A nonlinear geometric couple stress based strain gradient Kirchhoff–Love shell formulation for microscale thin-wall structures

doi:10.1016/j.ijmecsci.2021.106272

International Journal of Mechanical Sciences

Volume 196, 15 April 2021, 106272

https://doi.org/10.1016/j.ijmecsci.2021.106272 Get rights and content

Highlights

•
A geometric nonlinear couple stress based strain gradient Kirchhoff Love shell formulation is proposed.
•
The approach takes advantage of the higher order continuity of IGA.
•
The method is validated through several benchmark problems.

Abstract

We present a nonlinear Kirchhoff–Love micro-shell element based on isogeometric analysis (IGA) and couple stress theory. Higher-order NURBS functions are exploited for analyzing the strain gradient effect which automatically fulfill the higher-order continuity requirements. We express the strain gradient elastic formulation in natural curvilinear coordinates, which leads to an efficient numerical tool to examine geometric nonlinearities of thin micro-shell structures. The presented IGA formulation is verified through comparisons to analytical solution, experimental data as well as other popular benchmark problems of nonlinear geometric shells. We believe that the presented formulation is particularly suitable for analyzing two-dimensional materials at larger length scales, which are commonly studied at nanoscale.

Graphical abstract

Introduction

Conventional continuum theories are insufficient in accurately describing the mechanical behavior of microstructures. Evidences from experimental observations of thin metal films [1], thin copper wires [2], silver single crystals [3], thin foils [4], micro epoxy beams [5], CFRP composites [6] and silver nanowires [7] have indicated that the material behavior of micro-structures is size dependent. This size effect is insignificant for macroscopic structures but becomes dominant at lower scales. The phenomenon can be captured by considering the long-range inter-atomic attractions via introducing a material length scale into the classical elastic theory. Motivated by this concept, the theories of higher-gradient elasticity have been developed. Generally, it can be classified into three different approaches based on the chosen deformation metrics: couple stress elasticity proposed by Mindlin and Tiersten [8], Toupin [9] and Koiter [10]; strain gradient elasticity based on the early works of Mindlin [11], Eringen and Suhubi [12], Green and Rivlin [13], Kröner [14], Mindlin and Eshel [15] and Germain [16]; and surface stress elasticity pioneered by Mindlin [17], developed by Gurtin and Murdoch for continuum material surfaces [18], [19]. Although being general as including all the required higher-order terms, these early works are unfortunately too complicated. There are so many material parameters that cannot be experimentally determined. Hence there is a requirement of simplification to make these formulations more applicable. Early attempts of reformulating gradient elasticity were conducted by Eringen [20] with integral non-local theories. A very interesting stress-driven nonlocal integral model applied to nano-plates and nano-beams has been recently proposed by Barretta et al. [21], [22]. In [23], they presented a general variational framework for nano-beams based on the nonlocal gradient theory. Triantafyllidis and Aifantis [24] extended the theory for hyperelastic materials. From higher-order gradient model of Mindlin [11], Fleck and Hutchinson [25] and Fleck et al. [2] modified it to a second-order strain gradient version for plasticity. A general review for gradient elasticity and its applications in statics and dynamics can be found in Askes and Aifantis [26].

Surface elasticity models have been developed since the milestone paper of Mindlin [17]. A homogenization for rough surface elasticity was presented by Mohammadi et al. [27]. For thin-walled structures, a curvature-dependence of surface energy was proposed by Chhapadia et al. [28] and curvilinear based formulation for elasticity with boundary energies was presented by Javili et al. [29], [30]. A fundamental discrepancy between the surface elasticity and the strain gradient elasticity (also known as the first-order strain gradient formulation) is a third-order stress attached on the surface as a cohesive force in the former approach. Hence surface elasticity can be considered as a second-order strain gradient, relations of the elastic parameters between two approaches were studied in Cordero et al. [31]. On the other hand, based on the initial work of Mindlin and Tiersten [8], the couple stress model was modified in the context of the representative volume element by Yang et al. [32]. Via introducing the equilibrium of a couple moment, the rotational degrees of freedom are included in the same fashion of Cosserat theories or micropolar continuum. The isotropic couple stress based strain gradient model has three material parameters including two Lamé constants and one material length-scale, this enables practical applications of the model which has been extended based on Bernoulli-Euler theory [33], Timoshenko theory [34]. Besides numerous theoretical efforts, many experimental studies have been done to quantify size effects or measure important material parameters, see for instance the contributions in Lam et al. [5], Liebold and Müller [35], Lei et al. [36], Li et al. [37], Tang and Alici [38], Tang and Gursel [39], Wi and Sodemann [40], Li et al. [41], 42].

