Macromechanical Analysis of a Lamina 81 The strain energy in the body per unit volume,per Equation(2.20),is expressed as (2.30) Substituting Hooke's law,Equation(2.28),in Equation(2.30), w=∑∑cee (2.31) Now,by partial differentiation of Equation(2.31), aW=Cg de dej (2.32) and aW-Ci (2.33) de de Because the differentiation does not necessarily need to be in either order, Cij=Cjr (2.34) Equation(2.34)can also be proved by realizing that aW 6:202 Thus,only 21 independent elastic constants are in the general stiffness matrix [C]of Equation(2.25).This also implies that only 21 independent constants are in the general compliance matrix [S]of Equation(2.26). 2.3.1 Anisotropic Material The material that has 21 independent elastic constants at a point is called an anisotropic material.Once these constants are found for a particular point, the stress and strain relationship can be developed at that point.Note that 2006 by Taylor Francis Group,LLC
Macromechanical Analysis of a Lamina 81 The strain energy in the body per unit volume, per Equation (2.20), is expressed as (2.30) Substituting Hooke’s law, Equation (2.28), in Equation (2.30), (2.31) Now, by partial differentiation of Equation (2.31), (2.32) and (2.33) Because the differentiation does not necessarily need to be in either order, (2.34) Equation (2.34) can also be proved by realizing that Thus, only 21 independent elastic constants are in the general stiffness matrix [C] of Equation (2.25). This also implies that only 21 independent constants are in the general compliance matrix [S] of Equation (2.26). 2.3.1 Anisotropic Material The material that has 21 independent elastic constants at a point is called an anisotropic material. Once these constants are found for a particular point, the stress and strain relationship can be developed at that point. Note that W i i i = = ∑ 1 2 1 6 σ ε . W Cij j i i j = = = ∑∑ 1 2 1 6 1 6 ε ε . ∂ ∂ ∂ = W C i j ij ε ε , ∂ ∂ ∂ = W C j i ji ε ε . Cij ji = C . σ ε i i W = ∂ ∂ . 1343_book.fm Page 81 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
82 Mechanics of Composite Materials,Second Edition ，22 1,1 3 FIGURE 2.11 Transformation of coordinate axes for 1-2 plane of symmetry for a monoclinic material. these constants can vary from point to point if the material is nonhomoge- neous.Even if the material is homogeneous (or assumed to be),one needs to find these 21 elastic constants analytically or experimentally.However, many natural and synthetic materials do possess material symmetry-that is,elastic properties are identical in directions of symmetry because symme- try is present in the internal structure.Fortunately,this symmetry reduces the number of the independent elastic constants by zeroing out or relating some of the constants within the 6 x 6 stiffness [C]and 6 x 6 compliance [S] matrices.This simplifies the Hooke's law relationships for various types of elastic symmetry. 2.3.2 Monoclinic Material If,in one plane of material symmetry*(Figure 2.11),for example,direction 3 is normal to the plane of material symmetry,then the stiffness matrix reduces to C11 C12 C13 0 0 C16 C12 C22 C23 0 0 C26 0 0 [C]= C13 C23 C33 C36 0 0 (2.35) 0 C44 C45 0 0 0 0 C45 C5s 0 1C16 C26 C36 0 C66 as Material symmetry implies that the material and its mirror image about the plane of symmetry are identical. 2006 by Taylor Francis Group,LLC
82 Mechanics of Composite Materials, Second Edition these constants can vary from point to point if the material is nonhomogeneous. Even if the material is homogeneous (or assumed to be), one needs to find these 21 elastic constants analytically or experimentally. However, many natural and synthetic materials do possess material symmetry — that is, elastic properties are identical in directions of symmetry because symmetry is present in the internal structure. Fortunately, this symmetry reduces the number of the independent elastic constants by zeroing out or relating some of the constants within the 6 × 6 stiffness [C] and 6 × 6 compliance [S] matrices. This simplifies the Hooke’s law relationships for various types of elastic symmetry. 2.3.2 Monoclinic Material If, in one plane of material symmetry* (Figure 2.11), for example, direction 3 is normal to the plane of material symmetry, then the stiffness matrix reduces to (2.35) as FIGURE 2.11 Transformation of coordinate axes for 1–2 plane of symmetry for a monoclinic material. * Material symmetry implies that the material and its mirror image about the plane of symmetry are identical. 3 1, 1′ 2, 2′ 3′ [ ] C CCC C CCC C CCC = 11 12 13 16 12 22 23 26 13 23 33 0 0 0 0 0 0 000 0 000 0 0 0 36 44 45 45 55 16 26 36 66 C C C C C CCC C ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . 1343_book.fm Page 82 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
Macromechanical Analysis of a Lamina 83 C14=0,C15=0,C24=0,C25=0,C34=0,C35=0,C46=0,C56=0. The direction perpendicular to the plane of symmetry is called the principal direction.Note that there are 13 independent elastic constants.Feldspar is an example of a monoclinic material. The compliance matrix correspondingly reduces to S11 S12 S13 0 0 67 S12 S22 S23 0 0 526 [S= S13 S23 S33 0 0 536 (2.36) 0 0 0 S44 S45 0 0 0 0 S45 55 0 S16 S26 S36 0 0 S66] Modifying an excellent example2 of demonstrating the meaning of elastic symmetry for a monoclinic material given,consider a cubic element of Figure 2.12 taken out of a monoclinic material,in which 3 is the direction perpen- dicular to the 1-2 plane of symmetry.Apply a normal stress,o3,to the element.Then using the Hooke's law Equation(2.26)and the compliance matrix(Equation 2.36)for the monoclinic material,one gets e1=S1303 e2=S2303 E3=S3303 Y23=0 Y3别=0 Y12=S3603· (2.37a-f) The cube will deform in all directions as determined by the normal strain equations.The shear strains in the 2-3 and 3-1 plane are zero,showing that the element will not change shape in those planes.However,it will change 2006 by Taylor Francis Group,LLC
