Technical Description
Literature revision
Technical Description
Literature revision
In order to decrease associated delays and costs, mechanical design of parts/processes tends to be nowadays more virtual by using numerical simulation. The reduction of the development lead-time and the provision of robust solutions with highly improved quality is one of the major objectives of the automotive or aircraft industries (see Quote Q1, from Airbus), and the numerical solutions are opportunities for these goals. However, the quality of the simulation results directly depends on the quality of the input data, particularly on the choice of a well suited constitutive law with appropriate identification of its parameters for the studied material (see Quote Q2).
Therefore, the characterization of materials has received increasing attention due to the need of precise input data to computational analysis software described through constitutive equations and material parameters. A simulation software (e.g. FEM/FEA code), uses complex material constitutive models and its success reproducing the real behaviour is much dependent on the capability of these models to reproduce the material behaviour and on the material parameters.
In general, the parameters of those phenomenological models are determined by standard tests [1]. However, the homogeneous stress-strain fields generated in these relatively simple tests do not resemble the complex stress and strain fields which occur in many metal forming operations [2], particularly in warm forming. Since plastic deformation is strain path dependent, the validity of phenomenological models is limited to situations that are comparable to the range of experiments on which these are based. Therefore, the material behaviour obtained from standard tests and described by phenomenological models, is merely an approximation that in many cases doesn’t allow the reliable simulation of complex forming processes.
Therefore, the material characterization through complex/sophisticated constitutive models required for the use of simulation software present two major challenges/difficulties (see Quote Q3):
(i) the identification of the large number of parameters and
(ii) the quality, amount of information and number of the experiments performed.
A very recent alternative to circumvent the limitations of the standard approach is the use of experiments that lead to nominally heterogeneous strain states, which are measured through full-field experimental techniques such as Digital Image Correlation (DIC), Moiré interferometry, grid method, etc. and later processed with a suitable full-field inverse technique such as the Constitutive Equation Gap Method (CEGM, [34]) and its variants, the Constitutive Compatibility Method (CCM, [4]) and the Dissipative Gap method (DGM, [5]); the Equilibrium Gap Method (EGM, [6]), the Self-Optimizing Method (SOM), the Finite-Element Updating Method (FEMU, [7]) and the Virtual Fields Method (VFM, [8]) and its variants Eigenfunction VFM (EVFM, [9]), Fourier-VFM [10]. An overview of most of these identification techniques is presented in [11]. An appreciation and comparison of these methods was also recently made by the project team [PT1]. In FEMU, a finite element model of the actual test configuration is built up and the material parameters are iteratively tuned by repeatedly performing finite-element analyses until a close correspondence between experimental and numerical field variables is achieved. Although these techniques are quite popular, they incur high computational expense due to the large number of finite-element analyses required and present multiple solutions to the problem [7]. VFM is derived from the principle of virtual work [8], which is a statement of equations of equilibrium in weak form [8,11]. Although the VFM has been receiving increased attention due to the direct nature of material parameter estimation used herein, the problem of solution uniqueness is still not answered [12]. Additionally, it has never been applied in warm forming conditions, where the temperature field is also heterogeneous.
Merely obtaining heterogeneous strain fields from an experiment is not sufficient to ensure accurate computation of all the material parameters; unless the material parameter is strongly activated (i.e. has a strong influence on the measured kinematic fields), it cannot be uniquely ascertained using any inverse scheme. In order to ensure such activation, optimization of the geometry of the specimen [13-16] and loading profiles (including thermal loading) can be performed. This approach directly affects the well-posedness of the inverse problem and is an active area of research. In general, refinement of the experiment is done in order to ensure strain and strain-rate heterogeneity and thus ensure activation of all the material parameters.
In the last two years, some works have been presented to design a specimen in order to particularly improve the parameter identification process. One approach is using directly the identification process itself to verify which specimen geometry leads to accurate results [16,22]. Other approach comprehends the definition of a function/criterion that evaluates the richness of the strain field in the specimen [15,PT2,PT3] leading to a specimen with large heterogeneity and the maximum number of strain states. This approach, that designs the specimen through a shape optimization procedure, was conducted by the project team. Although the results of both approaches improve the identification procedure and the material characterization in a wide range of strain states, there are still limitations of these approaches.
The first limitation is related to the shape optimization procedure and its geometry definition. The use of a specimen boundary defined with NURBS or other curve definition always limit the design solutions. For instance, a solution with a specimen with a hole can be attractive. However, this solution cannot be taken in account unless the initial shape accounts for it.
The second limitation is related to the absence of imaging requirements in these both approaches. Indeed, DIC is a low pass spatial filter (i.e., gradients will be smoothed out by DIC) and its spatial resolution is a key issue, as well at its noise performance.
Other major problem of the large majority of the previous works is the absence of physical constraints when calibrating the constitutive models, leading to solutions that violate physical and phenomenological fundamentals [23,PT5]. The lack of proper physical constraints mostly lead the entirely identification process to non-admissible results.
Therefore, the stage of the definition of physical constraint should be of upmost importance for the development of parameter identification strategies.