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Thermomechanical behaviour of composite materials and structures under high temperatures: 2. Structures
Part A 28A (1997) 463-47 I ‘(I‘1997 Elsevier Science Limited
Composites
Printed in Great Britain. All rights reserved 1359-835x/97/$17.00
PII:S1359-835X(96)00145-5
ELSEVIER
Thermomechanical behaviour of composite materials and structures under high temperatures: 2. Structures
Yu. I. Dimitrienko ‘NPO
Mashinostroeniya’ R&D Industrial Corporation, region, 743952 Russia (Received 16 April 1996; revised 4 October 1996)
Gagarina
33a, Reutov,
Moscow
This paper is devoted to modelling of the thermomechanical behaviour of composite structures under hightemperature heating up to the temperature of charring of the composite. Attention is focused on the peculiarities of the behaviour of composite shells under high temperatures. It is also shown that the internal heat-mass transfer of pyrolysis gases has an essential effect on the stress state of the shell under high temperatures. Transverse stresses, which are usually neglected for composite shells, under high temperatures are indicated to be the most dangerous. These stresses can cause delamination of composite shells. A numerical analysis of the conditions of delamination for composite shells is conducted, and computed results are compared with experimental data on the thermomechanical destruction of cylindrical composite shells. 0 1997 Elsevier Science Limited (Keywords: high temperature; charring; pore pressure; delamination; loss of stability)
INTRODUCTION
The behaviour of composite structures of the thin-walled shell type under high temperatures (from 300 to lOOO”C, i.e. at temperatures higher than the temperature Bs of glass formation) differs in principle from the behaviour of the structures under relatively low temperatures (approximately lOO- 150°C). Under normal temperatures tangential stresses oI1, c722and g12 are, as a rule, the most important in thin-walled composite plates and shells (if the 0q3 axis is oriented along the normal to the shell surface) and sometimes considerable interlayer shear stresses 0~3 and ~~23 also occur. Transverse stresses 033 can be neglected in almost all cases owing to their smallness’. High temperatures lead to the appearance and development of intensive internal physico-chemical pyrolysis processes and charring in the composite matrix 2.3, which in turn leads to the formation of pores in the composite. The pores are filled with gaseous products of pyrolysis and the gas pressure p inside them can be very great (approximately 200atm). Moreover, the pore gas pressure under fast, non-stationary heating is distributed very non-uniformly throughout the composite shell thickness, leading to the appearance of important transverse stresses, ~7~~. These stresses prove to be so high that, under certain conditions, they can cause delamination of the composite shell. This phenomenon was investigated theoretically in refs 4 and 5.
The aim of the present paper is to elucidate the conditions for which delamination appears in composite shells. In addition, the overcritical behaviour of composite shells after the appearance of the first delaminations in them is investigated. As was established in experiments, under certain conditions the part of the shell that delaminates under high-temperature heating can lose its stability. Another dangerous phenomenon under hightemperature heating is cracking of the heated surface of the structure due to chemical shrinkage of the composite during pyrolysis6’7. Investigation of the thermomechanical conditions of cracking and stability loss is conducted in the paper.
PECULIARITIES OF THE THEORY ON COMPOSITE SHELLS UNDER HIGH TEMPERATURES Let us consider now the peculiarities that appear in thinwalled shell structures of composites under high temperatures when pyrolysis of the polymer matrix occurs. Main
assumptions
Let us consider a thin-walled shell of orthotropic composite with thickness h, where the line q3 coincides with the normal to the surface, -/l/2 5 q3 5 h/2, and ql
463
Thermomechanical behaviour under high temperature. 2: Yu. I. Dimitrienko
and q2 coincide with the lines of main curvature of the shell surface and with principal orthotropy axes. Then the following expressions can be given for the Lamet’s parameters: H3 = 1
H, = A,(1 +k,qs)
8H --E
= A,
= A,k,
as3
Strains and stresses in a shell On substituting expressions (2) into the kinematic relationship between strains ecrg and displacements u, given, for example, in ref. 1, the expression for the strains eaP in the shell is derived: cap = e,p +
E33 = 0
q36,fl
1,2
a,p=
~~3 = ea3
(1)
where A,(q) are coefficients of the first square form of the reduced surface (q3 = 0); k, are its main curvatures, and k,q3 ec 1. At present there exists a great number of theories for calculation of the stress-strain state of thin-walled shells. For laminated composite materials, Timoshenko’s shell theory’ is the simplest and sufficiently acceptable. However, this theory and also its analogies cannot be applied to shells at high temperatures because they do not consider the most dangerous delaminating stresses c33 arising from internal gas generation in the laminated composite under such conditions. In order to take account of this effect it is necessary to formulate a new system of hypotheses for a shell theory. Let us assume the following system of hypotheses.
(7)
Strains eafl, curvatures ~~~ of the middle surface and shear strain ecu3are determined by the formulae: e
1 dU 1 dA --Q+k,W eaa = A, & + AIA2 dqo
K
1) Distributions of the displacements u,, u3 through the shell thickness are described by three functions Ua(qp), W(qp) depending on coordinates qp, ,0 = 1,2: %I = UCk+
u3 =
437a
w
(2)
of shear oa3 and normal g33 stresses are chosen in the form of square functions of the coordinate q3: =
--
1 &I
aa= A,
g33=LpgP-P-+(P--P+)
q3 (T+;)+
1c, _!Yz=-
ea3
-h/2
s h/2
+
vwsl
kpp
-
$4
dq3
h/2
s h/2 -h/2
%a02(c33
-
i3)@3
(5)
-h/2
s h/2
wo3dq3
=
G+3,a+3
h’2
-h/2
1,2;
p =
as1 vGpq$%3
j
=
0,l
$1 + h’2 ae2cPs#dq3 J -h/2 1
wg2Ea3dq3
cl=
(10) h/2
h/2
Pg, =- (p _h,2 cpspfdq3
sf
ygl
s
=
_h,2fvw&~
h/2
pdq3
Similarly to the above, on substituting distribution (4) for stress c33 and (7) for strains E,, into the integral equation (5), we obtain a relationship connecting functions $J~to eflp and K~~. From this we can find:
J -h/2
c33
k= 1,2
J -h/2
-h/2
2 C3p /3=1
h/2
+
=
p=
_h12f3wdq3
X
h’2 (Psa&dq3 s -h/2
(4)
where p* are pressures at the surfaces q3 = fh/2 of and v,b3(q,) are new unknown the shell and $Jq,) functions of coordinates qb, p = 1,2. 3) The following integral relations are considered instead of constitutive relations (the second, fourth and fifth equations of (18) in Part 19):
5
(9)
Herein and below, the following notation for values averaged over the thickness of the shell is used:
o(i) ED
Q= 1,2
= -
cY= 1,2
h &+s,u+3’E)
6
(-$+a)$
cps
wdq3
a= I,2
(3)
G+3,o+3
h/2
1 W-t
dq, + Al A2 dq, ”
ai;) =
cps
a= 1,2
On substituting distributions (8) for stresses goa3and (7) for strains E,~ into integral equations (6), we obtain relationships for functions $J,,in terms of Ye, U, and W:
2) Distributions
ffa3
1 dW --+-ya-k,Ua a3 = A, dq,
1,2
Thermomechanical
If functions Ye, U, and W are known, then all the functions $,, qa and & can be found by formulae (9) and (ll), and, hence, stresses on3 and 033 can be determined by formulae (3) and (4). Stresses (T,,~in the shell layers are determined by the constitutive relations (10) from Part 19, whereinto strains (7) should be substituted:
behaviour
under high temperature.
Constitutive relations for a shell under high temperatures
On substituting expressions ( 12) into the equations of forces and moments (14), we obtain:
‘PST12
g12
=
&lC66(el2
(13)
+ q3&12)
Equilibrium equations
Equilibrium equations averaged over the thickness of the shell are the following:
2: Yu. I. Dimitrienko
A4412
=
N&l2
=
C66e12
+ 06642
+ N66n12
(FsQ,l = Ccr+3.,r+3en3
Equations (15) are constitutive relations for a composite shell under high temperatures. Here Cnrj, NaIj and D,,,j are elasticity coefficients of a composite shell: Co/J = C&7$)
D,,, = C’,,ga;’
N,,,j = Gab?
