10 Failure Theories of a Lamina 10.1 Basic Equations In this chapter we present various failure theories for one single layer of the composite laminate,usually called a lamina.We use the following notation throughout this chapter for the various strengths or ultimate stresses: of:tensile strength in longitudinal direction. of:compressive strength in longitudinal direction. :tensile strength in transverse direction. o:compressive strength in transverse direction. 2:shear strength in the 1-2 plane. where the strength means the ultimate stress or failure stress,the longitudinal direction is the fiber direction(1-direction),and the transverse direction is the 2-direction(perpendicular to the fiber). We also use the following notation for the ultimate strains: ultimate tensile strain in the longitudinal direction. ef:ultimate compressive strain in the longitudinal direction. ultimate tensile strain in the transverse direction. e8: ultimate compressive strain in the transverse direction. ultimate shear strain in the 1-2 plane. It is assumed that the lamina behaves in a linear elastic manner.For the longitudinal uniaxial loading of the lamina (see Fig.10.1),we have the following elastic relations: OT =E1eT (10.1) of =Eref (10.2)
10 Failure Theories of a Lamina 10.1 Basic Equations In this chapter we present various failure theories for one single layer of the composite laminate, usually called a lamina. We use the following notation throughout this chapter for the various strengths or ultimate stresses: σT 1 : tensile strength in longitudinal direction. σC 1 : compressive strength in longitudinal direction. σT 2 : tensile strength in transverse direction. σC 2 : compressive strength in transverse direction. τ F 12 : shear strength in the 1-2 plane. where the strength means the ultimate stress or failure stress, the longitudinal direction is the fiber direction (1-direction), and the transverse direction is the 2-direction (perpendicular to the fiber). We also use the following notation for the ultimate strains: εT 1 : ultimate tensile strain in the longitudinal direction. εC 1 : ultimate compressive strain in the longitudinal direction. εT 2 : ultimate tensile strain in the transverse direction. εC 2 : ultimate compressive strain in the transverse direction. γF 12 : ultimate shear strain in the 1-2 plane. It is assumed that the lamina behaves in a linear elastic manner. For the longitudinal uniaxial loading of the lamina (see Fig. 10.1), we have the following elastic relations: σT 1 = E1εT 1 (10.1) σC 1 = E1εC 1 (10.2)
184 10 Failure Theories of a Lamina 1 e e Fig.10.1.Stress-strain curve for the longitudinal uniaxial loading of a lamina where E is Young's modulus of the lamina in the longitudinal (fiber)direc- tion. For the transverse uniaxial loading of the lamina (see Fig.10.2),we have the following elastic relations: g=E2话 (10.3) =E2eS (10.4) where E2 is Young's modulus of the lamina in the transverse direction.For the shear loading of the lamina (see Fig.10.3),we have the following elastic relation: T12 G12712 (10.5) where G12 is the shear modulus of the lamina. 10.1.1 Maximum Stress Failure Theory In the marimum stress failure theory,failure of the lamina is assumed to occur whenever any normal or shear stress component equals or exceeds the corresponding strength.This theory is written mathematically as follows: o9<1<l (10.6) S<2< (10.7) Inal< (10.8)
184 10 Failure Theories of a Lamina Fig. 10.1. Stress-strain curve for the longitudinal uniaxial loading of a lamina where E1 is Young’s modulus of the lamina in the longitudinal (fiber) direction. For the transverse uniaxial loading of the lamina (see Fig. 10.2), we have the following elastic relations: σT 2 = E2εT 2 (10.3) σC 2 = E2εC 2 (10.4) where E2 is Young’s modulus of the lamina in the transverse direction. For the shear loading of the lamina (see Fig. 10.3), we have the following elastic relation: τ F 12 = G12γF 12 (10.5) where G12 is the shear modulus of the lamina. 10.1.1 Maximum Stress Failure Theory In the maximum stress failure theory, failure of the lamina is assumed to occur whenever any normal or shear stress component equals or exceeds the corresponding strength. This theory is written mathematically as follows: σC 1 < σ1 < σT 1 (10.6) σC 2 < σ2 < σT 2 (10.7) |τ12| < τ F 12 (10.8)
10.1 Basic Equations 185 62 e 6 e Fig.10.2.Stress-strain curve for the transverse uniaxial loading of a lamina 612 I 1 品 Y12 Fig.10.3.Stress-strain curve for the shear loading of a lamina where o1 and o2 are the maximum material normal stresses in the lamina, while T12 is the maximum shear stress in the lamina. The failure envelope for this theory is clearly illustrated in Fig.10.4.The advantage of this theory is that it is simple to use but the major disadvantage is that there is no interaction between the stress components
