Basis of the Ladevèze theory is the strain energy function of a damaged ply in a two dimensional formulation, shown in Equation (1):

In this equation, damage parameters, d_ij, are introduced to relate the elastic modulus to the damage state. and are initial Young’s modulus and shear modulus, respectively, σ_ij is stress, ν_ij is Poisson’s ratio, and subscripts 1 and 2 represents direction along fiber and transverse to the fiber, respectively. < >₊ and < >₋ are valid when the value is positive and negative, respectively (i.e. ₊= a when a is positive, and ₊ = 0 when a is negative). Note the crack closure under compressive transverse stress is considered. An increase of d_ij will result in a decrease of the modulus, resulting in the following strain-stress relationship of a damaged ply:

In the continuum damage mechanics model, thermodynamic forces, Y_ij, that drive the damage accumulation can be derived from the partial derivative of strain energy with respect to d_ij.

Figure 1 shows the typical relationship between the damage parameters, d_ij, and the thermodynamic forces, Y_ij of fiber-reinforced composites. In the fiber direction, d₁₁ reflects the brittle nature of fiber-dominated fractures; d₁₁ is set to be 0 at the initial stage, and a sudden jump to 1 takes place. The transverse and shear damages exhibit progressive accumulation; d₂₂ and d₁₂ are represented as a linear equation, polynomial form or other expressions of the thermodynamic forces. In general, transverse stresses and shear stresses induce matrix damages and fiber/matrix interfacial damages, which in turn results in transverse and

shear modulus reduction. Thus, the transverse and shear components of thermodynamic force and damage parameters should be coupled. The following coupling parameters (b₂ and b₃) are defined to account for this coupling effect:

Y is referred to as an equivalent thermodynamic force. The undamaged elastic properties and the damage curves (d₁₁-Y₁₁, and d₁₂-Y, d₂₂-Y) are identified from the tensile tests of the laminated composites, as explained in the next section. In addition, nonlinear elastic parameters in the fiber direction and plastic parameters are also to be determined [11] .

Ladevèze and Le Dantec [11] proposed to use [0/90]_mS, [45/−45]_mS, and [67.5/− 67.5]_mS laminates to identify the elastic, non-linear, and damage parameters. The overall longitudinal stress, σ_L, and the longitudinal and transverse strains, ε_L and ε_T, of three kinds of laminates are obtained from the monotonic and cyclic tensile tests. Elastic properties can be easily obtained based on initial slopes in stress-strain curves of [0/90]_mS, [45/−45]_mS, and [67.5/−67.5]_mS laminates [11] . This section emphasizes the identification process of damage parameters. Let the in-plane stiffness matrix of an undamaged unidirectional ply have the following form:

In the case of [0/90]_mS laminates in unidirectional tension, local stress and strain of 0-degree ply can be expressed in terms of overall stress and strain by

For the angle-ply laminates, [θ/−θ]_mS under uniaxial tensile loading, local transverse and shear stresses/strains in each ply is calculated by

where the following equations hold:

Specifically, in the case of [45/−45]_mS laminates, the following simple equations are derived for shear stress-strain relationship:

To identify the damage evolution curves, σ₁₂-γ₁₂ curves of [45/−45_]mS laminates and σ₁₂-γ₁₂ and σ₂₂-ε₂₂ curves of [67.5/−67.5]_mS laminates using Equations (8)-(11). Note that Equations (8)-(10) are affected by variations of elastic properties owing to intralaminar damage accumulation. We neglect the damage-induced variations and Equations (8)-(9) are used for identification of damage parameters as suggested by Ladevèze and Le Dantec [11] .

Cyclic tensile tests provide the relationship between the damage parameters (i.e. stiffness slope reduction) and the corresponding thermodynamic forces at maximum stress in the cycles using Equation (3). Three damage curves, d₁₂-Y₁₂ from [45/−45]_mS laminates, d₁₂-Y₁₂ from [67.5/−67.5]_mS laminates, and d₂₂-Y₂₂ from [67.5/−67.5]_mS laminates, are obtained from the experimental curves. The coupling parameters, b₂ and b₃, are determined such that three curves (d_ij-Y curves) are collapsed into a single master curve considering Equations (4) and (5). The fitted damage curve (i.e. d₁₂ = f(Y)) and coupling parameters are used for damage simulation in the Ladevèze model.

In the previous section, transverse stress and strain in each ply of [45/−45]_ms laminates are neglected. Actually, when θ is equal to 45degree, Equation (8) is expressed as

Thus, in the case of laminates made of carbon fiber-reinforced unidirectional plies, σ₂₂ and ε₂₂ can be approximately regarded as zero. However, transverse stress and strain exist, and these components are possibly taken into account in some cases (e.g. glass fiber-reinforced plastics). The present study investigates this effect on the identification of damage parameters. In the previous method (denoted as Case-A), three damage curves (d₁₂-Y₁₂ from [45/−45]_mS laminates, d₁₂-Y₁₂ from [67.5/−67.5]_mS laminates, and d₂₂-Y₂₂ from [67.5/−67.5]_mS) are utilized for the identification. If we consider Equation (12), Y₂₂ is also taken into account for [45/−45]_mS laminates in uniaxial tension, and one additional damage curve (i.e. d₂₂-Y₂₂ from [45/−45]_mS) is obtained. We need to consider the modified processes to determine the coupling parameters, b₂ and b₃, by finding a single master curve based on damage curves (d_ij-Y curves), which are discussed in the following sections.

The present study focuses on the damage curves. [0/90]_2S, [45/−45]_2S, and [67.5/67.5]_2S specimens were prepared using unidirectional glass fibers and epoxy matrix. The specimens are 120 mm in length (excluding the clamp area), 25 mm in width, and about 4 mm in thickness. Back-to-back strain gauges were attached in the longitudinal and transverse directions to the specimens to acquire ε_L and ε_T. First, quasi-static monotonic tensile tests of the three laminates were conducted to obtain the elastic parameters and the suggested load levels for the cyclic tension tests. Then, cyclic tension tests of [45/−45]_2S, and [67.5/67.5]_2S specimens were carried out to derive the damage parameters. All tensile tests were performed in reference to JIS K7161.

Typical stress-strain curves obtained by quasi-static monotonic tensile tests of three laminates are presented in Figure 2. The elastic parameters are then evaluated following the previous study [11] , and summarized in Table 1. Cyclic tensile results of [45/−45]_2S, and [67.5/67.5]_2S specimens were converted to the stress-strain relationships in the local direction using Equations (8)-(12). A typical in-plane shear stress-shear strain curve obtained by cyclic tests of [45/−45]_2S laminates is shown in Figure 3. The black solid lines indicate the apparent shear moduli of damaged laminates, from which we can evaluate the damage parameters d₁₂ (as defined in Equation (2)) as a function of the applied maximum stress (or the corresponding thermodynamic force) during each cycle. The d₁₂-Y₁₂

curve obtained from [45/−45]_2S laminates is presented in Figure 4. Similarly, other damage curves were obtained based on the cyclic stress-strain curves.

As explained in Section 2.1, transverse and shear damage curves are coupled and expressed by equivalent thermodynamic forces with use of coupling parameters defined in Equations (4) and (5). The coupling parameters, b₂ and b₃, are determined such that three curves (d₁₂-Y from [45/−45]_2S laminates, d₁₂-Y from [67.5/−67.5]_2S laminates, and d₂₂-Y from [67.5/−67.5]_2S) are collapsed into a single master curve based on the least-square method. Note that d₂₂ and Y₂₂ are neglected for the case of [45/−45]_2S laminates in the original method [11] . This is called Case-A in the present study.

If we consider Equation (12), Y₂₂ is also taken into account for [45/−45]_2S laminates in uniaxial tension. To investigate the transverse stress effects of [45/−45]_2S laminates, the following two methods are introduced in the present study. The Case-B includes Y₂₂ when the d₁₂-Y curve is obtained from [45/−45]_2S laminates, and coupling parameters are determined based on the three damage curves (d₁₂-Y from [45/−45]_2S laminates, d₁₂-Y from [67.5/−67.5]_2S laminates, and d₂₂-Y from [67.5/−67.5]_2S). The Case-C takes d₂₂ and Y₂₂ from [45/−45]_2S laminates into account, and four damage curves (d₁₂-Y from [45/−45]_2S laminates, d₂₂-Y from [45/−45]_2S, d₁₂-Y from [67.5/−67.5]_2S laminates, and d₂₂-Y from [67.5/−67.5]_2S) are utilized to identify the coupling parameters and damage master curve. Table 2 summarizes and compares the three cases investigated in the present study when the coupling parameters and the damage master curve are identified.

The damage master curves (d₂₂(=b₃d₁₂)-Y) obtained by three methods are presented in Figure 5. The fitted master curves are compared in Figure 6 for

three cases, and expressed as a function of Y, as seen in Table 3. It is confirmed that identified damage master curve based on the experimental data, which is incorporated into the damage simulation, depends on the identification methods as compared in Table 2. This infers the dependency of parameter identification methods on the damage simulation results of laminated composites. It is noted that identified b₃ is almost independent of identification methods while b₂ depends on the methods. b₃ reflects the influence of damages on the reduction of transverse and shear elastic modulus, which should be determined in a damage mechanics sense, and therefore, it does not depend on the consideration of transverse stress effects. On the other hand, b₂ accounts for the coupling degree of transverse and shear stresses which drive damage accumulations. It is justified that consideration of transverse stress effects of [45/−45]_2S laminates influences the identified values of b₂.

Finally, the degradation of longitudinal stiffness (i.e. apparent modulus in loading direction, E_L) of [45/−45]_2S laminates under uniaxial tensile loading is predicted using the identified damage parameters. Stiffness degradation is plotted as a function of applied stress in Figure 7, compared with experimental results. The predicted curves using Case-B identification (considering transverse stresses of [45/−45]_2S laminates) fit well with the experimental results. This demonstrates the importance of consideration of transverse stress of [45/−45]_2S laminates during the identification process. It is noted that the prediction based on Case-C identification overestimates the stiffness. This might results from the difficulty in obtaining the d₂₂-Y curve from [45/−45]_2S (because d₂₂ in [45/−45]_2S specimens is small during the uniaxial tensile loading), and the identified damage curve somewhat loses the accuracy, which is further to be investigated.