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1. Variable-order fracture mechanics and its application to dynamic fracture

   https://doi.org/10.1038/s41524-021-00492-x  

   Patnaik, Sansit; Semperlotti, Fabio (February 2021, npj Computational Materials)

   Abstract This study presents the formulation, the numerical solution, and the validation of a theoretical framework based on the concept of variable-order mechanics and capable of modeling dynamic fracture in brittle and quasi-brittle solids. More specifically, the reformulation of the elastodynamic problem via variable and fractional-order operators enables a unique and extremely powerful approach to model nucleation and propagation of cracks in solids under dynamic loading. The resulting dynamic fracture formulation is fully evolutionary, hence enabling the analysis of complex crack patterns without requiring any a priori assumption on the damage location and the growth path, and without using any algorithm to numerically track the evolving crack surface. The evolutionary nature of the variable-order formalism also prevents the need for additional partial differential equations to predict the evolution of the damage field, hence suggesting a conspicuous reduction in complexity and computational cost. Remarkably, the variable-order formulation is naturally capable of capturing extremely detailed features characteristic of dynamic crack propagation such as crack surface roughening as well as single and multiple branching. The accuracy and robustness of the proposed variable-order formulation are validated by comparing the results of direct numerical simulations with experimental data of typical benchmark problems available in the literature. 

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2. Applications of variable-order fractional operators: a review

   https://doi.org/10.1098/rspa.2019.0498  

   Patnaik, Sansit; Hollkamp, John P.; Semperlotti, Fabio (February 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling. 

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3. Fractional-Order Shell Theory: Formulation and Application to the Analysis of Nonlocal Cylindrical Panels

   https://doi.org/10.1115/1.4054677  

   Sidhardh, Sai; Patnaik, Sansit; Semperlotti, Fabio (August 2022, Journal of Applied Mechanics)

   Abstract We present a theoretical and computational framework based on fractional calculus for the analysis of the nonlocal static response of cylindrical shell panels. The differ-integral nature of fractional derivatives allows an efficient and accurate methodology to account for the effect of long-range (nonlocal) interactions in curved structures. More specifically, the use of frame-invariant fractional-order kinematic relations enables a physically, mathematically, and thermodynamically consistent formulation to model the nonlocal elastic interactions. To evaluate the response of these nonlocal shells under practical scenarios involving generalized loads and boundary conditions, the fractional-finite element method (f-FEM) is extended to incorporate shell elements based on the first-order shear-deformable displacement theory. Finally, numerical studies are performed exploring both the linear and the geometrically nonlinear static response of nonlocal cylindrical shell panels. This study is intended to provide a general foundation to investigate the nonlocal behavior of curved structures by means of fractional-order models. 

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4. Modeling metamaterials by second-order rate-type constitutive relations between only the macroscopic stress and strain

   Prusa, V; Rajagopal, K; Rodriguez, C; Trnka, L; Vejvoda, M (February 2025, arxiv_25_a)

   We propose a thermodynamically based approach for constructing effective rate-type constitutive relations describing finite deformations of metamaterials. The effective constitutive relations are formulated as second-order in time rate-type Eulerian constitutive relations between only the Cauchy stress tensor, the Hencky strain tensor and objective time derivatives thereof. In particular, there is no need to introduce additional quantities or concepts such as “micro-level deformation”,“micromorphic continua”, or elastic solids with frequency dependent material properties. Moreover, the linearisation of the proposed fully nonlinear (finite deformations) constitutive relations leads, in Fourier/frequency space, to the same constitutive relations as those commonly used in theories based on the concepts of frequency dependent density and/or stiffness. From this perspective the proposed constitutive relations reproduce the behaviour predicted by the frequency dependent density and/or stiffness models, but yet they work with constant—that is motion independent—material properties. This is clearly more convenient from the physical point of view. Furthermore, the linearised version of the proposed constitutive relations leads to the governing partial differential equations that are particularly simple both in Fourier space as well as in physical space. Finally, we argue that the proposed fully nonlinear (finite deformations) second-order in time rate-type constitutive relations do not fall into traditional classes of models for elastic solids (hyperelastic solids/Green elastic solids, first-order in time hypoelastic solids), and that the proposed constitutive relations embody a new class of constitutive relations characterising elastic solids. 

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5. Three-dimensional wave propagation and earthquake dynamic rupture simulations in complex poroelastic media

   https://doi.org/10.1093/gji/ggaf184  

   Wolf, Sebastian; Gabriel, Alice-Agnes; Galis, Martin; Moczo, Peter; Gregor, David; Bader, Michael (May 2025, Geophysical Journal International)

   SUMMARY Numerical simulations of earthquakes and seismic wave propagation require accurate material models of the solid Earth. In contrast to purely elastic rheology, poroelasticity accounts for pore fluid pressure and fluid flow in porous media. Poroelastic effects can alter both the seismic wave field and the dynamic rupture characteristics of earthquakes. For example, the presence of fluids may affect cascading multifault ruptures, potentially leading to larger-than-expected earthquakes. However, incorporating poroelastic coupling into the elastodynamic wave equations increases the computational complexity of numerical simulations compared to elastic or viscoelastic material models, as the underlying partial differential equations become stiff. In this study, we use a Discontinuous Galerkin solver with Arbitrary High-Order DERivative time stepping of the poroelastic wave equations implemented in the open-source software SeisSol to simulate 3-D complex seismic wave propagation and 3-D dynamic rupture in poroelastic media. We verify our approach for double-couple point sources using independent methods including a semi-analytical solution and a finite-difference scheme and a homogeneous full-space and a poroelastic layer-over-half-space model, respectively. In a realistic carbon capture and storage reservoir scenario at the Sleipner site in the Utsira Formation, Norway, we model 3-D wave propagation through poroelastic sandstone layers separated by impermeable shale. Our results show a sudden change in the pressure field across material interfaces, which manifests as a discontinuity when viewed at the length scale of the dominant wavelengths of S or fast P waves. Accurately resolving the resulting steep pressure gradient dramatically increases the computational demands, requiring high-resolution modelling. We show that the Gassmann elastic equivalent model yields almost identical results to the fully poroelastic model when focusing solely on solid particle velocities. We extend this approach using suitable numerical fluxes to 3-D dynamic rupture simulations in complex fault systems, presenting the first 3-D scenarios that combine poroelastic media with geometrically complex, multifault rupture dynamics and tetrahedral meshes. Our findings reveal that, in contrast to modelling wave propagation only, poroelastic materials significantly alter rupture characteristics compared to using elastic equivalent media since the elastic equivalent fails to capture the evolution of pore pressure. Particularly in fault branching scenarios, the Biot coefficient plays a key role in either promoting or inhibiting fault activation. In some cases, ruptures are diverted to secondary faults, while in others, poroelastic effects induce rupture arrest. In a fault zone dynamic rupture model, we find poroelasticity aiding pulse-like rupture. A healing front is induced by the reduced pore pressure due to reflected waves from the boundaries of the poroelastic damage zone. Our results highlight that poroelastic effects are important for realistic simulations of seismic waves and earthquake rupture dynamics. In particular, our poroelastic simulations may offer new insights on the complexity of multifault rupture dynamics, fault-to-fault interaction and seismic wave propagation in realistic models of the Earth’s subsurface. 

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