TTS_10 {TTS}

Time-Temperature Superposition (TTS) analysis

Description

This function provides an estimate of the Master Curve in a similar way to the TTS function, with the difference that, in this case, a thin plate spline fit is performed (instead the B-splines smoothing), within the framework of the application of generalized additive models (GAM), as implemented in the mgcv package.

Usage

TTS_10(
  x,
  reference.temperature = 40,
  n = 100,
  nB = 100,
  method = c("derived", "WLF", "Arrhenius")
)

Arguments

x

Matrix or data frame composed of three columns: a numeric column vector with the experimental frequencies (in logarithmic scale, base-ten), the response variable (E' or G' modulus, compliance, etc.) base-ten logarithm vector and, finally the corresponding temperatures vector.

reference.temperature

Value of the selected reference temperature at which the Master Curve of the response variable (E' or G' modulus, compliance, etc.) will be obtained, the default value of temperature is 40.

n

Number of partitions in the frequency domain in order to fit the B-spline basis. The default value is 100.

nB

Number of bootstrap replicates to estimate confidence intervals of Master Curve fitting. The default is 100.

method

A string vector composed of one of the following options: "derived" (by default), "WLF" and "Arrhenius".

Value

The function returns a list composed of the following outputs:

data

Input experimental data.

aT

Numerical vector of horizontal shifts between the modulus curves.

bT

Numerical vector of vertical shifts between the modulus curves.

TTS.data

Master Curve Data frame defined by three columns: Log10_time, Log10_Compliance and Temperature.

ref.temp

Reference temperature value.

TTS.gam

Data frame of the Generalized Additive Model with B-splines (GAM) estimate of the Master Curve. It contains two columns: time and Prediction.

I.lower

Lower limit of bootstrap confidence interval corresponding to the estimated B-splines Master Curve.

I.upper

Upper limit of bootstrap confidence interval corresponding to the estimated B-splines Master Curve.

residuals

Residuals corresponding to the GAM with B-splines Master Curve fitting.

Author(s)

Antonio Meneses [e-mail], Salvador Naya [e-mail] and Javier Tarrio-Saavedra [e-mail]

References

Naya, S., Meneses A., Tarrio-Saavedra, J., Artiaga R., Lopez-Beceiro, J. and Gracia-Fernandez C. (2013) New method for estimating shift factors in time-temperatura superposition models. Journal of Thermal Analysis and Calorimetry. ISSN 1388-6150. DOI 10.1007/s10973-013-3193-1.

Williams, M. L. (1964) Structural analysis of Viscoelastic materials. AIAA Journal, 785-808.

Artiaga R., Garcia A. Fundamentals of DMA. In: 'Thermal analysis. Fundamentals and applications to material characterization' (ed.: Artiaga R.) Publicaciones de la Universidade da Coruna, A Coruna, Spain, 183-206 (2005).

Wood, S.N. (2017) Generalized Additive Models: An Introduction with R (2nd edition). Chapman and Hall/CRC.

Examples

## Epoxy
data(Epoxy)
x=Epoxy
## TTS_10 function applied to Epoxy.
Q=TTS_10(x,reference.temperature=40, method=c("derived","WLF","Arrhenius"))
names(Q)
## Horizontal shifts vector of compliance versus time curves.
Q$aT
## Reference temperature
Q$ref.temp
PLOT <- PLOT.TTS(Q)
## Generic plots of: data, aT, bT, TTS.data and TTS.gam
PLOT$PLOT.data(main="PLOT: Data",xlab="Log_time",
ylab="Log_Compliance")
PLOT$PLOT.aT(main="PLOT: horizontal shift factors",
xlab="Temperature", ylab="aT")
PLOT$PLOT.bT(main="PLOT: vertical shift factors",
xlab="Temperature",ylab="bT")
PLOT$PLOT.TTS.data(xlab="Log_time",
ylab="Log_Compliance")
PLOT$PLOT.TTS.gam( xlab="Log_time",
ylab="Log_Compliance",
main = "Fitted gam, Bootstrap confidence intervals",
sub = "Reference temperature = 40 Celsius degrees")