Prediction of homogenized properties

In the current example, two representative volume elements are considered, the square and hexagonal arrays. However the self-consistent micromechanics method can be applied to other representative volume element geometries that meet the doubly periodic condition. A
schematic of each geometry is given in Figure 1. The first two columns in Table 1 are the input constituent material properties for an IM7/8552 carbon fiber, epoxy matrix composite. The final two columns in Table 1 give the predicted homogenized composite properties for the two representative volume elements.




Square Cell

Hex Cell

E1 (GPa)





E2 (GPa)





E3 (GPa)





G12 (GPa)







First, consider a general laminate to be analysed using a self-consistent micromechanics method. Presented graphically in Figure 3 is the self-consistent micromechanics method used herein. A complete set of material properties for both the fiber and matrix phase are required. Six canonical states of deformation, extending from Equations 10 and 11, are applied as boundary conditions to two representative volume elements, a square array and a hexagonal array shown in Figure 1. The fiber volume fraction of the representative volume elements is 60 percent. For this step the finite element program Abaqus is utilized. Six canonical states of deformation provide both the homogenized stiffness matrix (Cj ) and the enhancement matrix (Mj ) for the two micro-geometries.

Both domains are subjected to...


Representative volume element boundary conditions

The imposition of canonical states of strain upon the representative volume element utilizing finite-element analyses requires the development of a corresponding set of displacement boundary conditions. The representative volume element principal directions, (e1, e2, e3) are shown in Figure 1. Equation 10 defines the appropriate displacement boundary conditions for the prescribed extensional strain in the "1" direction with ui representing the displacement vector and xi the position vector.

U (0,X2,X3 ) = Txy (0,X2,X3 ) = Txz (0,X2,X3 ) = 0


U (L1, X2, X3 ) = S1L1; Txy (L1, X2, X3) = Txz (L1, X2, X3) = 0


v (X1,0,X3) = Tyx (X1,0,X3) = Tyz (X1,0,X3) = 0


v (X1, L2, X3 ) = Tyx ( X1, L2, X3 ) = Tyz ( X1, L2, X3) = 0


w(x1,x2,0) = Tzx (x1,x2,0)...



approach, there are several shortcomings, which, if overcome, may provide increasingly accurate predictions of ultimate properties. The most apparent shortcomings of a homogenized analysis are: the modeling of fictitious interfaces; stresses and strains in the homogenized continuum exist in neither the fiber phase nor the matrix phase and the loss of the residual micromechanical thermal stress field due to a temperature change. The current chapter will focus on the latter two shortcomings by predicting the strain state within the fiber and matrix phases using a process referred to as micromechanical enhancement (Gosse & Christensen, 2001; Buchanan et al., 2009).

The role of micromechanical enhancement is to provide a computationally efficient micromechanics analysis that includes congruen...



The homogenization process seeks to obtain equivalent homogenous continuum properties for a medium composed of multiple phases of varying constitutive properties. For the current discussion, we will limit ourselves to a heterogeneous medium consisting of collimated, continuous fibers within an isotropic matrix. Many methods and closed-form expressions have been developed to achieve this goal (Pindera et al., 2009). Among these, the most accurate in predicting the average response of an orthotropic medium is the finite element method (Daniel & Ishai, 2006). In the finite element approach, one would like to determine the relationship between the average stress and average strain as expressed in Equation 1.

ai = Cij (sj – ajAT) (i, j = 1 – 6) (1)

The overbar indicates an average or homogenize...


Self-Consistent Micromechanical Enhancement of Continuous Fiber Composites

Andrew Ritchey1, Joshua Dustin1, Jonathan Gosse2 and R. Byron Pipes1

1Purdue University 2The Boeing Company USA

1. Introduction

Much of the previous work in developing analytical models for high performance composite materials has focused on representations of the heterogeneous medium as a homogenous, anisotropic continuum. The development of the equivalent properties of the homogenous medium from the geometry of the microstructure and the fiber and matrix properties has been come to be known as "micromechanics" (Daniel & Ishai, 2006). The term "homogenization" has been applied to the process of determining the effective properties of the homogenous medium and for much of the past half century homogenization was the only task of micromechanics...


Comparison of fatigue data at room temperature obtained by the authors and Hirano

As stated in the introduction, Hirano reported S-N test results at RT for a G40-800/5260 CF/BMI composite material using a small number of specimens (Hirano, 2001). This section will compare his data with those obtained in the present study. He used OH specimens 38.1 mm wide, and with other dimensions such as the hole diameter, stacking sequence, and thickness the same as those in this study. The loading conditions of the fatigue tests were R=0.05 for tension fatigue, R=20 for compression fatigue, and R=-1 for tension-compression fatigue. The combination of R and the number of specimens tested, n, are {R=0.05, n=4(1)}, {R=20, n=3(1)}, and {R=-1, n=5}, where the figure in ( ) is the number of unbroken specimens...


Effect of compression load cycles on tension-compression fatigue strength

The repeated stress range, Srange, of compression fatigue tests (R=10) in this study is calculated by Srange=-0.9xSmin. If the Srange-N relationships in Fig. 5 are noticed, it can be understood intuitively that they are very close to the tension-compression Smax-N relationships in Fig. 6. These relationships are discussed in detail below.

Smax in tension-compression fatigue tests (R=-1) is the stress amplitude and indicates the tension stress range or compression stress range. Fig. 10 shows Fig. 6 itself together with the Srange-N relationships converted from the compression Smin-N relationships in Fig. 5. Thick lines are the tension-compression Smax-N relationships and fine lines represent the compression Srange-N relationships, whose parameters are given in Table 4. Fig...


Ratio of compression strength to tension strength in static and fatigue strengths

We now discuss the ratio of static compressive strength to static tensile strength in NH and OH specimens, and the ratio of compression fatigue strength (R=10) to tension fatigue strength (R=0.1) in OH specimens. That is, this section evaluates the weakness of the material tested for compression loading on the basis of the ratio of compression strength to tension strength.

Подпись: гст(N) Ratio of compression strength to tension strength in static and fatigue strengths Подпись: (5)

The fatigue strength ratio for arbitrary N is defined by

where Scomp min(N) is compression fatigue strength (R=10), and Stens max(N) tension fatigue strength (R=0.1). The static strength ratio is defined for N=1, rCT(1), where Scomp min(1) is the mean compressive strength and Stens max(1) the mean tensile strength. Fatigue strength is calculated by Eq. (2) and the parameters given in Table 4.

Table 7 indicat...


Comparison of RT and 150°C strengths subjected to static or fatigue loading

To examine the high-temperature performance of the material tested, we will discuss the ratio of 150°C strength to RT strength when the specimens are subjected to either static or fatigue loading. This strength ratio, r150(N), is defined as

r150(N) , 150°Cstrength(N) (4)

RT strength (N)

where static strength is given for N=1.

Table 6 indicates r150(N) values calculated by Eq. (4) for NH and OH specimens. Since the static strength ratio was described in Section 3.1, this section gives only numerical values in Table 6 for the sake of comparison. In the case of fatigue strength, the ratio was calculated by the S-N equation, Eq. (2), and the parameters in Table 4. For OH specimens the 150°C fatigue strength is adequately high in comparison with RT fatigue strength...