Copyrighted Materials i0pUyPress rm CHAPTER TWO Displacements,Strains, and Stresses We consider composite materials consisting of continuous or discontinuous fibers embedded in a matrix.Such a composite is heterogeneous,and the properties vary from point to point.On a scale that is large with respect to the fiber diam- eter,the fiber and matrix properties may be averaged,and the material may be treated as homogeneous.This assumption,commonly employed in macromechan- ical analyses of composites,is adopted here.Hence,the material is considered to be quasi-homogeneous,which implies that the properties are taken to be the same at every point.These properties are not the same as the properties of either the fiber or the matrix but are a combination of the properties of the constituents. In this chapter,equations are presented for calculating the displacements, stresses,and strains when the structure undergoes only small deformations and the material behaves in a linearly elastic manner. Continuous fiber-reinforced composite materials(and structures made of such materials)often have easily identifiable preferred directions associated with fiber orientations or symmetry planes.It is therefore convenient to employ two co- ordinate systems:a local coordinate system aligned,at a point,either with the fibers or with axes of symmetry,and a global coordinate system attached to a fixed reference point(Fig.2.1).In this book the local and global Cartesian coordinate systems are designated respectively by xi,x2,x3 and the x,y,z axes.In the x,y,z directions the displacements at a point Aare denoted by u,v,w,and in the x1,x2, x3 directions by u1,u2,u3 (Fig.2.2). In the x,y,z coordinate system the normal stresses are denoted by ox,oy,and o:and the shear stresses by tyz,txz,and try(Fig.2.3).The corresponding normal and shear strains are ,y and yy,respectively. In the xi,x2,x3 coordinate system the normal stresses are denoted by a1,o2, and o3 and the shear stresses by t23,t13,and 7i2(Fig.2.3).The corresponding normal and shear strains are e1,e2,e3,and y23,y13,yi2,respectively.The symbol y represents engineeringshear strain that is twice the tensorial shear strain,Yij=2eij (i,j=x,y,zori,j=1,2,3). 3
CHAPTER TWO Displacements, Strains, and Stresses We consider composite materials consisting of continuous or discontinuous fibers embedded in a matrix. Such a composite is heterogeneous, and the properties vary from point to point. On a scale that is large with respect to the fiber diameter, the fiber and matrix properties may be averaged, and the material may be treated as homogeneous. This assumption, commonly employed in macromechanical analyses of composites, is adopted here. Hence, the material is considered to be quasi-homogeneous, which implies that the properties are taken to be the same at every point. These properties are not the same as the properties of either the fiber or the matrix but are a combination of the properties of the constituents. In this chapter, equations are presented for calculating the displacements, stresses, and strains when the structure undergoes only small deformations and the material behaves in a linearly elastic manner. Continuous fiber-reinforced composite materials (and structures made of such materials) often have easily identifiable preferred directions associated with fiber orientations or symmetry planes. It is therefore convenient to employ two coordinate systems: a local coordinate system aligned, at a point, either with the fibers or with axes of symmetry, and a global coordinate system attached to a fixed reference point (Fig. 2.1). In this book the local and global Cartesian coordinate systems are designated respectively by x1, x2, x3 and the x, y, z axes. In the x, y, z directions the displacements at a point Aare denoted by u, v, w, and in the x1, x2, x3 directions by u1, u2, u3 (Fig. 2.2). In the x, y, z coordinate system the normal stresses are denoted by σx, σy, and σz and the shear stresses by τyz, τxz, and τxy (Fig. 2.3). The corresponding normal and shear strains are x, y, z and γyz, γxz, γxy, respectively. In the x1, x2, x3 coordinate system the normal stresses are denoted by σ1, σ2, and σ3 and the shear stresses by τ23, τ13, and τ12 (Fig. 2.3). The corresponding normal and shear strains are 1, 2, 3, and γ23, γ13, γ12, respectively. The symbol γ represents engineering shear strain that is twice the tensorial shear strain, γi j = 2i j (i, j = x, y, z or i, j = 1, 2, 3). 3
4 DISPLACEMENTS,STRAINS,AND STRESSES 2,E3个 Figure 2.1:The global x,y,z and local xi.x2.x3 coordinate systems. A stress is taken to be positive when it acts on a positive face in the positive direction.According to this definition,all the stresses shown in Figure 2.3 are positive. The preceding stress and strain notations,referred to as engineering notations, are used throughout this book.Other notations,most notably tensorial and con- tracted notations,can frequently be found in the literature.The stresses and strains in different notations are summarized in Tables 2.1 and 2.2. 2.1 Strain-Displacement Relations We consider a Ax long segment that undergoes a change in length,the new length being denoted by Ax'.From Figure 2.4 it is seen that u+△x'=△x+ u+ (2.1) where uandux are the displacements of points Aand B.respectively,in the x direction.Accordingly,the normal strain in the x direction is △x'-△xau Ex= (2.2) △x ax Similarly,in the y and z directions the normal strains are av y≠8y (2.3) aw e红202 (2.4) where v and w are the displacements in the y and z directions,respectively. Figure 2.2:The x,y,z and x,x2,x3 coordinate systems and the corresponding displacements
4 DISPLACEMENTS, STRAINS, AND STRESSES x1 x2 z x, 3 y x Figure 2.1: The global x, y, z and local x1, x2, x3 coordinate systems. A stress is taken to be positive when it acts on a positive face in the positive direction. According to this definition, all the stresses shown in Figure 2.3 are positive. The preceding stress and strain notations, referred to as engineering notations, are used throughout this book. Other notations, most notably tensorial and contracted notations, can frequently be found in the literature. The stresses and strains in different notations are summarized in Tables 2.1 and 2.2. 2.1 Strain–Displacement Relations We consider a x long segment that undergoes a change in length, the new length being denoted by x . From Figure 2.4 it is seen that u + x = x + u + ∂u ∂x x , (2.1) where u and u + ∂u ∂x x are the displacements of points Aand B, respectively, in the x direction. Accordingly, the normal strain in the x direction is x = x − x x = ∂u ∂x . (2.2) Similarly, in the y and z directions the normal strains are y = ∂v ∂y (2.3) z = ∂w ∂z , (2.4) where v and w are the displacements in the y and z directions, respectively. A w v u z y x x3 x2 x1 A u2 u1 u3 A' A' Figure 2.2: The x, y, z and x1, x2, x3 coordinate systems and the corresponding displacements.
