مدل سازی بلورهای مجزا: ادغام زمان، اپراتورهای مماس، تجزیه و تحلیل حساسیت و بهینه سازی شکل

In this paper, a comprehensive overview of the numerical analysis of single crystalline materials at high temperatures is presented, including implicit higher order time integration, consistent linearization and respective sensitivity analysis. Both a phenomenological and a crystallographic model are employed to simulate the mechanical isothermal behavior of the nickel-based superalloy CMSX-4 at 950°C in a finite element environment. A shape optimization methodology for testing specimen design based on extensive finite element computations is presented. The sensitivity analysis according to the chosen time integration scheme is performed leading to an algorithm for simultaneous analysis and design (SAND). Examples for the shape optimization of a cruciform specimen for biaxial tensile experiments at high temperatures are given and discussed in detail.

Single crystalline materials are typically used for high temperature applications, where polycrystalline alloys fail due to the weakening effects of grain boundaries acting as dislocation sources in highly loaded states. These so-called superalloys exhibit a pronounced anisotropic elastic and viscoplastic behavior under mechanical loading. Special material models of unified type are utilized for the description of such phenomena. The development of these alloys is mainly motivated from applications in gas turbines and aircraft engines. The energetic ratio of efficiency of these constructions is principally dependent on the maximum process temperature where consideration of the Carnot process allows for a simple estimate of a theoretical maximum efficiency. The higher the temperature in the first row of vanes is, the greater the theoretical maximum efficiency is. Material models for such alloys are intensively discussed in current research. The following list of publications is by far not complete, it is rather a brief sketch of selected research topics associated with single crystalline materials. The short time viscoplastic behavior of such alloys is for example investigated by Choi and Krempl, 1989, Sutcu and Krempl, 1990, Dame and Stouffer, 1988 and Jordan and Walker, 1992. All models are based on some kind of Hill-type plasticity and referred to as first generation models. Meric et al., 1991, Nouialhas and Culie, 1991 and Nouailhas and Cailletaud, 1995 proposed anisotropic models at small strains of the second generation with Chaboche-type hardening variables motivated from micromechanical observations on slip system levels on an atomistic length scale λ∗ (cf. Fig. 1). Forest et al. (1997) formulated an extension to large deformations in a Cosserat framework. However, Shu (1998) considers size effects in single crystalline structures. Creep deformations on a large time scale are modeled, for instance, by Brehm and Glatzel, 1998 and Qi and Bertram, 1997. Balke and Estrin (1994) concentrate on the development of shear bands leading to a macroscopic failure of the assembly. An approach towards an efficient numerical assessment of these models in a finite element context was given by Cuitiño and Ortiz (1992). A methodology for the determination of suitable sets of material parameters for these complex material models was proposed by Kunkel and Kollmann (1996), who concentrated on the Choi and Krempl (1989) model with not less than 30 free parameters. Full-size image (9 K) Fig. 1. Different length scales for modeling crystalline behavior. Figure options This short overview on research activities concerns mainly the mechanical modeling. Numerical approaches specially for single crystals are not yet that elaborated. Kirchner and Simeon (1999) present an implicit higher order time integration scheme and apply it successfully to a first generation model. Recently, Kirchner (1999) gave a generalization of this approach allowing for straightforward application to (almost) any kind of small strain plasticity models, provided an additively decomposable strain measure is used. Herein, the most important points of the argumentation will be reviewed to demonstrate the necessities to fulfil in order to apply this quite simple but nevertheless efficient method in plasticity. One problem for single crystalline materials at high temperature is the experimental testing and respective identification of the model parameters. Circular cross sections perpendicular to the specimen axis do not necessarily remain circular shaped under uniaxial tensile loading which is obviously in contradiction to the classical assumption usually met for isotropic materials (see, for instance, Anand and Kothari, 1997) that the uniaxial deformation in the measuring length of the specimen is homogeneous. It is therefore necessary to account for the resulting in-homogeneity in a sophisticated identification procedure. More elaborated two- and three-dimensional tests and respective identification procedures are thus needed. The problem of combined tension–torsion tests on thin tubular specimen is that the deformations are non homogeneous. “Weak” and “strong” zones