My colleague, Konstantin A. Lurie, began working in a novel paradigm of spatio-temporal composites or dynamic materials in the 1990s from an analytical point of view. I work on the computational end of things with supporting analysis. Our collaborators include William Sanguinet, Daniel Onofrei, Vadim Yakovlev, Mihhail Berezovski, and others.

Dynamic materials are formations assembled from material propertiess which are distributed on a microscale in space and in time. This material concept takes into consideration inertial, elastic, electromagnetic and other material properties that affect the dynamic behavior of various mechanical, electrical and environmental systems. In static or non-smart applications, the design variables, such as material density and stiffness, yield force and other structural parameters are position dependent but invariant in time. When it comes to dynamic applications, we also need temporal variability in the material properties in order to adequately match the changing environment. To this end, in dynamic material design, dynamic materials will take up the role played by ordinary composites in static material design. A dynamic disturbance on a scale much greater than the scale of a spatio-temporal microstructure will perceive this formation as a new material with its own effective properties.

By allowing spatio-temporal variability in the material constituents, we can create effects that are unachievable through purely spatial design. For example:

• Pulse Compression and Energy Accumulation in a Dynamic Checkerboard Material

• The Screening Effect Produced by a Dynamic Laminate

• Elimination of Cutoff Frequency in Waveguides and Dielectric Layers

• Activated Left-handed Medium, Negative Index Performance

Mathematically, the problem is formulated for linear, hyperbolic equations with spatio-temporally varying coefficients. Both analytic and computational means are applied to the analysis of the effective properties of dynamic materials generated by various microstructures.

Books

• An Introduction to the Mathematical Theory of Dynamic Materials (Advances in Mechanics and Mathematics), 2007

• An Introduction to the Mathematical Theory of Dynamic Materials, Second Edition, 2017

Pulse Compression and Energy Accumulation in a Dynamic Checkerboard Material

• William C. Sanguinet, PhD Dissertation, 2017, Various Extensions in the Theory of Checkerboard Materials with a Specific Emphasis on Checkerboard Geometry, advised by K. A. Lurie and S. L. Weekes

• Energy Accumulation in a Functionally Graded Spatial-Temporal Checkerboard, in IEEE Antennas and Wireless Propagation Letters 16, 1496-1499 (2017), K. A. Lurie, D. Onofrei, W. C. Sanguinet, S. L. Weekes, and V. V. Yakovlev

• Propagation of dilatation and shear waves through a dynamic checkerboard material in 1d-space+time. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f¨ur Angewandte Mathematik und Mechanik, 93(12):937–943, (2013), by W. C. Sanguinet and K. A. Lurie.

• Mathematical Analysis of the Waves Propagation through a Rectangular Material Structure in Space-Time, in Journal of Mathematical Analysis and Applications, 355 (2009), 180-194, K. A. Lurie, D. Onofrei, and S. L. Weekes

• Wave Propagation and Energy Exchange in a Spatio-Temporal Material Composite with Rectangular Microstructure in Journal of Mathematical Analysis and Applications, 314 (2006), 286-310, K. A. Lurie and S. L. Weekes

Other Work:

• Effective and Averaged Energy Densities in One-Dimensional Wave Propagation through Spatio-Temporal Dielectric Laminates with Negative Effective Values of epsilon and mu, in ``Nonlinear Analysis and Applications: To V. Lakshmikantham on His 80th Birthday'', edited by R. Agarwal and D. O'Regan, Kluwer Academic Publishers, Jan 2004, 767--789, K. A. Lurie and S. L. Weekes

• Some New Advances in the Theory of Dynamic Material, Proceedings of the 2003 International Conference "Physics and Control," St. Petersburg, Russia, K. A. Lurie and S. L. Weekes

• A Stable Scheme for the Numerical Computation of Long Wave Propagation in Temporal Laminates J. Computational Physics, 176 (2002) 345-362, S. L. Weekes

• A Dispersive Effective Equation for Wave Propagation through Dynamic Laminates, Wave Motion, Volume 38, Issue 1 , June 2003, 25-41, S. L. Weekes

• G-closures of Material Sets in Space-Time and Perspectives of Dynamic Control in the Coefficients of Linear Hyperbolic Equations , J. of Control and Cybernetics, (1998) Vol. 27, No. 2, 283-294, K. A. Lurie

• The Problem of Effective Properties of a Mixture of Two Isotropic Dielectrics Distributed in Space-Time and the Conservation Law for Wave Impedance in One-Dimensional Wave Propagation , Proc. R. Soc. Lond. A. (1998) 454, 1767-1779, K. A. Lurie

• Bounds for the Electromagnetic Material Properties of a Spatio-Temporal Dielectric Polycrystal with Respect to One-Dimensional Wave Propagation , Proc. R. Soc. Lond. A. (2000) 456, 1547-1557, K. A. Lurie

  • http://rspa.royalsocietypublishing.org/content/456/1999/1547

• Low Frequency Longitudinal Vibrations of an Elastic Bar Made of Dynamic Material and Excited at One End Journal Math. Anal. and Appl., (2000) 251, 364-375, K. A. Lurie

• On Homogenization of Activated Laminates in 1D-Space and Time, in Zeitschrift fuer Angewandte Mathematik und Mechanik, 89, #4, 233-340 (April 2009) , K. A. Lurie

• Dynamic Materials with Formation of Clots: Application to Optimal Mass Transport in One Spatial Dimension without Takeover, in Zeitschrift fuer Angewandte Mathematik und Mechanik, 90, #3, 241-256 (March 2010), K. A. Lurie

Some of our work was supported by the National Science Foundation under DMS-0204673.