Strain gradient plasticity

The physical and mechanical properties that can be obtained with metal matrix composites (MMCs) have made them attractive candidate materials for aerospace, automotive and numerous other applications. More recently, particulate reinforced MMCs have attracted considerable attention as a result of their relatively low costs and characteristic isotropic properties. Reinforcement materials include carbides, nitrides and oxides. In an effort to optimize the structure and properties of particulate reinforced MMCs various processing techniques have evolved over the last 20 years. The processing methods utilized to manufacture particulate reinforced MMCs can be grouped depending on the temperature of the metallic matrix during processing. Accordingly, the processes can be classified into three categories: (a) liquid phase processes, (b) solid state processes, and (c) two phase (solid-liquid) processes. Regarding physical properties, strengthening in metal matrix composites has been related to dislocations of a very high density in the matrix originating from differential thermal contraction, geometrical constraints and plastic deformation during processing.

Particulate reinforced metal matrix composites — a review

Metal Matrix Composites

Dislocation generation due to differences between the coefficients of thermal expansion

Liquid phase sintering (LPS) is a process for forming high performance, multiple-phase components from powders. It involves sintering under conditions where solid grains coexist with a wetting liquid. Many variants of LPS are applied to a wide range of engineering materials. Example applications for this technology are found in automobile engine connecting rods and high-speed metal cutting inserts. Scientific advances in understanding LPS began in the 1950s. The resulting quantitative process models are now embedded in computer simulations to enable predictions of the sintered component dimensions, microstructure, and properties. However, there are remaining areas in need of research attention. This LPS review, based on over 2,500 publications, outlines what happens when mixed powders are heated to the LPS temperature, with a focus on the densification and microstructure evolution events.

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Review: liquid phase sintering

Assuming ideal plastic behavior for an isotropic matrix containing a misfitting spherical precipitate, the total amount of work expended during elasto-plastic deformation is calculated and compared with the total strain energy in the corresponding pure elastic state. For precipitates larger than one micron (µm), the effective yield stress is taken as the macroscopic yield stress while for smaller precipitates, size-dependent yield stresses are obtained from the Ashby-Johnson model. In the case of coherent submicron precipitates, the effective yield stress becomes the theoretical yield strength and thus plastic relaxation is not possible unless the transformation stress is extremely large. For incoherent submicron precipitates, the effective yield stress is approximately inversely proportional to the precipitate radius,r. Hence plastic relaxation again is not possible whenr < 10 nm, but whenr ≃100 nm the strain energy can decrease by 10 ∼ 40 pct at a misfit of 3 pct. For supra-micron particles, however, the ratio of the effective yield stress to the shear modulus becomes 10−3 or less, and plastic relaxation can reduce the strain energy by factors of 3 to 15 at misfits of 1 to 3 pct. Under this circumstance, the plastic zone becomes wide, its radius ranging from 3 to 5r.

Plastic relaxation of the transformation strain energy of a misfitting spherical precipitate: Ideal plastic behavior

The dislocation generation due to a thermoelastic stress in 2024 Al/ceramic (SiC or TiC) composites was studied using transmission electron microscopy Composites containing different ceramic particulates, ceramic volume fraction, and particle size were investigated. Dislocation density profiles were measured as a function of the distance from an Al/ceramic interface and compared with those calculated from an elastoplasticity model which accounts for the volume fraction of the ceramic particles. The intensity of dislocation generation showed a strong particle size dependence: as the ceramic particle size became of the order of a micron, the intensity of dislocation generation increased significantly. With an increase in the volume fraction of the ceramic particles, the dislocation density also increased, and the dislocation structure became a more tangled arrangement. If heat dissipation was taken into account as part of the plastic work, the predicted dislocation densities of the elastoplasticity model were found to be in reasonable agreement with the measured dislocation densities of 109 to 1010 cm−2.

Plastic relaxation of thermoelastic stress in aluminum/ceramic composites

The strain energy of a coherent ellipsoidal precipitate

When alloys containing two ductile phases are heavily deformed, composite-like microstructures develop and strengths well in excess of either of the phases in single phase form may be exhibited as a result of microstructure/dislocation density effects. In this paper a previously-published model for such strengthening is reviewed, and its application in a predictive capacity discussed. Flow stressvs fabrication strain data for the two components in single phase form and for one two-phase alloy are necessary for this purpose. The model may then be applied to predict strength for any other two-phase alloy as a function of composition, fabrication strain, and interphase spacing. The approach is illustrated using existing data for several alloy systems. For Ag-Fe and Cu-Nb alloys (with very limited mutual solubility) strengths can be predicted within 15 to 20 pct of the experimental values over the entire range of strains and volume fractions for which data are available. In systems where the potential for precipitation hardening exists (e.g., Cu-Fe) thermal history is important. When such hardening becomes a significant factor, the model cannot be used in its present form due to uncertainty over how to “add” the strengthening from this effect. Such hardening may, however, be useful in further improving the properties of these materials.

J. K. Lee

Papers

Plastic relaxation of the transformation strain energy of a misfitting spherical precipitate: Ideal plastic behavior

Plastic relaxation of thermoelastic stress in aluminum/ceramic composites

The strain energy of a coherent ellipsoidal precipitate

Ductile two-phase alloys: Prediction of strengthening at high strains

Elastic strain energy and interactions of thin square plates which have undergone a simple shear