{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T04:48:33Z","timestamp":1785905313067,"version":"3.56.0"},"reference-count":55,"publisher":"MDPI AG","issue":"15","license":[{"start":{"date-parts":[[2026,8,3]],"date-time":"2026-08-03T00:00:00Z","timestamp":1785715200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSF","award":["# 2218063"],"award-info":[{"award-number":["# 2218063"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>This study empirically and numerically investigates sintering-induced shrinkage of 316L stainless steel cylinders fabricated via Bound Metal Deposition (BMD). Sintering shrinkages were measured for three cylindrical specimens, one each in x-, y-, and z-print orientations, via a dilatometer. Each specimen was horizontally sintered to 1360 \u00b0C via a prescribed temperature profile to maintain identical sintering conditions between specimens. Numerically, a two-dimensional axisymmetric finite element model was implemented in Abaqus using a user-defined viscoplastic constitutive model incorporating porosity- and temperature-dependent viscosities, grain-growth kinetics, and capillary-driven sintering stress. Model parameters were calibrated using particle swarm optimization by minimizing a combined error metric for temperature-dependent shrinkage responses and final relative densities. Results indicate: (1) final shrinkages of 13.5%, 13.7%, and 14.3% for the x-, y-, and z-oriented specimens, respectively, and (2) the optimized numerical model utilizing literature-informed material property constraints results in shrinkages and final densities that are in good agreement with empirical observations. Importantly, this work (1) suggests sintering shrinkage may depend on print orientation, in addition to furnace orientation, for BMD-fabricated 316L stainless steel, and (2) adds an initial dilatometer data set for BMD 316L to the literature. More broadly, this work supports future design activities via an experimental and numerical characterization of sintering behavior.<\/jats:p>","DOI":"10.3390\/ma19153294","type":"journal-article","created":{"date-parts":[[2026,8,3]],"date-time":"2026-08-03T14:16:27Z","timestamp":1785766587000},"page":"3294","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Effect of Print Orientation on Sintering Shrinkage of BMD 316L Stainless Steel: An Empirical and Numerical Study"],"prefix":"10.3390","volume":"19","author":[{"given":"Ayechew","family":"Aklilu","sequence":"first","affiliation":[{"name":"Mechanical Engineering, University of Maine, 75 Long Road, Orono, ME 04469, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2963-1108","authenticated-orcid":false,"given":"John","family":"Belding","sequence":"additional","affiliation":[{"name":"Advanced Manufacturing Center, University of Maine, 5796 AMC Building, Orono, ME 04469, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Joe","family":"Strauss","sequence":"additional","affiliation":[{"name":"HJE Company, 820 Quaker Road, Queensbury, NY 12804, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8801-3436","authenticated-orcid":false,"given":"Brett D.","family":"Ellis","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Technology, University of Maine, 5711 Boardman Hall, Orono, ME 04469, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1016\/j.procir.2022.02.064","article-title":"Environmental impacts assessment of Bound Metal Deposition 3D printing process for stainless steel","volume":"105","author":"Iacopo","year":"2022","journal-title":"Procedia CIRP"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1016\/j.jmapro.2022.10.060","article-title":"An overview of modern metal additive manufacturing technology","volume":"84","author":"Armstrong","year":"2022","journal-title":"J. Manuf. Proc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1177\/1687814018822880","article-title":"Additive manufacturing: Challenges, trends, and applications","volume":"11","author":"Abdulhameed","year":"2019","journal-title":"Adv. Mech. Eng."},{"key":"ref_4","unstructured":"(2025, March 01). Deep Dive: Bound Metal Deposition (BMD). Available online: https:\/\/www.desktopmetal.com\/resources\/deep-dive-bmd."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/j.matchar.2004.11.014","article-title":"Mixing and characterisation of 316L stainless steel feedstock for micro powder injection molding","volume":"54","author":"Liu","year":"2005","journal-title":"Mater. Charact."