Advantages of a 3-parameter reduced constitutive model for the measurement of polymers elastic modulus using tensile tests

Abstract

Exact measurements of the rheological parameters of time-dependent materials are crucial to improve our understanding of their intimate relation to the internal bulk microstructure. Concerning solid polymers and the apparently simple determination of Young’s modulus in tensile tests, international standards rely on basic protocols that are known to lead to erroneous values. This paper describes an approach allowing a correct measurement of the instantaneous elastic modulus of polymers by a tensile test. It is based on the use of an appropriate reduced model to describe the behavior of the material up to great strains, together with well-established principles of parameter estimation in engineering science. These principles are objective tools that are used to determine which parameters of a model can be correctly identified according to the informational content of a given data set. The assessment of the methodology and of the measurements is accomplished by comparing the results with those obtained from two other physical experiments, probing the material response at small temporal and length scales, namely, ultrasound measurements with excitation at 5 MHz and modulated nanoindentation tests over a few nanometers of amplitude.

Appendix:  Sensitivity analysis fundamentals

Let $y^{\mathrm{meas}} ( t )$ be the measured stress output of a system (our material specimen), and $y ( \boldsymbol{\upbeta},t )$ the theoretical stress output of the behavioral model with a parameter vector $\boldsymbol{\upbeta}$ of dimension $n$, representing in our case the $n$ constitutive material parameters. The output error $e ( t )$, or the residuals, can be defined as

$$ e(t) = y^{\mathrm{meas}}(t) - y ( \boldsymbol{\upbeta},t ). $$

(A.1)

The role of an estimator is to minimize this output error. A least squares criterion can be used classically and is written as follows:

$$ E_{\mathrm{LS}} = \sum_{i = 1}^{m} \bigl( y^{\mathrm{meas}} ( t_{i} ) - y ( \boldsymbol{\upbeta},t_{i} ) \bigr)^{2} $$

(A.2)

where the sum is over the successive acquisition times $t_{i}$, up to a total number of measurement data points $m$.

The minimization of the criterion is done when its derivatives with respect to parameters $\beta_{j}$ $(j = 1\ldots n)$ are null:

$$\begin{aligned} \forall j \in [1,n],\frac{\partial E_{\mathrm{LS}}}{\partial \beta_{j}} = 0 \quad \to\quad & \sum _{i = 1}^{m} \biggl( \frac{\partial y ( \boldsymbol{\upbeta},t_{i} )}{\partial \beta_{j}} \bigl[ y^{\mathrm{meas}} ( t_{i} ) - y ( \boldsymbol{\upbeta},t_{i} ) \bigr] \biggr) = 0 \\ \quad \to\quad & \sum_{i = 1}^{m} \bigl( X_{j} ( \boldsymbol{\upbeta},t_{i} ) \bigl[ y^{\mathrm{meas}} ( t_{i} ) - y ( \boldsymbol{\upbeta},t_{i} ) \bigr] \bigr) = 0. \end{aligned}$$

(A.3)

From this equation, the sensitivity coefficient vector component $X_{j}$ associated to the parameter $\beta_{j}$ can be clearly recognized as

$$ X_{j} ( \boldsymbol{\upbeta},t_{i} ) = \frac{\partial y(\boldsymbol{\upbeta},t_{i})}{\partial \beta_{j}}. $$

(A.4a)

Normalized sensitivity coefficients,

$$ \tilde{X}_{j} = \beta_{j}\frac{\partial y(\boldsymbol{\upbeta},t)}{\partial \beta_{j}}, $$

(A.4b)

are introduced to graphically check the level of sensitivity and possible correlations between parameters: different intervals of the time-independent variable $t$ may be advantageously considered. These coefficients are dimensionally homogeneous to the signal itself, the stress in MPa.

Sensitivity coefficients express how much a model reacts to some small variation of the parameters. Sensitivity coefficients play a fundamental role in the conditioning of the inverse parameter estimation process and, as a consequence, in the errors made in the estimates of confidence bounds. It is obvious that major sensitivities are sought for, when designing an experiment for metrological purposes. In almost all cases, sensitivity coefficients are nonlinear functions of the parameters themselves when the model $y$ is a nonlinear function of $\beta_{j}$.

