Dönel simetrik paraboloid yüzeyi ile sınırlı ortamda bir elastodinamik sınır değer problemi

Abstract

II ÖZET Bu çalışmada çözülmek istenen problem dönel simetrik bir paraboloidin tepe noktasına simetri ekseni doğrultusun da, başlangıç anında, ani olarak yüklenen tekil bir yükün paraboloid içerisinde meydana getirdiği gerilme ve yer de ğiştirme alanlarının elde edilmesidir. Malzemenin lineer elastik homogen ve izotrop olduğu kabul edilmektedir. Birinci Bölümde, kullanılan notasyonlar, gerekli mate matik ön bilgiler verilmiş ve temel kavramlar açıklanmış tır. İkinci Bölümde, birim elastodinamik hal tanımı yapıl mış ve problemin çözümünde kullanılacak olan parabolik koordinatlarda birim haller elde edilmiştir. Üçüncü Bölümde, çözülmek istenen problemin tanımı ya pılmış ve birim elastodinamik haller ve karşıtlık teoremi kullanılarak parabolik koordinatlarda yerdeğiştirme bileşen lerinin sağladığı iki integral denklem elde edilmiştir. Bu bolümde, ayrıca parabolik bölgede gerilme sınır değer prob lemi için Green halleri tanımlanmıştır. Green hali, aynı en- disli birim hal ile yardımcı bir elastodinamik halin topla mı olarak ifade edilmektedir. Yardımcı elastodinamik hal, sınır üzerinde kütle kuv vetleri olarak birim haldeki gerilmelerin alınması ile teş kil edilmiştir. Daha sonra bu hal ile problemdeki yerdeğiş tirme ve gerilme alanının oluşturduğu elastodinamik hale karşıtlık teoremi uygulanarak çözüm, değişken sınırlı yü zey integralleri şeklinde elde edilmiştir. Üçüncü Bölümde, integral formda elde edilen gerilme, yerdeğiştirme ifadelerinde integrandların yerdeğiştirmele- re ait olanları üçüncü bölümün sonunda, gerilmelere ait olanlar ise Ek. 1 de verilmiştir. Dördüncü Bölümde, sınır üzerindeki noktalarda sıfırdan farklı olan gerilmelerin zamanla değişimini inceleyebilmek için yapılan deneyler açıklanmış ve sınır üzerindeki bir noktada gerilme-zaman diyagramları verilmiştir.

