On discontinuities when computing the stress-field from the strain: a finite volume discretization

Recently, a widely applicable system of hyperbolic partial differential equations has been derived that enables the deterministic computation of a full heterogeneous stress field from a measured deformation field, for example, from a strain field obtained via digital image correlation. This information enables the determination of material properties, making this approach an alternative to finite element model updating or the virtual fields method. This article focuses on developing a finite volume discretization of this system of equations to address instabilities that arise from violations of the Courant-Friedrichs-Lewy condition. The developed discretization enables the system of equations to be applied to irregular geometries and finite deformation. We determine how, in general, one may translate knowledge of the traction at the boundary into boundary conditions, so that the numerical method can be applied to a variety of loading conditions. We analyse the solution structure in the case of deformation discontinuities, and the results are applied to account for discontinuities at the interfaces between finite volumes (this has relevance to other applications such as composite materials). Interestingly, the discontinuities cause reflection and transmission of the principal stresses. The finite volume discretization is validated using data output from commercial finite element software. Furthermore, the discretization is applied to an experimental uniaxial tension test with plastic deformation and necking. The strain field is obtained using digital image correlation, and the stress field is computed using the developed finite volume discretization. Together, these give the stress-strain behavior for each material element

Effects of production parameters on the structure of resol type phenolic resin/layered silicate nanocomposites

Polymer/layered silicate nanocomposites belong to one of the most promising group of materials of the past few decades and most probably for the near future. Following the pioneering works of Toyota Research Group in the 1980s, the interest on these materials increased rapidly and research is now being carried out world wide, using all kinds of polymers as base material. In this present study, the aim was to investigate the effects of several different production parameters; on the morphology of resol type phenol formaldehyde based layered silicate nanocomposites produced by mixing and casting. For this purpose; two different liquid resol type phenolic resins (PF76 and PF76TD), two different curing methods (heat cure route and acid cure route), two different montmorillonite clays (unmodified Cloisite Na+ and modified Rheospan), two different clay sources (Wyoming-USA and Tokat-Turkey), and five different clay amounts (0.5%, 1%, 1.5%, 3%, 10%) were used. XRD, SEM, TEM analyses and mechanical tests indicated that resol type phenolic resins lead to better structures when they were modified with ethylene glycol and cured by the use of an acidic curing agent. It was also observed that use of modified clay with no more than 1.5 wt% in the phenolic matrix lead to certain degree of exfoliation consequently better structure and higher mechanical performance

Partial differential equations to determine elasto-plastic stress–strain behavior from measured kinematic fields

A system of partial differential equations (PDEs) is derived to compute the full-field stress from an observed kinematic field when the flow rule governing the plastic deformation is unknown. These equations generalize previously proposed equations that assume pure plastic behavior without elasticity. A method to numerically solve these equations is also presented. In addition to force balance, the equations are derived from the elastic–plastic decomposition of the deformation gradient, the assumption of isotropy, and the assumption that the function mapping the elastic strain to stress is known. The system of equations can be directly applied to complex geometries, finite deformation, non-linear elasticity and plasticity, compressible materials, rate dependent materials, and a variety of hardening laws. This system of PDEs is non-linear and time dependent. Furthermore, it overcomes an important prior limitation: it can be directly applied to cases where some regions of a body are elastically deforming while others are elasto-plastically deforming. A two-dimensional case study of necking in a uniaxial tensile specimen is investigated to illustrate and validate the method. The governing equations are numerically solved using strain fields output from a finite element simulation and validated against this same simulation showing accurate results

Mechanical Performance of Resol Type Phenolic Resin/Layered Silicate Nanocomposites

Clay addition has been shown to affect polymer resins positively in terms of several physical and chemical properties, including mechanical performance, high temperature endurance and durability. These increases are limited only to relatively low concentrations of reinforcement phase, but at these low concentrations polymer/layered silicate nanocomposites (P/LS NC) have shown to exhibit higher mechanical performance than fiber reinforced polymer composites. This is among the several reasons that make P/LS NC's one of the most widely studied class of materials today. In this study, the mechanical performance of resol type phenolic resin/layered silicate nanocomposite specimens was examined by carrying out 3-point bending, Charpy impact and fracture toughness tests to couple the observations, microstructural analysis is done through X-ray diffraction and scanning electron microscopy. The effects of especially three factors; cure method, clay amount, and clay modification were investigated. It was concluded that highest mechanical performance was obtained by the acid curing of the phenolic resin with very low amounts (e.g., 0.5%) of either very hydrophobic or very hydrophilic Na-montmorillonite clay additions. Improvements as high as 7% in flexural strength, 11% in flexural strain at break, 16% in Charpy impact strength, and 66% in fracture toughness values were obtained. POLYM. COMPOS., 30:343-350, 2009. (C) 2008 Society of Plastics Engineer

