A comprehensive model represented by a set of equations governing the mechanics of planar hydraulic fracture propagation in a multi-layered reservoir is presented. A general-purpose integral formulation for the formation elasticity is developed along with a numerical scheme for mode I fracture response evaluation of an arbitrarily shaped planar pressurized crack in a layered medium. Non-Newtonian fluid flow in the hydraulically induced fracture is governed by a two-dimensional nonlinear partial differential equation. Finite element formulations for the governing equations as well as calibrative examples illustrating the computational features of the model are presented. Numerical schemes for determining the moving fracture front and coupling of the fluid flow and structural/fracture responses are also developed.

The forming behavior of austenitic stainless steels (types 201, 301, and 304) and ferritic stainless steels (types 437, 439, 444, and 468) was investigated. The tensile behavior and the forming-limit diagrams (FLDs) for these grades were determined. The ferritic alloys behave similarly to plain carbon steels and are relatively insensitive to small variations of strain rate and temperature. The formability of the austenitic alloys is influenced greatly by martensitic transformation during straining. The fraction of martensite transformed as a function of strain was found to be very sensitive to temperature, which, in turn, depends on the strain rate at typical testing rates (10 Ϫ3 to 10 Ϫ1 /s). At low rates (when the specimen remains near room temperature), the formability of the austenitic alloys is markedly improved by transformation strengthening. The enhancement of formability is largest on the biaxial side of the FLD, because the fraction martensite transformed was found to depend on the absolute thickness strain, which is maximized in the balanced biaxial strain state.

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SUMMARYIn this paper, a general variational principle for the initial boundary value problem of quasi-static thermoelastic consolidation is developed by assuming infinitesimal deformation and an incompressible fluid flowing through a linearly elastic solid. By manipulating the coupling operators, an extended form of the variational principle is derived. The associated finite element formulation based on this principle is presented and numerical applications for plane strain thermoelastic consolidation are revealed.

SUMMARYImplementation of the 3-D finite element method (FEM) of sheet forming operations has proceeded slowly. In particular, describing the arbitrary tool surfaces in an accurate and smooth manner, with well behaved first and second derivatives, has been a major obstacle. A new geometric method, suitable for representing any rigid tool surface, has been developed and shown to be superior to the usual schemes. The new method relies on the described tool surface for position data only. The spatial derivatives are obtained by considering only the sheet mesh nodal positions. This new scheme has been implemented with a rigid-viscoplastic FEM program using general surface descriptions based on B-splines and linear interpolation. Analytic representations of a hemispherical and rounded square punch were also compared. The comparisons show that the proposed method offers several advantages over other general formulations:1. The new representation is the proper one for formulating the equilibrium condition. Alternate forms can produce spurious results. 2. The proposed method introduces inherently smooth surface derivatives that improve numerical stability, even with crude surface resolution. Suitable test problems show convergence with the new method far past the divergence points of the simulations using the standard general surface description. 3. The method is reasonably efficient, with a time penalty of approximately 30-50 per cent (unoptimized) with respect to analytic surface descriptions.