# Maturity functions

### What is a maturity function?

A maturity function is a mathematical equation used to calculate the Maturity. This number describes the effect that temperature and time have on the strength development of concrete. The standard definition is as follows:

“A maturity function is a mathematical expression to account for the combined effects of time and temperature on the strength development of a cementitious mixture. The key feature of a maturity function is the representation of how temperature affects the rate of strength development.” (ASTM C1074)

### Types of functions

There are many different maturity functions, each having a slightly different approach and parameters. In Maturix, three of the most used maturity functions are available:

In the sections below, we will explain you each of these three functions in detail and we will conclude with a comparison table.

## The Nurse-saul function

### Background

The Nurse-Saul function is seen as the first maturity function as it was introduced already back in 1951. At that time, there was a need for a procedure to account for the combined effects of time and temperature on strength development – specifically related to the curing of concrete at different temperatures.  This lead to the basic concept of concrete maturity:

“Concrete of the same mix, at the same maturity, has approximately the same strength whatever combination of temperature and time go to make up that maturity.” (Saul 1951)

This idea of using the accumulated time and temperature to determine the strength development led to the Nurse-Saul maturity function. This was standardized by the ASTM C1074 in 1987, and is still today one of the most used functions for calculating concrete maturity (mainly in the USA and Canada).

### Function

The Nurse-Saul function is based on the assumption that the rate of strength development is a linear function of temperature:

$M=\sum_0^t(T-T_{0})\Delta t$
\begin{align*} \text{M =} \qquad &\text{maturity index in (°C-days)}\\ \text{T =} \qquad &\text{average temperature over the time interval (°C)}\\ {T_{0} =} \qquad &\text{datum temperature} \\ {\Delta t =} \qquad &\text{the time interval}\\ \end{align*}

The equation above is used to calculate the Maturity Index, which for this function is known under the term temperature-time factor and expressed in units as °C-hours or °C-days.

The diagram below shows a temperature history and the calculated temperature-time factor according to the above equation. The temperature-time factor (at some age t*) equals the area from under the concrete temperature curve until the datum temperature (T₀). Figure 1: Concrete temperature/Time graph. Extracted from Carino, Nicholas & Lew, H. (2001)

### Datum temperature

Datum temperature is the temperature at which concrete stops its strength development. The datum temperature was traditionally set to be -10 °C, however many use 0 °C as a conservative assumption, even though the development of strength might continue below these temperatures.

ASTM C1074 recommends a datum temperature of 0 °C for Type I cement without admixtures and a curing temperature range from 0 to 40 °C. The right datum temperature can be determined experimentally by following the procedure in Appendix X1 of ASTM C1074 when high accuracy is desired.

## The Freiesleben Hansen & Pedersen (Arrhenius) function

### Background

After the introduction of the Nurse-Saul function, some researchers found out that the function’s assumption that the rate of strength development was a linear function of temperature was not always applicable. This limitation was especially problematic if the temperature was outside the 0-40°C range. After this finding, many researchers proposed other alternative functions to overcome this limitation.

One of the new proposed functions was the Freiesleben Hansen and Pedersen function. This was introduced in 1977 and is based on the Arrhenius equation. This assumes that the rate of strength development is an exponential function of temperature, and is used to compute the equivalent age at a reference temperature

With its non-linear approach and with correct parameters, this function will be able to give you the most accurate prediction of the in-place strength (especially outside the 0-40°C range) compared to e.g. the Nurse-Saul function.

### Function

The Freiesleben Hansen & Pedersen (Arrhenius) function is based on the assumption that the rate of strength development has an exponential relationship with temperature. This means that the higher the temperature, the faster the strength development and vice versa.

${t_{e}=\sum_0^t{e} ^{\frac{-E}{R} \left( \frac{1}{T}\frac{1}{T_{r}} \right)}\Delta t\ }$
\begin{align*} { t_{e}=} \qquad &\text{the equivalent age at the reference temperature}\\ \text{E =} \qquad &\text{activation energy}\\ {R =} \qquad &\text{universal gas constant (8.314 J/mol-K)} \\ {T=} \qquad &\text{average temperature during Δt in Kelvin} \\ {T_{r}=} \qquad &\text{reference temperature in Kelvin} \\ {\Delta t =} \qquad &\text{the time interval}\\ \end{align*}

The equation above is used to calculate the maturity index, which for this function is known under the term equivalent age and expressed in units as hours at 20 °C or days at 20 °C. In North America, this would be hours at 23 °C or days at 23 °C.

