Group Overview

University of Wisconsin Theoretical & Computational Mechanics of Materials Group

Welcome to the site describing the work taking place in the Bronkhorst Group at the University of Wisconsin. We currently have openings for U.S citizen Ph.D. students and postdoctoral associates. If interested, please contact Prof. Bronkhorst directly at cbronkhorst@wisc.edu.

The design, use, and computational representation of structural components for applications involving extreme loading conditions remains a key research area. The basis for building this capability fundamentally rests on our physical understanding of the mechanistic processes involved, which then motivate the form of the mathematical framework. At these early stages of understanding and modeling structural evolution in materials for extreme structural loading, the nature of the work necessary to make significant advancements must be based upon a fundamental coupling of mathematics, materials science, and theoretical/computational mechanics of materials. It is recognized that structural evolution in materials in a spatial sense is an extreme-event process and must be described by the appropriate mathematics. Thus, mathematics must be the basis for adopting experimental and computational results and must assist in defining the results needed. This research can also form the fundamental basis for the design of materials for targeted performance depending upon application area.

The vision for our work here in the Theoretical & Computational Mechanics of Materials Group at the University of Wisconsin is to offer a new approach to the study and prediction of multi-physics events taking place within materials exposed to conditions of extreme loading. The physical theories which presume to represent coupled behavior are based upon hypotheses developed through physical insight gained from experiments and collected experience comparing simulations with experimental results. This continuous improvement process for advancing our ability to predict responses of materials to extreme loading conditions is the basis for a strategy of intense coupling of simultaneous advancements in theory, experiment, and computation. The nucleation, growth, and impingement/coalescence progression of both phase transformation and damage are challenging topics to explore under the loading conditions of interest. The rate of change of dislocation structure, deformation twinning, structural phase change and damage are intimately linked with the nature of the nucleated field. The nucleated field is very difficult to interrogate experimentally, and supplementation of insight gained via computational results are critical to enable tangible and timely progress. We propose to address the extreme events of plasticity, phase change and damage through the development of an extreme-event methodology and mathematics tools. With this, our goal is to eventually engage large datasets of experimental and computational information to rigorously guide our learning process and enable simultaneous use of different types of information self-consistently.

We invite you to reach out to us to discuss our work further via email to the addresses listed on the Group Members page.

Professor Curt A. Bronkhorst is Associate Chair for Undergraduate Studies in the Engineering Physics Department. He also has appointments in the Mechanical Engineering Department and Materials Science & Engineering Department. He is Guest Scientist within the Theoretical Division at Los Alamos National Laboratory. He is Honorary Commander for the Wisconsin Air National Guard 115th Fighter Wing. He is president of Northland Partners, LLC. He is fellow of the American Society of Mechanical Engineers. He is a member of the ASME Materials Division Executive Committee. He is also Associate Editor of the International Journal of Plasticity.

Bronkhorst_CV_public

email: cbronkhorst@wisc.edu

Sudip Kunda graduated with a bachelor’s degree in Civil Engineering at the Birla Institute of Technology and Science, Pilani, and worked for a year as a trainee engineer at Wadia Techno-Engineering Services in Kolkata. He earned a master’s degree in Civil Engineering with a specialization in Structural Engineering from the Indian Institute of Science, Bangalore in 2019. He joined UW-Madison in the fall of 2019 is a Engineering Mechanics Ph.D. student. He is working on capturing the effect of grain size on the flow stress of metal polycrystals. The theory he is developing is applicable to grains whose size is of the order of a few to tens of micrometers. The aim is to replace the empirical inverse square root law from the Hall-Petch effect that is currently in common use, with a theory in which the effect of grain size on flow stress is an emergent phenomenon.

email: skunda@wisc.edu

Akhilesh Pedgaonkar is currently a PhD student in Engineering Mechanics. He has done his B.Tech. in Mechanical Engineering from Indian Institute of Technology, Bombay and M.S. in Mechanical Engineering from University at Buffalo. He has also worked in Hero MotoCorp Ltd. and Ansys, Inc. before joining UW Madison. He is working on computational micromechanics of polycrystal materials which provide solutions to challenging problems such as prediction of material damage.

email: pedgaonkar@wisc.edu

Noah Schmelzer is an Engineering Mechanics Ph.D. student with a B.A. in Applied Physics from St. John’s University of Collegeville , MN. He is currently investigating the effects on anisotropic void growth on dynamic ductile damage within hyperelastic physical theories for computational prediction of structural failure of materials.

