with homotheticity ratio p+ (resp. p - ) . The part o f boundary o f the contracted domain where the outward normal D has positive (resp. negative) component along 2?xx coincides with a part of the boundary of Gf+ (resp. G f ) . The remaining part of boundary belongs to the cylinder with generatrices parallel to S = and tangent to the contracted domain at the points where the outward normal D is perpendicular to S xx (see Fig. 2).
l ~ik(i,k~x,x) t
G~ \
~
/
(1:)' I:) --"~"=o)
V"- ~'~ O+O'Im+
..~:.:.:.G. !::::::"~:~'.:.-:i::
,G;
_
"'
Fig. 2. Graphical construction of the domains Gf+ and Gf (see eqns (1 la,b)) in the macroscopic stress space. The dotted area is the intersection of these two domains (G7 n (:;7).
750
A. TALIERCIO
Thus, the scalar inequality eqn (10) is equivalent to the inclusion G h°m C (G7 N
G£). In order to improve the upper bound just defined, periodic failure mechanisms featured by rigid relative movements between blocks in the RVE will be considered. Only mechanisms with planes as slip surfaces will be considered; the planes will be supposed not to cross any fiber, their intersection with RVE will be denoted by S and their normal by ns. Recalling eqn (3') and denoting by V the jump in the velocity field across the plane of failure, for the considered mechanism the term (Tr(d)) is given by <~(d)> = [Sl~m(~s;_V). For further developments, it is convenient to associate any mechanism, with failure plane perpendicular to _ns, with a domain in the macroscopic stress space defined as follows:
c*(_~s) = I__sl__s:__D-< ~ m ( P ) l, where
z,~(D) = ~,~(es; V) = +oo
if D
=
_V (~ _ns;
otherwise.
IV. 1. Rectangular array If the fiber array is rectangular, a mechanism of the kind described above, with failure plane passing only through the matrix and perpendicular to ey (or _ez), can be defined for any percentage of reinforcement-see Figures 3a,b. Let Vy (resp. Vz) be the relative velocity between the two rigid blocks in movement and letsSy (resp. Sz) be the surface in the failure plane. Since D = e_y ~ Vy ISy I (resp. D = e z ® Vz ISz ]), by virtue of the definition of 7rh°m, eqn (3), the two following inequalities hold for any ~/ ( _< ~r) :
~hom(p = e_z Q ~ ) <- ~m(e_z; ~ ) .
n~
"~ Vy
¢)
Fig. 3. Periodic failure mechanisms for rectangular RVEs, featured by relative rigid movement of blocks.
Macroscopic strength domain of a fiber-reinforcedcomposite material
751
These inequalities amount at the inclusion between domains in the macroscopic stress space
G h°m C Gm(e_y) n G*(ez). Another failure mechanism featured by the required periodicity consists o f two parallel slip planes crossing each RVE, located at opposite sides with respect to the fiber and with traces in the (y, z) plane intersecting the edges of the RVE at their midpoints (see Fig. 3c). The unit vector perpendicular to these planes has the form n~ = +_e_yc~+_ ezS~. The triangular wedge of RVE with n, as inward normal is supposed to have relative velocity V~ with respect to the central block in which the fiber is located, so that by periodicity the triangular wedge with n~ as outward normal is required to have relative velocity - ~ with respect to the central block. Furthermore, in order to have failure planes passing only through the matrix, the fiber volume fraction ~ must not be greater than 7r/4s~c~ (i.e., ~ _< a-/8 = 0.39 for square RVEs). If this restraint is fulfilled,
7fh°m(D : V~ ~ n_13)~ 7fm(_V~;n~) that is, in terms of domains
G h°m C G*(n_/~). As a conclusion, an upper bound to G h°m for composites with rectangular array of fibers is given by the domain GI = Gf+ n G f n G*, where
G* = G* (ey) n G* (ez) n G*(_na) = G*(ey) n G*(ez)
if ~ < lr/4s~c,
otherwise.
IV.2. Hexagonal array Suitable periodic mechanisms featured by failure planes perpendicular to ey passing only through the matrix are possible provided that the trace o f any failure plane in the (y, z) plane intersects the skew edges o f the RVE at their midpoints (see Fig. 4a) and
b)
c)
Fig. 4. Periodic failure mechanisms for hexagonal RVEs, featured by relative rigid movement of blocks.
752
A. TALIERCIO
provided that any RVE be crossed by two parallel planes located at opposite sides with respect to the fiber. The failure planes pass through the only matrix if 7/_< 7r/Ssa/c~ (i.e., if ~/_< 7r%/-3/8 = 0.68 for regular hexagonal arrays). Denoting by Vy the relative velocity between the triangular wedge with ey as inward normal and the central block containing the fiber, by periodicity the velocity of the triangular wedge with _ey as outward normal relative to the central block must be -Vy. Similar considerations apply for mechanisms with failure planes perpendicular to _ez (Fig. 4b). These planes do not cross any fiber provided that ~/_< ~r/8c~/sa (i.e., ~ _< • -x/3/24 = 0.23 for regular hexagonal arrays). Furthermore, periodic are also mechanisms featured by two parallel failure planes per RVE, located at opposite sides with respect to the fiber and with traces in the (y, z) plane intersecting a skew edge o f the RVE and the adjacent edge parallel to the y-axis at the relevant midpoints (Fig. 4c). The equation of the unit vector perpendicular to these planes is n~ = _+eyC~_+ ezSa. In order to have failure planes passing through the matrix only, the fiber volume fraction 71 must not exceed ~l/2sac~ (that is, 0.68 for regular hexagonal fiber arrays). As pointed out in Section IV. l, denoting by V~ the relative velocity between the triangular wedge with na as inward normal and the central block containing the fiber, the velocity of the triangular wedge with n~ as outward normal relative to the central block must be -V~. As a conclusion, after having defined domains GTn(ey), G*(ez) and G * ( n a ) analogous to those defined for rectangular arrays, it is possible to state that G h°m C GI, where GI can be expressed as Gl = Gf+ n Gf- n G* and G*m = G*(ey) n G*(_ez) O G*(n~) if r / < 7r/8c~/s~ Gm(ey) O Gm(ne) if r/8c~/s~ < ~ < mini 7r/8s~/c~; 7r/2sec~l
G* (n~) if rc/8c~/s~ < rl < 7r/8s~/c~ and 3 -< 60 ° Gm*(ey) if ~r/2s~c~ < ~ < minlTr/8syc~;~e] and 3 -> 60 ° R 6 otherwise. V. APPLICATIONS The theoretical results presented up to here will now be applied to the evaluation of lower and upper approximations of the maximum solicitation that fiber reinforced composites subjected to particular states of stress can sustain. The material forming the matrix will be supposed to undergo Drucker-Prager strength criterion. This two-parameter criterion is an extension of Von Mises criterion; here it was chosen for the reasons discussed in Section I and because of the existence of experimental results showing that the ultimate strength of some polymeric matrices is well described by a Drucker-Prager criterion under some stress conditions (see, e.g., the results reported by HULL [1981]). As is well known, denoting by km the shear strength of the matrix and by ~m a parameter related to the "internal angle of friction" of the matrix, Drucker-Prager criterion reads (see, e.g. SALENqON [1983]):
O:mtr Om +
~/l (O=m:~=m -- ltr2 __am)--< km
and the relevant support function is
Macroscopic strength domain of a fiber-reinforcedcomposite material
Irm(D) =
=
km t r D
3arm
=
= +oo
if t r D >__OtmX/6(3_D:_D - tr2_D) = _ _ _
753
(12)
otherwise.
The parameters am, k,, defining Drucker-Prager criterion are related to the uniaxial strength values of the matrix by the relationships 1
-
a,.
-
a m -- am+
+, x/3 am + a..
km =
2
a m a i n+
+. x/3 am + am
Only the case am -> am+ (i.e., 0 _< ct,~ < l/x/-3) will be treated, for its greater interest in practical applications. When failure mechanisms with slip planes are considered, setting V @ _ns = D the condition tr D _ ct,,x/6(3D : D - tr 2 D) can be easily shown to be equivalent to 30/rn
_V'ns ~ x/1 - 3ot2m I N . In other words, finite values of a'm (and, consequently, significant upper bounds to ~rh°m) are obtained if the relative velocity _V between rigid blocks falls within the cone with opening angle 3'* = arccos3 otto/x/1 - 3at 2, having as axis the normal to the slip plane (see Fig. 5). Note that such cone degenerates if ctm > 0.5/4-3; thus, for Drucker-Prager matrices with excessively high compressive-to-tensile strength ratios (i.e. 0.5/x/3 < am --< l/x/3), no bound is obtained by this kind of mechanism. Drucker-Prager criterion reduces to Van Mises criterion if the matrix has equal strength in uniaxial compression and tension (am+ = am a m ~-- kmx[3, i.e., at,, = 0). The support function of Van Mises strength criterion can be deduced from eqn (12) noting that ctm ~ 0 implies tr D ---,0 if finite values for qrm (and consequently significant upper bounds to the macroscopic strength) are to be obtained. Thus (see also SAtENqON [1983]): =
r,,(D) = a , ~ ~ =+~
iftrD=0
iftrD~:0.
In the case o f Van Mises criterion, when failure mechanisms with slip planes are considered, setting D = _V ® _ns the condition tr __D= 0 is equivalent to _ns .1_ _V. In other words, for composites with Van Mises type matrix, the relative velocity between rigid blocks in movement has to be tangent to the failure plane. Also the material forming the reinforcing fibers will be assumed to undergo a DruckerPrager type criterion, defined by parameters Cry and ky. For the sake o f illustration, suppose that the ratios between the uniaxial strengths of fibers and matrix are such that + + af lain = a f / a m = r( >_ 1), so that eqns (9) under which the bilateral bounding of G h°m was obtained are fulfilled. Thus, o~y and Icy can be expressed in terms of am, km as otf=Ctm;
k f = r k m ( r > _ 1).
