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Abstract: This article presents numerical results for the issue ofcrack in piezoelectric sensor applied in robotics. Normalized force,normalized energy and stress intensity factor are analyzed for crack openingin mode I in the single material and between two materials. The linearelastic fracture mechanics is used in the analysis. Force and energydistribution around crack in piezoelectric PZT ceramic (PZT-4 and PZT-6B) arecalculated in the result section and interpreted in conclusion. Safetyintensity factor is calculated for mode I case. Importance of such analysisis in ability of robot to perform its task with smaller risk of failure.Failure due to sensor malfunction can endanger not just the task, but alsothe people in the proximity of robot manipulation area. Sensor's failurecan cause disability to avoid obstacles or to cause injuries (in the worstcase with mortal consequences).
Generally speaking, defects such as dislocation, crack, cavity andinhomogeneity may occur in piezoelectric materials both during manufactureand use. When the materials are subjected to mechanical or electrical loadsstress concentrations due to defects can give rise to critical growth andsubsequent mechanical failure. Therefore, it is of vital importance to studyelectro-elastic interaction of piezoelectric materials with defects.
Two important problems are addressed in this chapter, bothsafety-related: safety factor of PZT ceramic with some type of defect (nomatter whether it occurred during use or production) and distribution offorce around the defect. Forces can be used to calculate action or mechanicalenergy, and, finally, safety factor. Another analyzed case is problem of acrack developing between two dissimilar materials, which is important in caseof contact of different materials, which is very common in practice.
Stress intensity factor describes mathematical stress anddeformation distribution in the top of the crack. Physically, stressintensity factor is the measure the intensity of rise of stress in observedspace. In fracture mechanics a crack is an absence of bonds between twoneighboring atom layers. Stress distribution near the crack tip depends onthe form of the fracture surface formation. Fig. 1 shows types ofdeformation.
Mode I is defined by separation of the fracture surfacesymmetrically in regard to the primary crack plane ("crackopening"): u = (r, [phi]), v = v(r, [phi]), w = 0. Mode II is defined bythe movement of one crack surface to the other in the same plane, but inopposite directions ("shear"): u = u(r, [phi]), v = v(r, [phi]), w= 0. Mode III is defined by the movement of one fracture surface along thefront of the crack ("shear out of plane"): u = 0, v = 0, w = w(r,[phi]).
Every arbitrary deformation can be depicted by the combination oftheses modes, i.e. when crack front is a spatial curve, in an arbitrarypoint, all three modes can be present. In polar-cylindrical coordinate system(r, z), general expressions for components of the tensor of mechanical stressstate are expressed as (Hedrih & Peric, 1996):
provided that [K.sub.I],[K.sub.II], [K.sub.III] are stressintensity factors for modes I, II and III. In the vicinity of the crack,Hooke's law is inapplicable, because of the plastic area in whichnonlinear effects are dominant (Hellen, 1984). However, if the characteristiczone dimensions are very small compared to the size of the initial crack, asin case of small sensors for mobile robots, the linear elastic fracturemechanics (LEFM) in an analysis of stress and strain state of the crack tipis possible. In that case the stress intensity factor is a measure of thestress magnitude in the vicinity of the tip of the crack. It is defined for aknown stress state and geometry. For a fictitious infinite body with acentral crack of length 2a loaded along three perpendicular directions, it isexpressed as (Hellen, 1984, KrajSinovic & Sumarac, 1990):
The solution to the problem of mechanical stress and straindistribution in a stressed piezoelectric material can be found if therelationship between tensor of mechanical stress ([[sigma].sub.ij]), vectorof electric displacement ([D.sub.ij)], tensor of strain ([[epsilon].sub.kl]),and vector of electric field ([E.sub.k]) is established. In curvilinearsystem of coordinates, the governing linear piezoelectric equations, in theabsence of volume forces and free electric charges, may be written as:
For boundary conditions on the crack surface with normal incircular direction, when [phi] = [+ or -] [pi], it is assumed that:[[simga].sub.[phi]] = 0, [[tau].sub.[phi]r] = 0 (in radial direction),[[tau].sub.[phi]r] = 0 (in axial direction), [D.sub.[phi]] = 0 (in the spotsof crack surface). In this work, stress intensity factor is simulated formode I. Solution of derived equations is performed using analytical
Suppose that a piezoelectric material is transversely isotropicand containing a charged screw dislocation around a finite crack of length 2aas shown in Fig. 2. A set of Cartesian coordinates (x, y, z) is attached atthe center of the crack. The piezoelectric material has a hexagonal symmetrywith an isotropic basal plane of xy-plane and a poling direction of z-axes.The crack is situated along the plane y = 0. The piezoelectric material issubjected to far field antiplane mechanical and in plane electric loads. Inthis configuration, the piezoelectric boundary value problem is simplifiedconsiderably because only the out-of-plane displacement and in-plane electricfields exist. The constitutive relations for the piezoelectric material are(Kwon & Lee, 2002, Hirth & Lothe, 1982, Kulenovic, 2005):
Taking in consideration that the boundary conditions are: on thesurfaces of the crack ([absolute value of x]
where ), ( y x w is mechanical displacement. The solution can befound by letting w and [phi] be the complex analytic functions such that wW(Z), [phi] = [phi](Z), where Z = x + iy is a complex variable. Due tocomplex variable solution, a crack on the x-axis is constructed using themapping function [xi] = 1 / 1 [Z + [square root of ([Z.sup.2] - [a.sup.2)]]which transforms the circle [absolute value of ([xi])]= 1 in the Z-plane ontoa finite crack of length 2a along the real axis in the z-plane. In this casea screw dislocation subjected to a line force and a line charge, and thepotential functions W and [phi] has three terms, respectively: the firstcorresponds to the line force or charge, the second to the screw dislocation,the third to the uniform external loads. The strain, electric field, stress,and electric displacement can be expressed by these complex potentials (Wang,2001, Hirth, 1982), and the forces acting on a screw dislocation are givenwith:
Superscripts S and T represent internal domain in which a screwdislocation exists and external domain in which a crack subjected to themechanical and electrical loads exists, respectively (Meguid & Zhao,2002, Wang, 2001). Finally, after solving this symmetric problem, theequations which presented the forces on a screw dislocation located inarbitrary position around a crack, can be expressed in the forms:
Normalized force is particularly good for the comparison ofdifferent parameter changes or comparison of different materials, becausenormalization factor is chosen for purposes of reduction of experimentallyobtained constants. From Fig. 6. it is obvious that smaller crack produceslower stress intensity factor. Fig. 7. shows impact of electromechanicalcoupling factor to energy distribution around the crack.
In this work, force distribution around finite crack inpiezoelectric material has been analyzed in electric field presence.Decomposition of the spatial stress into three components of stress for thethree basic forms of crack deformation was needed not only to enable aneasier approach in determining analytical solutions, but also to individuallyidentify the influence of every component on crack propagation inpiezoelectric materials. These results can be useful for manufacturers andusers of PZT materials. The methods of spatial analysis presented in thiswork can be applied to other kinds of materials with cracks. The problemspresented in the article are solved by LEFM, because plastic area is toosmall to influence the final conclusions. From the obtained results, one canconclude that the existence of a crack in a stressed piezoelectric materialleads to a stress redistribution and concentration, and the tensors ofmechanical and piezoelectric stress at the crack tip become infinitely large.This "point of stress singularity" occurs in the most sensitivepoint to fracture. It is shown that the same conclusion can be reached byanalyzing both normalized force and stress intensity factor. Proximitysensors, sensors based for measuring of pressure, sound and vibration sensorsrequired in modern sensor fusion design can be relied on only if there is nodanger form failure. This study shows some problems in crack opening andpropagation which can occur in manufacturing process or in aging process.These problems can cause failure of the entire robot system or failure toperform its task. 2ff7e9595c