مدل ویسکوپلاستیک ناهمسانگرد برای کامپوزیت؛ تجزیه و تحلیل با حساسیت و برآورد پارامتر

Because of their potential in achieving many performance enhancements, composite material systems (e.g. fiber-reinforced composites) are presently called upon to operate under wide range of stresses, temperatures, and loading rates. This in turn requires the development of general material models to capture the significant effects of anisotropy on both elastic and inelastic responses. The starting point in the present contribution is the development of a class of such viscoplastic models. Furthermore, a number of robust, computationally efficient, algorithms are also presented for the development of an overall strategy to estimate the material parameters characterizing these complex models; i.e. rate-dependent plastic flow, non-linear kinematic hardening, thermal/static recovery, anisotropic viscoelastic and viscoplastic flow. The entire procedure is automated through an integrated software namely, COnstitutive Material PARameter Estimator, COMPARE, to enable the determination of an ‘optimum’ set of material parameters by minimizing the errors between the experimental test data and the predicted response. The key ingredients of COMPARE are (i) primal analysis, (ii) sensitivity analysis, (iii) a gradient-based optimization problem and a (iv) graphical user interface. The estimation of the material parameters is cast as a minimum-error, weighted multi-objective, non-linear optimization problem with constraints. Detailed derivations of the direct differentiation sensitivity expressions are presented. In addition, numerical comparisons of the sensitivities obtained by the more traditional finite difference approaches are given to assess accuracy. Results generated by applying the developed algorithms for anisotropic, strain-controlled tensile (with comparison to typical experimental data) and constant-stress creep tests are presented to demonstrate the ability of the present models to accurately capture time-dependent anisotropic material behavior.

The use of advanced materials, for example, metals, polymers and ceramics, are at the forefront of today's research. As a result, numerous computational models for predicting both deformation and life for these materials are under development. A key to the effective use of these advanced analysis techniques are accurate and computationally efficient constitutive models. These models must account for both reversible and irreversible time-dependent deformation. For example, the irreversible time-dependent response component becomes dominate for metals at high temperatures. On the other hand, polymers and rubbers have predominately a purely reversible viscous response. Significant progress has been made over the years in the development of theories for the phenomenological representations of the time-dependent viscoelastic and viscoplastic constitutive properties. In particular, at least considering the isotropic response case, the mathematical modeling of metal viscoplasticity is presently very well developed, based on the so-called internal variable formulism in the thermodynamics of irreversible processes [3], [4], [9], [11], [12], [17] and [19]. A number of specialized forms of these modern unified viscoplastic models (e.g. isotropic fully associative and non-associative, isothermal or non-isothermal, etc.) have been successfully applied to different metals and other material systems [4], [9], [12] and [17]. In addition, several attempts have also been made to account for material anisotropy in composites' inelasticity [13], [21], [22] and [24]. Here, we utilize, as a starting point, an extended formulation of the anisotropic viscoplastic model reported earlier in Ref. [24], with emphasis on two major enhancements. Firstly, adopting the notion of multiplicity of deformation mechanisms [28], we combine an aggregate of tensorial state variables to account for wider ranges of material relaxation spectra, in both domains of reversible and irreversible responses; i.e. viscoelastoplasticity. Secondly, we allow for anisotropy in both regimes; i.e. viscoelastic stiffness anisotropy, as well as orientational/directional-dependency, in the viscoplastic strengths. Of course, the improved accuracy and material representation capabilities in these models have often been acquired at the expense of greater mathematical complexity and a large number of material parameters (introduced to describe a host of physical phenomena and complicated deformation mechanisms). In addition, the experimental characterization of these material parameters is a major factor in the successful and effective utilization of the constitutive model. Material parameter estimation, expressed in the form of an inverse problem [8] involves the simultaneous identification of a large number of parameters from a variety of experimental tests, i.e. different loading conditions and control modes such as strain-, stress-, and mixed-controls). Such problems are known to be both mathematically and computationally challenging [7]. Adding to this difficulty is the fact that most of the material parameters lack an obvious or direct physical interpretation and they differ in scale for a given model. Also, even under load histories in simple laboratory tests, many parameters will highly interact with each other, affecting the model response predicted. Research work in the area of model parameter fitting is rather limited [7], [8], [14], [15], [23], [26] and [32]. In particular, specific guidelines for systematic determination of these material parameters are presently lacking. Therefore, an urgent and obvious need exists for a systematic development of a general methodology for constitutive material estimation and indeed this provides a major motivation for the work reported here.

In this work we have extended a general time-dependent model to the regimes of anisotropic viscoelasticity and viscoplasticity. All thermodynamic requirements (dissipation inequalities fullfilled in a straightforward manner. Details of several algorithmic developments have been presented as parts of an overall automated strategy to estimate the material parameters of a class of viscoelastoplastic models for anisotropic response. This model has been implemented into the COnstitutive Material PARameter Estimator, COMPARE. A general time-stepping algorithm based upon the implicit backward Euler integration scheme is used for updating the stress and state variables. The associated expressions for the iterative Jacobian and the consistent algorithmic tangent stiffness were derived in close form. This implicit integration scheme is utilized in the primal phase of the COMPARE optimization scheme. Expressions for the response sensitivities were derived on the basis of a direct differentiation/recursive approach which has been shown to be far more efficient than a finite difference scheme. In fact, the evaluation of the direct differentiation sensitivities (i.e. sensitivity phase), actually occurs at the end of the primal analysis phase since most of the necessary quantities are directly available from the integration algorithm. This makes for a very efficient primal/sensitivity phase of the COMPARE program. Suitable ‘scaled’ error functions were used as the basis for the optimization phase. Pertaining to the applications, the present viscoelastoplastic model was characterized for two different composite materials; i.e. anisotropic systems IM7/8552 and AS4/PEEK. The excellent correlations with the obtained experimental data were noted for both, consisting of various off-axis and on-axis specimens, over a range of strain rates, shows the robustness of the present model to predict anisotropic elastoviscoplastic material behavior. For IM7/8552 characterizations were performed using viscoelasticity only (five viscoelastic mechanisms) and using combined viscoelastoplasticity (one viscoelastic and two viscoplastic mechanisms). For the selected characterizations presented, the use of the combined viscoelastoplasticity was able to accurately capture the residual stress present at the end of the unloading period. This application demonstrated the capabilities of the present multi-mechanism viscoelastoplastic model to capture complex material behavior and the general utility of the COMPARE program to characterize complex load histories. For AS4/PEEK, several predictions were made for tensile tests, at a selected strain rate, at other off-axis angles, with the predicted response showing the as expected trends. In addition, constant-stress, creep tests were also predicted. Noting the lack of a primary creep region for the case of a single viscoplastic mechanism, additional predictions were made using two viscoplastic mechanisms. The improved creep response shows the flexibility of the present model. In all cases considered, the predicted response was consistent with expected results. Finally, the entire histories of material-parameters' sensitivities were also presented for different representative test cases. The varying degrees of relative importance of these sensitivities and the associated regions where the anisotropic material response becomes quite sensitive to several interacting parameters were identified. This can provide for a very useful tool in the actual design and optimization of the performance of composite systems in that the most impactive parameters can be easily determined corresponding to a given loading condition.