Computing stress intensity factors for curvilinear cracks

The use of the interaction integral to compute stress intensity factors around a crack tip requires selecting an auxiliary field and a material variation field. We formulate a family of these fields accounting for the curvilinear nature of cracks that, in conjunction with a discrete formulation of the interaction integral, yield optimally convergent stress intensity factors. We formulate three pairs of auxiliary and material variation fields chosen to yield a simple expression of the interaction integral for different classes of problems. The formulation accounts for crack face tractions and body forces. Distinct features of the fields are their ease of construction and implementation. The resulting stress intensity factors are observed converging at a rate that doubles the one of the stress field. We provide a sketch of the theoretical justification for the observed convergence rates, and discuss issues such as quadratures and domain approximations needed to attain such convergent behavior. Through two representative examples, a circular arc crack and a loaded power function crack, we illustrate the convergence rates of the computed stress intensity factors. The numerical results also show the independence of the method on the size of the domain of integration

Meat goat showmanship

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Sheep showmanship

The Oklahoma Cooperative Extension Service periodically issues revisions to its publications. The most current edition is made available. For access to an earlier edition, if available for this title, please contact the Oklahoma State University Library Archives by email at [email protected] or by phone at 405-744-6311

Livestock tagging

The Oklahoma Cooperative Extension Service periodically issues revisions to its publications. The most current edition is made available. For access to an earlier edition, if available for this title, please contact the Oklahoma State University Library Archives by email at [email protected] or by phone at 405-744-6311

Bio-Based Polyurethane Networks Derived from Liquefied Sawdust

The utilization of forestry waste resources in the production of polyurethane resins is a promising green alternative to the use of unsustainable resources. Liquefaction of wood-based biomass gives polyols with properties depending on the reagents used. In this article, the liquefaction of forestry wastes, including sawdust, in solvents such as glycerol and polyethylene glycol was investigated. The liquefaction process was carried out at temperatures of 120, 150, and 170 °C. The resulting bio-polyols were analyzed for process efficiency, hydroxyl number, water content, viscosity, and structural features using the Fourier transform infrared spectroscopy (FTIR). The optimum liquefaction temperature was 150 °C and the time of 6 h. Comprehensive analysis of polyol properties shows high biomass conversion and hydroxyl number in the range of 238–815 mg KOH/g. This may indicate that bio-polyols may be used as a potential substitute for petrochemical polyols. During polyurethane synthesis, materials with more than 80 wt% of bio-polyol were obtained. The materials were obtained by a one-step method by hot-pressing for 15 min at 100 °C and a pressure of 5 MPa with an NCO:OH ratio of 1:1 and 1.2:1. Dynamical-mechanical analysis (DMA) showed a high modulus of elasticity in the range of 62–839 MPa which depends on the reaction conditions.The authors would like to thank the National Science Centre of Poland (No. 2018/02/X/ST5/02784) for financial support

Swine showmanship

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Evidence of hydrological control of Sr behavior in stream water (Strengbach catchment, Vosges mountains, France)