The rapid development of micro-technology and additive manufacturing enables the fabrication of miniature structures, especially micro-thin sheets and membranes which keep an essential role in micro-devices due to their lightweight, flexibility, and low space occupation. Shell models can be categorized into thin shell models neglecting shear effects and thick shell models accounting for shear effects. The first type of shells is commonly based on Kirchhoff–Love theory while Mindlin-Reissner theory is adopted for thick shells. Thin shells have a wide range of applications in aircrafts, vessels, and automotives. Thin micro-shells, which is the focus of our studies, have emerged in engineering fields such as nano-electro-mechanical systems [43], [44] or optotronics [45], [46]. They have been employed to study bio-membranes [47], [48] and spherical shaped viruses [49]. There are numerous studies on the size effects of micro- and nano-structures. Mehralian et al. [50], Rahmanian et al. [51] studied the free vibration behavior of nano-shells using strain gradient models while Farokhi et al. [52], Dehrouyeh et al. [53], Krysko et al. [54] and Zeighampour et al. [55] for instance took advantage of couple stress theory for such applications. Papargyri et al. [56] and Sahmani et al. [57] focused on the stability analysis of such structures. A comprehensive review of nonlinear shells including micro-shells can be found in Alijani and Amabili [58]. Most contributions were based on analytical solutions of the equation of motion obtained from Navier’s series expansion or the perturbation technique. The key disadvantage of these analytical approaches are their limitation to very specific problems with simple geometries and specified loading cases. The emerging of isogeometric analysis (IGA) proposed by Hughes et al. [59], [60] has facilitated the application of $C^{1}$ continuous shell formulation. Over the last decades, the isogeometric based Kirchhoff–Love shell formulations have been exploited initially by Kiendl et al. [61]. The rotational free shell formulation requires a special treatment for patch boundaries or edge, the rotational degree of freedom can be constrained indirectly via the bending strip method [62], T-Splines [63], PHT-Splines [64], RHT-Splines [65], Nitsche’s method [66] or constraining directors of edge surfaces [67]. This formulation was extent to study a wide range of engineering problems including delamination in composites [68], thermal expansion [69], finite deformations [70], [71], fluid-structure interaction [72] and heart valve dynamics [73].

Computational methods for strain gradient theory have been presented for beams for instance in Balobanov and Niiranen [74], Niiranen et al. [75] or plates in Niiranen and Niemi [76]. Other contributions on functionally graded micro-plates based on couple stress theory, modified couple stress theory and nonlocal elasticity can be found in Thai et al. [77], Nguyen et al. [78] and references therein. These papers focused on the free vibration behavior of plates using first or second-order shear deformation theory exploiting the higher-continuity of B-splines. To our best knowledge, there is a lack of micro-shell elements, which can capture both, general size effects and geometric nonlinearities. The computational approaches in Thai et al. [77], Nguyen et al. [78] are about micro-plates, not suitable for arbitrary geometries and do not account for geometric nonlinearities. An interesting Kirchhoff–Love shell element for strain gradient problems has been proposed by Balobanov et al. [79], but for the case of linear loading. In this manuscript, we present a nonlinear Kirchhoff–Love micro-shell element based on isogeometric analysis (IGA) and couple stress theory. The kinematics of the gradient elasticity shell are derived including the higher-order equilibrium conditions. Through numerical examples, we show a significant influence of size effects on the buckling behavior of thin micro-shells.

