6 Elastic Constants Based on Global Coordinate System 6.1 Basic Equations The engineering properties or elastic constants were introduced in Chap.2 with respect to the lamina 1-2-3 coordinate system.Their evaluation was presented in Chap.3 based also on the 1-2-3 coordinate system.We can also define elastic con- stants with respect to the z-y-z global coordinate system.The elastic constants in the r-y-z coordinate system can be derived directly from their definitions,just as they were derived in Chap.3 for the 1-2-3 coordinate system. The elastic constants based on the r-y-z global coordinate system are given as follows [1]: E Er= (6.1) m+(品-2e）n2m2+n 2(m+m)-（1+悬-品）n2m2 (6.2) m4+(品-2e)n2m2+悬n2 E2 Ey= (6.3) m4+(品-22）n2m2+n4 2(m+m)-（1+景-品)n2m (6.4) m4+(品-22)n2m2+景n2 Gay= G12 (6.5) n4+m4+2(2g(1+212)+2-1)n2m2 B It is useful to define several other material properties for fiber-reinforced com- posite materials that can be used to categorize response [1].These properties have as their basis the fact that an element of fiber-reinforced composite material with its fiber oriented at some arbitrary angle exhibits a shear strain when subjected to a normal stress,and it also exhibits an extensional strain when subjected to a shear stress
6 Elastic Constants Based on Global Coordinate System 6.1 Basic Equations The engineering properties or elastic constants were introduced in Chap. 2 with respect to the lamina 1-2-3 coordinate system. Their evaluation was presented in Chap. 3 based also on the 1-2-3 coordinate system. We can also define elastic constants with respect to the x-y-z global coordinate system. The elastic constants in the x-y-z coordinate system can be derived directly from their definitions, just as they were derived in Chap. 3 for the 1-2-3 coordinate system. The elastic constants based on the x-y-z global coordinate system are given as follows [1]: Ex = E1 m4 + E1 G12 − 2ν12 n2m2 + E1 E2 n4 (6.1) νxy = ν12 n4 + m4 − 1 + E1 E2 − E1 G12 n2m2 m4 + E1 G12 − 2ν12 n2m2 + E1 E2 n2 (6.2) Ey = E2 m4 + E2 G12 − 2ν21 n2m2 + E2 E1 n4 (6.3) νyx = ν21 n4 + m4 − 1 + E2 E1 − E2 G12 n2m2 m4 + E2 G12 − 2ν21 n2m2 + E2 E1 n2 (6.4) Gxy = G12 n4 + m4 + 2 2G12 E1 (1 + 2ν12) + 2G12 E2 − 1 n2m2 (6.5) It is useful to define several other material properties for fiber-reinforced composite materials that can be used to categorize response [1]. These properties have as their basis the fact that an element of fiber-reinforced composite material with its fiber oriented at some arbitrary angle exhibits a shear strain when subjected to a normal stress, and it also exhibits an extensional strain when subjected to a shear stress
80 6 Elastic Constants Based on Global Coordinate System Poisson's ratio is defined as the ratio of extensional strains,given that the el- ement is subjected only to a simple normal stress.By analogy,the coefficient of mutual influence of the second kind is defined as the ratio of a shear strain to an extensional strain,given that the element is subjected to only a single normal stress. The coefficient of mutual influence of the first kind is defined as the ratio of an ex- tensional strain to a shear strain,given that the element is subjected to only a single shear stress (see 1). One coefficient of mutual influence of the second kind is defines as follows: Ty,Tty (6.6) Ex where o0 and all other stresses are zero.Another coefficient of mutual influence of the second kind is defined as follows: ney.v=Yty (6.7) Ey where oy0 and all other stresses are zero.It can be shown that the coefficients of mutual influence of the second kind can be written as follows: 江，x= 516 (6.8) 526 Tlzy.y= 522 (6.9) The coefficients of mutual influence of the first kind are defined as follows: nt,ty Ea (6.10) Yry Tyty Eu (6.11) Y红w whereT0 and all other stresses are zero.It can be shown that the coefficients of mutual influence of the first kind can be written as follows: nr,ty 516 S66 (6.12) 526 Thy.ty= 56 (6.13) 6.2 MATLAB Functions Used The nine MATLAB functions used in this chapter to calculate the constants based on the global coordinate system are Er(E1,E2,NU12,G12,theta)-This function calculates the elastic modulus E along the r-direction in the global coordinate system.Its input consists of five ar- guments representing the four elastic constants E,E2,v12,G12,and the fiber orientation angle 0
