A tutorial on the stability and bifurcation analysis of the electromechanical behaviour of soft materials

A tutorial on the stability and bifurcation analysis of the electromechanical behaviour of soft materials Shengyou Yang 1, 2 and Pradeep Sharma 2, 3* 1 Department of Engineering Mechanics, School of Civil Engineering, Shandong University, Jinan, 250061, China 2 Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA 3 Department of Physics, University of Houston, Houston, TX 77204, USA * E-mail: [email protected]; Fax: +1-713-743-4503; Tel: +1-713-743-4502 Contents 1 Introduction 4 2 Stability and bifurcation: the big picture and basic concepts 7 2.1 The big picture .......................................... 7 2.2 One-dimensional systems and geometric approach ....................... 12 2.3 Deﬁnition of stability and the importance of norm ...................... 14 2.3.1 Stability analysis by linearization: one-dimensional examples ............ 14 2.3.2 Principle of minimum energy: a conservative system ................. 15 2.3.3 Other stability criteria .................................. 16 2.4 The concept of bifurcation .................................... 16 2.4.1 Bifurcation points .................................... 16 2.4.2 Bifurcation in a continuum media ........................... 18 2.5 More examples .......................................... 18 2.5.1 Stability examples .................................... 18 2.5.2 Bifurcation examples ................................... 20 2.6 Closure on the basic concepts underlying stability and bifurcation .............. 22 3 One-dimensional electrostatics of deformable dielectrics 22 3.1 Kinematics ............................................ 22 3.1.1 Deformation of a body .................................. 22 3.1.2 Deformation gradient and strain ............................ 23 3.1.3 Diﬀerentiation and integration ............................. 24 3.2 Maxwell’s equations ....................................... 24 1

A tutorial on the stability and bifurcation analysis of the electromechanical behaviour of soft materials

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