The Study On Cracks With Electro-mechanical Nonlinear Zones In Piezoelectric Materials Of Finite Size

Piezoelectric materials play an important role in devices such as sensors,filters,ultrasonic generators,and actuators due to the inherent electromechanical coupling effect.In addition,piezoelectric materials are generally prone to mechanical or electrical damages because of brittleness and low fracture toughness.Therefore,the analysis for the fracture behavior of piezoelectric materials are still important and challenging tasks.Based on the linear electro-elasticity theory,numerous achievements have been made in the study of electro-elastic field of piezoelectric materials with a Griffith crack or a penny-shaped crack.However,the linear electro-elasticity theory believes that the stress at the crack tip is singular,which cannot explain some discrepancies between theory and experiments.This means that the nonlinear coupling effects need to be considered.For elastic materials,there would be a yielding zone in front of the crack tips due to the high concentration of stress,as proposed by Dugdale.In fact,Dugdale model is very effective and has been supported by experiments.Thus,the fracture analysis based on Dugdale model has been widely used in the Griffith crack or penny-shaped crack problems of elastic or piezoelectric materials.This paper deals with some crack problems in piezoelectric materials of finite size considering the electromechanical nonlinearity in front of the crack tips.For these mixed boundary value problems in finite domains proposed in this paper,we simplify problems into coupling dual integral equations by integral transformation technique and find Copson method has obvious advantages in the process of finding a concrete solution.The content of this article is as follows:(1)Based on nonlinear fracture theory,we study the problem of a Yoffe-type moving crack along the interface of an infinitely long piezoelectric bi-layer with the distinct strip-like electrical saturation and mechanical yielding zones in front of the crack tips.This mixed boundary value problem is simplified into coupling Fredholm integral equations of the second kind by applying the Fourier integral transformation technique and Copson method.With appropriate formula derivation and numerical discretization,we successfully obtained the numerical solutions for any layer-thickness.For the Yoffe-type crack,by assuming the crack propagates sub-sonically along the interface,and the crack surface is impermeable.Since the lengths of the electromechanical nonlinear zones are significantly affected by the loadings,three different cases,i.e.,the region of electrical saturation is longer,shorter than,or equal to the domain of mechanical yielding are respectively considered.The influences of the electro-mechanical loadings,the layer-thicknesses and crack velocity on the length of the electro-mechanical nonlinear zones as well as the energy release rate under small scale yielding are studied.When the layer-thicknesses of the piezoelectric bi-material approach infinity,the relationship obtained in this paper between the lengths of the electro-mechanical nonlinear zones and the loadings reduces to the analytical expression in literature.For the piezoelectric bi-material with finite layer-thicknesses,the numerical results show that the lengths of the strip-like yielding zones rely on both the loadings and the layer-thicknesses.The energy release rate under the small-scale yielding condition depends on the loadings,layer-thicknesses as well as crack velocity.(2)Based on nonlinear fracture theory,we deal with the problem of a central mode-I crack with the strip-like electrical saturation and mechanical yielding zones in front of the crack tips in a piezoelectric plate of finite width.For this central mode-I crack problem,we perform different Fourier transforms on the abscissa-coordinate and ordinate-coordinate respectively,and simplify this mixed boundary value problem into coupling Fredholm integral equations of the second kind by Copson method.In the same way,with appropriate formula derivation and numerical discretization,we successfully obtained the semi-analytical solutions for any width.When the width of the piezoelectric plate tends to be infinite,analytical expressions of the electromechanical nonlinear zones can be obtained.When the piezoelectric plate has a finite width,the numerical results show that the lengths of the nonlinear zones are not only influenced by the loadings,but also by the width of the piezoelectric plate,and the effect is significant.(3)Based on nonlinear fracture theory,an internal penny-shaped crack in a piezoelectric layer sandwiched by two outer elastic layers is proposed.It is assumed that the crack faces are electrically impermeable,and the thin annular electrical saturation and mechanical yielding zones appear around the crack front under the uniform distributed normal stress and electric displacement.For such an axisymmetric penny-shaped crack problem,we simplify the mixed boundary value problem into coupling Fredholm integral equations by employing Hankel transform technique and the Copson method.This problem is solved by constructing real general solutions in the real domain and the numerical solutions for any layer-thickness are derived with appropriate formula derivation and numerical discretization.When the outer elastic layers vanish and the piezoelectric layer-thickness approaches infinity,the relation between the nonlinear zone-lengths and the loadings obtained in this paper reduces to the analytical expression in literature.The influence of the electromechanical loads,the thicknesses of the piezoelectric layer and the elastic layer,and the material parameter ratios on the nonlinear zones is discussed.