Full-field stress computation from measured deformation fields: A hyperbolic formulation

© 2020 Elsevier Ltd Recent developments in imaging techniques and correlation algorithms enable measurement of strain fields on a deforming material at high spatial and temporal resolution. In such cases, the computation of the stress field from the known deformation field becomes an interesting possibility. This is known as an inverse problem. Current approaches to this problem, such as the finite element update method, are generally over-determined and must rely on statistical approaches to minimize error. This provides approximate solutions in some cases, however, implementation difficulties, computational requirements, and accuracy are still significant challenges. Here, we show how the inverse problem can be formulated deterministically and solved exactly in two or three dimensions for large classes of materials including isotropic elastic solids, Newtonian fluids, non-Newtonian fluids, granular materials, plastic solids subject to co-directionality, and some other plastic solids subject to associative or non-associative flow rules. This solution is based on a single assumption of the alignment of the principal directions of stress and strain or strain rate. No further assumptions regarding incompressibility, pressure independence, yield surface shape or the hardening law are necessary. This assumption leads to a closed, first order, linear system of hyperbolic partial differential equations with variable coefficients. The solution of this class of problems is well established and hence the equations can be solved to give the solution for any geometry and loading condition, enabling broad applicability to a variety of problems. We provide a numerical proof-of-principle study of the plastic deformation of a two-dimensional bar with spatially varying yield stress and strain hardening coefficient. The results are validated against the solution of the corresponding forward problem – solved with a commercial finite element solver – indicating the solution is exact up to numerical error (the normalized root mean square error of the stress is 1.63×10−4). No model calibration or material parameters are required. The sensitivity of the solution to error in the input data is also analyzed. Interestingly, this solution procedure lends itself to a simple physical interpretation of stress propagation through the material. Finally, we provide some examples showing how this approach may be analytically applied to both solid and fluid mechanics problems

SNG UPGRADING

This chapter aims at investigating and comparing the separation processes needed for upgrading synthetic natural gas (SNG) produced in second generation biomass conversion processes, which are run under operating conditions properly chosen in order to maximize the methane yield. Methane and ethane are obviously the most valuable components in SNG and their recovery is also important in order to avoid uncontrolled emissions to the atmosphere, which could be harmful considering for instance the high greenhouse potential of methane. The bulk separation of CH4 and CO2 may be achieved by applying one of the following process options: physical or chemical absorption, membrane separation, adsorptive separation, or low temperature separation. The chapter suggests that different processes are in principle applicable to the different separation steps required for SNG upgrading. More data on the different separation options and the associated costs are instead available for the upgrading of biogas or synthetic gas obtained from thermal gasification processes