MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis The compressive strength of the composite is plotted as a function of the fiber volume fraction,vr.in Fig- ure 5.2.3.2 for E-glass fibers embedded in an epoxy matrix.The compressive strength of glass-reinforced plastic,with a fiber volume fraction of 0.6 to 0.7,is on the order of 460 to 600 ksi (3100 to 4100 MPa). Values of this magnitude do not appear to have been measured for any realistic specimens.However, the achievement of a strength of half a million psi in a composite of this type would require an average shortening greater than 5%.For the epoxy materials used in this calculation,such a shortening would result in a decrease in the effective shear stiffness of the matrix material since the proportional limit of the matrix would be exceeded.Hence,it is necessary to modify the analysis to consider the inelastic defor- mation of the matrix.As a first approximation,the matrix modulus in Equations 5.2.3.2(a)and(b)can be replaced by a reduced modulus.A more general result can be obtained by modeling the matrix as an elastic,perfectly plastic material.For this matrix,the secant value at each axial strain value may be as- sumed to govern the instability.These assumptions (Reference 5.2.3.2(d))yield the following result for the shear mode: vfEf Fepl 5.2.3.2(c) 31-vf) where FP is the matrix yield stress level. 10.0 ELASTIC 50.0 INELASTIC "SHEAR"MODE HLON3ULS 10.0 1.0 "EXTENSION"MODE 5.0 0.5 1.0 0.1 0.0 0.2 0.4 0.6 0.8 1.0 FIBER VOLUME FRACTION FIGURE 5.2.3.2 Compressive strength of glass-reinforced epoxy composite. For the generally dominant shear mode,the elastic results of Equation 5.2.3.2(b)are independent of the fiber modulus,yet the compressive strength of boron/epoxy is much greater than that of glass/epoxy composites.One hypothesis to explain this discrepancy,is that use of the stiffer boron fibers yields lower matrix strains and less of a strength reduction due to inelastic effects.Thus,the results of Equation 5.2.3.2(c)show a ratio of 6 or 2.4 for the relative strengths of boron compared to glass fibers in the same matrix. All of the analytical results above indicate that compressive strength is independent of fiber diameter. Yet different diameter fibers may yield different compressive strengths for composites because large di- 5-16
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-16 The compressive strength of the composite is plotted as a function of the fiber volume fraction, vf, in Figure 5.2.3.2 for E-glass fibers embedded in an epoxy matrix. The compressive strength of glass-reinforced plastic, with a fiber volume fraction of 0.6 to 0.7, is on the order of 460 to 600 ksi (3100 to 4100 MPa). Values of this magnitude do not appear to have been measured for any realistic specimens. However, the achievement of a strength of half a million psi in a composite of this type would require an average shortening greater than 5%. For the epoxy materials used in this calculation, such a shortening would result in a decrease in the effective shear stiffness of the matrix material since the proportional limit of the matrix would be exceeded. Hence, it is necessary to modify the analysis to consider the inelastic deformation of the matrix. As a first approximation, the matrix modulus in Equations 5.2.3.2(a) and (b) can be replaced by a reduced modulus. A more general result can be obtained by modeling the matrix as an elastic, perfectly plastic material. For this matrix, the secant value at each axial strain value may be assumed to govern the instability. These assumptions (Reference 5.2.3.2(d)) yield the following result for the shear mode: 1 cu f f cpl f F = vEF 3(1- v ) 5.2.3.2(c) where Fcpl is the matrix yield stress level. FIGURE 5.2.3.2 Compressive strength of glass-reinforced epoxy composite. For the generally dominant shear mode, the elastic results of Equation 5.2.3.2(b) are independent of the fiber modulus, yet the compressive strength of boron/epoxy is much greater than that of glass/epoxy composites. One hypothesis to explain this discrepancy, is that use of the stiffer boron fibers yields lower matrix strains and less of a strength reduction due to inelastic effects. Thus, the results of Equation 5.2.3.2(c) show a ratio of 6 or 2.4 for the relative strengths of boron compared to glass fibers in the same matrix. All of the analytical results above indicate that compressive strength is independent of fiber diameter. Yet different diameter fibers may yield different compressive strengths for composites because large di-