For composites with Drucker-Prager matrix, the following form for the domain Go, defined in general by eqn (7), is obtained:
754
A. TALIERCIO
Fig. 5. For failure mechanisms with slip planes passing through a Drucker-Prager matrix, the relative velocity V between rigid blocks in movement must form an angle 3' with the normal _ns to the slip plane not greater 2 of the outlined cone. {han the opening angle 3'* = arccos 3Otm/~/1 - 3orm
Gb = {__SIc~m(tr__S- o) + ~/~(_S:_S- ~trZ__S) + a ( ~ t r _ _ Z - S ~ x ) -
k m < 0, -
6-
+ ~a2 (13)
< a < 6+}
whereas the e q u a t i o n s o f the d o m a i n s defined in Section IV are
Gm(ey)=
S=
Odin~,yy -- kin~3 ~_4o~2
\
1
2
"1"-(Sxy)2"}- (SYZ)2 ~ 0 "
~ -- 40t m
(14a) Olin S z z
Gm(ez) =
=27
~
--
kin~3
4Otm z
+ %/12012(OlmfZZ~km/3~ 2 ,~
\
~ _ 4otZm
(,~,yz)2
] +
(~zx)2 +
I <0
;
(14b)
~mS"" -~ _ n ~ ) = __2: } - 4 ~
km/3 +
f l 2 o t 2 { ° t m E _I n n - k m2/ 3 )
\
2 -I- (S-,)2-l- ( ~ , n x ) 2 ~ 0 ;
~ --4Otm
S "n = SYYc~ +_ 2SYZcas~ + SZZs~; S "x = +_L'XYct3 +_ SZXs~; S"'
=
(14c)
+_(,rez _ ,re~)cese + syZ(c~ - s~) 1 ;
G f f q G 7 = [__SIO~mtr__S+ x/~ (__S:__S-- ~tr2__S) _< [1 + o ( r -
l)]km}.
(14d)
Macroscopic strength domain of a fiber-reinforced composite material
755
V. 1. Bilateral bounding o f the macroscopic uniaxial strength Suppose that the composite is subjected to uniaxial tension (resp. uniaxial compression) acting in the ( x , y ) plane. Let 0(0 _ 0 < 90 ° ) be the orientation o f the uniaxial stress, S, to the fibers. Set, for brevity, So = sin 0 and Co = cos 0. T h e aim now is computing the functions S~(O) and 27~-(0) (resp. S~(O) and S;-(O)) constituting a lower and an u p p e r b o u n d , respectively, to the anisotropic macroscopic strength 27+(0) (resp. Z'-(0)) o f the composite. The lower b o u n d to the absolute values o f the tensile/compressive uniaxial strength is given by eqn (13) and reads: 27o-+(0) = s u p [27 10:m(27 -- (7) -['- ~jl (27 __ 0.)2 ..[_ 27(7S2 __ km ~ O; - 6 -
----- (7 ~--_ 6 + ]
0
that is
---6+ + 1--3---2 ~J ( ~ O t m 6 + S o 2 + k m ) 2 - ( 6 + ) 2 ( 1 - ' ~ 3 s 2 ) (1 - 3°t2)s2 1-6+$2 + o t m k m
=1=3 2
270±(0) =
-T-3~Otm
~/(1 - ~s~)(1 - 3~m2) ]
3 . 2 --74S0 3 2 "["
1 -
1 - 3c~2
(1
3"-'~-~2m ~ "~ ~So2-'~-_ )So
OtmO-So2 + km +
] km
if o>- 6 +
if - 5 _< o _< 5 +
- ( 6 - ) 2 1 - ~ s 2 (1 - 30~2)s 2
1 --3%m l
-T- 3
----
--~7
2
SO + o t m k m
1 -
3or 2
if o ~
-6-
where 17 - -
-1"3 0lm
km
1 -3a
2-
--
+
=
3 2 zso
1-3c~-
~sd3 2
x/1 - 3c~so 1 - 3~Zm - zSd32"
743s02
As for the upper b o u n d , note first o f all that G~,(e_z) does not impose any b o u n d to the uniaxial strength in the ( x , y ) plane, since 27= = 27yz = 27zx = 0. The definitions o f the other domains, eqns (14), yield 1. for rectangular RVEs with ~/ < 7r/4saca and for hexagonal RVEs with 7/ < mini 7r/8sa/ca; 7r/2saco} : I
27?(0) =km min
1 . (41 - 12t~2Co +_ 30lmSO)S 0 ' 1
-
--
. l+~(r-1)~.
+_ 3
(15a)
756
A. TALIERCIO
2. for rectangular RVEs with ~ > 7r/4s~c~ and for hexagonal RVEs with 7r/2soca < ~ < mini r/8s~/ce; ~e ] and 3 -> 60 ° :
S~(O)=kmmin
1 . 1 + r / ( r - 1) / " ( x / 1 - 12oe~Co +_ 3OemSo)So' + _ - - d ~ - i ~ ) '
(15b)
. for hexagonal RVEs with 7r/8s~/c~ < 71< 7r/2sac~ and 3 -< 60 ° :
S?(O) = kmmin
1
. l+~(r-1)
/"
(x/1 - 12c~2 x/1 - So2ca2 _ 3~,,SoC~)SoC ' _+~m---+-i-/~f~ )
(15c)
4. for hexagonal RVEs with fiber volume fraction different from those specified above, it is only possible to state that: 1+r/(rS?(O)
-
1)
++-Olm+ 1/ ~f3
kin.
(15d)
In Figure 6 plots of So~ and Z'~ for composites with Drucker-Prager type fibers and matrix (with r = 5) are shown. Figure 6a refers to a square reinforcing array and Figure 6b to a regular hexagonal array. First of all note that, apart from the cases with greater o/m values, the gap between the two bounds is relatively small. However, note that in Figure 6 the fiber volume fraction was implicitly assumed to be sufficiently lower than the maximum one compatible with each kind of reinforcing array, so that the strictest upper bound was used (see eqn (15a)). The gap between bounds may tend to increase at some 0 for larger volume fractions. Anyway, for any am and whatever the geometry (3, rt) of the composite may be, the lower and the upper bounds coincide at 0 = 0, so that 1 + r/(r -
S0-+(0) = ~'1-+(0) =
1)
k m -+-'~n'-~ 1-/~f~ ( = S ± ( 0 ) )
(16)
is the actual macroscopic strength of the composite along the fiber direction. Note that S ± (0) = (1 - T/)o,~ + ~of-+ is apparently the weighted average of the uniaxial strengths of fiber and matrix, the weights being the relevant volume fractions. Thus, eqn (16) is a rigourous validation of a well-known semiempirical formula widely used in practice, usually called "rule of mixtures" (see, e.g. HASrnN [1983]). V.2. Bilateral bounding o f the macroscopic biaxial strength I f the strength properties of the composite have to be estimated under stress conditions not as simple as the uniaxial ones considered in Section V. 1, the solution of the problem is likely to be obtained numerically. As a rule, this can be done by fixing a radial path in the stress space and by determining the m a x i m u m norm of the macroscopic stress tensor compatible with Go or GI. The lower and upper bounds to the macroscopic strength domains for the state of stress considered are thus derived point by point. In order to illustrate how the model described here makes it possible the description of the macroscopic strength properties of fiber composites subjected to any state of stress, the lower and upper bounds to the biaxial strength of composites with Drucker-
Macroscopic strength domain of a fiber-reinforced composite material
757
a) I0
"
6i
M
6~
"~
4i 2a
....
0
_
,-4-
...~,. •. ~ . . . :~... ~ . . . ~ .. ~
-8-
-6i I -a!
,"-'~
-'ll
/
4 _,o.
- l
O
-
1
~
t 2
~ ~
° ,"- --"] /
-10 -IB
- 12'
-14,
-14'
-16
x,,,,,,,.
:1
::lf/ 't
Fig. 6. Bounds to the uniaxial tensile (27+) and compressive (27-) strength for composites with o f / o + = of~ore = 5 and Von Mises (Ctrn = 0) or Drucker-Prager (am = 0.1,0.2) matrix. Solid line is lower hound; dashed line is upper hound. (a) Square reinforcing array (8 = 45°; ~ < 0.39); (h) regular hexagonal reinforcing array (/3 = 60°; ~/< 0.68).
Prager type matrix will be now derived. Denote by 27/, 27//the two nonvanishing principal stresses and by 0 the orientation of 27/with respect to the fibers. Suppose that the principal stresses act in the (x, y ) plane. The lower bound to the biaxial strength is obtained by making use of the definition o f Go, eqn (13), noting that in the present case t r ~ = 27/+ 2,H, ~ : ~ = 272 + 272 and 27~ = 27/c2 + 2~us2. According to the procedure mentioned above, let A be a positive parameter (measuring the norm of =27)and let ~ be an angle varying between 0 ° and 360 ° in the plane (27/, 27//). Setting 27/= A cos ~( = Ac~), 27/i = A sin ~k( = As~), for any prescribed orientation 0 of the principal stresses to the fibers the maximum value for A (q~) fulfilling eqn (13) has to be found:
758
A. TALIERCIO
FindAo(dp) = supIAa [/~Olm(C~ -1-S¢a) -- OlrnO
+~
1/
l
A2(1-c4~s4,)-~Aa[co+so+3(c4~-s,~)(coZ-sZ)]+°2-km<-O;
- 6-< o< 6+/.
(17)
Numerical solution of problem (17) yields the lower bound sought. As for the upper bound, formed by the intersection of all the domains defined by eqns (14) or some of them, again note that in the case considered here G~,(e_z) does not furnish any significant bound (since S z': = S yz = S zz = 0). Also note that S yy : ~1 $2 "l- ,~11C2 and
S xy = (271 - Su)CoSo.
Following the same procedure as for the lower bound, for any prescribed orientation 0 three values for the parameter A(_> 0), denoted by Ay, Aa and Af, are obtained by finding the maximum value of A such that 271 and 27//are compatible with G~,(ey), G,~ (n~) and G~ f) G f , respectively. In this case, unconstrained maximization problems have to be solved and their solutions read km
3Otm(C~S2 + $4,Cif)
+ X/1 -- 12c~21c~ - s~lSoCo
km 3Otm(CoS ~ + S6Cff)C~ + X/1 -- 12etm 2 ~/(c~ - s~,)2c~s~ + (c4,s~ + s4,coZ)2s~c~ [1 + ~(r - 1)]km Af = (i(1
-- ¢~S~) + 2m(C¢ +S~)"
Finally, the pairs of principal stresses at which the upper bound to the ultimate biaxial strength o f the composite is reached are computed as St = A 1c~, S I / = A i s~, where 1. for rectangular RVEs with ~ < 7r/4s~c~ and for hexagonal RVEs with ~/ < mini ~/8s~/c,; 7r/2s~c~ I : A~ = minlAy, As; Af};
(18a)
2. for rectangular RVEs with 7r/4s~c~ < ~/and for hexagonal RVEs with r/2saca < < minlTr/8sa/ca;~e} and/3 _> 60°: A~ = minlAy; As];
(18b)
3. for hexagonal RVEs with 7r/8sa/c~ < 71 < ~r/2st~c ~ and/5 < 60°: A~ = minlA~; As];
(18c)
Macroscopic strength domain of a fiber-reinforced composite material
759
4. for other types of hexagonal arrays: A, = A/.