(16) G+3.u+3
=
o!,,H
G+3.n+3ag)
=
I,2
for heat stresses ?‘a and heat moments M,: -AIAz’Ps(klTll
+kzT22) +
d&Q,& +WQz'Ps %71
-A1A2@+
-P-)
-
(kl
Ffi = &
&?2
+k2)A24PgPg
=
0
O=[
(13) Heat-mass
cu,p = 1.2
a#P
where Tap are forces in the shell; Q, are the shear forces; and M,, are moments determined by the formulae:
1
W
Mg = y‘pg .[-h/2
P, =A
apq&3
‘PPs
‘pg =
Ps
=
Pl
+
‘p2 +
‘p3
a,/3
h-2 _h,2 %pdq3
1=
(14)
9s
1,2
Equilibrium equations (13) for composite shells under high temperatures differ from the equations under normal conditions by additional terms, namely by forces P, and moments Mg of the pore pressure of pyrolysis gases and also by the appearance of the multiplier ‘ps at forces Tcrfiand Maa.
c,,,i;y
A& = 2
c,,,#
(17)
,i= I
transfer in a composite shell
Under the action of high temperatures on a composite shell, a large quantity of pores filled with gas is formed in the shell. The gases filter through the pores and partially pass to the surroundings. However, under certain conditions, the gases do not have enough time to filter out, resulting in the accumulation of a great intrapore pressure p of pyrolysis gases inside the composite material that can cause delamination of the shell. To take this phenomenon into consideration, it is necessary to consider both mechanics equations and the equations of internal heat and mass transfer for a composite, these equations being: a kinetic equation describing the change of concentration of the polymer phase during heating; a heat conductivity equation taking into account the heat effect of pyrolysis and cooling the composite at the expense of gas filtration in the pores; and a filtration equation for pyrolysis gases in the pores. Further external heating of the considered composite shell by heat fluxes q * is assumed to be low-changing along the shell surface, and thus the distributions of temperature 8, pore pressure p, gas density p4 and concentrations ‘p4, ‘p2 and ‘p3 can be assumed to be functions only of coordinate q3 and time t. Taking account of these assumptions, the equations of internal heat and mass transfer for the shell are written in
465
Thermomechanical
behaviour
under high temperature.
2: Yu. I. Dimitrienko
the form2:
+RcP+-P-)(a-
1)
Q= 1,2
hip,
where
de ‘“at
1 -d = A,A2 aq,
+
5‘1 =
(Ad2&3g)
ae
q&JG3 -apSe
A~OJ
1 -;[c3,(c22
-
c12)+~32(~ll
-cl,)1
A, = Ci, C22 - Ct2
52 =;+$(-c,,c,,+c32cll)
aq3 aq3
Here p = R,p,6 is the pore pressure of pyrolysis gases, ps is the gas density, Ae” is the heat of matrix pyrolysis, J is the intensity of the pyrolysis process defined in ref. 8 and cs is the gas heat capacity.
CYLINDRICAL
SHELL
Let us consider an orthotropic composite shell as being a cylindrical body with the symmetry axis z. The coordinate surface q3 = 0 is assumed to be coincident with the middle surface of the shell; the meridional arc s and the azimuth angle Q counted off from a certain point MO are chosen as coordinates q1 and q2; R is the radius of the external surface of the composite, r = R - (h/2) + q3 is the current radius of the shell. Derivatives d/dq3 in equations (18) are derivatives with respect to r: a/&. Principal curvatures and coefficients of the first squared form of a cylindrical shell are constants: ki = 0, k2 = R-‘, Al = 1, A2 = R. Under these conditions the system (13)-(15) has the solution: T,,
=Y
T22 =
cps
71 =
0
;
[(P,
-
P-)R + io,P,l
Kql =o
By substituting formula (19) into (15) we can find strains epp. Stresses in the cylindrical shell can be determined by the formulae: a,3
=
0
u12 =
0
Heat and mass transfer equations numerically.
COMPUTED
(18) are solved
RESULTS
The system (18) was solved by using a numerical method with a step-by-step method and schemes of matrix sweeping. The solution of this system is given by functions p,(r, t), t9(r, t) and jo2(r, t), with the help of which we can evaluate the pore gas pressure: ~(r, 4 = R,p,(r,
4e(r, 4
(21)
All computations were conducted for a composite shell with the following geometric parameters: R = 2m, h = 2 x 10e3 m, and with physical characteristics corresponding to those of a composite with a matrix of epoxy resin and glass-textile reinforcement. The characteristics of the glass/epoxy composite are given in Part 19. External and internal pressures are considered to be atmospheric: p- = p+ = 0.1 MPa, and the surface of the glass/epoxy shell is assumed to be heat-insulated. At the external surface q3 = h/2 of the shell a convective heat flux q+ = ~~(0, - 0) was given, and at the internal surface there was a condition of heat insulation: qi = 0, where aT = 0.1 kWme2 K-’ is the heat transfer coefficient and 13, is the temperature of the surroundings (e, = 400°C). The action of high temperature leads to heat propagation into the glass/epoxy shell. Figure la shows the temperature Q(r, t) distribution versus the shell thickness for different times t = 50, 100, 150, 200 and 250 s. Figure lb shows the distribution of pore gas pressure p(r, t) through the shell thickness for the same times. Figures 2a and 2b show distributions of radial 033 and tangential u22 stresses in the shell for different times. A peculiarity of the behaviour of composite shells under high temperatures is the presence of a specific profile of intrapore pressure p (Figure Ib). Intensive pyrolysis of the polymer matrix leads to growth of the pore gas pressure. Only gases near the external surface q3 = h/2 of the shell, where porosity and gas permeability of the composite are maximum, have time to filter out into the surroundings. Inside the shell wall porosity and gas permeability are small and gases accumulate to form a local peak of pore pressure (Figure lb). This peak
Thermomechanical
behaviour
under high temperature.
2: Yu. I. Dimitrienko
I
Ti 0.970
0.976
0.982
0.988
0.994
1.000
0.970
0.976
0.982
0.988
0.994
1.000
(a) w
! A \ /150 \I I/~250
OV 0.970
I 0.976
J/I
xi\I\
I
Jy/.y\\; 0.982
0.988
0.994
1.000
@)
Figure 1 Distributions of temperature 0 (a) and pore pressure p (b) versus thickness of a composite shell for different times t. Initial porosity of the composite (oi = 0.05, symbols on the curves are times (s)
moves towards the internal cold surface of the shell with time. The peak of pore pressure leads to the appearance of a local peak of tensile normal transverse (through-thickness) stresses ~7~~in the shell (Figure 2~2)that also moves with time towards the internal surface of the shell. These stresses are the most dangerous for laminated textile composite shells, as there appears a chance of their delamination. Tangential stresses u2? in the cylindrical composite shell under high temperatures have two local extrema: one in the domain of compressive stresses and another in the domain of tensile stresses (Figure 2b). The peak of the compressive stresses is connected with the presence of a peak of pore pressure, that also moves with time towards the cold internal surface. The peak of tensile tangential stresses is caused by shrinkage of the composite at temperatures of the matrix pyrolysis. As follows from formula (20) the tensile shrinkage stresses are: C?2 = -C&2 M E&. These stresses are localized near the most heated external surface q3 = h/2 of the shell. If the level of these stresses is sufficiently high and exceeds the strength limit in tension 02r, then longitudinal cracks can appear in the cylindrical shell that are directed along the axis Oyr of the shell (see Figure 6a below). The maximum values of the pore pressure p and also of the transverse g33 and tangential a22 stresses depend on the initial porosity of the composite, cpi. The higher the value of initial porosity cpi, the more intensive is
Figure 2 Distributions of transverse stresses q3 (a) and tangential stresses oz2 (b) versus thickness of a composite shell for different times r. Initial porosity of the composite ~2 = 0.05. symbols on the curves are times (s)
filtration of gases into the surroundings, and the lower the values of max p and max cr33. Figure 3-5 shows dependences of max p, max 022 and max 033 on heating time t for three values of cpi = 0.05, 0.07 and 0.1 (i.e. 5, 7 and 10% respectively). As can be seen, the kinetics of growth of functions max p and max cr33for all three values of cpi is the same: a sharp initial
0.6
0.0 0
50
100
150
200
t.s 250
Figure 3 Dependences of maximum values of functions in a composite shell: transverse stresses max crss, tangential stresses max ~2s. pore pressure max p. damage parameters max .q and max .zs upon heating time r. Initial porosity of the composite 9: = 0.05
467
Thermomechanical
behaviour
under high temperature.
