Series and parallel phase model; solids in hcp

Series and parallel models

Concrete is idealised as a composite of mortar (hcp + sand), the interface (ITZ) and coarse aggregate. Their arrangement under load decides how the stress is transferred.

 Parallel model        Series model
 load ↓ ↓ ↓            load ↓
 +----+---+----+       +---------+
 | M  | A | M  |       | mortar  |
 |    |   |    |       +---------+
 +----+---+----+       |aggregate|
 same strain           +---------+
 each phase            same stress,
 shares load           strains add

• Parallel (Voigt) model: the phases act side by side; each has the same strain and the load is shared in proportion to stiffness. Upper bound of modulus:

$$E_c = E_m V_m + E_a V_a$$

• Series (Reuss) model: the phases are in layers one after the other; each carries the same stress and strains add. Lower bound:

$$\frac{1}{E_c} = \frac{V_m}{E_m} + \frac{V_a}{E_a}$$

where $V$ is the volume fraction (subscripts $m$ = mortar, $a$ = aggregate). The actual modulus of concrete lies between the two bounds (a combination of both models, as in Hirsch's model); including the weak ITZ in series reduces it. Load transfer is by bond/shear at the interface, so the weak interface causes microcracks and nonlinearity.

Solids in hydrated cement paste

• C-S-H gel (50-60%): main binding product.
• Calcium hydroxide (20-25%): hexagonal crystals.
• Calcium sulphoaluminates (15-20%): ettringite, monosulphate.
• Unhydrated cement grains.