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Theory and computation of higher gradient elasticity theories based on action principles

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Abstract

In continuum mechanics, there exists a unique theory for elasticity, which includes the first gradient of displacement. The corresponding generalization of elasticity is referred to as strain gradient elasticity or higher gradient theories, where the second and higher gradients of displacement are involved. Unfortunately, there is a lack of consensus among scientists how to achieve the generalization. Various suggestions were made, in order to compare or even verify these, we need a generic computational tool. In this paper, we follow an unusual but quite convenient way of formulation based on action principles. First, in order to present its benefits, we start with the action principle leading to the well-known form of elasticity theory and present a variational formulation in order to obtain a weak form. Second, we generalize elasticity and point out, in which term the suggested formalism differs. By using the same approach, we obtain a weak form for strain gradient elasticity. The weak forms for elasticity and for strain gradient elasticity are solved numerically by using open-source packages—by using the finite element method in space and finite difference method in time. We present some applications from elasticity as well as strain gradient elasticity and simulate the so-called size effect.

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Acknowledgements

This work was completed while B. E. Abali was supported by a Grant from the Max Kade Foundation to the University of California, Berkeley.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of California, Berkeley, 6124 Etcheverry Hall, Mailstop 1740, Berkeley, CA, 94720, USA

    B. Emek Abali

  2. Institute of Mechanics, Chair of Continuum Mechanics and Theory of Materials, Technische Universität Berlin, Einsteinufer 5, 10587, Berlin, Germany

    Wolfgang H. Müller

  3. Dipartimento di Ingegneria Strutturale e Geotecnica, Sapienza Universita di Roma, Via Eudossiana 18, 00184, Roma, Italia

    Francesco dell’Isola

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  1. B. Emek Abali
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  2. Wolfgang H. Müller
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  3. Francesco dell’Isola
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Correspondence to B. Emek Abali.

Appendices

Appendix A: Convergence for linear elasticity

Standard h-convergence has been conducted by decreasing the mesh size in every tetrahedron. We use the same geometry and boundary conditions and compute \(L_2\) norm of the solution (displacement vector) over the whole mesh. We restrict the analysis to the first five time steps, \(t_1\), \(t_2\), \(t_3\), \(t_4\), and \(t_5\), and expect a linear relation for every time step between (log of) degrees of freedom (dofs) and (log of) \(L_2\) norm. The results are compiled in Table 2 and shown in Fig. 5.

Table 2 Convergence analysis for linear elastic transient FEM computation
Full size table
Fig. 5
figure 5

For the linear elastic transient FEM, plot of degrees of freedom versus \(L_2\) norm over the whole mesh in a log-log scale. Colors denote different time steps

Full size image

Appendix B. Convergence for nonlinear elasticity

Standard h-convergence has been conducted with the same geometry and boundary conditions for the first five time steps, \(t_1\), \(t_2\), \(t_3\), \(t_4\), and \(t_5\). The results are compiled in Table 3 and shown in Fig. 6.

Table 3 Convergence analysis for nonlinear elastic transient FEM computation
Full size table
Fig. 6
figure 6

For the nonlinear elastic transient FEM, plot of degrees of freedom versus \(L_2\) norm over the whole mesh in a log-log scale. Colors denote different time steps

Full size image
Table 4 Convergence analysis for linear strain gradient elastic transient FEM computation
Full size table

Appendix C: Convergence for linear strain gradient elasticity

Standard h-convergence has been conducted with the same geometry and boundary conditions for the first five time steps, \(t_1\), \(t_2\), \(t_3\), \(t_4\), and \(t_5\). The results are compiled in Table 4 and shown in Fig. 7.

Fig. 7
figure 7

For the linear strain gradient elastic transient FEM, plot of degrees of freedom versus \(L_2\) norm over the whole mesh in a log-log scale. Colors denote different time steps

Full size image

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Abali, B.E., Müller, W.H. & dell’Isola, F. Theory and computation of higher gradient elasticity theories based on action principles. Arch Appl Mech 87, 1495–1510 (2017). https://doi.org/10.1007/s00419-017-1266-5

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