(18d)
Figures 7 and 8 show the bounds computed according to the procedure just described. Only three orientations (i.e. 0 = 0 °, 22.5 °, and 45 °) have been considered for clarity of representation. Figure 7 refers to composites with square RVEs, Figure 8 to composites with regular hexagonal RVEs (~ = 60 ° ). The case o f Von Mises type matrix is considered in Figures 7a and 8a, whereas Figures 7b and 8b refer to Drucker-Prager type matrix with a,, = 0.1. As a rule, the two bounds are very close. Again note that the strictest upper bound (computed according to eqn (18a)) was used, so that at greater volume fractions a larger gap might be obtained. VI. C O N C L U D I N G
REMARKS
The homogenization theory applied to limit analysis has proved to be an effective tool for obtaining lower and upper bounds to the strength of composites under any stress condition in a relatively easy manner. In particular, by means of the simple failure mech-
4 Et
2¸
a)
,(-
9
n-, ¢I
;
/
-2
IIIllllll,llllllllllllllllllli'llllllllllllllllllllllll'll -8 -4 0
8
4
12
r,,Ik.
c e e e o #=0
l.b.
oeeoe =0 *¢*** #=22.5 *,,*** =22.5 e"=--= #=45 aeaoe =45
u.b. l.b. u.b. l.b. u.b.
\'l
[.,I .
.
.
.
•x,~,-.~ ~ .
.
.
.
--4I,,,.,,,,,l,.,,..,,,l,..,,.,.,l.,.,,,,,,l,,,,,,,,,l.i,.,.,
-16
-12
-8
-4
. , ,,
0
4
8
,',,"I
12
/I
,
I
x
X lr
Fig. 7. L o w e r a n d u p p e r b o u n d s to the b i a x i a l stress d o m a i n s o f c o m p o s i t e s with s q u a r e r e i n f o r c i n g array (with /~ = 45 ° a n d 7/ < 0.39), for d i f f e r e n t o r i e n t a t i o n s 0 o f 271 to the fibers. (a) Von Mises m a t r i x ; (b) D r u c k e r - P r a g e r m a t r i x .
760
A. TALIERCIO
a)
4 ~
2.
~11II
O' I /
•
//'*
_ _~ x ~ l . .
-4
iiiiiii
¢.... . . . . .
IIIIIIIIIIIIIIIIIIIIIIIlllllllllllllllllll|lllllllll
-8
-12
0
-4
I
8
4
12
Y:,/k=
e e e e o ~=0
l.b.
oe~oe ***** ***¢*
u.b. l.b. u.b. 1.b. u.b.
:0 11=22.5 =22.5 ===-" 4=45 oe~ =45
~
1 b) ............ /
::]'"
~
.
. . . . -.
•
iiiiiiiillllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
-18
-12
-8
-4
0
P.i/k=
4
8
12
.1i r<°,
Fig. 8. Lower and upper bounds to the biaxial stress domains of composites with hexagonal reinforcing array (with B = 60° and ~7< 0.68), for different orientations/9 of S I to the fibers. (a) Von Mises matrix; (b) Drucker-Prager matrix.
anisms for the RVE defined in Section IV, analytical equations for the upper bounds to the macroscopic strength exhibited by the composite under particular stress conditions were obtained (Section V). Provided that some conditions regarding the geometry of the composite and the strength properties of the components are fulfilled, these upper bounds do not excessively overestimate the lower bounds obtained by making use of the domain Go defined in Section III (and, in some cases, are coincident with the latter ones). If the geometry of the composite (defined by the percentage o f reinforcement 7/and by angle/3) is such that the discrepancy between the two bounds is excessive, an approach more complex than the one presented here might become necessary. First of all, periodic failure mechanisms for the RVE not as simple as those considered in Section IV could be taken into account, in order to improve the upper bound to the strength of the composite. Second, a structural analysis o f the RVE could be performed, in order to find with fair approximation the actual macroscopic strength of the composite, provided that a suitable computer program is available; however, this approach turns out to be quite cumbersome in terms of computational cost (as shown by similar nu-
Macroscopic strength domain of a fiber-reinforced composite material
761
merical studies performed by TURGEMAN and PASTOR [1987] for heterogeneous layered media and by MARIGO et ai. [1987] for perforated plates). Note that the approximations defined in Sections III and IV can be used for composites with components undergoing any kind o f strength criteria (provided that eqns (9) are fulfilled). If suitable experimental results are available, the definitions of Go and G~ can be specialized in order to properly take into account the actual strength properties of matrix and fibers. However, as mentioned in Section I, TAZlERCIO [1989] and DE BurIAN and TALIERCIO [1991] showed that, assuming that the matrix undergoes Von Mises strength criterion (which is a special case of that considered in Section V), the lower bound to the strength of the composite both under uniaxiai tension and under combined normal stress and shear is in many cases a close estimate of that measured by other authors on composite samples. Finally, it must be acknowledged that here the fiber-matrix interface was implicitly assumed to be indefinitely strong, or at least to be such that, in any case, failure never occurs along this surface. There are experimental results, however, that show that actually the fiber-matrix interface is the failure surface for composites subjected to particular load conditions. The bounds presented here can be quite easily modified if there is any experimental evidence for assuming that the interface has limited strength. In fact, homogenization theory applied to limit analysis allows also the formulation of macroscopic strength criteria taking into account interface strength criteria (see, e.g. DE BuHAN & TALIERClO [1988,1991]). Acknowledgements--This work was developed within the framework of a research program supported by the Italian Ministry for University and for Scientific and Technological Research (M.U.R.S.T.). The author is grateful to Professor Cesare Davini of Udine University (Italy) for his interest and most valuable suggestions.
REFERENCES 1950 1964 1965 1967 1970 1970 1971 1972 1974 1980 1981 1983 1983 1983a 1983b 1984 1987
HILL,R., "The Mathematical Theory of Plasticity," Clarendon Press, Oxford (GB). HASmN,Z., and ROSEN, B.W., "The Elastic Moduli of Fiber-Reinforced Materials," Trans. ASME, J. Appl. Mech., 31, 223. Azzi, V.D., and Ts~, S.W., "Anisotropic Strength of Composites," Experimental Mechanics, 5, 286. HOFFMANN,O., "The Brittle Strength of Orthotropic Materials," J. Compos. Mater., 1,200. McLAuor~IN, P.V., Jr., and BATTER~N, S.C., "Limit Behaviour of Fibrous Materials," Int. J. Solids Structures, 6, 1357. TVRRELLROCKAFELLAR, R., "Convex Analysis," Princeton University Press, Princeton, N.J. TsAI, S.W., and Wu, E.M., "A General Theory of Strength for Anisotropic Materials," J. Compos. Mater., 5, 58. McLAoGrtLIN, P.V., "Plastic Limit Behaviour and Failure of Filament Reinforced Materials," Int. J. Solids Structures, 8, 1299. Wu, E.M., "Phenomenological Anisotropic Failure Criterion," in B R O W , L.J., and KROCK, R.H. (eds.), Composite Materials, Vol. II, Academic Press, New York. HASmN,Z., "Failure Criteria for Unidirectional Fiber Composites," Trans. ASME, J. Appl. Mech., 47, 329. HULL,D., "An Introduction to Composite Materials," Cambridge Univ. Press. HASmN,Z., "Analysis of Composite M a t e r i a l s - A Survey," Trans. ASME, J. Appl. Mech., 50, 481. SALENqON,J., "Calcul ~ la Rupture et Analyse Limite," Presses de I'E.N.P.C., Paris. SUQtrET, P., "Plasticit6 et Homog6n6isation," Th~se d'Etat, Universit6 Pierre et Marie Curie, Paris. SUQUET, P., "Analyse Limite et Homog6n6isation," C.R. Acad. Sci., Paris, 296, S6rie II, 1355. BOErILER,J.P.,and RACLr~, J., "Failure Criteria for Glass-Fiber Reinforced Composites Under Confining Pressure," J. Struct. Mech., 13, 371. MARIOO,J.P., ML~,LO~, P., MICHEL, J.P., and SUQtr~T, P., "Plasticit6 et Homog~n6isation: Un Example de Pr~vision des Charges Limites d'une Structure H6t&og~ne P&iodique," J. M6c. Th. Appl., 6, 47.
762
1987 1988 1989 1991 1991
A. TALIERCIO
TURGEMAN,S., and PASTOR, J., "Comparaison des Charges Limites d'une Structure H6t6rog6ne et Homog6n6is6e", J. M6c. Th. Appl., 6, 121. DE BUI-IAN,P., and TAtIERCIO, A., "Crit6re de R6sistance Macroscopique pour les Mat6riaux Composites h Fibres," C.R. Acad. Sci., Paris, 307, S6rie If, 227. TALIERCIO,A., "Study of the Elastic and Ultimate Behaviour of Fiber Composite Materials and Composite Structural Elements" (in Italian), Ph.D. Thesis, Politecnico di Milano. DE BUI-IAN,P., SALENfON,J., and TALIERCIO,A., "Lower and Upper Bounds Estimates for the Macroscopic Strength Criterion of Fiber Composite Materials," in DVORAK,G.J. (ed.), Inelastic Deformation of Composite Materials, Springer-Verlag, New York, p. 563. DE BUHAN,P., and TALIERClO,A., "A Homogenization Approach to the Yield Strength of Composite Materials," Eur. J. Mech., A/Solids, 10, 129.
Dipartimento di Ingegneria Strutturale Politecnico di Milano Piazza Leonardo da Vinci, 32 20133 Milano, Italy
(Received 6 April 1991; in final revised form 17 September 1991 )
0749-6419/92 $5.00 + .00 Copyright © 1992PergamonPress Ltd.
Printed in the U.S.A.