Macromechanical Analysis of a Lamina 83 The direction perpendicular to the plane of symmetry is called the principal direction. Note that there are 13 independent elastic constants. Feldspar is an example of a monoclinic material. The compliance matrix correspondingly reduces to . (2.36) Modifying an excellent example2 of demonstrating the meaning of elastic symmetry for a monoclinic material given, consider a cubic element of Figure 2.12 taken out of a monoclinic material, in which 3 is the direction perpendicular to the 1–2 plane of symmetry. Apply a normal stress, σ3, to the element. Then using the Hooke’s law Equation (2.26) and the compliance matrix (Equation 2.36) for the monoclinic material, one gets . (2.37a–f) The cube will deform in all directions as determined by the normal strain equations. The shear strains in the 2–3 and 3–1 plane are zero, showing that the element will not change shape in those planes. However, it will change CCCCCCCC 14 15 24 25 34 35 46 ======= 0000000 ,,,,,,, 56 = 0. [ ] S SSS S SSS S SSS = 11 12 13 16 12 22 23 26 13 23 33 0 0 0 0 0 0 000 0 000 0 0 0 36 44 45 45 55 16 26 36 66 S S S S S SSS S ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ε σ 1 13 3 = S ε σ 2 23 3 = S ε σ 3 33 3 = S 23 γ = 0 31 γ = 0 γ σ 12 36 3 = S 1343_book.fm Page 83 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
84 Mechanics of Composite Materials,Second Edition 63 D H D E H & FIGURE 2.12 Deformation of a cubic element made of monoclinic material. shape in the 1-2 plane.Thus,the faces ABEH and CDFG perpendicular to the 3 direction will change from rectangles to parallelograms,while the other four faces ABCD,BEFC,GFEH,and AHGD will stay as rectangles.This is unlike anisotropic behavior,in which all faces will be deformed in shape, and also unlike isotropic behavior,in which all faces will remain undeformed in shape. 2.3.3 Orthotropic Material (Orthogonally Anisotropic)/Specially Orthotropic If a material has three mutually perpendicular planes of material symmetry, then the stiffness matrix is given by 2006 by Taylor Francis Group,LLC
84 Mechanics of Composite Materials, Second Edition shape in the 1–2 plane. Thus, the faces ABEH and CDFG perpendicular to the 3 direction will change from rectangles to parallelograms, while the other four faces ABCD, BEFC, GFEH, and AHGD will stay as rectangles. This is unlike anisotropic behavior, in which all faces will be deformed in shape, and also unlike isotropic behavior, in which all faces will remain undeformed in shape. 2.3.3 Orthotropic Material (Orthogonally Anisotropic)/Specially Orthotropic If a material has three mutually perpendicular planes of material symmetry, then the stiffness matrix is given by FIGURE 2.12 Deformation of a cubic element made of monoclinic material. G σ3 σ3 D H A B G′ F′ C′ B′ A′ H′ D′ E′ E F 1 2 3 1343_book.fm Page 84 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
Macromechanical Analysis of a Lamina 85 3 0 0 0 0 0 0 0 0 FIGURE 2.13 A unidirectional lamina as a monoclinic material with fibers,arranged in a rectangular array. Cu C12 C13 0 0 01 C12 C22 C23 0 0 0 0 0 0 [C]= C23 C33 0 (2.38) 0 0 C44 0 0 0 0 0 0 C55 0 0 0 0 0 0 C66】 The preceding stiffness matrix can be derived by starting from the stiffness matrix [C]for the monoclinic material(Equation 2.35).With two more planes of symmetry,it gives C16=0,C26=0,C36=0,C45=0. Three mutually perpendicular planes of material symmetry also imply three mutually perpendicular planes of elastic symmetry.Note that nine independent elastic constants are present.This is a commonly found material symmetry unlike anisotropic and monoclinic materials.Examples of an orthotropic material include a single lamina of continuous fiber composite, arranged in a rectangular array(Figure 2.13),a wooden bar,and rolled steel. The compliance matrix reduces to S11 S12 S13 0 0 0 S12 Sn S23 0 0 0 0 0 0 [S]= 513 S23 (2.39) 0 0 0 Su 0 0 0 0 0 0 S 0 0 0 0 0 0 Sos 2006 by Taylor Francis Group,LLC
Macromechanical Analysis of a Lamina 85 . (2.38) The preceding stiffness matrix can be derived by starting from the stiffness matrix [C] for the monoclinic material (Equation 2.35). With two more planes of symmetry, it gives . Three mutually perpendicular planes of material symmetry also imply three mutually perpendicular planes of elastic symmetry. Note that nine independent elastic constants are present. This is a commonly found material symmetry unlike anisotropic and monoclinic materials. Examples of an orthotropic material include a single lamina of continuous fiber composite, arranged in a rectangular array (Figure 2.13), a wooden bar, and rolled steel. The compliance matrix reduces to . (2.39) FIGURE 2.13 A unidirectional lamina as a monoclinic material with fibers, arranged in a rectangular array. 3 2 1 [ ] C CCC CCC C C C = 11 12 13 12 22 23 13 23 33 000 000 000 0 00 00 0000 0 00000 44 55 66 C C C ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ CCCC 16 26 36 45 ==== 0000 ,,, [ ] S SSS SSS S S S = 11 12 13 12 22 23 13 23 33 000 000 000 0 00 00 0000 0 00000 44 55 66 S S S ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 1343_book.fm Page 85 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC