- Email: [email protected]
Rheological characterization of anisotropic materials
Rheological characterization of anisotropic materials T. G. ROGERS (University of Nottingham, UK) Constitutive equations are considered to describe the behaviour of anisotropic materials produced by continuous reinforcement of a matrix with one or two families of relatively stiff fibres. In particular, the shear response of such materials is investigated, and it is shown how torsional tests on thin plate specimens can be used to directly determine the transverse and longitudinal shear moduli for elastic,
viscoelastic and elastic-plastic behaviours. Key words: shear response; torsional tests; anisotropic materials; thin plate specimens; fibre-reinforced material
The anisotropic nature of the mechanical response of fibre-reinforced composites is now well understood. It is also generally recognised that a composite consisting of an isotropic matrix reinforced in one or two directions by families of continuous fibres is not just anisotropic but highly anisotropic, with the extensional modulus in the fibre-directions being much greater than any other extensional or shear moduli. What appears not to be generally appreciated is that such high anisotropy in fact increases, rather than decreases, the importance of accurate evaluation of the shear moduli. By the same token, the need for accurate determination of the extensional modulus in a fibredirection decreases as its value relative to the shear moduli increases. These are natural consequences of the fact that in any deformation such a composite, if it can, will opt for the 'soft' options of shearing along and/or transverse to the fibres rather than extend in the much stiffer fibre-directions. /
Unfortunately, very little direct and accurate measurement of shear moduli has been reported in the 14iterature. A disproportionate amount of the available data on mechanical response has been determined from uniaxial tests, mainly in the fibre-directions. Whilst values of material parameters associated with shearing mechanisms can be deduced indirectly from 'off-axis' uniaxial tests, by their very nature they are usually subject to large relative experimental error. However, some recent experimental work 1' 2 carried out at the ICI Petrochemicals and Plastics Division has suggested that cyclic torsion of thin specimens could give significant direct information on the shear moduli of unidirectionally reinforced composites. At the same time, theoretical analyses 3-6 have confirmed that such torsion testing does appear to provide a direct and
accurate method of measuring the important shear moduli of such materials. Since the mechanical response is strongly influenced by the fibre-directions, it is sensible to formulate the constitutive equations in terms of intrinsic, 'preferred' directions which are defined by the fibre-orientations at any point in the composite. Such formulations also have the advantage of being independent of choice of reference axes. The characteristic material parameters are then independent of whether the fibres are straight or curved, and will naturally reflect the relative stiffness in different directions and for different shear mechanisms. In practice, these 'intrinsic' characterizations also highlight the marked differences in the responses of such composites when compared with those of isotropic materials. In the next section we outline the constitutive equations relevant to elastic, viscoelastic and plastic behaviours. The subsequent sections treat the stress analysis relevant for the torsion of thin plate specimens of such materials, with the fibre-reinforcement all lying within the plane of the plate. For elastic and viscoelastic composites, the analysis demonstrates how the shear moduli can be determined by using non-circular specimens of different shapes, when the axis of twist passes through the centre of mass of the specimen. The same analysis also shows that deliberate off-centering of specimens of the same shape provides an equally effective means of obtaining the required moduli. The elastic-plastic analysis is much more complicated. Nevertheless, for non-hardening materials the theory gives explicit expressions for the applied torque in terms of the two initial yield moduli for simple
0010-4361/89/010021-07 $3.00©1989 Butterworth & Co (Publishers) Ltd COMPOSITES. VOLUME 20. NUMBER 1. JANUARY 1989
21
shearing, for Mises and Tresca types of yield function. However, only the Tresca material gives a sufficiently simple relation in order to easily determine the yield moduli.
CONSTITUTIVE EQUATIONS FOR FIBREREINFORCED COMPOSITES The composite is treated as a continuum, so that the fibre-reinforcement is considered as being continuously distributed throughout the material. The local fibredirections are denoted by the unit vector a for unidirectional reinforcement and by unit vectors a and b for bidirectional reinforcement (as in balanced angle-ply and cross-ply composites). In the latter case we say that a and b are 'mechanically equivalent' if the response is unaltered when a and b are interchanged. The fibre directions can change with position x; a and b are constant only when the fibre reinforcement is straight.
Whilst the microstructure obviously dictates the directions of a and b, the continuum approach precludes any further micromechanics, so that detailed consideration of the interactions between individual fibres and the matrix is not possible. Nor are the relations between the mechanical properties of the composite and those of its constituent materials considered. The formulations which are described in this section have been derived and treated in much greater detail elsewhere. 7-10 The notation is standard, with all vector and tensor components referred to a system of rectangular Cartesian axes with position coordinates x~ (i = 1,2,3); the usual repeated index summation convention is used whenever necesary, and 6q represents the Kroneker delta (unity when i = j, zero otherwise). Components of the displacement u and velocity v are denoted by u~ and v~respectively, and those of the infinitesimal strain tensor e and rate-of strain d by @ and dij, with: (1)
The stress components are denoted by o~jand the components of the fibre-directions are a; and b~. It is also convenient to introduce the angle 29 between the fibre directions, so that: cos 2 9 = a.b = aib i
(2)
Other convenient quantities are the unit tensor I (with components 6ij) and the dyadic products A, B and C, where: A B
= bb
C = 1/ffab+ab) (a.b) with components Aij = aiay
Bq = bibj
22
o = (L tr e + a trAe)l + 2~re + 2([XL--IXT)(Ae+eA)
+ (oc tr e + 13tr Ae)A (4 Here tr e and tr Ae, for example, are the traces of e and Ae, with: tr e
= eii
tr Ae
= Aijeij = aiajei]
The component form of Equation (4) is: Oij = (~ekk-I-OCakalekl)Si]
+ 2~reij +2(~tL--IxT) (aiakeky+ayakeki) + (oCekk+13akalekt)azaj
(5)
The simplest form of Equations (4) and (5) is given when one of the coordinate directions, x~ for example, is chosen to coincide locally with the fibre-direction. In this case a = (1,0,0) and Equation (5) reduces to the usual form n of transverse isotropy with respect to the xl-axis:
o,1] [Cll c12 0
I
022/ 033 | = 023 [ O31 | O12 J
c12 C12 0 0 0
c22 C23 0 0 0
c23 C22 0 0 0
0 0 ] 0 0 0 0 0 0 0 2~r 0 0 2~tL 0 0 0 2~tL
[e,,] ] e22 [ [ e33 [ | e23 | (6) [ e31 [ L e12_]
where: Cll = ~. Jr- 2or + 4~,L -- 2~T + 13
c22 = k + 2~T C23 ~---~,
Of the five independent elastic constants involved, it is~ apparent (from Equation (6), for example) that ~L and, may be interpreted as the elastic moduli for simple shear along and transverse to the fibre-direction a; the I remaining constants ~, a and 13(or the equivalent @) i can be related 7' 8 to the Poisson ratios and extensional moduli of the material. Isotropic behaviour is described by two independent constants K and Ix = ~tL = IxTwith
~=13=0.
