The weight function for cracks in piezoelectrics |
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| Institution: | 1. Department of Mechanical and Environmental Engineering and Materials Department, University of California, Santa Barbara, Santa Barbara, CA 93106, USA;2. Institute of Mechanics and Fluid Dynamics, Freiberg University of Mining and Technology, 09596 Freiberg, Germany;1. Department of Civil Engineering, The University of Hong Kong, Hong Kong, China;2. Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, PR China;1. Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB T6G 1H9, Canada;2. School of Science, Harbin Institute of Technology, Shenzhen 518055, China;3. Department of Mechanical Engineering, University of Alberta, Edmonton, AB T6G 2G8, Canada;1. Laboratory of Physics of Materials, Faculty of Sciences of Sfax, BP 1171, 3000 University of Sfax, Tunisia;2. Sfax Preparatory Engineering Institute, Menzel Chaker Road 0.5 km, BP 1172, 3000 Sfax, Tunisia;1. School of Civil Engineering, Xi’an University of Technology, Xi’an 710048, China;2. State Key Laboratories of Transducer Technology, Chinese Academy of Sciences, Shanghai 200050, China;3. Piezoelectric Device Laboratory, School of Mechanical Engineering & Mechanics, Ningbo University, Ningbo 315211, China |
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| Abstract: | The weight function in fracture mechanics is the stress intensity factor at the tip of a crack in an elastic material due to a point load at an arbitrary location in the body containing the crack. For a piezoelectric material, this definition is extended to include the effect of point charges and the presence of an electric displacement intensity factor at the tip of the crack. Thus, the weight function permits the calculation of the crack tip intensity factors for an arbitrary distribution of applied loads and imposed electric charges. In this paper, the weight function for calculating the stress and electric displacement intensity factors for cracks in piezoelectric materials is formulated from Maxwell relationships among the energy release rate, the physical displacements and the electric potential as dependent variables and the applied loads and electric charges as independent variables. These Maxwell relationships arise as a result of an electric enthalpy for the body that can be formulated in terms of the applied loads and imposed electric charges. An electric enthalpy for a body containing an electrically impermeable crack can then be stated that accounts for the presence of loads and charges for a problem that has been solved previously plus the loads and charges associated with an unsolved problem for which the stress and electric displacement intensity factors are to be found. Differentiation of the electric enthalpy twice with respect to the applied loads (or imposed charges) and with respect to the crack length gives rise to Maxwell relationships for the derivative of the crack tip energy release rate with respect to the applied loads (or imposed charges) of the unsolved problem equal to the derivative of the physical displacements (or the electric potential) of the solved problem with respect to the crack length. The Irwin relationship for the crack tip energy release rate in terms of the crack tip intensity factors then allows the intensity factors for the unsolved problem to be formulated, thereby giving the desired weight function. The results are used to derive the weight function for an electrically impermeable Griffith crack in an infinite piezoelectric body, thereby giving the stress intensity factors and the electric displacement intensity factor due to a point load and a point charge anywhere in an infinite piezoelectric body. The use of the weight function to compute the electric displacement factor for an electrically permeable crack is then presented. Explicit results based on a previous analysis are given for a Griffith crack in an infinite body of PZT-5H poled orthogonally to the crack surfaces. |
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