LOWER AND UPPER BOUNDS TO THE MACROSCOPIC STRENGTH DOMAIN OF A FIBER-REINFORCED COMPOSITE MATERIAL
ALBERTO TALIERCIO Politecnico di Milano (Communicated by Cesare Davini, Universit~ degli Studi di Udine)
Abstract-The macroscopic strength domain of a composite material reinforced by long, parallel fibers is, in general, unknown but for its theoretical definition. In this note it is shown how a homogenization technique applied to yield design theory allows the derivation o f two domains (in the space o f macroscopic stresses) that are a lower and an upper b o u n d to the composite strength domain. The dependence of these domains on the fiber content and on the shape of the fiber array is pointed out. Analytical equations for the approximate uniaxial macroscopic strength o f composites with Drucker-Prager or Von Mises type matrix are derived. For more complex stress conditions, the relevant strength domains are numerically evaluated as well. The discrepancy between the two b o u n d s is in m a n y cases relatively small. In particular, the two bounds yield the same value for the uniaxial strength of the composite along the fiber direction, which, by consequence, is exactly determined.
I. INTRODUCTION
The use of fiber-reinforced composite materials, either with polymeric, metallic or ceramic matrix, is constantly expanding in a number of fields, such as aerospace, motorcraft, biomechanic, and civil engineering, especially in applications where high stiffness-to-weight and strength-to-weight ratios are required. As a consequence of their increasing use, a very large amount of theoretical and experimental work has been devoted, mainly in the last two decades, to the study of the mechanical behaviour of both composite materials and structures made of composite. In spite of the number of papers and books published on this subject, the problem of defining strength criteria, supported not simply by a good comparison with experimental results, but also by a clear mechanical interpretation, does not seem to be fully solved yet. In fact, the most widely used strength criteria for these materials are "phenomenological" criteria, such as those proposed by Azzi and TSAI [1965], HOFFMANN [1967], TSAI and Wv [1971], and they all are subsequent extensions of a criterion formulated by H~L [1950] for anisotropic materials. The strength properties of composites subjected to different kinds of stress are usually well matched by all of these criteria (see, e.g., the comparisons shown by Wtr [1974]). On the other hand, these criteria are defined by a rather large number of parameters (Tsai and Wu criterion, for instance, requires six parameters in plane stress conditions), some of which are not easily obtained by the experiments (biaxial or pure shear tests may be required). This difficulty in deriving the strength parameters is a direct consequence of the fact that phenomenologiA n abridged version of this note, in Italian, was presented at the X National Conference of the Italian Association o f Theoretical and Applied Mechanics, October 2-5, 1990, Pisa, Italy. 741
742
A. TALIERCIO
cal criteria merely aim at interpolating sets of experimental data, without considering the actual heterogeneous nature of the composite. The drawbacks related to a phenomenological approach can be avoided by directly taking into account the mechanical behaviour of fiber, matrix, and, possibly, the fibermatrix interface. This was done, for instance, by BOEHLERand RACLIN [1984], who proposed a criterion that allows for a distinction between failure modes, depending on the loading condition, and that involves parameters related to the strength properties of the components. Although their criterion allows an excellent reproduction of uni- and triaxial experimental tests, its use is limited by the number of required parameters, which amounts to 18. Similar considerations apply to the criterion proposed by HASmN [1980], which has been tested by the author under uniaxial conditions only. Another interesting criterion based on a micromechanical approach is that used by McLAUGHLIN and BATTERMAN[1970] and McLAUGHLIN [1972], who used limit analysis theory to predict the ultimate load domain of the Representative Volume Element (RVE) featuring composite members; this domain is used as strength domain for the composite. These authors do not operate within the framework of homogenization theory, which would provide a mathematical justification to their criterion, but their theoretical results are well validated by comparisons with experimental tests on unidirectional and angle-ply metal composites subjected to uniaxial tension. In order to derive criteria founded on a sound mathematical basis and defined by a reasonable number of parameters clearly related to the strength of fibers and matrix, an approach based on the application of limit analysis to homogenization theory can be used. Following McLaughlin's approach, RVEs are treated as "structures," and their ultimate loads, which are identified with the "macroscopic stress" in the composite structure, are determined. By applying the static and kinematic theorems of limit analysis, lower and upper bounds, respectively, to the carrying capacity of RVEs can be obtained. DE BUHAN and TALIERCIO[1988,1991] applied this approach to composites reinforced by one or more arrays of parallel fibers, embedded in a matrix conforming with Von Mises strength criterion. By defining suitable stress fields over the RVE, these authors obtained lower bounds to the macroscopic strength properties of the composite, which, in many cases, were in excellent agreement with available experimental results. However, these lower bounds had to be proved not to be overconservative when compared with upper bounds. Also, the use of a Von Mises matrix does not permit allowance for the effects of isotropic pressure on the composite strength, which, as shown by BOEHLER and RACLIN [1984], can be far from being negligible. This work is mainly devoted to the formulation of upper bounds to the macroscopic strength properties of fiber composites, by means of a kinematic limit analysis approach requiring the definition of suitable failure mechanisms for the RVE. This is done by extending the results of DE BUHAN et al. [1991], which actually were of interest only for composites in plane strain conditions. After having briefly reviewed the fundamental aspects of the theoretical approach used and recalled the definition of the lower bound (Section II), in Section I I I a simplified version of this bound particularly useful in applications is presented, and the assumptions under which the approximated lower bound is valid are discussed. Section IV is devoted to the definition of upper bounds for composites with any type of components. These theoretical results are specialized in Section V to composites with Drucker-Prager type matrix. The use of Drucker-Prager criterion permits us to obtain macroscopic strength criteria sensitive to the effects of hydrostatic pressure and, consequently, more
Macroscopic strength domain of a fiber-reinforced composite material
743
realistic than those derived with the Von Mises model. The lower bound of de Buhan and Taliercio is reformulated accounting for Drucker-Prager criterion, and upper bounds are computed as well. Both bounds are defined by only four strength parameters, for any stress state, which are related to the uniaxial tensile and compressive strengths of the components and can be rather easily obtained by means of experiments. In spite o f the simplicity o f the failure mechanisms used in the kinematic approach, lower and upper bounds tend to be rather close for composites with different fiber arrangement and volume fraction, subjected to uni- and biaxial stress.
11. PROBLEM FORMULATION
Consider a composite material consisting of an array of parallel, cylindrical fibers with circular cross-section, embedded in a matrix of bonding material. Suppose that the fibers are "long," in the sense that their length is comparable to the size of the structural member considered. If the fibers form a regular array, the composite can be regarded as heterogeneous material with periodic internal structure. Periodic materials are featured by a "representative volume element" ( R V E - s e e , e.g. I-IAsHIN [1983]). The volume of the RVE will be indicated by (~; by a suitable scale change, its measure can always be made equal to 1, so that in subsequent analysis It~l = 1 will be assumed for simplicity. Furthermore, (~f and (~m will denote the parts of the RVE relevant to the fiber and the matrix material, respectively. Let the volume fraction of fibers be ~ = [t~f[/[ (~1(= [(~f[), so that the volume fraction of matrix is 1 - ~/= [(~,, [. The points _xof the RVE will be referred to an orthogonal reference frame Oxyz, with the x-axis collinear with the fibers. Let e_x,ey, e~ be the base vectors. The length of the RVE along the x-axis is apparently arbitrary. As a consequence, subsequent analysis will be developed in any cross-section of the composite perpendicular to the x-axis (thus staying in the (y, z) plane) and, for the sake of brevity, "RVE" will actually denote the intersection of the representative element with this plane. Regular arrays o f fibers can be fundamentally o f two kinds: . Rectangular arrays, see Figure la, such that, in the (y, z) plane, the centers of the fibers coincide with the centers of rectangles with sides parallel to the axes y and z. In this case, dy and dz will denote the length o f the sides o f the RVE. . Hexagonal arrays, see Figure lb, such that the centers of the fibers, in any section perpendicular to the x-axis, are located at the centers of more or less regular hexagons. In this case, the reference frame Oxyz is chosen in such a way that z is the direction defined by the centers o f a row of fibers in any cross-section and y is perpendicular to z in this section, dy and dz will denote the distance between adjacent fibers along these axes. Reinforcing arrays will be featured by an angle/3, defined as 13 = arctg dJdz. The reference frame will be chosen in such a way that dy _> d z (so that 45 ° _
744
A. TALIERCIO
1
dy
4-
a)
Z
Fig. 1. Unidirectional fiber reinforced composite with (a) rectangular and (b) hexagonal reinforcing array. The relevant RVEs are also shown.
As pointed out in Section I, several authors have studied the problem of defining macroscopic strength domains for periodic fiber-reinforced materials, that is domains suitable for the description o f the overall strength properties of fiber composites subjected to any stress. To this end, it is customary to define suitable measures for the average stress at any point of the composite structure; these are called macroscopic stresses and are the volume average of the microscopic stresses over the RVE (see, e.g. HASmN 8, ROSEr~ [1964]). Thus, denoting by =a(x) the microscopic stress field in the RVE and by ( . ) volume average over RVE, the macroscopic stress =27is
____2:= (o) = fe e(x) d6t. Now, suppose that the strength domains of fibers and matrix are given convex domains in the stress space, denoted by Gf and Gin, respectively. SUQUET [1983a,b] proved that a convenient macroscopic strength domain can be defined as
[G h°m =
~[~
= (__o); div g = 0 ¥x E if; [[~] .ns = 0 v x E S;
o periodic over 6t; __a._nanti-periodic over 06~;
O)
g e GmVx_e ~m ; ~ 6 Gsvx e as I, and that this domain also is convex. Here S is any possible discontinuity surface for the microscopic stress field ~ in (t and _ns is the normal unit vector at any point of S. Remark. In the definition of G h°m, eqn (1), only the strength domains of matrix and fibers are involved. I f no further assumption is made about the constitutive law of the component materials, G h°" is called "domain of the potentially admissible macroscopic stresses" (SALEN?ON [1983]). This means that, if _27~ G h°'n, S__cannot be associated with any microscopic stress field being statically admissible and, at the same time, compatible with the strength properties of fibers and matrix. On the other hand, if the components are elastic-perfectly plastic, with associated flow rule, and if Gin, Gf are their
Macroscopic strength domain of a fiber-reinforcedcomposite material
745
elastic domains, Gh°m actually contains macroscopic stresses compatible with the strength properties o f the composite as a whole. For further details, see SAtENqON [1983] and SUQtrET [1983a]. Convexity allows the definition of G h°m in a form, dual to that of eqn (1), which will be used later. To this end, it is worth recalling the definition of support function 7r(d) o f a convex domain G C R N (see, e.g. TYRRELL ROCKArELLAR [1970]):
~r(_x) -- sup[_x._yl_y ~ G} where _x._ydenotes inner product o f two vectors _x,_yin R N. Convex domains coincide with the intersection in R N of the subspaces defined by means of their own support functions: O = [_yl_y._x__ < ~(_x)v_x E RN]. Refer now to the macroscopic stress domain G h°m, eqn (1), and denote by 7rh°m its support function. By identifying the macroscopic stress space with R 6, G h°m can be alternatively defined as O h°m =
I I :P
h°m( o)V O ~-" R 6 ] •
(2)
The definition of 7r h ° m w a s given by SUQUET [1983a], to which readers are referred for further details. Let v be a piecewise differentiable velocity field over the RVE, ~ d the relevant microscopic strain rate field, S any possible discontinuity surface for p, _ns the normal to this surface and Ilp]l the jump in p across this surface. If v can be expressed as
p=D._x+_u, where D is any symmetric second order tensor and u has the same kind o f periodicity as the considered medium, 2 it is possible to show that the support function of G h°m is
7rh°m(D) = inf I ( ~ r ( d ) ) I D = (_d), _D E R6}.
