A mixture theory for a thermoelastic laminated medium, with application to a laminated plate under impulsive loads

Journal of Sound and Vibration (1974) 33(2), 187-200

A MIXTURE THEORY FOR A THERMOELASTIC LAMINATED MEDIUM, WITH APPLICATION TO A LAMINATED PLATE UNDER IMPULSIVE LOADS J. ABOUDIt Department of Mathematics, University of Strathclyde, Glasgow, Scotland

(Received 10 July 1973, and hz revisedform 25 September 1973) A mixture theory of solids is evaluated for a thermoelastic laminated medium composed of two constituents in alternating layers. In this theory every constituent has its own motion and temperature, but is allowed to interact mechanically and thermally with the other. The resulting system of coupled equations of motion and heat conduction is then used to investigate the response of a laminated plate subjected to mechanical and thermal loadings. In addition, the corresponding thermoelastic effective modulus theory is derived and applied to the laminated plate. Comparisons between the results based on the two theories exhibit the effect of microstructure, which is very pronounced, and the effect of the reinforcement volume on the resulting field in the individual constituents.

1. INTRODUCTION Stress waves propagating in composite materials are dispersed and attenuated by the different phases which form the multiphase medium, so that any attempt to analyze wave propagation in such media must take into account the effect of the microstructure of the continuum. Nevertheless, satisfactory results for sumciently long wavelengths c a n be obtained when employing the effective modulus theory in which the composite is replaced by a homogeneous anisotropic medium with average elastic constants and density. Obviously any information on the field within each phase of the continuum is not available by this theory since effects of the microstructure are ignored, and only the average field within the whole medium is Obtained. An effective stiffness theory of the first order for laminated composite materials has been worked out by Sun, Achenbach and Herrmann [1]. This theory approximates the actual behaviour of the composite, includes microstructure and furnishes better dispersion curves when compared with those based on the exact theory of elasticity as applied in reference [2]. Later, the approximate theory has been extended by Grot and Achenbach [3] to include thermoelastic and viscoelastic effects. Another approach in an attempt to model a composite material is the mixture theory proposed by Bedford and Stern [4] in which the composite constituents are modeled by a mixture of superimposed interacting continua. In the solid mixture theory of reference [4] two or more media each having their individual motions, exist at each point of the space but are allowed to undergo thermal and mechanical interactions. These interactions provide a means of including the composite microstructure effects which give rise to dispersion and attenuation. For other forms of mixture theories see the work by Steel [5] and references cited there. t On leave from School of Engineering, Tel-Aviv University, Ramat Aviv, Israel. 187

188

J. ABOUOI

The main difficulty of the application of a mixture theory to composite materials is to express the parameters which appear in the constitutive and field equations in terms of the material constants and geometry of the individual phases. For an elastic laminated medium composed of two alternating layers, Stern and Bedford [6] evaluated these parameters by utilizing the analysis of Postma [7] in constructing the effective moduli for such a medium, and the interaction terms were determined on the basis of a simple static problem. The theory was then applied for the steady propagation of plane waves in a laminated medium, and dispersion curves for the first modes were obtained and compared with those based on elasticity theory. For a transient wave propagation problem in which the applied load acts in the layering direction, the response of an elastic laminated plate has been obtained [8] by solving the one-dimensional mixture equations of motion and comparing the results with those obtained by solving the complete dynamic equations of elasticity [9] and with the effective modulus theory results. It was shown in reference [8], by considering various applied load time durations, plate thicknesses and reinforcement ratios, that the mixture theory well predicts the dynamical behaviour of the plate. For elastic laminated and unidirectionally fiber-reinforced composites an asymptotic method can be applied to model the continuum. In this approach asymptotic expansion in a small parameter defined by the ratio of a typical composite micro-dimension to the dominant wavelength is performed. Truncation of the asymptotic series yields a hierarchy ofcontinuum equations, see Hegemier [10] and Ben-Amotz [11]. In reference [12] a similar method was applied for a transient heat conduction in a rigid laminated medium. For a first-order theory the asymptotic approach can be cast in a binary mixture form [13] with the same expression for the momentum transfer term as in-reference [6], but with constituent stress expressions which involve the strains in both constituents. On the other hand the general mixture theory as proposed in reference [4] assumes that the constitutive equations for every component depend solely on the appropriate field variables in that component. The dispersion relations for the two lowest modes in a thermoelastic laminated medium were investigated by Sve [14] by employing the complete theory of thermoelasticity which provides a twelfth-order determinant relating the frequency with the complex wave number. In this paper the mixture theory for the elastic laminated medium [13] is extended and applied to a thermoelastic laminated composite. Mixture stress-strain-temperature relations as well as the entropy expressions for each constituent are derived. Then the mixture stress and energy equations are obtained together with the expressions of the interaction terms for the momentum and heat transfer from one constituent to the other. The mixture field equations are given as a system of four coupled thermomechanical equations in the constituents' average displacements and temperatures. The effective modulus theory equations for the corresponding thermoelastic anisotropie medium are derived together with the appropriate average parameters: The theory is then applied to investigate thermoelastic waves propagating in a laminated plate excited by mechanical and thermal impulsive loads. Results for the stress and temperature fields obtained from the mixture theory and compared with those based on the effective modulus theory show clearly the effect of the microstructure. They also exhibit the effect of the reinforcement ratio on the stress and temperature in each phase of the plate. 2. BASIC EQUATIONS Consider a periodic array of two alternating isotropic thermoelastic layers of widths 2h~ and 2h2, respectively, with parallel plane boundaries (see Figure (la)). The Lam6 constants and density of each layer are denoted by 2~, it~ and p~ (~ = 1, 2), respectively, and/~, cr~ and k~ denote, respectively, the coefficient of linear expansion, the specific heat at constant volume

