## 5 Optimisation of Material Parameters
### Objective Functions  
An objective function describes the difference between the experimental data and the simulation data. Tensile tests and creep tests are described by using different datasets, so we use two different partial objective functions. When solving all experiments (tensile tests and creep tests), we use the so-called total objective function.  
Figure 14: Displacement boundary condition for (**a**) tensile test at rate 1 mm/min and (**b**) tensile tests at rate 5 and 10 mm/min (SACX0807 solder alloy).  
Figure 13: Initial parameters identification of Chaboche model directly in Ansys Workbench software: (**a**) experimental data as input; (**b**) resulting parameters as output.  
The tensile tests are typically described by the stress-strain curve, where strain values are the same for the experiment and simulation. Therefore, the objective function is based on the difference in stress, as follows:  
\[ERR^{T}=\frac{1}{N}{\sum_{i=1}^{N}}\frac{\sigma^{EXP}_{(i)}-\sigma^{FEM}_{(i)} }{\sigma^{EXP}_{(i)}}\, \tag{7}\]  
where \(ERR^{T}\) is a value of the objective function for a tensile test, \(\sigma^{EXP}_{(i)}\) are values of stress from the experiment, \(\sigma^{FEM}_{(i)}\) are values of stress from the simulation, \(i\) is the current point, and \(N\) is the number of points. Due to the method of measurement, the normal stress in the direction of specimen's axis was used in all cases (\(y\)-direction), which is here denoted in short as \(\sigma\).  
Creep test results are in the form of an axial strain-time curve, and the loading force is the same for the experiments and simulations. In the creep test, an hourglass specimen is used, which is evaluated at several points, see Figure 2b. Therefore, it is more appropriate to use it to describe the strain value at the selected points. Each point represents the complete data set for the creep simulation. The objective function is based on the difference of strains, as follows:  
\[ERR^{C}=\frac{1}{N}{\sum_{i=1}^{N}}\frac{\varepsilon^{EXP}_{(i)}-\varepsilon^ {FEM}_{(i)}}{\varepsilon^{EXP}_{(i)}}\, \tag{8}\]  
where \(ERR^{C}\) is a value of objective function for a creep test, \(\varepsilon^{EXP}_{(i)}\) are values of strain from the experiment, \(\varepsilon^{FEM}_{(i)}\) are values of strain form the simulation, \(i\) is the current point, and \(N\) is the number of points. Due to the measurement method, the equivalent strain values (\(\varepsilon_{HMH}\)) were used in all cases, here denoted in short as \(\varepsilon\).  
In Equation (8), the difference between the experimental and simulation strain values is related to the experimental data, and the resulting values are dimensionless. Therefore, the values of the partial objective functions can be added for all experiments. We assume that both types of experiments have the same effect on the result. The different number of tensile and creep tests is corrected using weights, as follows:  
\[ERR^{A}=\frac{1}{N^{T}+N^{C}}\sqrt{{\sum_{i=1}^{N^{T}}}\left(ERR^{T}_{(i)} \right)^{2}+w^{C}{\sum_{j=1}^{N^{C}}}\left(ERR^{C}_{(j)}\right)^{2}}\, \tag{9}\]  
where \(ERR^{A}\) is a value of the total objective function, \(N^{T}\) is the number of tensile tests, \(N^{C}\) is the number of creep tests, and \(w^{C}=\frac{N^{T}}{N^{C}}\) is the creep weight.