11 ELASTIC CONSTANTS OF A PLY ALONG AN ARBITRARY DIRECTION To study the behavior of a laminate made up of many plies with different orientations, it is necessary to know the behavior of each of the plies in directions that are different from the principal material directions of the ply.We propose to determine the elastic constants for this ply behavior using relatively simple calculations. 11.1 COMPLIANCE COEFFICIENTS The ply is already defined in Chapter 3.Let e,t and zbe the orthotropic axes of a ply shown in the Figure 11.1.For a thin laminate made up by a superposition of many plies,we assume that the stresses o are zero.It is then possible,for an orthotropic material,to write the stress-strain relation in the plane t,t starting from Equation 9.3 or 9.5 in the form: V E 飞列 (11.1) E E 0 6 0 0 Gu Problem:How can one transform this relation expressed in the coordinates t,t into a relation expressed in coordinates x,y inclined at an angle of 0 with the e,t coordinates (see Figure 11.1).3 First recall the following: 1 See Section 3.2. 2 The orthotropic axes 1.2.3 in Equation 9.3 are now called respectively. 3 What follows is treated more globally and completely in Section 13.2.2. 2003 by CRC Press LLC
11 ELASTIC CONSTANTS OF A PLY ALONG AN ARBITRARY DIRECTION To study the behavior of a laminate made up of many plies with different orientations, it is necessary to know the behavior of each of the plies in directions that are different from the principal material directions of the ply. We propose to determine the elastic constants for this ply behavior using relatively simple calculations. 11.1 COMPLIANCE COEFFICIENTS The ply is already defined in Chapter 3. 1 Let , t and z be the orthotropic axes of a ply shown in the Figure 11.1. 2 For a thin laminate made up by a superposition of many plies, we assume that the stresses szz are zero. It is then possible, for an orthotropic material, to write the stress–strain relation in the plane ,t starting from Equation 9.3 or 9.5 in the form: (11.1) Problem: How can one transform this relation expressed in the coordinates ,t into a relation expressed in coordinates x,y inclined at an angle of q with the ,t coordinates (see Figure 11.1).3 First recall the following: 1 See Section 3.2. 2 The orthotropic axes 1,2,3 in Equation 9.3 are now called l,t,z, respectively. 3 What follows is treated more globally and completely in Section 13.2.2. e et Óg t˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ l E ----- nt Et –------ 0 nt E ------ l Et – ---- 0 0 0 l Gt ------- Ó ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ s st Ót t˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = TX846_Frame_C11 Page 223 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC
Figure 11.1 Orthotropic Axes and Arbitrary Direction in the Plane of a Ply Recall 1:The stress acting on a surface with a normal vector n is given by {o}=[ol{n} (11.2) column matrix of stress column matrix of stress components matrix direction cosines Recall 2:The coordinates of the same vector in axesy as well as e,t,such that (x.)=cose,are =v,t+vi=c+V动 with the relation: 份-{5 (11.3) In axes t the stress acting on the surface with a normal can be expressed as follows,using Equation 11.2 above: toae-io,tou-iol母 where fo is the stress vector and [ol is the stress matrix and in axes x,y, following Equation 11.3: aw周 2003 by CRC Press LLC
