Despite extensive research over the past three decades into how indentation depth affects the hardness (H) of both crystalline and non-crystalline materials, a mechanistic understanding of this phenomenon remains elusive. Here, we report that the depth dependence of H is also present in bulk metallic glasses. Importantly, indentation depth dependence is observed not only in hardness but also in the reduced elastic modulus Er. We observed that H initially increases with increasing indentation depth ht up to the yielding point. Beyond this point, however, it decreases with further increase of ht, indicating the presence of an indentation depth dependence in the plastic regions. The evolution of Er follows a similar trend. Based on our findings, firstly, we established the relationship between indentation hardness and the ratio of contact radius to indentation depth using classical Hertzian contact mechanics. Then, we developed a model based on the atomic-scale cooperative shear mechanism to interpret the indentation size effects in bulk metallic glasses. Furthermore, we observed that H correlates with the cube of the ratio of indentation elastic depth he to total depth ht, or alternatively, with the ratio of indentation elastic work to total work. Our findings gave a scaling law that uncovers an inherent relationship of hardness with the mean pressure at the onset of plasticity, flow hardness, and the ratio. The work underscores that the indentation depth effect stems from the interplay between elasticity and plasticity, rather than being solely influenced by factors like indentation depth, contact area, or indenter radius. This highlights its crucial role in comprehending and evaluating the plastic deformation of bulk metallic glasses at the submicron scale.
尽管过去三十年来对压痕深度如何影响晶体和非晶体材料的硬度（H）进行了广泛研究，但对其现象的机理理解仍不明确。在此，我们报告了在大块金属玻璃中也存在硬度（H）对压痕深度的依赖性。重要的是，这种压痕深度依赖性不仅在硬度上有所体现，在归一化弹性模量 Er 上也有观察到。我们发现，硬度 H 在压痕深度 ht 增加到屈服点之前会随其增加而增大，然而超过该点后，硬度 H 会随着 ht 的进一步增加而减小，这表明在塑性区域存在压痕深度依赖性。Er 的变化趋势也类似。基于我们的发现，首先，我们利用经典的赫兹接触力学建立了压痕硬度与接触半径与压痕深度比值之间的关系。然后，我们基于原子尺度的协同剪切机制开发了一个模型来解释大块金属玻璃中的压痕尺寸效应。此外，我们还观察到 H 与压痕弹性深度 he 与总深度 ht 的比值的立方相关，或者与压痕弹性功与总功的比值相关。我们的研究结果揭示了一种标度律，它揭示了硬度与塑性变形起始时的平均压力、流动硬度以及该比值之间的内在关系。这项工作强调了压痕深度效应源于弹性与塑性的相互作用，而非仅仅受压痕深度、接触面积或压头半径等因素的影响。这突显了其在理解及评估大块金属玻璃在亚微米尺度下的塑性变形中的关键作用。
Journal of the Mechanics and Physics of Solids
The impacts of thermoelastic anisotropy and grain boundary misorientation on microcracking in ceramics
Andrew R. Ericks, Frank W. Zok, Daniel S. Gianola, Matthew R. Begley
doi:10.1016/j.jmps.2024.106024
热弹性各向异性和晶界取向偏差对陶瓷微裂纹的影响
This paper examines the role of thermoelastic anisotropy on grain boundary cracking in brittle materials using a highly efficient computational framework. Energy release rates (ERRs) are computed for 35 materials spanning all seven crystal systems. Two crack geometries are considered: short interface cracks in isolated bicrystals plates, and cracked grain boundaries in polycrystal plates comprising periodic hexagonal grains. Crack driving forces are computed for penetration through the plate thickness (for cracks of width equal to the length of a hexagonal grain boundary), extensions along bicrystal interfaces, transgranular cracks that emerge from triple junctions, and kinking into bulk materials and at grain triple junctions. The high throughput computational framework produces probably distributions for ERRs arising from randomly oriented grains; the distributions for cracks at grain edges in polycrystals are broader than those for short cracks along bicrystal interfaces. A broad study of different grain configurations also illustrates that only the first 5-6 rings of neighboring grains influence crack driving forces for a given interface. The implications for interpreting microcracking observations, quantifying the performance of textured ceramics, and designing two-phase ceramic composites are briefly discussed.
Hydromechanical field theory of plant morphogenesis
Hadrien Oliveri, Ibrahim Cheddadi
doi:10.1016/j.jmps.2025.106035
植物形态发生的流体力学场理论
The growth of plants is a hydromechanical phenomenon in which cells enlarge by absorbing water, while their walls expand and remodel under turgor-induced tension. In multicellular tissues, where cells are mechanically interconnected, morphogenesis results from the combined effect of local cell growths, which reflects the action of heterogeneous mechanical, physical, and chemical fields, each exerting varying degrees of nonlocal influence within the tissue. To describe this process, we propose a physical field theory of plant growth. This theory treats the tissue as a poromorphoelastic body, namely a growing poroelastic medium, where growth arises from pressure-induced deformations and osmotically-driven imbibition of the tissue. From this perspective, growing regions correspond to hydraulic sinks, leading to the possibility of complex non-local regulations, such as water competition and growth-induced water potential gradients. More in general, this work aims to establish foundations for a mechanistic, mechanical field theory of morphogenesis in plants, where growth arises from the interplay of multiple physical fields, and where biochemical regulations are integrated through specific physical parameters.
