1. Size effects in nanoscale and microscale metallic samples: from MD simulation to strain gradient plasticity George Z. Voyiadjis, M. Yaghoobi and Yooseob Song Computational Solid mechanics Laboratory Department of Civil and Environmental Engineering Louisiana State University Baton Rouge, Louisiana

2. Continuum Modelling

3. The goal is to develop a nonlocal continuum theory to address the plasticity and governing size effects mechanism for metallic materials of small volumes. continuum formulation Over the size scale range in which most of the experiments have been conducted, the number of dislocations is generally so large that a continuum formulation is required to describe the deformation. - the individual effect of microstructural interactions (i.e. dislocation motion) on macroscale observed behavior. - Thermo-Mechanichal Coupling: Thermal activation  heat generation due to plastic work Motivation : Continuum Modeling

4. Motivation: Size Effects Size effects: the dependency of material mechanical properties on the size of the structure. Size dependency of micro-bending of Nickel thin films Strength size dependence in micro-torsion of Copper wires McElhaney et al. (1998) Depth Dependence of Hardness of Copper Geometrically Necessary Dislocations Classical continuum theory

5. Wang, Askes, Sluys (1998) Khoei et al. (2005, 2007) Voyiadjis and Abu-Al Rub (2004) De Borst and Pamin (1996) Unique Solution Results Numerical (FE) Techniques: - Adaptive re-meshing - Meshless Method Material Constitutive Model: - Visco - plasticty - Nonlocal Gradient Plasticity shear zone in steel Lee et al. (2008) Motivation: Localization and Mesh Dependency Result

6. Strain Gradient Plasticity Theory Background  In the 1980s a scalar form of the strain gradient plasticity theories was proposed by Aifantis (1984,1987) to capture the size effects  Fleck and Hutchinson (2001) developed the higher order strain gradient plasticity theory based on the work of Aifantis (1984, 1987)  Gudmundson (2004) and Gurtin and Anand (2009) noted that the formulation of Fleck and Hutchinson (2001) violated the thermodynamic requirement on plastic dissipation under some non-proportional loadings  A modification of the theory of Fleck and Hutchinson (2001) is proposed to correct the thermodynamic deficiency by Hutchinson (2012)  Hutchinson (2012) and Fleck et al. (2014, 2015) pointed out that the theories proposed by Gudmundson (2004) and Gurtin and Anand (2009) have drawbacks as far as non-proportional loadings  Stress Jump

7. Strain Gradient Plasticity Theory

10. Validation of the proposed model Xiang, Y., Vlassak, J.J., 2006. Bauschinger and size effects in thin-film plasticity. Acta Mater 54, 5449-5460. Schematic representation of the plane strain bulge test technique Comparing the proposed model with Xiang and Vlassak (2006) on the stress-strain response of the electroplated Copper thin films Haque, M.A., Saif, M.T.A., 2003. Strain gradient effect in nanoscale thin films. Acta Mater 51, 3053-3061. SEM image of the MEMS-based tensile test chip Haque and Saif (2003) Comparing the proposed model with Voyiadjis and Faghihi (2014) and Haque and Saif (2003) on the stress-strain response of the sputter-deposited Aluminum thin films. Han, S., Kim, T., Lee, H., Lee, H., 2008. Temperature-dependent behavior of thin film by microtensile testing, Electronics System-Integration Technology Conference, 2008. ESTC 2008. 2nd, pp. 477-480. Micro tensile testing system Han et al. (2008) Comparing the proposed model with Voyiadjis and Faghihi (2014) and Han et al. (2008) on the stress-strain response of Nickel thin films.

11. Strain Gradient Plasticity Theory A finite stress jump due to infinitesimal changes in the plastic strain

12. Strain Gradient Plasticity Theory

14. Atomistic Modelling

15. Motivation: Size Effects All the previous experiments are showing size effects due to strain gradient. Greer and her coworkers conducted the micropillar compression test to investigate the size effects in the absence of strain gradient. Greer et al. (2015) They showed that the pillars show strong size effects even in the absence of strain gradient during the compression test.

16. Motivation: Size Effects Meng and his coworkers at LSU investigated the micropillar size effects using the compression experiment through the CIMM NSF project. In this research, the main target is to investigate the main mechanisms of size effects during the pillar compression test using atomistic simulations. Also, a continuum model will be developed to capture these mechanisms.

17. Outline:  Simulation details and methodology  Effects of pillar size on the deformation mechanism  Effects of pre-straining on the deformation mechanism  Visualization of dislocation nucleation and evolution  Summary  Conclusions

18. 18 The parallel code LAMMPS is incorporated to conduct the MD simulation of Ni pillars. Simulation details and methodology The crystal analysis tool is incorporated to visualize the dislocations. The boundary conditions for top and bottom surfaces are set free. The substrate is simulated using a prescribed potential wall, and the dislocations are able to pass through the sample bottom. The compressive displacement is then applied using a large flat indenter. The embedded-atom method (EAM) potential parameterized by Mishin et al. (1999) is incorporated to capture the Ni-Ni atomic interaction.

19. Outline:  Simulation details and methodology  Effects of pillar size on the deformation mechanism  Effects of pre-straining on the deformation mechanism  Visualization of dislocation nucleation and evolution  Summary  Conclusions

20. 20 Role of pillar diameter on the size effects governing mechanisms (H=45 nm) The elastic response is the same for all diameters. After that, increasing the sample diameter from 22.5 nm to 90 nm, sample strength decreases. Further increasing the diameter does not change the pillar strength. To study the governing mechanisms of size effects, the variation of dislocation density should be studied. For the smallest diameter (D=22.5 nm), the results show that the size effects in the pillar are controlled by dislocation starvation. Each time, the mobile dislocations are driven out of the sample which locally increases the stress until another dislocation nucleation and evolution occurs leading to a stress relaxation. In the case of pillar with D= 45 nm, the exhaustion hardening mechanism controls the size effects in a way that as the dislocation density decreases, the stress increases. Some small local jumps are observed in true stress which occur because some dislocations leave the pillar from the free surfaces. After the elastic response and following stress relaxation, there is no jump in the stress-strain curves in the case of pillars with the diameters of 90 nm and 135 nm, which is due to the smooth variation of dislocation density. The results show that both pillars eventually reach a steady response which is not size dependent.

21. 21 Role of pillar height on the size effects governing mechanisms (D=22.5 nm) The results show that the size effects governing mechanism is dislocation starvation for all samples, and the general trend is nearly independent of pillar height. when the dislocation starvation is the dominant mechanism of size effects, changing the sample height does not considerably alter the pillar response. The values of local stress peak are slightly different based on the density of dislocations remains in the sample during the starvation step.

22. 22 Role of pillar height on the size effects governing mechanisms (D=135 nm) The results show that as the pillar height decreases, the strength of the pillar increases. Also, the dislocation starvation does not occur. In the cases of samples with heights of 30 nm and 45 nm, pillars have nearly the similar dislocation density as the strain varies. However, the required stress to maintain the observed dislocation density increases as the pillar height decreases. It is due to the fact that as the pillar height decreases, dislocations leave the sample faster and more dislocations should be nucleated to maintain the plastic flow which increases the required stress. In the case of pillar with the height of 75 nm, the obtained dislocation density is higher than those of the pillars with the heights of 30 nm and 45 nm which leads to the less strength according to the exhaustion hardening mechanism.

23. 23 Size effects governing mechanisms for the largest simulated sample (D=0.3 m) In the absence of strain gradient and in the high rate deformations, there is no more size effect in the case of pristine pillars larger than the one with D=135 nm and H=75 nm. Compared to the experimental results obtained at quasi-static rates, such as Greer et al. (2005) and Meng et al. (2014), the results show that increasing the strain-rate decreases the size of pillars at which there is no more size effects.

24. Outline:  Simulation details and methodology  Effects of pillar size on the deformation mechanism  Effects of pre-straining on the deformation mechanism  Visualization of dislocation nucleation and evolution  Summary  Conclusions

25. 25 Effects of pre-straining on the deformation mechanism In the case of real experiments, the samples are not defect free. Up to know, all the simulated samples are started from a defect free structure. To study the effects of initial defects, the pre-straining procedure is conducted on some samples and the results are compared with those of the defect free. The results show that the controlling mechanisms of size effects are independent of sample initial structure in the cases of pillars with diameters of 22.5 nm and 135 nm and height of 45 nm. Only the initial phase of dislocation nucleation and evolution is changed.

26. Outline:  Simulation details and methodology  Effects of pillar size on the deformation mechanism  Effects of pre-straining on the deformation mechanism  Visualization of dislocation nucleation and evolution  Summary  Conclusions

27. 27 Visualization of dislocation nucleation and evolution Pillar with a diameter of 15 nm and height of 30 nm consists of around 500,000 atoms. The results show that each time, dislocation starvation occurs, the strength required to sustain the plastic flow starts increasing. New dislocations are then nucleated and elongated which releases the stress as shown in Figs. 9 (b), (c), (e), (g), and (i). Dislocations eventually leave the pillar which leads to another starvation, as shown in Figs. 9 (d), (f), and (h). Dislocation visualization at different strains.

28. 28 Pillar with the diameter of 0.15 μ m and height of 30 μ m consists of around 500,000,000 atoms. The pillar was pre-loaded and unloaded before the final simulation. The results show that forest hardening mechanism controls the size effects in which the strength increases as the dislocation density increases. In this mechanism, the dislocation interactions with each other controls the strength. Dislocation visualization at strain around 0.14 Visualization of dislocation nucleation and evolution

29. Outline:  Simulation details and methodology  Effects of pillar size on the deformation mechanism  Effects of pre-straining on the deformation mechanism  Visualization of dislocation nucleation and evolution  Summary  Conclusions

30. 30 The effects of pillar diameter and height are studied using MD simulation. A very large pillar consist of around 500 million atoms is also simulated. The effects of pillar diameter and height are studied using MD simulation. A very large pillar consist of around 500 million atoms is also simulated. The effects of pre-straining are investigated. The effects of pre-straining are investigated. Summary : The dislocation nucleation and evolution are studied by visualizing the dislocations. The dislocation nucleation and evolution are studied by visualizing the dislocations.

31. Outline:  Simulation details and methodology  Effects of pillar size on the deformation mechanism  Effects of pre-straining on the deformation mechanism  Visualization of dislocation nucleation and evolution  Summary  Conclusions

32. 32 Conclusions : The size effects are controlled by the dislocation starvation and source exhaustion. The size effects are controlled by the dislocation starvation and source exhaustion. In the case of very small samples, the dislocation starvation is a governing mechanism of size effects in which, each time, the mobile dislocations are driven out of the sample which locally increases the stress until another dislocation nucleation and evolution occurs leading to a stress relaxation. In the case of very small samples, the dislocation starvation is a governing mechanism of size effects in which, each time, the mobile dislocations are driven out of the sample which locally increases the stress until another dislocation nucleation and evolution occurs leading to a stress relaxation. Increasing the sample size, the governing mechanism changes to the source exhaustion hardening. Increasing the sample size, the governing mechanism changes to the source exhaustion hardening. During MD simulation, the loading rates are much higher than MD simulation due to the computational limitations. Increaseing the strain rates decreases the pillar diameter at which there no more size effects for pillars with larger diameters. During MD simulation, the loading rates are much higher than MD simulation due to the computational limitations. Increaseing the strain rates decreases the pillar diameter at which there no more size effects for pillars with larger diameters.

33. 33 Conclusions (cont.): In the case of the largest simulated pillar, D=0.3 μm, the pre-straining activates the forest hardening mechanism and consequently increases the sample strength. In the case of smaller samples, however, pre-straining only change the initial stages of dislocation nucleation. In the case of the largest simulated pillar, D=0.3 μm, the pre-straining activates the forest hardening mechanism and consequently increases the sample strength. In the case of smaller samples, however, pre-straining only change the initial stages of dislocation nucleation.

34. Voyiadjis and Yaghoobi, Computational materials science 117 (2016) 315–329. Yaghoobi and Voyiadjis, submitted to Acta Materialia, 2016.Publications Voyiadjis and Yaghoobi, Computational materials science 117 (2016) 315–329. Yaghoobi and Voyiadjis, submitted to Acta materialia, 2016. Voyiadjis, Song, and Park, submitted to JEMT, 2016. Zhang and Voyiadjis, Material science and engineering A, 2016.

35. 35 What is next? The next step will be developing a continuum model which incorporates the observed atomistic mechanisms of size effects including the effect of dislocation source size and source exhaustion hardening. The next step will be developing a continuum model which incorporates the observed atomistic mechanisms of size effects including the effect of dislocation source size and source exhaustion hardening.

36. Thank You! Questions? 36