Inelastic Behavior of Metal Matrix Composites

213 CHAPTER 8

I N E L A S T I C B E H A V I O R O F M E T A L M A T R I X COMPOSITES 8.1 INTRODUCTION A metal matrix composite consists of an elastoplastic matrix reinforced by fibers. The fibers are usually taken as perfectly elastic materials. The overall behavior of the composite exhibits an appreciable amount of inelastic deformation which is caused by plastic flow of the matrix. Although the existence of the fibers is the principal source for the composite high stiffness and strength, their presence has a relatively small effect on the overall stress level that causes the onset of plastic yielding. This fact was already noticed in the examples given on the initial yield surfaces of boron/aluminum composites. It was shown that the applied stresses which cause yielding of the composite were of the same order of magnitude as the yielding stress of the unreinforced matrix. Consequently, in order to take advantage of the high strength and stiffness of the metal matrix composites, it is necessary to admit working loads that exceed their elastic limits. However, the existence of an extensive plastic deformation does not imply that metal matrix composites may experience large plastic deformation before failure. In most cases the failure strain of the composite will be of the same order of magnitude as that of the elastic fiber. The latter are within the range of small strains. If, for example, the failure strain of the fiber is ten times the strain of the matrix at the yielding point, it follows that the plastic behavior of the composite will dominate its response as it occupies the major portions of the total strength range. A micromechanical model would allow the determination of the overall response of metal matrix composites from the properties of the constituents and their detailed interaction. The inelastic constituents are usually governed by incremental flow rules, and may be elastoplastic with isotropic, kinematic (directional) or mixed hardening. Accordingly, any a priori assumption about the resulting overall response of the composite might be un-realistic. Appropriate micromechanical theories would provide the initial and subsequent yield surfaces of the composite, its effective flow rule, hardening law, unloading paths and the effect of a hydrostatic loading on its inelastic behavior. In this chapter the method of cells will be employed for the prediction of the overall behavior of metal matrix composites. Closed form constitutive equations would be established for the effective behavior of unidirectional metal matrix composites under isothermal conditions and elevated temperatures. The behavior of metal matrix laminates will be determined from the micromechanically developed constitutive law.

214 8.2 CONSTITUTIVE EQUATIONS OF PLASTICITY In the present section, the relations between stress and strain when plastic flow is occurring, are discussed. Due to the dependence of the plastic strains on the loading path, it is necessary, in general, to compute the differentials or increments of plastic strain throughout the loading history and then obtain the total strains by integrations. A plausible constitutive relation would be one where the infinitesimal increments of plastic strain, measured from the current state, are given by AÎ

L

= f i (ja k )l Δλ

(8.1)

where fy are functions solely of the stresses a k^, while Δλ is an infinitesimal proportionality factor which is determined in the process of the solution and varies from point to point throughout the loading history. By employing the representation theorem for isotropic tensor-valued functions and the incompressibility of the plastic deformations, it can be shown (Hunter, (1983)) that the rate of plastic strain is given by

where Χλ is related to λ via a constant. Eqn. (8.2) is called the flow rule, and the function f is known as the plastic potential. The dots indicate dif­ ferentiation with respect to time. Since the flow rule is homogeneous in the rates of stress and strain, however, it is possible to replace time with any other parameter which increases monotonically with time, and is, therefore characteristic of the state of deformation. Drucker (1951) introduced the definition of a stable plastic workhardening material as one which satisfies both of the following conditions (postulates): (a) (b)

the plastic work done by the external agency during the application of the additional stresses is positive, and the net total work performed by the external agency during the cycle of adding and removing stresses is non-negative.

These definitions imply that (a) (b)

a plastic potential function exists, and it is identical to the yield function, which must represent a convex surface in stress space. For the particular case of the von Mises yield criterion, the resulting flow rule can be written in the form

215 Ç *

= À 5ϋ

(8.3)

where λ is related to λ ΐ5 and s^ denote the stress deviator. The PrandtlReuss equations follow from writing the total strain rate as a sum of an elastic (reversible) and plastic (irreversible), parts as follows L

έη = £ U

ij

L

+ £

ij

(8.4)

where the elastic part is given in terms of the rate of stresses by the Hooke's law for elastic materials. When an initial yield surface is known, the rule of work-hardening def­ ines its modification and changes during the process of plastic flow. A number of hardening rules have been proposed such as perfect plasticity, isotropic hardening, kinematic hardening and mixed hardening. For a perfectly plastic material the initial yield surface does not change at all, which is the simplest hardening assumption. The next simplest hardening assumption is that the yield surface maintains its shape, while its size expands uniformly without distortion and translation as plastic flow occurs, implying isotropic hardening. Usually, the increase in size is con­ trolled by a single scalar parameter that depends on the accumulated plastic strain or accumulated plastic work. Isotropic hardening is not always a good assumption since some materials develop anisotropic and Bauschinger effects that significantly change the size, shape and origin of the yield sur­ face. The kinematic hardening rule assumes that during plastic deforma­ tion, the yield surface does not change its size or its shape but merely tran­ slates as a rigid body in the stress space. A combination of kinematic and isotropic hardening would lead to the more general mixed hardening rule. L In the classical theory of plasticity the plastic strain, e?j , is dependent on the history of loading but is considered time independent. Time depen­ dency is introduced using the creep strain e?. The classical approach to inelude creep of the material is to incorporate the rate of creep strain

in

eqn. (8.4) as an additional term. The constitutive equation for the creep strain rate is defined to depend on the stress, strain and temperature. Such a constitutive creep law must express the constancy of volume and the lack of influence of the hydrostatic state of stress that have been observed experimentally during the creep process.

8.3 UNIFIED THEORIES OF VISCOPLASTICITY Since the classical plasticity theory is rate independent, time dependence is introduced through phenomenologically developed creep models. Thus, the

216 two terms, creep and plasticity, are independent and there is no influence of plasticity on creep or creep on plasticity. It is important, on the other hand, that constitutive equations have coupling between creep and plasticity terms. This appears to be difficult when using separate creep and plastic strains. The need to model the rate dependence inherent in nonrecoverable (plastic) deformation of many materials, particularly at elevated temperatures, led to the development of elastic-viscoplastic constitutive theories. Unified theories of viscoplasticity is a category of elastic-viscoplastic constitutive models which are based on the concept of considering both elastic and inelastic deformations to be generally nonzero at all stages of loading. The inelastic deformations include the creep effects of the material response. Thus, in the unified theories, the classical separation of strain into a time independent plastic strain and a time dependent creep strain is replaced by a total inelastic strain. Furthermore, there is no yield criteria that separates the inelastic part of deformation from the elastic part, since both occur simultaneously. The lack of a yield condition is one of the main advantages and attractiveness of the unified theories as compared with the classical theory of elastoplasticity. The latter involves different constitutive relations which are applied in the various stages of deformation. These relations include the elastic stress-strain equations before yielding, a yield criterion, a plastic flow rule and the elastic stress-strain equations again about a new reference configuration during unloading. It should be mentioned that a unified constitutive relation includes the more important manifestations of inelastic behavior in the same set of equations, e.g. strain rate dependent plastic flow, creep and stress relaxation. A review of various unified viscoplasticity theories has been presented by Chan et al. (1984). Some of these theories were evaluated at various circumstances, see Ramaswamy (1986), Eftis et al. (1989) and Hartmann (1990) for example.

8.4 BODNER-PARTOM VISCOPLASTIC EQUATIONS For an elastic-viscoplastic material, the total strain rate êy is assumed to be decomposable into elastic (reversible) and plastic (irreversible) components υ

ij

ij

where the elastic component is given by the time derivative of Hooke's law. For the inelastic strain rate component, the isotropic form of the PrandtlReuss flow law is assumed to be applicable, even as an approximation, under loading conditions generating anisotropic (directional) hardening. That equation is £

L

IJ

= As

(8.6) ij

217 where Sy is the deviatoric stress and A is the flow rule function.

Eqn. (8.6)

indicates plastic incompressibility, i.e., cj^* = 0. An approximate method of treating directional hardening is to include a scalar effective value of that parameter in the scalar function A of the isotropic flow law (8.6). That equation could then be considered as "incrementally isotropic" since A would depend on the loading history and the current stress state at each stage of the loading history, Bodner (1987). Squaring eqn. (8.6) leads to

L D^ =

l where D £

L = | £

A2

J2

(8.7)

L £

and J 2 =

|

SySy

are the plastic strain rate and

deviatoric stress invariants respectively. The relation governing the inelastic deformations is referred to as the kinetic equation which is taken as (Bodner and Partom (1975))

L DP

n

= D 5 e x p [ - ( A V J 2) ]

(8.8)

where

A2

=1

2 Z(

S ± I )l/n

In eqn. (8.8), D q is the limiting strain rate in shear for large J 2, η is a mat­ erial constant, and Ζ is interpreted as a load history dependent parameter that represents the hardened state of the material with respect to resistance to plastic flow, and considered to be an internal variable. Combining eqns. (8.6), (8.7) and (8.8) for a uniaxial stress condition gives

L £

=

V3

D 0 exp Γ- \ n , ( Z V ^ / l L

J

s g n ( a n)

(8.9)

where n x = (n+l)/n. It can be easily verified that the parameter η controls the strain rate sensitivity and also influences the overall level of the flow stress. Equation (8.9) has the desirable feature that ^ f / D 0 is small, essen­ tially negligible, at low values of ση/Ζ until a threshold value is reached and it then increases with a slope determined by n. The equation of evolution proposed for isotropic hardening is

218 (8.10)

Z(t) = m (Z1 - Z(t)) W p( t ) / Z 0 where Z 0, Z l5 m are additional inelastic parameters, and

Wp="iji

(8.11)

L

is the plastic work rate which is taken as a measure of hardening. eqn. (8.10) it follows that Z(t) = Ζλ - ( Z ^ Z o ) exp [-m W p( t ) / Z 0]

From

(8.12)

The incremental elastic-viscoplastic equations can be integrated for the case of uniaxial stress, constant plastic strain rate and isotropic hardening to give an analytical stress-strain relation which is a function of strain rate, Merzer and Bodner (1979). To this end, consider a constant plastic strain rate ^

= R x.

It follows from eqn. (8.9) that the term ση/Ζ

would be

constant, say K x, i.e., 2D êPL = i R _ =o n

p e

(8.13)

x

or that -|-l/2n f £ n ( 2 D / V 3 R j ) = 0

K1

(8.14)

From eqn. (8.12), it follows that σ η = K1Z

= K1 [ Z i - i Z j - Z o ) exp (-m W p/ Z 0) ]

Differentiating eqn. (8.15) with respect to

dail

L

d£

(8.15)

gives

m K ( Z ^ Z o ) exp [-m W / Z ] σ x 0 η o

(8.16)

dW pL = σ has been used. Combining (8.15) and (8.16) where the relation —Εη de gives

11

219 ά σ ι PL ι

d£

11

m_ =

0

(KZ xZ

^

^

(8.17)

An integration of eqn. (8.17) leads to an analytical stress-plastic strain rela­ tion 1 / σ η = 1/CKiZi) + Cx exp ( - K ^ m

€™7 Z 0)

(8.18)

where Cl9 the constant of integration, is determined from the limit of zero plastic strain at which eqn. (8.18) yields 1/σ 0 = l / i K ^ Z i ) + C x where σ 0 is the initial value of the stress level assuming the strain to be fully plastic, and could be interpreted as a yield stress. At this stress level = σ 0/ Ζ 0

(8.19)

It follows that Cx = ( Z i - Z o V O ^ Z o Z i )

(8.20)

It should be noted that in the calculated stress-strain curve based on the constitutive equations, both elastic and plastic strain components exist at all stages, and the total strain tends to the fully plastic state beyond a transi­ tion region. For a large plastic strain, eqn. (8.18) gives that Z 2 = σΒ/Κλ

(8.21)

where as is the saturation value of stress.