Multi-patch isogeometric analysis for smart plates with distributed piezoelectric patches
Tao Liu, Xiangrong Sun, Jinde Zheng, Lu Wang, Qingyun Liu, Tinh Quoc Bui
doi:10.1016/j.tws.2025.112937
分布式压电贴片智能板的多贴片等几何分析
The previous isogeometric analysis (IGA) on piezoelectric smart structures mainly focused on plates and shells that were fully covered with piezoelectric materials. However, the piezoelectric materials in smart structures commonly exist in the form of patches that are locally attached to the substrate structures, in practical engineering applications. Thus, this paper aims to apply IGA to analyze the electro-mechanical coupled behaviors of distributed piezoelectric smart plates. The Nitsche-based non-conforming multi-patch technology is adopted to deal with the precision limitations associated with single-patch IGA for distributed piezoelectric smart plates. In accordance with first-order shear deformation theory (FSDT) and NURBS-based IGA, the non-conforming multi-patch governing equations for piezoelectric smart plates are then derived. In particular, the Nitsche’s method is adopted for addressing the non-conforming meshes and ensuring the continuity of the field variables on the coupling boundary between two adjacent patches. The developed methodology is further extended to analyze the fully-covered and distributed piezoelectric smart plates. Meanwhile, to enhance the general applicability of the method, the piezoelectric smart plates integrated with traditional piezoelectric ceramics and macro-fiber composite (MFC) materials are designed in numerical examples. Finally, comprehensive assessments for natural frequency and static response of piezoelectric smart plates are carried out and then compared with the existing reference solutions or the results calculated by ABAQUS software to demonstrate the effectiveness and accuracy of the developed method. These numerical examples validate that the proposed method is capable of addressing the limited accuracy of single-patch IGA in distributed piezoelectric smart structures.
Thin-walled cylindrical shells are extensively used across various fields because of their exceptional load-carrying efficiency. In practical applications, these structures are typically subjected to localized axial compression rather than the uniform axial compression considered in traditional research. A reliable and efficient buckling design method for cylindrical shells under such localized loads has not been developed to date. To address this challenge, a machine learning (ML) approach is proposed in this study for predicting the lower-bound buckling design load of cylindrical shells under localized axial compression. The artificial neural network (ANN) is selected as the ML model. Based on the modified energy barrier approach (MEBA), 500 samples are obtained by numerical simulations and their results are used to train the ANN model. The ANN model takes six geometric parameters, three material parameters, and one localized axial compression parameter as the inputs, while the lower-bound buckling load and the knockdown factor are the outputs. The feasibility and accuracy of the proposed ANN model are demonstrated by comparison with existing design codes and experimental results. The results suggest that this ML-based approach can fully exploit the load-carrying capacity of shells under localized axial compression, enabling more efficient and lightweight designs.
AM-FEMU: An optimization method for additive manufacturing simulation parameters based on finite element model updating, utilizing three-dimensional deformation and melt pool temperature fields
Accurate model parameters are crucial for reliable metal additive manufacturing (AM) simulations, which are essential for understanding AM material formation mechanisms, designing AM components, and controlling manufacturing processes. This study addresses the discrepancy between AM simulations and experimental results by developing an Additive Manufacturing Finite Element Model Updating (AM-FEMU) method. The AM-FEMU method updates and optimizes the simulation parameters based on the temperature field of the melt pool and the deformation field of the substrate during the AM process. Online measurements of three-dimensional displacement and melt pool temperature were conducted using three-dimensional sampling moiré and multi-spectral colorimetric temperature measurement technologies. By comparing these measurements with finite element (FE) simulation, the heat source parameters and thermal expansion coefficient were updated successfully. Verification tests confirmed that the updated parameters significantly improved the accuracy of residual stress in AM simulations compared to the original parameters. This method promotes the application of FEMU in metal AM simulations, further providing a deeper understanding of the physical mechanism in metal AM process.
Thin-walled composite pressure vessels exhibit a promising potential for energy storage, but they are vulnerable to barely visible damage from random impacts. Yet, angle-dependent damage mechanisms remain unclear, challenging the structural design to resist impact loadings. Herein, this study elucidates the residual burst strength of composite vessels under low-velocity impacts, guiding impact-resistant design at varying impact angles. The impact damage model based on a segmented golden-section search algorithm enhances computational efficiency and accuracy. Results show that matrix damage intensifies within helical layers under small-angle oblique impacts, while fiber damage consistently concentrates in hoop layers, with delamination between the hoop and helical layers decreasing from the outer layers inward. Furthermore, impacts at different angles shift potential failure locations within the vessel, with small-angle oblique impacts resulting in a low peak impact force. Notably, increasing the proportion of the helical layer enhances resistance to oblique impacts, while thicker hoop layers improve resistance to vertical impacts; at high energies, both layers should be thickened regardless of impact angle. This work not only offers new insights into angle-dependent impact damage but also contributes to the design for impact-resistant enhancement of advanced composite pressure vessels.
