Department of Mathematics, Khalifa University of Science and Technology, Abu Dhabi, UAE.
Departamento de Matemática Aplicada a las TIC, ETS de Ingeniería de Sistemas Informáticos, Universidad Politécnica de Madrid, 28031, Madrid, Spain.
Sci Rep. 2023 Apr 21;13(1):6562. doi: 10.1038/s41598-023-33670-6.
In the past, to model fibre stiffness of finite-radius fibres, previous finite-strain (nonlinear) models were mainly based on the theory of non-linear strain-gradient (second-gradient) theory or Kirchhoff rod theory. We note that these models characterize the mechanical behaviour of polar transversely isotropic solids with infinitely many purely flexible fibres with zero radius. To introduce the effect of fibre bending stiffness on purely flexible fibres with zero radius, these models assumed the existence of couple stresses (contact torques) and non-symmetric Cauchy stresses. However, these stresses are not present on deformations of actual non-polar elastic solids reinforced by finite-radius fibres. In addition to this, the implementation of boundary conditions for second gradient models is not straightforward and discussion on the effectiveness of strain gradient elasticity models to mechanically describe continuum solids is still ongoing. In this paper, we develop a constitutive equation for a non-linear non-polar elastic solid, reinforced by embedded fibers, in which elastic resistance of the fibers to bending is modelled via the classical branches of continuum mechanics, where the development of the theory of stresses is based on non-polar materials; that is, without using the second gradient theory, which is associated with couple stresses and non-symmetric Cauchy stresses. In view of this, the proposed model is simple and somewhat more realistic compared to previous second gradient models.
在过去，为了对有限半径纤维的纤维刚度进行建模，先前的有限应变（非线性）模型主要基于非线性应变梯度（二阶导数）理论或 Kirchhoff 杆理论。我们注意到，这些模型描述了具有无穷多个零半径的纯柔性纤维的各向异性固体的力学行为。为了引入纤维弯曲刚度对零半径纯柔性纤维的影响，这些模型假设存在偶应力（接触扭矩）和非对称 Cauchy 应力。然而，在实际的非极性弹性固体中，这些应力并不存在于由有限半径纤维增强的变形中。除此之外，二阶导数模型的边界条件的实现并不简单，并且关于应变梯度弹性模型在机械上描述连续体的有效性的讨论仍在进行中。在本文中，我们为嵌入纤维增强的非线性非极性弹性固体开发了一个本构方程，其中纤维对弯曲的弹性抵抗通过连续体力学的经典分支来建模，该理论的应力发展基于非极性材料；也就是说，不使用与偶应力和非对称 Cauchy 应力相关的二阶导数理论。鉴于此，与以前的二阶导数模型相比，所提出的模型更简单，也更符合实际情况。
