The critical dimension for a 4th order problem with singular nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation \bi u=\f{\lambda}{(1-u)^2}, which models a simple Micro-Electromechanical System (MEMS) device on a ball B\subset\IR^N, under Dirichlet boundary conditions $u=\partial_\nu u=0$ on $\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \cite{LY} regarding the identification of a "pull-in voltage" \la^*>0 such that a stable classical solution u_\la with 0 exists for \la\in (0,\la^*), while there is none of any kind when \la>\la^*. Our main result asserts that the extremal solution $u_{\lambda^*}$ is regular $(\sup_B u_{\lambda^*} <1)$ provided $N \le 8$ while $u_{\lambda^*}$ is singular ($\sup_B u_{\lambda^*} =1$) for $N \ge 9$, in which case $1-C_0|x|^{4/3}\leq u_{\lambda^*} (x) \leq 1-|x|^{4/3}$ on the unit ball, where $C_0:= (\frac{\lambda^*}{\overline{\lambda}})^{1/3}$ and $\bar{\lambda}:= {8/9} (N-{2/3}) (N- {8/3})$.Comment: 19 pages. This paper completes and replaces a paper (with a similar title) which appeared in arXiv:0810.5380. Updated versions --if any-- of this author's papers can be downloaded at this http://www.birs.ca/~nassif

Alloy Design for Additive Manufacturing:Early-Stage Oxidation of Nickel-Based Superalloys

This body of work aims to inform alloy design for additive manufacturing by investigating the early-stage oxidation behavior of Ni-based superalloys processed by laser-powder bed fusion. The oxidation of 14 Ni-based superalloys—some novel and some heritage—at 1000 °C for 24 hours is studied through thermo-gravimetric analysis. The mass gain, oxide layer thickness, oxide scale composition, and depletion γ' zone size are measured. The influence of the alloy composition on these variables is assessed in order to elucidate how increasingly processable and oxidation resistant alloys can be developed. The alloy compositions with Al content greater than 9 at. pct form continuous Al2O3 scales at 1000 °C and display markedly lower parabolic rate constants, mass gain, oxide layer thickness, and γ' depletion zone size. The alloys of lesser Al content have reduced oxidation resistance and formed oxide scales of predominantly Cr2O3. Alloys with Ti content of 2.7 at. pct and greater formed Ti-rich oxide phases in their oxide scales as well as TiN subscale. A trade-off between alloy processability and oxidation resistance is identified, dictated by the deleterious effect of Al content on the ductility dip and the benefit of Al for oxidation resistance. A property space along the pareto front is highlighted which is ideal for having oxidation resistance and processability

Alloy Design for Additive Manufacturing:Early-Stage Oxidation of Nickel-Based Superalloys

This body of work aims to inform alloy design for additive manufacturing by investigating the early-stage oxidation behavior of Ni-based superalloys processed by laser-powder bed fusion. The oxidation of 14 Ni-based superalloys—some novel and some heritage—at 1000 °C for 24 hours is studied through thermo-gravimetric analysis. The mass gain, oxide layer thickness, oxide scale composition, and depletion γ' zone size are measured. The influence of the alloy composition on these variables is assessed in order to elucidate how increasingly processable and oxidation resistant alloys can be developed. The alloy compositions with Al content greater than 9 at. pct form continuous Al2O3 scales at 1000 °C and display markedly lower parabolic rate constants, mass gain, oxide layer thickness, and γ' depletion zone size. The alloys of lesser Al content have reduced oxidation resistance and formed oxide scales of predominantly Cr2O3. Alloys with Ti content of 2.7 at. pct and greater formed Ti-rich oxide phases in their oxide scales as well as TiN subscale. A trade-off between alloy processability and oxidation resistance is identified, dictated by the deleterious effect of Al content on the ductility dip and the benefit of Al for oxidation resistance. A property space along the pareto front is highlighted which is ideal for having oxidation resistance and processability

Alloy Design for Additive Manufacturing:Early-Stage Oxidation of Nickel-Based Superalloys

This body of work aims to inform alloy design for additive manufacturing by investigating the early-stage oxidation behavior of Ni-based superalloys processed by laser-powder bed fusion. The oxidation of 14 Ni-based superalloys—some novel and some heritage—at 1000 °C for 24 hours is studied through thermo-gravimetric analysis. The mass gain, oxide layer thickness, oxide scale composition, and depletion γ' zone size are measured. The influence of the alloy composition on these variables is assessed in order to elucidate how increasingly processable and oxidation resistant alloys can be developed. The alloy compositions with Al content greater than 9 at. pct form continuous Al2O3 scales at 1000 °C and display markedly lower parabolic rate constants, mass gain, oxide layer thickness, and γ' depletion zone size. The alloys of lesser Al content have reduced oxidation resistance and formed oxide scales of predominantly Cr2O3. Alloys with Ti content of 2.7 at. pct and greater formed Ti-rich oxide phases in their oxide scales as well as TiN subscale. A trade-off between alloy processability and oxidation resistance is identified, dictated by the deleterious effect of Al content on the ductility dip and the benefit of Al for oxidation resistance. A property space along the pareto front is highlighted which is ideal for having oxidation resistance and processability

