Instabilities in power law gradient hardening materials
Introduction
For thin metal plates or free standing thin films that play an important role in applications ranging from microelectro-mechanical systems to coatings the mechanical properties are not well described by conventional plasticity theory. The material models must account for observed size-effects (Fleck et al., 1994; Ma and Clarke, 1995; Fleck and Hutchinson, 1997; Stölken and Evans, 1998; Begley and Hutchinson, 1998; Haque and Saif, 2003), and a variety of such material models have been developed by incorporating gradient effects in the constitutive and equilibrium equations.
Instabilities under tensile or compressive loading are important limitations of the load-carrying capacity for thin plates or films. For the plane strain tension test (Hill and Hutchinson, 1975), or the plane strain compression test (Young, 1976), the critical stresses for bifurcation into diffuse modes have been determined, dependent on the ratio of the wave-length to the plate thickness, and the regimes have been identified, where the governing equations are elliptic, parabolic or hyperbolic. Size-effects have been incorporated into these bifurcation analyses by Benallal and Tvergaard (1995), who used a finite strain version of the gradient plasticity model proposed by Mühlhaus and Aifantis (1991). In addition to the two material moduli μ and μ∗ appearing in the previous bifurcation analyses, this gradient theory of plasticity also incorporates a characteristic material length scale l. It has been found that for very small values of the material length the value of the lowest bifurcation stress is practically unaffected by nonlocal effects, but when the wave-length decays towards the material length, the bifurcation is more and more delayed by the nonlocal effects.
The nonlocal plasticity model used in the present paper is a finite strain generalization recently developed by Niordson and Redanz (2004) for the strain gradient plasticity theory by Fleck and Hutchinson (2001). For plane strain specimens in tension or compression the focus here is on determining how much the onset of instability is delayed, or the load-carrying capacity increased, when the specimen size is as small as the characteristic material length, or even smaller. In these analyses the presence of initial geometrical imperfections is accounted for. The bifurcation analyses mentioned above have shown that both for tension and compression instabilities occur in symmetric modes as well as anti-symmetric modes. However, for longer wave-lengths symmetric modes give the lowest bifurcation stress in tension, leading to necking, and anti-symmetric modes give the first bifurcation in compression, leading to column buckling. Therefore, these will be the only cases considered in the present studies.
Necking in plane strain tension was also analyzed by Niordson and Redanz (2004) for a linear hardening solid. Their numerical implementation of the gradient plasticity theory did not give good convergence for power law hardening materials. This is resolved here by using higher order elements in the numerical implementation, and the results shown are for power law hardening.
Section snippets
Material model
The material behavior is modeled by a finite strain generalization proposed by Niordson and Redanz (2004) for the strain gradient plasticity theory by Fleck and Hutchinson (2001). An updated Lagrangian formulation is used to model the strain gradient effects at finite strains based on the work of McMeeking and Rice (1975) and Yamada and Sasaki (1995).
In the strain gradient plasticity theory by Fleck and Hutchinson (2001) gradient hardening is introduced through the gradient of the plastic
Numerical method
The numerical solutions are obtained using a finite element method where nodal effective plastic strain increments, , appear directly as unknowns on equal footing with the nodal displacement increments, . The structure of the finite element method as it is used in the present study has been used by Vardoulakis and Aifantis (1991); de Borst and Mühlhaus (1992), and de Borst and Pamin (1996), to model the gradient theory by Aifantis (1984). Niordson and Hutchinson (2003) have used it to
Results
The plane strain tension test and the plane strain compression test are here analyzed for specimens with dimensions in the micron range. Fig. 1a shows the geometry for the plane strain tension test. The specimen has the width 2a0 and the length 2b0, and an initial symmetric cosine imperfection of initial amplitude δ0. Along the horizontal edges displacements are prescribed to enforce tension in the vertical direction, while shear stresses vanish. The vertical edges are stress free. After some
Concluding remarks
Relative to the study of plane strain sheet-necking by Niordson and Redanz (2004) the only improvement of the numerical procedure here lies in the use of higher order elements. The procedure based on triangular elements worked very well for a linear hardening material, but did not show satisfactory converge for a power hardening material, even in the absence of a material length. This surprising problem, which is introduced due to the two-field finite element interpolation, is removed by using
Acknowledgment
This work is financially supported by the Danish Technical Research Council in a project entitled Modeling Plasticity at the Micron Scale.
References (22)
- et al.
Mechanics of size-dependent indentation
Journal of the Mechanics and Physics of Solids
(1998) - et al.
Nonlocal continuum effects on bifurcation in the plane strain tension–compression test
Journal of the Mechanics and Physics of Solids
(1995) - et al.
Strain gradient plasticity
- et al.
A reformulation of strain gradient plasticity
Journal of the Mechanics and Physics of Solids
(2001) - et al.
Strain gradient plasticity: theory and experiment
Acta Metallurgica et Materialia
(1994) - et al.
Strain gradient effect in nanoscale thin films
Acta Materialia
(2003) - et al.
Bifurcation phenomena in the plane tension test
Journal of the Mechanics and Physics of Solids
(1975) Post-bifurcation behavior in the plastic range
Journal of the Mechanics and Physics of Solids
(1973)- et al.
Finite-element formulations for problems of large elastic–plastic deformation
International Journal of Solids and Structures
(1975) - et al.
A variational principle for gradient plasticity
International Journal of Solids and Structures
(1991)