Şekil değiştiren elastik dielektriklerin kuvasi statik teorisi

Abstract

Bu çalışmada dış elektrik yük ve kuvvetler altında polarize olabilen elastik dielektrik bir ortamın kuvasistatik koşullar altında alan ve nonlineer bünye denklemleri elde edilmiş, büyük ön alanlar üzerine küçük yer değiştirme ve elektrik alam süperpoze edilerek artımsal alam ve bünye denklemleri türetilmiş, nihayet bütün bu teorik çalışmaların bir uygulaması olması için eksenel dış yüke ve elektrik alanına maruz dairesel ve sonsuz uzunluklu bir silindir içeri sinde küçük genlikli harmonik dalga yayıhmı problemi incelenerek dispersiyon bağıntısı elde edilmiş ve çeşitli özel haller tartılmıştır. Bu amaçla ikinci Bölüm'de denklik yasaları elde edilmiştir. Bunun için önce kuvazi-elektrostatik kuvvet,moment ve enerji (güç) yoğunlukları elde edilmiş,bunlar yardımıyla sürekli bir ortama ait denklik yasaları, şöyle ki lineer momentum denkliği,açısal momentumun denkliği,enerji denkliği ve entropi eşitsizliği ile elektrostatik alan denklemleri elde edilmiştir. Üçüncü Bölüm de böyle bir ortam için genel ve izotrop hale ait nonlineer bünye denklemleri elde edilmiştir. Büyük ön alan üzerine küçük alanların superpozisyonu sonucu oluşan artımsal alan ve bünye denklemleriyle sınır koşullarıda Dördüncü Bölüm de incelenmiştir. Nihayet 'Genelleştirilmiş Noe- Hookaen Malzemesi ' adım verebileceğimiz özel tipte bir malzemeye için nonlineer artımsal bünye ve alan denklemleri aynı bölümün sonunda incelenmiştir. Nihayet bütün bu teorik çalışmaların bir uygulaması olmak üzere eksenel dış ön yüke ve elektrik alana maruz sonsuz uzunluklu dairesel bir çubukta har monik dalga yayılımı incelenmiş, genel ve özel hallere ait dispersiyon bağıntısı elde edilmiş ve bazı yorumlar yapılmıştır.