80 6 Elastic Constants Based on Global Coordinate System Poisson’s ratio is defined as the ratio of extensional strains, given that the element is subjected only to a simple normal stress. By analogy, the coefficient of mutual influence of the second kind is defined as the ratio of a shear strain to an extensional strain, given that the element is subjected to only a single normal stress. The coefficient of mutual influence of the first kind is defined as the ratio of an extensional strain to a shear strain, given that the element is subjected to only a single shear stress (see [1]). One coefficient of mutual influence of the second kind is defines as follows: ηxy,x = γxy εx (6.6) where σx = 0 and all other stresses are zero. Another coefficient of mutual influence of the second kind is defined as follows: ηxy,y = γxy εy (6.7) where σy = 0 and all other stresses are zero. It can be shown that the coefficients of mutual influence of the second kind can be written as follows: ηxy,x = S¯16 S¯11 (6.8) ηxy,y = S¯26 S¯22 (6.9) The coefficients of mutual influence of the first kind are defined as follows: ηx,xy = εx γxy (6.10) ηy,xy = εy γxy (6.11) where τxy = 0 and all other stresses are zero. It can be shown that the coefficients of mutual influence of the first kind can be written as follows: ηx,xy = S¯16 S¯66 (6.12) ηy,xy = S¯26 S¯66 (6.13) 6.2 MATLAB Functions Used The nine MATLAB functions used in this chapter to calculate the constants based on the global coordinate system are : Ex (E1, E2, NU12, G12, theta) – This function calculates the elastic modulus Ex along the x-direction in the global coordinate system. Its input consists of five arguments representing the four elastic constants E1, E2, ν12, G12, and the fiber orientation angle θ.
6.2 MATLAB Functions Used 8 NUry(E1,E2,NU12,G12,theta)-This function calculates Poisson's ratio vry in the global coordinate system.Its input consists of five arguments representing the four elastic constants E1,E2,v12,G12,and the fiber orientation angle 0. Ey(E1,E2,NU21,G12,theta)-This function calculates the elastic modulus Ey along the y-direction in the global coordinate system.Its input consists of five ar- guments representing the four elastic constants E1,E2,v21,G12,and the fiber orientation angle 0. NUyr(El,E2,NU21,G12,theta)-This function calculates Poisson's ratio vur in the global coordinate system.Its input consists of five arguments representing the four elastic constants E,E2,v21,G12,and the fiber orientation angle 0. Gry(E1,E2,NU12,G12,theta)-This function calculates the shear modulus Gry in the global coordinate system.Its input consists of five arguments representing the four elastic constants E1,E2,v12,G12,and the fiber orientation angle 0. Etazyz(Sbar)-This function calculates the coefficient of mutual influence of the second kind ny.It has one argument-the transformed reduced compliance matrix [. Etaryy(Sbar)-This function calculates the coefficient of mutual influence of the second kind ny..It has one argument-the transformed reduced compliance matrix [. Etarxy(Sbar)-This function calculates the coefficient of mutual influence of the first kind n.y.It has one argument-the transformed reduced compliance matrix [. Etayxy(Sbar)-This function calculates the coefficient of mutual influence of the first kind ny.y.It has one argument-the transformed reduced compliance matrix [ The following is a listing of the MATLAB source code for each function: function y Ex(E1,E2,NU12,G12,theta) %Ex This function returns the elastic modulus % along the x-direction in the global 名 coordinate system.It has five arguments: E1 -longitudinal elastic modulus E2-transverse elastic modulus NU12-Poisson's ratio G12-shear modulus % theta -fiber orientation angle % The angle "theta"must be given in degrees % Ex is returned as a scalar m cos(theta*pi/180); n sin(theta*pi/180); denom=m4+(E1/G12-2*NU12)*n*n*m*m+(E1/E2)*n"4; y =E1/denom;