Effects of production parameters on the structure of resol type phenolic resin/layered silicate nanocomposites

Polymer/layered silicate nanocomposites belong to one of the most promising group of materials of the past few decades and most probably for the near future. Following the pioneering works of Toyota Research Group in the 1980s, the interest on these materials increased rapidly and research is now being carried out world wide, using all kinds of polymers as base material

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

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 ). 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

Towards physical insights on microstructural damage nucleation from data analytics

Variations in the chemical composition or thermomechanical processing of metallic materials result in a vast landscape of possible microstructural morphologies. While this creates ample opportunities for alloys with improved mechanical performance, the design process is challenging due to the morphological complexity and the range of deformation micro-mechanisms involved. Empirically based statistical approaches are well suited to address some of these challenges. Previously, a model based on n-point statistics and principal component analysis was successfully used for predicting damage nucleation based on the microstructural morphology of dual-phase steels. Here, we give an in-depth exploration and analysis of such algorithms as applied to experimental data. First, we investigate model architecture by implementing and testing over 1000 model variants. This leads to improved predictive ability and several alternate architectures including one with a Fourier transformation instead of a n-point statistics transformation. Second, we analyze the noise, resolution and data quantity impact to give guidelines on the necessary data required to train a predictive model. Third, we investigate which morphological features are utilized by the model to make predictions by inputting artificially-constructed microstructures, inverting the model, examination of the basis image, and variation of model hyperparameters. It is found that grain boundary fluctuations less than 1 μm are correlated to damage nucleation. This is consistent with observations in the literature on the effect of grain size and interconnected martensite regions on damage nucleation. Furthermore, it may give insights into the superior mechanical properties of alloys with bimodal grain size distributions. This demonstrates a unique approach of elucidating morphological effects from a single alloy by exploiting microstructural heterogeneity. It may be applied to other microstructures as well.<br/

Phase stability effects on hydrogen embrittlement resistance in martensite–reverted austenite steels

Earlier studies have shown that interlath austenite in martensitic steels can enhance hydrogen embrittlement (HE) resistance. However, the improvements were limited due to microcrack nucleation and growth. A novel microstructural design approach is investigated, based on enhancing austenite stability to reduce crack nucleation and growth. Our findings from mechanical tests, X-ray diffraction, and scanning electron microscopy reveal that this strategy is successful. However, the improvements are limited due to intrinsic microstructural heterogeneity effects

Laser-induced toughening inhibits cut-edge failure in multi-phase steel

The as-cut microstructures and the subsequent microstructural deformation characteristics of dual-phase steel specimens were analyzed using in-situ biaxial Marciniak tests, microscopic digital-image-correlation and nano-indentation, for two industrially relevant cutting processes: laser cutting and blanking. Interestingly, the strain-to-failure of the former is almost twice that of the latter, even though microstructural damage initiates twice as early (at 8% strain) in the ∼60 µm-thick, fully-martensitic surface layer of the laser-cut affected zone. However, its ∼145 µm-thick, tempered-martensite sub-surface layer provides the toughness to delay micro-damage propagation, arrest the crack growth, and ultimately provide the high strain-to-failure. These observations reveal guidelines to avoid cut-edge failure

Laser-induced toughening inhibits cut-edge failure in multi-phase steel

\u3cp\u3eThe as-cut microstructures and the subsequent microstructural deformation characteristics of dual-phase steel specimens were analyzed using in-situ biaxial Marciniak tests, microscopic digital-image-correlation and nano-indentation, for two industrially relevant cutting processes: laser cutting and blanking. Interestingly, the strain-to-failure of the former is almost twice that of the latter, even though microstructural damage initiates twice as early (at 8% strain) in the ∼60 µm-thick, fully-martensitic surface layer of the laser-cut affected zone. However, its ∼145 µm-thick, tempered-martensite sub-surface layer provides the toughness to delay micro-damage propagation, arrest the crack growth, and ultimately provide the high strain-to-failure. These observations reveal guidelines to avoid cut-edge failure.\u3c/p\u3