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A Strain Gradient Generalized Continuum Approach For Modelling Elastic Scale Effects

机译：弹性尺度效应建模的应变梯度广义连续谱方法

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摘要

This work follows a generalized continuum framework [C. Sansour, A unified concept of elastic-visco-plastic cosserat and micromorphic continua, J. Phys. IV Proc. 8 (1998) 341-348] to derive a first strain gradient formulation which features a generalized deformation description, new strain and stress measures. As a consequence of these new quantities a corresponding generalized variational principle is formulated and its underlying equilibrium equations are derived. The approach is completed by Dirichlet boundary conditions for the displacement field and its derivatives. The basic idea behind this generalized continuum theory is the consideration of a micro- and a macro-space which span together the generalized space. The approach is appealing from theoretical as well as numerical point of view as it allows for the consideration of classical material laws and circumvents the use of otherwise cumbersome representation theorems. The resulting expressions for first and second order stresses are obtained by numerical integration over the micro-space. In this way material information of the micro-space, which is here only the geometrical specifications of the micro-continuum, can naturally enter the constitutive law. Moreover, non-linear material behaviour can be considered in a straightforward manner. In this work conventional hyperelasticity will be considered.rnFour applications in the context of linear and non-linear hyperelasticity demonstrate the potential of the proposed method. In particular, the use of a moving least square-based meshfree method facilitates a pure displacement-based approximation scheme, as it can provide C1 continuity which is required for this strain gradient formulation.
机译：这项工作遵循通用的连续体框架[C. Sansour，弹性-粘塑性cosserat和微晶连续体的统一概念，J。Phys。 IV过程8（1998）341-348]推导了第一应变梯度公式，该公式具有广义的变形描述，新的应变和应力测量。这些新量的结果是，制定了相应的广义变分原理，并推导了其基本的平衡方程。该方法由Dirichlet位移场及其导数的边界条件完成。广义连续论的背后的基本思想是考虑将微观空间和宏观空间连在一起的广义空间。该方法从理论和数字的角度都具有吸引力，因为它允许考虑经典的材料定律，并规避了其他麻烦的表示定​​理的使用。通过在微空间上进行数值积分可以获得一阶和二阶应力的结果表达式。通过这种方式，微空间的材料信息（此处只是微连续体的几何规格）自然可以进入本构定律。此外，可以以直接的方式考虑非线性材料的行为。在这项工作中，将考虑常规的超弹性。在线性和非线性超弹性情况下的四个应用证明了所提出方法的潜力。特别地，使用基于移动最小二乘法的无网格方法有助于基于纯位移的近似方案，因为它可以提供此应变梯度公式所需的C1连续性。

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