6.2 MATLAB Functions Used 81 NUxy(E1, E2, NU12, G12, theta) – This function calculates Poisson’s ratio νxy in the global coordinate system. Its input consists of five arguments representing the four elastic constants E1, E2, ν12, G12, and the fiber orientation angle θ. Ey(E1, E2, NU21, G12, theta) – This function calculates the elastic modulus Ey along the y-direction in the global coordinate system. Its input consists of five arguments representing the four elastic constants E1, E2, ν21, G12, and the fiber orientation angle θ. NUyx (E1, E2, NU21, G12, theta) – This function calculates Poisson’s ratio νyx in the global coordinate system. Its input consists of five arguments representing the four elastic constants E1, E2, ν21, G12, and the fiber orientation angle θ. Gxy(E1, E2, NU12, G12, theta) – This function calculates the shear modulus Gxy in the global coordinate system. Its input consists of five arguments representing the four elastic constants E1, E2, ν12, G12, and the fiber orientation angle θ. Etaxyx (Sbar) – This function calculates the coefficient of mutual influence of the second kind ηxy,x. It has one argument – the transformed reduced compliance matrix [S¯]. Etaxyy(Sbar) – This function calculates the coefficient of mutual influence of the second kind ηxy,y. It has one argument – the transformed reduced compliance matrix [S¯]. Etaxxy(Sbar) – This function calculates the coefficient of mutual influence of the first kind ηx,xy. It has one argument – the transformed reduced compliance matrix [S¯]. Etayxy(Sbar) – This function calculates the coefficient of mutual influence of the first kind ηy,xy. It has one argument – the transformed reduced compliance matrix [S¯]. The following is a listing of the MATLAB source code for each function: function y = Ex(E1,E2,NU12,G12,theta) %Ex This function returns the elastic modulus % along the x-direction in the global % coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU12 - Poisson’s ratio % G12 - shear modulus % theta - fiber orientation angle % The angle "theta" must be given in degrees. % Ex is returned as a scalar m = cos(theta*pi/180); n = sin(theta*pi/180); denom = m^4 + (E1/G12 - 2*NU12)*n*n*m*m + (E1/E2)*n^4; y = E1/denom;
82 6 Elastic Constants Based on Global Coordinate System function y =NUxy(E1,E2,NU12,G12,theta) %NUxy This function returns Poisson's ratio % NUxy in the global % coordinate system.It has five arguments: % E1-longitudinal elastic modulus % E2-transverse elastic modulus % NU12 -Poisson's ratio % G12-shear modulus % theta-fiber orientation angle The angle "theta"must be given in degrees. 名 NUxy is returned as a scalar m cos(theta*pi/180); n sin(theta*pi/180); denom=m^4+(E1/G12-2*NU12)*n*n*m*m+(E1/E2)*n*n; numer=N012*(n4+m~4)-(1+E1/E2-E1/G12)*n*n*m*m; y =numer/denom; function y Ey(E1,E2,NU21,G12,theta) %Ey This function returns the elastic modulus % along the y-direction in the global % coordinate system.It has five arguments: % E1 -longitudinal elastic modulus % E2 transverse elastic modulus % NU21 -Poisson's ratio G12 shear modulus 名 theta fiber orientation angle The angle "theta"must be given in degrees 名 Ey is returned as a scalar m cos(theta*pi/180); n =sin(theta*pi/180); denom=m^4+(E2/G12-2*NU21)*n*n*m*m+(E2/E1)*n4; y=E2/denom; function y NUyx(E1,E2,NU21,G12,theta) %NUyx This function returns Poisson's ratio % NUyx in the global % coordinate system.It has five arguments: E1 -longitudinal elastic modulus E2 -transverse elastic modulus % NU21 -Poisson's ratio G12-shear modulus 名 theta-fiber orientation angle % The angle "theta"must be given in degrees. % NUyx is returned as a scalar m cos(theta*pi/180); n=sin(theta*pi/180); denom=m4+(E2/G12-2*NU21)*n*n*m*m+(E2/E1)*n*n; numer=NU21*(n4+m4)-(1+E2/E1-E2/G12)*n*n*m*m; y =numer/denom;
