Plain concrete as a composite material

Mechanics of Materials ! (1982) !39-150 North-Holland Publishing Company

139

PLAIN CONCRETE AS A C O M P O S I T E MATERIAL Miguel ORTIZ and Egor P. POPOV Division of Structural Engineeringand Structural Mechanics, Deportment of Civil Engineering, University of California, Berkelo, CA 94720, USA

Received 14 August 1981; revised 16 November 1981

The purpose of this paper is to study the consequences of the composite nature of concrete. A plausible energy balance equation is postulated and the Green-Rivlin invariance principle is applied to it to derive the linear and angular momentum balance laws. General constitutive equations are discussed with the aid of thermodynamic potentials and Coleman's method. The distribution of the applied stresses between mortar and aggregate is also studied in detail, showing for instance that substantial tensile lateral stresses may appear in mortar under uniaxial compressive loading. These results are used to derive a criterion for the onset of inelasticity in concrete. 1. Introduction

Most concrete models proposed in the past treat concrete as a simple material, Bazant and Parameshwara (1976), Neville (1974). However, in many respects this is unsatisfactory. It is a well-known fact, for instance, that when concrete is subjected to uniaxial compression it develops cracks that are parallel to the loading axis (see, for instance Wastiels (1979), where further references can be found). In some cases, these cracks become so large as to be the direct cause of failure of the specimen. Given that materials only crack in tension, one is led to the conclusion that a uniaxial compression has to induce transverse tensile stresses in mortar. The most widely accepted explanation of this effect idealizes the situation as that of an infinite medium (mortar) with a stiff spherical inclusion, Shah and Winter (1966), Taylor and Broms (1964). When an overall uniaxial compressive loading is imposed upon this system, a linear elastic analysis reveals that tensile stresses tend to appear around the particle, at right angles with the direction of the loading. This model highly idealizes the problem, as it entirely neglects the effect of the neighboring aggregate particles and, for this reason, its overall validity seems doubtful. It is shown in this paper that the appearance of transverse tensile stresses in mortar under an appfied load is a natural consequence of the composite nature of contrete. More generally, it is also shown how the theory of interacting continua can be applied to concrete to conclude, for instance, that the applied stresses distribute unequally between the two phases of the material, mortar and aggregate, and to derive expression for these 'phase-stresses' in terms of the applied ones. Once the correct stresses acting on mortar are known, one can, for instance, use the Griffith theory of cracks to study the propagation of microcracks in concrete. In any case, it seems reasonable to study in depth the consequences of this aspect of the material as a first step towards devising adequate material models for concrete. It would appear that the same approach applies also to a variety of other composite materials, such as sandstones in which the silica grains are cemented together by lime, iron oxides and clay-like substances. 2. Concrete as a composite material

The problem of two or more interacting continua was studied by Truesde!l and Toupin (1960), whose monograph also contains extensive references to previous work. Their results were developed further by 016%6636/82/0000-0000/$02.75 © 1982 North-Holland

140

M. Ortiz, E.P. Popov / Plain concrete as a composite material

Truesdell (1961, 1962), for linear systems by Adkins (1964) and Green and Adkins (1964), who discussed general nonlinear constitutive equations. Solid composites have also been studied by means of a variety of other techniques. For instance, the so called self-consistent methud has been used by Mackenzie (1950) to estimate the elastic and thermoelastic properties of a solid containing spherical voids, and by Kener (1956), Hill (1965) and Budiansky (1965) to estimate elastic moduli of composite materials, (see also Mura (to appear) for an extensive list of references). Other authors have studied solid composities by means of variational principles, Hashin and Strickman (1963), statiscal methods, Christoffersen (1973), Willis (1977), or by using exact elasticity solutions, Nemat-Nasser and Taya (1981), Nemat-Nasser et al. (1981). The approach followed here is due to Green and Naghdi (1965). Concrete is idealized throughout this paper as a composite material consisting of two simple components: mortar and aggregate. It is assumed, in the spirit of the theory of interacting contnua, that an arbitrarily small volume of concrete contains both mortar and aggregate in fixed proportions. This is a plausible assumption provided that the only processes that are considered are those whose characteristic length is large compared to the size of the aggregate. The volume fraction occupied by mortar is denoted by a~ and the one occupied by aggregate by a 2. For the analysis at hand, one can postulate the following energy balance equation: ~-~

