An Innovative Scheme to Make an Initial Guess for Iterative Optimization Methods to Calibrate Material Parameters of Strain-Hardening Elastoplastic Models

Abstract

Optimization can apply in almost every branch of science and technology. In particular, a gradient-based iterative method is a mathematical optimization procedure that can be used to make decisions. The gradient-based optimization method can only find a local minimum of the objective function if the algorithm starts with the appropriate initial data. The ambition of this article is to develop a new scheme to make an initial guess for iterative optimization methods to calibrate accurately the material parameters of strain-hardening elastoplastic constitutive models based on the test data. The elastoplastic models are Drucker–Prager and modified Cam-Clay, and the data obtained from triaxial, oedometric, and hydrostatic tests. The validity of proposed material parameters is evaluated using a home-made finite-element simulator. The results emphasize the ability of the proposed procedure to accurately calibrate material parameters.

Appendices

Appendix

Development of Analytical Equations for $\eta$ and $\xi _\mathrm{c}$

The analytical equations to compute the initial guess for parameters $\eta$ and $\xi _\mathrm{c}$ are developed. The proposed equations are obtained using experimental data that are at least two points. The points can be the extreme points, namely $pt_{1}=\left\{ P_1,\sqrt{J_{2_1}}\right\}$ and $pt_{2}=\left\{ P_2,\sqrt{J_{2_2}}\right\}$, where $P_2>P_1$ (Fig. 20a). The analytical equations are derived as:

$$\begin{aligned} \begin{array}{ccc} \sqrt{J_{2_{1}}}+\eta \,P_{1}-\xi _\mathrm{c}=0\\ &{} \implies &{} \sqrt{J_{2_{1}}}-\sqrt{J_{2_{2}}}+\eta \,\left( P_{1}-P_{2}\right) =0\\ \sqrt{J_{2_{2}}}+\eta \,P_{2}-\xi _\mathrm{c}=0. \end{array} \end{aligned}$$

(72)

Then, the parameter $\eta$ is estimated by:

$$\begin{aligned} \eta _\mathrm{est}=\frac{\sqrt{J_{2_{1}}}-\sqrt{J_{2_{2}}}}{P_{1}-P_{2}}. \end{aligned}$$

(73)

The parameter $\xi _\mathrm{c}$ is computed by:

$$\begin{aligned} \begin{array}{ccc} \sqrt{J_{2}}+\eta \,P-\xi _\mathrm{c}=0&\implies&\xi _{\mathrm{c}_\mathrm{est}}=\sum _{z=1}^{2}\left( \sqrt{J_{2_{z}}}+\eta \,P_{z}\right) .\end{array} \end{aligned}$$

(74)

Fig. 20

The selected points to estimate the initial guess for material parameters of: a Drucker–Prager, and b modified Cam-Clay, $P^{\circ }$, and $p_\mathrm{t}$

Development of Analytical Equations for $P^{\circ }$, $p_\mathrm{t}$, and eC

The analytical equations are developed to compute the initial guess for parameters $P^{\circ }$, $p_\mathrm{t}$, and eC. The proposed equations are obtained using hydrostatic test data that are at least three points. The points can be extremes and an intermediate point. The analytical equations are developed by considering $p_{zt}=P^{\circ }+p_\mathrm{t}$, and taking the derivative of $P_\mathrm{cc}=P$ with respect to $\epsilon _\mathrm{ev}$, as:

$$\begin{aligned} \begin{array}{ccc} P=-p_\mathrm{t}+p_{zt}\exp \left[ -\mathrm{eC}\;\epsilon _\mathrm{ev}\right]\Rightarrow & {} P^{'}=\frac{\mathrm{d}P}{\mathrm{d}\epsilon _\mathrm{ev}}=-\mathrm{eC}\,p_{zt}\exp \left[ -\mathrm{eC}\;\epsilon _\mathrm{ev}\right] . \end{array} \end{aligned}$$

(75)

The three selected points are called $pt_{1}=\left\{ \epsilon _{\mathrm{ev}_{1}}, P_{1}\right\}$, $pt_{2}=\left\{ \epsilon _{\mathrm{ev}_{2}},P_{2}\right\}$, and $pt_{3}=\left\{ \epsilon _{\mathrm{ev}_{3}},P_{3}\right\}$, where $\epsilon _{\mathrm{ev}_{3}}<\epsilon _{\mathrm{ev}_{2}}<\epsilon _{\mathrm{ev}_{1}}$ (Fig. 20b).

1.1 Estimation of eC

By taking two points pt$_{z}$ and pt$_{ w}$, the derivative $P^{'}$ for each point is obtained as:

$$\begin{aligned} \begin{array}{c} P_{z}^{'}=-\mathrm{eC}\,p_{zt}\exp \left[ -\mathrm{eC}\;\epsilon _{\mathrm{ev}_z}\right] \\ P_{ w}^{'}=-\mathrm{eC}\,p_{zt}\exp \left[ -\mathrm{eC}\;\epsilon _{\mathrm{ev}_{ w}}\right] . \end{array} \end{aligned}$$

(76)