EL

PL

Combining the elastic and plastic strain components e n = e^j + e™ where = ση/Ε (Ε is the Young's modulus), leads to the general uniaxial stress-strain relation σ

ni

/

ιι

χ

(8.22)

'11

A plot of eqn. (8.22) gives a typical stress-strain curve which would depend on the applied strain rate Rx.

220 Consider the case of a material directional hardening, i.e., deformation induced resistance to plastic flow due to nonproportional loading. Here, the isotropic form of the flow rule (8.6) is retained, but the scalar harden­ ing variable Z(t) in eqn. (8.8) is replaced by an effective scalar, Z e (f tf) given by (Aboudi (1983)) »t z e„ ( t ) = Z 0 + q

JO

3

ft

Z(r) dr + (1-q) Y ry Z(r) r ^ r ) dr /r-i JO i,J=l

(8.23)

where q is a new parameter, r i (t) j = a i (jt ) / [ a k (£t ) a k (£t ) ] i A

(8.24)

are the direction cosines of the current stress state, and Z(t) is defined by eqn. (8.10). According to eqn. (8.23) the hardening measure is divided into a sum of isotropic and directional parts with the material constant q deter­ mining the relative weight of each part. For q=l the special case of com­ plete isotropic hardening is obtained, whereas q=0 corresponds to fully directional hardening. Furthermore, under a proportional loading ry(t) in eqn. (8.24) are constants, and eqn. (8.23) implies that Z e (f tf) = Z(t) as it is required. The description of the six parameters which characterize, in the frame­ work of the unified theory of Bodner and Partom, the inelastic behavior of the material can be summarized as follows. The parameter Z 0 is related to the "yield stress" of the material in simple tension (eqn. (8.19)) and Zx is proportional to the ultimate stress (eqn. (8.21)). The material parameter m determines the rate of work-hardening (eqn. (8.12)), and the rate sensitivity of the material is controlled by the constant η (eqn. (8.9)). It should be noted however that with η = 10, (say), the elastoplastic response would be essentially rate independent for strain rates less than 10/sec. The constant D 0 is the limiting 4strain rate (eqn. (8.8)), and usually it can be arbitrarily chosen as D 0 = 10 /sec. Finally, the parameter q is the hardening measure (eqn. (8.23)). The material constants of several metals and metallic alloys are given in Bodner (1987). As an illustration of the characterization of an elastoplastic material by the unified theory, Fig. 8.1 exhibits the uniaxial stress-strain curve
221 TABLE 8.1 Material constants of a commercially pure titanium (isotropic in the elastic region with Ε, ν being the Young's modulus and Poisson's ratio, and aniso­ tropic hardening in the plastic region)

E(GPa)

ν

η

m

Z c( M P a )

Ζ ! (MPa)

q

_1 D 0 (sec)

120

0.34

1

350

1000

1400

0.05

ίο"

4

Let us consider the effect of temperature dependence on the inelastic parameters of the viscoplastic material. This was found (Merzer and Bodner (1979)) to be significant only for the rate dependence parameter η in eqn. (8.9). With η decreasing with increasing temperature, the flow stress level decreases and rate sensitivity of the material increases. It fol­ lows that as a result of this temperature dependence of n, the stress-plastic strain rate relations become a function of both strain rate and temperature. In Table 8.2 the elastic and inelastic constants of 2024-T4 aluminum alloy (assuming isotropic hardening) are given at five different temperatures ranging from 21.1°C (room temperature) to 371.1°C. At each temperature the inelastic properties are characterized in the framework of the unified theory. In Fig. 8.3 the computed uniaxial stress-strain curves of the alumi­ num alloy based on the constants of Table 8.2 are shown at the five tem­ perature levels for an imposed strain rate of 0.01/sec. For intermediate temperatures, the temperature-dependent properties (i.e. Ε and n) can be determined by a linear interpolation.

222 τ4-

TITANIUM

^"^^~

ί Iω

ι/ <-/—{

/ Q

-ί

/

Ι/

/ o . O I /s 1

ι 1-/-/1

I

I

l^^^^

7

„€( % )

1 -4

Fig. 8-1. Uniaxial stress-strain response of a commercially pure titanium for two values of strain rates: e,, = 10/s and 0.01/s.

223

TITANIUM ISOTROPIC HARDENING S

1

ι

I

"

/ io.O\/s

tT" / 2

«

/

ι

I I = r-H

-I

/ /

' • h-/-

/1

Ο

/ /

I

I

I / „I II

€ „ ( %)

ι

2/

'

I L*s

Fig. 8-2. Uniaxial stress-strain response of a commercially pure titanium for two values of strain rates: kn = 10/s and 0.01/s. Isotropic hardening (q = 1) is assumed.

224

2024 - T 4

400, 0

β

3

^l48S°C

0

o

j


^

^

—

-

"

—

/

UJ

w i-

21.1 °C

\00\-

204.4 C 2 6 0 °C

f

371.1 °C 0.4

0

0.8

1.2

STRAIN ( % )

Fig. 8-3. Uniaxial stress-strain response at several temperatures of an alu­ minum alloy (2024-T4) which is characterized in Table 8.2. The applied strain rate is 0.01/s. TABLE 8.2 Material constants of the aluminum alloy (2024-T4) (isotropic in the elastic region with Ε, ν being the Young's modulus and Poisson's ratio, a is the coefficient of thermal expansion, and isotropic work-hardening material in the plastic region).

Temp. Ε (°C) (GPa)

21.0 148.9 204.4 260 371.1

72.4 69.3 65.7 58.4 41.5

u

0.33 0.33 0.33 0.33 0.33

a6 (10- /°C)

22.5 22.5 22.5 22.5 22.5

1 ZQ Dj (sec) (MPa)

4 io- 4 ΙΟ" 4 IO"4 IO- 4 IO"

340 340 340 340 340

"Lx (MPa)

435 435 435 435 435

m

η

300 300 300 300 300

10 7 4 1.6 0.55

225 8.5 INELASTIC BEHAVIOR OF LAMINATED MEDIA Consider a periodically bilaminated medium in which every layer exhibits an elastic-viscoplastic behavior, Fig. 3.38. For this type of a composite material, it is possible to derive closed form constitutive equations for its overall elastoplastic behavior. The derivation is exact and the resulting relations which describe the average behavior of the laminated composite depend solely on the properties of the layers and their relative thicknesses. In the special case of perfectly elastic layers, the derived constitutive relations provide the average stresses and strains in terms of the effective elastic constants as derived by Postma (1955) and given by eqns. (3.118). The present derivation is somewhat different from the approach which was previously given by Aboudi and Benveniste (1981). A representative volume element of the laminated medium consists of the two layers (a = 1,2), at the interfaces of which the tractions must be continuous: a °H= n

i-1,2,3

(8.25)

Under homogeneous boundary conditions the stresses and strains are constants in each layer. The continuity of displacements at the interfaces are fulfilled if 1 2 2 fc( ) ( ) fc« . eW W = ( (8 26)U e = e £€ €6) 22 *2

2'

33

33 ' *3 2 *3

2 ^°^

'

The average stresses and strains are given by h - V i oW + v 2 of e "ij -

i

v+

v

f

2 f

where v{ = d i/ ( d 1+ d 2) . It follows that

and = €

€ }= £ 22

22

e 33

e

22

6 33

33

(8.27) 828 <- )

226 4? - 4? " « »

(8-30)

The assumption that the inelastic layers are isotropic in the elastic region leads to (see eqn. (8.5))

<#

} =

λ

«4ί ϋ δ

+

2

"« ^

- "« 2

i

L ( û

°

(8

·

31)

L where Xa and μ α are the Lamé constants and * j j ^ is the plastic strain in the layer a. Eqn. (8.25) with i=l provides in conjunction with (8.30) that

1 (Xx + 2 M) l eg) + XX{122 + €33) - 2μ χ

= (λ 2 + 2 μ 2)

e^ )

eg) + λ 2( 6 22 + c M) - 2μ 2 ef^ ) 2

(8.32)

which together with

ν ι

€

+ € Ν 2 4 ? - "ιι

η

83 3 ( · >

form two equations for eg) , eg) . Substitution of the solution for eg) , eg) in eqn. (8.29) with i = 1, provides the first constitutive equation of the laminated composite e + e H 83 4 + ^11 = n * n i 2 (^22 ^33) " n ( · ) where e^ are the effective elastic moduli given by eqns. (3.118), and H n is the inelastic contribution given by

2 H n = 2 d 2( V 2 M) l( μ 26 ^ where Δ = [ ( λ ^ μ ^

+ {Χ^ΐμ^ά^

) -

p μχ e W)

(dx d+2) / Δ + 2 μι

eJf-W

(dj+d^.

Similarly, the use of eqn. (8.27) with i=j=2, and the solution for eg), eg) gives the second constitutive relation

σ 22 where

= e 6 + e 6 + e € 12 11 22 22 23 33 " ^22

(8.35)

227

ζ^)/(ά ά )

+ μ2ά2

H 22 = 2 ( μ 1ά 1

ϊ+

2(μ 2 e*W - μ,

2

+

(Χ,-Χ2)

Similarly, σ 33

= eC + e6 + e C 1 2 l l 2 3 22 2 2 3 3 " ^ 3 3

where

H 33 is given by the same expression as H 22 but with ( a ) latter replaced by < ^ .

(8.36) n e * *

The overall behavior of the composite in shear is obtained as follows. Eqns. (8.25) with i = 2 and (8.28) with i = 1, j = 2, form two relations for the determination of

and e ^ .

sion of σ 12 yields

Substituting the solution in the expres­ 2

^12 =

e

H

83 7

4 4 *12 " 1 2

( ·

)

where L H 12 = 2 „ 1ά 2( μ 2 e™W - μ, £ W ) /

+ 2μχ cJJ-W

The constitutive relation for σ 13 - € 13 is similar to eqn. (8.37). Finally, σ =e e e 23 ( 22 " 2 3 ) 2 3 " ^23

(8.38)

where L)( 1 H 23 = 2(<11μ1 6 ^ 3 + d 2M 2

( d 1 d+2)

Eqns. (8.34)-(8.38) form the constitutive relations that represent the average behavior of the periodically laminated composite. For a given system of loading these equations are solved incrementally in a stepwise manner. The derived effective constitutive law which governs the effective behavior of the laminated composite can be written in the compact form P

σ = E (ê -

6

L )

(8.39)

228 where Ε = [ey] (whose structure is described by eqn. (3.58)) is the effective stiffness matrix which describe the overall transversely isotropic behavior of PL the composite ( x x is the axis of symmetry), and c is the average plastic strain of the composite which is given by

PL €

= Ε

1 Η

(8.40)

where Η = [ H n, H 2 , 2H 3 3 , H 1 , 2H 1 , 3 H 2 ]3. The plastic strain of the elastic-viscoplastic layer α is determined from the flow rule (see (eqn. (8.6))

L £ M = A a s.W

α

(8.41)

IJ

IJ

where ( A of) the inelastic βmaterial in this layer )a a is the flow rule function ( α ( )a and s. is the deviatoric part of σ. , i.e., s. = σί. > - cf$ ffi5/3. In the r ij

ij

ij

ij

kk

U'

framework of the unified theory of plasticity of Bodner and Partom, A a can be identified from eqns. (8.7)-(8.8) The step-by-step method of solution can be summarized as follows. Suppose that the laminated composite is uniaxially loaded in the x 2 - direc­ tion such that ση # 0 , and all other ay are zero, At any time t, the plastic strains in the layers,

eTM \t) a

y

are already known from integrating the evo­

lution equation (8.41) at the previous time step t - At (At is a time increment).

Thus it is possible to obtain the average strains l y (t) by solv­

ing the system (8.39) (ë n( t ) is known at any time). Consequently, the total strains in layers ei?) can be determined, from which the stresses aty in the layers can be obtained by using eqn. (8.31). The flow rule, eqn. (8.41), can be integrated to obtain the plastic strains at time t+At, to be used in the next time step. For example, the simple Euler method would yield the plastic strain in the form

L L 6*j M(t +At) = £? M(t) + At A e(t) sW(t)

(8.42a)

An improved procedure is given by the Euler-Cauchy method according to which the plastic strains are predicted at time t + At by using eqn. (8.42a), and then corrected by using the predicted values in the form

$**Ht + At)

= f-M(t) +

f

[ A a( t ) s W ( t )

7

229 + A e( t + At) W S ( t + At)]

(8.42b)

This timewise procedure can be continued to generate the average stressstrain response σ η - ë n of the composite. Other types of loading can be treated in the same manner. In Fig. 8.4 the response of a laminated composite ( d 1/ d 2 = 1) which consists of titanium and copper layers is shown for a uniaxial loading in the xx - direction, i.e., perpendicular to the layering direction. The figure exhibits also the stress-strain response of the titanium material only and the copper material only. In all cases the stress-strain curves were generated for an imposed strain rate of 0.01/sec. The material parameters of the titanium were given in Table 8.1, and those of the copper are given in Table 8.3. The response of the composite to other types of loading can be found in Aboudi and Benveniste (1981). 400

— 300

_/

d I

^

200

Itf 100

y

cu

I 0

0.5

I

I I

1.5

2

F„ (%) Fig. 8-4. Average uniaxial stress-strain response of an elastic-viscoplastic laminated medium made of titanium and copper. The response of the unreinforced materials are also shown for comparison. In all cases the applied strain rate is 0.01/s.

230 TABLE 8.3 Material constants of a copper (isotropic in the elastic region with Ε, ν being the Young's modulus and Poisson's ratio, and anisotropic hardening in the plastic region)

Dj'isec)

E(GPa)

ν

η

m

Z c( M P a )

Z x(MPa)

q

120

0.33

7.5

8.19

63

250

0.55

4

10"

8.6 INELASTIC BEHAVIOR OF FIBROUS COMPOSITES The elastoplastic response of unidirectional fiber-reinforced materials in which the matrix is inelastic, can be determined from the method of cells, Fig. 3.1. To this end, suppose that the continuous elastic fibers are transversely isotropic with the axis of symmetry oriented in the fiber direc­ tion (i.e., xx - direction). Their stress-strain-temperature relations are given by eqn. (3.3) with β = η = 1. The metal matrix is assumed to be iso­ tropic in the elastic region and inelastic with directional hardening in the plastic region. The plastic behavior is represented by the unified BodnerPartom theory. The average stresses

in the subcell (βη) are given by eqn. (3.7),

which in the present case provides C ll

< 11 + c j f )

WW

+ φ[ ) ~ T[^AT βΊ)

- 2μβΊ

Ljf)

C 12

ï 11

- ™= a

C 44 C

w - C44( ^ )

+ c( £ r> ^ τ )

c£»> +

-

m

. ι ^ )

lc

g7)L

2CJJT)

tjT)

L

ΔΤ

- 2,

βΊ

Ljf)

231 (8.43) In these equations the stress-strain relations are written for the general case of transversely isotropic inelastic constituents (the axis of symmetry is directed in the 1-direction), and cell.

are the inelastic strains in the sub-

For an initially isotropic phase, their time rates are given by the

Prandtl- Reuss rule (see eqns. (8.6)) L^> = A

(8.44)

ij

Ί where s . ^ are the deviators of the subcell average stress σ[? \

The flow

rule functions λβη are given by (see eqn. 8.8)

A „ - D ? * > expl-

2 7 ) (Ζ(^)) /[3γ |

where ή=Ο.5(η0 7+1)/η0 7 and J 2 ^ = s b e i n g

1/2 (8.45)

the second invariant

of the stress deviator s υ In the case of isotropic hardening matrix, the state variable Z ^ f ) is given by (see eqn. (8.12)) Z(fh) = Zf**> + ( Z ^ ) - Z[W)

exp

-m^w^/z^

7 where W ^ ^ is the plastic work in the subcell (βη).

(8.46)

Its time rate is given

by W^>«

i.^>

It shall be noted that for perfectly elastic fibers μη C = C 44^ 66^ for an isotropic matrix.

(8.47) = 0, and that

=

The overall constitutive behavior of the unidirectional composite is est­ ablished, as in the elastic case which was presented in Chapter 3, from the

232 continuity of displacements and tractions, in conjunction with the elasto­ plastic relations (8.43). The derivation leads to the following constitutive behavior (Aboudi (1987)) P L σ = E (e- € ) - U ΔΤ (8.48) PL and (3.61). The where Ε and U are given, respectively, by eqns. (3.58) and is given by average plastic strain of the composite is denoted by I , PL 1 6 = Β H (8.49)

Here, the matrix Β is given by eqn. (3.52), and ^ ~ [ H n, H 2 , 2 H33, H 1 , 2 H 1 , 3 H 2 ]3 where V H n = 2 Ql ^ C L J ? ) - L g » )

+2 Q 2 (μη L ^

1 + 2 Q 3 ( /m iL g » - μ( L g ) )

1 - μ, L g ) )

2 2 Q ( L £ ) g + 4 m M L> )

2 1 2 + 2[/xf v n L ^ + M( vm 12 g L> + v 21 L g ) + v 22 L g ) ) ] , 1 V H 22 = 2 Q1 ^ ( L g » ) - Lg»î) +2 Q 2( / , m L g ) - μ, l £ > ) 1 + 2 Q ^ m g L» - * L g ) ) + 2 Q 4 ^ ( L g ) - Lg»))1 + 2[μ, v u L g )

2 m g L> + (Mv 12

1 + 2Qs(M m Lg») - μ, L g ) )

1 +v 21 L g )

2 +v 22 g L) ) ]

1 2 2 Q ( L g ) L g )) + 4 m M

H33 = H 22 , 1 V H 12 = 2 [ v uc W L g )

2 c H ( v L g + 1 2 )

+v f Ll g » )

+v aL g » ) ) ]

1 f - 2 ( v nh 2c Î 4) - v 2 h1l H ) ( c ) L g ) - cHLffîVA C 4

.

,

233 V H 13 = 2 [ v l l2 C L^)

ο+ Η ( ν 1 Ι2^ >

v+ Ml £ ) + v 2 L2( f ) ]

- 2 ( v nh 2c 2 - W ^ X c g L ^ H 2s = 2 c g c H ( v u I # >

v+ 12 l £ >

- c&>L<«>)/A .

v+ 21 Lff)

v 23 ΐ £ > ) / Δ . , +

(8.50)

and where the various quantities were defined in Section 3.2. In the above equations μί and μτη stand for the shear moduli of the fibers and matrix respectively. The incremental procedure which is used to compute the composite response to a given type of loading is essentially similar to that described in Section 8.5 The inelastic micromechanical constitutive equations (8.48) were imple­ mented for the prediction of the response of unidirectional boron fibers reinforcing 6061-0 aluminum matrix (the fiber volume fraction is 0.46) under isothermal conditions. The obtained model prediction can be com­ pared with the experimental stress-strain response (Pindera et al. (1990)). The in-situ nonlinear response of the aluminum was backed-out using the experimentally obtained shear stress-strain response of the 10° off-axis ten­ sile coupon in conjunction with the Bodner-Partom plasticity model. The elastic constants and the inelastic parameters determined with the above methodology are given in Table 8.4. TABLE 8.4 Material parameters of the constituent phases of the boron/aluminum com­ posite used in the Bodner-Partom Model

E(GPa)

ν

Boron

400

0.20

Aluminum

72.5

0.33

1 D (sec) 0

Z 0(MPa)

Z a(MPa) m

η

100

190

10

4

10"

70

In Fig. 8.5, the predictions of the micromechanics model for tension and compression loading of 0° and 90° specimens are shown together with the experimentally generated data. A critical test of any micromechanical model is the accuracy of the transverse stress-strain response prediction. It is seen that the micromechanics model predicts the transverse response with

234 sufficient accuracy. The predicted shear stress-strain curves for selected off-axis tensile specimens are compared with measured data as shown in Fig. 8.6. Since the 10° off-axis coupon was employed to back-out the in-situ matrix res­ ponse, the experimental-analytical correlation for this coupon orientation is, not surprisingly, good. The correlation for the remaining off-axis configu­ rations is seen to be generally good. The model predicts, however, more softening than observed in the Iosipescu "pure shear" test. It seems that the employed Iosipescu shear specimen geometry produces results that contain components of both material and structural response of the specimen. Fig. 8.7 presents the comparison between theory and experiment for the normal stress-strain response of 10°, 15° and 45° off-axis coupons in the laminate coordinate system. The utility of the model is clearly demonstrated. In a recent study by Pindera and Lin (1989), the micromechanical con­ stitutive equations (8.48) (with ΔΤ=0) was employed to predict the response of two types of metal matrix unidirectional composites characterized by different microstructures, namely: boron/aluminum and graphite/aluminum. Comparisons with experimental data were given. The elastoplastic response of metal matrix composites discussed so far was temperature independent with ΔΤ=0. Consider a graphite/aluminum unidirectional fiber-reinforced material. The thermoelastic constants of the anisotropic T-50 graphite fibers are given in Table 8.5, and the properties of the thermoelastic-thermoinelastic aluminum matrix appear in Table 8.2 (the reinforcement volume ratio is 0.3). Assume that at the cure tempera­ ture T R=371.1°C the composite is at microscopically stress-free state. It is cooled from this initial state while it is kept at macroscopically stress-free conditions such that <^=0.

