^{1}

^{2}

^{2}

The purpose of the current work is the development and application of a new identification method of material parameters of elastoplastic damage constitutive model under large strains. A relationship relating the intrinsic and extrinsic parameters of a reference material is built and transformed in equivalence relation. Extrinsic parameters concern the shape of their experimental tensile force/elongation curve, however, intrinsic parameters deal with Swift hardening law coupled with an isotropic damage variable. The relationship is carried out from a statistical characterization of a material reference (standard-steel E24). It based on multiple linear regression of a data set obtained according to a full factor design of numerical simulations of mechanical tensile tests. All materials satisfying this equivalence relation belong to the same equivalence class. This is motivated by observing that gathered materials must behave somewhat like the reference material. The material parameters can be immediately identified by only one task by running the found relationship. The current method facilitates the identification procedure and offers a substantial savings in CPU time. However it just needs only one simulation for the identification of similar behavior instead of the few hundred required when using other methods.

The optimization of the forming process by plastic deformation leads to cost savings in manufacturing and improvement of the reliability of the formed parts. Numerical simulation of the mechanical behavior of thin shells is used as tool to predict and evaluate the risks and failures that maybe encountered in the forming process [

The material constitutive law must be able to capture anisotropy, strain hardening, damage evolution and forming limits. Continuum models for hardening plasticity coupled with damage help designers to evaluate manufacturability in the early design stage for mechanical parts production. Most of these models are based on a macroscopic consideration whose formulation is defined within the framework of irreversible thermodynamic processes [

Two modeling approaches are currently available in literature for the damage assessment. The first approach is based on the void growth rate inside a phase with elastoplastic behavior [

The development of different procedures for the characterization of constitutive laws of materials through the use of the finite element method [

In some cases a combination of methods were used to reach an optimal solution. For the most of these methods, the material parameters were obtained iteratively by minimizing an objective error function involving the constitutive law. The objective function is an expression of the deviation between experimental results and their counterparts obtained through the finite element method [

Parameter identification methods available in the literature are based on complex algorithms and time consuming calculation. In most optimization procedures, the uniqueness of the optimal solution is not guaranteed and this may be reflected in the dependence of the optimal solution on the initial guess for the parameters. Furthermore, additional constraints must be added in order to take into account certain physical phenomena that are mathematically impossible to uncouple. Given the complexity of these methods and the size of the experimental data, appropriate computing strategies are necessary.

To the authors’ best knowledge there is no method in the literature which would be able to identify material parameters easily and quickly. The aim of this paper is to pre- sent a new procedure of identifying material model parameters. It deals with establishing a linear relationship between material parameters and shape indexes extracted from the experimental force/elongation (tensile test) curves. This study is limited to the Swift hardening law coupled with an isotropic damage variable.

The idea is to gather materials having an equivalent behavior in the same class according to the shape of their experimental tensile force/elongation response. This is motivated by observing that gathered materials must behave somewhat like a reference material. An equivalence relation is built from a statistical characterization of a material reference and then extends to other materials belonging to this class. The equivalence relation relates the material parameters and shape indexes of experimental tensile force/ elongation curve.

In a previous study [

1) Hill’s equivalent stress s_{eq} de Hill48 as a function of the anisotropy parameters (F, G, H, N) of the in-plane stresses

2) The damage D as a function of the cumulative equivalent plastic strain e_{eq}:

where: D is the damage variable specified as 0 £ D £ 1; D = 0 (initial undamaged state), D = 1 (final fractured state); e_{u} is the total cumulated plastic strain up to failure; e_{s} is a threshold strain level under which no damage is incurred; g is an exponent that indicates the extent of damage in the material.

3) Swift’s isotropic hardening law coupled with damage expresses the equivalent stress as a function of the equivalent strain through the Equation (3).

In order to characterize the material law three sets of unknowns must be determined:

1) Four anisotropy parameters F, G, H and N in terms of the Lankford coefficients (r_{0}, r_{45} and r_{90}) are evaluated experimentally [

2) Three parameters from Swift’s hardening law (K, e_{0}, n) where the equivalent yield stress is given in terms of the cumulated equivalent plastic strain as:

3) Three parameters to identify the isotropic ductile damage law g, e_{s} and e_{u}.