email: njschmelzer@wisc.edu

 

 

Refereed Journal Articles Published Since 2010

Vogler, T. J., Brown, J. L., Bronkhorst, C. A., (2021). High-Pressure Dynamic Strength of Materials, J. Dyn. Behavior Mats. 7, 169. https://link.springer.com/article/10.1007/s40870-021-00308-5

Svolos, L., Mourad, H. M., Bronkhorst, C. A., Waisman, H. (2021). Anisotropic Thermal-Conductivity Degradation in the Phase-Field Method Accounting for Crack Directionality, Engr. Fracture Mech. 245, 107554. https://www.sciencedirect.com/science/article/pii/S0013794421000266

Foster, R. C., Vander Wiel, S. A., Anghel, V., Bronkhorst, C. A. (2021). Towards Random Generation of Microstructures of Spatially Varying Materials from Orthogonal Sections. Comp. Mats. Sci. 192, 110313. https://www.sciencedirect.com/science/article/pii/S0927025621000380

Gupta, S., Bronkhorst, C. A. (2021). Crystal Plasticity Model for Single Crystal Ni-based Superalloys: Capturing Orientation and Temperature Dependence of Flow Stress, Int. J. Plasticity 137, 102896. https://www.sciencedirect.com/science/article/pii/S0749641920304332

Bronkhorst, C. A., Cho, H., Marcy, P. W., Vander Wiel, S. A., Gupta, S., Versino, D., Livescu, V., Gray III, G. T. (2021). Local Micro-Mechanical Stress Conditions Leading to Pore Nucleation during Dynamic Loading, Int. J. Plasticity 137, 102903. https://www.sciencedirect.com/science/article/pii/S0749641920307488

Lieou, C. K. C., Bronkhorst, C. A. (2021). Thermomechanical Conversion in Metals: Dislocation Plasticity Model Evaluation of the Taylor-Quinney Coefficient, Acta Mat. 202, 170. https://www.sciencedirect.com/science/article/pii/S1359645420308351

Feng, B., Bronkhorst, C. A., Liu, Z., Morrow, B. M., Cerreta, E. K., Li, W. H., Daphalapurkar, N. P. (2020). Three-Dimensional Modeling and Simulations of Single-Crystal and Bi-Crystal Titanium for High-Strain-Rate Loading Conditions, Int. J. Plasticity 133, 102771. https://www.sciencedirect.com/science/article/pii/S0749641919307338

Lieou, C. K. C., Bronkhorst, C. A. (2020). Thermodynamic Theory of Crystal Plasticity: Formulation and Application to Polycrystal FCC Copper, J. Mech. Phys. Solids 138, 103905. https://www.sciencedirect.com/science/article/pii/S0022509620301411

Svolos, L., Bronkhorst, C. A., Waisman, H. (2020). Thermal-conductivity degradation across cracks in coupled thermo-mechanical systems modeled by the phase-field fracture method, J. Mech. Phys. Solids 137, 103861. https://www.sciencedirect.com/science/article/pii/S0022509619309317

Marcy, P. W., Vander Wiel, S. A., Storlie, C. B., Livescu, V., Bronkhorst, C. A. (2020). Modeling Material Stress using Integrated Gaussian Markov Random Fields. J. Appl. Statistics 47, 1616-1636. https://www.tandfonline.com/doi/full/10.1080/02664763.2019.1686131

Jin, T., Mourad, H. M., Bronkhorst, C. A., Beyerlein, I. J. (2019). Incorporation of Deformation Twinning in Single-Crystal Elasto-Viscoplasticity Model with Embedded Weak Discontinuity. J. Mech. Phys. Solids 133, 103723. https://www.sciencedirect.com/science/article/pii/S0022509619304466

Jin, T., Mourad, H. M., Bronkhorst, C. A., Livescu, V., Zhang, X., Linder, C., Regueiro, R. A. (2019). Three Dimensional Finite Element Formulation for Shear Localization with Global Tracking of Embedded Weak Discontinuities. Comp. Meth. Appl. Mech. Engr. 353, 416-447. https://www.sciencedirect.com/science/article/pii/S0045782519302804