The studies on the interactions of mechanical and electromagnetic fields had been received a great deal of interests since the mid of nineteen fifties (1950) Because of such properties of certain materials (such as some piozee- lectric and piezo-magnetic cyristals, bones etc.), it is possible to obtain mecha nical deformations by applying electrical or magnetic fields to such materials and vice-versa. The nonlinear field equations, constitutive relations and the boundary conditions of elastic dielectrics first developed by TOUPİN [1] and ERİNGEN [2] bye use of a variational principle. In the following years, the same scientists and their co-workers, (see TOUPİN [3] and DIXON and ERİNGEN [4] ) star ting from a micro model, had obtained the expressions of electromagnetic force, couple and energy and used them in deriving the balance laws of such an inte racting medium. However, because of their definition of generalized dielectric displacement vector(cf. DIXON and ERİNGEN [4] ) the quadrupole tensor remained constitutionally undetermined. The same subjects had been re in vestigated later by DEMİRAY and ERİNGEN [5] and obtained the quadrupole tensor by introducing the gradient of electric vector by one of the constitutive independent variables. KAFADAR [6] by employing another approach gene ralized our result to higher order moments. In this work, using the micro model propesed by Lorentz, we obtained the expressions of quasi-electrostatic force moment and energy (power) and used them in deriving the balance laws applicable to such a dielectric medium. The nonlinear constitutive relations for anisotropic and isotropic medium is pre sented in Chapter 3. The equations, governing the incremental displacement and electric field are given in Chapter 4. Finally the solution of an examples problem is presented in Chapter 5, where we have studied the small amplitude harmonic waves propagation in a cylindrical bar subjected to an initial stress and an electrostatic field in the axial direction ans obtained the dispersiyon relation. The master equations are summarized below. We assume that an elastic dielectric medium is subjected to external for- ces and electrical fields. The deformation of a representetive material particle may be given by x = x(X) or X = X(x) (1) This mapping is assumed to be one-to-one and onto. Having defined the deformation the balance laws are briefly given below. (i) Conservation of mass : The total mass of a body is unchanged with motion. The law of conser vation of mass states that the initial total mass of the body is the same as the total mass of the body at any other time,i.e J PodV= J pdv (2) Using the transformation law dv = JdV we may write this as PJ)dV = 0 (3) v (ii) Balance of Linear Momentum : The time rate of momentum is equal to the resultant force F acting on the body. This law is stated in local form as tji,j+pfi+9i = pvi (4) where t ij is the stress tensors,?; j is the velocity component, fi is the mechanical body force density and gi is the electrical force density of which the expression is given in the main text. Here dot is used to denote the material derivative and comma for partial differentiation. vi (iii) Ra.1a.nr.ft of angular momentum : The time rate of change of the momentum is equal to the resultant mo ment of all forces on the body. This balance law is stated in local form as follows Hij] + pliEj] = 0 (5) where Ei is the electrical field vector, P, the polarization vector and the indices enclosed in the brackets denote the skew-symmetric part of the second order tensor,i.e. *»i = Ö (*y ~ *i«) (6) This shows that the stress tensor is not symmetric. (iv) Balance of Energy The time of the sum of kinetic energy K and internal energy E is equal to the sum of the rate of work of all forces and couples W and all other energies Ua that enter and leave the body per unit time. This law reads in differential or local form as pe = tjiVj,i + EiP* - qifi + ph (7) where e is the internal energy density, qi the heat flux vector, h is the heat source and P* is defined by Pf = Pi + PiVr,r. (8) (v) Entropy inequality : This law is also known as the second law of thermodynamics and stated in the local form as follows vu *+*?-*& + $** w where rj is the entropy volume density and 6 is the absolute temperature. Eli minating ph between (7) and (9), and introducing the Helmholtz free energy as e = * + Br] + - (10) we have -po(* + r}6) + TKiFiK - JEiPi - Q^Z- > 0 (11) where we have defined TKi = JHKjtji, QK = JHKiqi, (hKİ = ^\ (12) (vi) Field Equations of electrostatics : These equations may be stated as a special case of Maxwell's equations and given by VXE = 0 (13) V.D = q0 (14) where D = eoE + P is the dielectric displacement vector,^ is the free charge density and eo is the dielectric constant of the vacuum. According to (13) the line integral of the field intensity E around any closed path is zero and the field is conservative. The conservative nature of the field is a necessary and sufficient condition for the existence of a scalar potential whose gradient is E. viii According to the (14) The total flux of the dielectric displacement across any closed surface S is equal to the total free charge q within the surface. Constitutive Equations The balance equations are not sufficiently in number for the determina tion of the unknown variables. Therefore the system must be supplemented bye a set of constitutive equations in order to obtain a complete system. We write additional relations between the unknowns, which is called the consti tutive equations. For the present problem we have selected the Temperature deformation gradient and electric field vector densitiy as independent consti tutive variables. After imposing the principle of objectivity and the entropy inequality the constitutive equations reduced to tij = jdc^;FiKFiL~PiEj i as as Pi = -jW^ Por} = -de> s = /?0* (15) where Ckl = F*K FiL is the Green deformation tensor. Further, assuming that the material under investigation is isotropic the constitutive equations reduce to Uj =My + hlC-l + h2c-? + hA{EicjiEk + EjcTfEk) +h5(c-k1Ekcj'El + \ (EiCJ^c-\Ek + c&cÛEuEj) (16) Pi = -(h3Ei + fuc^Ei + h5crfE{) (17) where ci j1 is the Finger deformation tensor and ho, hi, hi, h$, h^, h$ are some scalars of which the expressions are given in the main text. Small Fields Superposed on the Large Initial Fields : In this part Chapter 4 of the work we have obtained the governing field equations and incremental constitutive equations of a dielectric medium by ix superimposing small displacement and electrical field on given initial static field.