LAMINATEDPLATEUNDERIMPULSIVE LOADS

189

and ihe coetiicient of heat conductivity of each layer. Consider situations in which all quantities are independent of the Cartesian co-ordinate z (plane strain state) as well as m o t i o n a n d temperature yielding symmetric - U (~ ~ ) and antisymmetric ut2~) and Q~') with respect to the local co-ordinate y, (ct = 1, 2) measured from the mid-plane of each layer. Here n c') = (u]"~, ut2~)) = (u t'), v(=)) are the x and y components of the displacement vector a..~,), and Qt~) is the y component of the heat flux vector Qt=) = (Qt,), Qt,)). These situations can be obtained when uniformly distributed mechanical and thermal loadings independent o f y are applied in the layering direction x.

(a)

00

0

1 ~A

A

3A

4A

!

(b)

Figure 1. (a) Laminated composite medium of two alternating layers. (b) Temporal dependence of the applied loads. The constitutive equations for the stresses, entropies and heat fluxes in the linear thermoelastic isotropic layers are

oi~ ) - j ~t'~6u+ 2 1 t ~ -

7~(T~-

To)6 u,

i , j , k = 1,2,3, ct= 1,2,

(I)

where tr~ ~, e~l are the components of the stress and strain tensors, respectively, T t~ - To is the temperature perturbation from the equilibrium temperature To when the medium is undisturbed, 6u is the Kronecker delta and ),= = fl=(3).= + 2tt=). The entropy per unit volume in each layer is given by

tl t~' = 7~ e~l~ + P~ c,.~(T ~ - To)/To,

ct = 1,2.

(2)

For the heat flux,

Q~" = - k ~ rff'.

(3)

The stress equations, in the absence of body forces, are a2 P= at 2 '~'~

~

r

t=~.j y

(4)

190

j. ABOUDI

and the energy equations, in the absence of thermal sources, are 0 =

i.

(5)

Note that in equations (1)-(5) the Cartesian co-ordinates are denoted by

(xa,x2,x~)-

(x,r,z). Due to the previous specifications the following conditions are obtained: v ~') = 0 ,

a~7 = 0,

at ),, =

Q~=) = 0

0,

u = I, 2.

(6)

Further, the interface conditions between the layers are Id(l) ~ /1(2) U ( l ) ~ t.)(2) t r ( l ) - - ~r(2)

~12

- ~J2

a(l) ~22

- - rr(2) -- v22

Q

)

t

at Yl = lq, Y2 =

-h,.

(7)

These equations and conditions together with the initial and boundary conditions specify completely the motion and temperature distribution in the layers.

3. MIXTURE CONSTITUTIVE EQUATIONS In this section, approximate constitutive equations will be evaluated in terms of the average constituent displacement gradients and temperatures. These equations are based on the assumption that for a characteristic wavelength, L, and composite microdimension, h = ht + h2, the ratio e = h/L is a small parameter and terms of order r2 hence can be neglected. It is clear, therefore, that with this assumption the results based on the developed mixture theory are valid for long as well as moderate wavelengths as compared with the thickness Of two successive layers. This feature is similar to the first order effective stiffness theory as given in reference [1 ] for a laminated medium. Let the average of the quantityfC=~(x,y,t) in the ~ constituent be defined by 1

i'f~')(x,y,, t)dy,.