Recall 1: The stress acting on a surface with a normal vector is given by (11.2) Recall 2: The coordinates of the same vector in axes x,y as well as ,t, such that = cosq, are with the relation: (11.3) In axes ,t the stress acting on the surface with a normal can be expressed as follows, using Equation 11.2 above: where {s/x} is the stress vector and [sij] is the stress matrix and in axes x,y, following Equation 11.3: Figure 11.1 Orthotropic Axes and Arbitrary Direction in the Plane of a Ply s n { } σ σij = [ ]{ } n column matrix of stress components σ stress matrix column matrix of direction cosines n V ( ) x . V V Vt = = + t Vx x + Vyy Vx ÓVy ˛ Ì ˝ Ï ¸ c s –s c Ó ˛ Ì ˝ Ï ¸ V ÓVt ˛ Ì ˝ Ï ¸ c = cosq Ë ¯ c = sinq Ê ˆ = x { } s/x ,t sij [ ],t { }x ,t sij [ ],t c s Ó ˛ Ì ˝ Ï ¸ = = { } s/x x,y c s –s c sij [ ],t c s Ó ˛ Ì ˝ Ï ¸ = TX846_Frame_C11 Page 224 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC
In a similar manner,the stresses acting on the surface with the normal y are written in the x,y axes as: aiw- Therefore,the matrix of stresses in the x,y axes is: in setting: [P= c s -s c and observing that the matrix [P]is orthogonal ([p]=[P),one hast [oule.:[P][Guls>[P] where [P]is the transpose of matrix [P]. This expression can be developed to become 北 One can also rearrange the equation to be -2cs Ox 2cs 6y (11.4) Tet SC -SC (c2-s3J0 xy) Then: [o]=[T][o]y One has:Gsy=Pau 'P:Gn'P='Po Pou=dxP;au=PosP. 2003 by CRC Press LLC
In a similar manner, the stresses acting on the surface with the normal are written in the x,y axes as: Therefore, the matrix of stresses in the x,y axes is: in setting: and observing that the matrix [P] is orthogonal (t [P] = [P] –1), one has4 where t [P] is the transpose of matrix [P]. This expression can be developed to become One can also rearrange the equation to be (11.4) Then: 4 One has: sx,y = Ps,t t P; st t P = t Psxy; Ps,t = sx,yP; s,t = t Psx,yP. y { } s/y x,y c s –s c sij [ ],t –s c Ó ˛ Ì ˝ Ï ¸ = { } sij x,y s/x,s/y [ ] c s –s c sij [ ],t c s – s c = = [ ] P c s –s c = sij [ ], t [ ] Pt sij [ ] = x y, [ ] P s t t t t st c s – s c sx txy txy sy c s –s c = s st Ót t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ c 2 s 2 2– cs s 2 c 2 2cs sc sc c2 s 2 – ( ) – sx sy Ótxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = [ ] s ,t = [ ] T [ ] s x,y TX846_Frame_C11 Page 225 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC
withs 2 -2sc [T]= 2sc SC -SC (c2-s3 In a similar manner,the strain components can be transformed as: 2 s2 2cs s2 -2cs Exy -CS CS (c2-s3) or: s2 CS -CS E -2cs 2cs (c2-523) then: with: [T']= -CS -2cs 2cs (c2-s3 The stress-strain Equation 11.1 can then be expressed in the axes x,y since we have written: -Vie 0 月，：月 0()e ()t [TH(Gbs 0 0 5This matrix is readily established if one knows the relation that allows one to express the components of a tensor in one system in terms of the components of the same tensor in another system.Here this relation is:oy=cos"cos"with cos"=cos(m,1);see Section 13.1. 2003 by CRC Press LLC
with5 In a similar manner, the strain components can be transformed as: or: The stress–strain Equation 11.1 can then be expressed in the axes x,y since we have written: 5 This [T ] matrix is readily established if one knows the relation that allows one to express the components of a tensor in one system in terms of the components of the same tensor in another system. Here this relation is: sIJ = cos smn with = cos( ); see Section 13.1. I m cosJ m cosI m m,I [ ] T c 2 s 2 2– sc s 2 c 2 2sc sc sc c2 s 2 – ( ) – = ex e y Óexy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ c 2 s 2 2cs s 2 c 2 –2cs –cs cs c 2 s 2 ( ) – e et Óe t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = then: with: ex e y Óg xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ c 2 s 2 cs s 2 c 2 –cs –2cs 2cs c 2 s 2 ( ) – e et Óg t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = e gÓ ˛ Ì ˝ Ï ¸ x,y [ ] T ¢ e gÓ ˛ Ì ˝ Ï ¸ ,t = [ ] T ¢ c 2 s 2 cs s 2 c 2 –cs –2cs 2cs c2 s 2 ( ) – = e gÓ ˛ Ì ˝ Ï ¸ x,y [ ] T ¢ e gÓ ˛ Ì ˝ Ï ¸ ,t ; e gÓ ˛ Ì ˝ Ï ¸ ,t 1 E ----- nt – Et --------- 0 nt – E --------- 1 Et ---- 0 0 0 1 Gt ------- { } s ,t == = ; { } s ,t [ ] T { } s x,y TX846_Frame_C11 Page 226 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC
where after substitution: Ex 笑 07 Ox Ey [T] E 1E 0[T] Yxy. 0 0 Gu- Txy new matrix of elastic coefficients in x,y axes When all calculations are performed,one obtains the following constitutive relation, written in the coordinates x,y that make an angle 0 with the axes ,t.The elastic moduli and Poisson coefficients appear in these relations.One can also see the existence of the coupling coefficients n and u,which demonstrates that a normal stress can produce a distortion. Ex a Ox Ey Cy 是 莞 with: 1 Ex(θ)= 1 E,(0)= ++-增 (11.5) G(0)= 1 4(++2) Gu 0=+s-2+总 器=-2a怎-+-管动 急(0=-2c管--e-s管) Recall that the matrix of elastic coefficients is symmetric,meaning in particular: and Ho/Gsy=My/Ey 7 See example described in Section 3.1. 2003 by CRC Press LLC
where after substitution: When all calculations are performed, one obtains the following constitutive relation, written in the coordinates x,y that make an angle q with the axes ,t. The elastic moduli and Poisson coefficients appear in these relations. One can also see the existence of the coupling coefficients h and m, 6 which demonstrates that a normal stress can produce a distortion.7 (11.5) 6 Recall that the matrix of elastic coefficients is symmetric, meaning in particular: hxy /Gxy = hx/Ex and mxy /Gxy = my /Ey. 7 See example described in Section 3.1. ex e y Óg xy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ [ ] T ¢ 1 E ----- nt -Et -------- 0 nt E –------ 1 Et ---- 0 0 0 1 Gt ------- [ ] T sx sy Ótxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = new matrix of elastic coefficients in x,y axes Ï Ô Ô Ô Ô Ì Ô Ô Ô Ô Ó ex e y Óg xy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ 1 E -- x nyx Ey –------- hxy Gxy -------- n xy Ex –------- 1 E -- y mxy Gxy -------- hx Ex ----- my Ey ----- 1 Gxy -------- sx sy Ótxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = with: Ex( ) q 1 c E -----4 s 4 Et ---- c 2 s 2 1 Gt ------- 2 nt Et – ------ Ë ¯ Ê ˆ + + = ---------------------------------------------------------- Ey( ) q 1 s E -----4 c 4 Et ---- c 2 s 2 1 Gt ------- 2 nt Et – ------ Ë ¯ Ê ˆ + + = ---------------------------------------------------------- Gxy( ) q 1 4c 2 s 2 1 E ----- 1 Et ---- 2 nt Et + + ------ Ë ¯ Ê ˆ c 2 s 2 ( ) – 2 Gt + ---------------------- = ----------------------------------------------------------------------- nyx Ey -------( ) q nt Et ------ c 4 s 4 ( ) + c 2 s 2 1 E ----- 1 Et ---- 1 Gt + – ----- Ë ¯ Ê ˆ = – hxy Gxy --------( ) q 2cs c 2 E ----- s 2 Et – ---- c 2 s 2 + ( ) – nt Et ------ 1 2Gt – ----------- Ë ¯ Ê ˆ Ó ˛ Ì ˝ Ï ¸ = – mxy Gxy --------( ) q 2cs s 2 E ----- c 2 Et ---- c 2 s 2 ( ) – nt Et ------ 1 2Gt – ----------- Ë ¯ Ê ˆ – – Ó ˛ Ì ˝ Ï ¸ = – TX846_Frame_C11 Page 227 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC
11.2 STIFFNESS COEFFICIENTS When one inverts Equation 11.1 written in the coordinate axesl,t of a ply,one obtains VuEt (1-vevu) (1-vuvn) 0 vuE E o (1-vuvie) (1-vuvn) 0 0 0 GelYu where the "stiffness"coefficients appear,as opposed to the Equation 11.1 where the "compliance"coefficients appear.To simplify the writing,one can denote Ee VuEe 0 eEr E 0 (11.6) 0 0 An identical procedure can be followed to arrive at the stress-strain relation: Ox 2cs Oy s -2cs -CS (c2-s) [T] (11.7) -CS 2 Yu 2cs -2cs (c2-s) Yxy [T'] Recall that axes x,y are derived from the axes t,t by a rotation 0 about the third axis z.Substituting Equations 11.7 into 11.6,one obtains ViEe 0 = [T] E 0 Ey Tx 0 0 G Yxy which can be rewritten as: 6￥ E E12 E 6 En to) E 制 Yxy 2003 by CRC Press LLC
11.2 STIFFNESS COEFFICIENTS When one inverts Equation 11.1 written in the coordinate axes l,t of a ply, one obtains where the “stiffness” coefficients appear, as opposed to the Equation 11.1 where the “compliance” coefficients appear. To simplify the writing, one can denote (11.6) An identical procedure can be followed to arrive at the stress–strain relation: (11.7) Recall that axes x,y are derived from the axes ,t by a rotation q about the third axis z. Substituting Equations 11.7 into 11.6, one obtains which can be rewritten as: s st Ót t ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ E 1 ntntl ( ) – -------------------- ntE 1 ntnt ( ) – --------------------- 0 ntEt 1 ntnt ( ) – --------------------- Et 1 ntnt ( ) – --------------------- 0 0 0 Gt e et Óg t˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = sl st Ót t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ E ntE 0 ntEt Et 0 0 0 Gt e et Óg t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = sx sy Ótxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ c 2 s 2 2cs s 2 c 2 –2cs –cs cs c2 s 2 ( ) – s st Ót t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = : T1 [ ] Ï Ô Ô Ô Ì Ô Ô Ô Ó e et Óg t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ c 2 s 2 –cs s 2 c 2 cs 2cs –2cs c2 s 2 ( ) – ex e y Óg xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = T1 [ ]¢ Ï Ô Ô Ô Ô Ì Ô Ô Ô Ô Ó sx sy Ótxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ T1 [ ] E ntE 0 ntEt Et 0 0 0 Gt T1 [ ]¢ ex e y Óg xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = sx sy Ótxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ E11 E12 E13 E21 E22 E23 E31 E32 E33 ex e y Óg xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = TX846_Frame_C11 Page 228 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC
Once the calculations are performed,one obtains the following expressions for the stiffness coefficients Ey,where c=cose and s=sine. E12 E Exa E 正 with: E1(0)=cE+s4E,+2c22(yE+2G) E2(0)s'E+cEr+2c's(vuEc+2G) Es(0)=c2s(Ee+E-2vnE)+(c2-s)Gur (11.8) E2(0)=c2s2(Ee+B-4G)+(c*+s)y,E E3(0)=-cs{c2E-s2E-(c2-s)(yeE+2G)》 E23(0)=-cs{s2Ee-c2E,+(c2-s3)(eE+2G)} expressions in which: Et=E/(1-vervn):EE/(1-vervne) The rate of variation of stiffness coefficients Ey as functions of the angle 0 is repre sented in Figure 11.2 for a ply characterized by moduli E and E,with very different values,for example the case of unidirectional layers of fiber/resin. 11.3 CASE OF THERMOMECHANICAL LOADING 11.3.1 Compliance Coefficients When considering the temperature variations,one must substitute the stress-strain Equation 11.1 with Equation 10.9: 言 01oe de E 0 ,}+△Ta, Ye) 0 0 1 0 See characteristics of the fiber/resin unidirectionals in Paragraph 3.3.3. See Section 10.5. 2003 by CRC Press LLC
Once the calculations are performed, one obtains the following expressions for the stiffness coefficients , where c = cosq and s = sinq. (11.8) The rate of variation of stiffness coefficients as functions of the angle q is repre sented in Figure 11.2 for a ply characterized by moduli E and Et with very different values, for example the case of unidirectional layers of fiber/resin.8 11.3 CASE OF THERMOMECHANICAL LOADING 11.3.1 Compliance Coefficients When considering the temperature variations,9 one must substitute the stress–strain Equation 11.1 with Equation 10.9: 8 See characteristics of the fiber/resin unidirectionals in Paragraph 3.3.3. 9 See Section 10.5. Eij sx sy Ótxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ E11 E12 E13 E21 E22 E23 E31 E32 E33 ex e y Óg xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = with: E11( ) q c 4 E s 4 Et 2c 2 s 2 ntE + 2Gt = + + ( ) E22( ) q s 4 E c 4 Et 2c 2 s 2 ntE + 2Gt = + + ( ) E33( ) q c 2 s 2 ( ) E + Et – 2ntE c 2 s 2 ( ) – 2 = + Gt E12( ) q c 2 s 2 E + Et – 4Gt ( ) c 4 s 4 = + ( ) + ntE E13( ) q –cs c 2 E s 2 – Et c 2 s 2 ( ) – ntE + 2Gt = { } – ( ) E23( ) q –cs s 2 E c 2 – Et c 2 s 2 ( ) – ntE + 2Gt = { } + ( ) expressions in which: E E/ 1 – ntnt = ( ) : Et Et/ 1 – ntnt = ( ) Eij e et Óg t˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ 1 E ----- nt Et –------ 0 nt E –------ 1 Et ---- 0 0 0 1 Gt ------- s st Ót t˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = DT a at Ó 0 ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ + TX846_Frame_C11 Page 229 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC
En Er 45 90 459 90 E12 90 459 90 E23 Figure 11.2 Variation of Stiffness Coefficients for a Misaligned Ply in which ac and a,are the coefficients of thermal expansion of the unidirectional layer along the longitudinal direction e and transverse direction t,respectively. Following the same procedure as in Section 11.1 with the same notations,one can write 月=m月oia=moiw 2003 by CRC Press LLC
in which a and at are the coefficients of thermal expansion of the unidirectional layer along the longitudinal direction and transverse direction t, respectively. Following the same procedure as in Section 11.1 with the same notations, one can write Figure 11.2 Variation of Stiffness Coefficients for a Misaligned Ply e gÓ ˛ Ì ˝ Ï ¸ x,y [ ] T ¢ e gÓ ˛ Ì ˝ Ï ¸ ,t = = ;{ } s ,t [ ] T { } s x,y TX846_Frame_C11 Page 230 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC
Then,upon substituting, Ex 0 Ey =[T'] 0 (TK 〉+AT[T']H 0 0 One finds again in the first part of the second term a matrix of compliance coefficients,the terms of which are described in details in Equation 11.5.The second part of the second term is written as: ae ca+sar △T -CS △T sa+car -2cs 2cs (c2-s30 2cs(a;-ae) Therefore,the thermomechanical relation for a unidirectional layer written in the axes x.y,different from the e,t coordinates,can be summarized as follows: 「层 詈 cx】 (11.9) Ey 会 G o〉+AT (Yo) 是 气 1 x EEGare given by the relations [11.5] as=ca+s'ar ay sai+ca axy 2cs(a,-a) c=cos;s sin0 2003 by CRC Press LLC
