Indemnización

Concrete is a composite material mainly consisting of different sized aggre- gate particles embedded in a cement past matrix and its mechanical be- haviour is strongly affected by the microstructure properties. In order to give physical explanations to the experimental behaviour observed during tests, the knowledge of microstructure becomes fundamental. The main aspects to consider:

1. A large number of bond micro cracks exist at the interface between coarser aggregates and mortar.

2. The cement paste has a high porosity, the pores being filled with water or air.

3. At all dimensional levels, above the molecular level, air and or water voids exists.

Many of the microcracks in concrete are caused by segregation, shrink- age or thermal expansion in the mortar and therefore exist even before any load has been applied. Some of the microcracks can be developed during loading because of the difference in stiffness between aggregates and mortar. Therefore, the aggregate-mortar interface constitutes the weakest link in the composite system. This is the primary reason for the low tensile strength of concrete materials.

For example the propagation of micro cracks during loading contributes to the non linear behaviour of concrete at low stress levels and causes vol- ume increases near failure and uniaxial compressive state of stress. For high idrostatic pressure the intrusion of voids and paste pores becomes increas- ingly important in affecting the behaviour and strength of concrete. The conclusion is that in order to create an appropriate mathematical model it is essential to understand the behaviour of plain concrete under uniaxial, biaxial and triaxial states of stress. Typical tests results are illustrated in

the next paragraph and they all refer to normal weight concrete under short term quasi static loading conditions.

Uniaxial compression behaviour

Results from uniaxial compression tests are generally represented as stress strain curves as shown in Figure 3.1. The shape of stress strain curves is similar for low-, normal-, and high strength concretes as shown in Figure 3.2. The key observations are :

1. The concrete behaviour is nearly linear elastic up to about 30% of its maximum compressive strength f 0c. For stresses above 0.3f 0c concrete

begins to soften and the stress-strain curve shows a gradual increase in curvature up to about 0.75f 0c to 0.9f 0c, after which the curve bends

more sharply until it approaches the peak point at f 0c. Beyond this

point the curve has a descending part until crushing failure occurs at some ultimate strain value u Fig.3.1.

2. The volumetric strain v = 1 + 2 + 3 is almost linear up to about

0.75f 0c to 0.9f 0c. At this point the direction of the volumetric strain

is reversed and the material starts dilating Fig.3.1. The stress corre- sponding to the minimal volumetric strain is defined as critical stress (Richart et al., 1929).

3. Concrete with higher strength behave as linear to a higher stress level than low strength concrete, but seems to be more brittle on the de- scending portion of the stress strain curve. All peak point correspond approximately to a value of 0.002 Figure 3.2.

The first two points are associated with the mechanism of internal pro- gressive micro cracking. When the stress is still in the region of 0.3f 0c the

internal energy is not sufficient to create new micro crack surfaces and the cracking existing in the concrete before loading remain nearly unchanged. The stress level corresponding to 0.3f 0c has been defined as onset of local-

ized cracking and has been proposed as the limits of elasticity (Kotsovos and Newman, 1977). For stress between 30 and 50 % of f 0c the bond cracks start

to extend because of the stress concentration at the crack tips. Mortar cracks remain negligible and the available internal energy is approximately balanced by the required crack release energy. At this stage the crack propagation is stable in the sense that cracks rapidly reach their final lengths, if the applied stress is maintained constant.

For the next stress range 50-70 % of f 0c some cracks at nearby aggregates

Figure 3.1: Typical stress-strain curves for concrete in uniaxial compression test. (a) Axial and lateral strains. (b) Volumetric strain (v = 1+ 2+ 3)

cracks keep growing. If the load is maintained constant the crack continue to propagate with a decreasing rate to their final lengths.

For higher stress levels the system is unstable and complete disruption can occur even if the load is maintained constant. Microcracks through the mortar in the direction of the applied stress bridge together the bond microcracks at the surface of the nearby aggregates and form macroscopic cracks. This stage correspond to the descending portion of the concrete stress-strain curve (softening). The stress level of about 0.75f 0c is termed as

onset of unstable fracture propagation or critical stress since it corresponds to the minimum value of the volumetric strain v.

The initial modulus of elasticity E0 is generally correlated to the uniaxial

compressive strength and can be approximated with the empirical formula: E0 = 5700

p

Figure 3.2: Uniaxial compressive stress-strain curves for concrete with dif- ferent strengths

which gives a reasonable accuracy. The poissons ratio ν also varies with the compressive strength f 0c, with 0.19 or 0.20 being representative values.

Uniaxial tension behaviour

The stress-strain curves for uniaxial tension tests are similar in shape to those observed for uniaxial compression Figure 3.3. However the tensile strength f 0t is significantly lower than the corresponding strength in com-

pression f 0c, with a ratio of 0.05-0.1. The concrete behaviour is nearly linear

elastic up to about 60% of its maximum tension strength f 0t. The following

interval of stable crack propagation is very short and the system becomes un- stable around 0.75f 0t . The direction of cracks propagation is transversal to

the applied stress direction. The descending portion of the stress-strain curve is difficult to follow because the crack propagation is very rapid. The value of f 0tis difficult to measure experimentally and there are several formulae to

estimate it from the corresponding value of the compression strength. The modulus of elasticity and the poissons ratio under uniaxial tension are respectively higher and lower than the case of uniaxial compression.

Biaxial behaviour

The strength and ductility of the concrete under biaxial states depends on the nature of stress state: compressive type or tensile type. The biaxial strength envelope Fig.3.4 represented by Kupfer et al. (1969) [62] suggest that the biaxial compression strength of the material increases compared to the equivalent uniaxial state. Equally from the stress-strain curves it is possible to recognize that the tensile ductility of concrete is greater under biaxial compression state than uniaxial compression Fig.3.5 .

The biaxial tension strength is very similar to what measured for uniax- ial tension state. Under biaxial compression-tension state the compressive strength decreases almost linearly as the applied tensile strength is increased. Some studies suggest that the maximum strength envelope is almost in- dependent of load path (Nelissen,1972).

The growth of major microcracks is associated with an inelastic volume increase defined dilatancy. This phenomenon becomes visible when the fail- ure point is approached. The failure will occur along surfaces orthogonal to the direction of the maximum tensile stress or strain. In particular the tensile strains are critical in defining the failure criterion of concrete.

Figure 3.5: Stress-strain relationships for concrete under biaxial compression (Kupfer et al.,1969).

Typical stress-strain results from triaxial tests on concrete indicate that the ultimate axial strength increases considerably with the confining stress Figure 3.6. As the hydrostatic stress increases, the behaviour of the con- crete moves from quasi-brittle to plastic softening to plastic hardening. This happens because under higher hydrostatic stresses the possibility of bond cracking is reduce and the failure rather explained with the crushing of ce- ment paste. During hydrostatic compression tests the concrete behaves as nonlinear during the loadings stages, while upon unloading the slope of the curve is almost constant and approximately equal to the initial tangent of the loading curve Figure 3.7. The failure surface can be defined as a function of the three principal stresses. The elastic limit and failure surfaces of concrete representation in the three principal stresses space is shown indicatively in Figure 3.7, assuming the material is isotropic. For small hydrostatic pres- sures the deviatoric sections are convex and non circular, becoming more or less circular for increasing compressions (along the σ1 = σ2 = σ3 axis). Fi-

nally, the failure surface appears to be independent of the load path (Gerste et al,1978; Kotzovos,1979).