Feng, B., Bronkhorst, C. A., Addessio, F. L., Morrow, B. M., Li, W. H., Lookman, T., Cerreta, E. K. (2019). Coupled Nonlinear Elasticity, Plastic Slip, Twinning, and Phase Transformation in Single Crystal Titanium for Plate Impact Loading. J. Mech. Phys. Solids 127, 358-385. https://www.sciencedirect.com/science/article/pii/S0022509618309219

Livescu, V., Beyerlein, I. J., Bronkhorst, C. A., Dippo, O. F., Ndefru, B. G., Capolungo, L., Mourad, H. M. (2019). Microstructure Insensitive Twinning: A Statistical Analysis of Incipient Twins in High-Purity Titanium. Materialia 6, 100303. https://www.sciencedirect.com/science/article/pii/S2589152919300997

Lieou, C. K. C., Mourad, H. M., Bronkhorst, C. A. (2019). Strain Localization and Dynamic Recrystallization in Polycrystalline Metals: Thermodynamic Theory and Simulation Framework. Int. J. Plasticity 119, 171-187. https://www.sciencedirect.com/science/article/pii/S0749641918305461

Bronkhorst, C. A., Mayeur, J. R., Livescu, V., Pokharel, R., Brown, D. W., Gray III, G. T. (2019). Structural Representation of Additively Manufactured 316L Austenitic Stainless Steel. Int. J. Plasticity 118, 70-86. https://www.sciencedirect.com/science/article/pii/S0749641918307654

Jin, T., Mourad, H. M., Bronkhorst, C. A. (2019). A Comparative Study of Shear Band Tracking Strategies in Three-Dimensional Finite Elements with Embedded Weak Discontinuities. Finite Elements in Analysis & Design 155, 11-31. https://www.sciencedirect.com/science/article/pii/S0168874X18304438

Lieou, C. K. C., Bronkhorst, C. A. (2018). Dynamic Recrystallization in Adiabatic Shear Banding: Effective-Temperature Model and Comparison to Experiments in Ultrafine-Grained Titanium. Int. J. Plasticity 111, 107-121. https://www.sciencedirect.com/science/article/pii/S0749641918302821

Weaver, J. S., Li, N., Mara, N. A., Jones, D. R., Cho, H., Bronkhorst, C. A., Fensin, S., Gray III, G. T. (2018). Slip Transmission of High Angle Grain Boundaries in Body-Centered Cubic Metals: Micropillar Compression of Pure Ta Single and Bi-Crystals. Acta Mat. 156, 356-368. https://www.sciencedirect.com/science/article/pii/S1359645418305147

Feng, B., Bronkhorst, C. A., Addessio, F. L., Morrow, B. M., Cerreta, E. K., Lookman, T., Lebensohn, R. A., Low, T. (2018). Coupled Nonlinear Elasticity, Plastic Slip, and Twinning in Single Crystal Titanium Loaded by Split-Hopkinson Pressure Bar. J. Mech. Phys. Solids 119, 274-297. https://www.sciencedirect.com/science/article/pii/S0022509618309219

Cho, H., Bronkhorst, C. A., Mourad, H. M., Mayeur, J. R., Luscher, D. J. (2018). Anomalous Plasticity of Body-Centered-Cubic Crystals with Non-Schmid Effects. Int. J. Solids Struct. 139-140, 138-149. https://www.sciencedirect.com/science/article/pii/S0020768318300313

Pang, B., Case, S., Jones, I. P., Millett, J. C. F., Whiteman, G., Chiu, Y. L., Bronkhorst, C. A. (2018). The Defect Evolution in Shock Loaded Tantalum Single Crystals. Acta Mat. 148, 482-491. https://www.sciencedirect.com/science/article/pii/S1359645417310431

Versino, D., Bronkhorst, C. A. (2018). A Computationally Efficient Ductile Damage Model Accounting for Micro-Inertia. Comp. Meth. Appl. Mech. Engr. 333, 395-420. https://www.sciencedirect.com/science/article/pii/S0045782518300306

Kumar, A., Bronkhorst, C. A., Lookman, T. (2018). First-Principles Study of the α-ω Phase Transformation in Ti and Zr Coupled to Slip Modes. J. Appl. Phys. 123, 045903. https://aip.scitation.org/doi/10.1063/1.5007074