Then, upon substituting, One finds again in the first part of the second term a matrix of compliance coefficients, the terms of which are described in details in Equation 11.5. The second part of the second term is written as: Therefore, the thermomechanical relation for a unidirectional layer written in the axes x.y, different from the ,t coordinates, can be summarized as follows: (11.9) ex e y Óg xy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ [ ] T ¢ 1 E ----- nt Et –------ 0 nt E –------ 1 Et ---- 0 0 0 1 Gt ------- [ ] T sx sy Ótxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = DT T[ ]¢ a at Ó 0 ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ + DT c 2 s 2 cs s 2 c 2 –cs –2cs 2cs c2 s 2 ( ) – a at 0 Ó ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ DT c 2 a s 2 + at s 2 a c 2 + at 2cs at – a Ó ( )˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = ex e y Óg xy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ 1 Ex ----- nyx Ey –------- hxy Gxy -------- n xy Ex –------- 1 Ey ----- mxy Gxy -------- hx Ex ----- my Ey ----- 1 Gxy -------- sx sy Ótxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ DT ax ay Óaxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = + Ex, Ey,Gxy,nxy,nyx,hxy,mxy are given by the relations [11.5] ax c 2 a s 2 = + at ay s 2 a c 2 = + at axy 2cs at – a = ( ) c = = cosq ; s sinq TX846_Frame_C11 Page 231 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC
11.3.2 Stiffness Coefficients Inverting Equation 10.9 gives Er VuEe (1-vevi) (1-vuvn) 0 VuE E (1-vevie) (1-vevn) 0 tu) 0 0 Ge Yer] VuEt (1-vavn)ae+(1-vevn)ar -△T VeE -7% (1-Vuv 0 Following the procedure of Section 11.2,with the same notations,one can write: [Til(G)t where after substitution: Ox Ee VuEe 0 Ex Eae+vuEa Oy [Ti]veE, E 0 △T[T]水 vuE a+Era Txy 0 0 Yx 0 One finds again,in the first part of the second term,the matrix detailed in Equation 11.8.The second part of the second term can be developed as follows: c2 s2 2csEeae+vuEea -△7 C2 -2cs vuEac+Era -CS CS (c2-s) 0 c2Ee(ac+vra)+s Ei(verae+a) …-△T{s2i(a+yea)+c2Ei(va+a) cs[E(verae+a)-Ee(ae+vica ) 2003 by CRC Press LLC
11.3.2 Stiffness Coefficients Inverting Equation 10.9 gives Following the procedure of Section 11.2, with the same notations, one can write: where after substitution: One finds again, in the first part of the second term, the matrix detailed in Equation 11.8. The second part of the second term can be developed as follows: s st Ót t˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ E 1 ntnt ( ) – --------------------- ntE 1 ntnt ( ) – --------------------- 0 ntEt 1 ntnt ( ) – --------------------- Et 1 ntnt ( ) – --------------------- 0 0 0 Gt e et Óg t˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = º º DT E 1 ntnt ( ) – ---------------------a ntE 1 ntnt ( ) – + ---------------------at ntEt 1 ntnt ( ) – ---------------------al Et 1 ntnt ( ) – + ---------------------at Ó 0 ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ – { } s x,y Tl [ ]{ } s ,t ; e gÓ ˛ Ì ˝ Ï ¸ ,t T1 [ ]¢ e gÓ ˛ Ì ˝ Ï ¸ x,y = = sx sy Ótxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ Tl [ ] E ntE 0 ntEt Et 0 0 0 Gt T1 [ ]¢ ex e y Óg xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ DT Tl [ ] Ea + ntEat ntEta + Etat Ó 0 ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = – DT c 2 s 2 2cs s 2 c 2 –2cs –cs cs c2 s 2 ( ) – Ea + ntEat ntEta + Etat Ó 0 ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ – = º º DT c 2 E a + ntat ( ) s 2 Et nta + at + ( ) s 2 E a + ntat ( ) c 2 Et nta + at + ( ) cs Et nta + at ( ) – E a + ntat Ó [ ] ( ) ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ – TX846_Frame_C11 Page 232 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC