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Journal ArticleDOI

Indentation size effects in crystalline materials: A law for strain gradient plasticity

01 Mar 1998-Journal of The Mechanics and Physics of Solids (JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS)-Vol. 46, Iss: 3, pp 411-425
TL;DR: In this article, the indentation size effect for crystalline materials can be accurately modeled using the concept of geometrically necessary dislocations, which leads to the following characteristic form for the depth dependence of the hardness: H H 0 1+ h ∗ h where H is the hardness for a given depth of indentation, h, H 0 is a characteristic length that depends on the shape of the indenter, the shear modulus and H 0.
Abstract: We show that the indentation size effect for crystalline materials can be accurately modeled using the concept of geometrically necessary dislocations. The model leads to the following characteristic form for the depth dependence of the hardness: H H 0 1+ h ∗ h where H is the hardness for a given depth of indentation, h, H0 is the hardness in the limit of infinite depth and h ∗ is a characteristic length that depends on the shape of the indenter, the shear modulus and H0. Indentation experiments on annealed (111) copper single crystals and cold worked polycrystalline copper show that this relation is well-obeyed. We also show that this relation describes the indentation size effect observed for single crystals of silver. We use this model to derive the following law for strain gradient plasticity: ( σ σ 0 ) 2 = 1 + l χ , where σ is the effective flow stress in the presence of a gradient, σ0 is the flow stress in the absence of a gradient, χ is the effective strain gradient and l a characteristic material length scale, which is, in turn, related to the flow stress of the material in the absence of a strain gradient, l ≈ b( μ σ 0 ) 2 . For materials characterized by the power law σ 0 = σ ref e 1 n , the above law can be recast in a form with a strain-independent material length scale l. ( builtσ σ ref ) 2 = e 2 n + l χ l = b( μ σ ref ) 2 = l ( σ 0 σ ref ) 2 . This law resembles the phenomenological law developed by Fleck and Hutchinson, with their phenomenological length scale interpreted in terms of measurable material parametersbl].
Citations
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Journal ArticleDOI
13 Aug 2004-Science
TL;DR: Measurements of plastic yielding for single crystals of micrometer-sized dimensions for three different types of metals find that within the tests, the overall sample dimensions artificially limit the length scales available for plastic processes.
Abstract: When a crystal deforms plastically, phenomena such as dislocation storage, multiplication, motion, pinning, and nucleation occur over the submicron-to-nanometer scale. Here we report measurements of plastic yielding for single crystals of micrometer-sized dimensions for three different types of metals. We find that within the tests, the overall sample dimensions artificially limit the length scales available for plastic processes. The results show dramatic size effects at surprisingly large sample dimensions. These results emphasize that at the micrometer scale, one must define both the external geometry and internal structure to characterize the strength of a material.

2,113 citations

Journal ArticleDOI
TL;DR: In this paper, a mechanism-based theory of strain gradient plasticity is proposed based on a multiscale framework linking the microscale notion of statistically stored and geometrically necessary dislocations to the mesoscale notion of plastic strain and strain gradient.
Abstract: A mechanism-based theory of strain gradient plasticity (MSG) is proposed based on a multiscale framework linking the microscale notion of statistically stored and geometrically necessary dislocations to the mesoscale notion of plastic strain and strain gradient. This theory is motivated by our recent analysis of indentation experiments which strongly suggest a linear dependence of the square of plastic flow stress on strain gradient. While such linear dependence is predicted by the Taylor hardening model relating the flow stress to dislocation density, existing theories of strain gradient plasticity have failed to explain such behavior. We believe that a mesoscale theory of plasticity should not only be based on stress–strain behavior obtained from macroscopic mechanical tests, but should also draw information from micromechanical, gradient-dominant tests such as micro-indentation or nano-indentation. According to this viewpoint, we explore an alternative formulation of strain gradient plasticity in which the Taylor model is adopted as a founding principle. We distinguish the microscale at which dislocation interaction is considered from the mesoscale at which the plasticity theory is formulated. On the microscale, we assume that higher order stresses do not exist, that the square of flow stress increases linearly with the density of geometrically necessary dislocations, strictly following the Taylor model, and that the plastic flow retains the associative structure of conventional plasticity. On the mesoscale, the constitutive equations are constructed by averaging microscale plasticity laws over a representative cell. An expression for the effective strain gradient is obtained by considering models of geometrically necessary dislocations associated with bending, torsion and 2-D axisymmetric void growth. The new theory differs from all existing phenomenological theories in its mechanism-based guiding principles, although the mathematical structure is quite similar to the theory proposed by Fleck and Hutchinson. A detailed analysis of the new theory is presented in Part II of this paper.

