CHRISTIAN F. NIORDSON∗, JEFFREY W. KYSAR†
∗Department of Mechanical Engineering, Solid Mechanics,Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark e-mail: cn@mek.dtu.dk, web page: http://www.mek.dtu.dk/
†Department of Mechanical Engineering, Columbia University, New York, NY 10027, USA e-mail:
jk2079@columbia.edu- Web page: http://www.engineering.columbia.edu/
Key words: Size-effects, Strain Gradient Crystal Plasticity, Finite Element
Summary. In the present work the strain gradient formulation for isotropic plasticity, proposed by Fleck and Willis, is extended to crystal visco-plasticity. Size-effects are predicted by the theory due to the addition of gradient terms in both the free energy as well as through a dissipation potential. A robust numerical method applicable for crystal plasticity is presented.
Some plane deformation problems relevant for certain specific orientations of a face centered cubic crystal under plane loading conditions are studied. The problem of pure shear of a single crystal between rigid platens is studied, and convergence properties of the numerical method proposed are discussed.
1 INTRODUCTION
In metals strain gradient effects lead to significant strengthening on the micron scale. Experi-mental investigations on size-effects in metals have been carried out for a variety of materials and under different loading conditions such as bending1,2, torsion3, indentation and contact com-pression4, as well as the characterization of the density of geometrically necessary dislocations5. While some experiments suggest that the yield strength increases with decreasing size3,6, other experiments show that the size-effect mainly affects the material hardening behavior7. Some experiments even show size-effects on both yield strength and hardening behavior2.
Much research has been devoted to modeling observed size-effects. This includes modeling of the above mentioned experiments in addition to studies of size-effects in void growth, fiber reinforced materials and fracture problems.
In this paper variational principles for a strain gradient crystal plasticity theory are laid out along the lines developed for isotropic plasticity by Fleck and Willis8,9. Furthermore, the details of a robust finite element formulation of the strain gradient crystal plastic theory is presented.
The theory is closely related to that proposed by Gurtin and co-workers10 even though the numerical method is quite different.
Christian F. Niordson, Jeffrey W. Kysar
2 MATERIAL MODEL
Within a small strain framework an additive decomposition of the total strain, ǫij, into and elastic part, ǫeij, and a plastic part, ǫpij, is used
ǫij = ǫeij+ ǫpij (1)
The plastic strain rate is due to crystallographic slip on the slip planes α
˙ǫpij =X
(α)
˙γ(α)µ(α)ij (2)
with the Schmid orientation tensor given by µ(α)ij = 1
2(s(α)i m(α)j + s(α)j m(α)i ) (3) where s(α)i and m(α)j are the direction of slip and the slip plane normal, respectively.
The material model is based on the following form of the principle of virtual work Z
V
σijδǫij+X
α
(q(α)− τ(α))δγ(α)+X
α
ξ(α)s(α)i δγ,i(α)
! dV =
Z
S
Tiδui+X
α
r(α)δγ(α)
! dS
(4) Here, σij is the stress tensor, ǫij is the strain tensor, q(α) is the micro-stress on slip plane α and τ(α) = σijµ(α)ij is the Schmid stress. The higher order nature of the theory is due to the terms ξ(α), which are the higher order stresses work conjugate to the slip gradients, γ,i(α), where ( ),i signifies the gradient operator. With ni denoting the outward unit normal, the right hand side of the principle of virtual work includes the conventional traction vector Ti = σijnj, work conjugate to the displacement vector ui, and the higher order tractions r(α)= ξ(α)s(α)i n(α)i , which are work conjugates to the slips, γ(α).
Accounting for both dissipative and energetic gradient effects, the higher order stresses are decomposed into a dissipative part, ¯ξ(α) and an energetic part, ˜ξ(α)
ξ(α)= ¯ξ(α)+ ˜ξ(α) (5)
whereas the micro-stresses are assumed to have a dissipative part, ¯q(α), only.
2.1 Dissipative contributions
To account for dissipative gradient effects, a visco-plastic potential is used from which the following constitutive equations for the dissipative stress-quantities are derived
¯
q(α) = τe(α)˙γ(α)
˙γe(α)
and ξ¯(α) = τe(α)(L(α)d )2˙γ,i(α)s(α)i
˙γe(α)
(6)
Here, τe(α) is an effective stress, ˙γe(α) is an effective slip rate, and L(α)d are dissipative length parameters.
Christian F. Niordson, Jeffrey W. Kysar
2.2 Energetic contributions
Assuming that free energy, Ψ, is stored due to a decoupled quadratic form in the elastic strain and the gradients of slip, the energetic higher order stresses are derived according to
ξ˜(α)= ∂Ψ
∂(γ,i(α)s(α)i ) = G L(α)e 2
s(α)i γ,i(α) (7)
where G is the shear modulus and L(α)e are energetic length parameters.
2.3 Minimum principles used for generating solutions
To solve for the slip rate fields the following minimum principle is used within a finite element setting:
H[ ˙γ(α)] = Z
V
Φ[ ˙γ(α)∗] ˙γ(α)∗+ ˜ξ(α)si˙γ,i(α)∗− sijµ(α)ij ˙γ(α)∗ dV −
Z
ST
r(α)˙γ(α)∗dS (8) Here, Φ is a visco-plastic potential, and sij is the stress deviator
When knowing the solution for the slip rate fields, nodal velocities are found from minimizing J[ ˙u∗i] =
Z
V
1
2Lijkl ˙ǫ∗ij− X
α
µ(α)ij ˙γ(α)
!
˙ǫ∗kl− X
α
µ(α)kl ˙γ(α)
! dV −
Z
ST
T˙i˙u∗idS (9)
This two step solution method is a suitable modification for crystal plasticity of corresponding minimum principles for isotropic plasticity laid out by Fleck and Willis8,9.
3 RESULTS AND CONCLUSIONS
Results are obtained using a finite element discretization. An iterative algorithm is used to minimize the functional H[ ˙γ(α)] in order to obtain the slip rate fields. Then solution of a standard elastic-visco plastic finite element system yields the nodal velocities as a result of minimizing J[ ˙u∗i].
In Fig. 1a response curves in terms of shear stress versus shear strain are shown for pure shear of a long film of elastic-plastic material of thickness H between rigid platens. It is seen how increasing the dissipative length parameter relative to the thickness of the film increases the yield strength of the material system over that of conventional predictions (conv.). In Fig.
1b the number of iterations used to obtain converged solutions when minimizing H[ ˙γ(α)] are shown. It is seen that a large number of iterations are used in the transition from elastic to plastic behavior, and a small number of iterations are used otherwise.
REFERENCES
[1] St¨olken, J. S. & Evans, A. G. A microbend test method for measuring the plasticity length scale. Acta Materialia 46, 5109–5115 (1998).
[2] Haque, M. A. & Saif, M. T. A. Strain gradient effect in nanoscale thin films. Acta Materialia 51, 3053–3061 (2003).
Christian F. Niordson, Jeffrey W. Kysar
Ld/H = 1.00 Ld/H = 0.50conv.
increment no.
no.ofiterations
1500 1000
500 0
200 160 120 80 40 0 γ/γ0
τ/τ0
8 6 4 2 0 -2 -4 -6 -8 2
1
0 -1
-2
(a) (b)
Figure 1: (a) Size-dependent response curves in pure shear for different values of the dissipative length parameter normalized by the film thickness. (b) Number of iterations used to minimize H(α) per load increment.
[3] Fleck, N. A., Muller, G. M., Ashby, M. F. & Hutchinson, J. W. Strain gradient plasticity:
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[8] Fleck, N. A. & Willis, J. R. A mathematical basis for strain-gradient plasticity theory - Part I: Scalar plastic multiplier. Journal of the Mechanics and Physics of Solids 57, 161–177 (2009).
[9] Fleck, N. A. & Willis, J. R. A mathematical basis for straingradient plasticity theory -Part II: Tensorial plastic multiplier. Journal of the Mechanics and Physics of Solids 57, 1045–1057 (2009).
[10] Gurtin, M. E., Anand, L. & Lele, S. P. Gradient single-crystal plasticity with free energy dependent on dislocation densities. Journal of the Mechanics and Physics of Solids 55, 1853–1878 (2007).
23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds)
KTH, Stockholm, 2010c