On the stress state of a piezoceramic body with a flat crack under symmetric loads

Abstract

A static-equilibrium problem is solved for an electroelastic transversely isotropic medium with a flat crack of arbitrary shape located in the plane of isotropy. The medium is subjected to symmetric mechanical and electric loads. A relationship is established between the stress intensity factor (SIF) and electric-displacement intensity factor (EDIF) for an infinite piezoceramic body and the SIF for a purely elastic material with a crack of the same shape. This allows us to find the SIF and EDIF for an electroelastic material directly from the corresponding elastic problem, not solving electroelastic problems. As an example, the SIF and EDIF are determined for an elliptical crack in a piezoceramic body assuming linear behavior of the stresses and the normal electric displacement on the crack surface Keywords: piezoelectricity, flat crack, elliptical crack, stress intensity factor, electric-displacement intensity factorIntroduction. The wide use of piezoelectric ceramic materials, which are highly brittle, in various transducers (based on the coupling of mechanical and electric fields) necessitates a careful study into the concentration of mechanical and electric fields in electroelastic bodies with imperfections such as cavities, inclusions, and cracks. However, the solution of three-dimensional problems of electroelasticity involves severe mathematical difficulties since the original system of equations describing the electrostressed state of a body consists of complicated coupled differential equations [1,4]. This is why plane problems of electroelasticity have recently been studied in more detail. Noteworthy are the papers [2,11,14,17,18] that address the two-dimensional electroelastic state around a single cavity, inclusion, and crack and the interaction of concentrators of electric and mechanical fields. Three-dimensional problems of electroelasticity for an infinite medium with cavities, inclusions, and cracks are solved in [5-7, 9, 10, 13, 15, 16]. The papers [5,15,16] propose approaches to finding the general solutions of coupled equations of electroelasticity for a transversely isotropic body. The exact solutions of electroelastic problems for spheroidal and hyperboloidal cavities and inclusions have been found in [6,13]. The electrostressed state and stress intensity factors (SIFs) and electric-displacement intensity factors (EDIFs) for an infinite medium with penny-shaped and elliptic cracks are studied in [1, 9, 10] and [7, 15, 16], respectively.

Mathematical modeling and analysis of the stressed state in the orthotropic piezoelectric medium with a circle crack

“…Для апробации используемого подхода рассмотрим задачу о круговой трещине в трансверсально-изотропной электроупругой среде, расположен-ной в плоскости, перпендикулярной оси симметрии материала, при извест-ных значениях основного поля 0 [14,15] в этом случае коэффициент ин-тенсивности напряжения (КИН) определяются с помощью аналитических выражений:…”

“…где значение PIEZO  для пьезоэлектрических трансверсально-изотропных материалов зависит специальным образом от электроупругих постоянных материалов [14]. Согласно проведенным исследованиям для пьезоэлектри-ческих материалов PZT-4, PXE-5, PZT-7A, BaTiO3, PZT-5H, свойства кото-рых взяты из работ [1, 9,12,17], получаем следующие значения PIEZO  : 0,48513; 0,48815; 0,47324; 0,34369; 0,7867.…”

“…Для случая транс-версально-изотропных свойств электроупругого материала (представляют широкий класс пьезоэлектрических материалов) в работах [20,25] предло-жены подходы к построению общих решений системы связанных уравнений электроупругости, на основе которых получены решения ряда задач для пьезоэлектрического материала с полостями, включениями, трещинами, что специальным образом ориентированы относительно оси симметрии элек-троупругого материала. Предполагалось, что ось симметрии материала ори-ентирована вдоль оси вращения концентратора напряжений (для полости или включения), а для круговой или эллиптической трещин она расположе-на в плоскости, перпендикулярной оси симметрии материала [5,7,9,12,14,15,[17][18][19][20][21][22][23][24][27][28][29]. В случае же другой ориентации концентраторов напряже-ний (полостей, включений, трещин) в трансверсально-изотропном пьезо-электрическом материале упомянутые подходы не позволяют решать про-странственные задачи элетроупругости.…”

Contact interaction of two compressed electroelastic half-spaces

“…Methods for solving three-dimensional contact problems for elastic bodies were developed in [2, 3, 10, etc.]. The wide use of piezoceramic materials, which are very fragile, necessitates detailed study of the distribution of mechanical and electric fields in electroelastic bodies near cavities, inclusions, cracks, punches [5,[8][9][10][12][13][14][15][16][17][18][19][20]. However, the solution of three-dimensional problems of electroelasticity involves certain mathematical difficulties because the original system of equations for determining the electric and strain states is a complex coupled system of differential equations [5] (see [9,11] for its general solutions).…”

“…Note that the studies [16][17][18][19] are also based on the ideas of [4] but do not give any reference to [4]. We will study the contact interaction (without friction) of two electroelastic transversely isotropic half-spaces with different properties.…”