(3)
Because of the assumed periodicity, in eqn (3) ( d ) can be computed as
( d) = faa v_ ~) n_dS, where ~ denotes "symmetric part of dyadic product" (i.e. v ~ n = ½(_v ® n + _n ® p)) and n is the outward normal to the boundary o f 6L Finally
;
]
r(d=)dV + ,is 7r(n_s;[v])dS ,
(Y)
1The definition given by Suquet actually involves velocity fields more general than those used here. 21.e. U is Q-periodic.
746
A. TALIERCIO
a-(_d) being the support function of the convex strength domain of the material at each point n of the RVE. In eqn (3') the term computed on S is given by 7r(ns;l[v]) = supl_g.es. II_v]l,_oE G(_x)}. o_
At this stage, it is worth emphasizing that generally G h°m is known only through its definition, eqn (1) or eqn (2), but not explicitly. By defining suitable statically admissible microscopic stress fields over the RVE, DE BUICAN and TALIERCIO [1988,1991] proved that the domain Gs = [S= = o=,,,+ nofe_x ® e_,:; a,, E Gm; ~rn + ofe_x @ e_r E G f l
(4)
is contained in Gh°m; this domain is entirely known if G,,,, G f (and r/) are. The support function of G~, which will be used later, reads (see DE BUHAN & TALmRCIO [1991]) 7rs(__D) =
inf [(1 - ~ ) r m ( d m )
dm,df
+ ~Trf(__df)},
(5)
where 7r,~, Try are the support functions of Gm, Gf respectively, and tensors din, _df fulfill the following conditions: D = (1 - ,/)__d., + ~/__df;
D xx = d xx = d f x.
(6)
In particular, two tensors d~,,=d7 fulfilling eqns (6) exist, at which the infimum in eqn (5) is reached: 71-s(D ) :
( | - ~/)71m(__d~) + ~71"f(__d;).
(5')
In the next two sections, two domains constituting a lower and an upper bound to G h°m, denoted Go and G1, respectively, will be defined. Go will constitute a lower
bound stricter than Gs, defined by eqn (4), so that the set of inequalities Go C Gs C G h°m C G1
will be obtained. This will allow the bilateral bounding of the unknown domain G h°m and the use as macroscopic strength domain of either one of the domains defined, provided that the maximum gap between the two bounds is sufficiently small. Finally, in Section V the theoretical results obtained will be applied to composites with DruckerPrager matrix and the difference between the strength properties predicted by the two bounds will be quantified for some stress conditions. i!1. A LOWER BOUND TO G h°m Consider now a particular composite material with fibers reducing to filaments, that is to unidimensional reinforcing elements (7 -~ 0). Let $ + (resp. b - ) be the tensile (resp. compressive) strength of the filaments. As shown by DE BUHAN and TALmRCIO [1991], for this special case the definition of the lower bound Gs, eqn (4), here denoted Go, reads: GO :
[ ~ = ff=m -]- (Ye_x (~ C_x; ~=m ~- Gin, - 6 -
(7)
Macroscopic strength domain of a fiber-reinforced composite material
747
and its support function is 71"0(O ) -~- 7~m(D ) -~- m a x l # + D X X , - # - D ~ X l .
(8)
It is rather evident that the case considered now is mostly o f academic interest, since the assumption made about the fibers is unlikely to be met in real composites. The importance o f the domain now defined is that, under conditions to be specified later, Go can be proved to be contained in the macroscopic strength domain of a composite with fibers of any strength and volume fraction, provided that the parameters ~+ and # - be suitably defined. This feature makes Go more interesting to be used in comparison with G~; in fact, in the definition of Gs, eqn (4), the entire strength domain of the fiber is involved, whereas only the uniaxial strength values o f the fibers are required to define Go. Let us now prove what is stated above, namely that Go is contained in G h°m for any 7- To this end, for a composite with any 7/define the parameters #-, #+ as follows: ~-
n(a7
am);
~+
'7(al+ -
+
where am, + am- are the uniaxial strength values of the matrix along the fiber direction. By setting r + Uf+/am,+ r - = of~am (>- 1), assume that the strength domains of the components fulfill the following conditions: =
7 r i ( d ) _> r % r m ( _ d ) v _ d l d ~ _ 0,
(9)
7rf(d=) >_ r-Trm(d_)vd_ ldXX <_ O. In particular, eqns (9) are fulfilled if the strength domains o f the components are homothetic, for instance if fiber and matrix are both formed by Von Mises or Tresca materials. Under these assumptions, it is easy to show that Go C Gs(C Gh°m). The p r o o f is based on the inequality 7r0(D) --- ~rs(D) VD E R 6, which can be obtained by decomposing D into the two particular tensors __d~and __d] at which the infimum in eqn (5) is attained. By making use of Jenkin's inequality for convex functions, i.e. 91"m(D = (1 -- T/)d* + 7/__d7)) _< (1 - 1'/)71"re(d*) -~- l~71"m(_d;)
and by virtue o f the definition itself o f support function, which yields r m ( D ) --- maxl amD + ~", - a m-D xx }, from eqn (8) one gets: r 0 ( D ) < (1 - ~)a'm(d,~) + ~/Tr,,(__d~) + m a x l O + D ~ , - 6 - D ~1 ~/(r + - 1)Trm(=j), ifd~XX>0; d* _ 1)Trm(__tiT), if d]=___ 0,
~-~ ( l -- ~)71"m(=d~) "~" 7]Tl'm(=d;) + l r / ( r _
where eqns (6) have been accounted for. Finally, substitution o f the hypothesis concerning the strength domains o f fibers and matrix, expressed by eqns (9), into the latter inequality, yields:
748
A. TALIERCIO
• "o(O) -< (1
-
yl)"/rm(dr%)
Jr
~f(d;)
= "xs(D)VD E R 6
(see eqn (5')), q.e.d. Remarks: i. The inclusion Go C Gs has already been proved by DE BUHAN and TALIERCIO [1991], though under conditions more restrictive than eqns (9). ii. The definition of Go coincides with that given by MCLAUGHLIN [1972] for the strength domain of a fiber-reinforced composite. However, this author directly formulated his strength criterion with reference to composites with infinitely thin fibers and did not operate within the framework of homogenization theory, which actually shows that this is only a lower bound to the actual macroscopic strength domain of a composite with any fiber volume fraction. iii. The definition of Go can be rather easily extended to composites reinforced by N arrays of parallel fibers, each one having a certain orientation defined by the unit vector _e,t")(n -- 1 . . . N ) . Let G) m be the strength domain of the material forming the fibers of the n-th array, tr~ ~") the uniaxial tensile/compressive strength of the matrix along _elm (o ±(") = a ± for any n if the matrix is isotropic), a f (") the uniaxial tensile/compressive strength of the fibers of the n-th array and y, their volume fraction. Let r ±(") = o~(")/~,(")(>
1);
a +-(") = T / . ( o ~ (") -
a~.(")),
n = 1...N.
If the strength domains of the components fulfill the following conditions: r~
( d ) ->- r +<")
e_n
"u='gx
> O,
(9') 7rs(") ( = d ) >_ r - ( " ) T r m ( _ d ) v _ d
I ~- (. " ) .__u._~ . _(-) -0,
the domain Go
: a
=gm +
, ~m E Gin;
n=l
<--
-< a +(")
I
is a lower bound to the macroscopic strength domain of the composite. The proof is a trivial extension o f that given for unidirectional composites.
IV. AN U P P E R BOUND TO G h°m
The dual definition given for 7rh ° ' , eqn (3), shows that, if _/: is interpreted as microscopic velocity field, upper bounds to G h°m can be obtained by defining failure mechanisms for the RVE featured by velocity fields obtained by combination of a term linearly dependent on the position of any point x and another term (u) possessing the same kind of periodicity as the medium considered. Regardless of whether the reinforcing array is rectangular or hexagonal, the simplest
Macroscopic strength domain of a fiber-reinforced composite material
749
failure mechanism of this kind is featured by constant strain rate throughout the RVE (__d(_x) = =D v_x E (~). As a consequence, one immediately gets (see eqn (3))
7rh°m(D)
~--- ( T r ( d ) ) ~--
(1 - '0')'/rm(D ) "l- ' q T r f ( D ) .
Further, provided that the strength domains of matrix and fibers fulfill eqns (9), by setting p+ = 7/+ (1 - ~)/r +, p - = ~ + (1 - ~ ) / r - (p+- <_ 1), the inequality
•"lrh°m(D) <_ <71"(d)) _< p+'lrf(D) i f D ~ >
O; (10)
_< p - x f ( D ) if D x~ _< 0 holds. In order to obtain a relationship being equivalent to eqn (10), but involving domains and not scalar functions, define two domains G~, G f as follows:
o / = [__rl_r:_D_
I~I_~:_D___p-~-,(__D)vDID~_<0; ~=:_D_ <_ p-Tr1( ~ - O=e_x ® e_x)VDlD =
>_
01.
(lla)
(1 lb)
The geometrical meaning of these domains in the macroscopic stress space is as follows. Gy+ (resp. G f ) is obtained by first homothetically contracting GI with homotheticity ratio p+ (resp. p - ) . The part o f boundary o f the contracted domain where the outward normal D has positive (resp. negative) component along 2?xx coincides with a part of the boundary of Gf+ (resp. G f ) . The remaining part of boundary belongs to the cylinder with generatrices parallel to S = and tangent to the contracted domain at the points where the outward normal D is perpendicular to S xx (see Fig. 2).
l ~ik(i,k~x,x) t
G~ \
~
/
(1:)' I:) --"~"=o)
V"- ~'~ O+O'Im+
..~:.:.:.G. !::::::"~:~'.:.-:i::
,G;
_
"'
Fig. 2. Graphical construction of the domains Gf+ and Gf (see eqns (1 la,b)) in the macroscopic stress space. The dotted area is the intersection of these two domains (G7 n (:;7).
750
A. TALIERCIO
Thus, the scalar inequality eqn (10) is equivalent to the inclusion G h°m C (G7 N
G£). In order to improve the upper bound just defined, periodic failure mechanisms featured by rigid relative movements between blocks in the RVE will be considered. Only mechanisms with planes as slip surfaces will be considered; the planes will be supposed not to cross any fiber, their intersection with RVE will be denoted by S and their normal by ns. Recalling eqn (3') and denoting by V the jump in the velocity field across the plane of failure, for the considered mechanism the term (Tr(d)) is given by <~(d)> = [Sl~m(~s;_V). For further developments, it is convenient to associate any mechanism, with failure plane perpendicular to _ns, with a domain in the macroscopic stress space defined as follows:
c*(_~s) = I__sl__s:__D-< ~ m ( P ) l, where
z,~(D) = ~,~(es; V) = +oo
if D
=
_V (~ _ns;
otherwise.
IV. 1. Rectangular array If the fiber array is rectangular, a mechanism of the kind described above, with failure plane passing only through the matrix and perpendicular to ey (or _ez), can be defined for any percentage of reinforcement-see Figures 3a,b. Let Vy (resp. Vz) be the relative velocity between the two rigid blocks in movement and letsSy (resp. Sz) be the surface in the failure plane. Since D = e_y ~ Vy ISy I (resp. D = e z ® Vz ISz ]), by virtue of the definition of 7rh°m, eqn (3), the two following inequalities hold for any ~/ ( _< ~r) :
~hom(p = e_z Q ~ ) <- ~m(e_z; ~ ) .
n~
"~ Vy
¢)
Fig. 3. Periodic failure mechanisms for rectangular RVEs, featured by relative rigid movement of blocks.
Macroscopic strength domain of a fiber-reinforcedcomposite material
751
These inequalities amount at the inclusion between domains in the macroscopic stress space
G h°m C Gm(e_y) n G*(ez). Another failure mechanism featured by the required periodicity consists o f two parallel slip planes crossing each RVE, located at opposite sides with respect to the fiber and with traces in the (y, z) plane intersecting the edges of the RVE at their midpoints (see Fig. 3c). The unit vector perpendicular to these planes has the form n~ = +_e_yc~+_ ezS~. The triangular wedge of RVE with n, as inward normal is supposed to have relative velocity V~ with respect to the central block in which the fiber is located, so that by periodicity the triangular wedge with n~ as outward normal is required to have relative velocity - ~ with respect to the central block. Furthermore, in order to have failure planes passing only through the matrix, the fiber volume fraction ~ must not be greater than 7r/4s~c~ (i.e., ~ _< a-/8 = 0.39 for square RVEs). If this restraint is fulfilled,
7fh°m(D : V~ ~ n_13)~ 7fm(_V~;n~) that is, in terms of domains
G h°m C G*(n_/~). As a conclusion, an upper bound to G h°m for composites with rectangular array of fibers is given by the domain GI = Gf+ n G f n G*, where
G* = G* (ey) n G* (ez) n G*(_na) = G*(ey) n G*(ez)
if ~ < lr/4s~c,
otherwise.
IV.2. Hexagonal array Suitable periodic mechanisms featured by failure planes perpendicular to ey passing only through the matrix are possible provided that the trace o f any failure plane in the (y, z) plane intersects the skew edges o f the RVE at their midpoints (see Fig. 4a) and
b)
c)
Fig. 4. Periodic failure mechanisms for hexagonal RVEs, featured by relative rigid movement of blocks.
752
A. TALIERCIO
provided that any RVE be crossed by two parallel planes located at opposite sides with respect to the fiber. The failure planes pass through the only matrix if 7/_< 7r/Ssa/c~ (i.e., if ~/_< 7r%/-3/8 = 0.68 for regular hexagonal arrays). Denoting by Vy the relative velocity between the triangular wedge with ey as inward normal and the central block containing the fiber, by periodicity the velocity of the triangular wedge with _ey as outward normal relative to the central block must be -Vy. Similar considerations apply for mechanisms with failure planes perpendicular to _ez (Fig. 4b). These planes do not cross any fiber provided that ~/_< ~r/8c~/sa (i.e., ~ _< • -x/3/24 = 0.23 for regular hexagonal arrays). Furthermore, periodic are also mechanisms featured by two parallel failure planes per RVE, located at opposite sides with respect to the fiber and with traces in the (y, z) plane intersecting a skew edge o f the RVE and the adjacent edge parallel to the y-axis at the relevant midpoints (Fig. 4c). The equation of the unit vector perpendicular to these planes is n~ = _+eyC~_+ ezSa. In order to have failure planes passing through the matrix only, the fiber volume fraction 71 must not exceed ~l/2sac~ (that is, 0.68 for regular hexagonal fiber arrays). As pointed out in Section IV. l, denoting by V~ the relative velocity between the triangular wedge with na as inward normal and the central block containing the fiber, the velocity of the triangular wedge with n~ as outward normal relative to the central block must be -V~. As a conclusion, after having defined domains GTn(ey), G*(ez) and G * ( n a ) analogous to those defined for rectangular arrays, it is possible to state that G h°m C GI, where GI can be expressed as Gl = Gf+ n Gf- n G* and G*m = G*(ey) n G*(_ez) O G*(n~) if r / < 7r/8c~/s~ Gm(ey) O Gm(ne) if r/8c~/s~ < ~ < mini 7r/8s~/c~; 7r/2sec~l
G* (n~) if rc/8c~/s~ < rl < 7r/8s~/c~ and 3 -< 60 ° Gm*(ey) if ~r/2s~c~ < ~ < minlTr/8syc~;~e] and 3 -> 60 ° R 6 otherwise. V. APPLICATIONS The theoretical results presented up to here will now be applied to the evaluation of lower and upper approximations of the maximum solicitation that fiber reinforced composites subjected to particular states of stress can sustain. The material forming the matrix will be supposed to undergo Drucker-Prager strength criterion. This two-parameter criterion is an extension of Von Mises criterion; here it was chosen for the reasons discussed in Section I and because of the existence of experimental results showing that the ultimate strength of some polymeric matrices is well described by a Drucker-Prager criterion under some stress conditions (see, e.g., the results reported by HULL [1981]). As is well known, denoting by km the shear strength of the matrix and by ~m a parameter related to the "internal angle of friction" of the matrix, Drucker-Prager criterion reads (see, e.g. SALENqON [1983]):
O:mtr Om +
~/l (O=m:~=m -- ltr2 __am)--< km
and the relevant support function is
Macroscopic strength domain of a fiber-reinforcedcomposite material
Irm(D) =
=
km t r D
3arm
=
= +oo
if t r D >__OtmX/6(3_D:_D - tr2_D) = _ _ _
753
(12)
otherwise.
The parameters am, k,, defining Drucker-Prager criterion are related to the uniaxial strength values of the matrix by the relationships 1
-
a,.
-
a m -- am+
+, x/3 am + a..
km =
2
a m a i n+
+. x/3 am + am
Only the case am -> am+ (i.e., 0 _< ct,~ < l/x/-3) will be treated, for its greater interest in practical applications. When failure mechanisms with slip planes are considered, setting V @ _ns = D the condition tr D _ ct,,x/6(3D : D - tr 2 D) can be easily shown to be equivalent to 30/rn
_V'ns ~ x/1 - 3ot2m I N . In other words, finite values of a'm (and, consequently, significant upper bounds to ~rh°m) are obtained if the relative velocity _V between rigid blocks falls within the cone with opening angle 3'* = arccos3 otto/x/1 - 3at 2, having as axis the normal to the slip plane (see Fig. 5). Note that such cone degenerates if ctm > 0.5/4-3; thus, for Drucker-Prager matrices with excessively high compressive-to-tensile strength ratios (i.e. 0.5/x/3 < am --< l/x/3), no bound is obtained by this kind of mechanism. Drucker-Prager criterion reduces to Van Mises criterion if the matrix has equal strength in uniaxial compression and tension (am+ = am a m ~-- kmx[3, i.e., at,, = 0). The support function of Van Mises strength criterion can be deduced from eqn (12) noting that ctm ~ 0 implies tr D ---,0 if finite values for qrm (and consequently significant upper bounds to the macroscopic strength) are to be obtained. Thus (see also SAtENqON [1983]): =
r,,(D) = a , ~ ~ =+~
iftrD=0
iftrD~:0.
In the case o f Van Mises criterion, when failure mechanisms with slip planes are considered, setting D = _V ® _ns the condition tr __D= 0 is equivalent to _ns .1_ _V. In other words, for composites with Van Mises type matrix, the relative velocity between rigid blocks in movement has to be tangent to the failure plane. Also the material forming the reinforcing fibers will be assumed to undergo a DruckerPrager type criterion, defined by parameters Cry and ky. For the sake o f illustration, suppose that the ratios between the uniaxial strengths of fibers and matrix are such that + + af lain = a f / a m = r( >_ 1), so that eqns (9) under which the bilateral bounding of G h°m was obtained are fulfilled. Thus, o~y and Icy can be expressed in terms of am, km as otf=Ctm;
k f = r k m ( r > _ 1).
For composites with Drucker-Prager matrix, the following form for the domain Go, defined in general by eqn (7), is obtained:
754
A. TALIERCIO
Fig. 5. For failure mechanisms with slip planes passing through a Drucker-Prager matrix, the relative velocity V between rigid blocks in movement must form an angle 3' with the normal _ns to the slip plane not greater 2 of the outlined cone. {han the opening angle 3'* = arccos 3Otm/~/1 - 3orm
Gb = {__SIc~m(tr__S- o) + ~/~(_S:_S- ~trZ__S) + a ( ~ t r _ _ Z - S ~ x ) -
k m < 0, -
6-
+ ~a2 (13)
< a < 6+}
whereas the e q u a t i o n s o f the d o m a i n s defined in Section IV are
Gm(ey)=
S=
Odin~,yy -- kin~3 ~_4o~2
\
1
2
"1"-(Sxy)2"}- (SYZ)2 ~ 0 "
~ -- 40t m
(14a) Olin S z z
Gm(ez) =
=27
~
--
kin~3
4Otm z
+ %/12012(OlmfZZ~km/3~ 2 ,~
\
~ _ 4otZm
(,~,yz)2
] +
(~zx)2 +
I <0
;
(14b)
~mS"" -~ _ n ~ ) = __2: } - 4 ~
km/3 +
f l 2 o t 2 { ° t m E _I n n - k m2/ 3 )
\
2 -I- (S-,)2-l- ( ~ , n x ) 2 ~ 0 ;
~ --4Otm
S "n = SYYc~ +_ 2SYZcas~ + SZZs~; S "x = +_L'XYct3 +_ SZXs~; S"'
=
(14c)
+_(,rez _ ,re~)cese + syZ(c~ - s~) 1 ;
G f f q G 7 = [__SIO~mtr__S+ x/~ (__S:__S-- ~tr2__S) _< [1 + o ( r -
l)]km}.
(14d)
Macroscopic strength domain of a fiber-reinforced composite material
755
V. 1. Bilateral bounding o f the macroscopic uniaxial strength Suppose that the composite is subjected to uniaxial tension (resp. uniaxial compression) acting in the ( x , y ) plane. Let 0(0 _ 0 < 90 ° ) be the orientation o f the uniaxial stress, S, to the fibers. Set, for brevity, So = sin 0 and Co = cos 0. T h e aim now is computing the functions S~(O) and 27~-(0) (resp. S~(O) and S;-(O)) constituting a lower and an u p p e r b o u n d , respectively, to the anisotropic macroscopic strength 27+(0) (resp. Z'-(0)) o f the composite. The lower b o u n d to the absolute values o f the tensile/compressive uniaxial strength is given by eqn (13) and reads: 27o-+(0) = s u p [27 10:m(27 -- (7) -['- ~jl (27 __ 0.)2 ..[_ 27(7S2 __ km ~ O; - 6 -
----- (7 ~--_ 6 + ]
0
that is
---6+ + 1--3---2 ~J ( ~ O t m 6 + S o 2 + k m ) 2 - ( 6 + ) 2 ( 1 - ' ~ 3 s 2 ) (1 - 3°t2)s2 1-6+$2 + o t m k m
=1=3 2
270±(0) =
-T-3~Otm
~/(1 - ~s~)(1 - 3~m2) ]
3 . 2 --74S0 3 2 "["
1 -
1 - 3c~2
(1
3"-'~-~2m ~ "~ ~So2-'~-_ )So
OtmO-So2 + km +
] km
if o>- 6 +
if - 5 _< o _< 5 +
- ( 6 - ) 2 1 - ~ s 2 (1 - 30~2)s 2
1 --3%m l
-T- 3
----
--~7
2
SO + o t m k m
1 -
3or 2
if o ~
-6-
where 17 - -
-1"3 0lm
km
1 -3a
2-
--
+
=
3 2 zso
1-3c~-
~sd3 2
x/1 - 3c~so 1 - 3~Zm - zSd32"
743s02
As for the upper b o u n d , note first o f all that G~,(e_z) does not impose any b o u n d to the uniaxial strength in the ( x , y ) plane, since 27= = 27yz = 27zx = 0. The definitions o f the other domains, eqns (14), yield 1. for rectangular RVEs with ~/ < 7r/4saca and for hexagonal RVEs with 7/ < mini 7r/8sa/ca; 7r/2saco} : I
27?(0) =km min
1 . (41 - 12t~2Co +_ 30lmSO)S 0 ' 1
-
--
. l+~(r-1)~.
+_ 3
(15a)
756
A. TALIERCIO
2. for rectangular RVEs with ~ > 7r/4s~c~ and for hexagonal RVEs with 7r/2soca < ~ < mini r/8s~/ce; ~e ] and 3 -> 60 ° :
S~(O)=kmmin
1 . 1 + r / ( r - 1) / " ( x / 1 - 12oe~Co +_ 3OemSo)So' + _ - - d ~ - i ~ ) '
(15b)
. for hexagonal RVEs with 7r/8s~/c~ < 71< 7r/2sac~ and 3 -< 60 ° :
S?(O) = kmmin
1
. l+~(r-1)
/"
(x/1 - 12c~2 x/1 - So2ca2 _ 3~,,SoC~)SoC ' _+~m---+-i-/~f~ )
(15c)
4. for hexagonal RVEs with fiber volume fraction different from those specified above, it is only possible to state that: 1+r/(rS?(O)
-
1)
++-Olm+ 1/ ~f3
kin.
(15d)
In Figure 6 plots of So~ and Z'~ for composites with Drucker-Prager type fibers and matrix (with r = 5) are shown. Figure 6a refers to a square reinforcing array and Figure 6b to a regular hexagonal array. First of all note that, apart from the cases with greater o/m values, the gap between the two bounds is relatively small. However, note that in Figure 6 the fiber volume fraction was implicitly assumed to be sufficiently lower than the maximum one compatible with each kind of reinforcing array, so that the strictest upper bound was used (see eqn (15a)). The gap between bounds may tend to increase at some 0 for larger volume fractions. Anyway, for any am and whatever the geometry (3, rt) of the composite may be, the lower and the upper bounds coincide at 0 = 0, so that 1 + r/(r -
S0-+(0) = ~'1-+(0) =
1)
k m -+-'~n'-~ 1-/~f~ ( = S ± ( 0 ) )
(16)
is the actual macroscopic strength of the composite along the fiber direction. Note that S ± (0) = (1 - T/)o,~ + ~of-+ is apparently the weighted average of the uniaxial strengths of fiber and matrix, the weights being the relevant volume fractions. Thus, eqn (16) is a rigourous validation of a well-known semiempirical formula widely used in practice, usually called "rule of mixtures" (see, e.g. HASrnN [1983]). V.2. Bilateral bounding o f the macroscopic biaxial strength I f the strength properties of the composite have to be estimated under stress conditions not as simple as the uniaxial ones considered in Section V. 1, the solution of the problem is likely to be obtained numerically. As a rule, this can be done by fixing a radial path in the stress space and by determining the m a x i m u m norm of the macroscopic stress tensor compatible with Go or GI. The lower and upper bounds to the macroscopic strength domains for the state of stress considered are thus derived point by point. In order to illustrate how the model described here makes it possible the description of the macroscopic strength properties of fiber composites subjected to any state of stress, the lower and upper bounds to the biaxial strength of composites with Drucker-
Macroscopic strength domain of a fiber-reinforced composite material
757
a) I0
"
6i
M
6~
"~
4i 2a
....
0
_
,-4-
...~,. •. ~ . . . :~... ~ . . . ~ .. ~
-8-
-6i I -a!
,"-'~
-'ll
/
4 _,o.
- l
O
-
1
~
t 2
~ ~
° ,"- --"] /
-10 -IB
- 12'
-14,
-14'
-16
x,,,,,,,.
:1
::lf/ 't
Fig. 6. Bounds to the uniaxial tensile (27+) and compressive (27-) strength for composites with o f / o + = of~ore = 5 and Von Mises (Ctrn = 0) or Drucker-Prager (am = 0.1,0.2) matrix. Solid line is lower hound; dashed line is upper hound. (a) Square reinforcing array (8 = 45°; ~ < 0.39); (h) regular hexagonal reinforcing array (/3 = 60°; ~/< 0.68).
Prager type matrix will be now derived. Denote by 27/, 27//the two nonvanishing principal stresses and by 0 the orientation of 27/with respect to the fibers. Suppose that the principal stresses act in the (x, y ) plane. The lower bound to the biaxial strength is obtained by making use of the definition o f Go, eqn (13), noting that in the present case t r ~ = 27/+ 2,H, ~ : ~ = 272 + 272 and 27~ = 27/c2 + 2~us2. According to the procedure mentioned above, let A be a positive parameter (measuring the norm of =27)and let ~ be an angle varying between 0 ° and 360 ° in the plane (27/, 27//). Setting 27/= A cos ~( = Ac~), 27/i = A sin ~k( = As~), for any prescribed orientation 0 of the principal stresses to the fibers the maximum value for A (q~) fulfilling eqn (13) has to be found:
758
A. TALIERCIO
FindAo(dp) = supIAa [/~Olm(C~ -1-S¢a) -- OlrnO
+~
1/
l
A2(1-c4~s4,)-~Aa[co+so+3(c4~-s,~)(coZ-sZ)]+°2-km<-O;
- 6-< o< 6+/.
(17)
Numerical solution of problem (17) yields the lower bound sought. As for the upper bound, formed by the intersection of all the domains defined by eqns (14) or some of them, again note that in the case considered here G~,(e_z) does not furnish any significant bound (since S z': = S yz = S zz = 0). Also note that S yy : ~1 $2 "l- ,~11C2 and
S xy = (271 - Su)CoSo.
Following the same procedure as for the lower bound, for any prescribed orientation 0 three values for the parameter A(_> 0), denoted by Ay, Aa and Af, are obtained by finding the maximum value of A such that 271 and 27//are compatible with G~,(ey), G,~ (n~) and G~ f) G f , respectively. In this case, unconstrained maximization problems have to be solved and their solutions read km
3Otm(C~S2 + $4,Cif)
+ X/1 -- 12c~21c~ - s~lSoCo
km 3Otm(CoS ~ + S6Cff)C~ + X/1 -- 12etm 2 ~/(c~ - s~,)2c~s~ + (c4,s~ + s4,coZ)2s~c~ [1 + ~(r - 1)]km Af = (i(1
-- ¢~S~) + 2m(C¢ +S~)"
Finally, the pairs of principal stresses at which the upper bound to the ultimate biaxial strength o f the composite is reached are computed as St = A 1c~, S I / = A i s~, where 1. for rectangular RVEs with ~ < 7r/4s~c~ and for hexagonal RVEs with ~/ < mini ~/8s~/c,; 7r/2s~c~ I : A~ = minlAy, As; Af};
(18a)
2. for rectangular RVEs with 7r/4s~c~ < ~/and for hexagonal RVEs with r/2saca < < minlTr/8sa/ca;~e} and/3 _> 60°: A~ = minlAy; As];
(18b)
3. for hexagonal RVEs with 7r/8sa/c~ < 71 < ~r/2st~c ~ and/5 < 60°: A~ = minlA~; As];
(18c)
Macroscopic strength domain of a fiber-reinforced composite material
759
4. for other types of hexagonal arrays: A, = A/.