UdV+½

flVl .v~dV~+~~

.,P2V2• v2 dV2 =

=fj,.v, dV, +f../2-,~dV2+'IOBI f t,.v, dS, +~a8B 2t~..dS2+frdV+fo?dS

(1)

where B denotes the volume of concrete under consideration and 0B its boundary, B~ and B 2 denote the parts of B occupied by mortar and aggregate, respectively, and 0B I and 0B2 their respective boundaries, p denotes the mass density of concrete, p~ the mass density of mortar per unit volume of mortar and P2 the mass density of aggregate per unit volume of aggregate, U denotes the internal energy per unit volume of concrete, r and h are the body heat supply and heat flux of concrete, respectively, v~ and v2 denote the velocities of mortar and aggregate, respectively, I~ denotes the body force acting on mortar, per unit volume of mortar and f2 the body force acting on aggregate, per unit volume of aggregate, t I is the surface traction on mortar, per unit area of mortar and t 2 the surface traction on aggregate, per unit area of aggregate. Eq. (1) simply expresses that the rate of change of the sum of the internal and kinetic energies of B equals the sum of the external power and the external heat supply. Note that no attempt is made at this stage to decompose the thermodynamic variables U, r and h. The following relations follow from the previous definitions: dV~ = a~ dV,

dV2 = a 2 dV,

dS~ = a~ dS,

dS 2 = a 2 dS,

p=

a l p I -3L a 2 p 2 .

(2)

Moreover, since there is no diffusion between mortar and aggregate, the following compatibility condition holds:

v, = ~

-v.

(3)

Making use of eqs. (2) and (3), eq. (1) simplifies to d -~'i~[U+~pv'v]dV=~l'vdV+ fnt.vdS+ fnrdV+ fonhdS

(4)

where f'-

a l I I -I- a 2 f2,

t=

a,tI + a 2 t 2 .

(5)

Applying the Green-Rivlin invariance princip!e to eq. (4) (i.e., the invariance of the energy balance law with respect to superimposed translations and rotations, Green and Rivlin (1964)) and using the Cauchy's

M. Ortiz, E.P. Popov / Plain concrete as a composite material

141

tetrahedron method, the following balance laws ate readily found: Balance of linear momentum V.o+/=pa

inB,

t=n.o

onaB

(6)

where o denotes the stress tensor for concrete whose components % are defined, as usual as the j t h component of the traction t acting on surfaces which are perpendicular to the xi-axis, n denotes the unit normal to ~)B, and a = D v / D t = a v / a t + v- v t, is the acceleration vector. It follows from (5) that o admits the representation O "-- a l O I -~- 0[20"2

(7)

where o! and o z are defined componentwise in terms of t~ and t z, respectively, in the same manner as o was defined in terms of t. In particular, it follows that t~ = n . o I and t z = n . o z. Thus, it makes sense to interpret o I and o z as representing the st.resses acting on mortar and aggregate respectively. Note, however, that these individua| stress tensors have to be understood in the sense of the theory of interacting continua. In particular, o 2 does not represent the stresses within the aggregate particles but it rather accounts for the contact forces that there appear between them. Balance of angular,nomentum o=o

T

(8)

where ( )1" denotes transposition. Note that it does not automatically follow from balance of angular momentum that the individual stress tensors o! and o z are both symmetric. In fact, it may very well be the case that mortar and aggregate interact through stress couples. Balance of energy Making use of the linear and angular momentum balance eqs. (6) and (8) and applying the Cauchy's tetrahedron method to h, eq. (4) reduces to the simpler form U-r+

V .h--o.D=O

(9)

where D is the rate of deformation tensor, D = ½ ( V v + v Tv), and h is the heat flux vector whose ith component h i is defined as the heat flux through a surface perpendicular to the xi-axis. The inner product of two second rank tensors, like c - D in eq. (9), is taken to be oijDij. Moreover, the following entropy production inequality is postulated (Clausius-Duhem inequality, see, e.g., Lubliner, 1972)

1 VO~>O OS--r+ V .h--~h.

(10)

where S is the entropy per unit volume of concrete and 0 the absolute temperature.