10.1 Basic Equations 185 Fig. 10.2. Stress-strain curve for the transverse uniaxial loading of a lamina Fig. 10.3. Stress-strain curve for the shear loading of a lamina where σ1 and σ2 are the maximum material normal stresses in the lamina, while τ12 is the maximum shear stress in the lamina. The failure envelope for this theory is clearly illustrated in Fig. 10.4. The advantage of this theory is that it is simple to use but the major disadvantage is that there is no interaction between the stress components
186 10 Failure Theories of a Lamina 0 o Fig.10.4.Failure envelope for the maximum stress failure theory 10.1.2 Maximum Strain Failure Theory In the marimum strain failure theory,failure of the lamina is assumed to occur whenever any normal or shear strain component equals or exceeds the corre- sponding ultimate strain.This theory is written mathematically as follows: ef <E1<ET (10.9) 8<e2< T (10.10) mal< (10.11) where E1,E2,and Y12 are the principal material axis strain components.In this case,we have the following relation between the strains and the stresses in the longitudinal direction: 61= 02 E1 01一2E1 E1 (10.12) Simplifying (10.12),we obtain: 01-a1 02= (10.13) 12 Similarly,we have the following relation between the strains and the stresses in the transverse direction: E2= 02 01 E2E 一21 (10.14) Simplifying (10.14),we obtain:
186 10 Failure Theories of a Lamina Fig. 10.4. Failure envelope for the maximum stress failure theory 10.1.2 Maximum Strain Failure Theory In the maximum strain failure theory, failure of the lamina is assumed to occur whenever any normal or shear strain component equals or exceeds the corresponding ultimate strain. This theory is written mathematically as follows: εC 1 < ε1 < εT 1 (10.9) εC 2 < ε2 < εT 2 (10.10) |γ12| < γF 12 (10.11) where ε1, ε2, and γ12 are the principal material axis strain components. In this case, we have the following relation between the strains and the stresses in the longitudinal direction: ε1 = σT 1 E1 = σ1 E1 − ν12 σ2 E1 (10.12) Simplifying (10.12), we obtain: σ2 = σ1 − σT 1 ν12 (10.13) Similarly, we have the following relation between the strains and the stresses in the transverse direction: ε2 = σT 2 E2 = σ2 E2 − ν21 σ1 E2 (10.14) Simplifying (10.14), we obtain:
10.1 Basic Equations 187 02=2101+o (10.15) The failure envelope for this theory is clearly shown in Fig.10.5 (based on (10.13)and (10.15)).The advantage of this theory is that it is simple to use but the major disadvantage is that there is no interaction between the strain components. 02 slope Va stope-2 o.c Fig.10.5.Failure envelope for the maximum strain failure theory Figure 10.6 shows the two failure envelopes of the maximum stress the- ory and the maximum strain theory superimposed on the same plot for comparison. 10.1.3 Tsai-Hill Failure Theory The Tsai-Hill failure theory is derived from the von Mises distortional energy yield criterion for isotropic materials but is applied to anisotropic materials with the appropriate modifications.In this theory,failure is assumed to occur whenever the distortional yield energy equals or exceeds a certain value related to the strength of the lamina.In this theory,there is no distinction between the tensile and compressive strengths.Therefore,we use the following notation for the strengths of the lamina: of:strength in longitudinal direction. strength in transverse direction. 712:shear strength in the 1-2 plane. The Tsai-Hill failure theory is written mathematically for the lamina as follows:
10.1 Basic Equations 187 σ2 = ν21σ1 + σT 2 (10.15) The failure envelope for this theory is clearly shown in Fig. 10.5 (based on (10.13) and (10.15)). The advantage of this theory is that it is simple to use but the major disadvantage is that there is no interaction between the strain components. Fig. 10.5. Failure envelope for the maximum strain failure theory Figure 10.6 shows the two failure envelopes of the maximum stress theory and the maximum strain theory superimposed on the same plot for comparison. 10.1.3 Tsai-Hill Failure Theory The Tsai-Hill failure theory is derived from the von Mises distortional energy yield criterion for isotropic materials but is applied to anisotropic materials with the appropriate modifications. In this theory, failure is assumed to occur whenever the distortional yield energy equals or exceeds a certain value related to the strength of the lamina. In this theory, there is no distinction between the tensile and compressive strengths. Therefore, we use the following notation for the strengths of the lamina: σF 1 : strength in longitudinal direction. σF 2 : strength in transverse direction. τ F 12 : shear strength in the 1-2 plane. The Tsai-Hill failure theory is written mathematically for the lamina as follows: