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].

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

Couple stress based strain gradient theory for elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, bh. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, bh, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.

Experiments and theory in strain gradient elasticity

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.

http://science.sciencemag.org/content/sci/305/5686/986.full.pdf

Sample dimensions influence strength and crystal plasticity.

Strain gradient plasticity

The hardness of thick, high-purity, epitaxially grown silver on sodium chloride is found to be dependent on the size of the indentation for sizes below ≃10 μm. The measurement of the size effect has been made in two ways. In one, the hardness has been calculated from the load-displacement curve obtained from an instrumented microhardness testing machine and assuming a geometric self-similarity in the indenter shape. In the other measurement, the hardness was obtained from the load exerted by the microhardness tester divided by the indentation impression area as measured by atomic force microscopy. The observed variation in microhardness with indentation size is consistent with a simplified strain gradient plasticity model in which the densities of the geometrically necessary and statistically stored dislocations are fitting parameters. An equally good fit can also be made with a simple geometric scaling relationship. Transmission electron microscopy observations of a thin (≃50 nm) epitaxial gold film embedded in the silver layers revealed that the deformation was primarily restricted to the sharp edges of the indentation. In addition, deformation twinning within the indentation impression was observed on the {1H} planes.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0884291400077517

Size dependent hardness of silver single crystals

Adhesion and debonding of multi-layer thin film structures

A general methodology is developed for determining the state of stress and the numerical value of the stresses from observed shifts and broadening of optical fluorescence lines. The method is based on the piezospectroscopic properties of single crystals. We present general relationships between the measured fluorescence shifts and the stress state for a number of illustrative cases, pertinent to both polycrystalline and single-crystal ceramics under stress. These include measuring the stresses applied to polycrystalline ceramics, the residual stress distribution due to crystallographically anisotropic thermal expansion, and the stresses applied to single crystals. Using the recently implemented technique of performing the fluorescence measurements in an optical microprobe, we also provide experimental tests of the relationships derived.

Stress Measurement in Single-Crystal and Polycrystalline Ceramics Using Their Optical Fluorescence

A chip-on-flex package which includes at least one moisture barrier layer to prevent metal corrosion and delamination of flex component layers. An exemplary microelectronic package includes a microelectronic die having an active surface and at least one side, wherein the microelectronic die active surface includes at least one contact. A flex component is attached by a first surface to the microelectronic die active surface. At least one conductive trace is disposed on a second surface of the flex component and extends through the flex component to contact at least one of the contacts. An encapsulation material is adjacent the microelectronic die side and a bottom surface of the flex component. A moisture barrier is disposed on the flex component and the conductive trace(s). A second moisture barrier may be disposed on the encapsulation material. A heat dissipation device may also be incorporated into the chip-on-flex package.

Structures and processes for fabricating moisture resistant chip-on-flex packages

The residual stresses created in polycrystalline aluminum oxide as a result of its constrained anisotropic thermal contraction are measured with the technique of piezospectroscopy using the fluorescence from trace Cr[sup 3+] impurities. The average residual stresses in the crystallographic a and c directions are determined as a function of grain size for a high-purity alumina, as is the width of the stress distribution (assuming it to be Gaussian). Over the range of grain sizes investigated, from 2 to 16 [mu]m, the residual stresses exhibit a dependence on grain size consistent with the prediction of the Evans-Clarke model of thermal stress relaxation by grain boundary diffusion.

Qing Ma

Papers

Size dependent hardness of silver single crystals

Adhesion and debonding of multi-layer thin film structures

Stress Measurement in Single-Crystal and Polycrystalline Ceramics Using Their Optical Fluorescence

Structures and processes for fabricating moisture resistant chip-on-flex packages

Piezospectroscopic Determination of Residual Stresses in Polycrystalline Alumina