3. General constitutive equations In this section, Coleman's method (Coleman and Gurtin, 1967) is used to derive general forms of the constitutive equations in terms of thermodynamic potentials. These constitutive equations are then specialized further by bringing about the specific features of mortar and aggregate. The attention is restricted to the case of infinitesimal deformations. In this setting D can be identified with ~, where c denotes the strain tensor. Furthermore, the variables (S, c, q) with q representing some set of internal variables, are taken as the state variables, and a dependence of the type U:U(S,¢,q)

(ll)

34. Ortiz, E.P. Popov / Plain concreteas a compositematerial

142

is assumed of the internal energy density. Then, eqs. (9) and (10) can be combined to yield 1 0s- 0+~.~-~h. v0~>o and making use of eq. (11), this becomes

which in turn implies (see, e.g., Lubliner, 1972) 0

OU ~S'

~U o.= 8---~-,

OU 1 -@q • ¢ + _ h - V 0 < ~ 0 .

(12)

Eq. (12) can be interpreted as the most general constitutive relations that are compatible with the requirements of the Second Law of Thermodynamics. These relations state that the internal energy density U acts as a potential for the temperature and the stresses, and impose restrictions on the direction of the heat flux vector and on the dependence of the the internal energy on the internal variables. The conjugate thermodynamic potentials, like the Helmholtz free energy per unit volume,

A(O,¢,q)=U-OS,

(13)

of the Gibbs energy per unit volume,

G(O,er, q ) = , . ¢ - ~ ,

(14)

with dleir usual properties, Lubliner (1972), will also be needed in the sequel. Eqs. (12) place general restrictions on the constitutive equations for concrete. To make progress, however, one has to make ~urther assumptions on the form of the thermodynamic potentials. Denote by A!(0, ¢: q) and A2(0, ¢, q) the free energy functions for mortar and aggregate alone, respectively. Clearly, the following consistency conditions must hold A --,A !

as a I --, 1,

A -.A 2

as a 2 ~ 1.

(15)

The gimplest possible choice consistent with eqs. (15) is an additive decomposition

A(O,,.,q) =oqAl(O,,.,q) + a2A2(O,c,q )

(16)

where A mand A 2 are measured per unit volume of mortar and aggregate, respectively. Note this choice of free energy function for concrete is equivalent to assuming that there is no interaction energy between mortar and aggregate. Substituting eq. (16) into eq. (14) it is found that the Gibbs energy of the material also has an additive form

where

and It follows from eq. (16) that

aA = a ~Al + S=

~A

OAI

- - 0"-0 - - - - Q' " ~ "

~A2

"17)

OA~. -- Q2"-~

-" Q l s l

-[- Q 2 S 2 "

Thus, an additive decomposition of the entropy follows from eq. (16). Eq. (16) also results in stress tensors ~ and ~2 which are both symmetric, as was to be expected from the absence of interaction energy between mortar and aggregate, that prevents them from interacting by means of stress couples.

M. Ortiz, E.P. ?opov / Plain concrete as a cc~posite material

143

To make further progress, the specific features of mortar and aggregate have to be brought in. Let us momentarily consider mortar and aggregate as separate materials. Let qs be a subset of the set of all internal variables q, which describe the extent of microcracking in mortar. It is a well-known fact, for instance, that the elastic compliances of mortar degrade as a result of microcrack formation, Karsan and Jirsa (1969). It is therefore of interest to find the most general expression of thermodynamic potentials for a material such that ~)2G' - C,(qs),