},{"key":"ref_6","first-page":"102472","article-title":"Fused filament fabrication of stainless steel structures\u2014From binder development to sintered properties","volume":"49","author":"Wagner","year":"2022","journal-title":"Addit. Manuf."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Watson, A., Belding, J., and Ellis, B.D. (2020, January 23\u201327). Characterization of 17-4 PH Processed via Bound Metal Deposition (BMD). Proceedings of the Minerals, Metals & Materials Annual Meeting & Exhibition (TMS 2020), San Diego, CA, USA.","DOI":"10.1007\/978-3-030-36296-6_19"},{"key":"ref_8","unstructured":"Watson, A. (2022). Development and Characterization of Bound Metal Deposition Including Laser Ablation. [Master\u2019s Thesis, University of Maine]."},{"key":"ref_9","unstructured":"Campbell, Q., Filiault, M., Rooney, K., Belding, J., and Ellis, B.D. (2019, January 17\u201319). Design and Manufacture of a Functional 3D-Printed Stirling Engine: A Case Study Involving Bound Metal Deposition of 17-4 PH. Proceedings of the TechConnect World, Boston, MA, USA."},{"key":"ref_10","unstructured":"(2025, March 01). Thermal Debinding and Sintering 101. Available online: https:\/\/www.desktopmetal.com\/resources\/sintering-101."},{"key":"ref_11","unstructured":"German, R.M. (1996). Sintering Theory and Practice, Wiley."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Fang, Z.Z. (2010). Sintering of Advanced Materials: Fundamentals and Processes, Woodhead.","DOI":"10.1533\/9781845699949"},{"key":"ref_13","unstructured":"Kang, S.-J.L. (2005). Sintering: Densification, Grain Growth, and Microstructure, Elsevier Butterworth-Heinemann."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"070801","DOI":"10.1115\/1.4068066","article-title":"Sintering mechanisms in metal extrusion-based sintering-assisted additive manufacturing: State-of-the-art and perspectives","volume":"147","author":"Liu","year":"2025","journal-title":"J. Manuf. Sci. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/s11740-014-0574-7","article-title":"Comparison between different numerical models of densification during solid-state sintering of pure aluminium powder","volume":"9","author":"Kammler","year":"2015","journal-title":"Prod. Eng. Res. Devel."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2805","DOI":"10.1063\/1.1782127","article-title":"Coalescence of two spheres by surface diffusion","volume":"37","author":"Nichols","year":"1966","journal-title":"J. App. Phy."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"S\u014dmiya, S., and Moriyoshi, Y. (1990). Self-diffusion in sintering of metallic particles. Sintering Key Papers, Springer.","DOI":"10.1007\/978-94-009-0741-6"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1063\/1.364004","article-title":"Digital-image-based models of two-dimensional microstructural evolution by surface diffusion and vapor transport","volume":"81","author":"Bullard","year":"1997","journal-title":"J. App. Phy."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1146\/annurev.matsci.32.112001.132041","article-title":"Phase-field models for microstructure evolution","volume":"32","author":"Chen","year":"2002","journal-title":"Annu. Rev. Mater. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2857","DOI":"10.1111\/j.1151-2916.1990.tb06686.x","article-title":"Computer simulation of final-stage sintering: I, model kinetics, and microstructure","volume":"73","author":"Hassold","year":"1990","journal-title":"J. Am. Ceram. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/S0927-796X(98)00009-6","article-title":"Theory of sintering: From discrete to continuum","volume":"23","author":"Olevsky","year":"1998","journal-title":"Mater. Sci. Eng. R Rep."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1914","DOI":"10.1111\/j.1551-2916.2006.01054.x","article-title":"Multi-scale study of sintering: A review","volume":"89","author":"Olevsky","year":"2006","journal-title":"J. Am. Ceram. Soc."},{"key":"ref_23","unstructured":"Skorohod, V.V. (1972). Rheological Basis of the Theory of Sintering, Naukova Dumka."