Metallic interfaces, such as grain boundaries, phase bound- aries and
metallization layers, ... strain gradient crystal plasticity frameworks [1, 2]. As a
benchmark ...
Mechanics of Materials
Grain boundary interface model in strain gradient crystal plasticity P.R.M. van Beers, G. McShane, V.G. Kouznetsova and M.G.D. Geers
Metallic interfaces, such as grain boundaries, phase boundaries and metallization layers, play a dominant role in bulk materials, functional materials and metallic microdevices in defining their strength, reliability and life time properties. Current modeling approaches of metal interfaces lack the critical interaction between plasticity and interfaces. This interaction takes place at the level of individual dislocations that can be accumulated, transmitted, absorbed or nucleated at the interfaces. Molecular Dynamics
Continuum Interface
Discrete Dislocations + ++ ++ -+ -+---+ --- +- ---+ ++- -+ ++ -+- + +-+ +++ ++ +++++ ++ ++ - -++ - -++-- - +--
The BCs used in current models consider the limiting situations of either impenetrable or completely transparant BCs, shown in Figures 3 and 4 for the benchmark problem. 90 80
Engineering stress [MPa]
A multi-scale methodology for the analysis of metallic interfaces
Bicrystal: impenetrable GB
70 60
Single crystal Bicrystal: penetrable GB (interface model)
50 40 30 20 10
Applications
0 0
0.05
0.1
0.15
0.2
0.25
0.3
Engineering strain [%]
0.35
0.4
0.45
0.5
Figure 3: Stress strain response of a bicrystal in tension having an impenetrable (blue) or transparant (red) grain boundary (GB). 11
4 3
To date, very few models can account for the explicit presence of an interface. At the continuum scale, interfaces are sometimes accounted for, implicitly, in the high-order boundary conditions (BC) used in gradient enhanced or strain gradient crystal plasticity frameworks [1, 2]. As a benchmark problem a bicrystal in plane strain is taken, shown in Figure 2. L0 A B 2
U
Bicrystal: impenetrable GB
GB
1 0
Single crystal
−1 Bicrystal: impenetrable GB −2 −3
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
Dimensionless horizontal position [-]
0.3
0.4
0.5
Figure 4: Pile-up of geometrically necessary dislocations (GND) at the boundaries for a bicrystal in tension having an impenetrable (blue) or transparant (red) grain boundary (GB).
Future work The main reason for this simplification is the lack of knowledge of the mechanisms occurring at the underlying scale. The goals at the continuum scale are (1) to develop constitutive equations for plasticity through interfaces, including high-order BCs and (2) to define a computational homogenization scheme that may be used as a reference to bridge discrete dislocation to continuum simulations.
U
References
1 m1 s
f
e2 e1
Figure 2: Benchmark problem in tension. Grains A and B in the bicrystal each have two slip systems. These slip systems are defined by an angle φ with respect to the e1 direction having a slip plane normal m and a slip direction s. Periodic BCs are applied on top and bottom.
/
2
−4 −0.5
Problem description at continuum scale
s m2
GND density [m−2 ]
Figure 1: Scales and scale transitions up to the application level.
A better insight and quantification of these mechanisms at interfaces with different characteristics can only be gained through detailed analysis of these processes across the scales: from the atomistic via the level of discrete dislocations to the continuum interface level, see Figure 1.
x 10
department of mechanical engineering
[1] L.P. Evers, Strain gradient crystal plasticity based on dislocation densities, PhD Thesis, 2004. [2] C.J. Bayley, W.A.M. Brekelmans and M.G.D. Geers, A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity, IJSS 43, 7268–7286, 2006.