In this study an original method to identify the last two sets of parameters for hardening and damage is presented.

A procedure of identification by multiple regression is executed in three steps. In the first step data are collected according to a full factorial design. The independent variables consist in virtual material parameters of the constitutive law. Ultimately these material parameters are the target of the identification procedure. For each set of the virtual material parameters, the constitutive law is defined and then implemented in ABAQUS/Standard to simulate a characterization test. The dependent variables are represented by a set of shape indexes extracted from the simulated characterization tests.

In the second step, it comes to establish a relationship by multiple linear regression between material parameters and shape indexes. In fact, multiple regression provides a means to express a dependent variable (z) in terms of f independent variables (

where _{z}.

In the present study we are interested in the transposed problem stated as follows; given p dependent variables (^{th} variable x_{i} multiple linear regression leads to a linear equation of the form

This generates a system of f linear equations that can be written in a standard matrix form:

The criteria for the minimization of the residual vector

In the third step the material parameters are identified by substituting in the established relation (Equation (8)) the shape indexes extracted from the experimental tensile curves of a reference material.

The accuracy of the identified model is estimated by minimizing the objective functions [

where, F_{iexp}_{ }and F_{inum} are respectively the experimental and calculated tensile force. U_{iexp} and U_{inum} are respectively, the experimental and the calculated elongations and m is the total number of experimental points.

Once the identification of the material parameters is successful for the reference material, the relation (12) of the regression coefficent matrix derived from relaion (8) can be written as:

where the couple

where

It is easy to prove that the equality relation R satisfies: the reflexive, symmetric, and transitive properties. The set

So the reference material characterizes by

The shape indexes vector

The points A(d_{A}, F_{A}) and C(d_{C}, F_{C}) correspond, respectively, to the maximum force, the maximum elongation. E(d_{E}, F_{E}) corresponds to the minimum force of the hardening area on the tensile force/elongation curve. After selecting D_{1}, D_{2} and D_{3} are tangent lines to the tensile force/elongation curve that pass through the selected points A, C and E. These three tangent lines are plotted on excel to show the bell shaped tensile curve. B(d_{B}, F_{B}) and D(d_{D}, F_{D}) are the intersection points of the tangent lines D_{1} with D_{2} and D_{1} with D_{3}, respectively. The indexes p_{1}(D_{1}), p_{2}(D_{2}) and p_{3}(D_{3}) are the respective slopes of the tangent lines D_{1}, D_{2} and D_{3}.

The standard-steel specified as E24 in accordance with the NF A 35-573/4 (France Standard Structural steel) (United States Equivalent Grades: A283C and European Union Equivalent Grades: S235) is chosen as reference material. It is a structural grade steel with a minimum yield strength which widely used in the engineering and construction industries. With minimum yield strength, E24 structural steel is a common carbon structural steel that can be used in a very broad range of fabrication processes and its plate has excellent formability. It is often favored by the engineer trying to maximize strength or structure while minimizing its weight.

The procedure used to identify the material parameters of the anisotropic elastoplastic behavior coupled to ductile damage of the reference material E24 is described in a previous works [

The tensile tests were carried out at a strain rate of 10^{−3} s^{−1} at room temperature of 25˚C. The tensile tests of the 0˚ oriented specimens were taken as reference. _{0}, r_{45} and r_{90}) [

A three-dimensional finite element analysis (FEA) has been performed using the finite

element code ABAQUS/Standard to investigate the tensile test. The imposed boundary conditions on the 0˚ oriented specimens (NF A03-151) include a fixed end on the side of the stationary grip and a uniform displacement on the side of the moving grip. The finite element type “C3D8R” used, are eight-node three-dimensional continuum elements with reduced linear integration; the mesh size is 2391 elements. For a given constitutive law, the numerical model returns the computed response as a tensile force/ elongation curve.

To build the database for analysis the three steps above are applied on two identification phases: hardening phase and damage phase.

The collected data for hardening phase is made up from a set of simulations based on full factorial design. The independents variables consist in a virtual hardening material parameters (K, ε_{0}, n) of the uncoupled constitutive law (Swift model D = 0; Equation (4)). _{3}(3^{3}) (where 3^{3} = (levels number)^{factors number}). These levels were chosen within the variability limits of the hardening coefficients observed for steels.

Each combination of the full factorial design is used to implement the uncoupled constitutive law in ABAQUS/Standard to perform simulation of the tensile test. The dependent variables are the shape indexes of the computed tensile force/elongation curve of the simulation test. Eight shape indexes are retained for the identification of the hardening parameters; they are the coordinates of the points A, D and E and the slopes of the tangent lines D_{1} and D_{3} (

Making use of the shape indexes extracted from the experimental tensile curve, the relationships are used to get the results of the parameter identification of the uncoupled model of the reference material.

Mechanical properties | Lankford coefficients | Anisotropy parameters | |||||||
---|---|---|---|---|---|---|---|---|---|

Modulus of elasticity (GPa) | Poisson’s ratio | Yield stress (GPa) | r_{0} | r_{45} | r_{90} | F | G | H | N |

210 | 0.33 | 195 | 1.985 | 1.234 | 1.637 | 0.665 | 0.335 | 0.406 | 1.285 |

Influencing factors | Label | Levels | ||
---|---|---|---|---|

1 | 2 | 3 | ||

Hardening modulus (MPa) | K | 550 | 700 | 850 |

Initial strain (%) | ε_{0} | 0.575 | 1.05 | 1.525 |

Hardening coefficient | n | 0.2 | 0.25 | 0.3 |

The coefficients (K, ε_{0}, n) identified above are treated as constants and used to address the damage phase of identification. The collected data is made up from a new full factorial design P_{3}(3^{3}) where the simulations are conducted using the coupled constitutive law (D ≠ 0; Equation (3)). The independents variables consist in a virtual damage material parameters (γ, ε_{s}, ε_{u}). _{3}(3^{3}); they are chosen in the range available for steels.

Eight other shape indexes are retained for the identification of the damage parameters; they are the coordinates of the points A, B and C and the slopes of the tangent lines D_{1} and D_{2}. A relationship is built by multiple linear regression for each damage material parameter in function of the new selected shape indexes.

Finally, the shape indexes extracted from the experimental tensile curve are introduced to the relationships to get the results of the parameter identification of the uncoupled model of the reference material.

Influencing factors | Label | Levels | ||
---|---|---|---|---|

1 | 2 | 3 | ||

Damage threshold (%) | 0.5 | 0.625 | 0.75 | |

Total cumulated plastic strain (%) | 50 | 60 | 70 | |

Damage index | 5 | 7.25 | 9.5 |

The procedure involves running ABAQUS/Standard from a python script with input from an Excel file which holds the full factorial design. The Excel file can be easily parameterized and analyzed within the Matlab software.

Once results have been validated through both Equations (10) and (11), all materials verifying the equivalence relation (12) belong to the equivalence class

Twenty seven simulations of tensile test according to the full factorial design P_{3}(3^{3}) are run. The different combinations of the virtual hardening parameters are defined in accordance with levels in

Virtual hardening parameters | Shape indexes of simulation tensile force/elongation curves (Hardening case) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

P_{3}(3^{3}) | K (MPa) | ε_{0}_{ }(%) | n | d_{E} (mm) | F_{E} (N) | d_{D} (mm) | F_{D} (N) | d_{A} (mm) | F_{A} (N) | p_{3} (N/mm) | p_{1} (N/mm) |