### Activation energy

The activation energy (E) represents the minimum energy that a molecule needs before it can take part in the chemical reaction. The value of the activation energy will depend on several factors like:

• Cement composition
• Cement fineness
• Water/cement ratio
• Degree of hydration

In the graph below you can see the results of calculating the Age conversion factor according to the equation below using different values of activation energy. Figure 2: Age conversion factor calculated with different activation energy values. Extracted from Carino, Nicholas & Lew, H. (2001).

In Maturix we use the standard activation energy values proposed by Freiesleben Hansen and Pedersen. The activation energy is a function of the concrete temperature Tc as follows:

\begin{align} for\ T_{c} &\geq 20 °C \\ E(T_{c})&=33,500 \ \mathrm{J}/\mathrm{mol}\\ \\ for\ T_{c} &

ASTM C1074 recommends an activation energy of 38.000 to 45.000 J/mol for Type I cement without admixtures or additions. But in fact, this will vary from concrete mix to concrete mix and on the curing temperature. Therefore, when a higher accuracy is desired this can be determined experimentally by following the procedure in Appendix X1 of ASTM C1074.

## The Dutch “Weighted Maturity” function

### Background

The weighted maturity function was proposed by Papadakis and Bresson, and modified by Vree in 1979. This maturity function is widely used in the Netherlands and in some other European countries.

### Function

The equation proposed by Vree is as follows:

$M_{w}=\sum t_{k}T_{k}C^{nk}$
\begin{align*} { M_{w}=} \qquad &\text{the weighted maturity (°C-hr) }\\ {t_{k} =} \qquad &\text{hardening time of concrete (hrs)}\\ {T_{k} =} \qquad &\text{the hardening temperature interval} \\ {C=} \qquad &\text{the C-value of Cement } \\ {n_{k}=} \qquad &\text{the temperature-dependent parameter for} \ {T_{k}} \\ \end{align*}

### The weighted factor

This function also tries to address some of the limitations with the Nurse-Saul equation. To do that, two additional parameters are included:

• The C-value, which is specific for the different types of cement
• The “nk” value allows for a non-linear effect of temperature on the strength development

The “C” and “nk” values combined as C^(nk) make up the “weighted factor” which, for values of C greater than one, increases almost exponentially with temperatures above 12.45 °C. Some values for “C” have been recommended for certain cement types.

• C = 1.25 for CEM I 32.5R, CEM I 52.5, CEM I 52.5R, and CEM II/B-V 32.5R
• C = 1.65 for CEM III/B 42.5 LH HS
• C = 1.60 for CEM II/B 42.5 LH HS plus
• C = 1.0 for CEM III/A 52.5 and CEM V/A 42.5

These values can also be determined by carrying out some laboratory tests. For more information on how to conduct them refer to NEN 5790.

## Comparison of maturity functions

In the table below, we have summarized the key things of each of the three functions explained above. So that you can get a quick overview of their main characteristics and how these compare with each other.

Nurse-Saul function

Freiesleben Hansen & Pedersen function

Weighted maturity function

Standards

ASTM C1074

ASTM C1074,

DS/EN 206-1:2002,

BS EN 13670

NEN 5790

Mostly used in

North America

Europe

Netherlands and Europe

Function

$M=\sum_0^t(T-T_{0})\Delta t$

${t_{e}=\sum_0^t{e} ^{\frac{-E}{R} \left( \frac{1}{T}\frac{1}{T_{r}} \right)}\Delta t\ }$

$M_{w}=\sum t_{k}T_{k}C^{nk}$

Strength development assumption

Linear function of temperature

Exponential relationship with temperature

Exponential relationship with temperature

Maturity Index

expressed in

Temperature-Time factor (°C-hours or °C-days)

Equivalent age at a reference

temperature (e.g. X hours at 20°C)

Weighted maturity (°C-hours or °C-days)

Specific parameters

Datum temperature

Activation energy

Weighted factor (C and n- value)

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