Finite Element Modeling of Lattice Strainsin Polycrystalline Materials with Comparisons to Diffraction Experiments
Paul Dawson and Matthew Miller
Sibley School of Mechanical and Aerospace Engineering, Cornell University
Metallic alloys frequently are used to fabricate components that that will serve in the most demanding environments of high stress and elevated temperatures. To compound the demands on these components, they also must be extremely reliable. At the same time, the quest for better performance discourages the strategy of achieving reliability simply through employing high factors of safety in mechanical design. The frames and engine components of an aircraft offer a number of examples where performance and reliability are delicately balanced. To continue to improve such systems, a more comprehensive understanding of the performance of the alloys used is essential. In particular, we must advance beyond the practice of assuming that the macroscopic stress is an adequate indicator of how severely a material is loaded. Rather, the variations of stress that arise from heterogeneity of the microstructure must be factored into the design analyses. Spatial variations of the stress are a natural consequence of the di_erences in properties of the constituent phases. But stress variations also arise in simpler single phase alloys comprised of grains having anisotropic properties. When combined with the fact that the physical mechanisms that control the mechanical behavior are defined in domains that are small compared to the crystal size, the issue of deviations in stress associated with heterogeneous microstructure cannot be avoided short of using components fabricated of single crystals.
There are a number of aspects of the behavior of polycrystalline alloys that remain inadequately understood, but are central to using better materials or better using the materials we have. Stiffness and strength are strongly influenced by load sharing between phases and between crystals within a phase. The initiation of fatigue defects can occur under macroscopic loading that is nominally elastic. Alloys can change phases when subjected to changing temperature or deformation. These behaviors are difficult to understand without a more complete picture of the microstructural state and mechanical loading than has been available.
Such a picture is not likely to emerge from either modeling or experiment alone, but rather with the coordinated combination of the two. One combination that is particularly
effective is that of finite element modeling of polycrystals together with diffraction measurements under in situ loading. The finite element simulations can help complete the partial picture generated by experiments and aid in its physical interpretation.
In this presentation we will briefly describe the principal features of a finite element model of polycrystals. The intent will be to show what physical aspects are included in the model, as well as to indicate where the model is limited by inadequate constitutive theory or inability to resolve the microstructure su_ciently. We will then present several examples in which comparisons between simulation and experiment have been performed. The first comparison is on lattice strains measured grain-by-grain in a copper specimen subjected to tension. The second also is on lattice strains, but for all crystals along particular crystallographic fibers in a two-phase (Fe-Cu) alloy. Here the focus is on the progression of trends in interphase load sharing through the stages of yielding. Finally, we’ll discuss an application in which the stress varies at both the continuum and crystal scales with attention to the additional challenges faced when the gradients in stress exist over dimensions comparable to the di_raction volumes. Two figures from the first example are presented here as an illustration. Figure 1 shows a model copper
Figure 1: Finite element polycrystal before (left) and after deformation (right) with color shading indicating. Deformations exaggerated by a factor of 100.
Figure 2: The standard deviation of the stress in crystals with the 400 direction aligned with the tensile axis. Error bars indicate the uncertainty in determining a normal distribution from a finite sample size.
polycrystal comprised of dodecahedral crystals, with each crystal resolved with 48 tetrahedral finite elements. Sectional cuts through the polycrystal expose square patches within the individual crystals. The polycrystal is deformed according to the loading history in the experiment [Lienert, U., et al., Acta Materialia, in press] and the responses of crystals satisfying the experimental Bragg condition are extracted. The variation in the resulting stress is indicated in Figure 2, indicating reasonable agreement in the spatial variation in the stress between experiment and simulation for this material.
To address the challenges described above, the body of experimental data must be enlarged. In particular, we need to have more complete time records of stress for dynamic loading. Similarly, time resolution of the changing structure is essential to build a better understanding of processes associated with phase changes or substructure evolution under deformation. More highly resolved spatial information will also be needed. That is, measurements that give the multiaxial states of strain in a number of neighboring crystals are vital. At the same time, development of the finite element models must continue to accommodate more complete physical descriptions of the material. Finally, the capabilities to make direct comparisons must keep pace.
