A semi-analytical approach to Biot instability in a growing layer: Strain gradient correction, weakly non-linear analysis and imperfection sensitivity. (October 2015)

• Record Type: Journal Article
• Authors: Ciarletta, P.; Fu, Y.
• Is Part Of: International journal of non-linear mechanics. Volume 75(2015)
• Page Start: 38
• Page End: 45
• Publication Date: 2015-10
• Subjects: Buckling -- Biot instability -- Growth -- Weakly non-linear analysis -- Imperfection sensitivity; Nonlinear mechanics -- Periodicals; Mécanique non linéaire -- Périodiques; Nonlinear mechanics; Periodicals; 531
• DOI: 10.1016/j.ijnonlinmec.2015.03.002
• Languages: English
• ISSNs: 0020-7462

Abstract

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. Abstract : Highlights: We proposes a semi-analytical approach to study the non-linear buckling of the growing soft layer. The strain gradient effects provide a correction to the classical Biot thresholds. A weakly non-linear analysis is proposed using a multiple-scale method, highlighting a pitchfork bifurcation. A sensitivity analysis investigates the effect of an initial sinusoidal imperfection on the free surface. For a subcritical pitchfork, the results show the incipient formation of an elastic singularity, possibly a crease.