Atomistic Simulation studies of the Cement Paste Components

Atomistic Simulation studies of the Cement Paste Components
⎛σ1⎞ ⎛C11 C12 C13 C14 C15 C16 ⎞⎛ε1⎞
⎜ ⎟ ⎜ ⎟⎜ ⎟
⎜
σ2⎟ ⎜
C22 C23 C24 C25 C26 ⎟⎜
ε2⎟
⎜σ ⎟ ⎜
3
C33 C34 C35 C ⎟⎜
36
ε ⎟
3
⎜ ⎟=
⎜ ⎟⎜ ⎟
⎜σ4⎟ ⎜ C44 C45 C46 ⎟⎜ε4⎟
⎜σ ⎟ ⎜ C C ⎟⎜ε
⎟
5 55 56 5
⎜ σ ⎟ ⎜ 6
C ⎟⎜
66
ε ⎟
⎝ ⎠ ⎝ ⎠⎝ 6 ⎠
(7.5.2)
where the terms under the matrix diagonal are equal to those up by symmetry. A
material has, therefore, 21 independent elastic constants, that can be further reduced by
crystalline symmetry [273]. The unit cell of the crystal can be deformed in such a way
that there is only one strain component different from zero [282]. In that case, the
relationship between the applied strain ε j and the generated stress σ i depends only on
one elastic tensor coefficient C ij :
σ
= C ε
(7.5.3)
i ij j
In chapter 6 the elastic tensor coefficients of tricalcium aluminate were calculated
following this procedure. Different degrees of strain were applied, in steps of ±1%, from
–3% (contraction) to +3% (expansion) by modifying the lattice parameter of the
crystals. The SIESTA code calculates the corresponding stress components from energy
derivatives with respect to the strain [231]:
σ
i
2
∂ U
= (7.5.4)
∂ ε
i
Finally, the correspondent coefficient C ij was calculated from the linear fit of the
computed obtained data. This procedure was followed in reference [283] to calculate the
elastic constants of more than 40 ceramic materials. In chapter 5, the GULP code was
used to calculate the elastic properties of the main phases present in the cement paste, as
well as 21 hydrated calcium silicate crystals. The only difference with the previous
procedure is that GULP calculates directly the elastic tensor from the energy derivatives
[239]. Thus, there is no need to make any data fit.
188