1 | 1 | 1 | 1 | 0.25 | 1486 | 2.86 | 2475 | 15.37 | 2459 | 379 | −1.25 |

2 | 2 | 1 | 1 | 0.25 | 1884 | 2.93 | 3152 | 15.47 | 3129 | 472 | −1.78 |

3 | 3 | 1 | 1 | 0.25 | 2271 | 2.92 | 3827 | 15.44 | 3798 | 583 | −2.27 |

4 | 1 | 2 | 1 | 0.25 | 1670 | 3.09 | 2490 | 15.34 | 2470 | 289 | −1.62 |

5 | 2 | 2 | 1 | 0.25 | 2118 | 3.21 | 3181 | 15.32 | 3144 | 359 | −3.02 |

6 | 3 | 2 | 1 | 0.25 | 2561 | 3.24 | 3863 | 15.35 | 3817 | 436 | −3.76 |

7 | 1 | 3 | 1 | 0.25 | 1797 | 3.36 | 2512 | 14.76 | 2483 | 229 | −2.49 |

8 | 2 | 3 | 1 | 0.25 | 2278 | 3.38 | 3197 | 14.65 | 3159 | 293 | −3.33 |

9 | 3 | 3 | 1 | 0.25 | 2757 | 3.43 | 3882 | 14.86 | 3835 | 354 | −4.08 |

10 | 1 | 1 | 2 | 0.25 | 1163 | 3.06 | 2201 | 17.32 | 2280 | 370 | 5.55 |

11 | 2 | 1 | 2 | 0.25 | 1468 | 3.00 | 2786 | 17.41 | 2900 | 479 | 7.97 |

12 | 3 | 1 | 2 | 0.25 | 1779 | 3.05 | 3355 | 17.38 | 3522 | 562 | 11.62 |

13 | 1 | 2 | 2 | 0.25 | 1342 | 3.31 | 2218 | 17.28 | 2292 | 286 | 5.31 |

14 | 2 | 2 | 2 | 0.25 | 1698 | 3.34 | 2839 | 17.22 | 2916 | 369 | 5.57 |

15 | 3 | 2 | 2 | 0.25 | 2056 | 3.33 | 3413 | 17.31 | 3540 | 440 | 9.12 |

16 | 1 | 3 | 2 | 0.25 | 1465 | 3.45 | 2240 | 17.28 | 2304 | 242 | 4.64 |

17 | 2 | 3 | 2 | 0.25 | 1860 | 3.56 | 2855 | 17.16 | 2931 | 301 | 5.54 |

18 | 3 | 3 | 2 | 0.25 | 2250 | 3.57 | 3461 | 17.18 | 3557 | 365 | 7.09 |

19 | 1 | 1 | 3 | 0.25 | 909 | 3.16 | 1948 | 17.92 | 2109 | 357 | 10.92 |

20 | 2 | 1 | 3 | 0.25 | 1155 | 3.19 | 2451 | 17.85 | 2684 | 440 | 15.89 |

21 | 3 | 1 | 3 | 0.25 | 1390 | 3.21 | 3006 | 17.85 | 3256 | 546 | 17.08 |

22 | 1 | 2 | 3 | 0.25 | 1079 | 3.49 | 1977 | 17.84 | 2126 | 278 | 10.33 |

23 | 2 | 2 | 3 | 0.25 | 1367 | 3.48 | 2507 | 17.85 | 2704 | 352 | 13.74 |

24 | 3 | 2 | 3 | 0.25 | 1653 | 3.48 | 3030 | 17.85 | 3282 | 427 | 17.51 |

25 | 1 | 3 | 3 | 0.25 | 1197 | 3.62 | 1999 | 17.91 | 2139 | 238 | 9.81 |

26 | 2 | 3 | 3 | 0.25 | 1520 | 3.70 | 2546 | 17.84 | 2723 | 298 | 12.49 |

27 | 3 | 3 | 3 | 0.25 | 1839 | 3.69 | 3079 | 17.84 | 3304 | 360 | 15.89 |

As a predictive analysis, the Matlab stepwise linear regression is used to select the most important shape indexes that contribute to the hardening parameter variations. This stepwise method keeps the number of potential variables to a minimum. The coefficients (a_{1i}) of the regression Equation (16) are expressed in

where x_{h} expresses the hardening parameter (K, e_{0}, n); and (d_{E}, F_{E}, d_{D}, F_{D}, d_{A}, F_{A}, p_{3}, p_{1}) are the hardening shape indexes of the tensile force/elongation graph.