=aa--a~a
Cij = Vz(aibj+ ajbi)akbk
The simplest case is that of a linearly elastic composite • in which the reinforcement is provided everywhere by a single family of fibres. The constitutive equation is therefore that of a transversely isotropic elastic material with the response being rotationally symmetri, with respect to the fibre-direction a. A convenient, intrinsic form is:
C12 = ~ + 0¢
@ = 1/2(Oui/Oxj-l-~uj/~xi) dij : 1/2( ~vi/~xj-J-~vj/~xi)
Linearly elastic behaviour
(3)
Two important approximations which have been very useful in obtaining solutions in stress analysis problems involving fibre-reinforced materials are incompressibility and fibre-inextensibility. Whilst the former (kinematic) constraint is the more familiar one often being used in fluid mechanics and finite elasticity,, for example - the latter is at least as relevant for the deformation of fibre-reinforced components. In both instances the equations take a simpler form. COMPOSITES. JANUARY 1989
Assuming incompressibility means that volume change (tr ¢) is always zero and that the stress contains an arbitrary hydrostatic pressure term - p I , with Equation (4) replaced by: o = - p l + 2~t-re + 2(l~L--IJ~r) (Ae+eA) t
(7)
+ 13A tr Ae
so the number of independent elastic constants is reduced to three. Analogously, inextensibility in the fibre-direction means that: aiajeO = tr Ae = 0
(8)
and the stress contains an arbitrary tension T in the fibre direction, with Equation (4) replaced by: o = TA + k / t r e + 2~a-e + 2(~ti --~XT) (Ae+eA)
(9)
and number of independent constants is again three. If both incompressibility and fibre-inextensibility are assumed, then: o = - p l + TA + 2p~Te + 2(~tL--~tT) (Ae+eA)
(10)
There are now just two independent elastic constants ~r and P~L,and any stress analysis of boundary value problems would then require values of just these two quantities. Such a model has been termed 7 an ideal fibre-reinforced material', and many useful solutions have been obtained by adopting it. For a material with two fibre-directions a and b the stress-strain relations take the form: o = 2~e
+ 2~1 (Ae+eA) + 2 ~ (Be+eB) + 2~3 (Ce+eC)
(11)
orthogonal, then the material is orthotropic with its behaviour characterized by only the nine constants, X, ~t, I~1, ~2, ch, or2,131, [32,133. When a and b are not orthogonal but are mechanically equivalent (so that Equation (11) is unchanged when a and b are interchanged) then: 134 = 135
and again the behaviour is orthotropic. Putting a = (1,0,0) and b = (0,1,0) into Equation (11) gives the bonventional form ~ for orthotropic linear elasticity:
Ecllc12c13° I11 ° °][e111 0
0
0
033 / =
C13 c23 c33
0
0
0
023 | o31 ] o12 _1
0 0 0
0 0 0
0 2~tT 0 0 0 0 2~tL 0 0 0 0 21xi
e22 e33 e23 e31 e12
(12)
~L = ~ + ~tl,
~I =
COMPOSITES. JANUARY 1989
The relevant constitutive equations for isothermal viscoelasticity are a simple generalization 12 of the elastic equations, in which algebraic products of moduli and strains are replaced by convolutions of the relaxation moduli and the strain-rates; hence a term such as 2~e in Equation (14) is replaced by: gt(t-z)d(x,x) dr
~ + ~h + We
~t(t-x) de(x,z)
(15)
T=--oo
Such equations are applicable for problems involving small strain history e(x,x), - oo < ~ < t; they also require the assumption 12 that the relaxation modulus tensor is symmetric (otherwise more independent moduli are required). In these circumstances, the 'correspondence principle' of viscoelasticity11 usually holds, so that elastic solutions can be readily extended to give the solutions to the equivalent problem in viscoelasticity. The reaction stress terms involving p, Ta and Tb are unaffected, apart from being dependent on the time t as well as on position. The special case of elastic behaviour is obviously recovered by making the relaxation moduli constant. The equally special case of an anisotropic viscous fluid is obtained by replacing the moduli with impulse functions: ~t(t) = TIr(t)
~,a = ,~a
(16)
where 6(0 is the Dirac delta function and vl is the appropriate viscosity. Thus an incompressible transversely isotropic fluid may be characterized by: o = - p l + 2~1~/+ 2(~lL--~l"r) (Ad+dA)
where: ~l" = ~ -It- ~2,
Linearly viscoelastic response and viscous composites
d
with thirteen independent constants. When a and b are
c12 C22 c23
where ~t, ~1 and ~t3 are expressible in terms of ~tL, ~t-r and ~tl.
= 2~
+ (o~3tr e + 134trAe + ~5 trBe)C
022 [
(14)
d
+ (oc2tr e + 133trAe + ~2trBe + 135tr Ce)B
~1 = 132'
o = 2bte + 2~q (Ae+eA+Be+eB) + 2~t3 (Ce+eC) - p l + TaA + Tt,B
2~t*d ~ 2 /
+ (0¢ltre + ~1 trAe + 133trBe + 134tr Ce)A
0el = (X2'
If such a material is incompressible, it can be shown 8 that the number of independent elastic constants is reduced from nine to six and a reaction stress - p ! is added. If fibre-inextensibility is assumed (so that tr Ae = 0 = tr Be) then the number is reduced to five and two reaction stresses T~4 and TbB are added. If the material is both incompressible and fibre-inextensible, then there are only three independent elastic constants and Equation (11) reduces to:
l
+ (~. tr e + oq trAe + o~2trBe + ~3 tr Ce)l
~1 ~---~t2,
and cii (Lj = 1,2,3) are again simply related to the nine independent elastic constants. A similar form is obtained when the axes are chosen to coincide with the bisectors of the two fibre-directions when a and b are mechanically equivalent. We see from Equation (12) that ~L and 0x represent the moduli for shear along and transverse to the planes containing the fibres, while [xi is the in-plane shear modulus.
(13)
+ vA t r a a
(17) 23
Plastic and elastic-plastic response Plastic behaviour of a composite can occur when irreversible deformations are produced during loading, with a permanent change of shape being found even when the material is completely unloaded again. This anelastic response may be distinguished from viscoelastic irreversibility by the additional property of negligible dependence on 'real' time in quasi-static situations. Indeed, in plasticity theory the 'time' t represents any convenient parameter which determines the sequence of events as the deformation develops. Central to most plasticity theories is the existence of a yield criterion, together with an associated flow rule. These assume that a scalar yield functionf(o) exists, such that only stress states for which: f(o) <~ 1 (18) are admissible. I f f = 1 and the material time derivative D f / D t = 0, the material is in a plastic state; if f < 1, or if f = 1 and D f / D t < 0, then reversible elastic deformation is taking place. The associated flow rule relates the plastic strain-rate d p (= d - a ~) to f according to: 0
elastic loading or unloading
(19)
plastic state where ~. is a positive scalar multiplier, possibly dependent on the history of irreversible deformation. The elastic strain-rate d e is related to the stress-rate through the relevant elasticity equations. Most plasticity theories also assume plastic incompressibility so that tr dp = 0. For plastic deformations of fibre-reinforced composites, just as for elastic and viscoelastic behaviours, it is convenient to incorporate the fibredirections a and/or b into the form of f so that the yield criterion is formulated independently of any particular choice of axes of reference. The general theory of such formulations may be found elsewhere 7-1° but in this paper only the simplest are considered. =
~.~f/~Oi]
For transversely isotropic materials, with just one strong fibre direction a, the most general smooth quadratic form for fwhich is unaffected by the superposition of an arbitrary hydrostatic stress is: 2x /2 242 fM(O) = --~-x+--~L + y2
(20)
Here the stress invariants 21,/2 and/4 are defined (for reasons of convenience) 1°' 11 by /1 = 1/2 tr s 2 - tr A s 2 22 = trAs 2 /4 = 1/2 tr o'
(21)
where s and the deviatoric stress tensor o' are given by: s o'
= o - 1/2(tr o - t r A o ) I + l/2(tr o - 3 trAo)A =
o
-
g3(tr 0)2
In Equation (20), k-r and ke correspond to the yield stress in simple shear transverse to and along the
24
(22)
fibre-direction a, and are analogous to t~r and ktL in the elastic formulation; Y denotes the yield stress for simple tension in the fibre-direction, and is typically much greater than the shear yield stresses. This yield function is another form of Hill's generalization 13 of the yon Mises criterion for isotropic materials. A piecewise smooth quadratic form that corresponds to a generalization of Tresca's yield criterion is:
,1... A(o) =
I2/ ,
1221 1241 Y 1211 4 , 1241 Y (23)
2 /r2,
1Ii1<- g l241<-
