Modelling strength development of cement-stabilised clay and clay with sand impurity cured under varying temperatures

Abstract

Curing temperature has been reported to have significant effect on the early and long-term strength development of cementitious systems such as concrete, mortar, cement-stabilised granular soil, and cement-stabilised clay. For cement-stabilised clays, elevated curing temperature is reported to enhance both early and long-term strength, which is different from that of concrete, mortar, and cemented granular soil. Presently, long-term physio-chemical studies were limited in the literature to fully explain this behaviour. At the same time, sand impurities in clay, which are commonly encountered in the field, have not been considered thoroughly in previous studies. Discussion on methodologies to evaluate temperature sensitivity and its consequence on strength development of cement-stabilised soil is limited. This paper aims to address these knowledge gaps by conducting unconfined compressive and physio-chemical tests on Portland blast furnace cement (CEM III/C) and ordinary Portland cement (CEM I)-stabilised kaolin clay with and without sand impurities cured at different temperatures (cement classification is based on BS EN 197-1 (BSI 2011). It is found that the distinct temperature effects on long-term strength behaviour are mainly attributed to both increased strength-enhancing materials in the cement-soil system and the presence of fine-grained clay particles. A generic method of evaluating temperature sensitivity on cementitious systems with a novel approach to incorporate temperature effect on strength development of cement-stabilised clayey soil is proposed and validated with data obtained from published literature on similar materials.

Appendix

• Changing mechanism assumption

${t}_{r}$ and ${t}_{T}$ are the curing time required to reach a certain degree of reaction, $D$ under temperature ${T}_{r}$ and $T$.

$$\begin{array}{c}\frac{{D}_{r}^{{^{\prime}}}}{{D}_{T}^{{^{\prime}}}}=\frac{{k}_{{T}_{r}}\left(D\right)}{{k}_{T}\left(D\right)}=\frac{\frac{1}{\sqrt{2\pi }{\sigma }_{r}{t}_{r}}{exp}\left(-\frac{{\left(\mathit{ln}{t}_{r}-{\mu }_{r}\right)}^{2}}{2{{\sigma }_{r}}^{2}}\right)}{\frac{1}{\sqrt{2\pi }{\sigma }_{T}{t}_{T}}{exp}\left(-\frac{{\left(\mathit{ln}{t}_{T}-{\mu }_{T}\right)}^{2}}{2{{\sigma }_{T}}^{2}}\right)}\\ =\frac{{\sigma }_{T}}{{\sigma }_{r}}\frac{{t}_{T}}{{t}_{r}}{exp}\left(\frac{{\left({ln}{t}_{T}-{\mu }_{T}\right)}^{2}}{2{{\sigma }_{T}}^{2}}-\frac{{\left({ln}{t}_{r}-{\mu }_{r}\right)}^{2}}{2{{\sigma }_{r}}^{2}}\right)\\ =\frac{{\sigma }_{T}}{{\sigma }_{r}}{{t}_{r}}^{\frac{{\sigma }_{T}}{{\sigma }_{r}}-1}{exp}({\mu }_{T}-\frac{{\sigma }_{T}}{{\sigma }_{r}}{\mu }_{r})={exp}(\frac{{E}_{a}^{{^{\prime}}}\left(D\right)}{R}\left(\frac{1}{T}-\frac{1}{{T}_{r}}\right))\end{array}$$

(21)

• Single mechanism assumption

${t}_{r}$ and ${t}_{T}$ are the curing time required to reach a certain maturity, $M$ under temperature ${T}_{r}$ and $T$.

$$\begin{array}{c}M={k}_{{T}_{r}}{t}_{r}={k}_{T}{t}_{T}\\ \frac{{t}_{T}}{{t}_{r}}=\frac{{k}_{{T}_{r}}}{{k}_{T}}={exp}(\frac{{E}_{a}{^{\prime}}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{r}}\right))\end{array}$$

(22)

By taking logarithmic of both sides, the following relationship can be derived:

$${ln}{t}_{T}-{ln}{t}_{r}=\frac{{E}_{a}{^{\prime}}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{r}}\right)$$

(23)

Subtract ${\mu }_{r}$ to both sides and divide by $\sqrt{2}{\sigma }_{r}$:

$$\begin{array}{c}{ln}{t}_{T}-{ln}{t}_{r}-{\mu }_{r}=\frac{{E}_{a}{^{\prime}}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{r}}\right)-{\mu }_{r}\\ \frac{{ln}{t}_{T}-\left(\frac{{E}_{a}{^{\prime}}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{r}}\right)+{\mu }_{r}\right)}{\sqrt{2}{\sigma }_{r}}=\frac{{ln}{t}_{r}-{\mu }_{r}}{\sqrt{2}{\sigma }_{r}}\end{array}$$