82 6 Elastic Constants Based on Global Coordinate System function y = NUxy(E1,E2,NU12,G12,theta) %NUxy This function returns Poisson’s ratio % NUxy in the global % coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU12 - Poisson’s ratio % G12 - shear modulus % theta - fiber orientation angle % The angle "theta" must be given in degrees. % NUxy is returned as a scalar m = cos(theta*pi/180); n = sin(theta*pi/180); denom = m^4 + (E1/G12 - 2*NU12)*n*n*m*m + (E1/E2)*n*n; numer = NU12*(n^4 + m^4) - (1 + E1/E2 - E1/G12)*n*n*m*m; y = numer/denom; function y = Ey(E1,E2,NU21,G12,theta) %Ey This function returns the elastic modulus % along the y-direction in the global % coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU21 - Poisson’s ratio % G12 - shear modulus % theta - fiber orientation angle % The angle "theta" must be given in degrees. % Ey is returned as a scalar m = cos(theta*pi/180); n = sin(theta*pi/180); denom = m^4 + (E2/G12 - 2*NU21)*n*n*m*m + (E2/E1)*n^4; y = E2/denom; function y = NUyx(E1,E2,NU21,G12,theta) %NUyx This function returns Poisson’s ratio % NUyx in the global % coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU21 - Poisson’s ratio % G12 - shear modulus % theta - fiber orientation angle % The angle "theta" must be given in degrees. % NUyx is returned as a scalar m = cos(theta*pi/180); n = sin(theta*pi/180); denom = m^4 + (E2/G12 - 2*NU21)*n*n*m*m + (E2/E1)*n*n; numer = NU21*(n^4 + m^4) - (1 + E2/E1 - E2/G12)*n*n*m*m; y = numer/denom;
6.2 MATLAB Functions Used 83 function y Gxy(E1,E2,NU12,G12,theta) %Gxy This function returns the shear modulus Gxy in the global % coordinate system.It has five arguments: % E1 -longitudinal elastic modulus % E2-transverse elastic modulus % NU12-Poisson's ratio % G12 -shear modulus % theta -fiber orientation angle 4 The angle "theta"must be given in degrees. % Gxy is returned as a scalar m cos(theta*pi/180); n sin(theta*pi/180); denom=n4+m^4+2*(2*G12*(1+2*NU12)/E1+2*G12/E2-1) *n*n*m*m; y =G12/denom; function y =Etaxyx(Sbar) %Etaxyx This function returns the coefficient of % mutual influence of the second kind % ETAxy,x in the global coordinate system. % It has one argument -the reduced % transformed compliance matrix Sbar. Etaxyx is returned as a scalar y=Sbar(1,3)/Sbar(1,1); function y Etaxyy(Sbar) %Etaxyy This function returns the coefficient of 名 mutual influence of the second kind % ETAxy,y in the global coordinate system. % It has one argument -the reduced % transformed compliance matrix Sbar. Etaxyy is returned as a scalar y =Sbar(2,3)/Sbar(2,2); function y =Etaxxy(Sbar) ZEtaxxy This function returns the coefficient of % mutual influence of the first kind % ETAx,xy in the global coordinate system. % It has one argument-the reduced % transformed compliance matrix Sbar. % Etaxxy is returned as a scalar y =Sbar(1,3)/Sbar(3,3);
6.2 MATLAB Functions Used 83 function y = Gxy(E1,E2,NU12,G12,theta) %Gxy This function returns the shear modulus % Gxy in the global % coordinate system. It has five arguments: % E1 - longitudinal elastic modulus % E2 - transverse elastic modulus % NU12 - Poisson’s ratio % G12 - shear modulus % theta - fiber orientation angle % The angle "theta" must be given in degrees. % Gxy is returned as a scalar m = cos(theta*pi/180); n = sin(theta*pi/180); denom = n^4 + m^4 + 2*(2*G12*(1 + 2*NU12)/E1 + 2*G12/E2 - 1) *n*n*m*m; y = G12/denom; function y = Etaxyx(Sbar) %Etaxyx This function returns the coefficient of % mutual influence of the second kind % ETAxy,x in the global coordinate system. % It has one argument - the reduced % transformed compliance matrix Sbar. % Etaxyx is returned as a scalar y = Sbar(1,3)/Sbar(1,1); function y = Etaxyy(Sbar) %Etaxyy This function returns the coefficient of % mutual influence of the second kind % ETAxy,y in the global coordinate system. % It has one argument - the reduced % transformed compliance matrix Sbar. % Etaxyy is returned as a scalar y = Sbar(2,3)/Sbar(2,2); function y = Etaxxy(Sbar) %Etaxxy This function returns the coefficient of % mutual influence of the first kind % ETAx,xy in the global coordinate system. % It has one argument - the reduced % transformed compliance matrix Sbar. % Etaxxy is returned as a scalar y = Sbar(1,3)/Sbar(3,3);