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis ameter fibers such as boron(0.005 inch,0.13 mm D)are better collimated than small diameter fibers, such as glass(0.0004 in,0.010 mm D).For small diameter fibers,such as aramid and carbon,local out- of-straightness can introduce matrix shear stresses,cause fiber debonding,and produce lower instability stress levels (References 5.2.3.2(e)and 5.2.2.1(d)).Carbon and aramid fibers are anisotropic and have extremely low axial shear moduli.As a result,the elastic buckling stress in the shear mode is reduced to: Fccr= Gm 5.2.3.2(d 1-vf(1-Gm/Gif) where Gif is the fiber longitudinal shear modulus(Reference 5.2.3.2(e)).For high fiber shear moduli,this equation reduces to Equation 5.2.3.2(b). Another failure mechanism for oriented polymeric fibers such as aramid fibers(Reference 5.2.3.2(e)) is a kink-band formation at a specific angle to the direction of compressive stress.The formation of kink- bands is attributed to the fibrillar structure of the highly anisotropic fiber and poor fiber shear strength. Breakup of the fiber into very small diameter fibrils results in degradation of shear stiffness and hence the compressive strength. The results of the compressive strength analyses indicate that for the elastic case,the matrix Young's modulus is the dominant parameter.For the inelastic case,however,there are strength limitations which depend both upon the fiber modulus and upon the matrix strength.For some materials,performance is limited by a matrix yield strength at a given fiber modulus.For other materials,a gain in compressive strength can be achieved by improving the matrix modulus. 5.2.3.3 Matrix mode strength The remaining failure modes of interest are transverse tension and compression and axial shear.For each of these loading conditions,material failure can occur without fracture of the fibers,hence the termi- nology "matrix-dominated"or "matrix modes of failure".Micromechanical analyses of these failure modes are complex because the critical stress states are in the matrix,are highly non-uniform,and are very de- pendent upon the local geometry.As a result,it appears that the most fruitful approaches will be those that consider average states of stress. There are two types of shearing stresses which are of interest for these matrix-dominated failures: (1)in a plane which contains the filaments,and (2)in a plane normal to the filaments.In the first case, the filaments provide very little reinforcement to the composite and the shear strength depends on the shear strength of the matrix material.In the second case,some reinforcement may occur;at high volume fractions of filaments,the reinforcement may be substantial.It is important to recognize that filaments provide little resistance to shear in any surfaces parallel to them. The approach to shear failure analysis is to consider that a uniaxial fibrous composite is comprised of elastic-brittle fibers embedded in an elastic-perfectly plastic matrix.For the composite,the theorems of limit analysis of plasticity (e.g.,References 5.2.3.3(a)and(b))may be used to obtain upper and lower bounds for a composite limit load(Reference 5.2.3.3(c)).The limit load is defined as the load at which the matrix yield stress permits composite deformation to increase with no increase in load.The failure strength of a ductile matrix may be approximated by this limit load. 5.2.4 Strength under combined stress It is possible to apply the micro-mechanical models for failure described above,to combined stresses in the principal directions.Little work of this type has been done however.Generally the strengths in principal directions have been used in a failure surface for a homogeneous,anisotropic material for esti- mation of strength under combined loads.The understanding of failure mechanisms resulting from the above micro-mechanical models can be used to define the general form of failure surface to be utilized. This approach is outlined in the following sections. Knowledge of the different failure mechanisms and quantitative experimental data for a UDC under single stress components can be used to formulate practical failure criteria for combined stresses.Plane 5-17
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-17 ameter fibers such as boron (0.005 inch, 0.13 mm D) are better collimated than small diameter fibers, such as glass (0.0004 in, 0.010 mm D). For small diameter fibers, such as aramid and carbon, local outof-straightness can introduce matrix shear stresses, cause fiber debonding, and produce lower instability stress levels (References 5.2.3.2(e) and 5.2.2.1(d)). Carbon and aramid fibers are anisotropic and have extremely low axial shear moduli. As a result, the elastic buckling stress in the shear mode is reduced to: ccr m f m 1f F = G 1- v (1- G / G ) 5.2.3.2(d) where G1f is the fiber longitudinal shear modulus (Reference 5.2.3.2(e)). For high fiber shear moduli, this equation reduces to Equation 5.2.3.2(b). Another failure mechanism for oriented polymeric fibers such as aramid fibers (Reference 5.2.3.2(e)) is a kink-band formation at a specific angle to the direction of compressive stress. The formation of kinkbands is attributed to the fibrillar structure of the highly anisotropic fiber and poor fiber shear strength. Breakup of the fiber into very small diameter fibrils results in degradation of shear stiffness and hence the compressive strength. The results of the compressive strength analyses indicate that for the elastic case, the matrix Young's modulus is the dominant parameter. For the inelastic case, however, there are strength limitations which depend both upon the fiber modulus and upon the matrix strength. For some materials, performance is limited by a matrix yield strength at a given fiber modulus. For other materials, a gain in compressive strength can be achieved by improving the matrix modulus. 5.2.3.3 Matrix mode strength The remaining failure modes of interest are transverse tension and compression and axial shear. For each of these loading conditions, material failure can occur without fracture of the fibers, hence the terminology "matrix-dominated" or "matrix modes of failure". Micromechanical analyses of these failure modes are complex because the critical stress states are in the matrix, are highly non-uniform, and are very dependent upon the local geometry. As a result, it appears that the most fruitful approaches will be those that consider average states of stress. There are two types of shearing stresses which are of interest for these matrix-dominated failures: (1) in a plane which contains the filaments, and (2) in a plane normal to the filaments. In the first case, the filaments provide very little reinforcement to the composite and the shear strength depends on the shear strength of the matrix material. In the second case, some reinforcement may occur; at high volume fractions of filaments, the reinforcement may be substantial. It is important to recognize that filaments provide little resistance to shear in any surfaces parallel to them. The approach to shear failure analysis is to consider that a uniaxial fibrous composite is comprised of elastic-brittle fibers embedded in an elastic-perfectly plastic matrix. For the composite, the theorems of limit analysis of plasticity (e.g., References 5.2.3.3(a) and (b)) may be used to obtain upper and lower bounds for a composite limit load (Reference 5.2.3.3(c)). The limit load is defined as the load at which the matrix yield stress permits composite deformation to increase with no increase in load. The failure strength of a ductile matrix may be approximated by this limit load. 5.2.4 Strength under combined stress It is possible to apply the micro-mechanical models for failure described above, to combined stresses in the principal directions. Little work of this type has been done however. Generally the strengths in principal directions have been used in a failure surface for a homogeneous, anisotropic material for estimation of strength under combined loads. The understanding of failure mechanisms resulting from the above micro-mechanical models can be used to define the general form of failure surface to be utilized. This approach is outlined in the following sections. Knowledge of the different failure mechanisms and quantitative experimental data for a UDC under single stress components can be used to formulate practical failure criteria for combined stresses. Plane
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis stress failure criteria are discussed below with references also given for more complicated stress sys- tems.The stresses considered are averaged over a representative volume element.The fundamental assumption is that there exists a failure criterion of the form: F(011,022,012)=1 5.2.4(a) which characterizes the failure of the UDC.The usual approach to construction of a failure criterion is to assume a quadratic form in terms of stress or strain since the quadratic form is the simplest form which can adequately describe the experimental data.The various failure criteria which have been proposed all use coefficients based on experimental information such as ultimate stresses under single load compo- nents (References 5.2.4(a)-(d)).For example,the general quadratic version of Equation 5.2.4(a)for plane stress would be: C111+C2202+C662+2C12011022+2C1611O12 5.2.4(b) +2C26022012+C111+C2022+C612=1 The material has different strengths in uniaxial,longitudinal,and transverse tension and compression. Evidently the shear strength is not affected by the sign of the shear stress.It follows that all powers of shear stress in the failure criterion must be even.Consequently,the criterion simplifies to C11o11+C22052+C6612+2C1211022+C1011+C2022=1 5.2.4(c) The ultimate stresses under single component stress conditions for each ofo1,o22,and o12 determine the constants for the failure criterion. 1 1 C11= FF四 C22= FFu 11 11 C1= 5.2.4d) F地Fw C2= Fu Fgu 1 C66= (E0)2 However,C12 cannot be determined from the single component ultimate stresses.Biaxial stress tests must be performed to determine this coefficient.Frequently,the coefficient is established by relating Equation 5.2.4(c)to the Mises-Henky yield criterion for isotropic materials,yielding C2=7(C1C2)2 5.2.4(e) The above failure criterion is the two-dimensional version of the Tsai-Wu criterion(Reference 5.2.4(c)). Its implementation raises several problems;the most severe of these is that the failure criterion ignores the diversity of failure modes which are possible. The identification of the different failure modes of a UDC can provide physically more realistic,and also simpler,failure criteria(Reference 5.2.4(e)).Testing a polymer matrix UDC reveals that tensile stress in the fiber direction produces a jagged,irregular failure surface.Tensile stress transverse to the surface produces a smooth,straight failure surface (See Figures 5.2.4(a)and(b)).Since the carrying capacity deterioration in the tensile fiber mode is due to transverse cracks and the transverse stress oz2 has no effect on such cracks,it is assumed that the plane tensile fiber mode is only dependent on the stresses ou and 612. For compressive ou,failure is due to fiber buckling in the shear mode and the transverse stress o22 has little effect on the compressive failure.In this compressive fiber mode,failure again depends primar- ily on ou.The dependence on oiz is not known and arguments may be made for and against including it in the failure criterion. 5-18
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-18 stress failure criteria are discussed below with references also given for more complicated stress systems. The stresses considered are averaged over a representative volume element. The fundamental assumption is that there exists a failure criterion of the form: F(σ11,σ 22 ,σ12) = 1 5.2.4(a) which characterizes the failure of the UDC. The usual approach to construction of a failure criterion is to assume a quadratic form in terms of stress or strain since the quadratic form is the simplest form which can adequately describe the experimental data. The various failure criteria which have been proposed all use coefficients based on experimental information such as ultimate stresses under single load components (References 5.2.4(a) - (d)). For example, the general quadratic version of Equation 5.2.4(a) for plane stress would be: 11 11 2 22 22 2 66 12 2 12 11 22 16 11 12 26 22 12 1 11 2 22 6 12 C +C +C + 2C + 2C +2C + C + C + C = 1 σ σ σ σσ σσ σσ σ σ σ 5.2.4(b) The material has different strengths in uniaxial, longitudinal, and transverse tension and compression. Evidently the shear strength is not affected by the sign of the shear stress. It follows that all powers of shear stress in the failure criterion must be even. Consequently, the criterion simplifies to 11 11 2 22 22 2 66 12 2 C σ σ σ σσ σ σ +C +C + 2C12 11 22 1 11 2 22 +C + C = 1 5.2.4(c) The ultimate stresses under single component stress conditions for each ofσ11, σ 22 , and σ12 determine the constants for the failure criterion. 11 1 tu 1 cu 22 2 tu 2 cu 1 1 tu 1 cu 2 2 tu 2 cu 66 1 su 2 C = 1 F F C = 1 F F C = 1 F - 1 F C = 1 F - 1 F C = 1 (F ) 5.2.4(d) However, C12 cannot be determined from the single component ultimate stresses. Biaxial stress tests must be performed to determine this coefficient. Frequently, the coefficient is established by relating Equation 5.2.4(c) to the Mises-Henky yield criterion for isotropic materials, yielding 12 11 22 1/2 C = - 1 2 (C C ) 5.2.4(e) The above failure criterion is the two-dimensional version of the Tsai-Wu criterion (Reference 5.2.4(c)). Its implementation raises several problems; the most severe of these is that the failure criterion ignores the diversity of failure modes which are possible. The identification of the different failure modes of a UDC can provide physically more realistic, and also simpler, failure criteria (Reference 5.2.4(e)). Testing a polymer matrix UDC reveals that tensile stress in the fiber direction produces a jagged, irregular failure surface. Tensile stress transverse to the surface produces a smooth, straight failure surface (See Figures 5.2.4(a) and (b)). Since the carrying capacity deterioration in the tensile fiber mode is due to transverse cracks and the transverse stress σ22 has no effect on such cracks, it is assumed that the plane tensile fiber mode is only dependent on the stresses σ11 and σ12. For compressive σ11, failure is due to fiber buckling in the shear mode and the transverse stress σ22 has little effect on the compressive failure. In this compressive fiber mode, failure again depends primarily on σ11. The dependence on σ12 is not known and arguments may be made for and against including it in the failure criterion
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis FIGURE 5.2.4(a)Tensile fiber failure mode. CATASTROPHIC CRACK FIGURE 5.2.4(b)Tensile matrix failure mode. 5-19
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-19 FIGURE 5.2.4(a) Tensile fiber failure mode. FIGURE 5.2.4(b) Tensile matrix failure mode
MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis For tension transverse to the fibers,the tensile matrix mode,failure occurs by a sudden crack in the fiber direction as shown in Figure 5.2.4(b).Since stress in the fiber direction has no effect on a crack in the fiber direction,this failure mode is dependent only on o22 and oiz. For compressive stress transverse to the fibers,failure occurs on some plane parallel to the fibers, but not necessarily normal to o22.This compressive matrix mode is produced by normal stress and shear stress on the failure plane.Again,the stress ou does not effect this failure. Each of the failure modes described can be modeled separately by a quadratic polynomial(Refer- ence 5.2.4(e)).This approach provides four individual failure criteria.Note the choice of stress compo- nents included in each of these criteria,and the particular mathematical form used,are subjects which are not yet fully resolved.The following criteria appear to a reasonable set with which the different modes of failure can be handled separately. Fiber modes Tensile 012 F 5.2.4(0 F我 Compressive 5.2.4(g) F Matrix modes Tensile 021 012 -1 F 5.2.4(h) F Compressive 612 5.2.40 F哭 2F器 F Note that Fgu in Equation 5.2.4(i)should be taken as the absolute value.The ultimate transverse shear stress,23=F,is very difficult to measure.A reasonable approximation for this quantity is the ultimate shear stress for the matrix.For any given state of stress,one each of Equations 5.2.4(f)and(g)and Equations 5.2.4(h)and(i)are chosen according to the signs of ou and o22.The stress components are introduced into the appropriate pair and whichever criterion is satisfied first is the operative criterion. The advantages are Equations 5.2.4(f)-(i)are: 1.The failure criteria are expressed in terms of single component ultimate stresses.No biaxial test results are needed. 2.The failure mode is identified by the criterion which is satisfied first. The last feature is of fundamental importance for analysis of fiber composite structural elements,since it permits identification of the nature of initial damage.Moreover,in conjunction with a finite element analy- sis,it is possible to identify the nature of failure in elements,modify their stiffnesses accordingly,and pro- ceed with the analysis to predict new failures. 5-20
MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-20 For tension transverse to the fibers, the tensile matrix mode, failure occurs by a sudden crack in the fiber direction as shown in Figure 5.2.4(b). Since stress in the fiber direction has no effect on a crack in the fiber direction, this failure mode is dependent only on σ22 and σ12. For compressive stress transverse to the fibers, failure occurs on some plane parallel to the fibers, but not necessarily normal to σ22. This compressive matrix mode is produced by normal stress and shear stress on the failure plane. Again, the stress σ11 does not effect this failure. Each of the failure modes described can be modeled separately by a quadratic polynomial (Reference 5.2.4(e)). This approach provides four individual failure criteria. Note the choice of stress components included in each of these criteria, and the particular mathematical form used, are subjects which are not yet fully resolved. The following criteria appear to a reasonable set with which the different modes of failure can be handled separately. Fiber modes Tensile 2 11 1 tu 2 12 12 su F + F = 1 F σ σ H G I K J F H G I K J 5.2.4(f) Compressive 2 11 1 cu 2 12 12 su F + F = 1 F σ σ H G I K J F H G I K J 5.2.4(g) Matrix modes Tensile 2 22 2 tu 2 12 12 su F + F = 1 F σ σ H G I K J F H G I K J 5.2.4(h) Compressive 2 22 23 su 2 2 cu 23 su 22 2 cu 2 12 12 su 2 F + F 2 F -1 F + F = 1 F σ σσ H G I K J F H G I K J L N M M O Q P P F H G I K J F H G I K J 5.2.4(i) Note that 2 cu F in Equation 5.2.4(i) should be taken as the absolute value. The ultimate transverse shear stress, 23 23 su σ = F , is very difficult to measure. A reasonable approximation for this quantity is the ultimate shear stress for the matrix. For any given state of stress, one each of Equations 5.2.4(f) and (g) and Equations 5.2.4(h) and (i) are chosen according to the signs of σ11 and σ22. The stress components are introduced into the appropriate pair and whichever criterion is satisfied first is the operative criterion. The advantages are Equations 5.2.4(f) - (i) are: 1. The failure criteria are expressed in terms of single component ultimate stresses. No biaxial test results are needed. 2. The failure mode is identified by the criterion which is satisfied first. The last feature is of fundamental importance for analysis of fiber composite structural elements, since it permits identification of the nature of initial damage. Moreover, in conjunction with a finite element analysis, it is possible to identify the nature of failure in elements, modify their stiffnesses accordingly, and proceed with the analysis to predict new failures