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首页> 外文期刊>Theoretical and Applied Fracture Mechanics >Equilibrium mechanics model of multiscaling by segmentation: Asymptotic solution for macro-meso-micro damage in anti-plane shear deformation
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Equilibrium mechanics model of multiscaling by segmentation: Asymptotic solution for macro-meso-micro damage in anti-plane shear deformation

机译：分段多尺度平衡力学模型：反平面剪切变形中宏观-微观-微观损伤的渐近解

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A multiscaling model of cracking is developed whereby a closed form solution is obtained for the description of microscopic material damage corresponding to different constraints of the micro-crack. Coupling between the damage at the macro- and micro-scale is made via the mesoscopic zone to smooth out the transition. The range of each scale can be controlled to permit the application of equilibrium mechanics. In principle, the procedure can be applied to simulate the behavior of a non-equilibrium process by a series of segmented equilibrium processes. It is rarely possible to obtain closed form solution for multiscale damage models. Exception is found for the present model of macro-meso-micro damage by cracking under anti-plane shear deformation. Accuracies in numerical computations are thus reduced to the evaluation of algebraic equations. Examined, in particular, are the inhomogenities at the microscale arising from uneven stiffness of the material microstructures which can vary the constraints on the micro-crack. These geometrically created inhomogeneities are simulated by the free-free, fixed-fixed and free-fixed boundary conditions. The encouraging aspect of this work is that the results for anti-plane are found to be similar to those for in-plane extension. This suggests further exploration of the procedure where scale segmentation can indeed be reduced to any size at will, in the limit approaching zero. This would in principle correspond to the non-equilibrium process. Keeping in mind that the present modeling process involves segmentation and connection, the latter requires a scale invariant criterion. To this end, the volume energy density function is used with a length parameter or an area parameter so that the product can cross scale. The details of this procedure would be beyond the scope of this discussion. Nevertheless, ample evidence will be shown in the work to follow that the volume energy density function is fundamental not only to the scale shifting process but also in the ways how microscopic constraints can influence macroscopic behavior.
机译：建立了裂纹的多尺度模型，从而获得了封闭形式的解，用于描述与微裂纹的不同约束相对应的微观材料损伤。宏观和微观损伤之间的耦合是通过介观区进行的，以平滑过渡。可以控制每个刻度的范围，以允许应用平衡力学。原则上，该程序可用于通过一系列分段的平衡过程来模拟非平衡过程的行为。对于多尺度损伤模型，几乎不可能获得封闭形式的解决方案。发现了在反平面剪切变形下开裂造成的宏观-微观-微观损伤的本模型的例外。因此，数值计算的精度降低到了代数方程的评估。特别要检查的是材料微观结构刚度不均匀所引起的微观尺度上的不均匀性，这种不均匀性会改变对微裂纹的限制。通过自由，自由，固定和自由边界条件来模拟这些几何上产生的不均匀性。这项工作的令人鼓舞的方面是，发现反平面的结果与平面内扩展的结果相似。这表明需要进一步探索该过程，在该过程中，可以将比例分割确实减小到任意大小，且极限接近零。原则上，这将对应于非平衡过程。请记住，当前的建模过程涉及分段和连接，后者需要尺度不变标准。为此，将体积能量密度函数与长度参数或面积参数一起使用，以便产品可以跨比例使用。此过程的详细信息将超出此讨论的范围。然而，将在工作中显示大量证据表明，体积能量密度函数不仅对于尺度转换过程至关重要，而且对于微观约束如何影响宏观行为的方式也至关重要。

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