1,679 citations

Journal ArticleDOI
TL;DR: In this paper, a review of continuum-based variational formulations for describing the elastic-plastic deformation of anisotropic heterogeneous crystalline matter is presented and compared with experiments.

1,573 citations


Cites background from "Indentation size effects in crystal..."

  • ...Size effects can be introduced into CPFE frameworks by using phenomenological strain gradient theories which were developed by Fleck et al. [229], Fleck and Hutchinson [230], and Nix and Gao [231]....

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  • ...[229], Fleck and Hutchinson [230], and Nix and Gao [231]....

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Journal ArticleDOI
TL;DR: In this paper, the effects of the substrate on the determination of mechanical properties of thin films by nanoindentation were examined, and the properties of aluminum and tungsten films on the following substrates: aluminum, glass, silicon and sapphire.

1,410 citations

Journal ArticleDOI
TL;DR: In this article, the authors used uniaxial compression experiments on Au cylinders at the sub-micron scale, without stress/strain gradients, and determined compression stress, strain, and stiffness of the pillars.

1,387 citations

References
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01 Jan 1997

4,469 citations

Journal ArticleDOI
M.F. Ashby1
TL;DR: The geometrically necessary dislocations as discussed by the authors were introduced to distinguish them from the statistically storages in pure crystals during straining and are responsible for the normal 3-stage hardening.
Abstract: Many two-phase alloys work-harden much faster than do pure single crystals. This is because the two phases are not equally easy to deform. One component (often dispersed as small particles) deforms less than the other, or not at all, so that gradients of deformation form with a wavelength equal to the spacing between the phases or particles. Such alloys are ‘plastically non-homogeneous’, because gradients of plastic deformation are imposed by the microstructure. Dislocations are stored in them to accommodate the deformation gradients, and so allow compatible deformation of the two phases. We call these ‘geometrically-necessary’ dislocations to distinguish them from the ‘statistically-stored’ dislocations which accumulate in pure crystals during straining and are responsible for the normal 3-stage hardening. Polycrystals of pure metals are also plastically non-homogeneous. The density and arrangement of the geometrically-necessary dislocations can be calculated fairly exactly and checked by electr...

3,527 citations

Journal ArticleDOI
TL;DR: In this paper, a deformation theory of plasticity is introduced to represent in a phenomenological manner the relative roles of strain hardening and strain gradient hardening, which is a non-linear generalization of Cosserat couple stress theory.
Abstract: Dislocation theory is used to invoke a strain gradient theory of rate independent plasticity. Hardening is assumed to result from the accumulation of both randomly stored and geometrically necessary dislocation. The density of the geometrically necessary dislocations scales with the gradient of plastic strain. A deformation theory of plasticity is introduced to represent in a phenomenological manner the relative roles of strain hardening and strain gradient hardening. The theory is a non-linear generalization of Cosserat couple stress theory. Tension and torsion experiments on thin copper wires confirm the presence of strain gradient hardening. The experiments are interpreted in the light of the new theory.

3,266 citations

Journal ArticleDOI
TL;DR: In this paper, a strain gradient theory of plasticity is introduced, based on the notion of statistically stored and geometrically necessary dislocations, which fits within the general framework of couple stress theory and involves a single material length scale l.
Abstract: A Strain Gradient Theory of plasticity is introduced, based on the notion of statistically stored and geometrically necessary dislocations. The strain gradient theory fits within the general framework of couple stress theory and involves a single material length scale l. Minimum principles are developed for both deformation and flow theory versions of the theory which in the limit of vanishing l, reduce to their conventional counterparts: J2 deformation and J2 flow theory. The strain gradient theory is used to calculate the size effect associated with macroscopic strengthening due to a dilute concentration of bonded rigid particles; similarly, predictions are given for the effect of void size upon the macroscopibic softening due to a dilute concentration of voids. Constitutive potentials are derived for this purpose.

1,300 citations


"Indentation size effects in crystal..." refers background or methods in this paper

  • ... To account for these strain gradient effects, Fleck and Hutchinson (1993) have...

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  • ...where xi, is the so-called curvature tensor ( Fleck and Hutchinson, 1993 ) which is related to the third order strain gradient tensor by...

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  • ...where 1 is a phenomenological length scale, p is an exponent usually taken to be 2 ( Fleck and Hutchinson, 1993 ), but can be generalized to an arbitrary number larger than 1 without destroying the basic theoretical framework (Fleck and Hutchinson, 1997)....

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  • ...As discussed by Fleck and Hutchinson (1993, 1997) , conventional theories of plasticity...

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  • ...In order to compare this form of strain gradient plasticity with that developed by Fleck and Hutchinson (1993, 1997) , we consider the usual power hardening law in the absence of a strain gradient...

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