A semi-analytical approach to Biot instability in a growing layer: Strain gradient correction,weakly non-linear analysis and imperfection sensitivity |
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| Institution: | 1. CNRS and Institut Jean le Rond d׳Alembert, UMR 7190, Université Paris 6, 4 place Jussieu case 162, 75005 Paris, France;2. Department of Mathematics, Keele University, ST5 5BG, UK;3. Department of Mechanics, Tianjin University, Tianjin 300072, China;1. Zienkiewicz Centre for Computational Engineering, College of Engineering Swansea University, Bay Campus, SA1 8EN, United Kingdom;2. Institute of Mechanics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany;1. Institute of Biomechanics and Medical Engineering, AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China;2. School of Materials Science and Engineering, Tianjin University, Tianjin 300072, PR China;3. School of Engineering, Brown University, Providence, RI 02912, USA;4. CMM, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China;1. Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Republic of Korea;2. Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea;3. School of Engineering, Brown University, Providence, RI 02912, USA;1. Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d''Alembert, F-75005 Paris, France;2. MOX – Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy;3. School of Mathematics, Statistics and Applied Mathematics, NUI Galway, University Road, Galway, Ireland;4. School of Mechanical and Materials Engineering, University College Dublin, Belfield, Dublin 14, Ireland;5. School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK;6. Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK |
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| Abstract: | Many experimental works have recently investigated the dynamics of crease formation during the swelling of long soft slabs attached to a rigid substrate. Mechanically, the spatially constrained growth provokes a residual strain distribution inside the material, and therefore the problem is equivalent to the uniaxial compression of an elastic layer.The aim of this work is to propose a semi-analytical approach to study the non-linear buckling behaviour of a growing soft layer. We consider the presence of a microstructural length, which describes the effect of a simple strain gradient correction in the growing hyperelastic layer, considered as a neo-Hookean material. By introducing a non-linear stream function for enforcing exactly the incompressibility constraint, we develop a variational formulation for performing a stability analysis of the basic homogeneous solution. At the linear order, we derive the corresponding dispersion relation, proving that even a small strain gradient effect allows the system to select a critical dimensionless wavenumber while giving a small correction to the Biot instability threshold. A weakly non-linear analysis is then performed by applying a multiple-scale expansion to the neutrally stable mode. By applying the global conservation of the mechanical energy, we derive the Ginzburg–Landau equation for the critical single mode, identifying a pitchfork bifurcation. Since the bifurcation is found to be subcritical for a small ratio between the microstructural length and the layer׳s thickness, we finally perform a sensitivity analysis to study the effect of the initial presence of a sinusoidal imperfection on the free surface of the layer. In this case, the incremental solution for the stream function is written as a Fourier series, so that the surface imperfection can have a cubic resonance with the linear modes. The solutions indicate the presence of a turning point close to the critical threshold for the perfect system. We also find that the inclusion of higher modes has a steepening effect on the surface profile, indicating the incipient formation of an elastic singularity, possibly a crease. |
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| Keywords: | Buckling Biot instability Growth Weakly non-linear analysis Imperfection sensitivity |
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