exist in circumferential direction of the specimen due to the varying angle between shear stresses and crystal lattice orientation (cf. Nouailhas and Cailletaud, 1995). The combination of tension and torsion is appropriate to avoid problems with stability of the experimental setup for pure torsion. However, for these combined tests one has to solve an initial boundary value problem when using these two-dimensional tests for parameter identification. If a specimen behaves sufficiently homogeneous under multi-axial loading — at least in some measuring domain with being the domain in Euclidean space occupied by the specimen — one can simply expand the identification procedures from the one-dimensional testing regime to account for two or even more pairs of force–displacement data. Identification methods based on displacement field measurements and a corresponding finite element analysis as done by Mahnken and Stein, 1994 and Mahnken and Stein, 1996 are not applicable in high temperature testing. Chemical reactions taking place on the specimen surface prohibit a successful use of optical measuring methods based on some surface etched reference grid point net and respective (time dependent) trajectory recording. This papers focuses on numerics of such single crystalline material models in a nonlinear finite element context. Furthermore, an approach for high-temperature specimen design is presented including both formulation of a suitable objective function and respective sensitivities for the optimization procedure consistent with the time integration approach applied. One aim of this paper is to contribute to the development of an efficient two-dimensional experimental setup with a strong interaction between specimen shape and testing program. The discussion is restricted to monotonous biaxial tensile experiments and do comment on necessary modifications for other two-dimensional tests. However, to be able to use the proposed form-finding procedure material parameters for a specific constitutive model are necessarily needed. But these parameters are often available from a first series of one-dimensional tests performed on the same material and thus this requirement can be met in practical applications. The structure of this paper is as follows. In Section 2, the necessary aspects from geometrically linear continuum mechanics are reviewed and two sets of constitutive equations and respective material parameters are given for the examples shown in this paper. A description of the optimization problem setup is also given in this continuous context. Until now, no discretization or approximation is necessary in the description. Next, a general way how to assess small strain viscoplasticity models for an efficient higher order time integration including respective consistent linearization of the integration algorithm is shown and comment on finite element implementation aspects are made. The sensitivity analysis necessary for a deterministic solution of the optimization problem is discussed in Section 4 and the computation of the objective function and its gradient is sketched in an isoparametric finite element context. A major part of that section is devoted to the computation of the stress sensitivity including all history effects consistently. Examples for shape design of biaxial tensile specimen for the nickel-based high-temperature alloy CMSX-4 at 950°C are given in Section 5. Conclusions are drawn afterwards.

In the present paper, some new results concerning the numerical analysis of single crystalline materials are given. The theory discussed here is kept as short as possible to outline the principal steps in argumentation and implementation. The optimization problem considered is of significant interest for forthcoming sophisticated testing methods in high temperature applications where optical methods cannot be carried out by Mahnken and Stein (1994) for low temperature applications. If only two or three pairs of displacements and associated forces are accessible one has to develop a suitable specimen that guarantees sufficiently homogeneous deformation fields to reduce its behavior to analogous sets of forces and displacements to compare simple simulation results for a material point and experimental averaged data. We do hope that this paper contributes to the ongoing debate on multi-axial identification methods and their numerical solution. Furthermore, a novel approach to efficient higher order time integration is given that expands the arguments of Kirchner and Stein (1999) who have already proven the advantages of higher order methods compared to the standard first order Euler scheme. The method can be easily implemented in some finite element code once the ability for storing and up-dating the necessary history information is secured. In addition, the methods allows for adaptive time stepping schemes. Summarizing, a quite comprehensive analysis of single crystalline materials using state-of-the-art procedures is given in a very condensed form. More information is supplied in Kirchner (1999), the discussion considers there also large deformation problems.