Crack Extension Force in a Piezoelectric Material

1990 ◽  
Vol 57 (3) ◽  
pp. 647-653 ◽  
Author(s):  
Y. Eugene Pak
Keyword(s):  
Mechanical Load ◽  
Piezoelectric Materials ◽  
Closed Form Solution ◽  
Form Solution ◽  
Piezoelectric Medium ◽  
Electrical Load ◽  
Governing Equations ◽  
Mode Iii ◽  
Extension Force ◽  
Independent Integral

A conservation law that leads to a path-independent integral of fracture mechanics is derived along with the governing equations and boundary conditions for linear piezoelectric materials. A closed-form solution to the antiplane fracture problem is obtained for an unbounded piezoelectric medium. The path-independent integral is evaluated at the crack tip to obtain the energy release rate for a mode III fracture problem. For a fixed value of the mechanical load, it is shown that the crack growth can be either enhanced or retarded depending on the magnitude, the direction, and the type of the applied electrical load. It is also shown that, for certain ratios of the applied electrical load to mechanical load, crack arrestment can be observed.

Orthotropy Rescaling for the Fracture Problem in Anisotropic Piezoelectric Materials

2002 ◽  
Author(s):  
William S. Oates ◽  
Christopher S. Lynch
Keyword(s):  
Closed Form ◽  
Piezoelectric Materials ◽  
Stress Singularity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Governing Equations ◽  
Closed Form Solutions ◽  
Order Polynomial ◽  
Work Done ◽  
Elastic Dielectric

To date, much of the work done on ferroelectric fracture assumes the material is elastically isotropic, yet there can be considerable polarization induced anisotropy. More sophisticated solutions of the fracture problem incorporate anisotropy through the Stroh formalism generalized to the piezoelectric material. This gives equations for the stress singularity, but the characteristic equation involves solving a sixth order polynomial. In general this must be accomplished numerically for each composition. In this work it is shown that a closed form solution can be obtained using orthotropy rescaling. This technique involves rescaling the coordinate system based on certain ratios of the elastic, dielectric, and piezoelectric coefficients. The result is that the governing equations can be reduced to the biharmonic equation and solutions for the isotropic material utilized to obtain solutions for the anisotropic material. This leads to closed form solutions for the stress singularity in terms of ratios of the elastic, dielectric, and piezoelectric coefficients. The results of the two approaches are compared and the contribution of anisotropy to the stress intensity factor discussed.

Experimental Evaluation of Material Behavior in a Wire Under Transverse Impact

1967 ◽  
Vol 34 (2) ◽  
pp. 392-396 ◽  
Author(s):  
A. B. Schultz ◽  
P. A. Tuschak ◽  
A. A. Vicario
Keyword(s):  
Aluminum Alloy ◽  
Pure Copper ◽  
Closed Form Solution ◽  
Strain Curve ◽  
Simple Wave ◽  
Form Solution ◽  
Material Behavior ◽  
Governing Equations ◽  
Transverse Impact ◽  
Behavior Consistency

Results are reported from an experiment involving photographic observation of constant-velocity transverse impact on long wires of annealed pure copper, two pure aluminums, and an aluminum alloy. Predictions of deformation are made assuming the quasi-static stress-strain curve governs behavior. Consistency with experimental observations is examined. Predictions are based on a closed-form solution to the problem, which is shown to be a compounding of two simple wave solutions of the governing equations. Predictions are consistent with observations for the aluminum alloy even under conditions of moderate or high static prestrain. The two pure aluminums and the copper show consistency at low but not at high strain levels. Highest strain levels reached were in the range 0.06–0.14.

Axisymmetrical Turbulent Swirling Jet

1965 ◽  
Vol 32 (2) ◽  
pp. 258-262 ◽  
Author(s):  
Shao-Lin Lee
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Velocity Fields ◽  
Form Solution ◽  
Turbulent Jet ◽  
Governing Equations ◽  
Ambient Fluid ◽  
Swirling Jet ◽  
Experimental Findings ◽  
Circular Source

A simple closed-form solution has been obtained for an axisymmetrical turbulent swirling jet issuing from a circular source into a semi-infinite motionless ambient fluid by introducing the assumptions of similar axial and swirling velocity profiles and lateral entrainment of ambient fluid into the integrated governing equations. Results for the decays of the axial and swirling velocities and the spray of the jet agree closely with the existing experimental findings on the velocity fields of a swirling round turbulent jet of air generated by flow issuing from a rotating pipe into a reservoir of motionless air.

