2 Governing Equations

Transcription

1 2 Govening Equations This chapte develops the govening equations of motion fo a homogeneous isotopic elastic solid, using the linea thee-dimensional theoy of elasticity in cylindical coodinates. At fist, classical elationships between stess, stain, and displacement ae eviewed and implemented into the dynamic equilibium equations. The mathematical epesentations of the linea theoy of elasticity deived in this chapte will set the stage fo the development of the equied govening equations fo the possible modes of vibations in cylindical stuctues with any thickness. y σ σ θθ δ σ δ σ δθ σ σ θθ θ x FIGURE 2.1. Diect Stesses in Cylindical Coodinates. A detailed mathematical eview of elasto-dynamic poblems can be found in most classical text books on advanced mechanics of mateials and the theoy of elasticity. In paticula, efeences such as (Fod and Alexande 1963) and (Mal and Singh 1991) can povide the best extensive eviews H.R. Hamidadeh, R.N. Jaa, Vibations of Thick Cylindical Stuctues DOI / _2, Spinge Science+Business Media, LLC 2010

2 16 2. Govening Equations σθθ σ θθ + δθ θ τ τ θ θ + δθ θ δθ σ τ τ + δ τ τ θ θ + δ σ σ + δ τ θ τ θ σ θθ FIGURE 2.2. Stesses in the and θ Diections. of stess, stain, and displacement in cylindical coodinates. The following sections povide a succinct eview of essential topics needed fo the establishment of the govening elasto-dynamic equations. 2.1 State of Stesses at a Point A thee dimensional state of stess in an infinitesimal cylindical element is shown in the following thee figues. Figue 2.1 depicts such an element with diect stesses, dimensions, and diections of the cylindical coodinate. Figue 2.2 epesents the diect and shea stesses in the adial and tansvese diections ( and θ), and the vaiation of diect and shea stesses in these two diections. Figue 2.3 shows diect and shea stesses associated with the planes pependicula to the and diections, as well as thei vaiations along these diections. In the above gaphical epesentations the changes in diect and shea stesses ae given by consideing the fist ode infinitesimal tem used in Taylo seies appoximation. The seies appoximation has been tuncated afte the second tem. Futhe tems within the seies epesentation contain tems of an infinitesimal length squaed. Assuming that the second

3 σ δ τ θ 2. Govening Equations 17 σ σ + δ δ τ τ θ θ + δθ θ τ τ + δ F σ θθ δθ τ θ σ σ + δ FIGURE 2.3. Stesses in the plane pependicula to and diection. ode tems ae vey small, they can be neglected. Theefoe, the change in stess acoss the element is consideed vey small. 2.2 Equilibium Equations in Tems of Stess Utiliing Newton s second law and the gaphical epesentation of the state of stess, the equilibium equations fo an infinitesimal element in a cylindical coodinates will be developed. By examining the state of stess on the element shown in section 2.1, the following equilibium equation in the diection is given. µ σ + σ δ µ τ + τ δ µ ( + δ) δθδ + µ + δ 2 τ θ + τ θ θ δθ + δδθ + F δδθδ = σ δθδ + δτ θ δδ cos δθ µ 2 + τ + δ 2 µ + σ θθ + σ θθ θ δθ δδ sin δθ 2 + σ θθδδ sin δθ 2 δδθ + δδ cos δθ 2 (2.1)