Switching to a matrix formulation and defining vectors $\mathbf{Y}^{\mathrm{meas}}$ and $\mathbf{Y}$ as

$$ \mathbf{Y}^{\mathrm{meas}} = \begin{bmatrix} y^{\mathrm{meas}} ( t_{1} )\\ y^{\mathrm{meas}} ( t_{2} ) \\ \ldots\\ y^{\mathrm{meas}} ( t_{m} ) \end{bmatrix} \quad \mbox{and} \quad \mathbf{Y} = \begin{bmatrix} y ( \boldsymbol{\upbeta},t_{1} )\\ y ( \boldsymbol{\upbeta},t_{2} ) \\ \ldots \\ y ( \boldsymbol{\upbeta},t_{m} ) \end{bmatrix} $$

(A.5)

makes the minimization process expressed as

$$ \mathbf{X}^{T} \bigl( \mathbf{Y}^{\mathrm{meas}} - \mathbf{Y} \bigr) = 0 $$

(A.6)

with $\mathbf{X}$, the $m \times n$ rectangular sensitivity matrix, where $m$ is the number of rows (dimension of the experimental observation time vector $\mathbf{t}$) and $n$ is the number of columns (dimension of the parameter vector $\boldsymbol{\upbeta}$).

In the case of a linear model with respect to the parameters (linear estimation problem) for which the matrix of the sensitivity coefficients does not depend on the parameters, we have

$$ \mathbf{Y} = \mathbf{X}\boldsymbol{\upbeta}. $$

(A.7)

The estimated parameter vector, denoted by $\hat{\boldsymbol{\upbeta}}$, corresponds to the value reached by $\boldsymbol{\upbeta}$ when the criterion is minimized. Therefore, using (A.7), we can rewrite Eq. (A.6) as

$$ \mathbf{X}^{T} \bigl( \mathbf{Y}^{\mathrm{meas}} - \mathbf{X}\hat{\boldsymbol{\upbeta}} \bigr) = \boldsymbol{0}. $$

(A.8)

Relation (A.8) can be inverted to obtain the expression of $\hat{\boldsymbol{\upbeta}}$ in the case of a linear estimation problem:

$$ \hat{\boldsymbol{\upbeta}} = \bigl( \mathbf{X}^{T} \mathbf{X} \bigr)^{ - 1} \mathbf{X}^{T} \mathbf{Y}^{\mathrm{meas}}. $$

(A.9)

However, most parameter estimation problems are not linear and require an iterative linearizing procedure. This is obtained by developing the solution at rank $n+1$ in the neighborhood of the solution obtained for the prior iteration at rank $n$:

$$ \mathbf{Y}^{(n + 1)} = \mathbf{Y}^{(n)} + \mathbf{X}^{(n)} \bigl( \hat{\boldsymbol{\upbeta}}^{(n + 1)} - \hat{\boldsymbol{\upbeta}}^{(n)} \bigr). $$

(A.10)

If relation (A.6) is written at rank $n+1$ for parameters estimated at rank $n$, we obtain $\mathbf{X}^{T(n)} ( \mathbf{Y}^{\mathrm{meas}} - \mathbf{Y}^{(n + 1)} ) = \boldsymbol{0}$, and when this is combined with relation (A.10), we obtain the following relation of recurrence between estimated parameters at rank $n+1$ and rank $n$:

$$ \hat{\boldsymbol{\upbeta}}^{(n + 1)} = \hat{\boldsymbol{\upbeta}}^{(n)} + \bigl( \mathbf{X}^{T(n)} \ \mathbf{X}^{(n)} \bigr)^{ - 1} \mathbf{X}^{T(n)} \bigl( \mathbf{Y}^{\mathrm{meas}} - \mathbf{Y}^{(n)} \bigr) $$

(A.11)

which defines the iterative procedure that can be used to estimate the parameters (Gauss–Newton algorithm).

The statistical estimator’s properties depend on the noise $\varepsilon (t)$of the signal. If the theoretical model is assumed to be unbiased (perfect) then we have

$$ Y^{\mathrm{meas}}(t) = Y(t,\boldsymbol{\upbeta} ) + \varepsilon (t). $$

(A.12)

If classical statistical assumptions are made concerning the experimental noise $\varepsilon (t)$ in the measured stress signal (Beck and Arnold 1977), it is possible to obtain an approximation of the errors that can be made in the estimation process for the different parameters. These assumptions are:

• (a) Zero mean value of the signal in the absence of excitation, which corresponds to a zero expectancy for noise (expected value $E(\boldsymbol{\upvarepsilon} ) = 0$);

• (b) Constant variance or standard deviation of the noise, $V(\boldsymbol{\upvarepsilon} ) = \sigma_{0}^{ 2}$.

In the case of a 1st order linearized estimation, its expected value can be proved to be

$$ E(\hat{\boldsymbol{\upbeta}} ) = \boldsymbol{\upbeta}. $$

(A.13)

This means that there is no error or bias made on the identified parameters.