On the Influence of Alloy Composition on the Additive Manufacturability of Ni-Based Superalloys

The susceptibility of nickel-based superalloys to processing-induced crack formation during laser powder-bed additive manufacturing is studied. Twelve different alloys—some of existing (heritage) type but also other newly-designed ones—are considered. A strong inter-dependence of alloy composition and processability is demonstrated. Stereological procedures are developed to enable the two dominant defect types found—solidification cracks and solid-state ductility dip cracks—to be distinguished and quantified. Differential scanning calorimetry, creep stress relaxation tests at 1000 °C and measurements of tensile ductility at 800 °C are used to interpret the effects of alloy composition. A model for solid-state cracking is proposed, based on an incapacity to relax the thermal stress arising from constrained differential thermal contraction; its development is supported by experimental measurements using a constrained bar cooling test. A modified solidification cracking criterion is proposed based upon solidification range but including also a contribution from the stress relaxation effect. This work provides fundamental insights into the role of composition on the additive manufacturability of these materials

A new class of alumina-forming superalloy for 3D printing

A new class of crack-resistant nickel-based superalloy containing high γ ′ fraction is studied for the laser-powder bed fusion (L-PBF) process. The effects of the (Nb+Ta)/Al ratio is emphasized, a strategy that is shown to confer excellent low-temperature strength whilst maintaining oxidation resistance at high temperatures via stable alumina scale formation. The processability of the new alloys is characterized with respect to defect assessment by micro-focus X-ray computed tomography; use is made of a prototype turbine blade geometry and the heritage alloy CM247LC as a benchmark. In all cases, some processing-related porosity is present in thin wall sections such as the trailing edge, but this can be avoided by judicious processing. The cracking seen in CM247LC – in solid-state, liquation and solidification forms – is avoided. A novel sub-solvus heat treatment strategy is proposed which takes advantage of AM not requiring solutioning; super-solvus heat treatment is inappropriate since it embrittles the material by deterioration of the texture and coarsening of grain boundary carbides. The tensile strength of the new superalloy is greatest when the Nb+Ta content is highest and exceeds that of CM247LC up to ∼900°C. The oxidation resistance is best when Al content is highest, and oxidation-assisted cracking resistance maximized when the (Nb+Ta)/Al ratio is balanced. In all cases these are equivalent or superior to that of CM247LC. Nevertheless, the creep resistance of the new alloys is somewhat inferior to that of CM247LC for which the γ ′, C, and B contents are higher; this implies a processing/property trade-off which requires further clarification. </p

A new class of alumina-forming superalloy for 3D printing

A new class of crack-resistant nickel-based superalloy containing high γ ′ fraction is studied for the laser-powder bed fusion (L-PBF) process. The effects of the (Nb+Ta)/Al ratio is emphasized, a strategy that is shown to confer excellent low-temperature strength whilst maintaining oxidation resistance at high temperatures via stable alumina scale formation. The processability of the new alloys is characterized with respect to defect assessment by micro-focus X-ray computed tomography; use is made of a prototype turbine blade geometry and the heritage alloy CM247LC as a benchmark. In all cases, some processing-related porosity is present in thin wall sections such as the trailing edge, but this can be avoided by judicious processing. The cracking seen in CM247LC – in solid-state, liquation and solidification forms – is avoided. A novel sub-solvus heat treatment strategy is proposed which takes advantage of AM not requiring solutioning; super-solvus heat treatment is inappropriate since it embrittles the material by deterioration of the texture and coarsening of grain boundary carbides. The tensile strength of the new superalloy is greatest when the Nb+Ta content is highest and exceeds that of CM247LC up to ∼900°C. The oxidation resistance is best when Al content is highest, and oxidation-assisted cracking resistance maximized when the (Nb+Ta)/Al ratio is balanced. In all cases these are equivalent or superior to that of CM247LC. Nevertheless, the creep resistance of the new alloys is somewhat inferior to that of CM247LC for which the γ ′, C, and B contents are higher; this implies a processing/property trade-off which requires further clarification. </p

Alloys-By-Design: Application to New Superalloys for Additive Manufacturing

New grades of γ/γ′ nickel-based superalloy for the additive manufacturing process are designed using computational approaches. Account is taken of the need to avoid defect formation via solidification and solid-state cracking. Processing trials are carried out using powder-based selective laser melting, with the heritage alloys IN939 and CM247LC. Microstructural characterisation, calorimetry and hot tensile testing are used to assess the approach employed. The superior processability and mechanical integrity of the new alloys are demonstrated. Suggestions are made for refinements to the modelling approach