4 18 2. Govening Equations Canceling appopiate tems fom both sides of the equation and afte simplifying, it yields: σ + 1 τ θ θ + τ + σ σ θθ + F =0 (2.2) Similaly, the equilibium equation fo the θ diection yields: τ θ + 1 σ θθ θ + τ θ + 2 τ θ + F θ =0 (2.3) and finally, fo the diection one may wite: τ + 1 τ θ θ + σ + 1 τ + F =0 (2.4) In the above simplifications, due to vey small angle of δθ, thefollowing appoximations wee used: cos δθ 2 1 sin δθ 2 δθ 2 (2.5) In addition to the stesses, body foces acting thoughout the element have been consideed fo each diection. These ae denoted by F, F θ,andf which ae intoduced as foces in the, θ, and diection pe unit of volume. Due to the cancellation of the moments about each of the thee pependicula axes, the elations among the six shea stess components ae pesented by the following thee equations: τ θ = τ θ τ θ = τ θ τ = τ (2.6) Theefoe, the stess at any point in the cylinde may be accuately descibed by thee diect stesses and thee shea stesses. 2.3 Stess-Stains Relationships The constitutive elation between stesses and stains fo a homogeneous and isotopic mateial can be expessed by Hooke s law. By definition, a homogeneous and isotopic mateial has the same popeties in all diections. Fom this, the following thee equations fo diect stain in tems of stess ae pesented: e E = σ ν (σ θθ + σ ) (2.7) e θθ E = σ θθ ν (σ + σ θθ ) (2.8) e E = σ ν (σ + σ θθ ) (2.9)

5 2. Govening Equations 19 whee e, e θθ,ande ae the diect stain in the, θ, and diections espectively; E is the Young s modulus o the modulus of elasticity; and ν is a popotionality facto called Poisson s atio. The othe thee Hooke s law elations esult fom the following popotionality between shea stesses and shea stains: τ θ = Ge θ (2.10) τ = Ge (2.11) τ θ = Ge θ (2.12) whee, e θ is the shea stain along θ and pependicula to ; e is the shea stain along and pependicula to ; e θ is the shea stain along and pependicula to θ; andg is the shea modulus o the modulus of igidity. Though the geneal definition of shea stess and stain, the elationship between shea modulus, Young s modulus, and Poisson s atio is given as: G = E 2(1+ν) (2.13) Lame s elastic constant, λ, and volumetic stain, ε ae intoduced by the following equations: λ = νe (1 2ν)(1+ν) (2.14) ε = e + e θθ + e (2.15) Then, a diffeent fom of Hooke s law elating diect stesses and diect stains can be achieved by adding the diect stain equations (2.7)-(2.9). Eε 1 2ν = σ + σ θθ + σ (2.16) Reaanging equation (2.7), one may wite: σ θθ + σ = σ Ee ν Now, by substituting equation (2.17) into (2.16) it yields: Eε 1 2ν = σ + σ ν Ee ν and the esult can be aanged as: (2.17) (2.18) νeε 1 2ν =(1+ν) σ Ee (2.19)

6 20 2. Govening Equations fom which the diect stess in the adial diection is detemined to be: σ = νe (1 2ν)(1+ν) ε + E 1+ν e (2.20) Now using the definitions of the shea modulus and Lame s elastic constant, the diect adial stess is pesented as: σ = λε +2Ge (2.21) In a simila pocedue, the diect cicumfeential stess and the diect axial stess ae epesented in tems of the volumetic stain, Lame s elastic constant, the shea modulus, and the appopiate diect stains ae pesented in the following equations. σ θθ = λε +2Ge θθ (2.22) σ = λε +2Ge (2.23) 2.4 Stain-Displacement Relationships In Figue 2.4, a small element of an elastic homogenous and isotopic medium is epesented in cylindical coodinates. The element contains the point A, which epesents a given point having the coodinates of (, θ, ) and the point F, an infinitesimal distance away, having the coodinates ( + δ, θ + δθ, + δ). In this figue, the angle θ may be measued fom any abitay coodinate diection such as x. A typical small linea defomation of this element is depicted in Figue 2.5 whee displacement and the distoted shape of the enlaged element is outlined. As can be seen, the displacement of point A to A 0 is defined by the thee components of u, u θ,andu.wheeu, u θ,andu ae the displacements in the adial diection, tansvese diection, and axial diection, espectively. It should be noted that u θ is the actual linea displacement along a cicumfeential ac. The displacements of the point F to F 0 ae (u + δu ), (u θ + δu θ ),and(u + δu ). Consideing a hoiontal plane, in Figue 2.6, the face ACDB of the element peviously shown in Figue 2.5 moves to A 0 C 0 D 0 B 0 whee thee is a change in the length of the sides and the angles ae sheaed. Angle sheaing is esulted by the change of the angle δθ to (δθ + 4δθ).

7 2. Govening Equations 21 y δ G H B ( + δ, θ+δθ, +δ ) F A E D θ δθ C x δ FIGURE 2.4. An Element in Cylindical Coodinates. H F 1 F ' G F u θ u B E D' D 1 x A u A' C C ' C 1 D y FIGURE 2.5. Element Subjected to Small Defomation.

8 22 2. Govening Equations D ' B B 1 B ' D O δθ δθ + Δδθ uθ A A 2 A' C ' C 3 C 1 D C 2 FIGURE 2.6. Hoiontal Plane of Stained Element. Upon examining the adial stain at point A and ignoing the effects of stains in the diection, the stain in the side AC can be found. If the distance A 0 C 0 is tansfeed to the line AC by dawing acs, with cente O, though A 0 and C 0 to intesect the line OAC at points A 2 and C 2 then the adial stain can be defined as: e = A 2C 2 AC (2.24) AC Consideing the geomety of the Figue 2.6, the above equation can be witteninthefollowingfom: µ δ + u δ δ e = = u (2.25) δ In a simila manne, the diect cicumfeential stain may be defined as: e θθ = A0 B 0 AB (2.26) AB whee, AB = δθ and µ A 0 B 0 =( + u ) δθ + u θ θ δθ. (2.27) In the definition of A 0 B 0, the incease in the angle of the ac of A 0 B 0 is given by δu θ / ( + u ) which can be appoximated by ( u θ / ( θ)) δθ.

9 2. Govening Equations 23 Theefoe, by neglecting the second ode tems, the cicumfeential diect stain is given by: e θθ = u θ θ + u (2.28) The shea stain, e θ,isepesentedbythechangeoftheanglebac. By dawing A 0 C 1 paallel to AC, A 0 B 1 paallel to AB, and continuing line OA 0 to yield point C 3, the following pocedues will yield e θ which is a ate of change of the line A 0 C 1.NoticethatA 0 C 1 is paallel to line AC. whee, e θ = C 3 A 0 C 0 + B 1 A 0 B 0 (2.29) C 3 A 0 C 0 = C 1 A 0 C 0 C 1 A 0 C 3 (2.30) and C 1 A 0 C 3 = AOA 0 = A 2A 0 = u θ + u. (2.31) The definition of the length C 1 C 0 is: then, C 1 C 0 = u θ δ (2.32) C 1 A 0 C 0 = C 1C 0 AC = u θ. (2.33) Theefoe, the shea angle C 3 A 0 C 1 can be defined as: u θ C 3 A 0 C 0 = δ u θ δ = u θ u θ The shea angle B 1 A 0 B 0 can be defined as: (2.34) B 1 A 0 B 0 = B 1B 0 (2.35) δθ which is the adial displacement of B due to the angle δθ ove the initial length. In patial deivative fom this simplifies to: B 1 A 0 B 0 = 1 u δθ θ δθ = u (2.36) θ Theefoe, the shea stain e θ,isgivenas: e θ = u θ u θ + u θ (2.37)

10 24 2. Govening Equations H ' H G ' G 1 G δ B' B 1 B δθ A' A FIGURE 2.7. The (, θ) Plane of Stained Element. Consideing the diection, the diect axial stain can be defined similaly to the pocedue used in Catesian coodinates. Recall that the axial stain is defined as the atio of the change in length to the oiginal length of the element in the diection. Examining the stain in line AF in the diection, the diect axial stain is given as: e = A0 F 0 A F A F (2.38) which simplifies to: and finally to: δ + u δ δ e = δ e = u (2.39) (2.40) In Figue 2.7 the plane is shown as viewed fom the oigin. On the face ABHG the shea stain e θ causes the ight angle BAG to be displaced to B 0 A 0 G 0.NotethatA 0 B 1 is paallel to AB and A 0 G 0 is paallel to AG. Theefoe, the shea stain, e θ,isgivenas: which is equivalent to: e θ = G 1 A 0 G 0 + B 1 A 0 B 0 (2.41) and simplifies to: e θ = 1 u θ δ δ + 1 δθ e θ = u θ + u θ u δθ (2.42) θ (2.43)