/'='(x, t) = F,, o

(8)

Then as a result of the different thermoelastic properties of the two constituents (cr = 1,2) the average displacements and temperatures in the two constituents are different. This difference gives rise to a mechanical and thermal interaction at the interface between the two constituents as shown in the following. Applying equation (8) to a[] ) as given in equation (1), and utilizing the asymmetry of the v~') (equation (6)) and their continuity across the interface (equation (7)), gives

a E, Vx

aC"+

0C']/;., = -S(x, t),

0 a~2~+ Y20c2~]/).2 =

J

S(x, t),

(9)

191

LAMINATED PLATE UNDER IMPULSIVE LOADS

where

n, = hJh,

E, = 2~ + 2p,,

0c,) = T m - To and

hS(x, t) = v r (x, h2, t) = - v m ( x , h ,

t).

(10)

Similarly~ by applying equation (8) to o22 -c,)

,

a

]

HI [~22 -- "'1 ~X /~(1) .31_~1 0(1)~ / E l =

-S(x, l),

a fie,, + ~'20C2~] / E 2 = S ( x , t ) . n~ [ O~2] - & ~x

(11)

By expanding a~4~ and 0 m in terms of y,, utilizing their symmetry property, applying the fourth and fifth conditions ofexpressions of(7) and neglecting terms of order of~ z, the following relation is obtained: fret) 22 = O~

(12)

According to equations (I 1) and (12), S(x,t) takes the form

S(x,t)=

]

[

a f i t t ) - ) . 2 ~ox fi(2~--~, 0(t) + y20 r ).l-~x

/E,

(13)

where

E = (nl E2 + nz En)/nl nz. Substituting equation (13) in equation (9) yields the following relations between the partial stresses, o r = th~ir and the average displacement gradients and temperatures: ar

0 = c ~ fi(l) + c12

o'fi"

0 = c,=Wox a,', +

~2~ _ dn 0 (t) - dl20 m, 0

a,', - a,, 0,,, - ,t,, 0 %

(14)

where c = = n~ & - ;.~/E,

c~2 = :.~22/s

d~# = ).~,y~/E,

~ ~ B.

(15)

These equations differ from those proposed by Bedford and Stern [4] where the constituent stresses were assumed to be dependent solely on the displacement gradient and temperature of the constituent. This point will be discussed further in the sequel. In the isothermal case equations (14) reduce to the stress expressions given in reference [13]. The entropy of each constituent, qr is given by equation (2). The approximate constituent equations for the partial entropies are obtained by applying the average processes (8) to (2), yielding 0

n~ ;1c~) = p~ c~ n~ Oc~)]To + ~ n~ O'-xar + ~ vC*)(x' h~, t)/h,

(16)

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J. ABOUDI

which, according to equation (10), reduces to 0

To ~7(Ip) = nt p, c~a 0 (t) + To )'~ nl ~ x f~(') - To h S(x, t ), 0 To rlt2p) = n2 P2 ct,2 0(2~ + To )'2 n2 "~x (lt2l + TO )'2 S ( x , t ).

(17)

Then with equation (I0), one obtains, consequently T~ rl(+') = bll 0it) + bt20(2) + T ~

0

0

~ x fit1) + Tod2, ~x fi<2),

0 0 To r/(2p) = b,2 0(') + b22 0(2) + To dr2 ~xx t]tl) + To d22 OX u(2)'

(18)

with b,, ='n~ p~ c,~ + )'z To~E,

b,2 = -)'1 )'2 To/E.

(19)

According to equations (18), the entropy in each constituent involves the temperatures and displacement gradients of both constituents. Hence the constitutive equations (18) differ, too, from those proposed in the general theory in reference [4]. The partial heat fluxes are given by Qtx:'p) = ,1, Q~') = - n , k,, ~0 0 <').

(20)

4. MIXTURE STRESS EQUATIONS Applying the averaging process to the equation for u c~) in equations (4) gives the following stress equations" a2

Ill

Pl ~'~ 02

n2 P2 ~

0 /~(t) = _ax _ a (vI p )it

- p(x, t),

0

fi(z, = Oxx atx2p)+ p(x, t),

(21)

where (2) hp(x, t) _- -at2( i ) (x, . lq, t) _- trx2 (x, h2, t ).

