Modeling creep and anelasticity in particle strengthened alloys with strain gradient crystal plasticity

Modeling creep and anelasticity in particle strengthened alloys with strain gradient crystal plasticity For small material volumes, size effects, e.g. due to the interface constraints or heterogeneous strain ¿elds, may signi¿cantly affect the mechanical behavior of metals such that a deformation mechanism that is less important for the response in bulk form may become decisive for the performance of the material. Such second order effects were observed experimentally in the last two decades and form engineering challenges for the development and production of high-end modern technology. For example, creep and anelasticity observed in metallic thin ¿lm components of capacitive RF-MEMS switches may lead to time dependent deviations from the design speci¿cations of the device. The characterization and understanding of the mechanical behavior of the material is indispensable to overcome the reliability issues of these switches which hinder their full commercialization. In this thesis, a numerical framework is presented for modeling the time dependent mechanical behavior of thin ¿lms made of particle strengthened fcc alloys as an extension of a previously developed strain gradient crystal plasticity (SGCP) model (here referred to as Evers-Bayley type model) for pure fcc metals. A physically based ¿ow rule for crystallographic slip is developed based on the dislocation-dislocation and dislocation-particle interaction mechanisms. The extended SGCP framework is intrinsically able to capture the effect of an inhomogeneous distribution of geometrically necessary dislocation densities on the material behavior via the formulation of a back stress incorporating a material length scale. In chapter 2, the physically based Evers-Bayley type model and a thermodynamically consistent strain gradient theory of crystal plasticity by Gurtin are compared by deriving micro-stresses for the Gurtin type formulation based on the energetic back stresses of the Evers-Bayley type models, incorporating dislocation-dislocation interactions. It is shown that the defect energy function for a micro-stress that con¬tains the physical description of the interaction between dislocations of different slip systems has a more complicated form than those suggested in literature and is possibly non-convex. It is also shown that similar boundary conditions can be de¿ned for the Evers-Bayley type and Gurtin type models despite their differ¬ent additional ¿eld equations within the ¿nite element context. Thereafter, in chapter 3, the SGCP model is employed in electromechanical ¿nite element simulations of bending of polycrystalline thin beams made of a pure metal and a two phase alloy with a focus on the description of anelastic material behavior. Sim¬ulation results obtained with the SGCP model show a macroscopic strain recovery over time following the load removal. However, a detailed analysis demonstrates that the anelastic relaxation time and strength have no solid physical basis. A comparison of the results with experimental data implies that a single deformation mechanism may not be adequate for capturing the material response. Moreover, the slip law falls short in describing the behavior of a particle enhanced material. Subsequently, an extension of the SGCP model for a more realistic description of the time dependent mechanical behavior of two phase alloys, i.e. creep and anelasticity, is given in chapter 4 and its appli¬cation in multiphysical simulations of a capacitive RF-MEMS switch is presented in chapter 5. A new constitutive rule for crystallographic slip is developed by considering dislocation-dislocation interactions and three distinct dislocation-particle interactions: i) the Orowan process, ii) the Friedel process and iii) the climb of edge dislocations over particles. The new constitutive rule is obtained by the combination of separate slip laws for each type of interaction and is built based on the physically well-founded Orowan type rate equation. A ¿ow rule for the slip rate of mobile dislocations governed by dislocation-dislocation interactions is written by taking into account the jerky and continuous glide regimes of dislocations. Slip laws corresponding to the Orowan and Friedel processes are constructed by considering thermally activated dislocation motion. The climb of edge dislocations is described via a thermal detachment model. Results of ¿nite element simulations of bending of a single crystalline thin beam and a micro-clamp experiment with the extended SGCP model show that creep and anelastic behavior of a metallic thin ¿lm can be predicted with the extended SGCP framework. The amounts of the plastic deformation, anelastic recovery strength and associated relaxation times strongly rely on particle properties, the diffusional rate and the magnitude of internal stresses. The results of the simulations of the micro-clamp experiment imply that inhomoge-neous material diffusion may play an important role in the anelastic behavior of polycrystalline thin ¿lms. The results also suggest that the internal stress formulation of the extended SGCP may need to be extended by considering additional sources of internal stresses. The extended SGCP framework is applied to analyse the behavior of a capacitive RF-MEMS switch in multiphysical simulations. The electrodes of the switch are considered to be made of a metal thin ¿lm with incoherent second phases and have a polycrystalline structure with columnar grains through the thickness and passivated surfaces. The variation of the gap between the electrodes over time is analyzed. First, the in¿uences of particle size, volume fraction, surface constraints and ¿lm thickness on the performance of the switch after a loading and unloading cycle are studied. Then, the effects of cyclic loading and the duration of the unloaded state between sequential cy¬cles are investigated. The results show that the residual changes in the gap and the amount and rate of time dependent recovery after the load removal are highly sensitive to the microstructure and the ¿lm thickness. The smallest amounts of permanent deformation and anelastic recovery are obtained with an upper elec¬trode made of a relatively thin ¿lm which has a surface passivation and involves small incoherent particles with a relatively large volume fraction. Furthermore, the simulations revealed that the maximum residual change of the gap measured after completion of the unloading stage of each cycle saturates within a few cycles. A shorter duration of the unloaded state between successive loading-unloading cycles leads to a larger maximum residual gap change. Due to the decreasing gap, the pull-in voltage also decreases within a few cycles and shows a tendency to level off to a certain value. However, the release voltage does not seem to be as sensitive to the residual deformations as the pull-in voltage. Finally, in chapter 6, the conclusions and recommendations for a future work are given.