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Unified theory of structures based on micropolar elasticity

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Abstract

This paper intends to establish a unified theory of structures based on the micropolar elasticity (ME). ME allows taking into consideration the microstructure of the material, through the adoption of four additional material parameters. In this way, the size-effects of the structure can be caught. The proposed model is developed in the domain of the Carrera unified formulation (CUF), according to which theories of structures can degenerate into unknown kinematics that makes use of an arbitrary expansion of the generalized variables. CUF is a hierarchical formulation that considers the order of the structural model as input of the analysis, so that no specific approximation and manipulation is needed to implement refined theories. Different types of structures have been analyzed in the present work, and the results are compared and validated with benchmarks from the literature. The effects of the new material parameters are addressed too, along with the capability of the proposed model to deal with size-effects and high-order structural behaviors. Finally, stress analysis is detailed to further highlight the differences between micropolar and classical elasticity.

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Authors and Affiliations

  1. Mul2 Group, Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    R. Augello, E. Carrera & A. Pagani

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  1. R. Augello
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  2. E. Carrera
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  3. A. Pagani
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Appendix: Components of the secant stiffness matrix

Appendix: Components of the secant stiffness matrix

$$\begin{aligned} {\mathbf {K}}_{u u x x}^{\tau s i j }&= C_{11}\int _\Omega F_{\tau _{,x}} \, F_{s_{,x}} d\Omega \, \int _L N_i \, N_j dL + C_{55}^{M}\int _\Omega F_{\tau _{,z}} \, F_{s_{,z}} d\Omega \, \int _L N_i \, N_j dL \, \\&+C_{44}^{M}\int _\Omega F_{\tau } \, F_{s} d\Omega \, \int _L N_{i_{,y}} \, N_{j_{,y}} dL \\ {\mathbf {K}}_{u u x y}^{i j \tau s}&= C_{12}\int _\Omega {\textit{F}}_{\tau } \, {\textit{F}}_{s_{,x}}d\Omega \, \int _L N_{i_{,y}} \, N_j dL + C_{44}^{MT} \int _\Omega {\textit{F}}_{\tau _{,x}} \, {\textit{F}}_{s} d\Omega \, \int _L N_i \, N_{j_{,y}} dL \end{aligned}$$
$$\begin{aligned} {\mathbf {K}}_{u u x z}^{i j \tau s}&= C_{55}^{MT} \int _\Omega F_{\tau _{,x}} \, F_{s_{,z}} d\Omega \, \int _L N_{i} \, N_j dL + C_{13} \int _\Omega F_{\tau _{,z}} \, F_{s_{,x}} d\Omega \, \int _L N_i \, N_{j} dL \\ {\mathbf {K}}_{u u y x}^{i j \tau s}&= C_{44}^{MT} \int _\Omega F_{\tau _{,z}} \, F_{s_{,x}} d\Omega \, \int _L N_{i_{,y}} \, N_j dL + C_{12} \int _\Omega F_{\tau _{,x}} \, F_{s} d\Omega \, \int _L N_i \, N_{j_{,y}}dL \\ {\mathbf {K}}_{u u y y}^{\tau s i j }&= C_{44}^{M} \int _\Omega F_{\tau _{,x}} \, F_{s_{,x}} d\Omega \, \int _L N_i \, N_j dL + C_{66}^{M} \int _\Omega F_{\tau _{,z}} \, F_{s_{,z}} \, N_i \, N_j d\Omega \, \int _L \, \\&+C_{22} \int _\Omega F_{\tau } \, F_{s} d\Omega \, \int _L N_{i_{,y}} \, N_{j_{,y}} dL \end{aligned}$$
$$\begin{aligned} {\mathbf {K}}_{u u y z}^{i j \tau s}&= C_{66}^{MT} \int _\Omega F_{\tau } \, F_{s_{,z}} d\Omega \, \int _L N_{i_{,y}} \, N_j dL + C_{23} \int _\Omega F_{\tau _{,z}} \, F_{s} d\Omega \, \int _L N_i \, N_{j_{,y}} dL \\ {\mathbf {K}}_{u u z x}^{i j \tau s}&= C_{13} \int _\Omega F_{\tau _{,x}} \, F_{s_{,z}} d\Omega \, \int _L N_{i} \, N_j dL + C_{55}^{MT} \int _\Omega F_{\tau _{,z}} \, F_{s_{,x}} d\Omega \, \int _L N_i \, N_{j} dL \\ {\mathbf {K}}_{u u z y}^{i j \tau s}&= C_{23} \int _\Omega F_{\tau } \, F_{s_{,z}} d\Omega \, \int _L N_{i_{,y}} \, N_j dL + C_{66}^{MT} \int _\Omega F_{\tau _{,z}} \, F_{s} d\Omega \, \int _L N_i \, N_{j_{,y}} dL \end{aligned}$$
$$\begin{aligned} {\mathbf {K}}_{u u z z}^{\tau s i j }&= C_{55}^{MT} \int _\Omega F_{\tau _{,x}} \, F_{s_{,x}} d\Omega \, \int _L N_i \, N_j dL + C_{33} \int _\Omega F_{\tau _{,z}} \, F_{s_{,z}} d\Omega \, \int _L N_i \, N_j dL \, \\&+C_{66}^{MT} \int _\Omega F_{\tau } \, F_{s} d\Omega \, \int _L N_{i_{,y}} \, N_{j_{,y}} dL \\ {\mathbf {K}}_{u \omega x x}^{i j \tau s}&= 0 \end{aligned}$$
$$\begin{aligned} {\mathbf {K}}_{u \omega x y}^{i j \tau s}&= -C_{55}^{M} \int _\Omega F_{\tau } \, F_{s_{,z}} d\Omega \, \int _L N_{i} \, N_j dL + C_{55}^{MT} \int _\Omega F_{\tau } \, F_{s_{,z}} d\Omega \, \int _L N_i \, N_{j} dL \\ {\mathbf {K}}_{u \omega x z}^{i j \tau s}&= C_{44}^{M} \int _\Omega F_{\tau } \, F_{s} d\Omega \, \int _L N_{i} \, N_{j_{,y}} dL - C_{44}^{MT} \int _\Omega F_{\tau } \, F_{s} d\Omega \, \int _L N_{i} \, N_{j_{,y}} dL \\ {\mathbf {K}}_{u \omega y x}^{i j \tau s}&= C_{66}^{M} \int _\Omega F_{\tau } \, F_{s_{,z}} d\Omega \, \int _L N_{i} \, N_j dL - C_{66}^{MT} \int _\Omega F_{\tau } \, F_{s_{,z}} d\Omega \, \int _L N_{i} \, N_{j} dL \end{aligned}$$
$$\begin{aligned} {\mathbf {K}}_{u \omega y y}^{i j \tau s}&= 0 \\ {\mathbf {K}}_{u \omega y z}^{i j \tau s}&= C_{44}^{MT} \int _\Omega F_{\tau } \, F_{s_{,x}} d\Omega \, \int _L N_{i} \, N_j dL - C_{44}^{M} \int _\Omega F_{\tau } \, F_{s_{,x}} d\Omega \, \int _L N_{i} \, N_{j} dL \\ {\mathbf {K}}_{u \omega z x}^{i j \tau s}&= -C_{66}^{M} \int _\Omega F_{\tau } \, F_{s} d\Omega \, \int _L N_{i} \, N_{j_{,y}} dL + C_{66}^{MT} \int _\Omega F_{\tau } \, F_{s} d\Omega \, \int _L N_{i} \, N_{j_{,y}} dL \end{aligned}$$
$$\begin{aligned} {\mathbf {K}}_{u \omega z y}^{i j \tau s}&= C_{55}^{M} \int _\Omega F_{\tau } \, F_{s_{,x}} d\Omega \, \int _L N_{i} \, N_j dL - C_{55}^{MT} \int _\Omega F_{\tau } \, F_{s_{,x}} d\Omega \, \int _L N_{i} \, N_{j} dL \\ {\mathbf {K}}_{u \omega z z}^{i j \tau s}&= 0 \\ {\mathbf {K}}_{\omega u x x}^{i j \tau s}&= 0 \end{aligned}$$
$$\begin{aligned} {\mathbf {K}}_{\omega u x y}^{i j \tau s}&= C_{66}^{M} \int _\Omega F_{s} \, F_{t_{,z}} d\Omega \, \int _L N_{i} \, N_j dL - C_{66}^{MT} \int _\Omega F_{s} \, F_{t_{,z}} d\Omega \, \int _L N_i \, N_{j} dL \\ {\mathbf {K}}_{\omega u x z}^{i j \tau s}&= -C_{66}^{M} \int _\Omega F_{s} \, F_{t} d\Omega \, \int _L N_{i_{,y}} \, N_{j} dL + C_{66}^{MT} \int _\Omega F_{s} \, F_{t} d\Omega \, \int _L N_{i_{,y}} \, N_{j_{,y}} dL \\ {\mathbf {K}}_{\omega u y x}^{i j \tau s}&= -C_{55}^{M} \int _\Omega F_{s} \, F_{t_{,z}} d\Omega \, \int _L N_{i} \, N_j dL + C_{55}^{MT} \int _\Omega F_{s} \, F_{t_{,z}} d\Omega \, \int _L N_{i} \, N_{j} dL \end{aligned}$$
$$\begin{aligned} {\mathbf {K}}_{\omega u y y}^{i j \tau s}&= 0 \\ {\mathbf {K}}_{\omega u y z}^{i j \tau s}&= C_{55}^{M} \int _\Omega F_{s} \, F_{t_{,x}} d\Omega \, \int _L N_{i} \, N_j dL - C_{55}^{MT} \int _\Omega F_{s} \, F_{t_{,x}} d\Omega \, \int _L N_{i} \, N_{j} dL \\ {\mathbf {K}}_{\omega u z x}^{i j \tau s}&= C_{44}^{M} \int _\Omega F_{s} \, F_{t} d\Omega \, \int _L N_{i_{,y}} \, N_{j} dL - C_{44}^{MT} \int _\Omega F_{s} \, F_{t} d\Omega \, \int _L N_{i_{,y}} \, N_{j} dL \end{aligned}$$
$$\begin{aligned} {\mathbf {K}}_{\omega u z y}^{i j \tau s}&= -C_{44}^{M} \int _\Omega F_{s} \, F_{t_{,x}} d\Omega \, \int _L N_{i} \, N_j dL + C_{44}^{MT} \int _\Omega F_{s} \, F_{t_{,x}} d\Omega \, \int _L N_{i} \, N_{j} dL \\ {\mathbf {K}}_{\omega u z z}^{i j \tau s}&= 0 \\ {\mathbf {K}}_{\omega \omega x x}^{i j \tau s}&= 2C_{66}^{M} \int _\Omega F_{s} \, F_{t} d\Omega \, \int _L N_{i} \, N_j dL - 2C_{66}^{MT} \int _\Omega F_{s} \, F_{t} d\Omega \, \int _L N_i \, N_{j} dL + A_{11} \int _\Omega F_{\tau _{,x}} \, F_{s_{,x}} d\Omega \, \int _L N_i \, N_j dL + A_{55} \int _\Omega F_{\tau _{,z}} \, F_{s_{,z}} d\Omega \, \int _L N_i \, N_j dL + A_{44} \int _\Omega F_{\tau } \, F_{s} d\Omega \, \int _L N_{i_{,y}} \, N_{j_{,y}} dL\end{aligned}$$
$$\begin{aligned} {\mathbf {K}}_{u u x z}^{i j \tau s}&= A_{12} \int _\Omega F_{\tau } \, F_{s_{,x}} d\Omega \, \int _L N_{i_{,y}} \, N_j dL + A_{44}^{MT} \int _\Omega F_{\tau _{,x}} \, F_{s} d\Omega \, \int _L N_i \, N_{j_{,y}} dL \\ {\mathbf {K}}_{u u x z}^{i j \tau s}&= A_{55}^{MT} \int _\Omega F_{\tau _{,x}} \, F_{s_{,z}} d\Omega \, \int _L N_{i} \, N_j dL + A_{13} \int _\Omega F_{\tau _{,z}} \, F_{s_{,x}} d\Omega \, \int _L N_i \, N_{j} dL \\ {\mathbf {K}}_{u u y x}^{i j \tau s}&= A_{44}^{MT} \int _\Omega F_{\tau } \, F_{s_{,x}} d\Omega \, \int _L N_{i_{,y}} \, N_j dL + A_{12} \int _\Omega F_{\tau _{,x}} \, F_{s} d\Omega \, \int _L N_i \, N_{j_{,y}} dL \\ {\mathbf {K}}_{\omega \omega x x}^{i j \tau s}&= 2C_{55}^{M} \int _\Omega F_{s} \, F_{t} d\Omega \, \int _L N_{i} \, N_j dL - 2C_{55}^{MT} \int _\Omega F_{s} \, F_{t} d\Omega \, \int _L N_i \, N_{j} dL + A_{44} \int _\Omega F_{\tau _{,x}} \, F_{s_{,x}} d\Omega \, \int _L N_i \, N_j dL + A_{66} \int _\Omega F_{\tau _{,z}} \, F_{s_{,z}} d\Omega \, \int _L N_i \, N_j dL + A_{22} \int _\Omega F_{\tau } \, F_{s} d\Omega \, \int _L N_{i_{,y}} \, N_{j_{,y}} dL\end{aligned}$$
$$\begin{aligned} {\mathbf {K}}_{u u y z}^{i j \tau s}&= A_{66}^{MT} \int _\Omega F_{\tau } \, F_{s_{,z}} d\Omega \, \int _L N_{i_{,y}} \, N_j dL + A_{23} \int _\Omega F_{\tau _{,z}} \, F_{s} d\Omega \, \int _L N_i \, N_{j_{,y}} dL \\ {\mathbf {K}}_{u u z x}^{i j \tau s}&= A_{13} \int _\Omega F_{\tau _{,x}} \, F_{s_{,z}} d\Omega \, \int _L N_{i} \, N_j dL + A_{55}^{MT} \int _\Omega F_{\tau _{,z}} \, F_{s_{,x}} d\Omega \, \int _L N_i \, N_{j} dL \\ {\mathbf {K}}_{u u z y}^{i j \tau s}&= A_{66}^{MT} \int _\Omega F_{\tau _{,z}} \, F_{s} d\Omega \, \int _L N_{i} \, N_{j_{,y}} dL + A_{23} \int _\Omega F_{\tau } \, F_{s_{,z}} d\Omega \, \int _L N_{i_{,y}} \, N_{j} dL \\ {\mathbf {K}}_{\omega \omega x x}^{i j \tau s}&= 2C_{44}^{M} \int _\Omega F_{s} \, F_{t} d\Omega \, \int _L N_{i} \, N_j dL - 2C_{44}^{MT} \int _\Omega F_{s} \, F_{t} d\Omega \, \int _L N_i \, N_{j} dL + A_{55} \int _\Omega F_{\tau _{,x}} \, F_{s_{,x}} d\Omega \, \int _L N_i \, N_j dL + A_{33} \int _\Omega F_{\tau _{,z}} \, F_{s_{,z}} d\Omega \, \int _L N_i \, N_j dL + A_{66} \int _\Omega F_{\tau } \, F_{s} d\Omega \, \int _L N_{i_{,y}} \, N_{j_{,y}} dL\end{aligned}$$

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Augello, R., Carrera, E. & Pagani, A. Unified theory of structures based on micropolar elasticity. Meccanica 54, 1785–1800 (2019). https://doi.org/10.1007/s11012-019-01041-z

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