(22)

The interaction term, p ( x , t ) , is derived as in reference [13], yielding, for the present firstorder theory, p(x, t) = 3tq lt2(f~ (t) - f~(2))/[h(pth2 + 112 h l ) ] "

'

(23)

This expression for the momentum transfer is the same as that evaluated by Stern and Bedford [6].

LAMINATED PLATE UNDER IblPULSIVE LOADS

193

5. MIXTURE ENERGY EQUATIONS Applying equation (8) to the energy equations (5) yields a 1:, o

a ,:~"

= -

Ot I

--

Ox

a?'~

- a~(x,

h~, t)/h,

(24)

for which the vanishing of Q~') at y, = 0 was utilized. Accordingly, the following energy equations are deduced: 0

0

To ~'~ q ( " ) = - Ox--Qt, v)_ e(x, t), a 0 To ~ q(~P)-- -- ~Ox O~2p)+ e(x, t),

(25)

he(x, t) = Qt2t)(x,ht, t) = -Qt2Z)(x, 112,t),

(26)

and

which is the heat transfer from one constituent to the other due to their thermal interaction. The interaction term e(x, t) in equations (25) is evaluated by considering the relation a 0 (*) = 0,

which yields

':[

(27)

hz

~o

y~ Qt2~)+k~y~O(~

dye=0.

(28)

Since Q(=) is asymmetric iny=, it can be expressed in the form Q~,)

,n(~) v_~ + ~:~(') ~ 2 ( 1 ) -" 2 ( 3 ) J~ a /6

+"

"*

(29)

Hence by integrating the first term in equation (28), integrating by parts the second term in (28) and neglecting terms of order e2, one obtains h~

Qt2")(x,h,, t) "~ + k,O(')(x,h=, t) - k,O(*)(x, t) = 0.

(30)

By subtracting equation (30) for ct = 1, 2 and recalling the asymmetry of Qt2")with respect to y, = 0 and the continuity of Q~') and 0 (~) across the interface, one obtains

he(x, t) = 3kl k2[0 (l) - Ocu)]/[hlk2 + h2 kl].

(31)

According to equation (31) the thermal interaction depends on the difference between the average temperatures of the constituents, whereas the mechanical interaction (23) depends on the difference between the average displacements. These expressions for the interaction terms are special forms of those given for these terms in the general theory [4]. 6. MIXTURE FIELD EQUATIONS The equations of motion and heat conduction of the mixture theory are obtained by substituting the partial stresses (14) in the stress equations (21), and the partial entropies (18) and heat fluxes (20) in the energy equations (25), to yield the following thermomechanical

194

J. ABOUDI

system of coupled equations for ~
a2

02

a

a

,h pt ~-i t~"' = c,t Ox 2 ~ fi(" + ct2 ~-~ # 2 ' - dtt ~x 0(!' w dr2 ~x O(2'-p(x,t), a2

n2p~ ~

O

a2

a2

~'~' = c,, ~ x ~ ~"' + c,~

a--7

O

o

a

0"' - d , ~ x

0'

a2

(33)

fit2, = nt k~ ~x 2 0"' - e(x, t), (34) a2

bla ~ 0('' + b22 ~ 0 `2' + d~z To ~

0 '~' + ; ( x , t), az

b,t at O('' + btz at 0(2' + art To O-ff~t a"' + d2t To ~ a

a

~,', - d,, ~

a2

(32)

f~(" + d2z To ~

a2

fi(2, = n2 k2 ~

0 `2' + e(x, t). (35)

For a specific problem, these equations must be solved subject to the appropriate initial and boundary conditions. In the sequel, equations (32)-(35) will be applied to the problem of a thermoelastic laminated plate under impulsive mechanical and thermal loadings in order to investigate the resulting propagating stress waves and temperature behaviour within the plate. 7. EFFECTIVE MODULUS THEORY In the framework of this theory, the whole composite is replaced by a homogeneous anisotropic medium with average moduli and density. Consequently dispersion and attenuation caused by the inner boundaries and regions are neglected in this theory and the results provided are valid for long wavelengths (see reference [2]) for elastic laminated medium. In the present case, where the loads are applied in the direction of the layering, the thermoelastic laminated medium is governed by the following constitutive equations:

O'2~

O" ~ 0"(1~ +

___ ~70o, + ~(2p,,

Q = Q,O, + Q(2~,,

(36)

where a, t/and Q are, respectively, the effective stress, entropy and heat flux in the medium. For the single displacement component u(x, t) and temperature distribution O(x, t) a2

a

P 0 - ~ " = ~x ~ , 0

0

To aS n = - ~x Q'

(37)

with the effective density of the medium p = n~ Pl + n, P2"

(38)

LAMINATED PLATE UNDER IMPULSIVE LOADS

195

Substituting equation (36) in equation (37) gives the following thermomechanical coupled equations for u(x,t) and O(x,t): a2

a2

a

P ~iT u = C -~x2 U - d ~x O, a

a"

pc~ ~ O + Tod ~ - ~

a~

u = k -~x2 0 ,

(39)

where C = Cll + 2Ct2 q- C22,

d = d l l + d l , + d,l + d,,, k =

nl kl + n2 k2,

pco = bll + 2b12 +

(40)

b22.

Equations (37)-(40) form the effective modulus theory for the anisotropic thermoelastic medium. In the special isothermal case they reduce to the first equation of (39) as given by Postma [7] (without the thermal term). Consequently, the effective modulus can be regarded as a mixture theory of zero order in which one average displacement and temperature exist in the whole medium, i n t h e absence of any mechanical or thermal interactions between the constituents. 8. APPLICATION TO A LAMINATED PLATE SUBJECTED TO IMPULSIVE LOADS As an application of the mixture theory, consider a laminated plate of width I occupying the region 0 < x < l, -co < y < co and consisting of two alternating thermoelastic layers at rest at the equilibrium temperature To. At time t = 0, time-dependent uniform temperature and normal stress start to act on the whole surface x -- 0. In the framework ofthe mixture theory, the laminated plate is replaced by a single equivalent continuum composed of two mechanically and thermally interacting constituents which are permitted to possess their individual average displacements and temperatures at each point in the region. The equations of motion and heat conduction are given by expressions (32)-(35) with the interaction terms given by expressions (23) and (31). The initial conditions are 0 ~(~, = O/ fit~) = ~ t <~0, ct = l, 2.

)

0 (~, = 0

(41)

Two separate cases of mechanical and thermal loadings are considered, together with boundary conditions, as follows. (a) Mechanical loading. For this case, a]~P)(O, t) = - % G(t),

ct = 1,2,

(42)

where G(t) describes the time-dependence of the applied stress and ao is an amplitude factor which has the dimension of a stress. At x = / r i g i d l y clamped boundary conditions are chosen: i.e., fit,)(l, t) = 0,

ct = 1,2.

(43)

Both boundaries are kept at the equilibrium temperature, To, and hence O(')(x,t)=O

at

x = 0,/,ct = 1,2.

(44)

196

J. ABOUDI

(b) Thermal loading. Here t) = 0

a(~)(/, t) = 0 , ~ = 1, 2 ,

(45)

Oc=~(O,t) = OoG( t l Oc~(l, t) = 0 where 0o is an amplitude factor which has the dimension of a temperature. In reference [8] a detailed description was given for the numerical method of solution of the isothermal equations of motion together with discussion of its consistency, stability, convergence and its accuracy. In the present thermoelastic case one needs to solve the coupled system (32)-(35). In reference [15], a numerical solution to non-linear dynamic thermoelastic equations for a single material was presented. In the special case of the linear thermoelastic problem, the numerical scheme was checked in some special cases where analytical solutions are available and it was shown that the proposed numerical method furnishes good results. It is not necessary to describe here the numerical schemes appropriate to the present linear system (32)-(35) since, apart from the interaction terms, they are basically similar to the linear case as described in reference [15]. The interaction termsp(x, t) and e(x, t) of the mixture theory, add no essential difficulty to the method of solution. As in reference [8] the spatial increment Ax was chosen equal to h]50. The temporal function G(t) was also chosen as in reference [8]. It rises smoothly from zero at t = 0 to I at t = 2A and then falls back to zero at t = 4A (see Figure l(b)). The.parameter 4A determines the duration of the applied normal stress on the plate surface or the prescribed temperature (see equations (42) and (45), respectively). The various parameters of the reinforcement (~ = 1) and the matrix (~ = 2) were chosen as in reference [14]. They are as follows:

pl/lt2

= 50,

kffk2=lO,

Pl/P2 = 3, f12/fl~=l'5,

vl = 0"3, e~=0"03,

v2 = 0"35 e2=0"4,

where v, are the Poisson ratios and e~ are the coupling parameters defined by

e~=?lTo/[p~c~.~E,],

ct= 1,2.

(46)

8.1. RESULTS

Results for the stresses a t~ =- a ~ p~and temperatures 0 c~ (ct = I, 2) were obtained at x/h = 0.1 within the plate as functions of the non-dimensional time ct2t/h, where ct2 -- [(22 + 2~2)/p2] 1z2 is the compressional wave velocity in the matrix material. The pulse time duration and the plate thickness were chosen, respectively, as 4~2A/h = 0.3, l/h = 1, and results for the two reinforcement ratios hdh = 0-3, 0"8 were calculated. (a) Mechanicalloading. In the isothermal case the stress expressions are given by equations (14) with 0 c~ = 0. As indicated before these relations differ from those given by Stern and Bedford [6] and employed in reference [8]. The latter are a

0 atl~p) = (c12 + c2z) ~x •t2,.

(47)

LAMINATED

P L A T E U N D E R I M P U L S I V E l.J !

In equations (47) the partial stresses in the constituent depen 1 gradient in that constituent whereas in equations (14) they gradient in both constituents. In reference [8] satisfactory agree] between the mixture theory based on equations (47) and the ~ ~, matrix and reinforcement materials chosen here, a comparison 1 G. based on equations (14) and (47) was performed, showing sligh o .-: r It was found that the deviation between the corresponding n inforcement ratio, hl/h. Nevertheless, even in the case of high rei the deviation is still insignificant. For hJh = 0.3 the resulting c~ It can be concluded, therefore, on the basis of reference [8] . . . . . . . . . . . . . v......... yield normal stresses which are in good agreement with the complete theory of elasticity. I

i

,

f

l

! I

b

b -0-~

-I 000:

0-~

O.OOt

0-1

ko

,%

IQt) C

~ . . . .

-0-1

-0-001 0

0"5 a a t/h

0

I

05 aat/h

Figure 2. Mechanical loading. Stresses and temperatures variations obtained from the mixture theorY (solid lines) and the effectivemodulus theory (dashed lines) in a plate with the reinforcement ratio hJh = 0.3. In Figure 2 the stresses trt')/Go and the induced temperatures O(')/To in the two constituents are shown and compared with those based on the effective modulus theory (EMT). It is clearly seen that E M T yields different amplitudes, shapes and arrival times for the main pulses than those based on the mixture theory, especially in the matrix region (~ = 2). Moreover, even the order of magnitude of the excited temperature 0 c2> is not predicted by EMT which yields about one hundred times lower values. In the matrix region the temperature is about sixty times higher than in the reinforcement region (due to the volume ratio and contrastbetween the material properties). This shows clearly the effect of microstructure and the importance of a theory which yields information on the field in every constituent. In Figure 3 the stresses and temperatures are shown for hl[h = 0.8, together with those obtained from EMT. The effect of the interaction between the constituents is well seen in this case of high reinforcement ratio. Here, more appreciable values for the temperature in the matrix region are obtained which are again entirely unpredicted by EMT (which yields only 0.002 of the temperature there). In the reinforcement region (~ = 1) unsatisfactory results for the temperature are obtained by EMT as compared with the previous ease in Figure 2.

J. ABOUDI

198

I

I

0 I5 ~

05

/

/\

/

~

I "~

%

I I

~.

51- "~ ~ I I

o

-0.5

,

;1 ..... ,

\

;----~

I

~ ;

-I

o.21 ~

0.002

ko 0 -001

o-H

w

ol

0

-O-O01

O.It

I 0-5

O5

aatlh

a 2 t/h

Figure 3. As Figure 2 for the reinforcementratio h~lh=0"8.

20

IO

o b

%

t~-I0

-I

-20

-20

0.]

0"3

0"~

~o

0-1

0.2

0"1

0"5

a2t/h

O

jJ'r~~ 0"5 aa

t/h

Figure 4. Thermalloading. Stressesand temperaturesvariations obtained from the mixture theory (solid lines) and the effectivemodulus theory(dashed lines) in a plate with the reinforcementratio hllh = 0-3..