The results show that two shape indexes (_{0} and n, respectively are explained. These high values of R^{2} indicate that the models of hardening material parameters have a good fit. K, e_{0} and n can therefore be identified by introducing the shape indexes obtained from the experimental tensile force/elongation graph (

The results of the parameter identification are presented in

Coefficient | K | e_{0} | n |
---|---|---|---|

a_{1}_{0} | −18.247 | −0.044836 | 0.087771 |

a_{11} | 0 | 0 | 0 |

a_{12} | −0.25195 | 0 | 0 |

a_{13} | 27.393 | 0.017289 | 0 |

a_{14} | 0 | 0 | 0 |

a_{15} | −3.4644 | 0 | 0.0098491 |

a_{16} | 0.43379 | 0 | 0 |

a_{17} | −0.40557 | 0 | −7.47E−05 |

a_{18} | 0 | −0.00028498 | 0.0041557 |

R^{2} | 99.9% | 94.4% | 95.3% |

R^{2}adj | 99.9% | 93.9% | 94.7% |

RMSE | 2.98 | 0.000973 | 0.00962 |

E | D | A | D_{3} | D_{1} | |||
---|---|---|---|---|---|---|---|

d_{E}(mm) | F_{E}(N) | d_{D}(mm) | F_{D}(N) | d_{A}(mm) | F_{A}(N) | p_{3}(N/mm) | p_{1}(N/mm) |

0.2489 | 1525.8 | 3.6949 | 2172.1 | 16.3284 | 2193.8 | 187.5335 | 1.7213 |

Full factorial design | Parameters | Average error | ||
---|---|---|---|---|

K (MPa) | e_{0}_{ }(%) | n | x_{F }(%) | |

P_{3}(3^{3}) | 517.56 | 1.86 | 0.242 | 0.20 |

(25˚C and 10^{−3} s^{−1}) is presented. The average relative error x_{F} of 0.2% of the hardening portion of the graph (between 0.05 mm and 17 mm) was observed indicating good agreement between the two graphs.

The hardening material parameters (K = 517.56, ε_{0} = 1.86, n = 0.242) identified above (_{3}(3^{3}) are run. The different combinations of the virtual damage parameters (γ, ε_{s}, ε_{u}) are defined in accordance with levels in

The coefficients (a_{2i}) of the multiple regression Equation (18) are expressed in

where x_{d} is a damage parameter (e_{s}, e_{u} and g); d_{A}, F_{A}, d_{B}, F_{B}, d_{C}, F_{C}, p_{1} and p_{2} are the damage shape indexes of the tensile force/elongation graph.

Virtual damage parameters | Shape indexes of simulation tensile force/elongation curves (Damage case) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

P_{3}(3^{3}) | e_{s} (%) | e_{u} (%) | g | d_{C} (mm) | F_{C} (N) | d_{B} (mm) | F_{B} (N) | d_{A} (mm) | F_{A} (N) | p_{1}(∆_{1}) (N/mm) | p_{2}(∆_{2}) (N/mm) |