(18a)

au, aan a2G~

a,,,ao, - ,~,(vs),

(18b)

a2Gn 0|---~-12 - c,(qs),

(18c)

n

for some functions Cn(qs), ~l(qs) and %(q,), representing the elastic compliances, thermal e~pansion coefficients and specific heat under constant volume of mortar, respectively. The internal variables q~ on which these quantities are assumed to depend can be more generally thought of as some arbitrary set of softening parameters. Here, an and On denote the stress tensor and temperature of mortar, respectively. To find G, one can proceed as follows: integrating (18a) once it is found that ~'G n

aa, =

+,,(o,,q)

for some tensor valued function a ( 0 n, q). Differentiating the latter expression with respect to 0 s and making use of (18b) it follows that a2Gn

aa

aa,ao,- ao, (O,,q)=,8,(qs) and, therefore,

OG, aa, --- c, ----C,(qs ) .o., 4-~,(qs)(0 , -0o) + ~r(q),

(19)

for some tensor valued function cP(q) and some reference temperature 00, and with c, representing the strain tensor of mortar. Integrating eq. (19) once more, it is readily found that G , - ½a,. C,(q s) "an + a , . ,(l,(q,)(0 n -0o)

+a,'cP,(q)-Ain(Ou,q)

(20)

for some scalar function A~(On,q). To determine the form of this function, use can be made of eq. (18c) along with eq. (20) to yield

a2G' - ~ a2A~ (O,,q)=c,(qs) o, ao, -.., ao,2 and, therefore, by direct integration,

Ain(O,,q) :cn(q~)On l-ln.~o

-- SlP(q)(O, --0o) + a,(q),

for some scalar functions S~(q) and an(q). The corresponding expression for the free energy can be readily obtained from eq. (l 3), leading to

An_½[c,_~,(qs)(O,_Oo)_~p(q)].D,(q~).[~.,_~,(q,)(On_Oo)_~p(q)]

+Ai,(On,q)

(22)

144

M. Ortiz, E.P. Popov / Plain concrete as a composite material

where one takes D~(qs)= C~-l(qs). It follows from this expression that the function interpreted as the inelastic part of the free energy. Moreover

Si =

aA~

ao I -~ll(qs ) . ~ +c~(qs)ln~+ Sp(q).

A~(O~,q) can

be

(23)

Note that c['(q) in eq. (19) and S~(q) in eq. (23) are the parts of the strain tensor ¢~ and the entropy S~ that do not vanish upon setting % = 0 and 0~ = 00, and can therefore be interpreted a s the irrecoverable or plastic parts of the strain tensor and entropy, respectively. The fact that the elastic compliances and the plastic deformations depend on some common set of internal variables allows for the possibility of an interaction between elastic degradation and the plasticity of the material. This phenomenon has often been termed 'elastoplastic coupling', Dafalias (1978). Aggregate, on the other hand, is a granular material consisting of cohesionless linear elastic particles that interact on contact. A characteristic of such granular media is that of 'positive dilatancy', whereby a granular mass under shearing experiences an increase in volume (after an initial decrease, Nemat-Nasser, 1980, Christoffersen et al., 1981). This effect was first noted by Reynolds (1885) and has been considered from various points of view in the past (see, e.g., Home (1965, 1969), Matsouka (1974), Nemat-Nasser (1980), Newland and Allely (1957), Oda (1974), Rowe (1962, 1971) and Satake (1978), for a phenomenological plasticity approach and Cowin (1978) for a review of microstructural theories). Other distinct aspects of granular material behavior, like the 'noncoaxiality effect', or lack of alignment between the principal directions of the applied stresses and the stretch rate tensor, have also received the attention of numerous workers in the field, de Josselin de Jong (1971, 1977), Mandel (1947), Mehrabadi and Cowin (1978), Spencer (1964). A general overview of the presently available theories in the mechanics of granular materials can be obtained from Cowin and Satake (1978). A detailed study of the properties of mortar as a granular material is beyond the scope of this paper. For the discussion at hand, a simplified model is used. This model can be introduced as follows. Except for the difference in scale, large samples of aggregate are likely to behave in a manner similar to a cohesionless sand:Neglecting for the time being the friction between particles, it is apparent that the material must pose a very limited resistance to imposed deviatoric deformations. The only stabilizing effect present under these circumstances is the positive dilatancy property, in combination with superposed compressive hydrostatic pressures. If moderate deviatoric stresses are superposed on a previously applied large compressive pressure, the positive dilatancy effect results in an increase in volume which does work against the applied pressure. In order for flow to occur, it follows then that a certain activation energy has to be overcome equal to the w ?rk done against the applied pressure by the maximum volume increase due to dilatancy. For simplicity, though, this effect is neglected here. Consequently, it is assumed that aggregate can only sustain, without taking friction into account, purely hydrostatic compressive states of stress, i.e., the material does not pose any appreciable resistance to imposed deviatoric deformations. The relation between the applied pressure and the volumetric deformation of aggregate is assumed to be elastic, in terms of an overall bulk modulus k 2. The most general form of the free energy for a linear elastic material can be shown, Lubliner (1972), to be given by