This form effectively states that yield occurs when the component, in either the transverse direction or the longitudinal direction, of the shear stress on planes containing the fibres reaches a critical value, or that the tensile stress in the fibre-direction does. These forms (20) and (23) are of course extreme simplifications of the most general forms possible; n however, comparison with experimental data on initial! yield (ie, failure data) has shown 7 a certain gratifying agreement for both forms. In practice, parameters such as kT, kL and Y (and any others introduced for more complicated forms off) can, and often will, depend significantly on the strain history. The consequential added complexity in stress analysis should then be a considerable deterrent to the formulation of complicated forms offi Corresponding results are readily obtained for orthotropic composites. The yield function then depends on the two fibre-directions a and b in addition' to the stress field. The Mises-type form involves six yield parameters when a and b are mechanically equivalent or when they are orthogonal; when fibreinextensibility is incorporated, this number reduces to three, with the relevant moduli expressible 8' 9 in terms ; of the yield stresses for simple shear on and in the surfaces containing the two fibre families. When the material first yields, the yield criterionf = 1 assumes the role of a failure criterion, and the yield moduli are failure constants characteristic of the I particular composite involved. As loading is increased, the composite usually 'hardens' (especially when the t matrix is metallic) in that it becomes more resistant to continued yielding; this is reflected by the yield parameters increasing in a way that is dependent on the history (in 'plastic' time) of loading. To take this into account, most descriptions of hardening in anisotropic materials are generalizations of theories proposed for isotropic materials. Accordingly they fall within two I types of hardening, in both of which the form of the : initial yield function f is maintained, whilst being adapted to incorporate the varying deformation history. One of these is analogous to the 'isotropic' hardening rule 13 for isotropic plasticity, whilst the other corresponds to 'kinematic' hardening which takes some account of the plastic hysteresis that is often observed in cyclic loading conditions. These characterizations, together with alternate proposals, may be found elsewhere (as reviewed, for example, in Refs 9 and 10); unfortunately they lead to very complicated analysis. In
COMPOSITES. JANUARY 1989
practice, it is usual to assume 'perfectly plastic', or non-hardening, behaviour, so that the yield parameters are constant throughout any deformation. Such a property can result in useful analytical solutions being obtained, and can give meaningful predictions of the I~haviour of components in real-life problems. The torsiow of thin plate specimens provides an illustrative example of this.
apart from 'edge' effects localized to the edge of the specimen. If no 'squeezing' takes place, so that the displacement normal to the plane of the platens is zero, then the displacement field for small deformations is described by U 1 = -0X2X3 U 2 ~ 0X1X3,
u3 = 0
Torsion of thin plate specimens The experimental work ~' 2 considered at the begining of this paper involved high frequency cyclic torsion of thin plate specimens of unidirectional APe. The fibrereinforcement all lay in the plane of the plate. The specimens were clamped between two rigid platens and twisted about an axis perpendicular to its faces (Fig. 1). Accurate measurements of the applied twisting couple M and the angle of twist could be obtained, from which shear characteristics of the composite were deduced. Initially the specimens were discs of circular crosssection, with the centres coincident with the axis of ~otation. Since such a configuration does not allow a distinction to be made between isotropic and anisotropic responses, it was anticipated that further data would be needed and obtained by an independent test. However, the operating temperatures meant that the material behaviour was not purely elastic, but contained a substantial viscous element, rendering impracticable the standard extensional tests relevant to solids. Subsequent analyses of the problem 3-6 have shown how two simple modifications of the experimental method can obviate the need to perform such alternative tests. The first modification is to use a number of non-circular specimens of different shapes, such as rectangular plates of different aspect ratios, which are 'centred' such that the axis of twist passes through the mass-centre of the test specimen. The other is to use the same shape in each test but to deliberately 'off-centre' the specimens. The analyses are based on the observation that because the plate specimens are thin, and constrained by attachment to the rigid platens, then the deformation must be essentially pure torsion with zero warping,
(24)
Here, the coordinate axes are as shown in Fig. 1, with the origin on the axis of rotation and with the fibres parallel to the xl-axis. For composites with two-family fibre-reinforcement, the Xl- and x2-axes are chosen to coincide either with the two fibre-directions a and b if they are orthogonal, or with the bisectors 1/2(a+b)if a and b are not orthogonal but are mechanically equivalent; for definiteness, the former case is assumed for the remainder of the paper, so that: a = (1,0,0)
b = (0,1,0)
(25)
The angle 0 is the angle of twist per unit thickness of specimen. If 0 varies with time, as in cyclic loading or due to stress relaxation, then the velocity field is given by: V 1 ~-" --OXEX 3 V2 ---- 0 X I X 3
v3 = 0 where 0 denotes the time derivative of 0(t). The strain and strain-rate components are then: e l 3 = --1//20x2 ,
e23 =
(26)
1/20x1,
e l 1 ~__ e22 = e33 = e l 2 m_ 0 d13 =
--1/20X2,
d23 =
--1/20X 1
dll = d22 = d 3 3 = da2 = 0 (27) Straightforward substitution of Equation (27) into the constitutive equations described in the previous section shows that in all cases the only non-zero stress components are o13 and o23, satisfying the only non-trivial equilibrium equation: ~013/~X 1 + ~ O 2 3 / ~ X 2 = 0
(28)
assuming negligible inertial effects in the thin specimen. The applied torque M about the xa-axis is then given by:
x.
M=ff
x 2
bo
(X1023--X2013) dA
(29)
where 5e denotes the surface of the plate.
ELASTIC TORSION - ~ x
~x 1
1
M
It is straightforward to show 3' 6 that, for elastic response, Equation (29) leads to the crucial momentangle relation: M
b Fig. 1 Schematic diagram of torsion test: (a) physical set-up; (b) plan view of specimen
COMPOSITES. JANUARY 1989
=
(~LI1 -'{'-/J,TI2)0
(30)
where 11 and 12 are the second moments of area (or equivalently the moments of inertia) about the x l - and x2-axes respectively. This relation makes it clear that if 25
I1 = 12, as for a circular or square disc positioned centrally at the origin, then only data for ~tL+ ~T c a n be obtained. However, it is also clear from Equation (30) that ~tL and [~r can be determined separately if the moments of inertia are varied in specific ways. Thus I1 and 12 can be changed by altering the shape of the specimen and/or by varying its position relative to O. For a centred rectangular specimen of length 2L in the fibre-direction and 2H in the transverse direction, Equation (30) gives: M=
n3 LI-I3
[,2] + ~-~r
0
(31)
So by determining M/O for various aspect ratios L/H, a plot of 3M/(4LH3-O) against L2/H 2 should give a straight line with slope ~r and extrapolated value [~Las the intercept as L/H--> O. If it is not convenient or practicable to prepare specimens of different aspect ratios, or if it is advantageous to use the same specimen for a number of tests, then off-centering provides an alternative method of determining ~L and [XT.If the centre of mass G is not coincident with O, but is at a distance p from O with OG making an angle ot with the fibre-direction a (Fig. 1), then the following is obtained: 6 M = {~tL/1 + ILT/2 q- (I£L sin 2 0¢+ ~tT COS2 00132S}0(32) or equivalently: M = {P,L[1 + IXTI2"]'- 1/2(~L"{'-~I'T)O2S -- 1/2(~,L--IkT)pZS COS 20¢}0
(33)
Here [1 and [2 are the second moments of area about the relevant axes through G (not O), and S is the surface area of the specimen. Equation (33) shows that if a series of tests were carried out using the same shape and off-centre position of G, but with different orientations 0c, then a plot of M/O versus cos 2oc should give a straight line with slope proportional to I]J,T--I~L, and with the intercept of o¢ = 1/4n giving another independent relation between ~L and ~tT. Alternatively Equation (32) shows how keeping the orientation fixed but varying the distance p also gives a straight line plot between M/O and p2S; the constant slope ([LLsin 2 oc + ~T COS2 0~) and intercept IttLll + I£TI2 a t 13 = 0 again provide two simultaneous equations from which [XLand ~r may be determined.
By comparison with Equation (30), it is seen that all the results of Equations (31)-(33) again hold, except that ' now ~tL and I~r are time-dependent functions. Accordingly, the data can be analysed by following the same procedures as those described for elastic behaviour. In the more standard forced vibration test,l' 2 the material is subjected to a sinusoidally varying torque of frequency f~ and amplitude M0(f2), and the twist response 00(if2) and phase lag &(fl) are measured. Equation (34) shows that these quantities are related through: M0 cos 6 = (ItlxL+I21x~)00 Mo sin 6 = (IllxL'+I2PJr)Oo
where ~t[, ~t-~and ~t[', I~r'are the storage and loss moduli respectively. Again, comparison of Equation (36) with ! the form of Equation (30) shows how the subsequent results (31)-(33) may be reinterpreted for this test. : Thus M is replaced by M0 cos 6 for determining IX~.and IX~-,and by M0 sin 6 for Ix['and IX~'.
ELASTIC-PLASTIC TORSION If the deformation is elastic-plastic, the analysis is more complicated, even for the simplest yield functions given in the second section. For continued loading, a complete analysis has been obtained 5 using both fM and fT. Furthermore, for the Tresca-type behaviour (characterized by fT), the solution has also been determined 5 for the remaining portions of a cyclic test, namely unloading and reloading, in which reverse yielding can occur. The following gives only a brief outline of the relevant results for loading. As the applied torque M increases from zero, the specimen first deforms elastically until the yield function attains its critical value (here unity) at some point. Further increase in M then results in further increase in 0, which can thus take the role of 'plastic' time as discussed previously, so during loading 0 = 1, and similarly during unloading we have 0 = - 1. With o13 and 023 being the only non-zero stress components, the stress invariants take the simple forms: I 1 = 0"23,
VISCOELASTIC SPECIMENS The principles outlined previously (second section) for extending the elastic theory to include viscoelastic effects leads to Equation (30) being replaced by: 6 M(0 = / 1 D L * 0 + I212T • 0
(34)
where ~tL(0 and ~tT(t) are the time-dependent relaxation moduli of the composite.
26
12 = O~13,
(379
/4 = 0
Hence for the Mises criterionfM ~< 1, yielding first occurs when 0 = 0c, say, at those points on the boundary of the specimen for which the elastic stresses' satisfy Equations (20) and (37), with: -5 ac
+
27 = 1 bc
where
For a static torsion test, in which the test specimen is subjected to an initial twist 00 which is then held constant, Equation (34) shows that the torque relaxes according to
M(t) = { I, btL(O+ I2P,T(t) }O0
(36
(35)
a¢=
--~,bc =
--ff
(38)
As 0 increases with increasing M, so do both ac and bc decrease, so that the elastic-plastic boundary is a COMPOSITES. JANUARY 1989
contracting ellipse Fc described by Equation (38). If b~e denotes the portion of the plate which is within Fc and hence still deforming elastically, and 9pp is the remainder of the plate, then the analysis shows that for non-hardening materials the torque is given by: M =0
ff o (39)
+f
So for any given shape of specimen, a straightforward (if tedious) computation gives M as a function of kL and kT, assuming ~XLand ~tT are now known from the elastic analysis. For Tresca-type behaviour given byfT ~< 1 the loading analysis is even simpler, leading to the elastic-plastic boundary F¢ taking the form of a rectangle with the elastic region confined to: Ix, I ~
Ix21 ~
(40)
It is a simple matter to compute the torque, 5 and the results for a centred rectangular specimen (assuming without loss of generality that H~tL/kL > L~r/kT) are: 4/3LH3(~L +~tTL 2/H2)O, kL
O~O~--
H~L 2LH2kL + 4/3L3HIXL0 - ~3L k_~ 0_2, M=
_kL _~<0~< H~tL
2 L H ( L k T + HkL)
kT L~tT
-2/311-1k3 + L
k___L<~ 0 L~T
02,
Thus, as for ~L and lxT, it is possible that kL and kT may be obtained from a series of torsion experiments on a set of specimens of differing aspect ratios L/H.
ACKNOWLEDGEMENT Much of this work was obtained as a result of a research programme for an ICI joint research scheme (JRS), supported by a joint SERC/ICI Cooperative Research Grant. This support is gratefully acknowledged.
REFERENCES 1 Groves,D. ICI Wilton Materials Research Centre, private communication (1987) 2 Groves, D. 'A characterisation of shear flow in continuousfibre thermoplastic laminates' Composites 19 No 6 (1988)pp 3 Kaprielian, P. V. The measurement of the shear moduli of fibre-reinforcedmaterials by torsion tests' JRS 1/86, Dept Theor Mech, Universityof Nottingham (1986) 4 Kapdelian, P. V. and Rogers, T. G. 'The measurement of the shear moduli of viscoelasticfibre-reinforcedmaterials' JRS 2/86, Dept Theor Mech, Universityof Nottingham (1986) 5 Kaprielian, P. V. and Rogers, T. G. 'The measurement of the shear moduli of elastic-plasticfibre-reinforcedmaterials' JRS 1/87, Dept Theor Mech, Universtiyof Nottingham (1987) 6 Rogers, T. G. 'Off-centre torsion testing of fibre-reinforced materials' JRS 5/87, Dept Theor Mech, Universityof Nottingham (1987) 7 Spencer,A. J. M. 'Deformation of Fibre-reinforced Materials' (Clarendon Press, Oxford, 1972) 8 Spencer,A. J. M., et al. 'Continuum theory of the mechanicsof fibre-reinforcedcomposites' CISM Courses and Lectures No. 282 (Springer-Verlag, Wien - New York, 1984) 9 Spencer,A. J. M. 'Yield conditions and hardening rules for fibre-reinforcedmaterials with plastic response' In: Failure Criteria of Structured Media, edited by J. P. Boehler, Colloque International du CNRS, No. 351, Genoble (1983) (in press) 10 Rogers,T. G. 'Yield criteria, flowrules and hardening in anisotropic plasticity', IUTAM/ICM Symposiumon Yielding, Damage and Failureof Anisotropic Solids, Villard-de-Lans, 1988 (in press) 11 Christensen, R. M. 'Mechanics of Composite Materials' (John Wiley & Sons, New'York, 1979) 12 Rogers,T. G. and Pipkin, A. C. 'Asymmetric relaxation and compliance matrices in linear viscoelasticity'ZAMP 14 (1963) pp 334-343 13 Hill,R. "TheMathematical Theory of Plasticity' (Clarendon Press, Oxford, 1950)
(41)
A U THOR It is noted that as 0 increases, M asymptotes to the constant value M~, where M® = 2LHZ(kL +kTL/H)
COMPOSITES. JANUARY 1989
(42)
T. G. Rogers is R e a d e r in Theoretical Mechanics in the D e p a r t m e n t of Theoretical Mechanics, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
27
viscoelastic and elastic-plastic behaviours. Key words: shear response; torsional tests; anisotropic materials; thin plate specimens; fibre-reinforced material
The anisotropic nature of the mechanical response of fibre-reinforced composites is now well understood. It is also generally recognised that a composite consisting of an isotropic matrix reinforced in one or two directions by families of continuous fibres is not just anisotropic but highly anisotropic, with the extensional modulus in the fibre-directions being much greater than any other extensional or shear moduli. What appears not to be generally appreciated is that such high anisotropy in fact increases, rather than decreases, the importance of accurate evaluation of the shear moduli. By the same token, the need for accurate determination of the extensional modulus in a fibredirection decreases as its value relative to the shear moduli increases. These are natural consequences of the fact that in any deformation such a composite, if it can, will opt for the 'soft' options of shearing along and/or transverse to the fibres rather than extend in the much stiffer fibre-directions. /
Unfortunately, very little direct and accurate measurement of shear moduli has been reported in the 14iterature. A disproportionate amount of the available data on mechanical response has been determined from uniaxial tests, mainly in the fibre-directions. Whilst values of material parameters associated with shearing mechanisms can be deduced indirectly from 'off-axis' uniaxial tests, by their very nature they are usually subject to large relative experimental error. However, some recent experimental work 1' 2 carried out at the ICI Petrochemicals and Plastics Division has suggested that cyclic torsion of thin specimens could give significant direct information on the shear moduli of unidirectionally reinforced composites. At the same time, theoretical analyses 3-6 have confirmed that such torsion testing does appear to provide a direct and
accurate method of measuring the important shear moduli of such materials. Since the mechanical response is strongly influenced by the fibre-directions, it is sensible to formulate the constitutive equations in terms of intrinsic, 'preferred' directions which are defined by the fibre-orientations at any point in the composite. Such formulations also have the advantage of being independent of choice of reference axes. The characteristic material parameters are then independent of whether the fibres are straight or curved, and will naturally reflect the relative stiffness in different directions and for different shear mechanisms. In practice, these 'intrinsic' characterizations also highlight the marked differences in the responses of such composites when compared with those of isotropic materials. In the next section we outline the constitutive equations relevant to elastic, viscoelastic and plastic behaviours. The subsequent sections treat the stress analysis relevant for the torsion of thin plate specimens of such materials, with the fibre-reinforcement all lying within the plane of the plate. For elastic and viscoelastic composites, the analysis demonstrates how the shear moduli can be determined by using non-circular specimens of different shapes, when the axis of twist passes through the centre of mass of the specimen. The same analysis also shows that deliberate off-centering of specimens of the same shape provides an equally effective means of obtaining the required moduli. The elastic-plastic analysis is much more complicated. Nevertheless, for non-hardening materials the theory gives explicit expressions for the applied torque in terms of the two initial yield moduli for simple
0010-4361/89/010021-07 $3.00©1989 Butterworth & Co (Publishers) Ltd COMPOSITES. VOLUME 20. NUMBER 1. JANUARY 1989
21
shearing, for Mises and Tresca types of yield function. However, only the Tresca material gives a sufficiently simple relation in order to easily determine the yield moduli.
CONSTITUTIVE EQUATIONS FOR FIBREREINFORCED COMPOSITES The composite is treated as a continuum, so that the fibre-reinforcement is considered as being continuously distributed throughout the material. The local fibredirections are denoted by the unit vector a for unidirectional reinforcement and by unit vectors a and b for bidirectional reinforcement (as in balanced angle-ply and cross-ply composites). In the latter case we say that a and b are 'mechanically equivalent' if the response is unaltered when a and b are interchanged. The fibre directions can change with position x; a and b are constant only when the fibre reinforcement is straight.
Whilst the microstructure obviously dictates the directions of a and b, the continuum approach precludes any further micromechanics, so that detailed consideration of the interactions between individual fibres and the matrix is not possible. Nor are the relations between the mechanical properties of the composite and those of its constituent materials considered. The formulations which are described in this section have been derived and treated in much greater detail elsewhere. 7-10 The notation is standard, with all vector and tensor components referred to a system of rectangular Cartesian axes with position coordinates x~ (i = 1,2,3); the usual repeated index summation convention is used whenever necesary, and 6q represents the Kroneker delta (unity when i = j, zero otherwise). Components of the displacement u and velocity v are denoted by u~ and v~respectively, and those of the infinitesimal strain tensor e and rate-of strain d by @ and dij, with: (1)
The stress components are denoted by o~jand the components of the fibre-directions are a; and b~. It is also convenient to introduce the angle 29 between the fibre directions, so that: cos 2 9 = a.b = aib i
(2)
Other convenient quantities are the unit tensor I (with components 6ij) and the dyadic products A, B and C, where: A B
= bb
C = 1/ffab+ab) (a.b) with components Aij = aiay
Bq = bibj
22
o = (L tr e + a trAe)l + 2~re + 2([XL--IXT)(Ae+eA)
+ (oc tr e + 13tr Ae)A (4 Here tr e and tr Ae, for example, are the traces of e and Ae, with: tr e
= eii
tr Ae
= Aijeij = aiajei]
The component form of Equation (4) is: Oij = (~ekk-I-OCakalekl)Si]
+ 2~reij +2(~tL--IxT) (aiakeky+ayakeki) + (oCekk+13akalekt)azaj
(5)
The simplest form of Equations (4) and (5) is given when one of the coordinate directions, x~ for example, is chosen to coincide locally with the fibre-direction. In this case a = (1,0,0) and Equation (5) reduces to the usual form n of transverse isotropy with respect to the xl-axis:
o,1] [Cll c12 0
I
022/ 033 | = 023 [ O31 | O12 J
c12 C12 0 0 0
c22 C23 0 0 0
c23 C22 0 0 0
0 0 ] 0 0 0 0 0 0 0 2~r 0 0 2~tL 0 0 0 2~tL
[e,,] ] e22 [ [ e33 [ | e23 | (6) [ e31 [ L e12_]
where: Cll = ~. Jr- 2or + 4~,L -- 2~T + 13
c22 = k + 2~T C23 ~---~,
Of the five independent elastic constants involved, it is~ apparent (from Equation (6), for example) that ~L and, may be interpreted as the elastic moduli for simple shear along and transverse to the fibre-direction a; the I remaining constants ~, a and 13(or the equivalent @) i can be related 7' 8 to the Poisson ratios and extensional moduli of the material. Isotropic behaviour is described by two independent constants K and Ix = ~tL = IxTwith
~=13=0.