1 | 1 | 1 | 1 | 17.71 | 1950 | 17.51 | 2180 | 14.19 | 2179 | 0.37 | −1136 |

2 | 2 | 1 | 1 | 17.66 | 1512 | 17.59 | 2176 | 14.19 | 2178 | −0.64 | −10634 |

3 | 3 | 1 | 1 | 17.55 | 1998 | 17.28 | 2177 | 14.19 | 2178 | −0.17 | −669 |

4 | 1 | 2 | 1 | 20.79 | 1633 | 20.70 | 2187 | 15.51 | 2191 | −0.71 | −6095 |

5 | 2 | 2 | 1 | 20.74 | 1873 | 20.49 | 2191 | 15.46 | 2191 | 0.19 | −1318 |

6 | 3 | 2 | 1 | 20.63 | 1891 | 20.39 | 2191 | 15.46 | 2190 | 0.18 | −1300 |

7 | 1 | 3 | 1 | 24.04 | 1857 | 23.45 | 2195 | 16.34 | 2197 | −0.27 | −582 |

8 | 2 | 3 | 1 | 24.04 | 1821 | 23.57 | 2203 | 15.90 | 2196 | 0.91 | −818 |

9 | 3 | 3 | 1 | 23.87 | 1868 | 23.23 | 2203 | 15.90 | 2196 | 0.94 | −528 |

10 | 1 | 1 | 2 | 20.41 | 2000 | 20.16 | 2195 | 16.34 | 2198 | −0.69 | −799 |

11 | 2 | 1 | 2 | 20.35 | 2009 | 20.11 | 2200 | 16.06 | 2198 | 0.44 | −793 |

12 | 3 | 1 | 2 | 20.30 | 2009 | 20.06 | 2198 | 16.06 | 2198 | 0.22 | −813 |

13 | 1 | 2 | 2 | 23.65 | 1989 | 23.48 | 2195 | 17.44 | 2201 | −1.06 | −1228 |

14 | 2 | 2 | 2 | 23.65 | 1964 | 23.46 | 2195 | 17.44 | 2201 | −1.10 | −1205 |

15 | 3 | 2 | 2 | 23.65 | 1932 | 23.46 | 2194 | 17.44 | 2201 | −1.16 | −1387 |

16 | 1 | 3 | 2 | 27.34 | 1782 | 26.74 | 2201 | 17.44 | 2203 | −0.23 | −705 |

17 | 2 | 3 | 2 | 27.28 | 1754 | 26.79 | 2201 | 17.44 | 2203 | −0.19 | −905 |

18 | 3 | 3 | 2 | 27.23 | 1774 | 26.70 | 2200 | 17.44 | 2203 | −0.30 | −809 |

19 | 1 | 1 | 3 | 22.00 | 1999 | 21.78 | 2198 | 17.44 | 2202 | −0.87 | −910 |

20 | 2 | 1 | 3 | 21.89 | 2012 | 21.67 | 2198 | 17.44 | 2202 | −0.90 | −846 |

21 | 3 | 1 | 3 | 21.84 | 1997 | 21.62 | 2199 | 17.44 | 2202 | −0.81 | −953 |

22 | 1 | 2 | 3 | 25.52 | 1872 | 25.21 | 2204 | 17.44 | 2203 | 0.05 | −1054 |

23 | 2 | 2 | 3 | 25.47 | 1828 | 25.22 | 2204 | 17.44 | 2203 | 0.04 | −1515 |

24 | 3 | 2 | 3 | 25.41 | 1867 | 25.14 | 2205 | 17.44 | 2203 | 0.20 | −1251 |

25 | 1 | 3 | 3 | 29.15 | 1756 | 28.58 | 2204 | 17.44 | 2204 | 0.01 | −783 |

26 | 2 | 3 | 3 | 29.10 | 1727 | 28.68 | 2205 | 17.44 | 2204 | 0.12 | −1145 |

27 | 3 | 3 | 3 | 28.99 | 1756 | 28.48 | 2204 | 17.44 | 2204 | 0.07 | −885 |

The results of the analysis presented in _{s }does not depend of the shape indexes. In fact, it may be considered as a parameter intrinsic to the material and related to the initial damage in the sheet metal. However, none of the five shape indexes (F_{B}, d_{A}, F_{A}, p_{1}, p_{2}) has no effect on damage parameters. The coefficients of determination show that the linear regression relations obtained explain 84.6% of the variability in the cumulated total plastic deformation e_{u}(%) and 67% of the variability in the damage index g.

The identification of the damage material parameters e_{u} and g are now achieved by introducing the damage shape indexes, of the experimental tensile force/elongation curve presented in

The identification results of the damage material parameters are presented in _{U} between the elongations of the two graphs is on the order of 0.34% confirming the accuracy of the predicted response of the identified model.

To recap, the coupled Swift model of standard-steel E24 depicted by the anisotropic elastoplastic behavior coupled to ductile damage of the reference material E24 is described by the constitutive model (Equation (20)). The material parameters are identified above (

Coefficient | e_{s} | e_{u} | g |
---|---|---|---|

a_{20} | 0.00625 | 0.62376 | −15.294 |

a_{2}_{1} | 0 | 0.24571 | −6.8441 |

a_{2}_{2} | 0 | −0.00017488 | 0.0055914 |

a_{2}_{3} | 0 | −0.23597 | 7.4649 |

a_{2}_{4} | 0 | 0 | 0 |

a_{2}_{5} | 0 | 0 | 0 |

a_{2}_{6} | 0 | 0 | 0 |

a_{2}_{7} | 0 | 0 | 0 |

a_{2}_{8} | 0 | 0 | 0 |

R^{2} | 84.6% | 67% | |

R^{2}adj | 82.6% | 62.7% | |

RMSE | 0.104 | 0.0347 | 1.14 |

C | B | A | D_{1} | D_{2} | |||
---|---|---|---|---|---|---|---|

δ_{C}(mm) | F_{C}(N) | δ_{B}(mm) | F_{B}(N) | δ_{A}(mm) | F_{A}(N) | p_{1}(N/mm) | p_{2}(N/mm) |

25.3856 | 1.59E+03 | 25.1719 | 2.17E+03 | 18.0415 | 2.19E+03 | −3.2085 | −2.70E+03 |

Full factorial design | Parameters | Average error | ||
---|---|---|---|---|

e_{s} (%) | e_{u} (%) | g | x_{U} (%) | |

P_{3}(3^{3}) | 0.625 | 64.29 | 7.78 | 0.34 |

In

The material parameters of the reference material are now identified this means that the equivalence class {S} is now defined. For any other material whose the couple of parameters (

Based on Equation (17) and Equation (19) the matrix [A], where all values are extracted from

The validation of this technique has been extended is extended to the identification of the coefficients of the coupled Swift model of:

1) The behavior of standard-steel E24 under other conditions of temperature and strain rate

2) The behavior in tension of a material with characteristics different from standard-steel E24 such as 1050A aluminum.

The first implementation of this method is concerned with a tensile test of standard- steel E24 at a temperature of 25˚C and a strain rate of 1.66 × 10^{−3} s^{−1}.

The experimental shape indexes from the tensile force/elongation graph are presented in

Hardening | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

E | D | A | D_{3} | D_{1} | ||||||

d_{E}(mm) | F_{E}(N) | d_{D}(mm) | F_{D}(N) | d_{A}(mm) | F_{A}(N) | p_{3}(N/mm) | p_{1}(N/mm) | |||

1.33 | 1598.60 | 4.32 | 2480.02 | 19.33 | 2496.56 | 295.06 | 1.10 | |||

Damage | ||||||||||

C | B | A | D_{1} | D_{2} | ||||||

d_{C}(mm) | F_{C}(N) | d_{B}(mm) | F_{B}(N) | d_{A}(mm) | F_{A}(N) | p_{1}(N/mm) | p_{2}(N/mm) | |||

22.71 | 1957.0 | 22.5238 | 2503.70 | 19.33 | 2496.60 | 1.3427 | −2981.70 | |||

Hardening | Damage | ||||||
---|---|---|---|---|---|---|---|

Parameters | Average error | Parameters | Average error | ||||

K (MPa) | ε_{0}(%) | n | x_{F}(%) | ε_{s}(%) | ε_{u}(%) | γ | x_{U}(%) |

593.63 | 2.951 | 0.261 | 0.33 | 0.625 | 54.5 | 8.375 | 0.34 |

_{F} between the two responses is about 0.33% for elongations between 5.2 mm and 17.0 mm. The average relative error x_{U} for elongations between 17.0 mm and 23.0 mm is about 0.34%. For elongations below 5.2 mm a difference exists which seems to be affected by the experimental shape indexes due to uncertainty in the coordinates of point A as well as the slope p_{1} of the tangent line D_{1}.

The second application of the method is concerned with a tensile test of standard-steel E24 at 200˚C and a strain rate of 1.66 × 10^{−3} s^{−1}. ^{−3} s^{−1}); this makes it possible to identify the hardening and damage parameters directly for the tensile test of standard-steel E24 at 200˚C. The experimental shape indexes are presented in

The results of the identification of hardening and damage parameters by multiple regression at a temperature of 200˚C are presented in _{F} between the two responses is 0.22% for elongations below 14.6 mm whereas for elongations in the range 14.4 mm to 19.1 mm the average relative error x_{U} is about 0.10%.

