A new damage-based failure criterion for nonlinear behavior of fibrous composite materials

Abstract

In the present paper, a novel combined damage-based failure criterion is being proposed for predicting failure stresses in unidirectional fibrous composite laminas or laminates having a nonlinear material behavior. The present model incorporates the effect of a quantitative damage factor on the final stresses at failure. This is achieved through a new term called the quantitative directional damage-index (QDD-I) which assesses the contribution and effectiveness of damage in each principal material direction on the present failure criterion. From the QDD-I, it is proved that the principal material-direction with a linear or nonlinear stress-strain behavior showed a quantitative damage response on the proposed failure criterion. In a composite lamina, the contribution of fiber-damage and matrix transverse-damage are proved to have minor effects on the failure criterion, while in-plane shear-damage has the major effect. In order to verify the suitability and applicability of the criterion, results are tested using various theoretical and experimental data available from the literature. Furthermore, the model is compared with other failure criteria under both uniaxial and biaxial loading cases from a worldwide comparison, which showed reasonable accuracy and good agreement. Three types of fibrous composite materials are used; Graphite/Epoxy 4617/Modmore-II, Carbon/Epoxy AS4/3501-6, and Boron/Epoxy Narmco 5505.

Keywords

Damage-model failure-criterion quantitative-directional damage-index nonlinear composite-material behavior

Introduction

The fibrous composite laminate is widely used in structural engineering applications, such as for strengthening and retrofitting of structures. The massive demand for fibrous composite materials is mainly due to their high stiffness/density ratio, lightweight, and outstanding mechanical properties. However, most fibrous composite materials exhibit nonlinear behavior, at least in one or two of the principal material directions. This is one reason that encourages the researchers to examine, predict, and develop different models and methods to attain the failure mechanisms of a fibrous composite lamina. Furthermore, as a consequence of the increased stresses in the lamina, damage also developed in a composite material due to voids, debonding between fiber and matrix, matrix micro-cracks, and other deteriorations. However, enrolling the damaging effect on the failure criteria was and still a massive research area for the researchers.

Over the last decades, the failure accounting for the nonlinear behavior of composite materials has been a wide research area due to the complexity and non-uniformity in the behavior of these materials. And in order to reach a comprehensive model for predicting the failure-stress in a lamina made of a composite material, researchers examine various terms to achieve such purpose.

Tsai-Wu failure criterion (1971) is one of the well-known criteria for the researchers. It is extremely popular, and it was used for composite materials and gave reasonable results. However, it could not determine the different modes of failure for a composite material, which made the theory limited for a specific application.

Failure criteria based on the total strain-energy density (TSED) were developed starting in 1976. The basic assumption of these criteria was that failure occurs in a lamina when the summation of the ratio of strain-energies in each principal material direction (longitudinal, transverse, and shear) at a specific load to the corresponding maximum strain-energies of that material equals unity (Sandhu, 1976; Sandhu et al., 1982; Swanson et al., 1987).

A new material model for linear and nonlinear behavior of fibrous composite materials was developed by Abu-Farsakh (1989). In the model, the secant mechanical property was presented in in terms of the plastic strain-energy density (PSED) instead of the TSED as in the Jones model (1975). Moreover, both Sandhu model (1976) and Abu-Farsakh model (1989) provided valuable observations by using the TSED, where the fundamental difference between the two models was in selecting the area under the stress-strain diagram. In Sandhu (1976) the TSED was equal to the whole area under the stress-strain curve. In contrast, in Abu-Farsakh (1989), the area under the secant modulus line (triangular area) was considered as the TSED.

Abu-Farsakh and Abdel-Jawad (1994) developed a three-dimensional failure criterion for nonlinear composite materials to determine the failure envelope. It was based on TSED material model using the secant modulus approach (Abu-Farsakh, 1989). The model was comprehensive and compared well with Cole and Pipes (1974) experimental failure stresses and corresponding failure strains in the composite lamina.

Numerical and finite element models based on TSED (whole area under the stress-strain curve) were developed and expanded to predict the fatigue life of polymer composite lamina and laminate. The models consider the damage model from Sandhu et al. (1982) and the material property degradation in the polymer composite lamina. Furthermore, the models are capable of predicting the fatigue-damage under static and cycling loading. (Samareh-Mousavi et al., 2019; Shokrieh and Taheri-Behrooz, 2006, 2010).

Daniel et al. (2009) investigated experimentally the inter-laminar failure of Carbon/Epoxy fibrous composite laminates under a multi-axial state of stress (through-thickness and in-plane tensile, compressive, and shear) at various fiber-orientations. Furthermore, the authors developed a new failure criterion, which showed excellent agreement with the experimental results. Results using the model showed good agreement in the off-axis compressive behavior, especially, for combined shear and transverse compression parallel to the fiber-direction case.

Another new three-dimensional model (Camanho et al., 2015) to determine the failure- envelope of a fibrous composite material. The model has an invariant quadratic formulation based on structural tensors, and it was valid for both quasi-static and dynamic loading cases. However, it was verified using many experimental results from the literature such as; Swanson et al. (1987) and Daniel et al. (2009).