=aa--a~a
Cij = Vz(aibj+ ajbi)akbk
The simplest case is that of a linearly elastic composite • in which the reinforcement is provided everywhere by a single family of fibres. The constitutive equation is therefore that of a transversely isotropic elastic material with the response being rotationally symmetri, with respect to the fibre-direction a. A convenient, intrinsic form is:
C12 = ~ + 0¢
@ = 1/2(Oui/Oxj-l-~uj/~xi) dij : 1/2( ~vi/~xj-J-~vj/~xi)
Linearly elastic behaviour
(3)
Two important approximations which have been very useful in obtaining solutions in stress analysis problems involving fibre-reinforced materials are incompressibility and fibre-inextensibility. Whilst the former (kinematic) constraint is the more familiar one often being used in fluid mechanics and finite elasticity,, for example - the latter is at least as relevant for the deformation of fibre-reinforced components. In both instances the equations take a simpler form. COMPOSITES. JANUARY 1989
Assuming incompressibility means that volume change (tr ¢) is always zero and that the stress contains an arbitrary hydrostatic pressure term - p I , with Equation (4) replaced by: o = - p l + 2~t-re + 2(l~L--IJ~r) (Ae+eA) t
(7)
+ 13A tr Ae
so the number of independent elastic constants is reduced to three. Analogously, inextensibility in the fibre-direction means that: aiajeO = tr Ae = 0
(8)
and the stress contains an arbitrary tension T in the fibre direction, with Equation (4) replaced by: o = TA + k / t r e + 2~a-e + 2(~ti --~XT) (Ae+eA)
(9)
and number of independent constants is again three. If both incompressibility and fibre-inextensibility are assumed, then: o = - p l + TA + 2p~Te + 2(~tL--~tT) (Ae+eA)
(10)
There are now just two independent elastic constants ~r and P~L,and any stress analysis of boundary value problems would then require values of just these two quantities. Such a model has been termed 7 an ideal fibre-reinforced material', and many useful solutions have been obtained by adopting it. For a material with two fibre-directions a and b the stress-strain relations take the form: o = 2~e
+ 2~1 (Ae+eA) + 2 ~ (Be+eB) + 2~3 (Ce+eC)
(11)
orthogonal, then the material is orthotropic with its behaviour characterized by only the nine constants, X, ~t, I~1, ~2, ch, or2,131, [32,133. When a and b are not orthogonal but are mechanically equivalent (so that Equation (11) is unchanged when a and b are interchanged) then: 134 = 135
and again the behaviour is orthotropic. Putting a = (1,0,0) and b = (0,1,0) into Equation (11) gives the bonventional form ~ for orthotropic linear elasticity:
Ecllc12c13° I11 ° °][e111 0
0
0
033 / =
C13 c23 c33
0
0
0
023 | o31 ] o12 _1
0 0 0
0 0 0
0 2~tT 0 0 0 0 2~tL 0 0 0 0 21xi
e22 e33 e23 e31 e12
(12)
~L = ~ + ~tl,
~I =
COMPOSITES. JANUARY 1989
The relevant constitutive equations for isothermal viscoelasticity are a simple generalization 12 of the elastic equations, in which algebraic products of moduli and strains are replaced by convolutions of the relaxation moduli and the strain-rates; hence a term such as 2~e in Equation (14) is replaced by: gt(t-z)d(x,x) dr
~ + ~h + We
~t(t-x) de(x,z)
(15)
T=--oo
Such equations are applicable for problems involving small strain history e(x,x), - oo < ~ < t; they also require the assumption 12 that the relaxation modulus tensor is symmetric (otherwise more independent moduli are required). In these circumstances, the 'correspondence principle' of viscoelasticity11 usually holds, so that elastic solutions can be readily extended to give the solutions to the equivalent problem in viscoelasticity. The reaction stress terms involving p, Ta and Tb are unaffected, apart from being dependent on the time t as well as on position. The special case of elastic behaviour is obviously recovered by making the relaxation moduli constant. The equally special case of an anisotropic viscous fluid is obtained by replacing the moduli with impulse functions: ~t(t) = TIr(t)
~,a = ,~a
(16)
where 6(0 is the Dirac delta function and vl is the appropriate viscosity. Thus an incompressible transversely isotropic fluid may be characterized by: o = - p l + 2~1~/+ 2(~lL--~l"r) (Ad+dA)
where: ~l" = ~ -It- ~2,
Linearly viscoelastic response and viscous composites
d
with thirteen independent constants. When a and b are
c12 C22 c23
where ~t, ~1 and ~t3 are expressible in terms of ~tL, ~t-r and ~tl.
= 2~
+ (o~3tr e + 134trAe + ~5 trBe)C
022 [
(14)
d
+ (oc2tr e + 133trAe + ~2trBe + 135tr Ce)B
~1 = 132'
o = 2bte + 2~q (Ae+eA+Be+eB) + 2~t3 (Ce+eC) - p l + TaA + Tt,B
2~t*d ~ 2 /
+ (0¢ltre + ~1 trAe + 133trBe + 134tr Ce)A
0el = (X2'
If such a material is incompressible, it can be shown 8 that the number of independent elastic constants is reduced from nine to six and a reaction stress - p ! is added. If fibre-inextensibility is assumed (so that tr Ae = 0 = tr Be) then the number is reduced to five and two reaction stresses T~4 and TbB are added. If the material is both incompressible and fibre-inextensible, then there are only three independent elastic constants and Equation (11) reduces to:
l
+ (~. tr e + oq trAe + o~2trBe + ~3 tr Ce)l
~1 ~---~t2,
and cii (Lj = 1,2,3) are again simply related to the nine independent elastic constants. A similar form is obtained when the axes are chosen to coincide with the bisectors of the two fibre-directions when a and b are mechanically equivalent. We see from Equation (12) that ~L and 0x represent the moduli for shear along and transverse to the planes containing the fibres, while [xi is the in-plane shear modulus.
(13)
+ vA t r a a
(17) 23
Plastic and elastic-plastic response Plastic behaviour of a composite can occur when irreversible deformations are produced during loading, with a permanent change of shape being found even when the material is completely unloaded again. This anelastic response may be distinguished from viscoelastic irreversibility by the additional property of negligible dependence on 'real' time in quasi-static situations. Indeed, in plasticity theory the 'time' t represents any convenient parameter which determines the sequence of events as the deformation develops. Central to most plasticity theories is the existence of a yield criterion, together with an associated flow rule. These assume that a scalar yield functionf(o) exists, such that only stress states for which: f(o) <~ 1 (18) are admissible. I f f = 1 and the material time derivative D f / D t = 0, the material is in a plastic state; if f < 1, or if f = 1 and D f / D t < 0, then reversible elastic deformation is taking place. The associated flow rule relates the plastic strain-rate d p (= d - a ~) to f according to: 0
elastic loading or unloading
(19)
plastic state where ~. is a positive scalar multiplier, possibly dependent on the history of irreversible deformation. The elastic strain-rate d e is related to the stress-rate through the relevant elasticity equations. Most plasticity theories also assume plastic incompressibility so that tr dp = 0. For plastic deformations of fibre-reinforced composites, just as for elastic and viscoelastic behaviours, it is convenient to incorporate the fibredirections a and/or b into the form of f so that the yield criterion is formulated independently of any particular choice of axes of reference. The general theory of such formulations may be found elsewhere 7-1° but in this paper only the simplest are considered. =
~.~f/~Oi]
For transversely isotropic materials, with just one strong fibre direction a, the most general smooth quadratic form for fwhich is unaffected by the superposition of an arbitrary hydrostatic stress is: 2x /2 242 fM(O) = --~-x+--~L + y2
(20)
Here the stress invariants 21,/2 and/4 are defined (for reasons of convenience) 1°' 11 by /1 = 1/2 tr s 2 - tr A s 2 22 = trAs 2 /4 = 1/2 tr o'
(21)
where s and the deviatoric stress tensor o' are given by: s o'
= o - 1/2(tr o - t r A o ) I + l/2(tr o - 3 trAo)A =
o
-
g3(tr 0)2
In Equation (20), k-r and ke correspond to the yield stress in simple shear transverse to and along the
24
(22)
fibre-direction a, and are analogous to t~r and ktL in the elastic formulation; Y denotes the yield stress for simple tension in the fibre-direction, and is typically much greater than the shear yield stresses. This yield function is another form of Hill's generalization 13 of the yon Mises criterion for isotropic materials. A piecewise smooth quadratic form that corresponds to a generalization of Tresca's yield criterion is:
,1... A(o) =
I2/ ,
1221 1241 Y 1211 4 , 1241 Y (23)
2 /r2,
1Ii1<- g l241<-