2.1 STRAIN-DISPLACEMENT RELATIONS 5 0 x3 037 个 2 T2 2 Figure 2.3:The stresses in the global x,y,z and the local x,x2,x3 coordinate systems. For angular(shear)deformation the tensorial shear strain is the average change in the angle between two mutually perpendicular lines(Fig.2.5) +B Exy=- 2 (2.5) For small deformations we have a≈tana= o+器△x)-v-u (2.6) △x ax Similarly B =au/ay,and the xy component of the tensorial shear strain is 1 /au Exy (2.7) In a similar manner we obtain the following expressions for the eyz and ex components of the tensorial shear strains: du d (2.8) Table 2.1.Stress notations Normal stress Shear stress x,y,z coordinate system Tensorial stress Oyy Gy: Oxy Engineering stress Ox Oy 0: Tyz try Contracted notation Gg Os x1,x2,x3 coordinate system Tensorial stress 011022 033 023013012 Engineering stress 01 02 03 T23 T13 T12 Contracted notation 01 02 0304 0506
2.1 STRAIN–DISPLACEMENT RELATIONS 5 x3 x2 x1 σ3 σ2 σ1 τ13 τ12 τ23 τ21 τ32 τ31 z y x σz σy σx τxz τxy τyz τyx τ τzx zy Figure 2.3: The stresses in the global x, y, z and the local x1, x2, x3 coordinate systems. For angular (shear) deformation the tensorial shear strain is the average change in the angle between two mutually perpendicular lines (Fig. 2.5) xy = α + β 2 . (2.5) For small deformations we have α ≈ tan α = v + ∂v ∂x x − v x = ∂v ∂x . (2.6) Similarly β = ∂u/∂y, and the xy component of the tensorial shear strain is xy = 1 2 ∂u ∂y + ∂v ∂x . (2.7) In a similar manner we obtain the following expressions for the yz and xz components of the tensorial shear strains: yz = 1 2 ∂v ∂z + ∂w ∂y xz = 1 2 ∂u ∂z + ∂w ∂x . (2.8) Table 2.1. Stress notations Normal stress Shear stress x, y, z coordinate system Tensorial stress σxx σyy σzz σyz σxz σxy Engineering stress σx σy σz τyz τxz τxy Contracted notation σx σy σz σq σr σs x1, x2, x3 coordinate system Tensorial stress σ11 σ22 σ33 σ23 σ13 σ12 Engineering stress σ1 σ2 σ3 τ23 τ13 τ12 Contracted notation σ1 σ2 σ3 σ4 σ5 σ6
6 DISPLACEMENTS,STRAINS,AND STRESSES Table 2.2.Strain notations (the engineering and contracted notation shear strains are twice the tensorial shear strain) Normal strain Shear strain x,y,z coordinate system Tensorial strain Exx Eyz Ex2 Exy Engineering strain Ex Ey e Yyt Yus Yry Contracted notation Ex Ey Eg Es x1,X2,X3 coordinate system Tensorial strain e11e22633E23 613 e12 Engineering strain E2 E3 23M3 M12 Contracted notation 1 E2 E3 EA ES E6 The engineering shear strains are twice the tensorial shear strains: av aw Yy =2Ey= ay (2.9) 8z yit 2ex au aw (2.10) 8z'ax du dv y=2=)+x (2.11) In the x1,x2,x3 coordinate system the strain-displacement relationships are also given by Eqs.(2.2)-(2.4)and (2.9)-(2.11)with x,y,z replaced by x1,x2,x3, the subscripts x,y,z by 1,2,3,and u,v,w by u,u2,u3. 2.2 Equilibrium Equations The equilibrium equations at a point O are obtained by considering force and moment balances on a small AxAyAz cubic element located at point O.(The point O is at the center of the element,Fig.2.6.)We relate the stresses at one face to those at the opposite face by the Taylor series.By using only the first term of the Taylor series,force balance in the x direction gives -Ox△zAy-tx△x△y-tyr△x△z+ 0x+8x △z△y +(+股axa+(.+ayA x△z+f△x△y△z=0, (2.12) △x y B u+ 0u△正 Figure 2.4:Displacement of the AB line segment