Ghosh, S., Bronkhorst, C. A. (2018). Foreword – Special Issue “Integrated Structure – Material Modeling, Comput. Mech. 61, 1-2. https://link.springer.com/article/10.1007/s00466-017-1506-0

Jin, T., Mourad, H. M., Bronkhorst, C. A., Livescu, V. (2018). Finite Element Formulation with Embedded Weak Discontinuities for Strain Localization under Dynamic Conditions. Comp. Mech. 61, 3-18. https://link.springer.com/article/10.1007/s00466-017-1470-8

Bronkhorst, C. A., Ghosh, S. (2017). Integrated Computational Structure – Material Modeling of Deformation and Failure under Extreme Conditions, Int. J. Fract. 208, 1-3. https://link.springer.com/article/10.1007/s10704-017-0253-8

Versino, D., Alberto, T., Bronkhorst, C. A. (2017). Data Driven Modeling of Plastic Deformation. Comp. Methods Appl. Mech. Engr. 318, 981-1004. https://www.sciencedirect.com/science/article/pii/S0045782516314499

Francois, M. M., Sun, A., King, W. E., Henson, N. J., Tourret, D., Bronkhorst, C. A., Carlson, N. N., Newman, C. K., Haut, T., Bakosi, J., Gibbs, J. W., Livescu, V., Vander Wiel, S. A., Clarke, A. J., Schraad, M. W., Blacker, T., Lim, H., Rodgers, T., Owen, S., Abdeljawad, F., Madison, J., Anderson, A. T., Fattebert, J.-L., Ferencz, R. M., Hodge, N. E., Khairallah, S. A., Walton, O. (2017). Modeling of Additive Manufacturing Processes for Metals: Challenges and Opportunities. Current Opinion in Sol. State and Mats. Sci. 21, 198-206. https://www.sciencedirect.com/science/article/pii/S1359028616300833?via%3Dihub

Mourad, H. M., Bronkhorst, C. A., Plohr, J. N., Livescu, V. and Cerreta, E. K. (2017). Modeling and Simulation Framework for Dynamic Strain Localization in Elasto-viscoplastic Metallic Materials subject to Large Deformations. Int. J. Plasticity 88, 1-26. https://www.sciencedirect.com/science/article/pii/S0749641916301668

Vacchani, S., Trujillo, C., Mara, N., Livescu, V., Bronkhorst, C. A., Gray III, G. T., Cerreta, E. K. (2016). Local Mechanical Property Evolution during High Strain-Rate Deformation of Tantalum. J. Dyn. Behavior Mats. 2, 511-520. https://link.springer.com/article/10.1007/s40870-016-0085-z

Lieberman, E. J., Lebensohn, R. A., Bronkhorst, C. A., Rollett, A. D. (2016). Microstructural Effects on Damage Evolution in Shocked Copper Polycrystals. Acta Mat. 116, 270-280. https://www.sciencedirect.com/science/article/pii/S1359645416304852

Morrow, B. M., Lebensohn, R. A., Trujillo, C. P., Martinez, D. T., Addessio, F. L., Bronkhorst, C. A., Lookman, T., Cerreta, E. K. (2016). Characterization and Modeling of Mechanical Behavior of Single Crystal Titanium Deformed by Split-Hopkinson Pressure Bar, Int. J. Plasticity 82, 225-240. https://www.sciencedirect.com/science/article/pii/S0749641916300365

Bronkhorst, C. A., Gray III, G. T., Addessio, F. L., Livescu, V., Bourne, N. K., MacDonald, S. A., Withers, P. J. (2016). Response and Representation of Ductile Damage under Varying Shock Loading Conditions in Tantalum, J. Appl. Phys. 119, 085103. Cover Article. https://aip.scitation.org/doi/10.1063/1.4941823

Long, C. C., Zhang, D. Z., Bronkhorst, C. A., Gray III, G. T. (2016). Representing Ductile Damage with the Dual Domain Material Point Method, Comput. Methods Appl. Mech. Eng. 300, 611-627. https://www.sciencedirect.com/science/article/pii/S0045782515004016

Alleman, C., Luscher, D. J., Bronkhorst, C. A., Ghosh, S. (2015). Distributed-Enhanced Homogenization Framework and Model for Heterogeneous Elasto-Plastic Problems, J. Mech. Phys. Solids 85, 176-202. https://www.sciencedirect.com/science/article/pii/S0022509615301551

