Size Effects of Mechanical Properties

1. Size Effects of Mechanical Properties SumanGuha

2. Overview • Size Effects

3. Size Dependence – Experimental Observations • Bending of thin Ni Beams around a cylindrical mandrill. • The strain is dictated by the radius of the mandrill. • The bending moment is calculated from the elastic recovery on unloading. • The moment at point ‘b’ is given by- Stölken and Evans, Acta Mater. Vol 46 (1998), No. 14, pp. 5109-5115.

4. Size Dependence – Experimental Observations • Bending of thin beams. • Bending of Cu-single crystal beams with Hysitronnanoindenter. • Non-uniform deformation. • Flow stress exhibits an inverse relationship with the thickness. Flow Stress (MPa) Beam thickness (micron) Results of the bending of micro sized single crystal copper beams.1 1. C. Motz et al. / Acta Materialia 53 (2005) 4269–4279

5. Size Dependence – Experimental Observations • Tensile and torsion tests on thin Cu wires. • Diameter ranging from 12-170 μm. • Grain size ranging from 5 and 25 μm. Results of tension test on Cu wires of different diameter. Results of torsion test on Cu wires of different diameter. Fleck et. al., Acta Mater. Vol 42 (1994), No. 2, pp. 475-487.

6. Size Dependence - Observations • Nano indentation • Nano indentation on single crystalline and polycrystalline Cu. • Deformation is non-uniform. • With reducing depth of indentation hardness increases. William D. Nix and Huajian Gao, J. Mech. Phys. Solids, Vol 46, No. 3, pp 411- 425, 1998.

7. Size Dependence – Controlling factors • The observed phenomena are attributed to • Geometrically Necessary Dislocations (GNDs). • Non-uniform deformation is accommodated by GNDs. • Density of GNDs can be related to gradient of strain. • Introduction of strain gradient in the constitutive relation to account for the size effects.

8. Plastically Non-homogeneous Deformation • Plastic strain gradient • order of average shear strain/length scale of the deformation. • Length scales: Grain size, separation between the second phase particles etc. • A density ρG is required for compatible deformation which depends on the length scale of the deformation (λ).

9. Density of GNDs with length scale of the deformation • ρG = 4γ/bλ • Below a certain length scale ρG >> ρS . • Hardening becomes strongly dependant on strain gradient. Densities of geometrically necessary dislocations at various assumed length scale.1 1. Fleck et al. Acta metall. Mater. Vol 42, No.2, 1994, pp.475-487

10. Geometrically Necessary Dislocations • Indentation • GNDs are required for compatible deformation • Bending of beam

11. Density of GNDs • In case of an uniform strain distribution no GNDs are necessary. • For a uniform shear the displacement field is given by – u1 = kx2 , u2 = u3 = 0 • The number of dislocation slipping in each block is same.

12. Slip in a single crystal beam under non-uniform shear. • The displacement field • The density of GNDs : u1 = kx1x2 u2 = 0 u1 = - kx1x2 u2 = 1/2kx12

13. Density of GNDs • Slip distance in the first cell : • No of GNDs at the boundary : n1b n1 n2 δx2 δx1

14. Theory - Strain energy density • Conventional theory • Gradient theory Where - work conjugate to

15. Strain Energy Density Function • Linear isotropic hyper-elastic solid • Quadratic terms. • Scalar products only. • The constitutive law , higher order stress.

16. Variational Formulation • Principle of virtual work • Equilibrium Equation fk rk

17. Finite Element Formulation • Second order derivatives of displacement in principle of virtual work demands a displacement based elements of C1 continuity. • The formulations become simpler by treating uk,j as an independent DOF ψjk. • Kinematic constraints are imposed on ψjkand uk,j by Lagrange multipliers.

18. Finite Element Formulation • The weak form • The kinematic constraint • Modified virtual work equation

19. Numerical Results Shu et al. Int. J. Numer. Meth. Engng. 44, 373-391 (1999)

20. Numerical Results u1 , u2 , ψ11 , ψ12 , ψ21 , ψ22 , ρ1 , ρ2 , ρ3 , ρ4 u1 , u2 (a) (b) Distribution of normalized ε22 around a hole. (a) Hole radius a=l. (b) Hole radius a>>l. Where ‘l’ is the length scale.

21. Extension of J2 Flow theory • The work conjugate to plastic strain is considered to be a microstress, not the conventional deviatoric Cauchy stress. • The internal virtual work is expressed as • Equilibrium Equations

22. Constitutive Equations • The rate of dissipation Where, is the free energy and function of , and • For non-negative dissipation • A simplified situation arises, when ,

23. Constitutive Equations • The plastic dissipation , is similar to the local J2 theory. • The incremental stress strain relation is the effective stress. • Yield Criterion

24. Size effects without strain gradient • Compression Tests • Micro-compression of single crystal Au-pillars with a flat tip nanoindenter. • Uniform deformation. • Flow stress increases with decreasing diameter of the pillars. Stress – Strain behavior of <001> oriented, FIB milled Au pillars.1 1. Julia R Greer, W C Oliver, W D Nix, Acta Materialia, v 53, n 6, (2005), p 1821-30.

25. Size effects without strain gradient • Micro pillar compression shows strong size effects. • Strain gradient absent, gradient based plasticity theory fails to explain the behavior. • It is assumed that dislocation starvation causes this high strength.

26. Size effects without strain gradient • The probability of finding a source becomes lower in smaller samples. • The size of the sample puts a restriction on the size of the source. • Smaller sources require higher stress to operate. • Dislocations disappear at the surface quite faster, leading to very low dislocation density.

27. Size effects without strain gradient • The stress required to operate a Frank-Read source – • Strain-rate, velocity relationship. • For low dislocation density, average velocity should be high, causing higher flow stress.

28. Micro-compression Test • Micro-pillars are to be fabricated using Focused Ion Beam milling. • Material : Polycrystalline Aluminum, cold rolled annealed.

29. Micro-pillar fabrication

30. Micro-compression Test Load Cell Indenter Micro-pillar XY positioner stage Linear Actuator

31. Thank you!