Ill SUMMARY 1. Introduction. The theory of linear elastodynamics has developed very rapidly in the last decade. However, review of literature shows that the problems are solved for bodies of very simple shapes, such as half spaces, layers or cylinders. In this study it is attempted to give a linear elasto- dynamic solution, based on Wheeler & Sternberg's paper and Achenbach's book, for bodies having arbitrary curved boundaries. The method of the solution is applied to the domain of a paraboloid of revolution, where the surface traction exists only on the top of the paraboloid. According to the shape of the considered domain, the parabolic coordinate system is used. The parabolic coordinates u,v,0 are connected with the cartesian coordinates x,y,z as follows. x - uvGosQ y = uvSine (1>1) z = ± (u2-v2) 2 «>u>0, oo>V£0, O>0>2tt In this study, only physical components of the vectors and tensors will be used, x denotes the position vector of a point in the three dimensional Euclidean space. 2. Formulation of the problem. In this problem the domain V and boundary S are defined as follows : (2.1)IV V is the closure of V and contain all the points of V and S. The material is assumed to be linear elastic, homogeneous and isotropic. The constitutive equations are given for this material as follows, t..- A6..U,, + y(U.,+U...) (2.2) Where A and u are the Lame elastic constants, and JJ(x,t), t(x,t) are displacement vector and stress tensor respecti- 'vely. Equations of motion are given as follows. Tij;j+Pfi =<>Ui <2'3> where p and f denote mass density and body force density in the medium respectively. The initial conditions at the time t=0 are (2.4) (2.5) (2.6) where t,t,tfl denote the components of surface traction vector £. In this case these components actually corres pond to the three components of the stress tensor, x designates the top point of the paraboloid. It should be noted that two dimensional Dirac delta function is employed in (2.6). The aim of this study is to obtain the displacement vector U(x,t) and the stress tensor j(x,t) which satisfy the equations (2. 2), (2. 3), with initial conditions (2.4) and also boundary conditions (2.6) in the above mentioned domain V. 3. Singular elastodynamic states in parabolic coordinates. In this section some necessary concepts will be definedfor the solution. It is assumed that the reader is familiar with the tensor analysis, the concept of elastodynamic state which was introduced by Wheeler & Sternberg, also Riemann convolutions and the dynamic reciprocal identity. The convolution of two vectors a and b is defined as a *b = a.* b. (2.7) «. «. 11 where summation convention must be invoked. Let us define three special elastodynamic states which correspond to the Stokes states as used by Wheeler and Sternberg. g(t) is a regular twice differentiable function of time and g(t) =0 i &&*> n = l,2 dtr for t^O (2.8) It is considered a concentrated load of magnitude g(t) in the infinite medium as the body force density. The above load applied to an arbitrary point y,ı(uı,vı,eı) in the domain V and directed along the constant vector a which is one of the unit base vectors at point y^. Then due to this body force density displacement vector U(x,t) and stress field x(x,t) will be calculated. These Ji and t defines a singular elastodynamic state as shown by >ul(Uul(x,t;y1g(t)), TUl(ç,t;yıg(t)) In this elastodynamic state body force density can be written as fUl =6(x-yi)g(t)a (2.9) *» «t *» «*u In (2.9) Dirac delta function is three dimensional. After these definitions, following Achenbach's work step by step and utilizing some knowledge about the curvilinear coordinates, the displacement vector U * and the stress field x 1 can be calculated easily. In a similar manner the other two elastodynamic states &> '* and y> ^ can be constructed.VI The covariant derivatives of these singular states with respect to the parabolic coordinates of g,. are also new elastodynamic states and represented byj©*!3^ (x,,x, = ujyvı,6.ı.). These are called doublet elastodynamic states. It is obvious that there exists nine such doublet elasto dynamic states. Utilizing the above doublet elastodynamic states, two new kinds of elastodynamic states can be defined as follows. k Jl k S. Si k ^(xi.xj) _ 1_ ^xjx^x^j j (2 1Q) #>h ^lXl (2al) 4. Solution Let SP (U,t) be an elastodynamic state which corresponds to the problem which is formulated in (2. l)-(2. 6).Applying dynamic reciprocal identity to the pairs otiP 1, &> and SP, jo by employing shifting property of Dirac delta func tion, the following results are obtained. pg(t)*u (yi,t)= -HCO^Cj.tixrJgCt» -J Itui-(x.t;Zl.g(t))*.U-] dS pg(t)*U (yi,t)= -H(t)*u31(x,t;y1g(t)) - / ftvl(x,t;yig(t)) *U ] dS (2.12) From the doublet elastodynamic states, six new elasto dynamic states can be constructed in the following manner. ^VlL ^A+2y^(xlXl) (2.13) Applying dynamic reciprocal identity to these states and SP, following equality for the stress components can be obtained., a Pg(t)*T k/Yl,t)= -H(t) *Dw^l»xl](xft;yi.g(t)) xx ~ k I V [tXlxl(x,t;Xlg(t))*u] dS b (2,14)VII Now in our problem we can assume that the surface trac* tions are zero and the body force density are taken as fol lows `. f = -H(t)a 6(x-x )- (2.15) ~v - -o p in (2.15) the Dirac delta function is three dimensional. This definition does not make any difference in the form of the equations (2.12), (2.13) and (2,14). Three Green states will be defined and each of them corresponds to a singular elastodynamic state. These Green states are represented by iP 1 &,£? and similarly six new elastodynamic states are defined in the form of ^l^iXi] # xhe body force density in a Green state is equal to the body force in the corresponding singular or doublet state. But the Green states yield null tractions on the boundary surface S. A Green state like^ 1 can be considered to be the summation of a singular state J* 1 and a complemantary state 5>UI i.e. & ul =#>ul +S>U1 (2.16) In the complemantary state the body force density in the domain V is null and the surface tractions is defined as tUl= -^(^tjyilgXt)) (2.17) The application of dynamic reciprocal identity to5> 1 and &> yields -H(t) *U^ (xo,t;zlg(t)) = sil?1*?]' dS (2.18) If^U1 can be obtained, the displacement compo nents may be calculated from equations (2.11) and (2.18) as follows pg(t) * Uv(Xl,t) = -H(t)* U^(xo,t;xlg(t)) -H(t) *D`l(x `,t;yig(t)) (2.19) V **0.* Recalling commutative properties of convolutionsVIII is obtained. For construction of U *, the body force density [-t /x3,t' ;y_i g(t')- )dS3] is applied at a point x3 on the surface S when the time t' equals zero. Here t'=0 defines the arriving time of the motion to the point x3 which started from the point ^1 at the time t=0 due to the body force density g(t)g6(x-y1). Due to this body force the displacement field can be cal culated at an arbitrary point x in the domain V. (in infinite space) Now let that this displacement field be U2 which belongs to an elastodynamic state &>. At an arbitrary time t, a surface S' can be found which is undisturbed by the elastodynamic states &> and <J^2 in the infinite medium. If the dynamic reciprocal identity is applied to the SP and^2 in the closed domain V' which bounded by surface S' the following equation is obtained. P / (-^- a 6(x-x ) *U2(x,t)) dV v p ~v ~ :° ~6(v-v ) = p /[-tUl(x3,t;zlg(t)i-).° *U(x3,t)J. V' ` P /u2 + v2 3 o.uv(u2+v2)dudvde (2.21) From (2.21) it follows that -H(t) *U2(x.t) = /[-tUl(x3,t;y1g(t))*U(x,t)].dS3 v ~o S - ~ - ~ ~ (2>22) The following final equation is obtained from (2.22) and.. (.2. 18.) U2(xo,t)= U2(xo,t;Xlg(t)) (2.23) The other complemantary states are constructed in similar manner and displacements and stresses are obtained in the form of surface integrals. These integrals are calculated over surfaces which varies with time.