This form effectively states that yield occurs when the component, in either the transverse direction or the longitudinal direction, of the shear stress on planes containing the fibres reaches a critical value, or that the tensile stress in the fibre-direction does. These forms (20) and (23) are of course extreme simplifications of the most general forms possible; n however, comparison with experimental data on initial! yield (ie, failure data) has shown 7 a certain gratifying agreement for both forms. In practice, parameters such as kT, kL and Y (and any others introduced for more complicated forms off) can, and often will, depend significantly on the strain history. The consequential added complexity in stress analysis should then be a considerable deterrent to the formulation of complicated forms offi Corresponding results are readily obtained for orthotropic composites. The yield function then depends on the two fibre-directions a and b in addition' to the stress field. The Mises-type form involves six yield parameters when a and b are mechanically equivalent or when they are orthogonal; when fibreinextensibility is incorporated, this number reduces to three, with the relevant moduli expressible 8' 9 in terms ; of the yield stresses for simple shear on and in the surfaces containing the two fibre families. When the material first yields, the yield criterionf = 1 assumes the role of a failure criterion, and the yield moduli are failure constants characteristic of the I particular composite involved. As loading is increased, the composite usually 'hardens' (especially when the t matrix is metallic) in that it becomes more resistant to continued yielding; this is reflected by the yield parameters increasing in a way that is dependent on the history (in 'plastic' time) of loading. To take this into account, most descriptions of hardening in anisotropic materials are generalizations of theories proposed for isotropic materials. Accordingly they fall within two I types of hardening, in both of which the form of the : initial yield function f is maintained, whilst being adapted to incorporate the varying deformation history. One of these is analogous to the 'isotropic' hardening rule 13 for isotropic plasticity, whilst the other corresponds to 'kinematic' hardening which takes some account of the plastic hysteresis that is often observed in cyclic loading conditions. These characterizations, together with alternate proposals, may be found elsewhere (as reviewed, for example, in Refs 9 and 10); unfortunately they lead to very complicated analysis. In
COMPOSITES. JANUARY 1989
practice, it is usual to assume 'perfectly plastic', or non-hardening, behaviour, so that the yield parameters are constant throughout any deformation. Such a property can result in useful analytical solutions being obtained, and can give meaningful predictions of the I~haviour of components in real-life problems. The torsiow of thin plate specimens provides an illustrative example of this.
apart from 'edge' effects localized to the edge of the specimen. If no 'squeezing' takes place, so that the displacement normal to the plane of the platens is zero, then the displacement field for small deformations is described by U 1 = -0X2X3 U 2 ~ 0X1X3,
u3 = 0
Torsion of thin plate specimens The experimental work ~' 2 considered at the begining of this paper involved high frequency cyclic torsion of thin plate specimens of unidirectional APe. The fibrereinforcement all lay in the plane of the plate. The specimens were clamped between two rigid platens and twisted about an axis perpendicular to its faces (Fig. 1). Accurate measurements of the applied twisting couple M and the angle of twist could be obtained, from which shear characteristics of the composite were deduced. Initially the specimens were discs of circular crosssection, with the centres coincident with the axis of ~otation. Since such a configuration does not allow a distinction to be made between isotropic and anisotropic responses, it was anticipated that further data would be needed and obtained by an independent test. However, the operating temperatures meant that the material behaviour was not purely elastic, but contained a substantial viscous element, rendering impracticable the standard extensional tests relevant to solids. Subsequent analyses of the problem 3-6 have shown how two simple modifications of the experimental method can obviate the need to perform such alternative tests. The first modification is to use a number of non-circular specimens of different shapes, such as rectangular plates of different aspect ratios, which are 'centred' such that the axis of twist passes through the mass-centre of the test specimen. The other is to use the same shape in each test but to deliberately 'off-centre' the specimens. The analyses are based on the observation that because the plate specimens are thin, and constrained by attachment to the rigid platens, then the deformation must be essentially pure torsion with zero warping,
(24)
Here, the coordinate axes are as shown in Fig. 1, with the origin on the axis of rotation and with the fibres parallel to the xl-axis. For composites with two-family fibre-reinforcement, the Xl- and x2-axes are chosen to coincide either with the two fibre-directions a and b if they are orthogonal, or with the bisectors 1/2(a+b)if a and b are not orthogonal but are mechanically equivalent; for definiteness, the former case is assumed for the remainder of the paper, so that: a = (1,0,0)
b = (0,1,0)
(25)
The angle 0 is the angle of twist per unit thickness of specimen. If 0 varies with time, as in cyclic loading or due to stress relaxation, then the velocity field is given by: V 1 ~-" --OXEX 3 V2 ---- 0 X I X 3
v3 = 0 where 0 denotes the time derivative of 0(t). The strain and strain-rate components are then: e l 3 = --1//20x2 ,
e23 =
(26)
1/20x1,
e l 1 ~__ e22 = e33 = e l 2 m_ 0 d13 =
--1/20X2,
d23 =
--1/20X 1
dll = d22 = d 3 3 = da2 = 0 (27) Straightforward substitution of Equation (27) into the constitutive equations described in the previous section shows that in all cases the only non-zero stress components are o13 and o23, satisfying the only non-trivial equilibrium equation: ~013/~X 1 + ~ O 2 3 / ~ X 2 = 0
(28)
assuming negligible inertial effects in the thin specimen. The applied torque M about the xa-axis is then given by:
x.
M=ff
x 2
bo
(X1023--X2013) dA
(29)
where 5e denotes the surface of the plate.
ELASTIC TORSION - ~ x
~x 1
1
M
It is straightforward to show 3' 6 that, for elastic response, Equation (29) leads to the crucial momentangle relation: M
b Fig. 1 Schematic diagram of torsion test: (a) physical set-up; (b) plan view of specimen
COMPOSITES. JANUARY 1989
=
(~LI1 -'{'-/J,TI2)0
(30)
where 11 and 12 are the second moments of area (or equivalently the moments of inertia) about the x l - and x2-axes respectively. This relation makes it clear that if 25
I1 = 12, as for a circular or square disc positioned centrally at the origin, then only data for ~tL+ ~T c a n be obtained. However, it is also clear from Equation (30) that ~tL and [~r can be determined separately if the moments of inertia are varied in specific ways. Thus I1 and 12 can be changed by altering the shape of the specimen and/or by varying its position relative to O. For a centred rectangular specimen of length 2L in the fibre-direction and 2H in the transverse direction, Equation (30) gives: M=
n3 LI-I3
[,2] + ~-~r
0
(31)
So by determining M/O for various aspect ratios L/H, a plot of 3M/(4LH3-O) against L2/H 2 should give a straight line with slope ~r and extrapolated value [~Las the intercept as L/H--> O. If it is not convenient or practicable to prepare specimens of different aspect ratios, or if it is advantageous to use the same specimen for a number of tests, then off-centering provides an alternative method of determining ~L and [XT.If the centre of mass G is not coincident with O, but is at a distance p from O with OG making an angle ot with the fibre-direction a (Fig. 1), then the following is obtained: 6 M = {~tL/1 + ILT/2 q- (I£L sin 2 0¢+ ~tT COS2 00132S}0(32) or equivalently: M = {P,L[1 + IXTI2"]'- 1/2(~L"{'-~I'T)O2S -- 1/2(~,L--IkT)pZS COS 20¢}0
(33)
Here [1 and [2 are the second moments of area about the relevant axes through G (not O), and S is the surface area of the specimen. Equation (33) shows that if a series of tests were carried out using the same shape and off-centre position of G, but with different orientations 0c, then a plot of M/O versus cos 2oc should give a straight line with slope proportional to I]J,T--I~L, and with the intercept of o¢ = 1/4n giving another independent relation between ~L and ~tT. Alternatively Equation (32) shows how keeping the orientation fixed but varying the distance p also gives a straight line plot between M/O and p2S; the constant slope ([LLsin 2 oc + ~T COS2 0~) and intercept IttLll + I£TI2 a t 13 = 0 again provide two simultaneous equations from which [XLand ~r may be determined.