6 DISPLACEMENTS, STRAINS, AND STRESSES Table 2.2. Strain notations (the engineering and contracted notation shear strains are twice the tensorial shear strain) Normal strain Shear strain x, y, z coordinate system Tensorial strain xx yy zz yz xz xy Engineering strain x y z γyz γxz γxy Contracted notation x y z q r s x1, x2, x3 coordinate system Tensorial strain 11 22 33 23 13 12 Engineering strain 1 2 3 γ23 γ13 γ12 Contracted notation 1 2 3 4 5 6 The engineering shear strains are twice the tensorial shear strains: γyz = 2yz = ∂v ∂z + ∂w ∂y (2.9) γxz = 2xz = ∂u ∂z + ∂w ∂x (2.10) γxy = 2xy = ∂u ∂y + ∂v ∂x . (2.11) In the x1, x2, x3 coordinate system the strain–displacement relationships are also given by Eqs. (2.2)–(2.4) and (2.9)–(2.11) with x, y, z replaced by x1, x2, x3, the subscripts x, y, z by 1, 2, 3, and u, v, w by u1, u2, u3. 2.2 Equilibrium Equations The equilibrium equations at a point O are obtained by considering force and moment balances on a small x y z cubic element located at point O. (The point O is at the center of the element, Fig. 2.6.) We relate the stresses at one face to those at the opposite face by the Taylor series. By using only the first term of the Taylor series, force balance in the x direction gives −σx z y − τzx x y − τyx x z + σx + ∂σx ∂x x z y + τzx + ∂τzx ∂z z x y + τyx + ∂τyx ∂y y x z + fx x y z= 0, (2.12) u A B ∆x y x x∆x u u ∂ ∂ + ∆x′ A′ B′ Figure 2.4: Displacement of the AB line segment
2.2 EQUILIBRIUM EQUATIONS 7 个 + Figure 2.5:Displacement of the ABC segment. where f is the body force per unit volume in the x direction.After simplification, this equation becomes ++ 8题+f=0. (2.13) ax ay az By similar arguments,the equilibrium equations in the y and z directions are 0xy+2++fv=0, (2.14) ax ay +g+o+5=0. (2.15) ax ay az where f,and f:are the body forces per unit volume in the y and z directions. A moment balance about an axis parallel to x and passing through the center (point O)gives(Fig.2.7) gAra号-rAay +(+aya-(+aAa (2.16) 0:+ 00Ξ△z 0z T+ 0r型△2 8z 0r丝△z T江+ 8z 0 Ty+ rg△y T:+ 0r丝△r by Ox 0g+ do1△yy 0x+ 0c王△士 8 8 △x 0r型△E Tw+ arg△y by Ox I △y Figure 2.6:Stresses on the AxAyAz cubic element
2.2 EQUILIBRIUM EQUATIONS 7 y x ∆x v A B β α C x∆x v v ∂ ∂ B′ A′ C′ + Figure 2.5: Displacement of the ABC segment. where fx is the body force per unit volume in the x direction. After simplification, this equation becomes ∂σx ∂x + ∂τyx ∂y + ∂τzx ∂z + fx = 0. (2.13) By similar arguments, the equilibrium equations in the y and z directions are ∂τxy ∂x + ∂σy ∂y + ∂τzy ∂z + fy = 0, (2.14) ∂τxz ∂x + ∂τyz ∂y + ∂σz ∂z + fz = 0, (2.15) where fy and fz are the body forces per unit volume in the y and z directions. A moment balance about an axis parallel to x and passing through the center (point O) gives (Fig. 2.7) τyz x z y 2 − τzy x y z 2 + τyz + ∂τyz ∂y y x z y 2 − τzy + ∂τzy ∂z z x y z 2 = 0. (2.16) σz σy σx τxz τxy τyz τyx τzy τzx z y x O y∆ x∆ x∆ x τ τ x∆ x σ σ x∆ x τ τ xy xy x x xz xz ∂ ∂ + ∂ ∂ + ∂ ∂ + z∆ z τ τ z∆ z τ τ z∆ z σ σ zx zx zy zy z z ∂ ∂ + ∂ ∂ + ∂ ∂ + y∆ y τ τ y∆ y σ σ y∆ y τ τ yx yx y y yz yz ∂ ∂ + ∂ ∂ + ∂ ∂ + z∆ Figure 2.6: Stresses on the x y z cubic element