(24)

According to maturity theory, the same degree of reaction is achieved as the same maturity:

$$\frac{1}{2}+\frac{1}{2}\mathrm{erf}\left[\frac{\mathrm{ln}{t}_{T}-{\mu }_{T}}{\sqrt{2}{\sigma }_{T}}\right]=\frac{1}{2}+\frac{1}{2}\mathrm{erf}\left[\frac{\mathrm{ln}{t}_{r}-{\mu }_{r}}{\sqrt{2}{\sigma }_{r}}\right]$$

(25)

Both sides are the cumulative distribution function of lognormal distribution and are thus monotonically increasing. The following relationships can be obtained:

$$\frac{\mathrm{ln}{t}_{T}-{\mu }_{T}}{\sqrt{2}{\sigma }_{T}}=\frac{\mathrm{ln}{t}_{r}-{\mu }_{r}}{\sqrt{2}{\sigma }_{r}}$$

(26)

For Eq. (24) and (26) to be true, the following criteria are derived:

$$\begin{array}{c}{\sigma }_{T}={\sigma }_{r}\\ {\mu }_{T}=\frac{{E}_{a}{^{\prime}}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{r}}\right)+{\mu }_{r}\end{array}$$

(27)

The above derivation can be validated by taking ratio of rate of reaction. Since $k$ is independent of $D$, a constant ratio $\mathrm{c}$ is derived for $D\in [\mathrm{0,1}]$:

$$\frac{{D}_{r}^{^{\prime}}}{{D}_{T}^{^{\prime}}}=\frac{{k}_{{T}_{r}}}{{k}_{T}}=\mathrm{exp}\left(\frac{{{E}_{a}}^{^{\prime}}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{r}}\right)\right)=\mathrm{c}$$

(28)

At the same degree of reaction, Eqs. (25) and (26) can be used to obtain the relationship between ${t}_{r}$ and ${t}_{T}$:

$${t}_{T}={t}_{r}^{\frac{{\sigma }_{T}}{{\sigma }_{r}}}\mathrm{exp}({\mu }_{T}-\frac{{\sigma }_{T}}{{\sigma }_{r}}\times {\mu }_{r})$$

(29)

Substitute corresponding $\frac{dD}{dt}$ into Eq. (28):

$$\begin{array}{c}\frac{{D}_{r}^{{^{\prime}}}}{{D}_{T}^{{^{\prime}}}}=\frac{\frac{1}{\sqrt{2\pi }{\sigma }_{r}{t}_{r}}{exp}\left(-\frac{{\left(\mathit{ln}{t}_{r}-{\mu }_{r}\right)}^{2}}{2{{\sigma }_{r}}^{2}}\right)}{\frac{1}{\sqrt{2\pi }{\sigma }_{T}{t}_{T}}{exp}\left(-\frac{{\left(\mathit{ln}{t}_{T}-{\mu }_{T}\right)}^{2}}{2{{\sigma }_{T}}^{2}}\right)}\\ =\frac{{\sigma }_{T}}{{\sigma }_{r}}\frac{{t}_{T}}{{t}_{r}}{exp}\left(\frac{{\left({ln}{t}_{T}-{\mu }_{T}\right)}^{2}}{2{{\sigma }_{T}}^{2}}-\frac{{\left({ln}{t}_{r}-{\mu }_{r}\right)}^{2}}{2{{\sigma }_{r}}^{2}}\right)=c\end{array}$$

(30)

Substitute Eq. (29) in Eq. (30):

$$\frac{{D}_{r}^{{^{\prime}}}}{{D}_{T}^{{^{\prime}}}}=\frac{{\sigma }_{T}}{{\sigma }_{r}}{{t}_{r}}^{\frac{{\sigma }_{T}}{{\sigma }_{r}}-1}{exp}({\mu }_{T}-\frac{{\sigma }_{T}}{{\sigma }_{r}}{\mu }_{r})={exp}\left(\frac{{E}_{a}{^{\prime}}}{{R}}\left(\frac{1}{T}-\frac{1}{{T}_{r}}\right)\right)=c$$

(31)

Eq. (31) is true when Eq. (27) is met.

Thus, the apparent activation energy ${E}_{a}^{^{\prime}}$ is $R$ times of the gradient of the linear relationship between the difference of reciprocal of temperature and $\mu$ difference:

$${\mu }_{r}-{\mu }_{r}=\frac{{\mathrm{E}}_{a}^{^{\prime}}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{r}}\right)$$

(32)