Mayeur, J. R., Beyerlein, I. J., Bronkhorst, C. A., Mourad, H. M. (2015). Incorporating Interface Affected Zones into Crystal Plasticity, Int. J. Plasticity 65, 206-225. https://www.sciencedirect.com/science/article/pii/S0749641914001806

Zong, H., Ding, X., Lookman, T., Li, J., Sun, J., Cerreta, E. K., Escobedo, A. P., Addessio, F. L., Bronkhorst, C. A. (2014). Collective Nature of Plasticity in Mediating Phase Transformation under Shock Compression, Phys. Rev. B 89, 220101 – 1-5. https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.220101

Mayeur, J. R., Beyerlein, I. J., Bronkhorst, C. A., Mourad, H. M. (2013). The Influence of Grain Interactions on the Plastic Stability of Heterophase Interfaces, Materials 7, 302-322. https://www.mdpi.com/1996-1944/7/1/302

Cerreta, E. K., Escobedo, J. P., Rigg, P. A., Trujillo, C. P., Brown, D. W., Sisneros, T. A., Clausen, B., Lopez, M. F., Lookman, T., Bronkhorst, C. A., Addessio, F. L. (2013). The Influence of Phase and Substructural Evolution during Dynamic Loading on Subsequent Mechanical Properties of Zirconium, Acta Mat. 61, 7712-7719. https://www.sciencedirect.com/science/article/pii/S1359645413006940

Mourad, H. M., Bronkhorst, C. A., Addessio, F. L., Cady, C. M., Brown, D. W., Chen, S.-R., Gray III, G. T. (2014). Incrementally Objective Implicit Integration of Hypoelastic-viscoplastic Constitutive Equations Based on the Mechanical Threshold Strength Model, Comp. Mech. 53, 941-955. https://link.springer.com/article/10.1007/s00466-013-0941-9

Alleman, C., Ghosh, S., Luscher, D. J., Bronkhorst, C. A. (2013). Evaluating the Effects of Loading Parameters on Single Crystal Slip in Tantalum Using Molecular Mechanics, Phil. Mag. 94, 92-116. https://www.tandfonline.com/doi/abs/10.1080/14786435.2013.843795

Luscher, D. J., Bronkhorst, C. A., Alleman, C. N., Addessio, F. L. (2013). A Model for Finite-Deformation Nonlinear Thermodynamical Response of Single Crystal Copper under Shock Conditions, J. Mech. Phys. Solids 61, 1877-1894. https://www.sciencedirect.com/science/article/pii/S0022509613000914

Bronkhorst, C. A., Mayeur, J. R., Beyerlein, I. J., Mourad, H. M., Hansen, B. L., Mara, N. A., Carpenter, J. S., McCabe, R. J., Sintay, S. D. (2013). Meso-scale Modeling the Orientation and Interface Stability of Cu/Nb Layered Composites by Rolling, JOM 65(3), 431-442. Invited. https://link.springer.com/article/10.1007/s11837-012-0541-8

Mayeur, J. R., Beyerlein, I. J., Bronkhorst, C. A., Mourad, H. M., Hansen, B. L. (2013). A Crystal Plasticity Study of Heterophase Interface Character of Cu/Nb Bicrystals, Int. J. Plasticity 48, 72-91. https://www.sciencedirect.com/science/article/pii/S0749641913000430

Hansen, B. L., Carpenter, J. S., Sintay, S. D., Bronkhorst, C. A., McCabe, R. J., Mayeur, J. R., Mourad, H. M., Beyerlein, I. J., Mara, N. A., Chen, S. R., Gray, G. T. (2013). Modeling the Texture Evolution of Cu/Nb Layered Composites during Rolling, Int. J. Plasticity 49, 71-84. https://www.sciencedirect.com/science/article/pii/S0749641913000636

Lebensohn, R. A., Escobedo, J. P., Cerreta, E. K., Dennis-Koller, D., Bronkhorst, C. A., Bingert, J. F. (2013). Modeling Void Growth in Polycrystalline Materials, Acta Mat. 61, 6918-6932. https://www.sciencedirect.com/science/article/pii/S1359645413005909

Hansen, B. L., Beyerlein, I. J., Bronkhorst, C. A., Cerreta, E. K., Dennis-Koller, D. (2013). A Dislocation-based Multi-rate Single Crystal Plasticity Model, Int. J. Plasticity 44, 129-146. https://www.sciencedirect.com/science/article/pii/S0749641912001957

Cerreta, E. K., Binger, J. F., Gray, G. T., Trujillo, C. P., Lopez, M. F., Bronkhorst, C. A., Hansen, B. L. (2013), Microstructural Examination of Quasi-Static and Dynamic Shear in High-Purity Iron, Int. J. Plasticity 40, 23-38. https://www.sciencedirect.com/science/article/pii/S074964191200099X

Escobedo, J. P., Cerreta, E. K., Dennis-Koller, D., Trujillo, C. P., Bronkhorst, C. A. (2013). Influence of Boundary Structure and Near Neighbor Crystallographic Orientation on the Dynamic Damage Evolution during Shock Loading, Phil. Mag. 93, 833-846. https://www.tandfonline.com/doi/abs/10.1080/14786435.2012.734638

Luscher, D. J., Bronkhorst, C. A., McDowell, D. L. (2012). Effects of Local and Nonlocal Substructure Spin on Localization in Tantalum Top-Hat Specimen, Technische Mechanik 323, 393. https://www.semanticscholar.org/paper/Effects-of-Local-and-Nonlocal-Substructure-Spin-on-Luscher-Bronkhorst/51cf79b82485d5329d98a9e2372e37b51a031ca8

Luscher, D. J., McDowell, D. L., Bronkhorst, C. A. (2012). Essential Features of Fine Scale Boundary Conditions for Second Gradient Multiscale Homogenization of Statistical Volume Elements, Int. J. Multiscale Comp. Engr. 10, 1543-1649. http://www.dl.begellhouse.com/en/journals/61fd1b191cf7e96f,58ae1c03228f5d43,22758e010b9fed16.html

Escobedo, J. P., Cerreta, E. K., Dennis-Koller, D., Patterson, B. M., Bronkhorst, C. A., Hansen, B. L., Tonks, D., Lebensohn, R. A. (2011). Effects of Grain Size and Boundary Structure on the Dynamic Tensile Response of Copper, J. Appl. Phys. 110, 033513. https://aip.scitation.org/doi/10.1063/1.3607294

Bronkhorst, C. A., Kalidindi, S. R., Zavaliangos, A., Thamburaja, P. (2010). Editorial – Special Issue in honor of Lallit Anand, Int. J. Plasticity 26, 1071-1072. https://www.sciencedirect.com/science/article/pii/S074964191000080X

Patterson, B. M., Cerreta, E. K., Dennis-Koller, D., Bronkhorst, C. A., Hansen, B. L. (2010). Characterization of Dynamic Damage of Copper Targets using X-ray Micro Tomography, Microscopy and Microanalysis 16, 726-727. https://www.cambridge.org/core/journals/microscopy-and-microanalysis/article/characterization-of-dynamic-damage-of-copper-targets-using-xray-micro-tomography/3A12A5035CA0D84F506014977FE27095#

Cerreta, E. K., Gray III, G. T., Dennis-Koller, D., Bronkhorst, C. A., Trujillo, C. P., Hansen, B. L. (2010). The Influence of Shock Loading on Material Properties and Dynamic Damage Evolution, J. Acoustical Soc. America 127, 1948. https://asa.scitation.org/doi/10.1121/1.3384924

Bronkhorst, C. A., Ross, A. R., Hansen, B. L., Cerreta, E. K., Bingert, J. F. (2010). Modeling and Characterization of Grain Scale Strain Distribution in Polycrystalline Tantalum, Computers, Materials, & Continua 17, 149-174. https://www.techscience.com/cmc/v17n2/33468

Hansen, B. L., Bronkhorst, C. A., Ortiz, M. (2010). Dislocation Subgrain Structures and Modeling the Plastic Hardening of Metallic Single Crystals, Modelling Simulation Mater. Sci. Eng., 18, 055001. https://iopscience.iop.org/article/10.1088/0965-0393/18/5/055001/meta

Luscher, D. J., McDowell, D. L., Bronkhorst, C. A. (2010). A Second Gradient Theoretical Framework for Hierarchical Multiscale Modeling of Materials, Int. J. Plasticity 26, 1248-1275. https://www.sciencedirect.com/science/article/pii/S0749641910000719