11 2. Govening Equations 25 Finally by examining the (, ) plane, the shea stain, e,isdefined as: e = G 1 A 0 G 0 + C 1 A 0 C 0 (2.44) which yields: and simplifies to: e = u δ + δ u δ δ e = u + u (2.45) (2.46) Thee ae now six stain components given in tems of the cylinde displacements. This completes the development of the equied stain-displacement elationships. 2.5 Stess-Displacement Relationships In this section, the stess-displacement elationships ae developed by building upon Hooke s law and stain displacement elationships. Beginning with the diect adial stess in tems of stain and substituting the equations fo diect stains, the adial stess in tems of displacement can be pesented as: µ u σ = λ + u θ θ + u + u +2G u (2.47) The substituted diect stains ae in tems of displacements and the volumetic stain. Similaly, the diect cicumfeential stess and diect axial stess, in tems of displacement, may be given as: µ u σ θθ = λ σ = λ + u θ θ + u + u µ u + u θ θ + u + u µ uθ +2G θ + u +2G u (2.48) (2.49) Similaly, the thee shea stesses in tems of shea stains ae given by equations (2.10)-(2.12) and the shea stains, in tems of displacement components, ae povided by equations (2.37), (2.43), and (2.46). Theefoe,

12 26 2. Govening Equations these shea stesses ae: µ uθ τ θ = G u θ + u θ µ uθ τ θ = G + u θ µ u τ = G + u (2.50) (2.51) (2.52) 2.6 Equations of Motion In this section, the govening equations of motion in tems of a displacement vecto ae geneated. The displacement vecto is given as: u = u î + u θ î θ + u î (2.53) whee î, î θ,andî denote unit vectos diected along the (, θ,and) axes, espectively. Substituting Hooke s law equations into the dynamic equilibium equations and intoducing stain-displacement elationships yield the govening equations of motion: μ 2 u +(λ + μ) ε μ 2 u θ +(λ + μ) ε θ μ 2 u +(λ + μ) ε = ρ 2 u t 2 (2.54) = ρ 2 u θ t 2 (2.55) = ρ 2 u t 2 (2.56) whee μ is the same as shea modulus G, ε is the volumetic stain, and the 2 is the thee dimensional Laplacian opeato in cylindical coodinates defined by: 2 = θ (2.57) Multiplying equation (2.54) by î, equation (2.55) by î θ, equation (2.56) by î, and adding these thee equations, the vecto fom of the govening equation of motion is given by: μ 2 u +(λ + μ) ( u) =ρ 2 u t 2 (2.58) whee is the cylindical gadient: = µ + 1 î + θ îθ + î (2.59)

13 2. Govening Equations Key Symbols A, B, A 0,B 0 point label e stain e, e θθ, e diect stain in, θ, diections e θ shea stain along θ and pependicula to e shea stain along and pependicula to e θ shea stain along and pependicula to θ E Young s modulus F, F θ, F foces in the, θ, diections pe unit of volume G, μ shea modulus, modulus of igidity î, î θ, î unit vectos along the axes, θ, adial diection u, u θ, u displacements in, θ, diections u displacement vecto axial diection δ vaiation ε volumetic stain θ tansvese diection λ Lame s elastic constant μ, G shea modulus, modulus of igidity ν Poisson s atio σ nomal stess σ, σ θθ, σ diect stess in the, θ, diections τ shea stess τ θ shea stess along θ and pependicula to τ shea stess along and pependicula to shea stess along and pependicula to θ τ θ 2 angle Laplacian opeato gadient