2: Yu. I. Dimitrienko
FRACTURE OF COMPOSITE STRUCTURES UNDER HIGH TEMPERATURES Conditions for the appearance of fracture
0.6
0.2
0.0
6s
0
50
100
150
200
250
Figure 4 Dependences of maximum values of functions in a composite shell: transverse stresses max q3, tangential stresses max q2, pore pressure max p, damage parameters max z2 and max z3 upon heating time t. Initial porosity of the composite ‘pj = 0.07
As shown above, in composite shells under intensive non-stationary heating to the high pyrolysis temperature of the material, important thermostresses appear: both normal through-thickness 033 and tangential a22 stresses occur even without external mechanical loads. To estimate the danger of these thermostresses (when they reach the ultimate strength of the composite and the composite structure is destroyed) it is convenient to introduce parameters Z, called damage parameters. The parameters z, are determined as ratios of the thermostresses a,, and the corresponding limits of strength cCYT of the composite in tension: Za(q3, t) = lga;b” cT,,
a: = 2,3
(22)
LIT
growth (approximate duration 50 s), at time t = 100-180 s, and finally a the values that is connected with intensive stage of matrix pyrolysis. max oZ2(t) at all stages and for all grow monotonically, behaviour that the increasing content of coke phase surface and, hence, with increasing
then a stabilization sharp reduction of completion of the Tangential stresses three values of cpi is connected with ‘p3 near the heated shrinkage stresses
022.
However, the decreasing initial porosity cpj, the absolute values of max p and max g33 essentially increase (Figures 3-5): for cpt = 0.1 the greatest values of max p and max u33 are 3 and 0.2 MPa; for cpj = 0.07 they are 20 and 1.5 MPa; while for ‘pt = 0.05 they are 25 and 2.3 MPa, respectively. The absolute values of max cr22 show less variation: they are 15, 26 and 50 MPa, respectively, as they are determined only by the degree of charring of the composite matrix cp3 and are independent of the level of pore pressure p.
where )a,, 1is the modulus of stresses o,, . If stresses fraa are absent in the shell (gas = 0) or they are negative (o,,(q3, t) < 0) at a given time t in a zone of the shell surface with coordinate q3, then the corresponding damage parameter z, (q3, t), introduced with the help of (22), is equal to zero within the zone of the shell. If a,,(q3, t) > 0 there, then za(q3, t) > 0. If at a certain time ti within a zone of the shell with coordinate q3 the following condition is satisfied:
C) = 1
_(h,2m;&$&131
Q=
273
(23)
gas 0”
fibres
(4
mination t=t;
0.6
A
gas 0"
q3f .
heatflur t=t
0.4
**
0.2
(cl 0.0 0
50
100
150
zoo
fS 250
Figure 5 Dependences of maximum values of functions in a composite shell: transverse stresses max ass, tangential stresses max oaa, pore pressure max p, damage parameters max z2 and max zj upon heating time t. Initial porosity of the composite +$ = 0.1
468
2
Figure 6 A scheme of three types of fracture of a composite shell under high-temperature heating: (a) cracking of the heated surface of the composite at t = t; and q = 1; (b) delamination of the composite at t = t; and zj = 1; (c) appearance of several delaminations and loss of their stability at t = t,,
Thermomechanical
then fracture of the shell occurs. This means that the corresponding stresses a,, reach the ultimate strength gnT of the composite in tension. As was shown above (see Figure Zh), the maximum tensile stresses ~22 are always near the external surface of the shell r = R (i.e. q3 = h/2). Therefore, if at the certain time t; condition (23) for cr = 2 is satisfied, then during destruction of the shell longitudinal cracks appear on the external surface q3 = h/3 (Figure 6a), as the destruction at Q:= 2 occurs due to tensile tangential stresses (~~2 at the surface q3 = h/2 of the shell. As seen from Figure Za, the stresses 033 can reach their maximum values at any surface with coordinate q;: -h/2 < q: <: h/2. Therefore if at the certain time ti condition (23) for (Y= 3 is satisfied, then the shell is destroyed by delamination-type damage (Figure 6b) and the delamination appears within the shell at a surface with coordinate q3 = q;. If condition (23) is satisfied for both Q = 2 and Q = 3, then both types of fracture of the shell take place. Thus, the damage parameter zLy is a convenient indicator of the danger of thermostresses appearing within the shell during heating: the closer the value of zz or z3 is to 1, the higher the level of thermostresses and the greater the danger of destruction of the shell by type 2 (longitudinal cracking) or type 3 (delamination) fracture, respectively. Below, calculated damage parameters zn will be given for a composite cylindrical shell for the considered case of external heating. Figures 3-5 show the kinetics of the maximum values of damage parameters, max z,,(l), in the composite shell for three values of initial porosity: cpt = 0.05, 0.07 and 0.1. As seen from these figures, the function max z2( t) does not change considerably with & but max z3(t) changes dramatically. This can be explained by the fact that z2 is determined mainly by shrinkage of the composite during pyrolysis, which is independent of porosity cpi, while the parameter z3 on the other hand is defined by internal mass transfer and pore pressure which depend significantly on the porosity of the composite. For cpi = 0.05 and 0.07 the condition max ~1 = I is satisfied earlier than max z2 = 1; this means that delamination of the shell occurs earlier under high temperatures. For ‘py = 0.1 the condition z3 = 1 is not realized at all, i.e. the composite shell is destroyed only as a result of cracking on its external heated surface. Behaviour of a composite of fracture
shell after the appearance
Composite structures are usually examined only for the time interval 0 < t < tz, until the time tz when fracture first appears: for example, until composite delamination appears due to the accumulating intrapore pressure of gaseous pyrolysis products. This restriction is justified for many heat-shield structures because, after delamination has arisen in a piece of hardware having (for example) a plane shape (plates, unclosed shells, panels). it fails and loses completely its exploitational
behaviour
under
high
temperature.
2: Yu. 1. Dimitrienko
thermal-protective properties. However, for some types of structure there exist some exploitational conditions when the hardware still performs its functional protective purpose for some time interval t; I t 2 t,, after delamination has taken place. This situation occurs under external heating of a shell with a closed contour made of textile composite material; e.g. under the action of high temperatures on a cylindrical fuel container made of glass-reinforced plastic. The action of a temperature field uniformly distributed over the surface of the cylindrical shell of the fuel container leads to the formation of ring delaminations having a closed contour. Because the cylindrical shell has a closed contour, it does not fail after the first delamination and continues to perform its functional purposes. Upon further heating new ring delaminations appear. the formation of which is directed from the outer surface of the shell to the inner surface. Pore pressure is accumulated in each ring crack so that the pressure difference pd(rI, t) -pd(ri_, , t) (where i = 1.. N; r, is the radius of ith delamination, ri = R - h/2 + q.Si, where qJi is the coordinate of a surface of the ith delamination), uniformly distributed, acts upon each of the cylindrical layers stripped off. The other non-delaminated part of the cylindrical shell of the fuel container, with thickness hd(t). proves to be the most loaded as it is subjected to the action of the maximum pressure difference pd(rN, t) -p_, where 1~ = rN - R + h. The process of delamination continues until time t,, when, at a certain critical thickness hd( f,,), the inner part of the shell loses its stability under the action of the external pressure difference pd(r,y. t,,) - p_ > 0. After that the structure stops performing its functional purpose and the fuel container fails completely. However. the time interval t,, - r; from the appearance of first delamination until the loss of stability of the fuel container is sufficiently long: t,, - t3 >> t,, therefore the method of stresses and heat-mass transfer calculations developed above for t < t; should be continued for the time interval t; < t _< t,,. The condition for the appearance of delamination in the composite under high temperatures has the form (23) for o. = 3. We will assume that the appearance of delamination does not change the picture of heat-mass transfer in the shell. The critical external pressure p*, for which the loss of stability of the non-delaminated shell section occurs, was obtained in ref. 10 and. for the considered problem, is written as follows: p*(t) = 0.92E,a0,(t)
(24)
where L is the shell length, and the stability condition has the form: Pd(?V?
l) -
Pp
< p*ct)
r;
5
t 5
t,,
(25)
Now,
let us derive the expression for the pressure of gaseous pyrolysis products accumulated within the ring crack. For this, the state equation of a perfect
pd
469
Thermomechanical behaviour under high temperature. 2: Yu. I. Dimitrienko
gas is used: Ed = R,P,&
(26)
where R, is the gas constant. In order to determine the density psd of the gas within the crack, let us apply the equation of gas mass conversion before and after formation of the crack: &dvd
=
t;I\
I
(27)
&.$g
where pg is the gas density in the pores; Vg is the gas volume in the pores coming to the surface of the ring crack: Vg = 27rRLlgcp4
(28)
where 1, is the characteristic diameter of the pores ( 10P4lop3 m); vd is the crack volume opened by the action of pressure pd (29)
R( 1 + e,g)= R
Here ee = IY~/E!~.Q~is the tangential deformation of the shell, and a8 is the tangential stress of the shell, connected to the pressure pd in the crack by the relation*: ge = PdR/hdHaving substituted formula (27) and (28) into (26), we can find the gas density within the crack, &j. Then from (25) we obtain an expression for the pressure of the pyrolysis gases pd within the crack that has the form: Pd(%‘,
d=
(30)
On substituting (24) into (30), we obtain the condition for the absence of destruction in the composite cylindrical
hdlh 1.0
0.6
0
0
Figure 7 Distributions of non-delaminated thickness h,_+of a composite shell wrsus heating time t for different values of initial porosity of the composite & = 0.05, 0.07, 0.08 and 0.1; A are experimental values for (p4 = 0.05
470
0 I
I
0
0.05
/
J
Q.
0.10
Times to the appearance of three types of fracture, ts, t; and
t**’versusinitial porosity pj of the composite
2 81
where R and Rdef are the radii of the delamination surface in the non-opened and opened (by deformation) states respectively: Rdef =
.
Figure 8
Vd = xL(R2 - R&J = 2rL-&$$ d
50
shell by stability loss under high temperatures. If at time t = t,, there is an equality in (30), then there occurs a loss of stability of the internal layer of the composite shell. Computed results Figure 7 shows a graph of the function (hd(t)/h) for the glass/epoxy cylindrical shell with the characteristics given before for four values of cpi = 0.05,0.07, 0.08 and 0.1. For cpj = 0.05, during the first stage of heating t 5 50 s there is no delamination in the shell and (hd/h) = 1. At time t = t; M 50 s the condition (23) is realized for the first time and the first delamination occurs near the external surface of the shell. Figure 6c presents the picture for the appearance of new delaminations; this graph is characterized by the thickness hd(t) of the nondelaminated material. From Figure 7 the dependence (hd( t)/h) is seen to be practically linear. At time t = t,, FZ220 s, when hd/h reaches the value 0.05, the non-delaminated shell thickness becomes so small that the pore pressure pd of gas accumulated in delamination cavities exceeds the stability limit (25) and the internal layer of the composite shell loses its stability. In this case complete failure of the cylindrical composite shell occurs. As seen from the computations conducted, the time interval t,, - t; is equal to approximately 170 s, which exceeds t; by 3 times. Thus, the calculation of internal heat-mass transfer and stresses is necessary up to time t,, as the shell still performs its designated purpose at these times. For a composite shell with initial porosity cpi = 0.07 the first delamination appears at t; M 90 s, and delaminations continue to arise up to t = t,, = 170 s, when a loss of stability of the internal layer of the shell occurs (Figure 7). For cpi = 0.1 delaminations to not appear at all, and (hd/h) = 1 during the whole heating time due to
Thermomechanical
a low level of pore gas pressure pp; hence, there is a low level of normal transverse stresses ~733. Figure 8 shows the dependences of the times for appearance of the three types of failure, 1;, t; and t,,, on the initial porosity of the composite, &. As can be seen from this figure, for cpi 2 0.08 there are no delaminations and loss of stability of the shell during heating (as t; and t,. -+ OO),and there is only the occurrence of cracking due to chemical shrinkage at time t = t; M 150 s. For ‘pj 5 0.08 delamination of the composite shell occurs first at f = t; x 50-100 s, then there is shrinkage cracking of the shell surface at t = t; = 100-l 30 s, and finally there is a loss of stability of the delaminated internal part of the shell at t = t,, M 160-200s. Experimental
results
To test the accuracy of the developed model for calculation of the fracture of composite shells under high temperatures, heat testing was conducted for cylindrical glass/epoxy composite shells with the following geometrical parameters: L = 0.8 m, h/R = 0.03, and the results were compared with the corresponding computed ones. Heating was performed with the aid of infra-red heaters so that the heat flux q+ to the composite shell and the surface temperature 8(R, t) would be the same as the values used in the computations, namely 10 kW m-* and 300°C. The initial porosity cpj of the textile glass/epoxy composite shell, which was manufactured by pressing in a press mould, was approximately 0.04-0.05. During testing, acoustic emission was used to determine the time of delamination onset t; in the shell; it was detected easily by the peak of intensity in the acoustic signal and also by the sharp growth of gas generation from the shell, which was noticeable with the naked eye. The time of stability loss t,, of the internal layer of the shell was also detected easily by the peak in the acoustic signal and also by the typical audible click as well. After testing, the thickness hd of the shell part that lost its stability was measured. Three composite shells with the same parameters were tested. The points in Figure 7 show experimental values of the times t; to the beginning of delamination and t,, to the stability loss, and also the corresponding experimental values of thickness hd at t; and t,,. As can be seen from these graphs, the agreement of experimental and computed results is sufficient to allow us to confirm the adequacy of the model developed for description of the thermomechanical behaviour of composite structures under high temperatures.
behaviour
under high temperature.
2: Yu. I. Dimitrienko
CONCLUSIONS The theory developed for thin-walled composite shells adequately describes the phenomena observed in experiments, namely the appearance of intrapore gas pressure under high-temperature heating and the generation of transverse stresses. The theory allows us also to determine the condition for the appearance of delamination in composite shells under high temperatures and the condition for stability loss of the delaminated composite. These conditions are shown to depend essentially on the initial porosity of composite shells; in particular, for a textile epoxy-based composite with initial material porosity of more than lo%, delamination in the shell does not occur. The theory also enables determination of the time at which shrinkage cracks appear on the composite shell surface heated under high temperatures. It has been shown that the appearance of shrinkage cracks is practically independent of the initial porosity of the composite.
REFERENCES I. 2.
3.
Whitney, J. M., Structural Analysis of Laminated Anisotropic Plates. Technomic, Lancaster, PA, 1987. Dimitrienko, Yu. I., Thermal stresses and heat-mass-transfer in ablating composite materials. ht. J. Heat Mass Transfer, 1995. 38(l), 139-146. Griffis, C. A., Nemes, J. A., Stonesifer, F. R. and Chang, C. I.. Degradation in strength of laminated composites subjected to intense heating and mechanical loading. J. Compos. Mater.. 1986,20,216-235.
4.
5.
6.
I.
Dimitrienko, Yu. I. and Epifanovski, I. S., Deforming and strength of ablating thermal-protective materials under high temperatures. Mech. Comp. Mater., 1990, No. 3, 460-468 (translated from Russian by Consultants Bureau, New York). Dimitrienko, Yu. I., Efremov. G. A. and Epifanovski, I. S., Reusable re-entry vehicles with reclaimable ablating thermal protection. In Proc. 19th Int. Symp. on Space Technology and Science, May 1994, Yokohama, Japan, 94-b-26, Dimitrienko, Yu. I., Ultra-light thermal protective composite materials. In Proc. Int. Conf. on Composite Engineering !ICCE/2). 1995, New Orleans, LA, pp. 189-190. Dimitrienko, Yu. I., Efremov, G. A., et al., Theory and synthesis of advanced thermal-protective composite materials. J. Appl. Compos. Mater., 1995, 2(6), 367-384.
8. 9.
Timoshenko, S. P. and Gere, J., Theory ofElastic Stability, New York, 1961. Dimitrienko, Yu. I., Thermomechanical behaviour ofcomposite materials and structures under high temperatures: 1. Materials. Composites, Part A: Appl. Sci. Manuf., Z&453-461
10.
Mises, R., Der Kritische AuDendruck fiir Allseits Belastete Cylindrischer Rohre. In Festschr. Zum. 70 Geburtstag van Prqf. A. Stodola. Ziirich, 1929.
471
Composites
Printed in Great Britain. All rights reserved 1359-835x/97/$17.00
PII:S1359-835X(96)00145-5
ELSEVIER
Thermomechanical behaviour of composite materials and structures under high temperatures: 2. Structures
Yu. I. Dimitrienko ‘NPO
Mashinostroeniya’ R&D Industrial Corporation, region, 743952 Russia (Received 16 April 1996; revised 4 October 1996)
Gagarina
33a, Reutov,
Moscow
This paper is devoted to modelling of the thermomechanical behaviour of composite structures under hightemperature heating up to the temperature of charring of the composite. Attention is focused on the peculiarities of the behaviour of composite shells under high temperatures. It is also shown that the internal heat-mass transfer of pyrolysis gases has an essential effect on the stress state of the shell under high temperatures. Transverse stresses, which are usually neglected for composite shells, under high temperatures are indicated to be the most dangerous. These stresses can cause delamination of composite shells. A numerical analysis of the conditions of delamination for composite shells is conducted, and computed results are compared with experimental data on the thermomechanical destruction of cylindrical composite shells. 0 1997 Elsevier Science Limited (Keywords: high temperature; charring; pore pressure; delamination; loss of stability)
INTRODUCTION
The behaviour of composite structures of the thin-walled shell type under high temperatures (from 300 to lOOO”C, i.e. at temperatures higher than the temperature Bs of glass formation) differs in principle from the behaviour of the structures under relatively low temperatures (approximately lOO- 150°C). Under normal temperatures tangential stresses oI1, c722and g12 are, as a rule, the most important in thin-walled composite plates and shells (if the 0q3 axis is oriented along the normal to the shell surface) and sometimes considerable interlayer shear stresses 0~3 and ~~23 also occur. Transverse stresses 033 can be neglected in almost all cases owing to their smallness’. High temperatures lead to the appearance and development of intensive internal physico-chemical pyrolysis processes and charring in the composite matrix 2.3, which in turn leads to the formation of pores in the composite. The pores are filled with gaseous products of pyrolysis and the gas pressure p inside them can be very great (approximately 200atm). Moreover, the pore gas pressure under fast, non-stationary heating is distributed very non-uniformly throughout the composite shell thickness, leading to the appearance of important transverse stresses, ~7~~. These stresses prove to be so high that, under certain conditions, they can cause delamination of the composite shell. This phenomenon was investigated theoretically in refs 4 and 5.
The aim of the present paper is to elucidate the conditions for which delamination appears in composite shells. In addition, the overcritical behaviour of composite shells after the appearance of the first delaminations in them is investigated. As was established in experiments, under certain conditions the part of the shell that delaminates under high-temperature heating can lose its stability. Another dangerous phenomenon under hightemperature heating is cracking of the heated surface of the structure due to chemical shrinkage of the composite during pyrolysis6’7. Investigation of the thermomechanical conditions of cracking and stability loss is conducted in the paper.
PECULIARITIES OF THE THEORY ON COMPOSITE SHELLS UNDER HIGH TEMPERATURES Let us consider now the peculiarities that appear in thinwalled shell structures of composites under high temperatures when pyrolysis of the polymer matrix occurs. Main
assumptions
Let us consider a thin-walled shell of orthotropic composite with thickness h, where the line q3 coincides with the normal to the surface, -/l/2 5 q3 5 h/2, and ql
463
Thermomechanical behaviour under high temperature. 2: Yu. I. Dimitrienko
and q2 coincide with the lines of main curvature of the shell surface and with principal orthotropy axes. Then the following expressions can be given for the Lamet’s parameters: H3 = 1
H, = A,(1 +k,qs)
8H --E
= A,
= A,k,
as3
Strains and stresses in a shell On substituting expressions (2) into the kinematic relationship between strains ecrg and displacements u, given, for example, in ref. 1, the expression for the strains eaP in the shell is derived: cap = e,p +
E33 = 0
q36,fl
1,2
a,p=
~~3 = ea3
(1)
where A,(q) are coefficients of the first square form of the reduced surface (q3 = 0); k, are its main curvatures, and k,q3 ec 1. At present there exists a great number of theories for calculation of the stress-strain state of thin-walled shells. For laminated composite materials, Timoshenko’s shell theory’ is the simplest and sufficiently acceptable. However, this theory and also its analogies cannot be applied to shells at high temperatures because they do not consider the most dangerous delaminating stresses c33 arising from internal gas generation in the laminated composite under such conditions. In order to take account of this effect it is necessary to formulate a new system of hypotheses for a shell theory. Let us assume the following system of hypotheses.
(7)
Strains eafl, curvatures ~~~ of the middle surface and shear strain ecu3are determined by the formulae: e
1 dU 1 dA --Q+k,W eaa = A, & + AIA2 dqo
K
1) Distributions of the displacements u,, u3 through the shell thickness are described by three functions Ua(qp), W(qp) depending on coordinates qp, ,0 = 1,2: %I = UCk+
u3 =
437a
w
(2)
of shear oa3 and normal g33 stresses are chosen in the form of square functions of the coordinate q3: =
--
1 &I
aa= A,
g33=LpgP-P-+(P--P+)
q3 (T+;)+
1c, _!Yz=-
ea3
-h/2
s h/2
+
vwsl
kpp
-
$4
dq3
h/2
s h/2 -h/2
%a02(c33
-
i3)@3
(5)
-h/2
s h/2
wo3dq3
=
G+3,a+3
h’2
-h/2
1,2;
p =
as1 vGpq$%3
j
=
0,l
$1 + h’2 ae2cPs#dq3 J -h/2 1
wg2Ea3dq3
cl=
(10) h/2
h/2
Pg, =- (p _h,2 cpspfdq3
sf
ygl
s
=
_h,2fvw&~
h/2
pdq3
Similarly to the above, on substituting distribution (4) for stress c33 and (7) for strains E,, into the integral equation (5), we obtain a relationship connecting functions $J~to eflp and K~~. From this we can find:
J -h/2
c33
k= 1,2
J -h/2
-h/2
2 C3p /3=1
h/2
+
=
p=
_h12f3wdq3
X
h’2 (Psa&dq3 s -h/2
(4)
where p* are pressures at the surfaces q3 = fh/2 of and v,b3(q,) are new unknown the shell and $Jq,) functions of coordinates qb, p = 1,2. 3) The following integral relations are considered instead of constitutive relations (the second, fourth and fifth equations of (18) in Part 19):
5
(9)
Herein and below, the following notation for values averaged over the thickness of the shell is used:
o(i) ED
Q= 1,2
= -
cY= 1,2
h &+s,u+3’E)
6
(-$+a)$
cps
wdq3
a= I,2
(3)
G+3,o+3
h/2
1 W-t
dq, + Al A2 dq, ”
ai;) =
cps
a= 1,2
On substituting distributions (8) for stresses goa3and (7) for strains E,~ into integral equations (6), we obtain relationships for functions $J,,in terms of Ye, U, and W:
2) Distributions
ffa3
1 dW --+-ya-k,Ua a3 = A, dq,
1,2
Thermomechanical
If functions Ye, U, and W are known, then all the functions $,, qa and & can be found by formulae (9) and (ll), and, hence, stresses on3 and 033 can be determined by formulae (3) and (4). Stresses (T,,~in the shell layers are determined by the constitutive relations (10) from Part 19, whereinto strains (7) should be substituted:
behaviour
under high temperature.
Constitutive relations for a shell under high temperatures
On substituting expressions ( 12) into the equations of forces and moments (14), we obtain:
‘PST12
g12
=
&lC66(el2
(13)
+ q3&12)
Equilibrium equations
Equilibrium equations averaged over the thickness of the shell are the following:
2: Yu. I. Dimitrienko
A4412
=
N&l2
=
C66e12
+ 06642
+ N66n12
(FsQ,l = Ccr+3.,r+3en3
Equations (15) are constitutive relations for a composite shell under high temperatures. Here Cnrj, NaIj and D,,,j are elasticity coefficients of a composite shell: Co/J = C&7$)
D,,, = C’,,ga;’
N,,,j = Gab?
(16) G+3.u+3
=
o!,,H
G+3.n+3ag)
=
I,2
for heat stresses ?‘a and heat moments M,: -AIAz’Ps(klTll
+kzT22) +
d&Q,& +WQz'Ps %71
-A1A2@+
-P-)
-
(kl
Ffi = &
&?2
+k2)A24PgPg
=
0
O=[
(13) Heat-mass
cu,p = 1.2
a#P
where Tap are forces in the shell; Q, are the shear forces; and M,, are moments determined by the formulae:
1
W
Mg = y‘pg .[-h/2
P, =A
apq&3
‘PPs
‘pg =
Ps
=
Pl
+
‘p2 +
‘p3
a,/3
h-2 _h,2 %pdq3
1=
(14)
9s
1,2
Equilibrium equations (13) for composite shells under high temperatures differ from the equations under normal conditions by additional terms, namely by forces P, and moments Mg of the pore pressure of pyrolysis gases and also by the appearance of the multiplier ‘ps at forces Tcrfiand Maa.
c,,,i;y
A& = 2
c,,,#
(17)
,i= I
transfer in a composite shell
Under the action of high temperatures on a composite shell, a large quantity of pores filled with gas is formed in the shell. The gases filter through the pores and partially pass to the surroundings. However, under certain conditions, the gases do not have enough time to filter out, resulting in the accumulation of a great intrapore pressure p of pyrolysis gases inside the composite material that can cause delamination of the shell. To take this phenomenon into consideration, it is necessary to consider both mechanics equations and the equations of internal heat and mass transfer for a composite, these equations being: a kinetic equation describing the change of concentration of the polymer phase during heating; a heat conductivity equation taking into account the heat effect of pyrolysis and cooling the composite at the expense of gas filtration in the pores; and a filtration equation for pyrolysis gases in the pores. Further external heating of the considered composite shell by heat fluxes q * is assumed to be low-changing along the shell surface, and thus the distributions of temperature 8, pore pressure p, gas density p4 and concentrations ‘p4, ‘p2 and ‘p3 can be assumed to be functions only of coordinate q3 and time t. Taking account of these assumptions, the equations of internal heat and mass transfer for the shell are written in
465
Thermomechanical
behaviour
under high temperature.
2: Yu. I. Dimitrienko
the form2:
+RcP+-P-)(a-
1)
Q= 1,2
hip,
where
de ‘“at
1 -d = A,A2 aq,
+
5‘1 =
(Ad2&3g)
ae
q&JG3 -apSe
A~OJ
1 -;[c3,(c22
-
c12)+~32(~ll
-cl,)1
A, = Ci, C22 - Ct2
52 =;+$(-c,,c,,+c32cll)
aq3 aq3
Here p = R,p,6 is the pore pressure of pyrolysis gases, ps is the gas density, Ae” is the heat of matrix pyrolysis, J is the intensity of the pyrolysis process defined in ref. 8 and cs is the gas heat capacity.
CYLINDRICAL
SHELL
Let us consider an orthotropic composite shell as being a cylindrical body with the symmetry axis z. The coordinate surface q3 = 0 is assumed to be coincident with the middle surface of the shell; the meridional arc s and the azimuth angle Q counted off from a certain point MO are chosen as coordinates q1 and q2; R is the radius of the external surface of the composite, r = R - (h/2) + q3 is the current radius of the shell. Derivatives d/dq3 in equations (18) are derivatives with respect to r: a/&. Principal curvatures and coefficients of the first squared form of a cylindrical shell are constants: ki = 0, k2 = R-‘, Al = 1, A2 = R. Under these conditions the system (13)-(15) has the solution: T,,
=Y
T22 =
cps
71 =
0
;
[(P,
-
P-)R + io,P,l
Kql =o
By substituting formula (19) into (15) we can find strains epp. Stresses in the cylindrical shell can be determined by the formulae: a,3
=
0
u12 =
0
Heat and mass transfer equations numerically.
COMPUTED
(18) are solved
RESULTS
The system (18) was solved by using a numerical method with a step-by-step method and schemes of matrix sweeping. The solution of this system is given by functions p,(r, t), t9(r, t) and jo2(r, t), with the help of which we can evaluate the pore gas pressure: ~(r, 4 = R,p,(r,
4e(r, 4
(21)
All computations were conducted for a composite shell with the following geometric parameters: R = 2m, h = 2 x 10e3 m, and with physical characteristics corresponding to those of a composite with a matrix of epoxy resin and glass-textile reinforcement. The characteristics of the glass/epoxy composite are given in Part 19. External and internal pressures are considered to be atmospheric: p- = p+ = 0.1 MPa, and the surface of the glass/epoxy shell is assumed to be heat-insulated. At the external surface q3 = h/2 of the shell a convective heat flux q+ = ~~(0, - 0) was given, and at the internal surface there was a condition of heat insulation: qi = 0, where aT = 0.1 kWme2 K-’ is the heat transfer coefficient and 13, is the temperature of the surroundings (e, = 400°C). The action of high temperature leads to heat propagation into the glass/epoxy shell. Figure la shows the temperature Q(r, t) distribution versus the shell thickness for different times t = 50, 100, 150, 200 and 250 s. Figure lb shows the distribution of pore gas pressure p(r, t) through the shell thickness for the same times. Figures 2a and 2b show distributions of radial 033 and tangential u22 stresses in the shell for different times. A peculiarity of the behaviour of composite shells under high temperatures is the presence of a specific profile of intrapore pressure p (Figure Ib). Intensive pyrolysis of the polymer matrix leads to growth of the pore gas pressure. Only gases near the external surface q3 = h/2 of the shell, where porosity and gas permeability of the composite are maximum, have time to filter out into the surroundings. Inside the shell wall porosity and gas permeability are small and gases accumulate to form a local peak of pore pressure (Figure lb). This peak
Thermomechanical
behaviour
under high temperature.
2: Yu. I. Dimitrienko
I
Ti 0.970
0.976
0.982
0.988
0.994
1.000
0.970
0.976
0.982
0.988
0.994
1.000
(a) w
! A \ /150 \I I/~250
OV 0.970
I 0.976
J/I
xi\I\
I
Jy/.y\\; 0.982
0.988
0.994
1.000
@)
Figure 1 Distributions of temperature 0 (a) and pore pressure p (b) versus thickness of a composite shell for different times t. Initial porosity of the composite (oi = 0.05, symbols on the curves are times (s)
moves towards the internal cold surface of the shell with time. The peak of pore pressure leads to the appearance of a local peak of tensile normal transverse (through-thickness) stresses ~7~~in the shell (Figure 2~2)that also moves with time towards the internal surface of the shell. These stresses are the most dangerous for laminated textile composite shells, as there appears a chance of their delamination. Tangential stresses u2? in the cylindrical composite shell under high temperatures have two local extrema: one in the domain of compressive stresses and another in the domain of tensile stresses (Figure 2b). The peak of the compressive stresses is connected with the presence of a peak of pore pressure, that also moves with time towards the cold internal surface. The peak of tensile tangential stresses is caused by shrinkage of the composite at temperatures of the matrix pyrolysis. As follows from formula (20) the tensile shrinkage stresses are: C?2 = -C&2 M E&. These stresses are localized near the most heated external surface q3 = h/2 of the shell. If the level of these stresses is sufficiently high and exceeds the strength limit in tension 02r, then longitudinal cracks can appear in the cylindrical shell that are directed along the axis Oyr of the shell (see Figure 6a below). The maximum values of the pore pressure p and also of the transverse g33 and tangential a22 stresses depend on the initial porosity of the composite, cpi. The higher the value of initial porosity cpi, the more intensive is
Figure 2 Distributions of transverse stresses q3 (a) and tangential stresses oz2 (b) versus thickness of a composite shell for different times r. Initial porosity of the composite ~2 = 0.05. symbols on the curves are times (s)
filtration of gases into the surroundings, and the lower the values of max p and max cr33. Figure 3-5 shows dependences of max p, max 022 and max 033 on heating time t for three values of cpi = 0.05, 0.07 and 0.1 (i.e. 5, 7 and 10% respectively). As can be seen, the kinetics of growth of functions max p and max cr33for all three values of cpi is the same: a sharp initial
0.6
0.0 0
50
100
150
200
t.s 250
Figure 3 Dependences of maximum values of functions in a composite shell: transverse stresses max crss, tangential stresses max ~2s. pore pressure max p. damage parameters max .q and max .zs upon heating time r. Initial porosity of the composite 9: = 0.05
467
Thermomechanical
behaviour
under high temperature.
2: Yu. I. Dimitrienko
FRACTURE OF COMPOSITE STRUCTURES UNDER HIGH TEMPERATURES Conditions for the appearance of fracture
0.6
0.2
0.0
6s
0
50
100
150
200
250
Figure 4 Dependences of maximum values of functions in a composite shell: transverse stresses max q3, tangential stresses max q2, pore pressure max p, damage parameters max z2 and max z3 upon heating time t. Initial porosity of the composite ‘pj = 0.07
As shown above, in composite shells under intensive non-stationary heating to the high pyrolysis temperature of the material, important thermostresses appear: both normal through-thickness 033 and tangential a22 stresses occur even without external mechanical loads. To estimate the danger of these thermostresses (when they reach the ultimate strength of the composite and the composite structure is destroyed) it is convenient to introduce parameters Z, called damage parameters. The parameters z, are determined as ratios of the thermostresses a,, and the corresponding limits of strength cCYT of the composite in tension: Za(q3, t) = lga;b” cT,,
a: = 2,3
(22)
LIT
growth (approximate duration 50 s), at time t = 100-180 s, and finally a the values that is connected with intensive stage of matrix pyrolysis. max oZ2(t) at all stages and for all grow monotonically, behaviour that the increasing content of coke phase surface and, hence, with increasing
then a stabilization sharp reduction of completion of the Tangential stresses three values of cpi is connected with ‘p3 near the heated shrinkage stresses
022.
However, the decreasing initial porosity cpj, the absolute values of max p and max g33 essentially increase (Figures 3-5): for cpt = 0.1 the greatest values of max p and max u33 are 3 and 0.2 MPa; for cpj = 0.07 they are 20 and 1.5 MPa; while for ‘pt = 0.05 they are 25 and 2.3 MPa, respectively. The absolute values of max cr22 show less variation: they are 15, 26 and 50 MPa, respectively, as they are determined only by the degree of charring of the composite matrix cp3 and are independent of the level of pore pressure p.
where )a,, 1is the modulus of stresses o,, . If stresses fraa are absent in the shell (gas = 0) or they are negative (o,,(q3, t) < 0) at a given time t in a zone of the shell surface with coordinate q3, then the corresponding damage parameter z, (q3, t), introduced with the help of (22), is equal to zero within the zone of the shell. If a,,(q3, t) > 0 there, then za(q3, t) > 0. If at a certain time ti within a zone of the shell with coordinate q3 the following condition is satisfied:
C) = 1
_(h,2m;&$&131
Q=
273
(23)
gas 0”
fibres
(4
mination t=t;
0.6
A
gas 0"
q3f .
heatflur t=t
0.4
**
0.2
(cl 0.0 0
50
100
150
zoo
fS 250
Figure 5 Dependences of maximum values of functions in a composite shell: transverse stresses max ass, tangential stresses max oaa, pore pressure max p, damage parameters max z2 and max zj upon heating time t. Initial porosity of the composite +$ = 0.1
468
2
Figure 6 A scheme of three types of fracture of a composite shell under high-temperature heating: (a) cracking of the heated surface of the composite at t = t; and q = 1; (b) delamination of the composite at t = t; and zj = 1; (c) appearance of several delaminations and loss of their stability at t = t,,
Thermomechanical
then fracture of the shell occurs. This means that the corresponding stresses a,, reach the ultimate strength gnT of the composite in tension. As was shown above (see Figure Zh), the maximum tensile stresses ~22 are always near the external surface of the shell r = R (i.e. q3 = h/2). Therefore, if at the certain time t; condition (23) for cr = 2 is satisfied, then during destruction of the shell longitudinal cracks appear on the external surface q3 = h/3 (Figure 6a), as the destruction at Q:= 2 occurs due to tensile tangential stresses (~~2 at the surface q3 = h/2 of the shell. As seen from Figure Za, the stresses 033 can reach their maximum values at any surface with coordinate q;: -h/2 < q: <: h/2. Therefore if at the certain time ti condition (23) for (Y= 3 is satisfied, then the shell is destroyed by delamination-type damage (Figure 6b) and the delamination appears within the shell at a surface with coordinate q3 = q;. If condition (23) is satisfied for both Q = 2 and Q = 3, then both types of fracture of the shell take place. Thus, the damage parameter zLy is a convenient indicator of the danger of thermostresses appearing within the shell during heating: the closer the value of zz or z3 is to 1, the higher the level of thermostresses and the greater the danger of destruction of the shell by type 2 (longitudinal cracking) or type 3 (delamination) fracture, respectively. Below, calculated damage parameters zn will be given for a composite cylindrical shell for the considered case of external heating. Figures 3-5 show the kinetics of the maximum values of damage parameters, max z,,(l), in the composite shell for three values of initial porosity: cpt = 0.05, 0.07 and 0.1. As seen from these figures, the function max z2( t) does not change considerably with & but max z3(t) changes dramatically. This can be explained by the fact that z2 is determined mainly by shrinkage of the composite during pyrolysis, which is independent of porosity cpi, while the parameter z3 on the other hand is defined by internal mass transfer and pore pressure which depend significantly on the porosity of the composite. For cpi = 0.05 and 0.07 the condition max ~1 = I is satisfied earlier than max z2 = 1; this means that delamination of the shell occurs earlier under high temperatures. For ‘py = 0.1 the condition z3 = 1 is not realized at all, i.e. the composite shell is destroyed only as a result of cracking on its external heated surface. Behaviour of a composite of fracture
shell after the appearance
Composite structures are usually examined only for the time interval 0 < t < tz, until the time tz when fracture first appears: for example, until composite delamination appears due to the accumulating intrapore pressure of gaseous pyrolysis products. This restriction is justified for many heat-shield structures because, after delamination has arisen in a piece of hardware having (for example) a plane shape (plates, unclosed shells, panels). it fails and loses completely its exploitational
behaviour
under
high
temperature.
2: Yu. 1. Dimitrienko
thermal-protective properties. However, for some types of structure there exist some exploitational conditions when the hardware still performs its functional protective purpose for some time interval t; I t 2 t,, after delamination has taken place. This situation occurs under external heating of a shell with a closed contour made of textile composite material; e.g. under the action of high temperatures on a cylindrical fuel container made of glass-reinforced plastic. The action of a temperature field uniformly distributed over the surface of the cylindrical shell of the fuel container leads to the formation of ring delaminations having a closed contour. Because the cylindrical shell has a closed contour, it does not fail after the first delamination and continues to perform its functional purposes. Upon further heating new ring delaminations appear. the formation of which is directed from the outer surface of the shell to the inner surface. Pore pressure is accumulated in each ring crack so that the pressure difference pd(rI, t) -pd(ri_, , t) (where i = 1.. N; r, is the radius of ith delamination, ri = R - h/2 + q.Si, where qJi is the coordinate of a surface of the ith delamination), uniformly distributed, acts upon each of the cylindrical layers stripped off. The other non-delaminated part of the cylindrical shell of the fuel container, with thickness hd(t). proves to be the most loaded as it is subjected to the action of the maximum pressure difference pd(rN, t) -p_, where 1~ = rN - R + h. The process of delamination continues until time t,, when, at a certain critical thickness hd( f,,), the inner part of the shell loses its stability under the action of the external pressure difference pd(r,y. t,,) - p_ > 0. After that the structure stops performing its functional purpose and the fuel container fails completely. However. the time interval t,, - r; from the appearance of first delamination until the loss of stability of the fuel container is sufficiently long: t,, - t3 >> t,, therefore the method of stresses and heat-mass transfer calculations developed above for t < t; should be continued for the time interval t; < t _< t,,. The condition for the appearance of delamination in the composite under high temperatures has the form (23) for o. = 3. We will assume that the appearance of delamination does not change the picture of heat-mass transfer in the shell. The critical external pressure p*, for which the loss of stability of the non-delaminated shell section occurs, was obtained in ref. 10 and. for the considered problem, is written as follows: p*(t) = 0.92E,a0,(t)
(24)
where L is the shell length, and the stability condition has the form: Pd(?V?