Closed-Form Solution of Natural Frequencies of a Cantilever Beam Under Longitudinal Rotation of the Support

2005 ◽  
Author(s):  
Mehdi Esmaeili ◽  
Mohammad Durali ◽  
Nader Jalili
Keyword(s):  
Closed Form ◽  
Cantilever Beam ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Longitudinal Direction ◽  
Form Solution ◽  
Governing Equations ◽  
Well Drilling ◽  
General Base ◽  
Tip Mass

This paper presents the modeling steps towards development of frequency equations for a cantilever beam with a tip mass under general base excitations. More specifically, the beam is considered to vibrate in all the three directions, while subjected to a base rotational motion around its longitudinal direction. This is a common configuration utilized in many vibrating beam gyroscopes and well drilling systems. The governing equations are derived using Extended Hamilton’s Principle with general 6-DOF base motion. The natural frequency equations are then extracted in closed-form for the case where the base undergoes longitudinal rotation. For validation purposes, the resulting natural frequencies are compared with two example case studies; one with a beam on a stationary base and the other one with a rotor having flexible shaft.

scholarly journals Best theory diagrams for multilayered plates considering multifield analysis

2017 ◽  
Vol 28 (16) ◽  
pp. 2184-2205 ◽  
Author(s):  
M Cinefra ◽  
E Carrera ◽  
A Lamberti ◽  
M Petrolo
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Governing Equations ◽  
Equilibrium Equations ◽  
Material Parameters ◽  
Stress Components ◽  
Minimum Number ◽  
Multilayered Plates ◽  
Carrera Unified Formulation ◽  
Definition Of

This work presents the best theory diagrams (BTDs) for multilayered plates involved in multifield problems (mechanical, thermal and electrical). A BTD is a curve that reports the minimum number of terms of a refined model for a given accuracy. The axiomatic/asymptotic technique is employed in order to detect the relevant terms, and the error is computed with respect to an exact or quasi-exact solution. The models that belong to the BTDs are constructed by means of a genetic algorithm and the Carrera Unified Formulation (CUF). The CUF defines the displacement field as an expansion of the thickness coordinate. The governing equations are obtained in terms of few fundamental nuclei, whose form does not depend on the particular expansion order that is employed. The Navier closed-form solution has been adopted to solve the equilibrium equations. The analyses herein reported are related to plates subjected to multifield loads: mechanical, thermal and electrical. The aim of this study is to evaluate the influence of the type of the load in the definition of the BTDs. In addition, the influence of geometry, material parameters and displacement/stress components are considered. The results suggest that the BTD and the CUF can be considered as tools to evaluate any structural theory against a reference solution. In addition, it has been found that the BTD definition is influenced to a great extent by the type of load.

scholarly journals Influence of Variable Nonlocal Parameter and Porosity on the Free Vibration Behavior of Functionally Graded Nanoplates

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Pham Van Vinh ◽  
Le Quang Huy
Keyword(s):  
Free Vibration ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Closed Form Solution ◽  
Shear Deformation Theory ◽  
Form Solution ◽  
Functionally Graded ◽  
Governing Equations ◽  
Vibration Behavior ◽  
Special Cases

This paper studies the influence of the variable nonlocal parameter and porosity on the free vibration behavior of the functionally graded nanoplates with porosity. Four patterns of distribution of the porosity through the thickness direction are considered. The classical nonlocal elasticity theory is modified to take into account the variation of the nonlocal parameter through the thickness of the nanoplates. The governing equations of motion are established using simple first-order shear deformation theory and Hamilton’s principle. The closed-form solution based on Navier’s technique is employed to solve the governing equations of motion of fully simply supported nanoplates. The accuracy of the present algorithm is proved via some comparison studies in some special cases. Then, the effects of the porosity, the variation of the nonlocal parameter, the power-law index, aspect ratio, and the side-to-thickness ratio on the free vibration of nanoscale porous plates are investigated carefully. The numerical results show that the porosity and nonlocal parameter have strong effects on the free vibration behavior of the nanoplates.

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