A small-deformation strain-gradient theory for isotropic viscoplastic materials

Abstract

This study develops a thermodynamically-consistent small-deformation theory of strain-gradient viscoplasticity for isotropic materials based on: (i) a scalar and a vector microstress consistent with a microforce balance; (ii) a mechanical version of the two laws of thermodynamics for isothermal conditions, that includes via the microstresses the work performed during viscoplastic flow; and (iii) a constitutive theory that allows:

•	the free energy to depend on ∇γ p , the gradient of equivalent plastic strain γ p , and this leads to the vector microstress having an energetic component;
•	strain-hardening dependent on the equivalent plastic strain γ p , and a scalar measure π p related to the accumulation of geometrically necessary dislocations; and
•	a dissipative part of the vector microstress to depend on , the gradient of the equivalent plastic strain rate.

The microscopic force balance, when augmented by constitutive relations for the microscopic stresses, results in a non-local flow rule in the form of a second-order partial differential equation for the equivalent plastic strain γ ^p . The flow rule, being non-local, requires microscopic boundary conditions. The theory is numerically implemented by writing a user-element for a commercial finite element program. Using this numerical capability, the major characteristics of the theory are revealed by studying the standard problem of simple shear of a constrained plate. Additional boundary-value problems representing idealized two-dimensional models of grain-size strengthening and dispersion strengthening of metallic materials are also studied.

Keywords:

• strain-gradient plasticity
• viscoplasticity
• Constitutive theory
• finite elements

Acknowledgements

This study owes much to Morton E. Gurtin. In particular, LA thanks Gurtin for his constructive comments on a previous version of this paper, and his specific suggestions concerning the definition of π ^p (cf. Section 2.2) based on the Burgers tensor G, and the general constitutive Equations (5.12) for the microstresses π and ξ _dis. This work was supported by the National Science Foundation CMS-0555614.

Notes

Notes

1. We mostly use direct notation, but whenever convenient we also use standard indicial notation with summation convention. Specifically: ∇ and div denote the spatial gradient and divergence; a superposed dot denotes the material time-derivative. We write tr A, sym A, skw A, A ₀, and sym₀ A, respectively, for the trace, symmetric, skew, deviatoric, and symmetric-deviatoric parts of a second-order tensor A. Also, the inner product of tensors A and B is denoted by A : B, and the magnitude of A by .The divergence and curl of a tensor field A are defined by We also make brief use of third-order tensors. The inner product of third-order tensors 𝕂 and 𝔸 is defined in the natural manner, 𝕂 ⋮ 𝔸 = K _ijk A _ijk . The gradient of a second-order tensor A is the third-order tensor (∇A) _ijk = A _{ij, k} .

2. For ease of presentation of the theory, we do not use here constant numerical factors of and , as commonly used in the definitions of the equivalent plastic shear strain rate and the equivalent shear stress, respectively.

3. Equation (15) of Fleck and Hutchinson 18, but written in our notation. See Section 3 for a derivation of this result from the principle of virtual power.

4. Also see, Fredriksson and Gudmundson 25.

5. The transpose of G is often referred to as Nye's tensor, although Nye's result (1953) involves elastic rotations, neglecting elastic strains.

6. Note that π ^p ≠ ∇γ ^p .

7. We emphasize that Nix, Gao, Huang and coworkers use a quantity η ^p instead of our constitutive variable π ^p to define the GND density. Their constitutive variable η ^p is not based on the Burgers tensor G; cf. e.g. Equation (2.11) in the recent paper by Zhang et al. 24.

8. A variant of the finite element procedure that was used for the one-dimensional gradient theory of Anand et al. (2005), was used here for implementing a two-dimensional version of the present theory as a user element (UEL) subroutine in ABAQUS. Briefly, for the present case, the element residual are,

while the Jacobian matrices are

The implicit time-integration scheme built-in ABAQUS/Standard was used. Since the coupled-problem is very non-linear and stiff, convergence of this implicit scheme is extremely difficult. Standard solution control parameters in ABAQUS had to be substantially modified, and several iterations were required in some time-steps.

9. The elastic shear and Lamé moduli are calculated using standard relations from linear elasticity. Also, MPa, and s⁻¹, where 0.02 s⁻¹ is the imposed shear strain rate in simple shear problem.

10. Indeed, there is no need for the values of ℓ₁/h and ℓ₂/h to be of similar magnitudes, because ℓ₁ and ℓ₂ enter the theory multiplying different scaling parameters. Additionally, it is important to note that in contrast to the theory presented here, the theories of Nix, Gao, Huang et al. 21–24 are ‘lower-order’ strain-gradient theories, in that the strain-gradient effects are accounted for by modifying the strain-hardening rate of a conventional non-gradient theory, and the flow rule in such theories does not involve a partial differential equation, with attendant boundary conditions. In our theory, the physical ideas of the ‘mechanism-based’ theory of Nix, Gao, Huang and coworkers are included as a part of our complete ‘higher-order’ theory, and hence do not suffer from non-uniqueness problems due to lack of boundary conditions; cf. e.g. 30,31.

11. We have intentionally chosen not to perform the numerical simulation by imposing constraints on the edges of a single-grain to produce simple shear. Displacement boundary conditions on the top and bottom edges together with periodic displacement-boundary conditions on the side-edges were applied on a nine-grain aggregate to model simple shear, and not on the boundaries of every grain. Hence, not all grains in the aggregate have the same boundary conditions, and this gives some grain-to-grain variation of the relevant fields. This was done in order to study a case slightly more complicated than a single grain subjected to simple shear.

12. The microhard boundary condition was not applied in baseline case with ℓ₃/a = 0.

13. Our theory, like other recent gradient theories (cf. Evers et al. 32; Borg 33), does not predict the classical Hall-Petch square-root dependence of the yield strength on the grain-size.

14. Note that distribution of γ ^p is not the same in all the grains because the ‘simple-shear’ constraints were only applied to the outer edges of the aggregate of nine-grains, and not to each grain individually.

15. For simplicity, we focus on only one value of particle radius and one volume-fraction of the dispersion.

16. Micro-hard boundary conditions at the particle/matrix interface were not applied in the baseline case.