In Figs. 8.8-8.9 the average strains e n in the axial direction, and e 22 =e 33 in the transverse direction, as predicted by the model and a finite element solution (Hashin and Humphreys (1981)) are shown when the composite is cooled from the assumed cure temperature to room temperature (21.1°C) and then reheated to the reference temperature Tft. It is clearly seen that satisfactory agreement exists between the model prediction and the numerical solution. It can be observed that plastic deformation of the composite occurs during cooling at about 150°C due to the yielding of the matrix. The re-heat curves are almost linear indicating similarity to the thermoelastic prediction.

235

BORO N / A L U M I N UM

1

0 2 ~

0THEORY

/

EXPERIMENT

/

8 0 0- /

jt+—0°

SPECIMEN

4 0 0-/

/^^y~

'd 2 ^90 ~

° Ο

-

SPECIMEN

/

- 4 0 0-/

- 8 0 0- /

-1200- / -1. 0 - 0 .

/

/ 5 0.

0 0. ?

5 1.

0

x( % x)

Fig. 8-5. Comparison between predicted and measured axial and transverse response of a unidirectional boron/aluminum lamina. The constituents pro­ perties are given in Table 8.4.

236

100-

i^y ~„ 5 0 - / t e lb |

b BORON/ALUMINU

Ρ

M

Experiment

Ρ 0

Theory

° Iosipescu

•

2 5 jx - 10

° off-axis

•

ή 15

° off-axis

ο

Κ 45

ol

° off-axis Ο

I 0 .5 1.

I

I

0 1.

5 2.

I 0 2.

I 5 3.

I 0

€" (%)

| 2

Fig. 8-6. Comparison between predicted and measured axial shear response of a unidirectional boron/aluminum lamina. The constituents properties are given in Table 8.4.

237 BORON / ALUMINUM

6 0 0 -

Theory Experiment

500 a.

10°

4 0 0 -

~

0

/ //

300-

^

«κ

f

l b

" aoo«Λ

/

5

loo-

ε

ο

——

5

4

/ ^ " ^ j

Ζ

/

^^^y

-ιοο- 2 0 0 -

45°

- 3 0 0 -1.0

-0.5 Normal

0 strain

0.5 €

1.0

χχ (%)

Fig. 8-7. Comparison between predicted and measured off-axis response of a unidirectional boron/aluminum lamina. The constituents properties are given in Table 8.4.

238

-0.3r-

GRAPHITE /ALUMINUM

-0.25 01

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ S

0

—

Theory

^^^^^^

Finite element

O.L 0

^ *^^-^^^ ^ ^ ^ ^

~k 100

J— 200

L 300

i

TEMPERATURE (°C)

Fig. 8-8. Average axial strain developed in the free thermal expansion of a unidirectional graphite/aluminum composite, due to a temperature cycle of cooling and re-heating.

-I.Or

GRAPHITE / ALUMINUM

Γ x

-0.8-

^Ss^Hx

^s\. ^ \ ^

-04-

^ ^ ^ O

-0.2Theory Finite element

ol

0

1 100

^\>CN

1 200

e

1 300

— ^

ι

400

TEMPERATURE ( C)

Fig. 8-9. Average transverse strain developed in the free thermal expan­ sion of a unidirectional graphite/aluminum composite, due to a temperature cycle of cooling and re-heating.

239 TABLE 8.5 Materials properties for elastic T-50 graphite fibers (transversely isotropic).

(GPa)

(GPa)

388.2

GA

0.41

7.6

(GPa)

0.45

14.9

" A6

(io- /°c)

-0.68

dtp6 (io- /°c)

9.74

E A and vA denote the axial Young's modulus and Poisson's ratio, E T and ντ are the transverse Young's modulus and Poisson's ratio, G A is the axial shear modulus, and α Α, α τ are the axial and transverse coefficients of thermal expansion. When the composite is cooled from the cure temperature, residual microstresses are developed in the constituents due to the different thermal expansion coefficients of the fiber and matrix. As a result of these residual stresses, considerable plastic deformation may occur. In Fig. 8.10 the aver­ age residual stresses which develop in the fiber and matrix regions are shown in the cycle of cooldown from the cure temperature to room tem­ perature and re-heat to the cure temperature. It is readily observed from this figure that the axial residual stresses in the graphite fibers and alumi­ num matrix have the value -770 MPa (compressive) and 330 MPa (tensile), respectively. These are very high stresses which would influence the beha­ vior of the composite under subsequent mechanical loading. A detailed study of this effect can be found in Aboudi (1985a,b,c). In Aboudi (1985a,b), the responses to several types of mechanical load­ ing of unidirectional composites are given and discussed. In all cases the composite was supposed to be cooled down from its curing temperature to room temperature at which it was subsequently loaded mechanically. In this situation the effect of residual stresses on the subsequent behavior of the loaded composite can be studied by comparison with the corresponding case in which the composite is loaded from a free initial state with no resi­ dual stresses. We choose to present the results for a transverse normal loading of the unidirectional graphite/aluminum composite. Let the initial state of the composite be the previous room temperature residual stress and strain field which was developed after cooling down from the reference temperature. The composite response to transverse normal loading (in the x 2-direction, say) at room temperature is determined from eqn. (8.48) with

240 σ22 Φθ and all other ây are zero. In Fig. 8.11 the predicted average stressstrain curves in tension and compression are shown together with the corresponding finite element solution. The composite response to transverse normal loading at room temperature shown in Fig. 8.11 includes the effect of cooling from the reference temperature to room temperature. The resulting residual field at room temperature leads to different response of the composite when it is subjected to tensile or compressive normal loading at this temperature. This difference turns out to be pronounced in particular in the σ 22 - I n response curves, Fig. 8.12. It should be noted that the curves have been shifted such that zero average strains are depicted at room temperature at the initial stage of mechanical loading. When the effect of residual field at room temperature is disregarded, the resulting response of the composite to transverse normal loading corresponds to a loading from a stress-free initial state. This response (in tension or compression) is exhibited in Figs. 8.11-8.12.

GRAPHITE/ALUMINUM

o-

/

-200-

/

î

/

o-

/

-20-

-600-

/ /

/

/

î

/ /

—-400lb /

/ ,

/

^

l

/ /

-40b /

/

/

/ /

/

-60-

1 -800 0300r £ 200ΕΓ 100lb \ 0-

L

100D 2 0 0 3 0 0 •4 0 0 TEMPERATURE C O

- 8 0030i

I

\. \

100) 200 300 400 TEMPERATURE CO

20-

Ίε cm 10lb

o-

-I00

Fig. 8-10. Fiber and matrix stresses developed in the course of the free thermal expansion of a unidirectional graphite/aluminum composite, due to a temperature cycle of cooling and re-heating.

241

2 0G4R A P H I T E / A L U M I N U

M

200-

yf

TENSION

/Y

^ ^ C O M P R E S S I ON 160 ~

/ / '

σ 5

// 120 -

/ /

-

/ 80 -

//

40 -

/ / Theor J Finit

I 0

QL

1 0.2

y e elemen t 1 0.4

I 0.6

I 0.8

1.0

I ^ 2 2 I (%) Fig. 8-11. Comparison between the average stress-strain response c r 2 -2e 22 for a tensile and compressive transverse normal loadings at room tempera­ ture. (—) Method of cells, ( — ) finite element solution.

242

2 4 0 r-

GRAPHITE/ALUMINUM

2 0 0 Λ

TENSION

Z-FREE

V

/

\ \ 160-

^ 2

\

V \

/'

/ s

STATE

/

i

'

χ ί

'

À\

Vi /

120-

INITIAL

/

—

ι 80 -

/

///

X'

COMPRESSION

/

I7 fV Β /V

40 -

J// W lL 0 0

--Theory 1 0.02

1 0.04

Finite 1 0.06

element 1 0.08

1 0.1

lij, I (%)

Fig. 8-12. Comparison between the average stress-strain response σ 22 - ë n for a tensile and compressive transverse normal loadings at room temperature. (—) Method of cells, ( — ) finite element solution.

The predicted response and finite element solution of the unidirectional graphite/aluminum composite to axial shear loading is shown in Fig. 8.13 at various temperatures. In all cases the composite is assumed to be initially at microscopically stress-free state at any given temperature, i.e., without initial residual stresses. The response of a composite with initial field of residual stresses, on the other hand, is completely different. Suppose that the unidirectional composite at room temperature includes the initial field of stress and strain obtained by cooling from the reference temperature. Due to the existence of the initial field, a pure axial shear deformation does not exist. The res-

243 ulting σ 12 versus e 12 response at room temperature of the graphite/alumi­ num is shown in Fig. 8.14a where the corresponding response of the com­ posite with free initial state is given for comparison. In both cases the shear strain loading is applied at a rate of 0.01/s. The effect of the initial residual stresses is clearly exhibited. Dimensional changes of metal matrix composites which include initial residual stresses may result when they are subjected to axial shear loading. For the considered unidirectional graphite/aluminum composite, the obta­ ined dimensional changes are illustrated in Fig. 8.14(b). This figure shows the variation of the axial and transverse strains with the applied shear strain at room temperature. For a composite at a free initial state, this type of a dimensional change is zero.

Γ

1 8

Theory Finite e l e m e n t

2I.I°C

^ /

150-

/ / ^ ^

120 -

/ / ^ 2 0 4 . 4 °C ; / / /

ο

/

lb"

f//y

g/

GRAPHITE/ALUMINUM

///

IZ 1 0

260°C^

,ί/τ'

6 0 -

0

/

FREE

1 0.1

INITIAL

1 0.2

1 0.3

T

STATE

1 0.4

0.5

| (% 2)

Fig. 8-13. Comparison between the average axial shear stress-strain res­ ponse at several temperatures. In all cases the unidirectional composite is at free initial state. (—) Method of cells, ( — ) finite element.

244

Graphite /Aluminum 250rFree initial 200-

ο a.

\

1501

0 "

lb"

50-

/

/

/

^

^

state ^

^

^

/

0

f /

/ ( a )

L 0

I

I

I

I

I

0.2

0.4

0.6

0.8 0.8

I .0

ë | 2 (%) Fig. 8-14. (a) Average axial shear stress-strain response at room temperature when the unidirectional composite is initially at room temperature with a state of internal deformation developed by cooling. Also shown is the corresponding response when the composite is at free initial state.

245

-I.Op

^ - 0 . 9=

!

- ^ ^ ^

^

1

1

1

1

I

I

I

ι

6 0.

8 1.

- 0 . 8' - 0 . 2 r-

g

0

1

(b) 0 0.

2 0.

4 0.

ι

0

€ ,2 (% )

(b) The variation of the average longitudinal and transverse strain with the applied axial shear strain.

8.7 M A T R I X M E A N - F I E L D AND LOCAL-FIELD APPROACHES IN THE ANALYSIS OF METAL MATRIX COMPOSITES In Section 7.2 the use of the average matrix stress in determining the initial yield surfaces of metal matrix composites was discussed. Specifically, the micromechanics method of cells was employed to generate initial yield sur­ faces of unidirectional and multidirectional laminates under a variety of loading conditions using two different approaches. In the first approach, overall yielding of the composite was assumed to take place when the yield condition in the matrix phase was fulfilled locally in one of the matrix subcells of the representative volume element used to model the composite. The von Mises criterion was the chosen yield condition. The yield surfaces generated in this fashion correlated very well with finite-element predic­ tions obtained by Dvorak and co-workers (1973) for most loading direc­ tions. In the second approach, initial yielding of the composite was assumed to occur when the average stress in the entire matrix phase was fulfilled.

