Abstract Mechanical metamaterials feature engineered microstructures designed to exhibit exotic, and often counter-intuitive, effective behaviour such as negative Poisson’s ratio or negative compressibility. Such a specific response is often achieved through instability-induced transformations of the underlying periodic microstructure into one or multiple patterning modes. Due to a strong kinematic coupling of individual repeating microstructural cells, non-local behaviour and size effects emerge, which cannot easily be captured by classical homogenization schemes. In addition, the individual patterning modes can mutually interact in space as well as in time, while at the engineering scale the entire structure can buckle globally. For efficient numerical predictions of macroscale engineering applications, a micromorphic computational homogenization scheme has recently been developed (Rokos et al., J. Mech. Phys. Solids 123, 119–137, 2019). Although this framework is in principle capable of accounting for spatial and temporal interactions between individual patterning modes, its implementation relied on a gradient-based quasi-Newton solution technique. This solver is suboptimal because (i) it has sub-quadratic convergence, and (ii) the absence of Hessians does not allow for proper bifurcation analyses. Given that mechanical metamaterials often rely on controlled instabilities, these limitations are serious. Addressing them will reduce the dependency of the solution on the initial guess by perturbing the system towards the correct deformation when a bifurcation point is encountered. Eventually, this enables more accurate and reliable modelling and design of metamaterials. To achieve this goal, a full Newton method, entailing all derivations and definitions of the tangent operators, is provided in detail in this paper. The construction of the macroscopic tangent operator is not straightforward due to specific model assumptions on the decomposition of the underlying displacement field pertinent to the micromorphic framework, involving orthogonality constraints. Analytical expressions for the first and second variation of the total potential energy are given, and the complete algorithm is listed. The developed methodology is demonstrated with two examples in which a competition between local and global buckling exists and where multiple patterning modes emerge. The numerical results indicate that local to global buckling transition can be predicted within a relative error of 6% in terms of the applied strains. The expected pattern combinations are triggered even for the case of multiple patterns.

中文翻译：

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用于微形计算均质化的牛顿求解器，能够对模式转换超材料进行多尺度屈曲分析

摘要 机械超材料具有工程微结构，旨在表现出奇特的、通常违反直觉的有效行为，例如负泊松比或负压缩率。这种特定的响应通常是通过不稳定引起的潜在周期性微观结构转变为一种或多种图案化模式来实现的。由于单个重复微结构细胞的强运动学耦合，出现了非局部行为和尺寸效应，经典均质化方案无法轻易捕捉到这些效应。此外，各个图案模式可以在空间和时间上相互影响，而在工程规模上，整个结构可以全局弯曲。对于宏观工程应用的有效数值预测，最近开发了一种微形态计算均质化方案（Rokos 等人，J. Mech. Phys. Solids 123, 119–137, 2019）。尽管该框架原则上能够解释各个模式模式之间的空间和时间相互作用，但其实现依赖于基于梯度的准牛顿求解技术。这个求解器是次优的，因为 (i) 它具有次二次收敛，并且 (ii) 没有 Hessians 不允许进行适当的分叉分析。鉴于机械超材料通常依赖于可控的不稳定性，这些限制非常严重。解决它们将通过在遇到分叉点时使系统朝着正确的变形方向进行扰动来减少解对初始猜测的依赖性。最终，这可以实现更准确、更可靠的超材料建模和设计。为了实现这一目标，本文详细提供了一种完整的牛顿方法，其中包含切线算子的所有推导和定义。由于与微观框架相关的潜在位移场分解的特定模型假设，宏观切线算子的构造并不简单，涉及正交性约束。给出了总势能的第一次和第二次变化的解析表达式，并列出了完整的算法。开发的方法通过两个例子来展示，其中存在局部和全局屈曲之间的竞争以及出现多种模式模式。数值结果表明，就施加的应变而言，可以在 6% 的相对误差内预测局部到全局屈曲的转变。即使对于多个模式，也会触发预期的模式组合。