LAMINATEDPLATEUNDER IMPULSIVELOADS

199

By tracing the induced temperature (not shown here) in an isotropic plate made of the thermoelastie material c( = 2, and those obtained when the reinforcement ratios are 0.3 and 0.8, one can conclude that the effect of the reinforcement is to increase the resulting temperature in the matrix region by increasing lh/h. (b) Thermal loading. In Figure 4 the temperatures and stresses are shown for the thermal loading boundary conditions (45) when lh/h = 0.3, together with the corresponding EMT results. Whereas the orders of magnitude of the stress and temperature are well predicted by E M T in the reinforcement region, their behaviour in the matrix region (occupying 70 ~o of the plate) is completely different. Indeed EMT yields about 8 times higher amplitudes for the stress and only 0.2 of the temperature as obtained by the mixture theory. This shows the effect of microstructure in the present case. Note that the scale factor EI/E2 is 40.3.

tO

-I0

'

U

,o[-

A

1

% tu- -io

-2C

-20

0-~

03

0-2

02

ill(i) 0-I

0"1

0

0

0-5

0

0"5

a2t/h

a z tlh

Figure 5. As Figure 4 for the reinforcement ratio ht/h = 0-8. In Figure 5 the resulting stresses and temperatures are shown for hJh = 0.8. Here, good agreement exists between the mixture theory and E M T in the reinforcement region. In the matrix region, on the other hand, E M T predicts about 48 times larger amplitudes for the stress and only 0.01 of the temperature, which are even worse than the previous case. Finally, the effect of the reinforcement is to decrease the stress in the matrix and to increase it in the reinforcement, as the value of the reinforcement ratio increases. ACKNOWLEDGMENTS I wish to thank Professor G. Eason for the visiting appointment at the University of Strathclyde where this work was performed. The computations reported in this paper were carried out on the IBM 370/155 of the Edinburgh Regional Computing Center through the University of Strathclyde Computing Center. This research was performed under contract AFOSR-71-2143 of the U.S. Air Force.

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J. ABOUDI REFERENCES

1. C.T. SUN, J. D. ACHENBACHand G. HERRMANN 1968 Journal of,4pplied Mechanics 35, 467-475. Continuum theory for a laminated medium. 2. C.T. SON, J. D. ACHENBACHand G. HERRMANN 1968 Journal of Applied Mechanics 35, 408-411. Time-harmonic waves in a stratified medium propagating in the direction of the layering. 3. R . A . GROT and J. D. ACHENaACH 1970 Acta Mechanica 9, 245-263. Linear anisothermal theory for a viscoelastic laminated composite. 4. A. BEDFORD and M. STERN 1972 Acta Mechanica 14, 85-102. A multi-continuum theory for composite elastic materials. 5. T. R. STEEL 1968 International Journal of Solids and Structures 4, 1149-1160. Determination o f the constitutive coefficients for a mixture theory of solids. 6. M. STERNand A. BEDFORD 1972 Acta Mechanica 15, 21-38. Wave propagation in elastic laminates using a multi-continuum theory. 7. G. W. POSTMA 1955 Geophysics 20, 780-806. Wave propagation in a stratified medium. 8. J. ABOUDI 1973 Journal of Sound and Fibration 29, 355-364. A mixture theory of the rseponse of a laminated plate to impulsive loads. 9. J. ABotJD1197361ternationalJournalofSolidsandStructures9,217-232. Stress wave propagation in a laminated plate under impulsive loads. 10. G. A. HEGEMIER 1972 in Dynamics of Composite Materials. On a theory of interacting continua for wave propagation in composites. New York" American Society of Mechanical Engineers. 11. M. BEN-AMoTZ 1973 International Journal of Engineering Science l l , 385-396. Continuum theory of wave propagation in laminated composites. 12. M. BEN-Ar,IOTZ 1971 International Journal of Engineering Science 9, 1075-1085. Continuum model o f heat conduction in laminated composites. 13. G. A. HEGEMIER, G. A. GURTMAN and A. H. NAYFEH 1973 International Journal of Solids and Structures 9, 395-414. A continuum mixture theory o f wave propagation in laminated and fiber re-inforced composites. 14. C. S w 1971 International Journal of Solids and Structures 7, 1363-1373. Thermoelastic waves in a periodically laminated medium. 15. J. ABOtJDI and Y. BENVENISTE1974 (To appear) International Journal of Solids and Structures. Finite amplitude one-dimensional wave propagation in a thermoelastic half-space.