Abstract
The mechanics of solids is increasingly shaped by the need to model and predict phenomena that span multiple scales and involve the interplay of different physical processes. Real-world materials and structures rarely operate under single-field conditions; instead, mechanical, thermal, acoustic, and electromechanical effects interact in ways that strongly influence their overall behaviour. The contributions collected in this Special Collection of Science Progress illustrate recent progress in addressing such complexity. Together, they demonstrate how phase-field modelling, isogeometric analysis, micromechanical homogenization, and machine learning can be brought to bear on some of the most challenging problems in solid mechanics and materials science.
A central topic in the collection is fracture under thermal loading, a longstanding problem in engineering applications ranging from ceramics to composites. Yang and co-workers [doi:10.1177/00368504251325746] propose a phase-field framework for simulating thermal cracking in which a length-scale-insensitive degradation function is introduced. This formulation alleviates the mesh dependence that has often undermined predictive fidelity in thermo-fracture simulations. By ensuring that thermal gradients and crack propagation are consistently coupled, the method opens the way for more reliable modelling of failure in systems where temperature effects are decisive.
Wave propagation in solids provides another example where multiphysics considerations are crucial. Gao and colleagues [doi:10.1177/00368504251357783] focus on two-dimensional acoustic models with ground reflection, a problem of practical importance in environmental acoustics and vibroacoustic design. They integrate Taylor expansion approximations with deep neural network surrogates into an isogeometric framework. The resulting approach maintains the geometric accuracy and spectral properties of isogeometric discretization while significantly accelerating computation. Their work exemplifies a growing trend in the field: combining rigourous numerical techniques with data-driven accelerators to make high-fidelity simulation feasible for large and complex systems.
The collection also highlights the role of machine learning in predicting functional properties of advanced materials. Ferroelectrics and piezoelectrics present particular challenges because their dielectric and electromechanical responses depend sensitively on microstructure and composition. Wang et al. [doi:10.1177/00368504251320846] demonstrate that deep neural networks can be trained to predict dielectric properties of ferroelectric materials with accuracy and efficiency, thereby providing a powerful surrogate for computationally expensive first-principles or continuum approaches. Building on this theme, Zhang and co-authors [doi:10.1177/00368504251359081] apply a similar strategy to sodium bismuth titanate (NBT)-based ceramics, predicting their piezoelectric properties through neural network modelling. These two studies, taken together, show how data-driven approaches are increasingly capable of bridging the gap between complex material descriptors and macroscopic functional performance, enabling faster exploration of design spaces for next-generation lead-free ceramics.
Complementing these methodological and data-driven contributions is a study of micromechanical interphases in bio-based composites. Grant, Hadavinia, Williams, and Koutsonas [doi:10.1177/00368504251324044] address the challenge of characterizing the elastic properties of interphase regions in cellulose nanocrystal/epoxy nanocomposites. Using a combination of analytical homogenization and finite element simulations, they quantify the differences between untreated and esterified interphases. Their results demonstrate how subtle chemical modifications can significantly influence interphase stiffness and thereby alter the effective mechanical properties of the composite. This work underscores the importance of interphases in multiscale mechanics and highlights the potential of bio-nanocomposites as sustainable high-performance materials.
Although these contributions address problems that range from thermal fracture and acoustic wave propagation to dielectric and piezoelectric functionality and composite micromechanics, several common threads emerge. Each study underscores the need for methods that can faithfully transmit information across scales, whether from microstructure to macroscopic response or from detailed boundary interactions to large-scale system behaviour. Each also emphasizes efficiency alongside rigour, whether through length-scale-insensitive formulations, surrogate modelling with deep learning, or hybrid analytical–numerical approaches. Above all, the works show how advances in modelling are driven by application challenges, ensuring that theoretical and computational developments remain connected to practical engineering needs.
In presenting these five open-access contributions, this Special Collection provides a timely overview of the challenges and opportunities at the intersection of scales and physics in solids. The works collected here will serve as both practical resources and conceptual inspiration for researchers in mechanics, materials science, and applied mathematics. We thank the authors, reviewers, and editorial staff whose efforts have made this collection possible, and we hope it will stimulate further advances in the predictive modelling of complex solid systems.