Section snippets

Weak formulation

The governing equation of the strain gradient problem and corresponding boundary conditions are obtained by employing the variational method. The detailed derivations are expressed in Deng et al. [80], in here we summarize the resulting equations ${\begin{matrix} - \frac{\partial}{\partial X_{K}} [\frac{\partial ψ}{\partial F_{k K}} - \frac{\partial}{\partial X_{L}} (\frac{\partial ψ}{\partial R_{k K L}})] = f_{k} in \hat{Ω}, (1 a) \\ (\frac{\partial ψ}{\partial F_{k K}} - \frac{\partial}{\partial X_{L}} \frac{\partial ψ}{\partial R_{k K L}}) {\hat{N}}_{K} = τ_{k} + t_{k} on \hat{Γ} or δ u_{k} = 0 on {\hat{Γ}}_{u^{*}}, (1 b) \\ \frac{\partial ψ}{\partial R_{k K L}} {\hat{N}}_{L} {\hat{N}}_{K} = 0 on {\hat{Γ}}_{D} or {\hat{N}}_{I} δ u_{k, I} = 0 on {\hat{Γ}}_{u^{* *}} . (1 c) \end{matrix}$ where $ψ$ is the total energy density, $τ_{k} = \frac{\partial}{\partial X_{I}} [\frac{\partial ψ}{\partial R_{k K L}} {\hat{N}}_{L} (δ_{K I} - {\hat{N}}_{K} {\hat{N}}_{I})]$ is referred as the higher order traction. The

Strain tensor and rotation strain gradient tensor

The Green-Lagrange tensors is defined in the Cartesian coordinates as $E = \frac{1}{2} (F^{T} F - I) .$ The elastic energy can be expressed as $ψ^{e} (F) = {\hat{ψ}}^{e} (E)$ . Applying the chain rule we have $\frac{\partial ψ^{e} (F)}{\partial F} : δ F = \frac{\partial {\hat{ψ}}^{e} (E)}{\partial E} \frac{\partial E}{\partial F} : δ F = \frac{\partial {\hat{ψ}}^{e} (E)}{\partial E} : δ E = S : δ E,$ where $S$ is the second Piola–Kirchhoff stress tensor. In the strain gradient approach, the strain gradient tensor is usually defined as $\nabla E$ . In the couple stress approach, only the gradient of the anti-symmetric part of the deformation gradient is considered since it fulfills the balance

Membrane strain and bending strain

Thin shell geometry can be described by its mid-surface as illustrated in Fig. 1. Kirchhoff–Love theory assumes that structural cross-sections are still straight and the director vectors are normal to the mid-surface under deformations, this neglects the effect transverse shear strain. Let us denote the curvilinear coordinates as ${ξ^{1}, ξ^{2}},$ and $ξ^{3}$ as the over thickness direction. The shell geometry $X (ξ^{1}, ξ^{2}, ξ^{3}),$ $x (ξ^{1}, ξ^{2}, ξ^{3})$ are presented via the mid-surface $T (ξ^{1}, ξ^{2}),$ $t (ξ^{1}, ξ^{2})$ and the director

Elastic tensors in 3D and 2D under plane stress assumption

Saint-Venant model is employed for describing the elastic behavior in this work, the elastic energy density $ψ^{e}$ and the second Piola–Kirchhoff stress are expressed respectively as $\begin{matrix} ψ^{e} & = & \frac{λ}{2} E_{M M} E_{N N} + μ E_{M N} E_{M N}, \\ S & = & \frac{\partial ψ^{e}}{\partial E} = \frac{\partial ψ^{e}}{\partial E_{I J}} A_{I} \otimes A_{J} = S^{I J} A_{I} \otimes A_{J}, \end{matrix}$ where $λ$ and $μ$ are the Lamé constants and $S^{I J} = \frac{\partial ψ^{e}}{\partial E_{I J}} = λ A^{I J} E_{M M} + μ (A^{M I} A^{N J} + A^{M J} A^{N I}) E_{M N} .$ The elastic tensor is computed as $C = \frac{\partial^{2} ψ^{e}}{\partial E^{2}} = \frac{\partial S}{\partial E} = \frac{\partial S^{I J}}{\partial E_{K L}} A_{I} \otimes A_{J} \otimes A_{K} \otimes A_{L} = C^{I J K L} A_{I} \otimes A_{J} \otimes A_{K} \otimes A_{L},$ where $C^{I J K L} = λ A^{I J} A^{K L} + μ (A^{I K} A^{J L} + A^{I L} A^{J K}) .$ The elastic energy density function can be rewritten in the quadratic