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1930","DOI":"10.1111\/j.1551-2916.2006.01041.x","article-title":"An Arrhenius-type viscosity function to model sintering using the Skorohod\u2013Olevsky viscous sintering model within a finite-element code","volume":"89","author":"Reiterer","year":"2006","journal-title":"J. Am. Ceram. Soc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1442","DOI":"10.1111\/j.1551-2916.2009.03008.x","article-title":"Verification, performance, validation, and modifications to the SOVS continuum constitutive model in a nonlinear large-deformation finite element code","volume":"92","author":"Reiterer","year":"2009","journal-title":"J. Am. Ceram. Soc."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"719","DOI":"10.5937\/fme2103719P","article-title":"Skorohod-Olevsky viscous sintering model sensitivity to temperature distribution during the sintering process","volume":"49","author":"Buljak","year":"2021","journal-title":"FME Trans."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/0001-6160(74)90167-9","article-title":"A first report on sintering diagrams","volume":"22","author":"Ashby","year":"1974","journal-title":"Acta Metall."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1016\/S0955-2219(03)00222-X","article-title":"Numerical simulation of solid state sintering; model and application","volume":"24","author":"Kraft","year":"2004","journal-title":"J. Eur. Ceram. Soc."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"927","DOI":"10.1016\/0001-6160(86)90066-0","article-title":"Viscoelastic stresses and sintering damage in heterogeneous powder compacts","volume":"34","author":"Hsueh","year":"1986","journal-title":"Acta Metall."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1002\/nme.1620250116","article-title":"Finite element simulation of hot isostatic pressing of metal powders","volume":"25","author":"Abouaf","year":"1988","journal-title":"Numer. Methods Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"118822","DOI":"10.1016\/j.actamat.2023.118822","article-title":"Analytical models for initial and intermediate stages of sintering of additively manufactured stainless steel","volume":"249","author":"Olevsky","year":"2023","journal-title":"Acta Mater."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1002\/zamm.19290090104","article-title":"Berechnung der flie\u00dfgrenze von mischkristallen auf grund der plastizit\u00e4tsbedingung f\u00fcr einkristalle","volume":"9","author":"Reuss","year":"1929","journal-title":"Z. Angew. Math. Mech."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Chawla, K.K. (1987). Micromechanics of composites. Composite Materials, Springer.","DOI":"10.1007\/978-1-4757-3912-1"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"37268","DOI":"10.1016\/j.ceramint.2024.04.427","article-title":"Phenomenological sintering model and experimental validation of gravity-induced distortions in binder-jetted stainless steel components","volume":"50","author":"Persson","year":"2024","journal-title":"Ceram. Int."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2414","DOI":"10.1111\/j.1551-2916.2012.05096.x","article-title":"Fundamental aspects of spark plasma sintering: II. Finite element analysis of scalability","volume":"95","author":"Olevsky","year":"2012","journal-title":"J. Am. Ceram. Soc."},{"key":"ref_36","unstructured":"(2023, January 16). 316L v.2 Stainless Steel: Material Data Sheet. Available online: https:\/\/www.desktopmetal.com\/uploads\/BMD-MDS-316L-v2-210129.pdf."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/S0955-2219(97)00108-8","article-title":"A rule-of-mixtures model for sintering of particle-reinforced ceramic-matrix composites","volume":"18","author":"Edrees","year":"1998","journal-title":"J. Eur. Ceram. Soc."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Cabo Rios, A., Hryha, E., Olevsky, E., and Persson, M. (2022, January 9\u201313). Modelling of delta-ferrite transformation effect on the sintering behavior of 316L binder jetting components. Proceedings of the World PM2022 Conference, Lyon, France.","DOI":"10.59499\/WP225371818"},{"key":"ref_39","unstructured":"German, R.M. (2014). Sintering: From Empirical Observations to Scientific Principles, Elsevier Butterworth-Heinemann. [1st ed.]."},{"key":"ref_40","unstructured":"Zhang, R. (2005). Numerical Simulation of Solid-State Sintering of Metal Powder Compact Dominated by Grain Boundary Diffusion. [Ph.D. Dissertation, Pennsylvania State University]. Available online: https:\/\/etda.libraries.psu.edu\/catalog\/6871."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"110528","DOI":"10.1016\/j.mtcomm.2024.110528","article-title":"Anisotropic sintering behavior of stainless steel 316L printed by binder jetting additive manufacturing","volume":"41","author":"Esmati","year":"2024","journal-title":"Mater. Today Commun."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"L13","DOI":"10.1016\/0921-5093(92)90392-E","article-title":"Grain growth behaviour of type 316L stainless steel","volume":"149","author":"Kashyap","year":"1992","journal-title":"Mater. Sci. Eng. A"},{"key":"ref_43","unstructured":"Kennedy, J., and Eberhart, R. (December, January 27). Particle swarm optimization. Proceedings of the International Conference on Neural Networks (ICNN\u201995), Perth, WA, Australia."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1109\/TEVC.2004.826071","article-title":"Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients","volume":"8","author":"Ratnaweera","year":"2004","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_45","unstructured":"Shi, Y., and Eberhart, R.C. (1999, January 6\u20139). Empirical study of particle swarm optimization. Proceedings of the 1999 Congress on Evolutionary Computation (CEC\u201999), Washington, DC, USA."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"10031","DOI":"10.1109\/ACCESS.2022.3142859","article-title":"Particle swarm optimization: A comprehensive survey","volume":"10","author":"Shami","year":"2022","journal-title":"IEEE Access"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"100718","DOI":"10.1016\/j.swevo.2020.100718","article-title":"Population size in particle swarm optimization","volume":"58","author":"Piotrowski","year":"2020","journal-title":"Swarm Evol. Comput."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s00500-016-2474-6","article-title":"Particle swarm optimization algorithm: An overview","volume":"22","author":"Wang","year":"2018","journal-title":"Soft Comput."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1080\/00401706.2000.10485979","article-title":"A comparison of three methods for selecting values of input variables in the analysis of output from a computer code","volume":"42","author":"Mckay","year":"2000","journal-title":"Technometrics"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"392","DOI":"10.2118\/105797-PA","article-title":"Adjoint-based well-placement optimization under production constraints","volume":"13","author":"Zandvliet","year":"2008","journal-title":"SPE J."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"106193","DOI":"10.1016\/j.asoc.2020.106193","article-title":"Influence of initialization on the performance of metaheuristic optimizers","volume":"91","author":"Li","year":"2020","journal-title":"Appl. Soft Comput."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"157","DOI":"10.3390\/make1010010","article-title":"Particle swarm optimization: A survey of historical and recent developments with hybridization perspectives","volume":"1","author":"Sengupta","year":"2019","journal-title":"Mach. Learn. Knowl. Extr."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1137\/13S012510","article-title":"A comparison of common methods for optimal well placement","volume":"7","author":"Minton","year":"2014","journal-title":"SIAM Undergrad. Res. Online"},{"key":"ref_54","unstructured":"(2024, January 16). BMD Design Guide, Rev03. Available online: https:\/\/www.desktopmetal.com\/uploads\/20230912_R03_Studio-System-BMD-Design-Guide_2023-09-13-123253_cupr.pdf?utm_source."},{"key":"ref_55","unstructured":"(2025, March 01). 316\/316L Stainless Steel Datasheet. Available online: https:\/\/www.carpentertechnology.com\/hubfs\/7407324\/Material%20Saftey%20Data%20Sheets\/316.pdf."}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/19\/15\/3294\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T04:16:52Z","timestamp":1785903412000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/19\/15\/3294"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,3]]},"references-count":55,"journal-issue":{"issue":"15","published-online":{"date-parts":[[2026,8]]}},"alternative-id":["ma19153294"],"URL":"https:\/\/doi.org\/10.3390\/ma19153294","relation":{},"ISSN":["1996-1944"],"issn-type":[{"value":"1996-1944","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,3]]}}}