شناسایی پارامترهای مواد الاستو ویسکوپلاستیک توسط تست دندانه و مدل سازی عناصر محدود ترکیب شده و بهینه سازی عددی

Using nanoindentation experiments, the material parameters of an elasto-viscoplastic material model with non-linear isotropic and kinematic hardening are determined by minimizing the least-squares difference between experimental data and results from a finite element model of the indentation test. The objective function is minimized by a gradient-based numerical optimization algorithm. The gradient of the objective function with respect to the material parameters is calculated using the direct differentiation method. The parameter coupling occurring in the indentation test is systematically analyzed, and virtual indentation tests are used to assess the reliability of the material parameter identification method. Experimental nanoindentation curves obtained on an aluminium alloy and annealed copper are analyzed. Objective functions consisting of load versus displacement-into-surface curves are used. The efficiency of the use of residual imprint data resulting from nanoindentation testing in the objective function is investigated in order to assess their appropriateness at such small scales.

Indentation testing is widely used for determining the hardness of materials through Brinell, Vickers or Rockwell hardness tests. Some simple considerations have also been used for relating the hardness to the initial yield stress [1]. The major interest of indentation and nanoindentation testing lies in the fact that due to the small sampling volume size, local material properties, or coating properties, may be investigated. With the development of depth-sensing instrumented indentation and nanoindentation testing, where the full history of applied load versus indenter displacement-into-surface is monitored, methods have been put forward for determining material parameters. First of all, the pioneering work of [2] has presented a method for determining Young's modulus from indentation testing. In the late nineties, methods have been put forward for identifying more material parameters. In the work of [3], an inverse method based on neural networks has been presented for determining the elasto-plastic material parameters of a constitutive law including metal plasticity with both non-linear isotropic and kinematic hardening. Using a similar approach [4] also developed a method for determining Poisson's ratio. Later on, the neural network-based method was extended to elasto-viscoplastic material behaviour [5]. This method, which relies on the construction of dimensionless functions relating indentation curve characteristics to dimensionless variables defined from the material parameters – in this case by the use of neural networks – has also been applied to simpler material constitutive laws, like simple power law hardening laws, either via the construction of analytical dimensionless functions [6], [7] and [8] or neural networks [9]. Other methods rely on numerical optimization with analytically approximated indentation curves, either the loading segment of the curves [10] or unloading compliances [11]. In a different approach, an objective function, which quantifies the difference between numerically modelled and experimental indentation curves, is minimized. Hamasak et al. [12] used a multipoint response surface analysis for identifying material parameters of an elasto-viscoplastic constitutive law this way. In [13], a method is presented for determining the material parameters of the Norton–Hoff law for metallic materials using the minimization by gradient-based numerical optimization. Bolzon et al. [14] improved the optimization-based parameter identification method by including residual imprint data into the objective function, i.e. using an aggregate objective function in a multi-objective optimization problem. This method has also been used to identify parameters of anisotropic material behaviour [15] and residual stresses [16]. Concerning the optimization-based methods for minimizing an objective function, it is generally advised to use globally convergent optimization algorithms whenever possible. They can take the form of evolutionary algorithms, like simulated annealing or genetic algorithms, or deterministic algorithms like the Simplex method. Genetic or evolutionary algorithms are globally convergent and are the only meaningful choice in multi-objective optimization, which comes however at the price of a large number of objective function evaluations. Gradient-based optimization algorithms, like the Levenbergh–Marquardt or Gauss–Newton algorithm, should only be used in case the objective function can be shown to be convex. However, in the case of indentation testing, gradient-based optimization algorithms are preferred despite this drawback, because in general, they involve less function evaluations as gradient-free optimization methods. In fact, because of the highly non-linear nature of indentation testing, involving geometric, material and contact non-linearities, finite element computations of the indentation test are time-consuming. This is also the reason why the required gradients are not computed using time-consuming finite difference schemes, which involve an additional one or two non-linear finite element calculations per optimization variable. Their main advantage, the possibility to easily use them with black-box finite element solvers, is countered by the large computing effort involved. In order to be computationally more efficient, use is made of fast calculation algorithms of the gradients, either through direct differentiation method or through the adjoint state method. These methods have been developed starting with [17] and [18], and developed to maturity in the framework of material parameter identification by [19], [20], [21] and [22] or of shape optimization by [23], [24], [25], [26], [27], [28], [29], [30] and [31]. Sensitivity analysis in the case of contact modelling is well described in [32] and [33]. In these methods, either the continuum mechanical equations [23], [24] and [25] or the discretized equations [19] and [27] are differentiated. The derivatives of state variables and displacements with respect to some parameters, like material parameters, for example, are coupled via linear relationships, which can be motivated by the chain rule in differentiation, and the gradient can be calculated by a linear update scheme. This is in contrast to the finite difference method, where the calculation of gradient information requires the solution of additional non-linear finite element models. This way, gradient calculations involving the direct differentiation or adjoint state method are much faster than finite difference schemes. In addition to that, the fast differentiation methods are more accurate, as they do not depend on a numerical perturbation like the finite difference method. However, it should be noted that finite difference and the fast differentiation methods may produce locally poor results in case of non-differentiable evolution equations, as shown in [34], [35] and [36]. A different aspect of material parameter identification using inverse methods is parameter correlation. It has for example been shown in [21] that parameter correlation can impede a reliable parameter identification of mixed isotropic and kinematic plastic hardening laws if no reverse plastic flow is induced in uniaxial testing by applying a compressive load after the tensile cycle. The correlation of the material parameters can be assessed through the correlation or cosine matrices [37], which can be determined from the Hessian or approximated Hessian matrix. A strong correlation leads to near-singular matrices in the optimization algorithms, which also has a strong effect on convergence rates and on the accuracy with which the location of the minimum of the objective function can be determined. In this paper, the constitutive equations typical for metal plasticity are presented in the first section. In the next section, the objective function and the solution procedure for solving the inverse problem of material parameter identification using numerical optimization are presented. The parameter correlation is analyzed and material parameter identification is assessed using synthetic indentation curves, generated by finite element modelling of the nanoindentation test. In the final section, the described inverse method, which has already been used in the framework of depth-sensing microindentation [16] and [38], is used at a smaller length scale for identifying material parameters of two metallic materials by the use of real experimental nanoindentation data. In addition to the load-versus-displacement indentation curves, the use of experimental residual imprint data for improving the reliability of the method, reported in [14] and [16], is used in nanoindentation testing, and the suitability of residual imprint data at such a small length scale is investigated.

The assessment of the parameter identification procedure based on the minimization of an objective function has revealed that for synthetic experimental curves constructed with known material parameters, affected by various levels of noise but without systematic errors, the identification of material parameters is reasonably accurate. However, it was found that the parameters of viscosity exhibit large errors. It was also observed that even in the absence of noise, the true material parameters may not be found. On the one hand, this may be due to the coexistence of several local minima of the objective function. In fact, the convexity of the objective function has not been proved, and for this reason, the possible existence of several local minima has to be kept in mind. On the other hand, the strong parameter correlation, which leads to objective function surfaces with a very low curvature, may lead to a multitude of purely numerical minima, caused by issues related to round-off errors. On the other hand, it cannot be excluded that the objective function may not be continuously differentiable because of the numerical and incremental nature of the finite element modelling. In order to reach the global minimum of the objective function, other optimization algorithms are available, like genetic algorithms or simulated annealing. However, these methods require a large number of objective function evaluations, which disqualifies them for the use in indentation testing because of the large computing times required in the highly non-linear finite element model of the indentation test. Concerning the parameter identification using true experimental data from nanoindentation, it has to be admitted that globally speaking, the quality of the material parameters obtained is rather poor, on the one hand compared to the few reference values available in the case of Dural, on the other hand for the comparison of tensile curves in the case of annealed copper. The only parameter that was identified with a reasonable level of accuracy is Young's modulus. However, this is not satisfactory because the material parameter identification method used does not provide an advantage over the purely experimental methods, like the method put forward in [2]. Concerning the results obtained with the indenter tip radius of View the MathML source, it is clear from Table 6 that the strains occurring in the experiment are too low for allowing a reliable identification of the hardening parameters. A major reason of the poor results is the fact that in indentation testing, the parameter coupling is very strong, which leads to the possibility of distinct sets of parameters leading to nearly identical indentation curves, as has been put forward in [59]. This implies that a significant change in some material parameters may yield only very small changes in the indentation curves, and vice versa, modest changes in indentation curves, caused either by noise or by systematic errors due to numerical issues or the omission of some physical phenomena in the finite element model, may lead to large differences in the material parameters obtained in material parameter identification. This parameter coupling is to a large extent related to the compressive nature of the indentation test, where only very limited reverse plastic flow, necessary for separating isotropic and kinematic hardening [21], takes place. It is worth noticing that the parameters related to viscosity are very high, and the viscous deformation may be overestimated, which may adversely affect the identification of plastic hardening parameters. Another feature of the indentation test, the inhomogeneous stress field, also renders material parameter identification difficult. Unlike uniaxial tensile test, where the local stress and strain may be determined accurately until the occurrence of necking, the indentation test yields load and displacement-into-surface, which are integrals of stress and strain, respectively, over a geometric domain, and as such are global characteristics of the indentation test. The integration acts as a filter, which eliminates information necessary for accurate parameter identification. In nanoindentation, the sources of systematic differences between the numerical model and the experiment are numerous, and these systematic errors will often lead to large errors in the identified material parameters, as explained above. On the one hand, any constitutive law is always an approximation of the true material behaviour, and its reliability diminishes the more the usage conditions differ from the stress state of the experiment used for determining the material parameters. In addition to that, the assumption of a frictionless contact may be an over-simplification. This may be especially true if residual imprint data are used, as they are known to be affected by friction [14] and [60]. Additionally, it has been reported that friction may affect indentation curves much more than previously assumed, especially in case of deep indentation depths [61]. On the other hand, several physical aspects have been neglected in the numerical model. An ideally flat specimen surface was modelled, whereas it is known that surface roughness impacts the hardness measurements [62] and [63], and may at least in part be responsible for the indentation size effect [64], and that it can considerably shift the indentation curves [65]. In addition to that, it is straight-forward that a rough surface, with an increasing number of asperities contacting the indenter with increasing load, will yield a different initial indentation curve than a flat surface, a typical assumption in the finite element simulations of the indentation, whereas at high loads, the asperities are plastically flattened and will conform to the indenter shape, in better agreement with the finite element model. This may have a high impact on the accuracy of material parameter identification, especially in case of nanoindentation, where the ratio between surface roughness and imprint depth may be significant. The indenter tip surface in the model was assumed to be spherical, whereas the true indenter tip shape may deviate significantly from this assumption. It is known since the beginning of instrumented indentation testing [2] that the assumption of an ideal indenter geometry leads to substantial errors, thus leading to the need to calibrate each indenter tip in order to obtain the true contact surface as a function of indentation depth. In most articles relating to parameter identification from indentation testing and numerical optimization, the indenter geometry was considered to be ideal [14] and [13]. It seems that in the case of nanoindentation testing, the assumption of a perfectly spherical indenter tip shape is inappropriate, and an indenter geometry reflecting the true projected contact surface versus indentation depth has to be used. Only recently has a neural network-based method for correcting the indentation curves with respect to indenter tip imperfections been proposed [66]. In the model, the unloaded specimen is assumed to be free of stress, whereas in the real specimen, residual stresses due to processing and specimen preparation may exist. It should be noted that indentation testing may be used for determining residual stresses via the change in yield stress, for example [67]. This may be important in nanoindentation testing, where the surface layer affected by residual stresses may be in the same order of magnitude than the indentation depth. As the imprint and process zone size in nanoindentation are of an order similar to grain size for Dural and smaller than grain size in the case of annealed copper, the assumption of an isotropic continuum is a strong abstraction, and it would be more realistic to explicitly model the polycrystalline structure of the analyzed materials, together with the inherent anisotropy. However, such finite element models, which would have to be three-dimensional in most cases, would be too large to be repetitively modelled in an optimization algorithm in a reasonable period of time. Another conclusion of the experiments is that in nanoindentation testing, the use of residual imprint measurement data does not lead to an improved accuracy of the identification method. The main reason for this is the highly irregular shape of the residual imprint, which is highly affected by surface defects or the polycrystalline nature of the material. These effects, which may be small in macro- and microindentation, render the imprint data rather useless in nanoindentation, unless specimens with a very small surface roughness were used, for example. To resume, it seems that many of the issues described above may very well be neglected in microindentation testing, whereas in nanoindentation testing, their omission leads to significant systematic errors which render a reliable material parameter identification at such small scales rather difficult. One should also keep in mind that the iterative nature of non-linear numerical optimization simply inhibits the use of finite element models which are either three-dimensional or highly refined because of the computing times involved. It is interesting to compare the parameter identification methods based on numerical optimization, which rely on an objective function involving the whole indentation curve, with other methods reported in the literature. In a number of papers [5], [9], [6] and [11], material parameters obtained from physical indentation curves via inverse methods are presented. Concerning the method relying on numerical optimization, most articles rely on the use of synthetic experimental curves, produced by numerical modelling, where the material parameters are known and do not present any experimental results [13], [14], [15] and [49], with the exception of [16] and [38], where only a small number out of all material parameters are identified. It is striking to note that the inverse methods which rely only on specific indentation curve characteristics, like the neural network-based methods [3], [4], [5] and [9], the dimensionless functions [6], [7] and [8], the loading and unloading compliance method [11] yield rather good agreement between reference and identified material parameters. On the one hand, with the exception of [5], rather simple material models for representing metal plasticity, like power law hardening, for example, are used. On the other hand, whereas the optimization-based identification procedures are based on a rather basic objective function, i.e. sum of differences between modelled and experimental values, the methods based on neural networks or dimensionless functions manage to use specific parts of the indentation curve for identifying specific material parameters. This way, material parameters may be identified sequentially, in a staggered way [3], and only features of the identification curve known to be influenced by these parameters are used. For example, the initial unloading is used for Young's modulus [3], the hysteresis between unloading and reloading is used for identifying the kinematic hardening parameters [3]. In inverse analysis using dimensionless functions, use is made of specific characteristics of the indentation curve, like the initial unloading slope [6] and [7], the maximum load (or displacement-into-surface) [6], the ratio of plastic to total energy [6] and [9] and the constant of Kick's law representing the loading curve's curvature [6], [9] and [7]. The unloading compliance at various levels of unloading is used in [11]. It may be concluded that the use of prior knowledge about what material parameters have a high impact on specific parts of the indentation curve or on other indentation curve characteristics, as mentioned above, is crucial to a reliable identification of material parameters from indentation testing. The absence of this prior knowledge, which is totally neglected in the rather basic objective function in the optimization-based identification procedure, may be one important reason for the poor performance of this identification method. Finally, it has to be mentioned that most of the aforementioned methods are relying on indentation testing at micro- or macroscale, where most of the physical effects related to the nanoscale, like surface roughness, residual stress layers and the polycrystalline structure, do not lead to significant systematic errors. It has also to be stated that in contrast to nanoindentation testing, where residual imprint data are of little use, they may well be used effectively with indentation testing at larger scales, as reported in [14] and [16].