2 Macromechanical Analysis of a Lamina Chapter Objectives Review definitions of stress,strain,elastic moduli,and strain energy. Develop stress-strain relationships for different types of materials. Develop stress-strain relationships for a unidirectional/bidirec- tional lamina. Find the engineering constants of a unidirectional/bidirectional lam- ina in terms of the stiffness and compliance parameters of the lamina. Develop stress-strain relationships,elastic moduli,strengths,and thermal and moisture expansion coefficients of an angle ply based on those of a unidirectional/bidirectional lamina and the angle of the ply. 2.1 Introduction A lamina is a thin layer of a composite material that is generally of a thickness on the order of 0.005 in.(0.125 mm).A laminate is constructed by stacking a number of such laminae in the direction of the lamina thickness(Figure 2.1).Mechanical structures made of these laminates,such as a leaf spring suspension system in an automobile,are subjected to various loads,such as bending and twisting.The design and analysis of such laminated structures demands knowledge of the stresses and strains in the laminate.Also,design tools,such as failure theories,stiffness models,and optimization algorithms, need the values of these laminate stresses and strains. However,the building blocks of a laminate are single lamina,so under- standing the mechanical analysis of a lamina precedes understanding that of a laminate.A lamina is unlike an isotropic homogeneous material.For example,if the lamina is made of isotropic homogeneous fibers and an 61 2006 by Taylor Francis Group,LLC
61 2 Macromechanical Analysis of a Lamina Chapter Objectives • Review definitions of stress, strain, elastic moduli, and strain energy. • Develop stress–strain relationships for different types of materials. • Develop stress–strain relationships for a unidirectional/bidirectional lamina. • Find the engineering constants of a unidirectional/bidirectional lamina in terms of the stiffness and compliance parameters of the lamina. • Develop stress–strain relationships, elastic moduli, strengths, and thermal and moisture expansion coefficients of an angle ply based on those of a unidirectional/bidirectional lamina and the angle of the ply. 2.1 Introduction A lamina is a thin layer of a composite material that is generally of a thickness on the order of 0.005 in. (0.125 mm). A laminate is constructed by stacking a number of such laminae in the direction of the lamina thickness (Figure 2.1). Mechanical structures made of these laminates, such as a leaf spring suspension system in an automobile, are subjected to various loads, such as bending and twisting. The design and analysis of such laminated structures demands knowledge of the stresses and strains in the laminate. Also, design tools, such as failure theories, stiffness models, and optimization algorithms, need the values of these laminate stresses and strains. However, the building blocks of a laminate are single lamina, so understanding the mechanical analysis of a lamina precedes understanding that of a laminate. A lamina is unlike an isotropic homogeneous material. For example, if the lamina is made of isotropic homogeneous fibers and an 1343_book.fm Page 61 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
62 Mechanics of Composite Materials,Second Edition Fiber cross-section Matrix material FIGURE 2.1 Typical laminate made of three laminae. isotropic homogeneous matrix,the stiffness of the lamina varies from point to point depending on whether the point is in the fiber,the matrix,or the fiber-matrix interface.Accounting for these variations will make any kind of mechanical modeling of the lamina very complicated.For this reason,the macromechanical analysis of a lamina is based on average properties and considering the lamina to be homogeneous.Methods to find these average properties based on the individual mechanical properties of the fiber and the matrix,as well as the content,packing geometry,and shape of fibers are discussed in Chapter 3. Even with the homogenization of a lamina,the mechanical behavior is still different from that of a homogeneous isotropic material.For example,take a square plate of length and width w and thickness t out of a large isotropic plate of thickness t(Figure 2.2)and conduct the following experiments. Case A:Subject the square plate to a pure normal load P in direction 1. Measure the normal deformations in directions 1 and 2,and respectively. Case B:Apply the same pure normal load P as in case A,but now in direction 2.Measure the normal deformations in directions 1 and 2, δs andδ2B,respectively. Note that δ1A=δ2B, (2.1a,b) δ2A=δ1B· However,taking a unidirectional square plate (Figure 2.3)of the same dimensions wx w x f out of a large composite lamina of thickness t and conducting the same case A and B experiments,note that the deformations 2006 by Taylor Francis Group,LLC
62 Mechanics of Composite Materials, Second Edition isotropic homogeneous matrix, the stiffness of the lamina varies from point to point depending on whether the point is in the fiber, the matrix, or the fiber–matrix interface. Accounting for these variations will make any kind of mechanical modeling of the lamina very complicated. For this reason, the macromechanical analysis of a lamina is based on average properties and considering the lamina to be homogeneous. Methods to find these average properties based on the individual mechanical properties of the fiber and the matrix, as well as the content, packing geometry, and shape of fibers are discussed in Chapter 3. Even with the homogenization of a lamina, the mechanical behavior is still different from that of a homogeneous isotropic material. For example, take a square plate of length and width w and thickness t out of a large isotropic plate of thickness t (Figure 2.2) and conduct the following experiments. Case A: Subject the square plate to a pure normal load P in direction 1. Measure the normal deformations in directions 1 and 2, δ1A and δ2A, respectively. Case B: Apply the same pure normal load P as in case A, but now in direction 2. Measure the normal deformations in directions 1 and 2, δ1B and δ2B, respectively. Note that (2.1a,b) However, taking a unidirectional square plate (Figure 2.3) of the same dimensions w × w × t out of a large composite lamina of thickness t and conducting the same case A and B experiments, note that the deformations FIGURE 2.1 Typical laminate made of three laminae. Fiber cross-section Matrix material 1A 2B 2A 1B = = δ δ δ δ , . 1343_book.fm Page 62 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
Macromechanical Analysis of a Lamina 63 Case A Case B Undeformed state Undeformed state w+δ2B w+δ2A w+81A w+δ1B Deformed state p Deformed state FIGURE 2.2 Deformation of square plate taken from an isotropic plate under normal loads. 81A≠δ2B, (2.2a,b) δ2A≠δ1B· because the stiffness of the unidirectional lamina in the direction of fibers is much larger than the stiffness in the direction perpendicular to the fibers. Thus,the mechanical characterization of a unidirectional lamina will require more parameters than it will for an isotropic lamina. Also,note that if the square plate(Figure 2.4)taken out of the lamina has fibers at an angle to the sides of the square plate,the deformations will be different for different angles.In fact,the square plate would not only have 2006 by Taylor Francis Group,LLC
Macromechanical Analysis of a Lamina 63 (2.2a,b) because the stiffness of the unidirectional lamina in the direction of fibers is much larger than the stiffness in the direction perpendicular to the fibers. Thus, the mechanical characterization of a unidirectional lamina will require more parameters than it will for an isotropic lamina. Also, note that if the square plate (Figure 2.4) taken out of the lamina has fibers at an angle to the sides of the square plate, the deformations will be different for different angles. In fact, the square plate would not only have FIGURE 2.2 Deformation of square plate taken from an isotropic plate under normal loads. w 2 w 1 t t w w w Undeformed state Deformed state Undeformed state w + δ2A w + δ1A p p w + δ2B w + δ1B w Case A Case B p p Deformed state 1A 2B 2A 1B δ δ δ δ ≠ ≠ , . 1343_book.fm Page 63 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
64 Mechanics of Composite Materials,Second Edition Fiber cross section Case】 Case B Undeformed state Undeformed state w+δ2A w+δ2B D w+δ1B Deformed state ↓p Deformed state FIGURE 2.3 Deformation of a square plate taken from a unidirectional lamina with fibers at zero angle under normal loads. deformations in the normal directions but would also distort.This suggests that the mechanical characterization of an angle lamina is further complicated Mechanical characterization of materials generally requires costly and time-consuming experimentation and/or theoretical modeling.Therefore, the goal is to find the minimum number of parameters required for the mechanical characterization of a lamina. Also,a composite laminate may be subjected to a temperature change and may absorb moisture during processing and operation.These changes in temperature and moisture result in residual stresses and strains in the lam- inate.The calculation of these stresses and strains in a laminate depends on the response of each lamina to these two environmental parameters.In this chapter,the stress-strain relationships based on temperature change and moisture content will also be developed for a single lamina.The effects of temperature and moisture on a laminate are discussed later in Chapter 4. 2006 by Taylor Francis Group,LLC
64 Mechanics of Composite Materials, Second Edition deformations in the normal directions but would also distort. This suggests that the mechanical characterization of an angle lamina is further complicated. Mechanical characterization of materials generally requires costly and time-consuming experimentation and/or theoretical modeling. Therefore, the goal is to find the minimum number of parameters required for the mechanical characterization of a lamina. Also, a composite laminate may be subjected to a temperature change and may absorb moisture during processing and operation. These changes in temperature and moisture result in residual stresses and strains in the laminate. The calculation of these stresses and strains in a laminate depends on the response of each lamina to these two environmental parameters. In this chapter, the stress–strain relationships based on temperature change and moisture content will also be developed for a single lamina. The effects of temperature and moisture on a laminate are discussed later in Chapter 4. FIGURE 2.3 Deformation of a square plate taken from a unidirectional lamina with fibers at zero angle under normal loads. w w p p p p w + δ2B w + δ1B w + δ1A w + δ2A w w w w 2 1 Fiber cross section Case A Case B Undeformed state Deformed state Deformed state Undeformed state t t 1343_book.fm Page 64 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC
Macromechanical Analysis of a Lamina 65 ● Fiber cross section Undeformed state Deformed state FIGURE 2.4 Deformation of a square plate taken from a unidirectional lamina with fibers at an angle under normal loads. 2.2 Review of Definitions 2.2.1 Stress A mechanical structure takes external forces,which act upon a body as surface forces (for example,bending a stick)and body forces (for example, the weight of a standing vertical telephone pole on itself).These forces result in internal forces inside the body.Knowledge of the internal forces at all points in the body is essential because these forces need to be less than the strength of the material used in the structure.Stress,which is defined as the intensity of the load per unit area,determines this knowledge because the strengths of a material are intrinsically known in terms of stress. Imagine a body(Figure 2.5)in equilibrium under various loads.If the body is cut at a cross-section,forces will need to be applied on the cross-sectional area so that it maintains equilibrium as in the original body.At any cross- 2006 by Taylor Francis Group,LLC
Macromechanical Analysis of a Lamina 65 2.2 Review of Definitions 2.2.1 Stress A mechanical structure takes external forces, which act upon a body as surface forces (for example, bending a stick) and body forces (for example, the weight of a standing vertical telephone pole on itself). These forces result in internal forces inside the body. Knowledge of the internal forces at all points in the body is essential because these forces need to be less than the strength of the material used in the structure. Stress, which is defined as the intensity of the load per unit area, determines this knowledge because the strengths of a material are intrinsically known in terms of stress. Imagine a body (Figure 2.5) in equilibrium under various loads. If the body is cut at a cross-section, forces will need to be applied on the cross-sectional area so that it maintains equilibrium as in the original body. At any crossFIGURE 2.4 Deformation of a square plate taken from a unidirectional lamina with fibers at an angle under normal loads. w t t w w p p 2 1 Fiber cross section Undeformed state Deformed state 1343_book.fm Page 65 Tuesday, September 27, 2005 11:53 AM © 2006 by Taylor & Francis Group, LLC