Extended to the Tsai-Wu failure-criterion, Li et al. (2017) rationalized the determination of the interactive coefficient (F₁₂). In the Tsai-Wu failure-criterion, the factor was obtained from the experimental results, while Li et al. (2017) estimated the factor based on logical circumstances to eject the empiricism, as far as it was applicable to unidirectional composites.

Daniel et al. (2018) developed a new yield and failure theory for the linear and nonlinear behavior of fibrous composite materials under both static and dynamic loads. They derived the hypothesis to predict the yield and failure under a multi-axial state of stress.

Abu-Farsakh and Asfa (2018) proposed a macro-mechanical damage model for accounting and evaluating damage in the load-direction (directional-damage) and in the principal material directions. The model was extended for the Ghazi-Farid model (Abu-Farsakh et al., 1999) with more generalization accounting for the off-axis loading cases. Since, Ghazi-Farid model was derived to predict damage in the principal material directions from uniaxial loading cases only. They also proposed a quantitative-damage (QD) factor, which helped in the assessment of the quantity of damage that occurred in a composite material lamina subjected to a specific load.

Moreover, Abu-Farsakh and Asfa (2020) presented a unified-damage model (UDM). The model can determine the QD-factor in a certain composite material as being independent of the fiber-orientation angle and case of loading. Also, they obtained a relation between Abu-Farsakh (1989) material model and the QD-factor utilizing the Ghazi-Asfa damage model from Abu-Farsakh and Asfa (2018).

On the other hand, Okabe et al. (2018) considered a continuum-damage approach as the main factor including the transverse cracks, for predicting the stiffness reduction in the composite material laminate. The model was shown to be effective at early stages of damage formation due to transverse cracks in the composite laminas.

A micro-mechanical damage model was developed by Abu-Farsakh and Al-Jarrah (2018) to predict the matrix cracking effect in fibrous composite laminas. The model was comprehensive for predicting damage through a new term called volumetric crack-density (VCD) that occurred in the fibrous composite lamina or laminates. The model was compared with various damage models available in the literature, where the results showed a good agreement in damage prediction. Also, Lou et al. (2020) developed a micro-mechanical damage model to investigate the stress behavior of composite laminates under impact loading. The model considered stress amplification factors to calculate the micro-stress in fiber and matrix to obtain the structural response, residual compressive strength and failure mechanism.

A comprehensive review paper on the nonlinear mechanical behavior of fibrous composite material was published by Fallahi et al. (2020). The paper reviewed the unloading characteristics, the tension-compression asymmetry, and the interaction between stress components of the fibrous composite materials. Also, the effects of environmental factors and viscous behavior on mechanical properties are briefly reviewed. Furthermore, four models were reviewed (such as elastic-plastic models and Damage-Plasticity models) and categorized according to the complexity of the models in predicting the stress-strain behavior of the fibrous composite materials.

In a recent work, Khayyam et al. (2020) determined the ultimate failure strength of composite pipes made of woven composite laminas (with E-glass fabrics) by considering the progressive-damage modeling. The finite element analysis was carried out through using a representative volume-element model to form the equivalent stiffness matrix due to material-degradation, which was implemented utilizing the ABAQUS computer-software. The model compared well with both the experimental results and the finite element model.

Scope and objective

As a consequence of increased stresses in a lamina, damage will develop due to voids, debonding between fiber and matrix, matrix micro-cracks, and other deteriorations. Most of researchers in the literature mainly studied the damaging effect aside from the failure strength of a lamina. Due to the lack of combined damage-failure models in the literature, the present research is devoted to obtaining such a model. In order to achieve this objective, a certain damage-parameter (known as quantitative directional damage-index (QDD-I)) is incorporated in a modified failure-criterion. Thereafter, to study its effect on failure stresses and failure envelops of fibrous-composite laminas.

The failure criterion

Description of failure criterion

The three-dimensional failure criterion model proposed by Abu-Farsakh and Abdel-Jawad (1994) was the starting point to develop a new model to predict the failure stress in the fibrous composite material by considering the effect of the QD-factor. As aforementioned, the Abu-Farsakh-Jawad (AF-AJ) (1994) model depends on the TSED approach, and it was represented as;

\sum {(\frac{U_{s i}}{U_{sim}})}_{c o r t} + \sum {(\frac{U_{sij}}{U_{sijm}})}_{c o r t} = 1.0

(1)

where c or t is either compression or tension, respectively.

U_{s i} = \frac{{σ_{i}}^{2}}{2 M_{i s}}

(2)

U_{sij} = \frac{{τ_{i j}}^{2}}{2 M_{ijs}}

(3)

U_{sim} = \frac{{σ_{i m}}^{2}}{2 M_{i s}}

(4)

U_{sijm} = \frac{{τ_{ijm}}^{2}}{2 M_{ijs}}

(5)

The $U_{s i}$ (i = 1, 2, 3) terms represent the quantities of the total equivalent extensional strain-energy densities in the principal material directions 1, 2, and 3 (triangular area under the secant- line of stress-strain curve) of the fibrous composite material lamina (Figure 1). The $U_{sij}$ (i,j = 1, 2, 3, i ≠ j) terms represent the quantities of the total equivalent shear strain-energy densities. The subscript m in the terms $U_{sim}$ and $U_{sijm}$ indicates the corresponding maximum value of the TSED.

Figure 1. Typical nonlinear stress-strain curve for the i-th mechanical property showing all the energy terms.

Considering the in-plane stress state (1–2 plane) case, equation (1) will be represented as:

{(\frac{U_{s 1}}{U_{s 1 m}})}_{c o r t} + {(\frac{U_{s 2}}{U_{s 2 m}})}_{c o r t} + {(\frac{U_{s 12}}{U_{s 12 m}})}_{c o r t} = 1.0

(6)

The principal material directions in equation (6) represent the fiber direction (direction-1) and transverse direction (direction-2) and in-plane shear (plane 1–2). It should be noted that, for an orthotropic material the principal-stress directions are different than stresses in the principal material directions (1 and 2).

Determination of strain-energy densities

The quantities U_sim and U_sijm which represent the maximum TSED are determined experimentally through the uniaxial test (tension, compression, shear) of the composite lamina. If the fibrous composite material has the same behavior in tension and compression, then one set of data will be used for all cases.

The quantities U_si and U_sij which represent TSED at any position on the stress-strain curve. In a general in-plane stress state test (σ_x, σ_y, and τ_xy) for a fibrous composite lamina having a fiber-orientation angle (θ), stresses in the principal material directions σ₁, σ₂, and τ₁₂ are obtained using the following transformation relation;

[\begin{matrix} σ_{1} \\ σ_{2} \\ τ_{12} \end{matrix}] = [\begin{matrix} \cos^{2} θ & \sin^{2} θ & 2 \sin θ \cos θ \\ \sin^{2} θ & \cos^{2} θ & - 2 \sin θ \cos θ \\ - \sin θ \cos θ & \sin θ \cos θ & {(\cos}^{2} θ - \sin^{2} θ) \end{matrix}] [\begin{matrix} σ_{x} \\ σ_{y} \\ τ_{x y} \end{matrix}]

(7)

Where, x and y indicate the body geometric axes.

The M_is -term in equations (2) to (4) indicate the secant mechanical property; E_1s, E_2s, or G_12s. The value of the secant modulus is determined by using Abu-Farsakh (1989) model which was developed to account for the nonlinear material behavior and was given as:

\frac{M_{i s}}{M_{i o}} = (1 - B_{i} {U_{p i}}^{C_{i}} + D_{i} U_{p i})

(8)

where B_i, C_i, and D_i are the material property constants.

The B_i, C_i, and D_i constants are determined from the experimental stress-strain curves for the principal material directions 1, 2, and 12. The corresponding constants and the material mechanical properties for Boron/Epoxy Narmco 5505 (B-E), Carbon/Epoxy AS4/3501-6 (C-E), and Graphite/Epoxy 4617/Modmore-II (G-E) are listed in Tables 6 to 8 in the Appendix 1.

The U_pi -term in equation (8) indicates the plastic strain energy density (PSED) (Figure 1), which is determined using the relation;

U_{p i} = U_{s i} - U_{e i} i = 1, 2, or 12

(9)

The U_ei -term in equation (9) indicates the elastic strain-energy density.

U_{e i} = \frac{{σ_{i}}^{2}}{2 M_{i o}}

(10)

Where, the M_io indicates the initial modulus in the principal material direction (i).

Then, by rearranging equations (2), (9) and 10 the PSED (U_pi) can be represented as follows;

U_{p i} = \frac{{σ_{i}}^{2}}{2} [\frac{M_{i o} - M_{i s}}{M_{i o} M_{i s}}]

(11)

Damage model

The unified damage model (UDM) from Abu-Farsakh and Asfa (2020) is modified and then incorporated in the failure criterion developed in this research. The UDM determines the QD-factor ( $ϕ^{*}$ ) in the fibrous composite lamina due to stiffness-degradation (M_is/M_io). Furthermore, the UDM can predict damage independent of fiber-orientation angle and loading case. The relation between QD-factor and the plastic-strain-energy density is given as:

{Q D = ϕ}^{*} = 1 - \sqrt{1 - B_{d} {{(U}_{p})}^{C_{d}} + D_{d} U_{p}}

(12)

Or, alternatively, the modified damage model can be expressed as:

{(1 - ϕ^{*})}^{2} = (1 - B_{d} {U_{p}}^{C_{d}} + D_{d} U_{p})

(13)

In the present work, as a modification of the UD-model, the Quantitative-Directional-Damage (QDD) factor is determined through the equation below:

{QDD = (1 - {ϕ^{*}}_{i})}^{2} = (1 - B_{d} {U_{p}}^{C_{d}} + D_{d} U_{p}), U_{p} \leq U_{pim}

(14)

The material damage constants B_d, C_d, and D_d are different from those used in equation (8); the d-subscript indicates damage. The constants in equation (13) are determined by plotting ${(1 - ϕ^{*})}^{2}$ versus the PSED (U_p) for different fiber-orientation angles. Then the least-square fitting method is used to obtain the corresponding damage constants, as indicated in Table 1.

Table 1.

Damage constants of the UD-model for B-E, C-E, and G-E laminas (Abu-Farsakh et al., 2020).

Material type	Unified damage model constants
Material type	B_d	C_d	D_d
B-E	0.735196	0.089745	0.001171
C-E	0.496790	0.206952	0.00243
G-E	0.422605	0.168749	−0.00678

The QDD-factor is different from the QD-factor, in the sense that the QDD-factor does not depend on the total PSED of a lamina. Such that, the total PSED in equation (14) (U_p) induced in a lamina should not exceed the maximum plastic strain-energy (U_pim) for the same principal material direction. In other words, ${ϕ^{*}}_{i}$ has the same form as function $ϕ^{*}$ with a maximum substituted value of U_p not exceeding U_pim in the principal material direction-i.

The values of QD-factor ( $ϕ^{*}$ ), which are used in the present failure criterion, are illustrated in Table 2 for the B-E, C-E, and G-E fibrous composite laminas. The $ϕ^{*} - values$ are exhibited in Table 2 for the three composite materials.

Table 2.

The values of QD-Factor ( $ϕ^{*}$ ) for B-E, C-E, and G-E laminas.

Material type	Damage ( $ϕ^{*}$ )
	θ (degree)
	0°	15°	30°	45°	60°	75°	90°
B-E	0	0.906	0.796	0.679	0.590	0.539	0.498
C-E	0	0.892	0.670	0.491	0.344	0.189	0
G-E	0	0.438	0.344	0.269	0.207	0.143	0

Present failure criterion

In order to improve the AF-AJ (1994) failure-criterion by relating it to damage, the QD-factor ( $ϕ^{*}$ ) is incorporated into the failure-criterion to produce a new damage-based failure criterion. The derivation of this model starts with enrolling a new Quantitative Directional Damage-Index (QDD-I) (d_i) into the failure criterion. Accordingly, the modified form of the failure-criterion will be represented as:

{(\frac{U_{s 1}}{U_{s 1 m}} d_{1})}_{c o r t} + {(\frac{U_{s 2}}{U_{s 2 m}} d_{2})}_{c o r t} + {(\frac{U_{s 12}}{U_{s 12 m}} d_{12})}_{c o r t} = 1.0

(15)

By combining secant modulus of the material model from Abu-Farsakh (1989) (equation (8)) with the modified QDD model (equation (14)), one obtains:

E_{i s} = {E_{i o} (1 - {ϕ^{*}}_{i})}^{2}, i = 1, 2, 12

(16)

The relation between the QD-factor and the QDD-factor can be represented as:

{{(1 - {ϕ^{*}}_{i})}^{2} = d}_{i} {(1 - ϕ^{*})}^{2}

(17)

From equation (16), the new QDD-I (d_i) will become as:

d_{i} = \frac{{(1 - {ϕ^{*}}_{i})}^{2}}{{(1 - ϕ^{*})}^{2}}

(18)

The new QDD-I (d_i) factor represents the contribution of damage in each principal material direction on the failure criterion as indicated in equation (15). Figures 2 to 4 represent the inverse-value of QDD-I for B-E, C-E, and G-E fibrous composite materials, respectively, in the principal material directions 1, 2, and 12.

Figure 2.

The QDD-I for B-E composite material.

Figure 3.

The QDD-I for C-E composite material.

Figure 4.

The QDD-I for G-E composite material.

In order to illustrate the significance of the QDD-I term, it has been shown that from Figures 2 to 4 that the elastic linear stress-strain behavior direction has a damage quantity (d_i) that affects the failure-stress of a lamina. Whilst in the literature it was stated that there was no damage resulting from the elastic linear stress-strain behavior. The value of the QDD-I is set between 0 and 1. The value of one (1) indicates that the damage in that direction has a dominant effect on the failure-criterion, while on the other hand the value of zero (0) indicates no effect. With decreasing the QDD-I value, the impact of damage will be decreased on the failure-criterion, as illustrated in Figures 2 to 4. Thus, since shear has a high non-linearity stress-strain behavior, the QDD-I value for shear is 1 in all the three composite materials. Directions 1 and 2 have a linear elastic stress-strain behavior in both C-E and G-E materials, hence, that the two curves have the same behavior.

For a further grasp of the QDD-I terms, the contribution and effect on the failure-criterion (equation (15)) for the three materials for B-E, C-E, and G-E are presented in Figures 5 and 6 as a percentage using the bar-form for the angles $15 °$ and $45 °$ , respectively. It can be noticed from Figure 5 the effect of shear-term on the failure criterion had a 100% impact on all the composite materials, and any change in the shear stresses and properties will affect the final failure stress. Furthermore, according to Figure 5, the contribution of damage in direction-1 for B-E material is less than 1%, while it increases for the $45 °$ -angle up to 10%, which proves that there is a damage influence on the linear elastic stress-strain behavior. Moreover, the effect in the two other materials is around 10% for G-E and 3% for C-E at the angle of $15 °$ .

Figure 5.

The percentage of the quantitative directional damage-index for θ = $15 °$ .

Figure 6.

The percentage of the quantitative directional damage-index for θ = $45 °$ .

By substituting equation (18) into equation (15), the QDD-factor will be replaced with QD-factor, then the relation can be expressed as:

E_{i s} = E_{i o} d_{i} {(1 - ϕ^{*})}^{2}

(19)

The TSED ( $U_{s i}$ ) in the longitudinal direction (direction-1) after substituting equation (18) can be presented as:

U_{s 1} = \frac{{σ_{1}}^{2}}{2 E_{1 o} d_{1} {(1 - ϕ^{*})}^{2}}

(20a)

Similarly, the same procedure can be applied to determine U_s2 and U_s12;

U_{s 2} = \frac{{σ_{2}}^{2}}{2 E_{2 o} d_{2} {(1 - ϕ^{*})}^{2}}

(20b)

U_{s 12} = \frac{{τ_{12}}^{2}}{2 G_{12 o} d_{12} {(1 - ϕ^{*})}^{2}}

(20c)

Consequently, the PSED for directions 1, 2, and 12 will be represented as:

U_{1 p} = \frac{{σ_{1}}^{2}}{2 E_{1 o}} [\frac{2 ϕ^{*} - ϕ^{* 2}}{{(1 - ϕ^{*})}^{2}}]

(21a)

U_{2 p} = \frac{{σ_{2}}^{2}}{2 E_{2 o}} [\frac{2 ϕ^{*} - ϕ^{* 2}}{{(1 - ϕ^{*})}^{2}}]

(21b)

U_{12 p} = \frac{{τ_{12}}^{2}}{2 G_{12 o}} [\frac{2 ϕ^{*} - ϕ^{* 2}}{{(1 - ϕ^{*})}^{2}}]

(21c)

Finally, the present failure criterion in terms of the quantitative damage factor (QD) will appear as:

{(\frac{{σ_{1}}^{2}}{{2 E_{1 o} U}_{s 1 m}})}_{c o r t} + {(\frac{{σ_{2}}^{2}}{{2 E_{2 o} U s}_{s 2 m}})}_{c o r t} + {(\frac{{τ_{12}}^{2}}{2 G_{12 o} U_{s 12 m}})}_{c o r t} = {(1 - ϕ^{*})}^{2}

(22)

where c or t is either compression or tension, respectively.

Methodology

The present model is valid for a generally orthotropic composite material (whether it has linear or nonlinear behavior) subject to an in-plane stress-state (σ_x, σ_y, and τ_xy) as a case of loading. At a specific shear stress (τ_xy) and certain fiber-orientation angle (θ), the failure-envelope can be obtained according to the following procedure;

From the uniaxial experimental data for the stress-strain curves (σ₁-ε₁, σ₂-ε₂, and τ₁₂-γ₁₂), one can determine the maximum corresponding TSED terms (U_s1m, U_s2m, U_s12m). Furthermore, the corresponding maximum PSED in terms of QD-factor will be determined, accordingly, as:

{{(U}_{p 1})}_{m} = \frac{({σ_{1 m})}^{2}}{2 E_{10}} [\frac{2 ϕ^{*} - ϕ^{* 2}}{{(1 - ϕ^{*})}^{2}}]

(23a)

{{(U}_{p 2})}_{m} = \frac{{{(σ}_{2 m})}^{2}}{2 E_{20}} [\frac{2 ϕ^{*} - ϕ^{* 2}}{{(1 - ϕ^{*})}^{2}}]

(23b)

{{(U}_{p 12})}_{m} = \frac{({τ_{12 m})}^{2}}{2 G_{12 o}} [\frac{2 ϕ^{*} - ϕ^{* 2}}{{(1 - ϕ^{*})}^{2}}]

(23c)

Express the stress in y-direction (σ_y) as a ratio (R_y) to the stress in x-direction (σ_x) i.e.

σ_{y} = R_{y} σ_{x}

(24)

To determine the stress in the x-direction (σ_x) which satisfies equation (22) (at a certain shear stress τ_xy = τ, and using equation (24) one obtains:

σ_{x} = \frac{(- b + d)}{2 a}

(25)

where

b = (\frac{2 m_{1} n_{1}}{F_{1}}) + (\frac{2 m_{2} n_{2}}{F_{2}}) + (\frac{2 m_{12} n_{12}}{F_{12}})

m_{1} = \cos^{2} θ + R_{y} \sin^{2} θ

m_{2} = \sin^{2} θ + R_{y} \cos^{2} θ

m_{12} = \cos θ \sin θ (R_{y} - 1)

n_{1} = 2 τ \cos θ \sin θ

n_{2} {= - n}_{1}

n_{12} = τ (\cos^{2} θ - \sin^{2} θ)

F_{1} = 2 E_{1 o} {(1 - ϕ^{*})}^{2} U_{s 1 m}

F_{2} = 2 E_{2 o} {(1 - ϕ^{*})}^{2} U_{s 2 m}

F_{12} = 2 G_{12 o} {(1 - ϕ^{*})}^{2} U_{s 12 m}

The discriminant (d) in equation (25) obtained by

d = \sqrt{b^{2} - 4 a c}

(26)

where

a = (\frac{{m_{1}}^{2}}{F_{1}}) + (\frac{{m_{2}}^{2}}{F_{2}}) + (\frac{{m_{12}}^{2}}{F_{12}})

c = (\frac{{n_{1}}^{2}}{F_{1}} + \frac{{n_{2}}^{2}}{F_{2}} + \frac{{n_{12}}^{2}}{F_{12}})

4. After determining σ_x, σ_y, and τ_xy, we can find stresses in the principal material directions σ₁, σ₂, and τ₁₂ by using equation (7).

5. Determine the QD-factor ( $ϕ^{*}$ ). In this step, one determines the PSED (U_p) from the material mechanical properties, as given in Tables 6 to 8 in the Appendix 1. Then, substitute U_p in equation (13) using the corresponding material damage constant (B_d, C_d, and D_d) from Table 1.

6. Determine the PSED in each principal material direction (U_p1, U_p2, U_p12) using equation (23).

7. Determine the secant modulus in each principal material direction (E_1s, E_2s, or G_12s) from equation (8).

8. Substitute the new value of the secant modulus (E_1s, E_2s, or G_12s) from step 7 into step 3, in order to determine a new value for σ_x.

9. Repeat steps 3 to 8 in order to satisfy the convergence of failure-criterion depending on the PSED (U_p) of the system. In a mathematical form, the convergence criterion can be represented as:

|\frac{U_{p n} - U_{p o}}{U_{p n}}| \leq ϵ

(27)

Where, the subscripts o and n indicate old and new values, respectively. The term- $ϵ$ indicates a very small value and may be taken as $10^{- 8}$ . Moreover, the PSED (U_p) is equal to the summation of the directional PSED in each principal material direction (U_p1, U_p2, U_p12).

10. Now, at a certain shear-stress (τ_xy = τ) and from σ_x and σ_y that were obtained from the previous steps, we have a point on the failure-envelope. To determine other points on the envelope, change the R_y -ratio.

Model validation and discussion

In order to validate the current failure-criterion in predicting the failure stresses, several experimental and theoretical data of uniaxial off-axis and biaxial loading on a lamina are obtained from the literature. Furthermore, a unidirectional laminate with various fiber-orientation angles subject to uniaxial tension (or compression) is, examined using the new model.

Uniaxial off-axis loading

Two materials are considered in this part; the first material is G-E, and the second one is B-E.

The material mechanical properties of the Graphite/Epoxy 4617/Modmore-II (G-E) are given in Table 7 of Appendix 1. The material exhibits the same linear stress-strain behavior and the same failure stresses in both tension and compression in the principal material direction-1 and -2; failure stresses are 204 ksi for direction-1 and 9.05 ksi for direction-2. However, for the in-plane material direction-12, it has a nonlinear stress-strain behavior with a failure shear-stress equals to 11.5 ksi.

As mentioned before, the failure stresses produced from the present model are compared with those obtained from various theories and experimental ones. Table 3 represents the predicted failure stresses from the present model, Tsai (1968) and Hill (1948) and AF-AJ (1994), in addition to the experimental values from Cole and Pipes (1974) tests for the different fiber-orientation angles ( $θ = 0 °, 15 °, 30 °, 45 °, 60 °, 75 °, and 90 °$ ).

Table 3.

Comparison between experimental results and various failure criteria for G-E for the case of uniaxial off-axis tension load.

θ (degree)	Failure stress (ksi)
θ (degree)	Experimental (1974)	Tsai (1968) and Hill (1948)	AF-AJ (1994)	Present
0	204	204	204	204
15	48.4	43.79	44.46	42.92
30	22.5	22.04	22.46	21.43
45	14.4	14.59	14.73	14.23
60	–	11.14	11.17	11.07
75	–	9.53	9.53	9.52
90	9.05	9.05	9.05	9.05

According to Table 3, there is an excellent agreement between the present study and both Tsai-Hill and AF-AJ failure criteria. The absolute percentage-error between the failure criteria starts with 0% and maximum at the angle of $30 °$ with a value of 4.59%. As well as with the experimental results, failure stresses are very close to those obtained from the present study, except at the angle of $15 °$ . The variation at the angle $15 °$ between the experimental results and all other failure criteria is mainly produced due to the differences in the nonlinear shear-stress relationship obtained from different resources. Furthermore, Figure 7 illustrates the close agreement between the failure stresses obtained from various theories and the experimental results.

Figure 7.

Comparison between experimental results and various failure criteria for G-E; uniaxial off-axis tension load.

In order to further verify the validity of the present failure criterion, a comparison with another experimental tensile off-axis loading from Tsai-Hahn (1980) is performed. As indicated in Figure 8, the present criterion, also, compares well with the other failure criteria from Serpieri (2005) and AF-AJ (1994). According to Figure 8, one can notice the validity of the present failure-criterion in predicting failure stresses at smaller angles less than $20 °$ .

Figure 8.

Comparison between experimental results and various failure criteria for G-E; uniaxial off-axis tension load.

For Boron/Epoxy-Narmco 5505 (B-E), the mechanical material properties are as illustrated in Table 8 in the Appendix 1. The B-E material has a linear stress-strain behavior for the material principal direction-1 only, with a tensile failure-stress equals to 194.5 ksi and 295 ksi for the compressive failure-stress. The principal material directions -2 and -12 exhibit nonlinear stress-strain behavior. The failure stresses are 7.8 ksi, 37.5 ksi, and 9.3 ksi for the tensile, compressive, and shear stresses, respectively.

From Table 4, there is an excellent agreement between the present study and both Tsai-Hill and AF-AJ failure criteria. The maximum absolute percentage-error is 12% occurred at the angle of $45 °$ between the present study and the AF-AJ (1994) failure-criterion. As well as, with the experimental results, the failure-stress results are approximately similar to those predicted using the present criterion. The differences at angles $15 °$ and $75 °$ between the experimental results and other considered failure criteria came from the effect of nonlinear transverse and in-plane shear stress-strain behaviors, see also Figure 9.

Table 4.

Comparison between experimental results and various failure-stress criteria for B-E for the case of uniaxial off-axis tension load.

θ (degree)	Failure stress (ksi)
θ (degree)	Experimental (1974)	Tsai (1968) and Hill (1948)	AF-AJ (1994)	Present
0	194.5	194.5	194.5	194.5
15	36	34.96	36.46	34.97
30	18.2	17.66	19.74	17.66
45	12.9	11.95	13.65	11.95
60	9.45	9.36	10.07	9.53
75	7.25	8.15	8.31	8.25
90	7.8	7.8	7.8	7.8

Figure 9.

Comparison between experimental results and various failure criteria for B-E; uniaxial off-axis tension load.

For a further validation of the present model, a comparison of Carbon/Epoxy AS4/3501-6 (C-E) laminate with unidirectional fiber-orientation angles has been considered, the mechanical material properties are illustrated in Table 6 of the Appendix 1. As shown in Figure 10, a comparison is carried out between the experimental results and various failure criteria for a unidirectional fibrous composite laminate subject to a uniaxial off-axis tension load. A good agreement between Tsai and Wu (1971), Daniel et al. (2009) and the present criterion is found. Results of Camanho et al. (2015) failure criteria showed an increased discrepancy from the other models, because of the inclusion of the fiber-misalignment angle as a material property in their model. Moreover, as illustrated in Figure 10, the Maximum-stress failure-criterion represented an upper-bound curve, while the Camanho model and experimental results showed a lower-bound curve. The purpose of the present comparison is to show the reliability and validity of the present model with respect to other available models.

Figure 10.

Comparison between experimental results and various failure criteria for C-E; uniaxial off-axis tension load.

Biaxial loading

At a certain angle and zero shear stress (θ = 0, τ_xy = 0), the failure envelope of B-E lamina under a biaxial stress state has been determined. A world-wide comparison between the present failure criterion and six different failure criteria, as illustrated in Figure 11, which showed that the present failure criterion compares well with the other failure criteria.

Figure 11.

World-wide comparison between failure-envelopes of various failure criteria for B-E.

In a further study, a biaxial failure-stress comparison between the present failure-criterion and experimental results from Cole and Pipes (1974) are represented in Table 5. The experiments were performed using tubes made of a unidirectional lamina of B-E fibrous composite material. Moreover, various stress ratios were tested in the experiment, as illustrated in Table 5. According to Table 5, there is an improvement in results of the present criterion compared to AF-AJ (1994) and with respect to the experimental ones.

Table 5.

Comparison between experimental results and various failure-stress criteria for B-E for the case of biaxial loading.

Nominal stress ratio σ₁: σ₂	Actual stress ratio σ₁: σ₂ (1974)	Experimental (1974)		AF-AJ (1994)		Present
Nominal stress ratio σ₁: σ₂	Actual stress ratio σ₁: σ₂ (1974)	σ₁ (ksi)	σ₂ (ksi)	σ₁ (ksi)	σ₂ (ksi)	σ₁ (ksi)	σ₂ (ksi)
1:1	1.11:1	7.37	6.62	8.35	7.53	7.79	7.02
3:1	2.7:1	19.78	7.22	20.25	7.50	19.75	7.31
4:1	4.38:1	29.00	6.10	29.85	7.46	29.08	7.27
6:1	6:1	39.40	6.57	44.26	7.38	43.02	7.17

Failure envelopes

In order to further investigate and validate the effect of damage-factor on the present failure- criterion, the failure envelopes for G-E were determined. The failure envelopes from the present study are compared with those of AF-AJ (1994) failure-criterion. At zero shear stress (τ_xy = 0) and different fiber-orientation angles (θ = $15 °, 30 °, 45 °$ , $and 60 °$ ), the failure envelopes are illustrated in Figures 12 to 14.

Figure 12. Failure envelopes for G-E at fiber-orientation angle θ = $15 °$ and τ_xy = 0.

Figure 13.

Failure envelopes for G-E at fiber-orientation angle θ = $30 °$ and τ_xy = 0.

Figure 14.

Failure envelopes for G-E at fiber-orientation angle θ = $45 °$ and τ_xy = 0.

As expected, all failure envelopes have a closed shape. The effect of damage is noticed from Figures 12 to 14 as a reduction in the strength of the fibrous composite material. Moreover, the reduction in the second quadrant in all envelopes was the highest, which is the case of compressive stress in the x-direction and tensile stress in the y-direction, while the three other quadrants have a slightly reduced strength.

Conclusions

A novel damage-based failure-criterion was developed. It incorporated a newly proposed quantum damage factor (QD) into an energy-based failure-criterion. Moreover, a new term called the quantitative directional damage-index (QDD-I) was introduced, which indicated the damage contribution and effectiveness of each principal material direction on the present failure-criterion. Nevertheless, it was proved that the linear elastic stress-strain behavior of a composite lamina had a minor damage impact on the failure- criterion.

The proposed model was proved to be applicable to fibrous composite laminas or laminates and compared well with the corresponding experimental results and other available failure criteria from literature. The new model can be used for generally orthotropic fibrous composite laminas, whatever the case of stress (uniaxial, biaxial, or general). The composite material behavior might be linear or nonlinear and, also, might have different mechanical properties in tension and compression.