246 Comparison of the initial surfaces generated in this fashion with the corres­ ponding predictions based on local matrix yielding and finite-element ana­ lysis illustrated that significant differences may occur in the predicted sur­ faces for certain loading directions. In general, the initial yield surfaces obtained on the basis of the average stress in the entire matrix phase pred­ icted higher overall yield stresses than those obtained on the basis of local matrix yielding. For certain other directions however, the use of the aver­ age matrix stress in calculating overall yield surfaces appeared to be a good approximation. At elevated temperatures, on the other hand, the use of the average field in the matrix for the prediction of the composite initial yield surfaces appeared to be useless. In the present section, we extend the aforementioned micromechanical investigation to the prediction of inelastic response of metal matrix compo­ sites using the two different approaches to calculate inelastic strains in the matrix phase. The investigation is motivated by the recent development of approximate three-dimensional micromechanical models for the initial yielding and inelastic response of metal matrix composites based on the assumption that the entire matrix phase is uniformly strained (Wakashima et al. (1979), Pindera and Herakovich (1982), Dvorak and Bahei-El-Din (1982)). In such models, the effect of local stresses is neglected, and yield­ ing and subsequent inelastic response is thus governed in the first approxi­ mation by the average stress in the matrix. The present investigation was undertaken to quantify the effect of this assumption on the elastoplastic response of two types of metal matrix composites under selected loading paths using the micromechanics model of cells. It is shown that the use of the mean-field stress in the matrix phase to compute the effective behavior of metal matrix composites may produce results that significantly deviate from the response based on a more rigorous basis, and may lead to errone­ ous conclusions. Eqn. (8.44) is the flow rule for the plastic strains in the individual subcells of the matrix phase {β + η Φ 2). Alternatively, a flow rule based on the average stress in the entire matrix phase can be formulated as follows. Let us define the average stresses in the matrix constituent in the form

σΗ = v IJ

σ.(. ) + v A σί ) 12

12

*

21

21

lj

^

lj

+ V.22

°P

v v v / ( il2 + 2 i + 2 2)

(8.51)

where v 1 , 2 v 21 and v 22 are the volumes of the matrix subcells (i.e. v y = hjhj). The flow rule based on the average matrix stress that is the counter­ part of eqn. (8.44) thus becomes

m ij

v i

ij

β +

ηφ2

(8.52)

247 where

are the stress de viators of σ|ί"\ and A{m) is the flow function

of the matrix based on the average stresses σ~^· eqn. (8.45) but with j H = \ i ^ s H

Thus A^mj is given by

and

Z ( )m= Z1 + ( Z 0 - Z x) exp - m W H / Z 0

(8.53)

where Z 0, Zx and m are the inelastic parameters of the matrix material, and is the plastic work per unit volume of the matrix material such that its rate is given by m 1 > s H s!" ' (8.54) W< ) = A f m ρ K) ij ij 1 )2 2 )2 (22) We note that eqn. (8.52) implies that l i . = LJ = L . , contrary to the

previous formulation given by eqn. (8.44) in which the plastic strains in the matrix subcells are independent of each other. In Aboudi and Pindera (1990), stress-strain curves of two types of uni­ directional metal matrix composites were generated using the local-field and mean-field approaches outlined above for loading by different combi­ nations of externally applied stresses. In particular, the response to loading by longitudinal and transverse normal stresses, longitudinal shear stress and axisymmetric normal stresses applied separately in the principal material coordinate system was investigated. The response to combined stresses was also investigated by considering the stress-strain behavior of off-axis speci­ mens. The generated results are subsequently compared with numerical calculations based on the finite element method and experimental data obtained by various investigators. For example, the results for the transverse normal and longitudinal shear response of unidirectional boron/aluminum (0.5 fiber volume fraction) are presented in Fig. 8.15 and 8.16, respectively, using the material parameters given in Table 8.6 and noting that the aluminum matrix was assumed to be elastic perfectly-plas­ tic. In this case, the results obtained with the micromechanics model using the local-and mean-field approaches are compared with the finite element solution obtained by Foye (1973) for both loading situations. We note that as the finite element solution has been generated by a square array of fibers, the averaging procedure given by eqn. (3.57) to obtain the effective response of a transversely isotropic continuum has not been applied in this case. The correlation between the transverse response obtained with the finite element analysis and the micromechanics model based on the localfield approach, Fig. 8.15 is very good. The prediction of the microme-

248 chanics model based on the mean-field approach presented in the same figure is drastically different from the latter. Similar observations can be made regarding the longitudinal shear response, Fig. 8.16.

TABLE 8.6 Material parameters of boron fibers and aluminum matrix

Material

Material

E(GPa)

ν

Boron fibers

413.7

0.20

Ε

Z0

ν

(GPa)

(sec) [sec)

(MPa)

Zj

m

η

-

10

(MPa)

4

Aluminum matrix

55.16

0.30

10"

103.42

103.42

249

400|—

BORON/ALUMINUM 300 —

a. 2E

£

|b

2 0 0 -

.•'

^ — —

//

100—

-/

/ / Matri f Matri Finit

/ 0

x Subcel l x Averag e e elemen t

1 0.1

I

I

0.2_ e

00..33 (%)

I

I

0.4

0.5

2 2 Fig. 8-15. Transverse normal response of a unidirectional boron/aluminum composite. 200

BORON/ALUMINUM Matrix Matrix

Finite elemen t

I

a a.

Subcell Average

Έ

-

100 CVJ

lb"

ι 0

0.1

0.2

ι

ι

_

0.3

0.4

2 6

| (% 2 )

ι 0.5

Fig. 8-16. Axial shear response of a undirectional boron/aluminum compo­ site.

250 The predictions of the micromechanics model have also been compared with experimentally generated results obtained by Pindera et al. (1990) for unidirectional boron/aluminum composite (0.46 fiber volume fraction) whose constituent properties were given in Table 8.4. The longitudinal and transverse response in tension and compression obtained from the 0° and 90° specimens is illustrated in Fig. 8.17. The figure gives the comparison between the analytical predictions based on the two approaches and the measured data. It can be clearly observed that the mean-field prediction significantly underestimates the extent of plastic flow in the case of transverse loading. The longitudinal response, on the other hand, is predicted with sufficient accuracy by the two approaches. This is not surprising since it has been shown by Mulhern et al. (1967) that the longitudinal response of an elastoplastic composite can be accurately predicted by assuming that the entire matrix phase yields uniformly (cf. Hill (1967)). It should be noted however, that a true measure of a micromechanics model's predictive capability is given by the accuracy of the prediction for the matrix dominated response such as the transverse shear behavior. Thus, it can be concluded that the mean-field approach is not useful in this particular case. The off-axis response of 10°, 15° and 45° of the boron/aluminum composite discussed above is given in Fig. 8.18, where comparison between the measured data and the two micromechanics approaches is shown. The dramatic differences between the predictions based on the local-field and mean-field calculations are clear. Whereas the local-field predictions correlate very well with the experimentally observed behavior, the mean-field data exhibits unacceptable large deviations. It should be noted that the state of stress in the principal material directions of the employed off-axis specimens includes significant transverse normal and axial shear stress components. These stresses result in a matrix dominated response of the composite (either in shear and/or transverse tension). Further examples about the use of the average stresses and local stresses in the matrix for the prediction of elastoplastic response of metal matrix composites are given in Aboudi and Pindera (1990). In particular, it is shown that the effect of residual stresses produced by temperature changes in the case an axisymmetric loading magnify the extent of the deviation between the elastoplastic responses obtained with the two methods.

251

1

02

0

~ Matri

x Subcel l Matrix Average Experiment /

800-

y /

BORON/ALUMINU M/ 400-

I

/

—f-

ο

^ i f ^ V

lb

2

-400-

90°—r

-800-

/'

/ /

-1200-

JI

I

/1

-1.0

-0.5

I

0.0 €

x

0.5

1.0

( % )

x

Fig. 8-17. Axial and transverse normal response of a unidirectional boron/aluminum composite.

252

8 0 0 r—

BORON/ALUMINU M

640—

_ 480

S

—/

/ /

1

320—

.y

^

A- z _ ^ 3 < ^ ^ Matrix Subcel l x Averag e

:// //S Matri

:![// 16 0

—J/ .

A

L

I

00

.2 0.

Experiment

I

I

4 0. €

x

6 0.

I

8 1.

1 0

( % )

x

Fig. 8-18. Off-axis response of a unidirectional boron/aluminum composite.

253 8.8 SUBSEQUENT YIELD M A T R I X COMPOSITES

SURFACES

PREDICTION

OF

METAL

In Chapter 7, the initial yield surfaces of elastoplastic fibrous composites were generated by employing the micromechanical method of cells. The initial yield surfaces were predicted from the knowledge of the elastic properties of the phases (fiber and matrix) and the reinforcement volume ratio. The initial yielding of the metal matrix was assumed to obey the von Mises criterion. Suppose that an elastoplastic (unreinforced) material is loaded in the plastic region. At each stage of plastic deformation, a new yield surface, called subsequent yield surface, is established. If the state of stress is now changed such that the stress point representing it in a stress space moves inside the new yield surface, the behavior of the material is again elastic, and no plastic deformation will take place. Thus, contrary to the prediction of initial yield surfaces, the generation of the subsequent yield surfaces at the current plastic state requires the knowledge of the plastic strain flow rule and the hardening law in addition to the elastic constants. Consequently, the prediction of the subsequent yield surfaces of metal matrix composites is certainly more complicated because it is necessary to establish the overall inelastic constitutive equations of the composite. Further complication results from the existence of constraint hardening caused by the interaction effects in addition to the hardening of the metal matrix phase. Dvorak and Bahei-El-Din (1982) predicted subsequent yield surfaces of metal matrix composites by adopting a kinematic hardening rule for the aluminum matrix (according to which the yield surface moves without changing its shape or size) and applying the same hardening rule for the composite as well. This was performed in conjunction with the vanishing fiber diameter micromechanical model for the establishment of the overall constitutive behavior of the unidirectional composite. The constraint imposed on the matrix by the elastic fibers in this model causes kinematic hardening in the overall space. The prediction of the current yield surfaces enables the determination of the shakedown limits of metal matrix composites under various loading situations (Dvorak and Johnson (1980)). In the present section, the method of cells is employed to generate the current yield surfaces obtained after loading a metal matrix composite to any desired state in the plastic domain. As in Section 8.6, the prediction of these surfaces is based on the local field of the elastoplastic matrix rather than the average field there. The flow rule and hardening law of the matrix phase are given by the unified viscoplasticity theory of Bodner and his co-workers (Section 8.3). Since no yield criterion is used in this unified elastoplasticity theory, the rate of the plastic work is utilized in the present paper, in the process of generating subsequent yield surfaces, for the determination of whether the inelastic matrix has been yielded. This criterion is applied in the matrix regions of the model and its first fulfillment indicates the occurrence of yielding.

254 Consider an aluminum alloy 6061-0 whose in-situ properties are given in Table 8.4 (the corresponding yield stress in simple tension is Y = 84 MPa). The directional hardening parameter q=0.4 (eqn. (8.23)) was esti­ mated from the measured cyclic response of this material given by Lin and Pindera (1988). This value of q is also consistent with that given by Bodner et al. (1979) for a different type of aluminum. Notice that for the value of the parameter n=10, the material is practically rate insensitive. In Fig. 8.19 the cyclic stress-strain response of the aluminum alloy subjected to uniaxial stress loading (in the χλ -direction, say) is shown. The Bauschinger effect (e.g. reduction of hardening in the compressive part of the cycle which follows stressing in the preceding tension part) is clearly observed. In Fig. 8.20 the corresponding variation of the rate of plastic work with the amount of axial straining is shown. It is clear that the sudden change of the rate of plastic work from a vanishingly small quantity is a convenient indicator that the material passes from elastic to an inelastic loading state. Accordingly, initial yield surfaces as well as subsequent yield surfaces from any state of loading can be generated by examining the change of the rate of plastic work. Since in the present model for unidi­ rectional composites there are three subcells (β+ηφ2) which are occupied by the matrix material, it is necessary to examine the values of the rate of Ί plastic work \>Ι^ \β+ηφ2)

at any stage of loading. The first matrix subcell

which exhibits a sudden change in the rate of plastic work from a vanish­ ingly small quantity indicates that the composite enters locally to an inelas­ tic loading state. The values of σν and e y at this stage of loading deter­ mine a point on the composite yield surface in the stress and strain space, respectively. In Fig. 8.21 the initial yield surface of the aluminum alloy as well as =l% the subsequent yield surface obtained after loading to a strain e22 (which corresponds to uniaxial stressing of σ 2 =122 MPa) are shown. The 2 effect of anisotropic hardening on the subsequent yield surface which is translated and distorted is clearly exhibited.

255

Αί-6061-0

Ασ,,(ΜΡα)

160 — /

80—1

/

ι—/—ι—I—ι—t—i—I—ι—U—t

/-.θ

-.4

.4

Α8

— -Ι6 0

Fig. 8-19. Cyclic uniaxial stress-strain response of the aluminum alloy which is characterized by Table 8.4 and q = 0.4.

256

AL-6061-0

À W

p

(ΜΡα/sec)

160-

8 0 -

/ ^ ^ ^

40--

ι

J

ι

-.8

«

U

ιι

- 4

0 €„(%)

L

ι

.4

.

ι

.

.8

Fig. 8-20. Variation of the rate of plastic work (per unit volume) of the aluminum alloy at the various stages of loading and unloading.

257

Α°ϊ,

Αί-6061-0 I n i t i al Subsequen t

-2

/Y

+ 2

Η

ψ /-I

/

Ι

/2

1

sσ /ν/Υ 2 2

- L - -2

Fig. 8-21. Initial and subsequent (after loading to point L) yield surfaces of the aluminum alloy.

Consider a unidirectional boron/aluminum composite with a fiber volume fraction of 0.46. The properties of the elastic-plastic matrix and elastic fibers are given in Table 8.4 Furthermore, the aluminum matrix is characterized by a directional hardening parameter q = 0.4. In Fig. 8.22 the initial yield surface of the unidirectional composite for loading by axial and transverse normal stresses, σ η and σ 22 are shown. This surface can be of course predicted by using the elastic constants of the fiber and matrix only in conjunction with von Mises criterion (Chapter 7). Here the initial yield surface was constructed by the method presented in the present section which utilizes the inelastic properties of the matrix. The initial yield surfaces predicted by the two methods coincide. Fig. 8.22 exhibits also the subsequent yield surface which results after the application of a uniaxial stress loading of the unidirectional composite to e 2 =20 . 7 5 % in the transverse direction (which corresponds to σ 2 /2Υ = 3 . 1 ) . Significant shifting and distortion, as compared with the initial yield surface, can be observed which are caused by the hardening of the matrix as well as by constraint hardening (if the matrix does not harden, the composite exhibits only constraint hardening).

258 /Y BORON/ALUMINUM ^

°il

INITIAL

Subsequent

1 -6

__4

1 -4 -

2'

// ί ι

1

/

/

1

2

ι % 4

/ Y/Γ

2 2

/-/ / — 4

/

Fig. 8-22. Initial and subsequent (after loading to point L) yield surfaces of a unidirectional boron/aluminum composite.

Whereas Fig. 8.22 exhibits the initial and subsequent yield surfaces of the composite in the stress space, it is also possible to present the corres­ ponding yield surfaces in the strain space. In Fig. 8.23 the initial and sub­ sequent yield surfaces are shown in the axial strain e n - transverse strain e 22 space. Here, the significant effect of the resulting shifting with respect to the initial yield surface is clearly demonstrated.

Σ

259 AΈ

to, y BORON/ALUMINU Initial —.25 Subsequent

M

N*V/) € Λ.75

1

|.ο 22

-^--.25 Fig. 8-23. Initial and subsequent (after loading to point L) yield surfaces in the average strain space of a unidirectional boron/aluminum composite.

Figs. 8.22 and 8.23 were generated on the basis of the yielding of the first matrix subcell, implying local yielding. The prediction of initial yielding according to the average stress in the entire matrix phase led to underestimation of the yield stresses, Chapter 7. Let us examine the result­ ing initial and current yield surfaces based on the average matrix stresses for loading in the axial and transverse normal stresses σ η, σ 2 . 2 Fig. 8.24 exhibits the initial and subsequent yield surface, where the latter was obta­ ined, as in Fig. 8.22, after a uniaxial transverse stress loading to e 2 =20 . 7 5 % (which corresponds presently to σ 2 /2Υ = 7 . 2 ) . Comparison with Fig. 8.22 reveals higher composite yield stresses predicted on the basis of average matrix stresses. In addition, it can be readily observed that the effect of anisotropic hardening of the matrix leads to a uniform expansion and trans­ lation of the current yield surface as compared with the initial one. The subsequent yield surface of Fig. 8.22, on the other hand, which was based on local considerations was translated and significantly distorted. This is due to the fact that every matrix subcell has its own translated and expanded yield surface, and the overall yield surface is generated from the inner envelope of these separate surfaces. For an elastic-perfectly plastic matrix, the effect of phase hardening does not exist and only constraint hardening is present. For this type of matrix, the current yield surface of the composite generated on the basis of the matrix average stresses, after uniaxial transverse stress loading to σ 2 /2Υ = 7 . 2 , is also shown in Fig. 8.24. The resulting pure translation (without expansion) of the initial yield sur­ face to the current one is clearly observed in this special case. It can be concluded that the effect of constraint hardening leads to a translation of the initial yield surface when the average stress of the entire matrix is used. The combined effect of anisotropic hardening of the matrix together with

260 constraint hardening results, on the other hand, into a translation combined with a uniform expansion. Thus, in general, one cannot infer about the shape or size of the subsequent yield surface from the knowledge of the initial yield surface whenever the local stress in the matrix is employed. This statement is valid for both a hardening or nonhardening matrix.

11 ACT

/Y BORON/ALUMINUM Initial

--6

Subsequent

ft y '

ι—r -2

;

/

ι

\

/

i\ '

/

/

M—/—hf 2/

4

L

1—t—ι

/

6

8

f ·

σ

2 / Υ2

/ ;j

-1—6 Fig. 8-24. Initial and subsequent (after loading to point L) yield surfaces of a unidirectional boron/aluminum composite generated by employing the stresses in the entire matrix phase. The dotted line (...) represents the com­ posite yield surface assuming an elastic-perfectly plastic matrix.

261 In conclusion, it is shown that initial and subsequent yield surfaces of metal matrix composites can be easily generated by employing the flow rule of the ductile matrix in conjunction with the micromechanical analysis. The resulting composite yield surfaces generally exhibit translation as well as distortion, as compared with the corresponding initial yield surface. It is illustrated again that the yield surfaces based on the average behavior of the entire matrix might lead to unrealistic prediction, as compared with those based on local considerations. Further results can be found in Aboudi (1990), where the effect of temperature change on the composite subsequent yield surfaces is examined. An experimental study of yield sur­ faces of a fibrous boron/aluminum composite was recently presented by Dvorak et al. (1988). 8.9 METAL M A T R I X COMPOSITE LAMINATES 8.9.1 A direct formulation The overall inelastic constitutive behavior of unidirectional laminae (eqn. (8.48)) can be generalized for the prediction of the average behavior of metal matrix composite laminates. This can be performed by employing P L the classical lamination theory in which the average plastic strain term, It can be shown c , of the unidirectional composite is incorporated. (Arenburg (1988)) that the resulting inelastic thermomechanical constitutive relations of the metal matrix composite laminate are given (in terms of the laminate coordinates x,y,z) in the form

Î N] T AB \ MJ

[B D

]J>1

JN 1

J1 l * J

1 mP L J

-

p l

v.

J

T JN 1 mT * κ

(8.55)

J

Here Ν , M are the resultant force and moment, respectively and c°, ic are the midplane strains and curvatures. The extensional stiffness matrix A , the coupling stiffness matrix B , and the bending stiffness matrix D , are given by

rH / 2 (A, B , D )=

J-H/2

Q k( l , z , Z2 ) dz

(8.56)

where Q k is the reduced elastic stiffness matrix of the kth lamina, and H T L Tof the laminate. TheP thermal PL and moments are is the total thickness forces denoted by N , M and, similarly, N , and M denote the plastic forces and moments. P L The evaluation of the constitutive relations (8.55) is complicated due Pto L the inclusion of the plastic stress resultants and moments N , M .

262 P Lintegration through the thickness of the These quantities are based on the laminate of the plastic strains e . The latter are functions of the subcell plastic strains L ( ^ ) . These strains are evaluated by integrating the plastic flow rule (eqn. (8.44)) in addition to the plastic work expression (eqn. (8.47)). The integration of these equations with respect to time form a typ­ ical initial value problem. Due to the lack of simple analytical expressions for the plastic strains, numerical integration is used for both the temporal integration of the plastic subcell strains and the spatial integration of the plastic strains resultants and moments. At each integration point, an expli­ cit time integration scheme is employed such that the plastic subcell strains are computed at time t+At (At is a time increment), using the current field values at time t. Given N(t), M(t) and AT(t), the computations required for a typical time step can be summarized as follows (Arenburg (1988)):

1. 2. 3. 4. 5. 6.

T T Evaluate the stress resultants and moments Ν , M. Evaluate the thermal stress resultants and moments N , M . P L P L Evaluate the integrations required to obtain the plastic stress resultants and moments N , M . Solve eqn. (8.55) for c° and ic. Compute as required the laminae stresses and fiber and matrix stresses. For each integration point (in/ ?the 7 )laminate:

(a) Evaluate L ( ^ ) and W (b) Integrate L ( ^ ) and

. to obtain their values at t+At.