The parameter eC is computed by dividing $P_{z}^{'}$ by $P_{ w}^{'}$

$$\begin{aligned} \begin{array}{c} \frac{P_{z}^{'}}{P_{w}^{'}}=\frac{\exp \left[ -\mathrm{eC}\;\epsilon _{\mathrm{ev}_z}\right] }{\exp \left[ -\mathrm{eC}\;\epsilon _{\mathrm{ev}_w}\right] }=\exp \left[ -\mathrm{eC}\;(\epsilon _{\mathrm{ev}_z}-\epsilon _{\mathrm{ev}_w})\right] \\ \ln \left( \frac{P_{z}^{'}}{P_{w}^{'}}\right) =-\mathrm{eC}\;(\epsilon _{\mathrm{ev}_z}-\epsilon _{\mathrm{ev}_w})\\ \mathrm{eC}_\mathrm{est}=-\frac{\ln \left( {P_{z}^{'}}/{P_{w}^{'}}\right) }{\epsilon _{\mathrm{ev}_z}-\epsilon _{\mathrm{ev}_w}}, \end{array} \end{aligned}$$

(77)

where

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{c} P_{z}^{'}\cong \frac{P_{1}-P_{2}}{\epsilon _{\mathrm{ev}_{1}}-\epsilon _{\mathrm{ev}_{2}}}\\ \epsilon _{\mathrm{ev}_{z}}=\frac{\epsilon _{\mathrm{ev}_{1}}+\epsilon _{\mathrm{ev}_{2}}}{2}\\ P_{w}^{'}\cong \frac{P_{3}-P_{2}}{\epsilon _{\mathrm{ev}_{3}}-\epsilon _{\mathrm{ev}_{2}}}\\ \epsilon _{\mathrm{ev}_{w}}=\frac{\epsilon _{\mathrm{ev}_{3}}+\epsilon _{\mathrm{ev}_{2}}}{2}, \end{array}&\,\end{array}\right. } \end{aligned}$$

(78)

where the numbers 1, 2, 3 are indices of three hydrostatic points.

1.2 Estimation of $p_{zt}$

By taking two points pt$_{z}$ and pt$_{w}$, the P for each point is obtained as:

$$\begin{aligned} \begin{array}{c} P_{z}=-p_\mathrm{t}+p_{zt}\exp \left[ -\mathrm{eC}\;\epsilon _{\mathrm{ev}_z}\right] \\ P_{w}=-p_\mathrm{t}+p_{zt}\exp \left[ -\mathrm{eC}\;\epsilon _{\mathrm{ev}_w}\right] . \end{array} \end{aligned}$$

(79)

The parameter $p_{zt}$ is gotten by subtracting $P_{w}$ from $P_{z}$, as:

$$\begin{aligned} \begin{array}{c} \left( P_{z}-P_{w}\right) =p_{zt}\left( \exp \left[ -\mathrm{eC}\;\epsilon _{\mathrm{ev}_z}\right] -\exp \left[ -\mathrm{eC}\;\epsilon _{\mathrm{ev}_w}\right] \right) \\ p_{zt}=\frac{P_{z}-P_{w}}{\exp \left[ -\mathrm{eC}\;\epsilon _{\mathrm{ev}_z}\right] -\exp \left[ -\mathrm{eC}\;\epsilon _{\mathrm{ev}_w}\right] }. \end{array} \end{aligned}$$

(80)

By selecting any pairs of three points pt$_{1}$, pt$_{2}$, and pt$_{3}$, the parameter $p_{zt}$ can be computed using Eq. (81). Here, the parameter $p_{zt}$ is obtained from two extreme points, as:

$$\begin{aligned} p_{{zt}_\mathrm{est}}=\frac{P_{1}-P_{3}}{\exp \left( -\mathrm{eC}\,\epsilon _{\mathrm{ev}_{1}}\right) -\exp \left( -\mathrm{eC}\,\epsilon _{\mathrm{ev}_{3}}\right) }. \end{aligned}$$

(81)

1.3 Estimation of $p_\mathrm{t}$

The parameter $p_\mathrm{t}$ is estimated using the below expression

$$\begin{aligned} \begin{array}{ccc} P=-p_\mathrm{t}+p_{zt}\exp \left[ -\mathrm{eC}\;\epsilon _\mathrm{ev}\right]\Rightarrow & {} p_\mathrm{t}=p_{zt}\exp \left[ -\mathrm{eC}\;\epsilon _\mathrm{ev}\right] -P. \end{array} \end{aligned}$$

(82)

Then

$$\begin{aligned} p_{\mathrm{t}_\mathrm{est}}=\sum _{z=1}^{3}\left( p_{zt}\exp \left( -\mathrm{eC}\, \epsilon _{\mathrm{ev}_{z}}\right) -P_{z}\right) . \end{aligned}$$

(83)

Results of the Norm Error and the Quantities of Material Parameters

The results of the $L_2$ norm error and the quantities of material parameters of nonlinear elasticity using different initial guesses, such as: $\left( 0,0,0\right)$, $\left( 0.5,0.5,0.5\right)$, $\left( 1.0,1.0,1.0\right)$, and $\left( 14.856,-0.566,6.157\right)$ are given in Tables 13, 14, 15, and 16.

Table 13 The results of the method when the initial guess is $\left( 0,0,0\right)$

Table 14 The results of the method when the initial guess is $\left( 0.5,0.5,0.5\right)$

Table 15 The results of the method when the initial guess is $\left( 1.0,1.0,1.0\right)$

Table 16 The results of the method using the proposed initial guess $\left( 14.856,-0.566,6.157\right)$