A2=½[¢2--1112(02-00)]

"02"[¢2-1112(02-00) ] + c202(1- 1n(02/0o) )

(24)

.14_

where, for the case at ~''~"'~,,,,ll~,¢2 aJ-~d 02 are UlV strain tensor ,tliu ---' temperature of aggregate, respectively. Under the assumptions introduced above, the coefficients D 2 and/~12 in eq. (24) take the form

ja,,,,

a,;,

(25)

with A 2 = (t2)kk -- 3fl2(02 -- 00) , d ----the identity tensor, k 2 and 132 denoting the bulk modulus and thermal expansion coefficient of aggregate, respectively, 00 some reference temperature, which for simplicity is

M. Ortiz, E.P. Popov

/

145

Plain concrete as a composite material

taken to be the same one as the reference temperature for mortar. The Heaviside step function H in eq. (25) brings in the compression-only character of aggregate. It follows from eqs. (24) and (25) that 0"2- ~)A2 0¢2 - D2" [¢2 - ~2(02 - 0 0 ) ] = [ k 2 H ( - A 2 )

s2-

A 2 + 3fl2(02 --0o)] ~,

t)A2 a02 -- [ k E H ( - A : ) A2 + 3flz(02 -- 0o)]3fl2 + c2 1n(02/0o).

(26a) (26b)

Eq. (26a), however, neglects the fact that frictional forces may develop between the aggregate particles. To account for this effect, one can simply add to (26a) some 'frictional stresses', say o2t, i.e., 0.2 = 0.~q + 0.2t

(27)

where the 'equilibrium stresses', 0"~q, are given by eq. (26a). This situation is similar to that encountered in the case of fluids, where the state of stress is composed of an equilibrium part, which is hydrostatic and derivable from the free energy potential, and a viscous part of a dissipative nature. In order to have a complete set of constitutive equations for concrete, specific expressions have to be supplied for the functions C|(qs), ~l(qs), cu(qs), ¢lP(q) and S~(q) and for the frictional stresses 0"J, as well as rate equations for the evolution of the internal variables q and the heat flux vector h. This task will be undertaken in a follow-up paper.

4. The distribution of the applied stresses between moitar and aggregate The general stress-strain relations for mortar and aggregate that were proposed in the preceding section are used here to derive expressions relating the phase stresses o n and o 2 to the externally applied stress o. The uniaxial case is considered in detail to illustrate the nature of these relations. Substituting eqs. (19), (26a) and (27) into eq. (7) and making use of the compatibility conditions El --" q[2 ' : - - E and 0 n = 02 = 0, one obtains 0.:

,~t,0"n -I- a20"2 = [azDt(qs ) + ol2D:, ] -c

-[azDt(qs ) •/$n(qs) +

°t2D2" J~9.]( 0 - 0 o )

-- ~ l a , ( qs)" ¢•( q ) + a20"J.

(28)

Denote

Dtqs)=atDt(qs)+a2D 2, C(qs)=D-t(qs), ¢P(q)-ancP(q).On(qs).C(qs), 0.r = a20.f.