For the first time step, the process is begun with the initial conditions of zero plastic field. Details of some specific integration procedures can be found in Arenburg (1988). An illustration of the above procedure for the prediction of the stressstrain response of metal matrix composite laminate is given by Arenburg (1988). He utilized the experimental data of Shukow (1978) for the axial stress-strain response ( ^ ^ - 6 ° ^ of a [±45] 2g boron/aluminum composite laminate to back-out the parameters of the constituent parameters. These parameters are given in Table 8.7, and the results of this characterization is shown in Fig. 8.25. The validity of the model parameters were examined by predicting the behavior of a [0/±45/90] 8 quasi-isotropic laminate, see Fig. 8.26. The agreement between the micromechanical model and the experimental data is good. Another comparison of the micromechanical prediction with experi­ mental data for a boron/aluminum [0/90] 2s cross-ply laminate is shown in Fig.8.27. The properties of the constituents are slightly different from those in Table 8.7, and can be found in Biglow et al. (1989), from which the measured response curve was taken.

263 TABLE 8.7 Material constants of boron and aluminum constituents backed-out from [±45] 2s measured data. The fiber volume ratio is v f = 0.44.

Ε ν (GPa)

η

m

1 Z0 Zx q Dj a6 A (MPa)(MPa) (sec) (10' /°C)

Boron

400

0.2

-

-

-

-

-

-

Aluminum

60

0.25

10

60

96.5

179.2 1 10"

4

6 (10- /°C)

6.3

8.28

21.06

21.06

Arenburg (1988) and Arenburg and Reddy (1989) incorporated the derived micromechanical inelastic constitutive relations into a first-order shear deformation plate theory. The resulting boundary value problem was solved by utilizing a finite element method. Experimental test data for boron/aluminum notched test coupons reported by Shukow (1978) was used to evaluate the accuracy of the developed micromechanically based inelastic finite element model. To this end, the behavior of a notched test coupon e two t a laminates e [±45] e c and [0/±45/90] were examined. The axial strain for 2e g °xx * S of the hole and at the far field strain gages as a function of the applied load is shown in Figs. 8.28 and 8.29 for the [±45] 2g and [0/±45/90] 8 laminates, respectively. The experimental data presented in these figures is taken from Shukow (1978). For the [±45] 2e laminate, the finite element solution for the far field strain is in excellent agreement with the experimental data. The predicted results for the strain at the edge of the hole closely agrees with the experimentally measured strain up to a value of 0.6%. At this point, the finite element solution begins to under predict the axial strain for a given applied load. The probable reason for the observed discrepancy is the presence of 3-dimensional effects at the free-edge of the hole which are not included in the current analyses. Add­ itionally, some of the discrepancy could be attributed to geometrically non­ linear effects especially for the larger strains or possibly delaminations. For the quasi-isotropic laminate, [0/±45/90] 8, the finite element pred­ ictions are in excellent agreement with the experimental data. At the far field strain gage the results are very similar to those of the unnotched test coupon shown in Fig. 8.26. The axial strain computed at the edge of the hole follows the experimental data very well all the way to the failure of

264 the coupon at about 276 MPa. Further applications of the finite element procedure can be found in Arenburg (1988).

r *H [ i 4 5 ]

1

1

2 | ^s „

x

2 00 ρ

Boron/Aluminum 150 -

'σ

o_

Jt-^"^

—

~x 1 0 0 -

lb* jr —

Micromechanic

5 0 — /x

L 0 0.

Experimenta

.

I

2 0.

ι

1

ι

I

4 0.

6 0.

<= °

χχ

ι

I

8 1.

s l

i

1

0

(%)

Fig. 8-25. Predicted and measured (Shukow (1978)) response of a [±45]: boron/aluminum laminate.

265

500 ρ

S^-\ Γθ/±45/9θ1,|^ν 1 L

*x

J s

Boron/Aluminum 4 00 -

1 ^ 2 0 0-

yS Micromechanics

10 0 -

li 0 0.

χ Experimenta

S* l

I

ι

2 0

L_ 4 0.

.

l

J 6

€«(%) Fig. 8-26. Predicted and measured (Shukow (1978)) response of a quasi-isotropic boron/aluminum laminate.

266

r «•^1 6 0 0

Γ

[ 0 / 9 0 ] 2 | s^ .

„x

Boron/Aluminum

//' /

s

500 'o

//

û- 4 0 0 -

/ '

,£ 300-

/ '

A' Micromechanic 200 100 -

/ 0

s

//

//Experimenta

r

l

//

/ /

I

!

I

I

0.2

0.4

06

0.8

€ * ( %) Fig. 8-27. Predicted and measured (Biglow et al. (1989)) of a [0/90], boron/aluminum laminate.

267 yJ /

C XX

r- Strain Gage at Edge of Hole ^ - F a r Field Strain Gage

/

/

^xx^

B o r o n / A l u m i n um l oo — Far Field x / ^

"3

, 0

°-

^

/

Edge of

a lb

Ο

^

—

"

^ r ^ *

Finite Element

lj*

ί

Hole

χ

ι

l

I

0.5

1.0 6 ^ x( % )

Experimental

.

l

1.5

.

l

2.0

Fig. 8-28. Predicted and measured (Shukow (1978)) response of a notched [±45] 2s boron/aluminum laminate.

268

' ^-Strai /

σ

n Gag e a t Edg e o f Hole /-Far Fiel d Strai n Gag e

χχ [ 0 / t 4 5 / 9 0 s]

3 00

6

ρχ

e o f Hol e

y

"3 2 0 0 -

0 — Finit

e Elemen t

x

Experimen t

i* #C

X

•r

Boron/Aluminum

Far Fiel d A Edg

^ 10

—-

/

.. ,

I

j

0.2 0.

I

4 0.

l

I

.

6 0.

J

8

€ v y (%)

Fig. 8-29. Predicted and measured (Shukow (1978)) response of a notched quasi-isotropic boron/aluminum laminate.

8.9.2 Series expansion formulation The Love-Kirchhoff theory is the simplest approach which is frequently used in analyzing laminated plates. According to this theory, the strains in the plate are assumed to vary linearly across its thickness. To this end suppose that every lamina has a width of h k with k=l,2,...K where Κ represents the total number of the layering. For com­ bined bending and stretching deformations, the displacements can be assumed to be in the following form according to the Love-Kirchhoff theory:

V

269 / α , 3w (x,y,t) ι^ίχ,γ,ζ,ί) = u 0(x,y,t) - ζ A 0—

,

dw 0(x,y,t) —

,

ιι^χ,γ,ζ,Ο « v 0(x,y,t) - ζ

u B(x,y,z,t) = w 0(x,y,t) .

(8.57)

Here the functions u 0, v 0 and w 0 are the displacements at the midplane (z=0) of the laminate, expressed with respect to a coordinate system (x,y,z) with ζ perpendicular to the layering and t denotes the time. A local coordinatek system located at the mid-plane of every lamina is now defined as (x,y,z( )) with k=l,2,...,K. In terms of this local coordinate system the displacements in eqn. (8.57) can now be written as:

«00 = v 0 - (zW

+

l)

,

k

( )k u

= w0 ,

Ζ

(8.58)

with £ k being the ordinate of the mid-plane of the kth layer from the plane z=0, and where the dependence of the displacements on x,y and t has been omitted in the notation for simplicity. The infinitesimal strains with m,n=x,y are now given in every layer by

CW = £« mn

mn(0)

+ Z W WC

,

(8.59)

mn(l)

where £( )k _ au^ xx(0) -

9x

£ajw^ " k

2

_

) '

'-(I) - "

a x

2

w

„

yy(o) c xy(o)-

(

θν η

. a w„

_ ι u nk

ay

"â7 â r +

) / 2 _

( k 2

d'

x

2

oo

c

2

'a yyy(i)

^

âxây" '

aM w,

ο

:

2

'

d

y 6 )0 ( 8

âxây" ·

·

270 Introducing the nondimensionalization f

(k) . W / ( h / 2 ) , 2

k

the following form for the strains in the kth lamina is obtained

eW = ejjj + V3 e« fM ,

(8.61)

where L

J xx

C eW-rcW £ (0)

yy

xy

( )l t

Α) ι J

k

xx(0) '

η(0)

'

xy(0)

'

and no summation is implied on k. The constitutive behavior of the unidirectional metal matrix composite is given by eqn. (8.48), which is described in the material coordinates system ( x 1, x 2, x 3) in which the fibers are oriented in the x x -direction. In a plane stress situation (x x- x 2 plane) considered here for laminated plates, the following constitutive law for the fiber-reinforced kth lamina (x 2 is the direction of the fibers) can be obtained from eqn. (8.48) PL

ê

22 σ

~ 12

e

PL

*11

σ

= [QM] k

^l" 12 ^2 / 22

-ê

ë

-ë

£

e

-

22 22 ) 2(^i2-^

e

U2-e U /e 23

L

k

0

2

2

ΔΤ

22

k

(8.62)

where QW is the stiffness matrix of the lamina. In the coordinate system (x,y,z) of the laminate, in which x 3 = z, this law takes the form

σ« = QflO (eW - FW) -WW ΔΤ

(8.63)

where

1 xx

yy

xy

271 k k PL and F ( ) , W( ) are obtained, respectively, from c and the thermal term in eqn. (8.62) by using the standard transformation between ( x 1, x 2, x 3) and k (x,y,z) coordinate systems. The description of the strains e( ) is given in the framework of the Love-Kirchhoff theory by equation (8.61) which exhibits a linear variation across the thickness. In order to proceed with the formulation of a lami­ nated plate theory, a description of the stresses across the plate is needed. In a perfectly elastic lamina, the constitutive equation readily provides a linear variation across the thickness. In an inelastic lamina the stress distri­ bution is certainly nonlinear in the general case when the lamina is sub­ jected to a general loading. Thus, although the Love-Kirchhoff hypothesis may be adequate as a first approximation to the deformation field, a linear dependence of the stresses across the inelastic laminae is certainly not suffi­ cient, and a higher order distribution needs to be considered. A useful approach in the analysis of metal matrix laminate plates is to describe the distribution of the stresses across the thickness of the lamina by the Legendre polynomials (Aboudi and Benveniste (1984), Aboudi (1985c)). Thus the following expansion for the stresses is assumed:

σ « = (l +2n)V2 τ £ ) P Bf r M )

,

(8.64)

where rPQ , defined by, Γ >)

T

f

J

1 χχ(η) ' yy(n) ' x y ( n )

k are the coefficients of the expansion in the Legendre polynomials P n( f ( ) ) . In the above equation summation is implied by the repeated index n=0,l,2,...,N and there is no sum on k. Equation (8.64), together with k (8.61), is now introduced in eqn. (8.63), each side is multiplied by P m( f ^ ) and then integrated between -1 and +1 with respect to f ( ) . Making use of the orthogonality property of the Legendre polynomials yields: k)

= <*

(ejg, - R g ) - WW Δ Τ 6m0

m = 1 A....N

(8.65)

where

M

1 [Qm+l) /*/2]

1 -1

F(OP m(fOO)dr(O

,

(8.66)

272 k in which e ( l = 0 for m > 2. Again no sum is implied on k and m. P L At any time t, ë ( t ) are known from the time integration of the evolutionaryk equation at the previous time step (see eqn. (8.42) for example).