By comparison with Equation (30), it is seen that all the results of Equations (31)-(33) again hold, except that ' now ~tL and I~r are time-dependent functions. Accordingly, the data can be analysed by following the same procedures as those described for elastic behaviour. In the more standard forced vibration test,l' 2 the material is subjected to a sinusoidally varying torque of frequency f~ and amplitude M0(f2), and the twist response 00(if2) and phase lag &(fl) are measured. Equation (34) shows that these quantities are related through: M0 cos 6 = (ItlxL+I21x~)00 Mo sin 6 = (IllxL'+I2PJr)Oo
where ~t[, ~t-~and ~t[', I~r'are the storage and loss moduli respectively. Again, comparison of Equation (36) with ! the form of Equation (30) shows how the subsequent results (31)-(33) may be reinterpreted for this test. : Thus M is replaced by M0 cos 6 for determining IX~.and IX~-,and by M0 sin 6 for Ix['and IX~'.
ELASTIC-PLASTIC TORSION If the deformation is elastic-plastic, the analysis is more complicated, even for the simplest yield functions given in the second section. For continued loading, a complete analysis has been obtained 5 using both fM and fT. Furthermore, for the Tresca-type behaviour (characterized by fT), the solution has also been determined 5 for the remaining portions of a cyclic test, namely unloading and reloading, in which reverse yielding can occur. The following gives only a brief outline of the relevant results for loading. As the applied torque M increases from zero, the specimen first deforms elastically until the yield function attains its critical value (here unity) at some point. Further increase in M then results in further increase in 0, which can thus take the role of 'plastic' time as discussed previously, so during loading 0 = 1, and similarly during unloading we have 0 = - 1. With o13 and 023 being the only non-zero stress components, the stress invariants take the simple forms: I 1 = 0"23,
VISCOELASTIC SPECIMENS The principles outlined previously (second section) for extending the elastic theory to include viscoelastic effects leads to Equation (30) being replaced by: 6 M(0 = / 1 D L * 0 + I212T • 0
(34)
where ~tL(0 and ~tT(t) are the time-dependent relaxation moduli of the composite.
26
12 = O~13,
(379
/4 = 0
Hence for the Mises criterionfM ~< 1, yielding first occurs when 0 = 0c, say, at those points on the boundary of the specimen for which the elastic stresses' satisfy Equations (20) and (37), with: -5 ac
+
27 = 1 bc
where
For a static torsion test, in which the test specimen is subjected to an initial twist 00 which is then held constant, Equation (34) shows that the torque relaxes according to
M(t) = { I, btL(O+ I2P,T(t) }O0
(36
(35)
a¢=
--~,bc =
--ff
(38)
As 0 increases with increasing M, so do both ac and bc decrease, so that the elastic-plastic boundary is a COMPOSITES. JANUARY 1989
contracting ellipse Fc described by Equation (38). If b~e denotes the portion of the plate which is within Fc and hence still deforming elastically, and 9pp is the remainder of the plate, then the analysis shows that for non-hardening materials the torque is given by: M =0
ff o (39)
+f
So for any given shape of specimen, a straightforward (if tedious) computation gives M as a function of kL and kT, assuming ~XLand ~tT are now known from the elastic analysis. For Tresca-type behaviour given byfT ~< 1 the loading analysis is even simpler, leading to the elastic-plastic boundary F¢ taking the form of a rectangle with the elastic region confined to: Ix, I ~
Ix21 ~
(40)
It is a simple matter to compute the torque, 5 and the results for a centred rectangular specimen (assuming without loss of generality that H~tL/kL > L~r/kT) are: 4/3LH3(~L +~tTL 2/H2)O, kL
O~O~--
H~L 2LH2kL + 4/3L3HIXL0 - ~3L k_~ 0_2, M=
_kL _~<0~< H~tL
2 L H ( L k T + HkL)
kT L~tT
-2/311-1k3 + L
k___L<~ 0 L~T
02,
Thus, as for ~L and lxT, it is possible that kL and kT may be obtained from a series of torsion experiments on a set of specimens of differing aspect ratios L/H.
ACKNOWLEDGEMENT Much of this work was obtained as a result of a research programme for an ICI joint research scheme (JRS), supported by a joint SERC/ICI Cooperative Research Grant. This support is gratefully acknowledged.
REFERENCES 1 Groves,D. ICI Wilton Materials Research Centre, private communication (1987) 2 Groves, D. 'A characterisation of shear flow in continuousfibre thermoplastic laminates' Composites 19 No 6 (1988)pp 3 Kaprielian, P. V. The measurement of the shear moduli of fibre-reinforcedmaterials by torsion tests' JRS 1/86, Dept Theor Mech, Universityof Nottingham (1986) 4 Kapdelian, P. V. and Rogers, T. G. 'The measurement of the shear moduli of viscoelasticfibre-reinforcedmaterials' JRS 2/86, Dept Theor Mech, Universityof Nottingham (1986) 5 Kaprielian, P. V. and Rogers, T. G. 'The measurement of the shear moduli of elastic-plasticfibre-reinforcedmaterials' JRS 1/87, Dept Theor Mech, Universtiyof Nottingham (1987) 6 Rogers, T. G. 'Off-centre torsion testing of fibre-reinforced materials' JRS 5/87, Dept Theor Mech, Universityof Nottingham (1987) 7 Spencer,A. J. M. 'Deformation of Fibre-reinforced Materials' (Clarendon Press, Oxford, 1972) 8 Spencer,A. J. M., et al. 'Continuum theory of the mechanicsof fibre-reinforcedcomposites' CISM Courses and Lectures No. 282 (Springer-Verlag, Wien - New York, 1984) 9 Spencer,A. J. M. 'Yield conditions and hardening rules for fibre-reinforcedmaterials with plastic response' In: Failure Criteria of Structured Media, edited by J. P. Boehler, Colloque International du CNRS, No. 351, Genoble (1983) (in press) 10 Rogers,T. G. 'Yield criteria, flowrules and hardening in anisotropic plasticity', IUTAM/ICM Symposiumon Yielding, Damage and Failureof Anisotropic Solids, Villard-de-Lans, 1988 (in press) 11 Christensen, R. M. 'Mechanics of Composite Materials' (John Wiley & Sons, New'York, 1979) 12 Rogers,T. G. and Pipkin, A. C. 'Asymmetric relaxation and compliance matrices in linear viscoelasticity'ZAMP 14 (1963) pp 334-343 13 Hill,R. "TheMathematical Theory of Plasticity' (Clarendon Press, Oxford, 1950)
(41)
A U THOR It is noted that as 0 increases, M asymptotes to the constant value M~, where M® = 2LHZ(kL +kTL/H)
COMPOSITES. JANUARY 1989
(42)
T. G. Rogers is R e a d e r in Theoretical Mechanics in the D e p a r t m e n t of Theoretical Mechanics, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
27