The variance–covariance matrix $\boldsymbol{\Delta}$ of the estimated parameters (generalization of the scalar-valued variance to a higher dimension) involves the sensitivity coefficients. It is calculated as $\boldsymbol{\Delta} = E [ ( \hat{\boldsymbol{\upbeta}} - E ( \hat{\boldsymbol{\upbeta}} ) ) ( \hat{\boldsymbol{\upbeta}} - E ( \hat{\boldsymbol{\upbeta}} ) )^{T} ] = \sigma_{0}^{2} ( \mathbf{X}^{T} \mathbf{X} )^{ - 1}$ which in the expanded form gives

$$ \boldsymbol{\Delta} = \begin{bmatrix} V(\beta_{1}) & {\mathop{\mathrm{cov}}} (\beta_{1},\beta_{2}) &\ldots & \mathrm{cov} (\beta_{1},\beta_{p})\\ {\mathop{\mathrm{cov}}} (\beta_{1},\beta_{2}) & V(\beta_{2}) &\ldots & \mathrm{cov} (\beta_{2},\beta_{p}) \\ \ldots &\ldots &\ldots &\ldots \\ {\mathop{\mathrm{cov}}} (\beta_{1},\beta_{p}) & {\mathop{\mathrm{cov}}} (\beta_{2},\beta_{p}) &\ldots & V(\beta_{p}) \end{bmatrix}. $$

(A.14)

A stochastic analysis can be made that consists of calculating the variance–covariance matrix through Eq. (A.14) theoretically according to a given noise and set of parameters. This symmetric square matrix has dimension equal to the number of parameters. The diagonal terms correspond directly to the variance of each parameter $V(\beta_{j})$. They can be used to determine the error made for each parameter. This error can be expressed as a percentage

$$ \mathit{Err}(\beta_{j}) = \frac{\sqrt{V(\beta_{j})}}{\beta_{j}}. $$

(A.15)

The off-diagonal terms can be used to calculate the correlation coefficients $\rho_{mn}$ which express the degree of correlation of the parameters,

$$ \rho_{rs} = \frac{\mathrm{cov} (\beta_{r},\beta_{s})}{\sqrt{V(\beta_{r})} \sqrt{V(\beta_{s})}}. $$

(A.16)

The $\vert \rho_{rs} \vert$ values lie between 0 and 1. In the case of a model with strongly correlated parameters, the correlation coefficients are close to 1, which means that two columns of the sensitivity matrix $X$ are nearly proportional to each other. The resulting confidence bounds interval for two correlated parameters are therefore generally very high. This means that a large number of solutions exist for these two parameters to allow for a good fit to the experimental curve. A deterministic algorithm like the steepest gradient technique, used for the minimization process, is consequently very sensitive to the initial estimate made for the parameters. A strategy to produce first approximations using the physical background is highly recommended. But even so, the estimation problem is ill-conditioned and indicates to the experimentalist that the physical description involved is probably not appropriate and must be changed.

In the following considerations, the identifiability of the model parameters will be analyzed through the $\tilde{\boldsymbol{\Delta}}$ array, which combines the variance–covariance matrix and the correlation matrix. $\tilde{\boldsymbol{\Delta}}$ puts the main diagonal of $\boldsymbol{\Delta}$ in place and the correlation coefficients on the off-diagonal terms,

$$ \tilde{\boldsymbol{\Delta}} = \begin{bmatrix} \mathit{Err}(\beta_{1}) & \rho_{12} &\ldots & \rho_{1p}\\ \rho_{12} & \mathit{Err}(\beta_{2}) &\ldots & \rho_{2p} \\ \ldots &\ldots &\ldots &\ldots \\ \rho_{1p} & \rho_{2p} &\ldots & \mathit{Err}(\beta_{p}) \end{bmatrix}. $$

(A.17)

When there is no model bias, or perfect agreement between the conditions of the experiment and the model based on it, the relation between the estimated parameter vector for the nonlinear estimation problem and its ‘exact’ value can be given at convergence by the following formula:

$$ \hat{\boldsymbol{\upbeta}} = \boldsymbol{\upbeta} + \bigl( \mathbf{X}^{T} \mathbf{X} \bigr)^{ - 1} \mathbf{X}^{T} \boldsymbol{\upvarepsilon}. $$

(A.18)

In this case, the residuals $e ( t )$ that are the difference between the model and the data correspond exactly to the noise. Their standard deviation corresponds to the experimental standard deviation of the noise, and the residuals remain unsigned with no large fluctuation around the zero level.

The matrix $\tilde{\boldsymbol{\Delta}}$ is a good tool to investigate the identification conditions of a parameter estimation problem.