Thus F( )(t) and R ^ ( t ) are known. provides e j j ^ , and vice versa.

Consequently, the knowledge of r j j ^

The advantage of the Legendre series

expansion formalism can be readily recognized by considering a laminated metal matrix composite plate subjected to a prescribed time dependent force resultant and moment which are applied gradually. Here k r^Lx xx(0)

and τ^\Λ. are known functions of time while r^ l rail)

m > 2, are unk-

xx(m)

'

nowns. The use of eqn. (8.65) at ktime t provides all the unknown stress and strain coefficients for all m ( e ( l = 0, m > 2). The strains and stresses M in the kth lamina are given by eqns. (8.61) and (8.64), respectively. Let us illustrate the derived theory by the simplest possible loading of a laminated inelastic plate, that is, pure cylindrical bending. Each lamina consists of a unidirectional fiber-reinforced material (graphite/aluminum) whose constituents properties are given in Tables 8.2 and 8.5 (the volume fraction of the graphite is 0.3). In Fig. 8.30, the moment-curvature depen­ dence of three-layered cross-ply ([0/90/0] and [90/0/90]) laminated plates in pure cylindrical bending is shown under isothermal conditions (ΔΤ=0). Here the curvature is given by 2 a w0 86 7 * " - £ Γ · <· > k so that the coefficients e^ l . are given by xx(m)

2Jo)

e

•

'

0

So)-*«« ·

e

J

°

h

-(D

e

e

2 (i )

*

=

= h K/ 3)( "

h ^

/

'

2

)(

V 2

'

V

5

6 ) 8 ·

where the superindex k=l represents the middle layer, k=2 stands for the extreme layers of the symmetric laminate (the plus and minus signs denote the upper and lower layers, respectively), and h is the thickness of the lamina. The resultant moment is obtained by integrating through the thickness of the a „ times z. The result is

273

2 Hoc

- h

J 4 rxx(0) ^ l+/ 2

[ ( £ > ( )1 +2 t§0) )/V1

(8.69)

2 The nondimensional moment M* is related to in the form: M * = M x /x[ ( E ) A£ h / 2 ] where ( E ) A^ is the Young's modulus of the alumi­ num. 160 r-

G R A P H I T E / A L U M I N UM

ΔΤ*0 / /

/

120 -

/

// [ θ ° / 9 0 ° / 0 ° ] // /

Λ o •

/

8 0 /

/

/

/

/

40 -

/

/ /

[90°/0°/90°J

0.2

0.4

0.6

( h / c x/ 2 M 0

0.8

1.0

5>

Fig. 8-30. Comparison between the moment-curvature relations for two cross-ply graphite/aluminum laminates. Fig. 8.30 has been produced incrementally by prescribing the curvature and computing the corresponding value of the moment M ^ . This can be performed directly by computing the stress from eqn. (8.63) without the use of the Legendre expansion formalism, eqn. (8.64). Such a direct scheme cannot, however, be used in the inverse problem in which the moment is given and the curvature is desired, and thus the Legendre expansion (eqn. (8.64)) becomes necessary. Boundary value problems of plates under bending correspond to the latter situation. Similarly, problems which involve the determination of residual stresses created in inelastic laminated plates subjected to a temperature excursion necessitate the expan-

274 sion (8.64) as well. Such a problem is analyzed in the following. Consider a laminated plate in which each lamina is a graphite/aluminum unidirectional fiber-reinforced material. The properties of the thermoinelastic aluminum matrix are given in Table 8.2, while the thermoelastic gra­ phite fibers are described in Table 8.5, and their volume ratio is 0.3. At the reference (curing) temperature T R=371.1°C the composite is assumed to be at a microscopically stress-free state. The plate is cooled from this in­ itial state while it is kept at free conditions,i.e. N xx = N yy = Ν ^ = 0 , = Myy =

= 0 .

(8.70)

Here N ^ , Ν and are the resultant forces obtained by integrating respectively, the stresses of the laminae, σ ^ , and through the thickness of the laminate. Similarly, M ^ , Myy and are the resultant moments obtained by integration through the thickness of the correspond­ ing stresses time the moment arm with respect to the midplane of the lami­ nate. With m,n=x,y, Κ N mn - Y

h/2 f h 2*W dz

kt—i= i -J '- h// 2

β

(8.71a)

and Κ

, h / 2 ( )k = > a ζ dz , f-ί J-h/2 k=l - V 2

(8.71b)

where h is the width of each lamina. By substituting eqn. (8.64) in (8.71), we obtain from eqn. (8.70) that Κ Χ ΐ(ο) k=l

-

and Κ

Κ

k=l

k=l

0 ·

87 2 a < - >

275 Conditions (8.72) together with eqn. (8.65) provide a system k ofk linear alge­ braic equations for the determination of the unknowns ej j, e | | and rj*j in the laminae, from which the various field quantities of the laminated struc­ ture can be obtained. In Figs. 8.31-8.32 the extension strains e° = e° at the mid-plane of a °

xx

yy

[60/-60] laminated plate and the curvature / c ^ are shown in the course of a temperature cycle of cooling from the cure temperature T R to the room temperature and reheating back to T R. It should be noted that in the pre­ sent case € ^ = κ χ = *y = 0. The effect of deviation from linearity appears clearly in the graph of curvature developed in the cooling and reheating cycle. The residual stresses in the [60/-60] graphite/aluminum laminated plate which develop during the above temperature excursion are shown in Fig. 8.33. The figure shows the variation of the residual stresses σ η , across the 60° lamina. The effects of nonlinearity due to the presence of inelasti­ city effect can be noticed. Further results can be found in Aboudi (1985c).

276 7 ' Γ Ν

GRAPHITE/ALUMINUM

\ -.6 -.5-

[60/-60] \ \

-.4-

\

2

\\

κ

-3 —

\ \

-.2-

I Ο

\

I ΙΟΟ

1 I 200 300 TEMP. (°C)

I 400

Fig. 8-31. Midplane strain of a [±60] graphite/aluminum plate developed in the course of a temperature cycle of cooling and re-heating.

277

r GRAPHITE/ALUMINU

M

[60/-60] 0.2

-

CM

\

ν

\

ο

\

\ Χ

Χ

\

\ ο . \ ν :

0

L 0

χ Χ \

\

^ 1 0 0 2 0 0 3 0 0 TEMR(°C)

4 0 0

Fig. 8-32. Curvature of a [±60] graphite/aluminum plate developed in the course of a temperature cycle of cooling and re-heating.

278

GRAPHITE/ALUMINU M [ 6 0 / - 6 0]

- 4 00 -

I -1.0 0

I

I

I

I

I

I

1.

I

1

I

I 0

ζ Fig. 8-33. Variation of the stress σ η across the 60° lamina of a [±60] gra­ phite/aluminum plate. The stress is shown during cooling from cure tem­ perature (371. loc) at (a) 196.1°C; (b) 21.1°C (room temperature); and (c) subsequent re-heating from room temperature at 196.1°C.

279

8.10 SHORT-FIBER M E T A L - M A T R I X COMPOSITES So far the analysis presented in this chapter dealt with metal matrix com­ posites with continuous (long) fibers. This analysis can be extended for the study of the average elastoplastic behavior of short-fiber composites by est­ ablishing the appropriate continuum equations (Aboudi (1986)). Referring to the short-fiber composite model in Fig. 3.16, let us assume that both inclusion and matrix phases to be elastic-viscoplastic mat­ erials. The total strain rate of the material in the subcell (αβη) is decom­ posed into elastic, thermal and plastic composites in the form = έ(αβΊ)

ρ{αβ )

ÈE L ( A / ?7) +

Ί

+

i?H<*fil)

(8.73)

The elastic strains are related to the stresses by the Hooke's law, the plastic strains are controlled by the viscoplastic law (see eqn. (8.6)) ÉP L ( A / 37)

= ^A (o0 7s)

( 8 74) >

and the thermal strains are related to the temperature deviation Δ Τ via the coefficients of thermal expansion. Consequently, the following constitutive law of the material in subcell (αβι) can be established P La a M f ) = C W eWi) - rMf) Δ Τ - 2 μαβη e ( ^ ) (8.75) where μαβΊ is the shear modulus of the material in subcell (&βη). This constitutive relation generalizes eqn. (3.98) by incorporating the plastic effects of the material in accordance to eqn. (8.73), in which isotropy and plastic incompressibility are assumed. Thus, eqns. (8.75) are applicable either for a perfectly elastic anisotropic material, or for an elastoplastic material which is isotropic in the elastic region. α The average stresses σ ( ^ ) in the subcell {αβη) are determined from eqn. (3.96). Consequently, the following relation, which generalizes eqn. (3.99) in the elastic case, can be established from (8.73) (see the counterpart eqns. (8.43) for the long-fiber case): fjW'ï) = C ( ° ^ ) xMf)

α - Γ ( ^ ) Δ Τ - 2μαβΊ L ( ° ^ )

(8.76)

where L ( ° ^ ) is the plastic strain. The evolution law of the latter is Ι>/*7) = λ

α Ί βs

(8.77)

α where s ( ° ^ ) is the stress de viator of the average stress σ ( ^ ) . In the framework of Bodner-Partom unified viscoplasticity theory, the flow rule

280 function λαβΊ is given by eqn. (8.45) but with (£7) replaced by (αβη). Continuity of displacements, eqns. (3.90), and continuity of tractions, eqns. (3.95), form, in conjunction with (8.76), a set of constitutive continuum relations which govern the average behavior of metal matrix composites with short fibers. As in the continuous fibers case, these relations represent the effect of the transition from a medium with periodic microstructure to an equivalent homogeneous medium. The derived constitutive equations can be applied for the prediction of the overall behavior of a porous 2024-0 aluminum (Aboudi (1984)). The elastoplastic response of this material, with a porosity of 22%, was meas­ ured by Schock et al. (1976)). The solid work-hardening 2024-0 aluminum alloy is characterized by a Young's modulus of 73 GPa, a yield stress in simple tension of 75.8 MPa, an ultimate strength of 186.1 MPa and 20% elongation. In Table 8.8 the material properties are given in the framework of the unified Bodner-Partom theory. In Fig. 8.34 the response of the porous aluminum as predicted by the method of cells is shown when the material is subjected to a hydrostatic loading, and compared with the experimental results of Schock et al. (1976). The agreement between the theoretical and measured deformation is seen to be satisfactory. Further comparisons can be found in Aboudi (1984). TABLE 8.8 Material properties of the 2024-0 aluminum alloy (isotropic in the elastic region and isotropic work-hardening in the plastic region)

E(GPa)

ν

D 0\ s e c )

73

0.33 10~

Z 0(MPa)

Z x(MPa)

m

η

90

200

20

10

4

281

_

I20r—

" - S

\

ib

0

Ε

ZL

I

i I

Zi_

24

?

k k

(% )

Fig. 8-34. Comparison between the predicted ( ) and measured (Schock et al. (1976)) ( — ) response of a porous aluminum to hydrostatic loading and unloading.

282

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