The model exhibited reasonably failure stresses in all the considered cases and compared well utilizing a worldwide comparison for a biaxial-stress state for Boron/Epoxy composite material. Incorporating damage into the failure-criterion was shown to further enhance and improve the prediction of failure-stress and failure-envelope and to be closer to the experimental results.

Footnotes

Acknowledgements

This paper is part of the Master thesis of the second author.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

GA Abu-Farsakh

References

Abu-Farsakh

(1989) New material models for nonlinear stress-strain behaviour of composite materials. Composites 20(4): 349–360.

Abu-Farsakh

Al-Jarrah

(2018) Micro-mechanical damage model accounting for composite material nonlinearity due to matrix-cracking of unidirectional composite laminates. Composites Science and Technology 167: 268–276.

Abu-Farsakh

Asfa

(2018) Macro-mechanical damage modeling of fibrous composite materials accounting for non-linear material behavior. Composites Science and Technology 156: 287–295.

Abu-Farsakh

Asfa

(2020) A unified damage model for fibrous composite laminae subject to in-plane stress-state and having multi material-nonlinearity. International Journal of Damage Mechanics 29(9): 1329–1344.

Abu-Farsakh

Abdel-Jawad

(1994) A new failure criterion for nonlinear composite materials. Journal of Composites Technology and Research 16(2): 138–145.

Abu-Farsakh

Barakat

Abed

(1999) A macromechanical damage model of fibrous laminated composites. Applied Composite Materials 6(2): 99–119.

Cole

Pipes

(1974) Filamentary Composite Laminates Subjected to Biaxial Stress Fields. Chicago, IL: IIT Research Institute.

Camanho

Arteiro

Melro

, et al. (2015) Three-dimensional invariant-based failure criteria for fibre-reinforced composites. International Journal of Solids and Structures 55: 92–107.

Daniel

Fenner

(2018) A new yield and failure theory for composite materials under static and dynamic loading. Int J Solids Struct 148: 79–93.

10.

Daniel

Luo

Schubel

, et al. (2009) Interfiber/interlaminar failure of composites under multi-axial states of stress. Composites Science and Technology 69(6): 764–771.

11.

Fallahi

Taheri-Behrooz

Asadi

(2020) Nonlinear mechanical response of polymer matrix composites: A review. Polymer Reviews 60(1): 42–85.

12.

Hill

(1948) A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 193: 281–297.

13.

Jones

(1975) Mechanics of Composite Materials. Washington, DC: Scripta Book Co., p.367.

14.

Khayyam

Mazaheri

Taheri-Behrooz

. (2020) Strength prediction of woven composite rings using progressive damage modeling. International Journal of Damage Mechanics 28(6): 851–873.

15.

Sitnikova

Liang

, et al. (2017) The Tsai-Wu failure criterion rationalised in the context of UD composites. Composites Part A: Applied Science and Manufacturing 102: 207–217.

16.

Lou

Han

Cai

(2020) A micromechanics-based damage model for compressive behavior analysis of impacted composite laminates. International Journal of Damage Mechanics 29(3): 369–387.

17.

Okabe

Onodera

Kumagai

, et al. (2018) Continuum damage mechanics modeling of composite laminates including transverse cracks. International Journal of Damage Mechanics 27(6): 877–895.

18.

Samareh-Mousavi

Mandegarian

Taheri-Behrooz

(2019) A nonlinear FE analysis to model progressive fatigue damage of cross-ply laminates under pin-loaded conditions. International Journal of Fatigue 119: 290–301.

19.

Sandhu

(1976) Nonlinear behavior of unidirectional and angle ply laminates. Journal of Aircraft 13(2): 104–111.

20.

Sandhu

Gallo

Sendeckyj

(1982) Initiation and accumulation of damage in composite laminates. ASTM STP 787: 163–182.

21.

Serpieri

(2005) A novel constitutive model of composite materials with unidirectional long fibers: Theoretical aspects and computational issues. PhD Thesis, Università degli Studi di Napoli Federico II, Italy.

22.

Shokrieh

Taheri-Behrooz

(2006) A unified fatigue life model based on energy method. Composite Structures 75(1–4): 444–450.

23.

Shokrieh

Taheri-Behrooz

(2010) Progressive fatigue damage modeling of cross-ply laminates, I: modeling strategy. Journal of Composite Materials 44(10): 1217–1231.

24.

Soden

Hinton

Kaddour

(1998) Lamina properties, lay-up configurations and loading conditions for a range of fibre reinforced composite laminates. Composites Science and Technology 58(7): 1011–1022.

25.

Swanson

Messick

Tian

(1987) Failure of carbon/epoxy lamina under combined stress. Journal of Composite Materials 21(7): 619–630.

26.

Tsai

(1968) Strength theories of filamentary structure. In: Schwartz

Schwartz

(eds) Fundamental Aspects of Fiber Reinforced Plastic Composites. New York: Wiley Interscience, pp.3–12.

27.

Tsai

(1971) A general theory of strength for anisotropic materials. Journal of Composite Materials 5(1): 58–80.

28.

Tsai

Hahn

(1980) Introduction to composite materials. Connecticut, USA: Technomic Publishing Co.