/](qs) = C(q~)-[ anDt(qs)-~n(qs) + a2D2" ~'z], (29)

The quantities C(qs), D(qs), ~(qs), cP(q) and o f in eq. (29) may be regarded as the overall flexibility and stiffness compliances, thermal expansion coefficients, plastic strains and frictional stresses of concrete, respectively. Substituting these definitions into eq. (28) and solving for c one finds c= C(qs). (o-a

t ) + [ J ( q s ) ( 0 - 0 0 ) + ¢P(q)

(30)

and combining this expression with eqs. (19) and (27) and making use of eq. (26a) one finally obtains

°'-'Be(qs)'(°-0.f)-Dt(qs)'[~t(qs)-O(qs)]tO-O°)-an(qs)'[cP(q)-cP(q)] 0"2 = Bz(qs)" (0"-- 0.t) -- D2" [i~12 - / ] ( q s ) ] ( 0 - 0 0 ) + D2" ¢P(q) ,

(31)

M. Ortiz, E.P. Popov / Plain concrete as a conwosite material

146

where the influence coefficients Bi(qs ) and B2(qs ) are given by

B,(qs):Dt(qs)'C(qs),

B.z(qs)=O2"C(qs).

(32)

These coefficients determine how the externally applied stress o distributes between mortar and aggregate. It is seen that the influence coefficients depend on the internal variables q.,, i.e., on the extent of microcracking or, in general, on the value of the softening parameter. It can be readily checked also, from eqs. (22), (24) and (29), that the free energy for concrete takes the form

A(O,e,q) : a t A t ( O , c , q ) -F a2A:,(O,c,q) : ~ [ ¢ - ~(qs)(O-Oo) - ¢P(q)] "D(qs) " [ ¢ - ~ ( q s ) ( O - O o ) - ¢P(q)] + Ai(O,q)

(33)

and from the identity (3 : ( a - af) - c - A and eq. (33) the following expression for the Gibbs energy of concrete is obtained:

G( O, o, q ) = aiG,( O, ~t, q ) + a2G2(0,or2,q)

-'-½(o-orf)'f(qs)'(o-of)"l-(or--orf)'#(qs)(O--Oo)"F(cY--orf)'£P(q)--Ai(O,q) (34) where the inelastic free energy A i is given by

Ai( O, q ) : C( qs)O( l - In(0/0o) ) - SP( q )( O - Oo ) + a( q ),

(35)

C(qs)=a,ct(qs)-t-a2c2,

(36)

with

SP(q):a,SP(q),

a(q):atat(q).

From the form of eqs. (20) and (34) it follows that the generalized forces conjugate to eP and c p are o I and ( o r - or¢), respectively. In other words, the plastic deformation can be alternatively considered to occur in mortar and be driven by o n or to occur in concrete altogether and be driven by ( a - o f). In order to make more apparent the significance of these relations, a simple example is written out below.-Consider a 'virgin' concrete specimen, i.e., one for which c P = 0 and where the extent of microcracking reduces to small flaws in the mortar-aggregate interface, so that the mortar elastic compliances can be taken as being isotropic: D , - X,~,j 8,, + ~,(8,, ~j,+ 8 , 8 : )

(37)

with >'n and fan denoting the Lame constants of mortar. For simplicity, the attention is confined to the isothermal, frictionless case, i.e., 0 - - 0 o and o f-. 0. In this particular situation, eq. (31) reduces to kn o, = T p ~ + __i o', 4XI

o2 = p2 ~,

k2 p2 = T p'

(38)

where k I and k 2 a r e the bulk moduli of mortar and aggregate, respectively, and k = ank ~+ a 2 k 2, P2 is the hydrostatic pressure acting on the aggregate and p/~ and or' are the hydrostatic and deviatoric parts of or. In deriving eq. (38), the assumption has been made that the hydrostatic pressure acting on concrete be p ~<0, so that the aggregate be active. The problem can be further simplified by considering only a uniaxiai state of stress e=

o

0

0

0 10

0 0

0 0

(39)

with o ~
a2k2

s,o.

(40)

M. Ortiz, E.P. Popov 6.0 [ - -

1---

T....

l

- -1 . . . .

/

Plain concrete as a composite material

147

F7

1

=

L

nO ¢: O

4.5 k

IZIW

F-

Z

i

0.0 ~ 0.0

.... ~ 0.2

0.4

0.6

~

i

0.8

I.O

"~

AGGr~.EGAT E V O L U M E T R I C

~ _ ~ " ~

FRACTION ((Z z)

Fig. I.

AGGREGATE

'~o"

Fig. 2.

The influence coefficient B t hi eq. (40) is depicted in fig. 1 as a function of the parameters involved. It is seen that an overall state of stress of uniaxial compression can result in significantly large transverse tensile stresses in mortar. This in turn explains the development of rmcrocracks that are observed parallel to the axis of loading. Physically, the situation is illustrated in fig. 2. The tendency of the aggregate particles to move apart sideways is the cause for the tensile lateral stresses appearing in mortar. Under uniaxial tensile loading, however, aggregate does not resist any forces, due to its compression-only character, and all the applied stress is taken by mortar alone. This results in a uniaxiai tensile stress in mortar that amounts to olaf. Thus, it is seen that the applied stress o gets effectively magnified by a factor of 1 / a m.This in turn provides an explanation for the experimentally observed fact that concrete is weaker in tension than mortar alone.

5. A criterion for the onset of inelasticity in concrete As an application of the results of the preceding section, a criterion for the onset of inelasticity of concrete is derived below. For this purpose, it is assumed that inelasticity first appears in concrete as a result of the extension of the microcracks that develop in the mortar-aggregate interface during the process of hardening, Wastiels (1979). Since these microcracks are randomly oriented, Wastiels (1979), it follows that concrete will cease to behave elastically as soon as the maximum principal stress in mortar reaches the value of the critical tensile stJ'ess o~ for the extension of the microcracks. For the example at hand, it is advantageous to choose the coordinate frame coinciding with the principal directions of the applied

Roy Carlson. Milos Polivka, private communication.

M. Ortiz, E.P. Popov / Plain concrete as a composite material

148

stresses, so that oI

~=

0

0

o

o2 o

0

0

(41)

03

Under the same assumptions as in the uniaxial example developed in the previous section, it then follows from substituting eq. (41) into eq. (38) that ,

a~ ° l -

a~

Or I ---

0

k

0

P ~

,

(, k)

0 2 --

0

k

0

(42)

p

!

li

¢,,

[ ~,

__03_

-

k,~

k, ] p

if p ~O, aggregate does not become active and one simply has, from eq. (7), that

er,=(l/a,)~.

(43)

By virtue of eqs. (42) and (43), the criterion for the onset of inelasticity for concrete can now be expressed as % ~~O,

o~

1

a]

a~

kl k

P<~

ifP ~<0"

The elastic domain resulting from eq. {44) is depicted in figs. 3 and 4. Note the relations resulting from eq. (44):

3Oc ~ = (l/~,-k,/k)'

f, =~,oc,

(45)

where ft and f~ denote the tensile strength and elastic limit of concrete under uniaxial loading. Taking

PLANE Oil +CT2+O"3 =0

O-z -30-,. I/a

~

- k I/k

o

'

l

- ~"="

v Fig. 3. Biaxial elastic d o m a i n .

Fig. 4.

M. Ortiz, E.P. Popov / Plain concrete as a composite material

149

a I ~ 0.5, k I / k ~ 0.5 and ft ~ 0.1 fc, with f~ denoting the uniaxial compressive strength of concrete, it turns out that fe ~ 0.4 f~, which is in agreement with the experimentally observed values, Neville (1974), Wastiels (1979).

Acknowledgment This paper was motivated by general studies of Seismic Behavior of Structural Components being conducted at the University of California, Berkeley, and is prepared with financial assistance from N S F Grant CEE 81-07217. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation. The paper is based on part of the first author's dissertation done under the supervision of the second author.

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