Strontium and particularly 87Sr/86Sr ratios in stream water have often been used to calculate weathering rates in catchments. Nevertheless, in the literature, discharge variation effects on the geochemical behavior of Sr are often omitted or considered as negligible. A regular survey of both Sr concentrations and Sr isotope ratios of the Strengbach stream water draining a granite (Vosges mountains, France) has been performed during one year. The results indicate that during low water flow periods, waters contain lower Sr concentrations and less radiogenic Sr isotope ratios (Sr=11.6 ppb and 87Sr/86Sr=0.7246 as an average, respectively) than during high water flow periods (Sr= 13 ppb and 87Sr/86Sr=0.7252 as an average, respectively). This is contrary to expected dilution processes by meteoric waters which have comparatively lower Sr isotopic ratios and lower Sr concentrations. Furthermore, 87Sr/86Sr ratios in stream water behave in 3 different ways depending on moisture and on hydrological conditions prevailing in the catchment. During low water flow periods (discharge < 9 l/s), a positive linear relationship exists between Sr isotope ratio and discharge, indicating the influence of radiogenic waters draining the saturated area during storm events. During high water flow conditions, rising discharges are characterized by significantly less radiogenic waters than the recession stages of discharge. This suggests a large contribution of radiogenic waters draining the deep layers of the hillslopes during the recession stages, particularly those from the more radiogenic north-facing slopes. These results allow one to confirm the negligible instantaneous incidence of rainwater on stream water chemistry during flood events, as well as the existence in the catchment of distinct contributive areas and reservoirs. The influence of these areas or reservoirs on the fluctuations of Sr concentrations and on Sr isotopic variations in stream water depends on both moisture and hydrological conditions. Hence, on a same bedrock type, 87Sr/86Sr ratios in surface waters can be related to flow rate. Consequently, discharge variations must be considered as a pre-requisite when using Sr isotopes for calculating weathering rates in catchments, particularly to define the range of variations of the end-members

Convergence of domain integrals for stress intensity factor extraction in 2-D curved cracks problems with the extended finite element method

The aim of this study is the analysis of the convergence rates achieved with domain energy integrals for the computation of the stress intensity factors (SIF) when solving 2-D curved crack problems with the extended FEM (XFEM). Domain integrals, specially the J-integral and the interaction integral, are widely used for SIF extraction and provide high accurate estimations with FEMs. The crack description in XFEM is usually realized using level sets. This allows to define a local basis associated with the crack geometry. In this work, the effect of the level set local basis definition on the domain integral has been studied. The usual definition of the interaction integral involves hypotheses that are not fulfilled in generic curved crack problems, and we introduce some modifications to improve the behavior in curved crack analyses. Despite the good accuracy of domain integrals, convergence rates are not always optimal, and convergence to the exact solution cannot be assured for curved cracks. The lack of convergence is associated with the effect of the curvature on the definition of the auxiliary extraction fields. With our modified integral proposal, the optimal convergence rate is achieved by controlling the q-function and the size of the extraction domain.This work has been carried out within the framework of the research projects DPI2007-66995-C03-02 and DPI2010-20990 financed by the Ministerio de Economia y Competitividad. The support of the Generalitat Valenciana, Programme PROMETEO 2012/023 is also acknowledged.González Albuixech, VF.; Giner Maravilla, E.; Tarancón Caro, JE.; Fuenmayor Fernández, FJ.; Gravouil, A. (2013). Convergence of domain integrals for stress intensity factor extraction in 2-D curved cracks problems with the extended finite element method. International Journal for Numerical Methods in Engineering. 94(8):740-757. https://doi.org/10.1002/nme.4478S740757948Mo�s, N., Dolbow, J., & Belytschko, T. (1999). A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46(1), 131-150. doi:10.1002/(sici)1097-0207(19990910)46:13.0.co;2-jSukumar, N., Mo�s, N., Moran, B., & Belytschko, T. (2000). Extended finite element method for three-dimensional crack modelling. International Journal for Numerical Methods in Engineering, 48(11), 1549-1570. doi:10.1002/1097-0207(20000820)48:113.0.co;2-aMoës, N., Gravouil, A., & Belytschko, T. (2002). Non-planar 3D crack growth by the extended finite element and level sets-Part I: Mechanical model. International Journal for Numerical Methods in Engineering, 53(11), 2549-2568. doi:10.1002/nme.429Gravouil, A., Moës, N., & Belytschko, T. (2002). Non-planar 3D crack growth by the extended finite element and level sets-Part II: Level set update. International Journal for Numerical Methods in Engineering, 53(11), 2569-2586. doi:10.1002/nme.430Sukumar, N., & Prévost, J.-H. (2003). Modeling quasi-static crack growth with the extended finite element method Part I: Computer implementation. International Journal of Solids and Structures, 40(26), 7513-7537. doi:10.1016/j.ijsolstr.2003.08.002Huang, R., Sukumar, N., & Prévost, J.-H. (2003). Modeling quasi-static crack growth with the extended finite element method Part II: Numerical applications. International Journal of Solids and Structures, 40(26), 7539-7552. doi:10.1016/j.ijsolstr.2003.08.001Rice, J. R. (1968). A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks. Journal of Applied Mechanics, 35(2), 379-386. doi:10.1115/1.3601206Moran, B., & Shih, C. F. (1987). Crack tip and associated domain integrals from momentum and energy balance. Engineering Fracture Mechanics, 27(6), 615-642. doi:10.1016/0013-7944(87)90155-xWalters, M. C., Paulino, G. H., & Dodds, R. H. (2005). Interaction integral procedures for 3-D curved cracks including surface tractions. Engineering Fracture Mechanics, 72(11), 1635-1663. doi:10.1016/j.engfracmech.2005.01.002Duflot, M. (2007). A study of the representation of cracks with level sets. International Journal for Numerical Methods in Engineering, 70(11), 1261-1302. doi:10.1002/nme.1915Osher, S., & Fedkiw, R. P. (2001). Level Set Methods: An Overview and Some Recent Results. Journal of Computational Physics, 169(2), 463-502. doi:10.1006/jcph.2000.6636Nahta, R., & Moran, B. (1993). Domain integrals for axisymmetric interface crack problems. International Journal of Solids and Structures, 30(15), 2027-2040. doi:10.1016/0020-7683(93)90049-dGosz, M., Dolbow, J., & Moran, B. (1998). Domain integral formulation for stress intensity factor computation along curved three-dimensional interface cracks. International Journal of Solids and Structures, 35(15), 1763-1783. doi:10.1016/s0020-7683(97)00132-7Gosz, M., & Moran, B. (2002). An interaction energy integral method for computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions. Engineering Fracture Mechanics, 69(3), 299-319. doi:10.1016/s0013-7944(01)00080-7Sukumar N Element partitioning code in 2-D and 3-D for the extended finite element method 2000 http://dilbert.engr.ucdavis.edu/suku/xfemLaborde, P., Pommier, J., Renard, Y., & Salaün, M. (2005). High-order extended finite element method for cracked domains. International Journal for Numerical Methods in Engineering, 64(3), 354-381. doi:10.1002/nme.1370Eshelby, J. D. (1975). The elastic energy-momentum tensor. Journal of Elasticity, 5(3-4), 321-335. doi:10.1007/bf00126994Eriksson, K. (2000). International Journal of Fracture, 106(1), 65-80. doi:10.1023/a:1007646823223Simpson, R., & Trevelyan, J. (2011). Evaluation of J1 and J2 integrals for curved cracks using an enriched boundary element method. Engineering Fracture Mechanics, 78(4), 623-637. doi:10.1016/j.engfracmech.2010.12.006Tarancón, J. E., Vercher, A., Giner, E., & Fuenmayor, F. J. (2009). Enhanced blending elements for XFEM applied to linear elastic fracture mechanics. International Journal for Numerical Methods in Engineering, 77(1), 126-148. doi:10.1002/nme.2402Béchet, E., Minnebo, H., Moës, N., & Burgardt, B. (2005). Improved implementation and robustness study of the X-FEM for stress analysis around cracks. International Journal for Numerical Methods in Engineering, 64(8), 1033-1056. doi:10.1002/nme.1386Stazi, F. L., Budyn, E., Chessa, J., & Belytschko, T. (2003). An extended finite element method with higher-order elements for curved cracks. Computational Mechanics, 31(1-2), 38-48. doi:10.1007/s00466-002-0391-2Chessa, J., Wang, H., & Belytschko, T. (2003). On the construction of blending elements for local partition of unity enriched finite elements. International Journal for Numerical Methods in Engineering, 57(7), 1015-1038. doi:10.1002/nme.77