Hardening | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

E | D | A | D_{3} | D_{1} | |||||||

d_{E}(mm) | F_{E}(N) | d_{D}(mm) | F_{D}(N) | d_{A}(mm) | F_{A}(N) | p_{3}(N/mm) | p_{1}(N/mm) | ||||

0.41 | 1216.07 | 3.18 | 2197.54 | 16.75 | 2198.59 | 355.07 | 0.08 | ||||

Damage | |||||||||||

C | B | A | D_{1} | D_{2} | |||||||

d_{C}(mm) | F_{C}(N) | d_{B}(mm) | F_{B}(N) | d_{A}(mm) | F_{A}(N) | p_{1}(N/mm) | p_{2}(N/mm) | ||||

18.71 | 2049.6 | 18.5208 | 2205.80 | 15.5817 | 2198.60 | 2.4421 | −839.56 | ||||

Hardening | Damage | ||||||
---|---|---|---|---|---|---|---|

parameters | Average error | parameters | Average error | ||||

K(MPa) | ε_{0}(%) | n | x_{F}(%) | ε_{s}(%) | ε_{u}(%) | γ | x_{U}(%) |

514.137 | 1.01 | 0.2265 | 0.22 | 0.625 | 49.1 | 6.390 | 0.10 |

The third application is concerned with annealed 1050A aluminum. This alloy has good plastic deformation properties.

_{F} = 0.38% in the elongation range between 0.05 mm and 23.7 mm and x_{U} = 0.25% for elongation in the range 23.7 mm to 33.1 mm.

Hardening | |||||||
---|---|---|---|---|---|---|---|

E | D | A | D_{3} | D_{1} | |||

d_{E}(mm) | F_{E}(N) | d_{D}(mm) | F_{D}(N) | d_{A}(mm) | F_{A}(N) | p_{3}(N/mm) | p_{1}(N/mm) |

0.3 | 324.83 | 2.629 | 748.10 | 22.50 | 764.29 | 181.76 | 0.815 |

Damage | |||||||

C | B | A | D_{1} | D_{2} | |||

d_{C}(mm) | F_{C}(N) | d_{B}(mm) | F_{B}(N) | d_{A}(mm) | F_{A}(N) | p_{1}(N/mm) | p_{2}(N/mm) |

33.5313 | 375.3 | 33.3895 | 772.687 | 22.4987 | 764.29 | 0.771 | −2.80E+03 |

The presented work demonstrates the possibilities available through the multiple regression method to the parameter identification problem. The procedure was proven to be efficient in determining the parameters of the Swift hardening model coupled with an isotropic damage variable.

Once the relationship between the material parameters and the shape indexes of the reference material is carried out, it will be used to identify directly the material behavior for a class of materials having the same attributes.

However the extended use of these results to other materials under different test conditions requires their behavior similarity. Hence, the materials may be grouped into equivalence classes according to the shape of their experimental response.

This method provides significant gains in computer time necessary for a complete identification procedure for the reference material. It needs very few simulations compared to other identification methods. However it just needs only one simulation for the identification of similar behavior.

Future efforts will be directed to:

1) Introduction on nondimensional variable in the analysis which will broaden the utility range of the correlations.

Hardening | Damage | ||||||
---|---|---|---|---|---|---|---|

Parameters | Average error | Parameters | Average error | ||||

K(Mpa) | ε_{0}(%) | n | x_{F}(%) | ε_{s}(%) | ε_{u}(%) | g | x_{U}(%) |

151.80 | 0.038 | 0.299 | 0.38 | 0.625 | 91.82 | 6.56 | 0.25 |

2) Generalization to constitutive models which have a large number of parameters.

3) Sensitivity analysis of the parameter to identify with respect to the shape indexes.

Rezgui, M.-A., Nasri, M.-T. and Ayadi, M. (2016) Predictive Elastoplastic Damage Constitutive Law: Establishment of Equivalence Relation between Intrinsic and Extrinsic Material Parameters. Materials Sciences and Applications, 7, 730-753. http://dx.doi.org/10.4236/msa.2016.711058