Linearization and tangent operators

Linearizing Eq. (70), the incremental form of the governing equation is obtained as $δ Δ W^{e} + δ Δ W^{n} = δ W^{e x} - δ W^{e} - δ W^{n}$ where $\begin{matrix} δ Δ W^{e} & = & \int_{A} δ ϵ_{α β} Δ n^{α β} d A + \int_{A} n^{α β} δ Δ ϵ_{α β} d A + \int_{A} δ κ_{α β} Δ m^{α β} d A \\ + \int_{A} m^{α β} δ Δ κ_{α β} d A, \\ δ Δ W^{n} & = & \int_{A} δ {\tilde{ϵ}}_{I J K} Δ {\tilde{n}}^{I J K} d A + \int_{A} {\tilde{n}}^{I J K} δ Δ {\tilde{ϵ}}_{I J K} d A + \int_{A} δ {\tilde{κ}}_{I J K} Δ {\tilde{m}}^{I J K} d A \\ + \int_{A} {\tilde{m}}^{I J K} δ Δ {\tilde{κ}}_{I J K} d A, \\ δ W^{e x} & = & \int_{A} h {δ q}^{T} f d A + \int_{\partial A} h δ q^{T} t d A + \int_{\partial A} h δ q^{T} τ d A, \\ δ W^{e} & = & \int_{A} n^{α β} δ ϵ_{α β} d A + \int_{A} m^{α β} δ κ_{α β} d A, \\ δ W^{n} & = & \int_{A} {\tilde{n}}^{I J K} δ {\tilde{ϵ}}_{I J K} d A + \int_{A} {\tilde{m}}^{I J K} δ {\tilde{κ}}_{I J K} d A, \end{matrix}$ where $\begin{matrix} Δ n^{α β} & = & h C_{p}^{α β γ δ} Δ ϵ_{γ δ}; Δ m^{α β} = \frac{h^{3}}{12} C_{p}^{α β γ δ} Δ κ_{γ δ}; \\ δ Δ n^{α β} & = & h C_{p}^{α β γ δ} δ Δ ϵ_{γ δ}; δ Δ m^{α β} = \frac{h^{3}}{12} C_{p}^{α β γ δ} δ Δ κ_{γ δ}, \end{matrix}$ and $\begin{matrix} Δ {\tilde{n}}^{I J K} & = & h Q^{I J K A B C} Δ {\tilde{ϵ}}_{A B C}; δ Δ {\tilde{n}}^{I J K} = h Q^{I J K A B C} δ Δ {\tilde{ϵ}}_{A B C}; \\ Δ {\tilde{m}}^{I J K} & = & \frac{h^{3}}{12} Q^{I J K A B C} Δ {\tilde{κ}}_{A B C}; δ Δ \end{matrix}$

Cantilever beam under shear force - experiments vs. numerical results

Let us consider a cantilever beam subjected to shear force with boundary conditions as described in Fig. 2. Such beams have also been studied experimentally by Liebold et al. [35] using different materials, i.e. epoxy and SU-8, respectively. The associated material parameters are: Young’s modulus $E = 3.90$ GPa for epoxy and $E = 4.10$ GPa [35] for SU-8 photoresist. The other parameters are the same for both materials, i.e.: Poisson’s ratio $ν = 0.38,$ length $L = 20 h,$ width $w = 2 h$ and force $P = 10^{- 4}$ N where $h$ is

Conclusion remarks

We present an isogeometric nonlinear Kirchhoff–Love micro-shell element based on couple stress theory. Higher-order NURBS functions are exploited which automatically fulfill the higher-order continuity requirements to capture strain gradient effects. The proposed formulation is verified and validated by comparisons to analytical solutions, experimental data and other numerical solutions. We also demonstrate for several benchmark problems that the higher-order term related to the bending strain

CRediT authorship contribution statement

Tran Quoc Thai: Conceptualization, Writing - original draft, Investigation, Formal analysis, Validation, Visualization, Methodology. Xiaoying Zhuang: Methodology. Timon Rabczuk: Methodology, Conceptualization, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors Tran Quoc Thai and Xiaoying Zhuang would like to acknowledge the financial support from the Sofja Kovalevskaja Prize of the Alexander von Humboldt Foundation (Germany). The authors would like to thank the anonymous reviewers for their valuable comments that help to improve our manuscript.