l) -
Pp
< p*ct)
r;
5
t 5
t,,
(25)
Now,
let us derive the expression for the pressure of gaseous pyrolysis products accumulated within the ring crack. For this, the state equation of a perfect
pd
469
Thermomechanical behaviour under high temperature. 2: Yu. I. Dimitrienko
gas is used: Ed = R,P,&
(26)
where R, is the gas constant. In order to determine the density psd of the gas within the crack, let us apply the equation of gas mass conversion before and after formation of the crack: &dvd
=
t;I\
I
(27)
&.$g
where pg is the gas density in the pores; Vg is the gas volume in the pores coming to the surface of the ring crack: Vg = 27rRLlgcp4
(28)
where 1, is the characteristic diameter of the pores ( 10P4lop3 m); vd is the crack volume opened by the action of pressure pd (29)
R( 1 + e,g)= R
Here ee = IY~/E!~.Q~is the tangential deformation of the shell, and a8 is the tangential stress of the shell, connected to the pressure pd in the crack by the relation*: ge = PdR/hdHaving substituted formula (27) and (28) into (26), we can find the gas density within the crack, &j. Then from (25) we obtain an expression for the pressure of the pyrolysis gases pd within the crack that has the form: Pd(%‘,
d=
(30)
On substituting (24) into (30), we obtain the condition for the absence of destruction in the composite cylindrical
hdlh 1.0
0.6
0
0
Figure 7 Distributions of non-delaminated thickness h,_+of a composite shell wrsus heating time t for different values of initial porosity of the composite & = 0.05, 0.07, 0.08 and 0.1; A are experimental values for (p4 = 0.05
470
0 I
I
0
0.05
/
J
Q.
0.10
Times to the appearance of three types of fracture, ts, t; and
t**’versusinitial porosity pj of the composite
2 81
where R and Rdef are the radii of the delamination surface in the non-opened and opened (by deformation) states respectively: Rdef =
.
Figure 8
Vd = xL(R2 - R&J = 2rL-&$$ d
50
shell by stability loss under high temperatures. If at time t = t,, there is an equality in (30), then there occurs a loss of stability of the internal layer of the composite shell. Computed results Figure 7 shows a graph of the function (hd(t)/h) for the glass/epoxy cylindrical shell with the characteristics given before for four values of cpi = 0.05,0.07, 0.08 and 0.1. For cpj = 0.05, during the first stage of heating t 5 50 s there is no delamination in the shell and (hd/h) = 1. At time t = t; M 50 s the condition (23) is realized for the first time and the first delamination occurs near the external surface of the shell. Figure 6c presents the picture for the appearance of new delaminations; this graph is characterized by the thickness hd(t) of the nondelaminated material. From Figure 7 the dependence (hd( t)/h) is seen to be practically linear. At time t = t,, FZ220 s, when hd/h reaches the value 0.05, the non-delaminated shell thickness becomes so small that the pore pressure pd of gas accumulated in delamination cavities exceeds the stability limit (25) and the internal layer of the composite shell loses its stability. In this case complete failure of the cylindrical composite shell occurs. As seen from the computations conducted, the time interval t,, - t; is equal to approximately 170 s, which exceeds t; by 3 times. Thus, the calculation of internal heat-mass transfer and stresses is necessary up to time t,, as the shell still performs its designated purpose at these times. For a composite shell with initial porosity cpi = 0.07 the first delamination appears at t; M 90 s, and delaminations continue to arise up to t = t,, = 170 s, when a loss of stability of the internal layer of the shell occurs (Figure 7). For cpi = 0.1 delaminations to not appear at all, and (hd/h) = 1 during the whole heating time due to
Thermomechanical
a low level of pore gas pressure pp; hence, there is a low level of normal transverse stresses ~733. Figure 8 shows the dependences of the times for appearance of the three types of failure, 1;, t; and t,,, on the initial porosity of the composite, &. As can be seen from this figure, for cpi 2 0.08 there are no delaminations and loss of stability of the shell during heating (as t; and t,. -+ OO),and there is only the occurrence of cracking due to chemical shrinkage at time t = t; M 150 s. For ‘pj 5 0.08 delamination of the composite shell occurs first at f = t; x 50-100 s, then there is shrinkage cracking of the shell surface at t = t; = 100-l 30 s, and finally there is a loss of stability of the delaminated internal part of the shell at t = t,, M 160-200s. Experimental
results
To test the accuracy of the developed model for calculation of the fracture of composite shells under high temperatures, heat testing was conducted for cylindrical glass/epoxy composite shells with the following geometrical parameters: L = 0.8 m, h/R = 0.03, and the results were compared with the corresponding computed ones. Heating was performed with the aid of infra-red heaters so that the heat flux q+ to the composite shell and the surface temperature 8(R, t) would be the same as the values used in the computations, namely 10 kW m-* and 300°C. The initial porosity cpj of the textile glass/epoxy composite shell, which was manufactured by pressing in a press mould, was approximately 0.04-0.05. During testing, acoustic emission was used to determine the time of delamination onset t; in the shell; it was detected easily by the peak of intensity in the acoustic signal and also by the sharp growth of gas generation from the shell, which was noticeable with the naked eye. The time of stability loss t,, of the internal layer of the shell was also detected easily by the peak in the acoustic signal and also by the typical audible click as well. After testing, the thickness hd of the shell part that lost its stability was measured. Three composite shells with the same parameters were tested. The points in Figure 7 show experimental values of the times t; to the beginning of delamination and t,, to the stability loss, and also the corresponding experimental values of thickness hd at t; and t,,. As can be seen from these graphs, the agreement of experimental and computed results is sufficient to allow us to confirm the adequacy of the model developed for description of the thermomechanical behaviour of composite structures under high temperatures.
behaviour
under high temperature.
2: Yu. I. Dimitrienko
CONCLUSIONS The theory developed for thin-walled composite shells adequately describes the phenomena observed in experiments, namely the appearance of intrapore gas pressure under high-temperature heating and the generation of transverse stresses. The theory allows us also to determine the condition for the appearance of delamination in composite shells under high temperatures and the condition for stability loss of the delaminated composite. These conditions are shown to depend essentially on the initial porosity of composite shells; in particular, for a textile epoxy-based composite with initial material porosity of more than lo%, delamination in the shell does not occur. The theory also enables determination of the time at which shrinkage cracks appear on the composite shell surface heated under high temperatures. It has been shown that the appearance of shrinkage cracks is practically independent of the initial porosity of the composite.
REFERENCES I. 2.
3.
Whitney, J. M., Structural Analysis of Laminated Anisotropic Plates. Technomic, Lancaster, PA, 1987. Dimitrienko, Yu. I., Thermal stresses and heat-mass-transfer in ablating composite materials. ht. J. Heat Mass Transfer, 1995. 38(l), 139-146. Griffis, C. A., Nemes, J. A., Stonesifer, F. R. and Chang, C. I.. Degradation in strength of laminated composites subjected to intense heating and mechanical loading. J. Compos. Mater.. 1986,20,216-235.
4.
5.
6.
I.
Dimitrienko, Yu. I. and Epifanovski, I. S., Deforming and strength of ablating thermal-protective materials under high temperatures. Mech. Comp. Mater., 1990, No. 3, 460-468 (translated from Russian by Consultants Bureau, New York). Dimitrienko, Yu. I., Efremov. G. A. and Epifanovski, I. S., Reusable re-entry vehicles with reclaimable ablating thermal protection. In Proc. 19th Int. Symp. on Space Technology and Science, May 1994, Yokohama, Japan, 94-b-26, Dimitrienko, Yu. I., Ultra-light thermal protective composite materials. In Proc. Int. Conf. on Composite Engineering !ICCE/2). 1995, New Orleans, LA, pp. 189-190. Dimitrienko, Yu. I., Efremov, G. A., et al., Theory and synthesis of advanced thermal-protective composite materials. J. Appl. Compos. Mater., 1995, 2(6), 367-384.
8. 9.
Timoshenko, S. P. and Gere, J., Theory ofElastic Stability, New York, 1961. Dimitrienko, Yu. I., Thermomechanical behaviour ofcomposite materials and structures under high temperatures: 1. Materials. Composites, Part A: Appl. Sci. Manuf., Z&453-461
10.
Mises, R., Der Kritische AuDendruck fiir Allseits Belastete Cylindrischer Rohre. In Festschr. Zum. 70 Geburtstag van Prqf. A. Stodola. Ziirich, 1929.
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