(18d)
Figures 7 and 8 show the bounds computed according to the procedure just described. Only three orientations (i.e. 0 = 0 °, 22.5 °, and 45 °) have been considered for clarity of representation. Figure 7 refers to composites with square RVEs, Figure 8 to composites with regular hexagonal RVEs (~ = 60 ° ). The case o f Von Mises type matrix is considered in Figures 7a and 8a, whereas Figures 7b and 8b refer to Drucker-Prager type matrix with a,, = 0.1. As a rule, the two bounds are very close. Again note that the strictest upper bound (computed according to eqn (18a)) was used, so that at greater volume fractions a larger gap might be obtained. VI. C O N C L U D I N G
REMARKS
The homogenization theory applied to limit analysis has proved to be an effective tool for obtaining lower and upper bounds to the strength of composites under any stress condition in a relatively easy manner. In particular, by means of the simple failure mech-
4 Et
2¸
a)
,(-
9
n-, ¢I
;
/
-2
IIIllllll,llllllllllllllllllli'llllllllllllllllllllllll'll -8 -4 0
8
4
12
r,,Ik.
c e e e o #=0
l.b.
oeeoe =0 *¢*** #=22.5 *,,*** =22.5 e"=--= #=45 aeaoe =45
u.b. l.b. u.b. l.b. u.b.
\'l
[.,I .
.
.
.
•x,~,-.~ ~ .
.
.
.
--4I,,,.,,,,,l,.,,..,,,l,..,,.,.,l.,.,,,,,,l,,,,,,,,,l.i,.,.,
-16
-12
-8
-4
. , ,,
0
4
8
,',,"I
12
/I
,
I
x
X lr
Fig. 7. L o w e r a n d u p p e r b o u n d s to the b i a x i a l stress d o m a i n s o f c o m p o s i t e s with s q u a r e r e i n f o r c i n g array (with /~ = 45 ° a n d 7/ < 0.39), for d i f f e r e n t o r i e n t a t i o n s 0 o f 271 to the fibers. (a) Von Mises m a t r i x ; (b) D r u c k e r - P r a g e r m a t r i x .
760
A. TALIERCIO
a)
4 ~
2.
~11II
O' I /
•
//'*
_ _~ x ~ l . .
-4
iiiiiii
¢.... . . . . .
IIIIIIIIIIIIIIIIIIIIIIIlllllllllllllllllll|lllllllll
-8
-12
0
-4
I
8
4
12
Y:,/k=
e e e e o ~=0
l.b.
oe~oe ***** ***¢*
u.b. l.b. u.b. 1.b. u.b.
:0 11=22.5 =22.5 ===-" 4=45 oe~ =45
~
1 b) ............ /
::]'"
~
.
. . . . -.
•
iiiiiiiillllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
-18
-12
-8
-4
0
P.i/k=
4
8
12
.1i r<°,
Fig. 8. Lower and upper bounds to the biaxial stress domains of composites with hexagonal reinforcing array (with B = 60° and ~7< 0.68), for different orientations/9 of S I to the fibers. (a) Von Mises matrix; (b) Drucker-Prager matrix.
anisms for the RVE defined in Section IV, analytical equations for the upper bounds to the macroscopic strength exhibited by the composite under particular stress conditions were obtained (Section V). Provided that some conditions regarding the geometry of the composite and the strength properties of the components are fulfilled, these upper bounds do not excessively overestimate the lower bounds obtained by making use of the domain Go defined in Section III (and, in some cases, are coincident with the latter ones). If the geometry of the composite (defined by the percentage o f reinforcement 7/and by angle/3) is such that the discrepancy between the two bounds is excessive, an approach more complex than the one presented here might become necessary. First of all, periodic failure mechanisms for the RVE not as simple as those considered in Section IV could be taken into account, in order to improve the upper bound to the strength of the composite. Second, a structural analysis o f the RVE could be performed, in order to find with fair approximation the actual macroscopic strength of the composite, provided that a suitable computer program is available; however, this approach turns out to be quite cumbersome in terms of computational cost (as shown by similar nu-
Macroscopic strength domain of a fiber-reinforced composite material
761
merical studies performed by TURGEMAN and PASTOR [1987] for heterogeneous layered media and by MARIGO et ai. [1987] for perforated plates). Note that the approximations defined in Sections III and IV can be used for composites with components undergoing any kind o f strength criteria (provided that eqns (9) are fulfilled). If suitable experimental results are available, the definitions of Go and G~ can be specialized in order to properly take into account the actual strength properties of matrix and fibers. However, as mentioned in Section I, TAZlERCIO [1989] and DE BurIAN and TALIERCIO [1991] showed that, assuming that the matrix undergoes Von Mises strength criterion (which is a special case of that considered in Section V), the lower bound to the strength of the composite both under uniaxiai tension and under combined normal stress and shear is in many cases a close estimate of that measured by other authors on composite samples. Finally, it must be acknowledged that here the fiber-matrix interface was implicitly assumed to be indefinitely strong, or at least to be such that, in any case, failure never occurs along this surface. There are experimental results, however, that show that actually the fiber-matrix interface is the failure surface for composites subjected to particular load conditions. The bounds presented here can be quite easily modified if there is any experimental evidence for assuming that the interface has limited strength. In fact, homogenization theory applied to limit analysis allows also the formulation of macroscopic strength criteria taking into account interface strength criteria (see, e.g. DE BuHAN & TALIERClO [1988,1991]). Acknowledgements--This work was developed within the framework of a research program supported by the Italian Ministry for University and for Scientific and Technological Research (M.U.R.S.T.). The author is grateful to Professor Cesare Davini of Udine University (Italy) for his interest and most valuable suggestions.
REFERENCES 1950 1964 1965 1967 1970 1970 1971 1972 1974 1980 1981 1983 1983 1983a 1983b 1984 1987
HILL,R., "The Mathematical Theory of Plasticity," Clarendon Press, Oxford (GB). HASmN,Z., and ROSEN, B.W., "The Elastic Moduli of Fiber-Reinforced Materials," Trans. ASME, J. Appl. Mech., 31, 223. Azzi, V.D., and Ts~, S.W., "Anisotropic Strength of Composites," Experimental Mechanics, 5, 286. HOFFMANN,O., "The Brittle Strength of Orthotropic Materials," J. Compos. Mater., 1,200. McLAuor~IN, P.V., Jr., and BATTER~N, S.C., "Limit Behaviour of Fibrous Materials," Int. J. Solids Structures, 6, 1357. TVRRELLROCKAFELLAR, R., "Convex Analysis," Princeton University Press, Princeton, N.J. TsAI, S.W., and Wu, E.M., "A General Theory of Strength for Anisotropic Materials," J. Compos. Mater., 5, 58. McLAoGrtLIN, P.V., "Plastic Limit Behaviour and Failure of Filament Reinforced Materials," Int. J. Solids Structures, 8, 1299. Wu, E.M., "Phenomenological Anisotropic Failure Criterion," in B R O W , L.J., and KROCK, R.H. (eds.), Composite Materials, Vol. II, Academic Press, New York. HASmN,Z., "Failure Criteria for Unidirectional Fiber Composites," Trans. ASME, J. Appl. Mech., 47, 329. HULL,D., "An Introduction to Composite Materials," Cambridge Univ. Press. HASmN,Z., "Analysis of Composite M a t e r i a l s - A Survey," Trans. ASME, J. Appl. Mech., 50, 481. SALENqON,J., "Calcul ~ la Rupture et Analyse Limite," Presses de I'E.N.P.C., Paris. SUQtrET, P., "Plasticit6 et Homog6n6isation," Th~se d'Etat, Universit6 Pierre et Marie Curie, Paris. SUQUET, P., "Analyse Limite et Homog6n6isation," C.R. Acad. Sci., Paris, 296, S6rie II, 1355. BOErILER,J.P.,and RACLr~, J., "Failure Criteria for Glass-Fiber Reinforced Composites Under Confining Pressure," J. Struct. Mech., 13, 371. MARIOO,J.P., ML~,LO~, P., MICHEL, J.P., and SUQtr~T, P., "Plasticit6 et Homog~n6isation: Un Example de Pr~vision des Charges Limites d'une Structure H6t&og~ne P&iodique," J. M6c. Th. Appl., 6, 47.
762
1987 1988 1989 1991 1991
A. TALIERCIO
TURGEMAN,S., and PASTOR, J., "Comparaison des Charges Limites d'une Structure H6t6rog6ne et Homog6n6is6e", J. M6c. Th. Appl., 6, 121. DE BUI-IAN,P., and TAtIERCIO, A., "Crit6re de R6sistance Macroscopique pour les Mat6riaux Composites h Fibres," C.R. Acad. Sci., Paris, 307, S6rie If, 227. TALIERCIO,A., "Study of the Elastic and Ultimate Behaviour of Fiber Composite Materials and Composite Structural Elements" (in Italian), Ph.D. Thesis, Politecnico di Milano. DE BUI-IAN,P., SALENfON,J., and TALIERCIO,A., "Lower and Upper Bounds Estimates for the Macroscopic Strength Criterion of Fiber Composite Materials," in DVORAK,G.J. (ed.), Inelastic Deformation of Composite Materials, Springer-Verlag, New York, p. 563. DE BUHAN,P., and TALIERClO,A., "A Homogenization Approach to the Yield Strength of Composite Materials," Eur. J. Mech., A/Solids, 10, 129.
Dipartimento di Ingegneria Strutturale Politecnico di Milano Piazza Leonardo da Vinci, 32 20133 Milano, Italy
(Received 6 April 1991; in final revised form 17 September 1991 )