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:
188 10 Failure Theories of a Lamina Maximum Stress Theory 一一一-Maximum Strain Theory 01 Fig.10.6.Comparison of the failure envelopes for the maximum stress theory and maximum strain theory 0102 吃 a)- 十 (10.16) The failure envelope for this theory is clearly shown in Fig.10.7.The advantage of this theory is that there is interaction between the stress com- ponents.However,this theory does not distinguish between the tensile and compressive strengths and is not as simple to use as the maximum stress theory or the maximum strain theory. 0 0 Fig.10.7.Failure envelope for the Tsai-Hill failure theory
188 10 Failure Theories of a Lamina Fig. 10.6. Comparison of the failure envelopes for the maximum stress theory and maximum strain theory σ2 1 σF 1 2 − σ1σ2 σF 1 2 + σ2 2 σF 2 2 + τ 2 12 τ F 122 ≤ 1 (10.16) The failure envelope for this theory is clearly shown in Fig. 10.7. The advantage of this theory is that there is interaction between the stress components. However, this theory does not distinguish between the tensile and compressive strengths and is not as simple to use as the maximum stress theory or the maximum strain theory. Fig. 10.7. Failure envelope for the Tsai-Hill failure theory
10.1 Basic Equations 189 Maximum Stress Theor 0 一一·Tai-Hill Theory -·-。-。Maximum Strain Theory Fig.10.8.Comparison between the three failure envelopes Figure 10.8 shows the three failure envelopes of the maximum stress theory, the maximum strain theory,and the Tsai-Hill theory superimposed on the same plot for comparison. 10.1.4 Tsai-Wu Failure Theory The Tsai-Wu failure theory is based on a total strain energy failure theory.In this theory,failure is assumed to occur in the lamina if the following condition is satisfied: F1o7+F22o+F66T2+Fo1+F2+2012≤1 (10.17) where the coefficients F11,F22,F66,F1,F2,and F12 are given by: y F1= (10.18) 1 F2= (10.19) F=- 11 7 (10.20) 11 = 昭 (10.21) 1 F66= () (10.22) and F12 is a coefficient that is determined experimentally.Tsai-Hahn deter- mined F12 to be given by the following approximate expression:
10.1 Basic Equations 189 Fig. 10.8. Comparison between the three failure envelopes Figure 10.8 shows the three failure envelopes of the maximum stress theory, the maximum strain theory, and the Tsai-Hill theory superimposed on the same plot for comparison. 10.1.4 Tsai-Wu Failure Theory The Tsai-Wu failure theory is based on a total strain energy failure theory. In this theory, failure is assumed to occur in the lamina if the following condition is satisfied: F11σ2 1 + F22σ2 2 + F66τ 2 12 + F1σ1 + F2σ2 + F12σ1σ2 ≤ 1 (10.17) where the coefficients F11, F22, F66, F1, F2, and F12 are given by: F11 = 1 σT 1 σC 1 (10.18) F22 = 1 σT 2 σC 2 (10.19) F1 = 1 σT 1 − 1 σC 1 (10.20) F1 = 1 σT 2 − 1 σC 2 (10.21) F66 = 1 τ F 122 (10.22) and F12 is a coefficient that is determined experimentally. Tsai-Hahn determined F12 to be given by the following approximate expression:
190 Failure Theories of a Lamina Fig.10.9.A general failure ellipse for the Tsai-Wu failure theory Fe=-克 (10.23) The failure envelope for this theory is shown in general in Fig.10.9.The advantage of this theory is that there is interaction between the stress com- ponents and the theory does distinguish between the tensile and compressive strengths.A major disadvantage of this theory is that it is not simple to use. Finally,in order to compare the failure envelopes for a composite lamina with the envelopes of isotropic ductile materials,Fig.10.10 shows the failure envelopes for the usual von Mises and Tresca criteria for isotropic materials. Problems Problem 10.1 Determine the maximum value of a >0 if stresses of ox=30,oy=-20, and Try=50 are applied to a 60-lamina of graphite/epoxy.Use the maxi- mum stress failure theory.The material properties of this lamina are given as follows: Vf=0.70 =1500MPa E1=181 GPa o9=1500MPa E2=10.30GPa =40 MPa 12=0.28 of 246 MPa G12=7.17GPa T2=68 MPa
190 Failure Theories of a Lamina Fig. 10.9. A general failure ellipse for the Tsai-Wu failure theory F12 ≈ −1 2 #F11F22 (10.23) The failure envelope for this theory is shown in general in Fig. 10.9. The advantage of this theory is that there is interaction between the stress components and the theory does distinguish between the tensile and compressive strengths. A major disadvantage of this theory is that it is not simple to use. Finally, in order to compare the failure envelopes for a composite lamina with the envelopes of isotropic ductile materials, Fig. 10.10 shows the failure envelopes for the usual von Mises and Tresca criteria for isotropic materials. Problems Problem 10.1 Determine the maximum value of α > 0 if stresses of σx = 3α, σy = −2α, and τxy = 5α are applied to a 60◦-lamina of graphite/epoxy. Use the maximum stress failure theory. The material properties of this lamina are given as follows: V f = 0.70 σT 1 = 1500 MPa E1 = 181 GPa σC 1 = 1500 MPa E2 = 10.30 GPa σT 2 = 40 MPa ν12 = 0.28 σC 2 = 246 MPa G12 = 7.17 GPa τ F 12 = 68 MPa
Problems 191 Tresca criterion ---von Mises criterion Fig.10.10.Two failure criteria for ductile homogeneous materials Problem 10.2 Repeat Problem 10.1 using the maximum strain failure theory instead of the maximum stress failure theory. Problem 10.3 Repeat Problem 10.1 using the Tsai-Hill failure theory instead of the maxi- mum stress failure theory. Problem 10.4 Repeat Problem 10.1 using the Tsai-Wu failure theory instead of the maxi- mum stress failure theory. MATLAB Problem 10.5 Use MATLAB to plot the four failure envelopes using the strengths given in Problem 10.1. Problem 10.6 Determine the maximum value of a>0 if stresses of o=30,oy=-20,and Ty=5a are applied to a 30-lamina of glass/epoxy.Use the maximum stress failure theory.The material properties of this lamina are given as follows:
Problems 191 Fig. 10.10. Two failure criteria for ductile homogeneous materials Problem 10.2 Repeat Problem 10.1 using the maximum strain failure theory instead of the maximum stress failure theory. Problem 10.3 Repeat Problem 10.1 using the Tsai-Hill failure theory instead of the maximum stress failure theory. Problem 10.4 Repeat Problem 10.1 using the Tsai-Wu failure theory instead of the maximum stress failure theory. MATLAB Problem 10.5 Use MATLAB to plot the four failure envelopes using the strengths given in Problem 10.1. Problem 10.6 Determine the maximum value of α > 0 if stresses of σx = 3α, σy = −2α, and τxy = 5α are applied to a 30◦-lamina of glass/epoxy. Use the maximum stress failure theory. The material properties of this lamina are given as follows:
192 Failure Theories of a Lamina Vf=0.45 oT 1062 MPa E1=38.6GPa of =610MPa E2=8.27GPa o =31MPa 12=0.26 =118MPa G12=4.14GPa 72 MPa Problem 10.7 Repeat Problem 10.6 using the maximum strain failure theory instead of the maximum stress failure theory. Problem 10.8 Repeat Problem 10.6 using the Tsai-Hill failure theory instead of the maxi- mum stress failure theory. Problem 10.9 Repeat Problem 10.6 using the Tsai-Wu failure theory instead of the maxi- mum stress failure theory. MATLAB Problem 10.10 Use MATLAB to plot the four failure envelopes using the strengths given in Problem 10.6
192 Failure Theories of a Lamina V f = 0.45 σT 1 = 1062 MPa E1 = 38.6 GPa σC 1 = 610 MPa E2 = 8.27 GPa σT 2 = 31 MPa ν12 = 0.26 σC 2 = 118 MPa G12 = 4.14 GPa τ F 12 = 72 MPa Problem 10.7 Repeat Problem 10.6 using the maximum strain failure theory instead of the maximum stress failure theory. Problem 10.8 Repeat Problem 10.6 using the Tsai-Hill failure theory instead of the maximum stress failure theory. Problem 10.9 Repeat Problem 10.6 using the Tsai-Wu failure theory instead of the maximum stress failure theory. MATLAB Problem 10.10 Use MATLAB to plot the four failure envelopes using the strengths given in Problem 10.6