Sage Journals: Discover world-class research

Abstract

Meso-scale numerical simulation of composite coupons is often defined using a ply-by-ply discretisation, where a large number of material properties are required to define the orthotropic elastic and failure behaviour of each ply. The interrelations observed between Tsai’s modulus and ply (intralaminar) material properties of several carbon-epoxy systems are used to devise a procedure that enables the fast estimation of meso-scale parameters. The predicted intralaminar strength and fracture toughness values are in good agreement with the experimental values. Furthermore, the utilisation of Tsai’s modulus paves the way for a unified approach, where a minimum set of experiments allows for estimating the material card required to achieve meso-scale simulations (e.g., open-hole tension) with good accuracy.

Keywords

Invariant-based design carbon fiber reinforced polymer analysis strength stiffness toughness material characterization fracture analytical modelling

Introduction

When the first philosopher, Thales of Miletus, posited,“All things are from water, and all things are resolved into water.”,¹ he pioneered a shift towards explaining the nature of matter through naturalistic-secular and rational arguments.² This shift laid the foundational groundwork for the development of both natural and applied sciences, influencing methodologies and perspectives that continue to shape scientific inquiry today.

While his theory about water is no longer valid, Thales’ approach to explaining one material property through the interrelationships of other material properties still resonates in the domain of engineering. This is particularly true for the characterisation of composite laminates, especially carbon fibre-reinforced plastics (CFRPs). CFRPs are comprised of different materials, carbon fibre and epoxy resin, to achieve a unified structure with a distinctive blend of properties, including a high strength-to-weight ratio, modulus-to-weight ratio, and exceptional corrosion resistance. This enables the design of lightweight structures, enhancing fuel efficiency and reducing emissions.³ Consequently, they are becoming widely used in the aircraft industry. However, the mechanical characterisation and the generation of design allowables for CFRPs are challenging, expensive, and time-consuming,⁴ which has been a recurring impediment for the cost-effective and fast characterisation and certification of composite aerospace structures. Therefore, there is an increasing interest in methods that rely on computational simulations to reduce the amount of experimental tests.

Finite element models (FEMs) are widely employed for computational simulations in the aircraft industry for design purposes.⁵

Meso-scale models are typically employed to predict the progressive failure of coupons. They rely on ply-by-ply finite element models, where each ply is modelled as a homogeneous material with appropriate intralaminar constitutive laws, and delamination is predicted by cohesive elements in between.

Continuum damage models^6–10 or smeared crack approaches^5,11,12 have been developed to model the intralaminar behaviour. It should be noted that these models rely on a large number of properties, and acquiring these properties from experiments or through inverse calibration is challenging. These inputs comprise not only the elastic properties but also the strength values for different stress states (longitudinal tension and compression, transverse tension and compression, in-plane and out-of-plane shear) and toughness values (longitudinal and transverse tension and compression). Obtaining these values often includes extensive experimental investigations and complex data reduction techniques.¹³ Consequently, exploring alternative approaches to compute material parameters without relying on a large amount of experiments or inverse calibration has become a crucial aspect of characterisation within the aircraft industry.

This issue is tackled in the present contribution by exploring the potential of invariant-based approaches to estimate material input parameters. The concept of invariance has proven useful in laminate design, as invariant properties remain unaffected by ply orientation, and because it is a quick way to make estimations with only a few experimentally determined properties. In the literature, Tsai and Pagano¹⁴ presented stiffness transformation equations for ply rotation in a laminate based on invariants. Subsequently, Grédiac et al.¹⁵ proposed a method for measuring invariants that describe the elastic response of anisotropic plates in bending.

More recently, Tsai and Melo¹⁶ proposed a novel invariant-based approach to describe the stiffness of composite plies and laminates that uses the trace (or Tsai’s modulus, as renamed by Arteiro et al.¹⁷) of the plane stress stiffness matrix of a unidirectional ply, $tr (Q)$ , as a material property. The trace, $tr (Q)$ , is invariant to coordinate transformations, material systems, number of layers, layup sequence, and loading condition (in-plane or flexural). The value of $tr (Q)$ can be determined using the longitudinal Young’s modulus of a 0° unidirectional lamina, E₁, or the longitudinal Young’s modulus of a multi-directional laminate, E_x, by dividing it by the respective trace-normalised laminate factor (Master Ply concept), $E_{1}^{*}$ or $E_{x}^{*}$ , whose trace-normalised coefficients vary within 4% for CFRPs.¹⁸

The Master Ply concept can significantly simplify laminate stiffness characterisation by reducing the number of tests required to fully characterise the orthotropic in-plane elastic properties of unidirectional composites from three tests at the ply level to 1 test at the ply level (to obtain E₁) or 1 test at the laminate level (to obtain E_x).¹⁹

This work is an extension of the concept developed by Tsai et al.^16–20 over the past decade, extending the potential of the proposed invariant-based model (determination of laminae and laminates stiffness values) to include the estimation of intralaminar strength and longitudinal toughness parameters, which will further reduce experimental characterisation of CFRP laminae/laminate and enable the explanation of intralaminar toughness and strength parameters in terms of Tsai’s modulus. By reducing variables, this concept will simplify composite laminate design, analysis, and optimisation, resulting in lighter, stronger, and better parts. Additionally, reducing the number and type of tests for the characterisation of composite laminates will reduce the regulatory certification burdens (e.g., aerospace industry).

A new invariant-based model for predicting intralaminar strength and toughness

In this section, the proposed invariant-based (Tsai’s modulus-based) model is explained using quantitative (data-driven) analyses of intralaminar strength and toughness values in relation to the changes in Tsai’s modulus. CFRP materials were found to demonstrate common interrelationships with respect to their Tsai’s modulus, $tr (Q)$ , in which these interrelationships were defined in terms of either linear or non-linear functions. Constant values in linear and non-linear functions were defined as Master Parameter.

This study defined its Master Parameter concept within the domains of quasi-static loading, room temperature, and dry specimens (the effect of environmental conditions in cold-dry and hot-wet was not included in this study). The selected material type was unidirectional tape carbon fibre-reinforced plastic (CFRP). For fibre volume fractions in the range 0.5 < V_f < 0.7 and longitudinal fibre/matrix stiffness ratios γ₁ ≳ 50 (typical of advanced CFRPs⁶⁷), the trace-normalised ply elastic moduli coefficients (Master Ply-based lamina/laminate factors) tend to stay constant.¹⁹ Glass fibre-reinforced polymers (GFRPs) were therefore excluded from material selection. The research was limited to conventional, standard-thickness plies − considering the effect of ply thickness as a future research topic. Additionally, it was noted that the relationships between Tsai’s modulus and intralaminar strengths changed between CFRP materials with high and intermediate modulus. As a result, this study focused solely on intermediate modulus CFRPs, while the relationships of high modulus CFRPs were decided as the subject for future research. That said, it was noted that equation (10) (representing calculated longitudinal intralaminar fracture toughness) successfully captured the fracture toughness of high modulus, thick, thin, and woven laminae using an equation derived solely from CFRPs defined within the aforementioned domains.

The intralaminar strength, related longitudinal Young’s modulus and Tsai’s modulus values are shown in Table 1, and the longitudinal tensile intralaminar fracture toughness values and corresponding longitudinal lamina/laminate Young’s modulus are shown in Table 2.

Table 1.

Intralaminar strength and elastic modulus measurements of various CFRP laminae.

CFRP	Tsai’s modulus (GPa)	E₁ (GPa)	(MPa)	X_C (MPa)	S_L (MPa)	Y_T(≈S_T) (MPa)	Y_C (MPa)	Ref.
Primary Sources
IM10/8552	216	190	3310	1793	-	-	-	²¹
T1100G/Epoxy	210	185	3460	1870	-	80	-	²²
IM7/M56	207	182	2730	1550	81	-	-	²³
IM8/8552	206	181	2963	1781	-	-	-	²⁴
IMA/M21	202	178	3050	1500	94	-	-	²⁵
IM10/M91	200	176	3520	1880	105	-	-	²⁶
IMA/8552	200	176	2935	1903	-	-	-	²⁷
T1100G/3960	199	175	3941	2048	107	68	258	²⁸
IM7/8552	195	172	2326	1200	92	62	200	²⁹
T800S/M21	195	172	3039	1669	95	75	-	³⁰
IM5/8552	191	168	2579	1627	-	-	-	³¹
T800H/2592	191	168	2920	1550	-	-	-	³²
IM7/M91	188	165	2980	1860	110	-	-	²⁶
IM7/8552*	186	164	2724	1690	120	64	-	³³
T1100G/3960*	185	163	3687	1977	90	-	-	²⁸
IM7/M21	182	160	2860	1790	110	-	-	²⁵
IM7/977-3	181	159	2510	1680	128	-	-	³⁴
IM6/8552	181	159	2758	-	-	-	-	³⁵
IM7/8552**	180	158	2500	1716	91	64	-	³⁶
T40/5320	177	156	2700	1737	110	81	-	³⁷
T800H/3633	177	156	2700	-	120	81	-	³⁸
T1100G/3960**	176	155	3282	1387	95	71	254	²⁸
AS7/M21	168	148	2350	1560	109	-	-	²⁵
AS7/M91	161	142	2580	1350	115	-	-	²⁶
AS4/8552	160	141	2207	1531	114	81	-	³³
AS7/M56	160	141	2330	1308	115	-	-	²³
AS4/3501-6	160	141	2137	1723	-	-	-	³⁹
T700/M21	159	140	2200	-	-	85	-	⁴⁰
T300/2500	159	140	1820	1470	-	76	-	⁴¹
T700S/Epoxy	153	135	2550	1470	-	69	-	⁴²
AS4/8552*	153	135	2137	1889	-	-	268	⁴³
T700S/Epoxy*	152	134	2860	1450	136	81	-	⁴⁴
T700S/G-94	152	134	2810	-	132	53	-	⁴⁵
AS7/8552	151	133	2344	1790	-	-	-	⁴⁶
T700/M21*	148	130	2190	-	158	85	-	⁴⁷
AS4/8552**	145	128	1996	1397	-	64	200	⁴⁸
T700G/2510	142	125	2172	1448	154	-	199	⁴⁹
Literature- derived Sources
IMA/M21E	202	178	-	-	-	-	200	⁵⁰
IM7/977-3	184	162	-	-	-	-	186	⁵¹
HTS40/977-2	174	153	-	-	-	-	236	⁵²
Graphite/Epoxy	163	143	-	-	-	-	240	⁵³
T700/3234	159	140	-	-	-	-	229	⁵⁴
AS4/8552**	156	137	-	-	-	-	322	⁵⁵
G1157/RTM6	148	130	-	-	-	-	241	⁵⁶
T700/M21**	148	130	-	-	-	-	290	⁵⁷
T700/M21***	148	130	-	-	-	-	220	⁵⁸
HS300/ET223	139	122	-	-	-	-	290	⁵⁹
G1157/RTM6*	133	117	-	-	-	-	170	⁶⁰
T700/3234	125	110	-	-	-	-	198	⁶¹

Identical materials from different sources are separated using symbols *, ** and ***.

Table 2.

Intralaminar longitudinal tensile fracture toughness, elastic moduli, and strength measurements of various CFRP materials.

CFRP	Tsai’s modulus (GPa)	E₁ (GPa)	E_x (cross-ply) (GPa)	X_T (MPa)	t (mm)	R_ss,T (MPa mm)	Ref.
High Modulus Carbon Fiber Prepreg
HR40/736	219	193	-	2158	0.023	24.7	⁶²
Intermediate Modulus Carbon Fiber Prepreg
T800S/M21	194	-	91	3039	0.125	283	⁶²
IM7/8552	194	-	91	2724	0.131	205	⁶²
T800/736	169	148	-	2924	0.053	192	⁶³
T700/M21	184	162	-	2299	0.268	252	⁶⁴
T700/M21*	167	147	-	2299	0.138	131	⁶⁴
T700/M21**	167	147	-	2182	0.075	84	⁶⁴
Carbon Fiber- reinforced Thermoplastic
AS4/PEKK	151	-	71	2209⁽¹⁾	0.138	114	⁶³
Woven Carbon Fiber Prepreg
IM7/E745	132	-	62	1967⁽²⁾	0.15	93	⁶⁵

Identical materials from different sources are separated using symbols * and **.

(1): The longitudinal strength of the material defined using equation (3).

(2): The longitudinal strength of the material defined using unit circle failure criteria.^16,66

Fiber-dominated strengths

Using data-driven analyses, CFRP materials were found to share common longitudinal strength factors if the longitudinal tensile and compressive strengths are normalised by their respective Tsai’s modulus. The median values of these factors defined Master Parameter values to gain an understanding of the laminate behaviour of these composites.

In Figures 1(a) and (b), the relationship between tensile and compressive strength along the fibre direction (longitudinal direction) (X_T and X_C) with the respective Tsai’s modulus values are shown using the dataset provided in Table 1. Figures 1(a) and (b) show clear linear correlations between longitudinal strength and modulus values. Based on this relationship, the definition of Master Parameter for longitudinal strength parameters were given as:

\begin{align} X_{T}^{*} = \frac{X_{T}}{tr (Q)}, \end{align}

(1)

\begin{align} X_{C}^{*} = \frac{X_{C}}{tr (Q)} . \end{align}

(2)

Figure 1.

Distribution and trends of longitudinal tensile (a) and compressive (b) strengths. Gaussian distribution and histogram of the Master Parameter values for longitudinal tensile (c) and compressive (d) strengths.

Additionally, Figures 1(a) and (b) show that some longitudinal strength values deviated from the general trend line. These outlier data points are either notably lower or higher compared to similar CFRPs with equivalent modulus values. To systematically identify these outliers, boundaries representing 1.5 times the standard deviation from the trend line were highlighted in Figures 1(a) and (b). Values outside these ranges (outliers) were highlighted in red and excluded from the calculation of Master Parameter values.

The histogram and the probability density functions corresponding to the defined Master Parameter values are presented in Figures 1(c) and (d). As the probability density function, the Gaussian distribution was used, where the gradient of the linear relationships in Figures 1(a) and (b) correspond to the maximum probability density, while the Master Parameter values represent the median values of the strength values within the acceptable statistical boundaries (i.e., within 1.5 standard deviations of the mean trend line). Using the defined Master Parameter a universal relation between the components of the longitudinal tensile/compressive strengths and Tsai’s modulus, $tr (Q)$ , of any CFRP was characterised as:

\begin{align} X_{T} = X_{T}^{*} \cdot tr (Q), \end{align}

(3)

\begin{align} X_{C} = X_{C}^{*} \cdot tr (Q) . \end{align}

(4)

The defined Master Parameter values are shown in Table 4. The comparison between the predicted longitudinal strength parameters using Equations (1) and (2) and the corresponding experimental measurements is shown in a bar graph in Figures 2 and 3. Figure 2 shows an average error of 5.9% with a coefficient of variation (CV) of 7%, highlighting the overall effectiveness of the longitudinal tensile strength predictive model. Although there are minor deviations, the model provided predictions sufficiently close to the experimental values, allowing it to be used confidently in most cases.

Figure 2.

Longitudinal tensile strength predictions were compared with the corresponding experimental measurements.

Figure 3.

Longitudinal compressive strength predictions were compared with the corresponding experimental measurements.

In Figure 3, the predicted compressive strength values have a 8.1% mean error with a CV of 9.5%, which is within the acceptable range in many engineering applications.

It was worth noting that the calculated longitudinal strengths can be used to determine the strength of laminates in the longitudinal direction of loading based on the relationship between stiffness (or engineering constants) and strength of the lamina/laminates, which can be derived using the Master Parameter concept.⁶⁶

Matrix-dominated strengths

The matrix-dominated stiffness moduli contribute to the Tsai’s modulus of a CFRP lesser than the longitudinal elastic modulus of lamina/laminate.¹⁶ However, the variations of matrix-related Master Ply values are larger because of different matrices and curing processes. Furthermore, there are testing uncertainties where test standards have some room for interpolation in the result analysis.

For example, tests such as in-plane shear, due to fibre re-orientation (fibre scissoring), instead of the ultimate strength, the shear strength is approximated as the stress at the 5% strain displacement. Furthermore, multiple standards were proposed to observe more uniform in-plane shear stress concentration, where the ±45° in-plane (ASTM D3518⁶⁸), Iosipescu (ASTM D5379-12⁶⁹), and V-notched rail shear tests (ASTM D7078-12⁷⁰) are recognised as the most promising among the various ASTM shear test standards.⁷¹ However, these three standards can yield different strength values.⁷¹ Therefore, when creating the strength dataset in Table 1, the ±45° in-plane testing standard (ASTM D3518⁶⁸) was selected as the single preferred standard to maintain dataset consistency. Additionally, the ±45° in-plane test provides the largest dataset for data-driven analysis due to its widespread use in material characterisation (i.e., because of its straightforward testing procedure).

For the compilation of the transverse compression dataset, due to the scarcity of established experimental sources (e.g., HEXCEL, TORAY, and NIAR), the dataset included values from numerical studies that use the transverse compression of CFRPs. The literature-derived sources were separated from the experimental dataset − referred to as primary sources − in Table 1. Although this reduces the reliability of the data-driven analysis, it provides a general insight into the distribution of transverse compression values for CFRP materials. Consequently, no Master Parameter values were calculated for transverse compression. Instead, a dummy model (based on the average of the distribution) was provided.

During the out-of-plane shear, mode I failure is observed due to the stress transformation, leading to a similar failure mechanism as the transverse tensile stress. This similarity also allows their strength values to be comparable. Hence, out-of-plane strength and tensile strength are accepted as approximately equal in Table 1.

Regarding transverse tensile and in-plane shear strengths, the dataset in Table 1 reveals that they can be approximated as a function of Tsai’s modulus (c.f. Figure 4). Two different functions, (i) a linear approximation and (ii) an inverse relationship are fitted and their suitability is compared in Table 3, with their standard deviations, CVs, and error percentages. Both functions show acceptable accuracy, but the second function has the most accurate representation of the dataset (lower error). Furthermore, inverse relationship requires only one coefficient to be determined, whereas the linear function requires two. Hence, based on accuracy and simplicity, the function of type $f (tr (Q)) = a / tr (Q)$ is employed to approximate both the in-plane shear and transverse tension strengths.

Figure 4.

Distribution and trends of in-plane shear (a), transverse tensile (b), and mean transverse compressive (c) strengths. Gaussian distribution and histogram of the Master Parameter values for in-plane shear (d) and transverse tensile (e) strengths.

Table 3.

Comparison of different functions to approximate matrix dominated strength properties from Tsai’s modulus.

Mathematical model $(f (tr (Q)))$	S _L	Y_T (≈S_T)
$f (tr (Q)) = a \cdot tr (Q) + b$
Coef. (a)	−0.824	0.008
Coef. (b)	255.32	72.96
Error (%)	8.8	9.3
Std dev	0.062	0.005
CV (%)	7.57	111.5
$f (tr (Q)) = a / tr (Q)$
Coef. (a)	1.9196e + 07	1.2845e + 07
Coef. (b)	-	-
Error (%)	7.1	7.5
Std dev	1.561e + 06	1.121e + 06
CV (%)	8.13	8.93

Figure 4(a)–(c) show the relation between in-plane shear, transverse tensile, and transverse compression (S_L, Y_T (≈S_T), and Y_C) and their corresponding Tsai’s modulus values, along with the aforementioned functions employed for the approximation. The Master Parameter values for matrix-dominated strength parameters were defined based on this relationship as:

\begin{align} S_{L}^{*} = S_{L} \cdot tr (Q), \end{align}

(5)

\begin{align} Y_{T}^{*} = Y_{T} \cdot tr (Q) . \end{align}

(6)

Using the same method for finding outliers (as in Section 2.1), data points that are more than 1.5 times the standard deviation were marked in red and left out of the Master Parameter calculations in Figure 4(a)–4(c).

In Figure 4(c), the experimental strengths are compared with the mean strength (i.e., 236 MPa) with a maximum deviation of 54 MPa and a CV of 14.1% which indicates a crucial need for additional experimental data. Expanding the dataset would enhance the model’s accuracy and reliability across a wider range of materials. Furthermore, more data would help minimise potential biases and ensure that the model remains accurate when applied to new materials.

Figures 4(d)–(e) display the histograms and probability density functions for the matrix-dominated Master Parameter values, using the Gaussian distribution. The Master Parameter values for these parameters represented the median values within the statistical boundaries of 1.5 times the standard deviation. Using the defined Master Parameter, a universal relation between the components of the intralaminar in-plane shear and transverse tensile strengths and Tsai’s modulus, $tr (Q)$ , of any CFRP was defined as:

\begin{align} S_{L} = \frac{S_{L}^{*}}{tr (Q)}, \end{align}

(7)

\begin{align} Y_{T} = \frac{Y_{T}^{*}}{tr (Q)} . \end{align}

(8)

Table 4 summarises all the defined Master Parameter values. Figures 5 and 6 present comparisons between the predicted matrix-dominated strength parameters using Equations (5) and (6) and their experimental counterparts.

Table 4.

Defined Master Parameter.

Master Parameter	$X_{T}^{*}$ [−]	$X_{C}^{*}$ [−]	$S_{L}^{*}$ (MPa²)	$Y_{T}^{*}$ (MPa²)
0	0.0149	0.0091	1.9141e+7	1.2845e+7
Std dev	0.0010	0.0009	0.1561e+7	0.1121e+7
CV (%)	7.0	9.5	8.1	8.7

Figure 5.

In-plane shear strength predictions are compared with the corresponding experimental measurements.

Figure 6.

Transverse tensile strength predictions are compared with the corresponding experimental measurements.

In Figure 5, the 7.1% mean error with a maximum deviation of 18.9 MPa and CV of 8.1%, indicates that the in-plane shear strength predictions are generally reliable, especially given the complexity of modelling shear strength due to the intricacies of material behaviour. This level of accuracy further suggests that the model can be used in design and analysis.

In Figure 6, the 7.5% mean error with a maximum deviation of 15 MPa and CV of 8.7%, when estimating the transverse tensile strengths of composite materials, reflects the robust predictive capability of the invariant-based model.

Fiber-dominated toughness

The longitudinal intralaminar tensile fracture toughness does not have a standardised testing method. However, the double-edge notch tensile (DENT) test combined with the size-effect law, proposed by Catalanotti et al.,⁶² was used in industry (e.g., by AIRBUS) for material characterisation.⁷² Therefore, data-driven analyses were conducted using experimental data obtained from DENT specimens.⁷³

Additionally, the measured apparent fracture toughness was reported as a function of the ply thickness.^74,75 The study of Furtado et al.⁶⁴ found that effective tensile fracture toughness increases linearly with the ply thickness (due to thick plies observing split cracking in the vicinity of the notch, resulting in a stress redistribution that blunts the notch, leading to higher notched strengths). Thus, in the data-driven analysis for tensile fracture toughness, a reference ply thickness value (i.e., t_ref = 0.135 mm) was defined for all materials, and their respective fracture toughness values were adjusted to this reference thickness in Figure 7(a).

Figure 7.

(a) Adjusted tensile intralaminar fracture toughness and linear trend. (b) Comparison of measured tensile intralaminar fracture toughness and predictions.

The adjusted tensile intralaminar fracture toughness values of the unidirectional, intermediate modulus CFRPs (see Table 2) with reference thicknesses exhibited a linear relationship with their respective longitudinal tensile strengths, as shown in Figure 7(a), which was formulated as:

{(R_{s s, T})}_{adjusted} = (a \cdot X_{T} - b)

(9)

where the a and b represent the slope and intercept, respectively, of the linear trend observed in Figure 7(a) (i.e. 0.1929 and 304.5, respectively). When this linear relationship was re-established by incorporating the thickness effect defined by Furtado et al.,⁶⁴ the predictions presented in Figure 7(b) were obtained. The formulation for these predictions in Figure 7(b) was defined as follows:

R_{s s, T} = (a \cdot X_{T} - b) - m (t_{ref} - t)

(10)

where m is the slope associated with the thickness effect, as defined by Furtado et al.,⁶⁴ and is equal to 806.3.

A similar framework can also be applied to the compressive intralaminar fracture toughness. However, to implement this framework, the following shortcomings in the literature must be addressed: an adequate amount of experimental data for a data-driven understanding (e.g., Dalli et al.⁷³ proposed a more stable version of the double-edge notch compression (DENC) where the test fixture combined with combined loading compression (CLC), and only one experimental measurement was made for unidirectional CFRP using DENCLC by Danzi et al.³), and the influence of thickness on compressive fracture toughness must be clarified.

Figure 8 shows a bar graph comparing experimental and predicted intralaminar tensile fracture toughness values using the experimentally measured longitudinal tensile strengths and the calculated longitudinal tensile strengths based on the Master Parameter from equation (1). When experimental strength values were used, equation (10) successfully captured the intralaminar fracture toughness for all materials listed in Table 2, including woven and high-modulus composite materials. However, when strength values calculated from Tsai’s modulus were used, only materials within the scope of this study were considered (i.e., high-modulus, thin or thick ply, and woven materials were excluded). In future work, with a sufficient dataset, the current boundaries of this research will be extended further.

Figure 8.

Intralaminar tensile fracture toughness predictions are compared with the experimental measurements.

In Figure 8, an average error of 7%, with a maximum deviation of 10 kJ/mm², indicates that the predicted tensile intralaminar fracture toughness values closely align with the experimental results when using the measured tensile strength values. Conversely, when using tensile strength values calculated based on $tr (Q)$ , an average error of 21%, with a maximum deviation of 44.3 kJ/mm², suggests that small deviations in the calculation of strength can lead to significant discrepancies in the prediction of intralaminar tensile fracture toughness (e.g., in the case of T800S/M21, the ratio of predicted-to-measured strength values is 0.96, while the corresponding ratio of intralaminar tensile fracture toughness values is 0.88).

Assessment of the estimated material cards using finite element models

In this section, numerical simulations were performed to assess the accuracy of the invariant-based model in characterising intralaminar strengths and longitudinal tensile fracture toughness parameters.

A 2.5D (quasi-3D) infinitesimal continuum damage model (CDM) was used in this study to simulate the intralaminar degradation mechanisms of the composite ply. This constitutive model is based on the work of Maimí et al.,⁶ which developed a CDM for predicting the onset and evolution of intralaminar failure mechanisms and the collapse of composite laminates.

The main features of the model are summarised in Table 5, while the experimentally measured and predicted intralaminar properties of T800S/M21 (by Toray®) are shown in Table 6. Good accuracy is observed when comparing the ratio of predicted material properties to experimentally measured for T800S/M21. The rest of the intralaminar, interlaminar, and other CDM-related properties obtained from Furtado et al.⁹ were shown in black, while the predicted intralaminar properties were highlighted in a different colour. Similarly, material cards for four composites − IM10/8552, IM7/8552, AS7/M21, and AS4/8552 (by Hexcel®) − in Table 8 are completed using the proposed invariant-based model. The remaining intralaminar, interlaminar, and other CDM-related properties were obtained from Furtado et al.⁹ Furthermore, the dimensions and tensile strength of the open-hole coupons are provided in Table 7.

Table 5.

Physical capabilities of the continuum damage model.

Model		Continuum Damage Model
Version		2024
Failure criteria		2.5D
Solver		Abaqus explicit
Strain theory		Infinitesimal
Behaviour	Longitudinal (fiber) direction	Linear
	Transverse (matrix) direction	Linear
	Viscoelasticity	✗
	Plasticity	Bi-linear (only in shear)
Failure initiation	Fiber tension	✓
	Fiber compression	✓
	Fiber kink-band formation	✓
	Shear-driven fiber compression	✓
	Coupling between 3D fiber & matrix failure modes	✓
	Matrix fracture plane prediction	✓
Damage	Damage initiation energy	✓
	Energy-based damage propagation	✓
	Stress-strain hardening in longitudinal direction	✓
	Stress-strain hardening under in-plane shear	✓
	Stress-strain softening in longitudinal direction	✓
	Stress-strain softening in transverse direction	✓
	Stress-strain softening under in-plane shear	✓
	Coupling between plasticity and damage	✗

Hardening = non-linear increase in stress-strain behaviour. Softening = non-linear decrease in stress-strain behaviour.

Table 6.

Material properties of T800S/M21 (by Toray®).

Erçin³⁰ and Furtado et al.⁹ do not have information on the fibre volume ratio of T800S/M21. However, the same resin from the same manufacturer, when used with different fibres,²⁵ consistently has a fibre volume ratio of 0.59.

Table 7.

Properties of the open-hole specimens.

Carbon/Epoxy	X_OHT (MPa)	t_ply (mm)	Layup	W/D	D (mm)	Ref.
T800S/M21	534	0.125	${[90 / 45 / 0 / - 45]}_{3 s}$	4	5	³⁰
IM10/8552	593	0.131	${[\pm 45 / 0 / 90]}_{3 s}$	6	6	^21,77
IM7/8552	428	0.131	${[\pm 45 / 0 / 90]}_{3 s}$	6	6	^33,77
AS7/M21	365	0.184	${[\pm 45 / 0 / 90]}_{2 s}$	6	6	^25,77
AS4/8552	328	0.188	${[45 / 0 / - 45 / 90]}_{2 s}$	6	6	⁴⁸

While choosing suitable validation materials, the following criteria were taken into consideration: the availability of data at hand and the representativeness of the full spectrum of intermediate modulus materials. T800S/M21 was chosen for validation because, based on the works of Furtado et al.⁹ and Erçin,³⁰ the full material card and open-hole tension (OHT) characterisation had been conducted. Hence, initial numerical simulations were performed using material cards with fully measured intralaminar parameters and predicted counterparts, which would help to evaluate the general accuracy of the invariant-based model when compared to fully characterised intralaminar properties and provide a comparison for the predictions of open-hole tension strength. Furthermore, detailed information on open hole tensile tests and intralaminar longitudinal fracture toughness tests conducted for T800S/M21 (including the number of test specimens and standard deviation values) can be found in the works of Erçin et al.³⁰ and Catalanotti et al.⁶²

The four Hexcel® CFRPs were selected to cover the entire spectrum of intermediate modulus CFRPs. IM10/8552 and AS4/8552 represent the extremes of this range, as they have the highest and lowest longitudinal Young’s modulus values, respectively (as shown in Table 1), along with available open-hole tensile characterisation data. The other two CFRPs, IM7/8552 and AS7/M21, represent the middle of the spectrum, with IM7/8552 having slightly higher than average longitudinal Young’s modulus and AS7/M21 having slightly lower than average longitudinal Young’s modulus.

Furthermore, Hexcel® materials with −/8552 and −/M21 Fibre/Epoxy combinations were specifically chosen for this study, because of the availability of complete material cards for IM7/8552 and T800S/M21 in the source material⁹ that allowed for accurate representation of interlaminar and other matrix-dominated properties in the numerical analysis by filling gaps in the material cards (highlighted in black) with values closely representing the actual properties. This approach ensured that the numerical simulations were conducted with the most reliable and representative material data available.

Figure 9 presents a comparison between the predicted intralaminar properties and fully measured experimental results for the open-hole tensile strength of a T800S/M21 quasi-isotropic laminate. It was observed that the error recorded for the fully measured material card is 0.5% overestimation, while the material card with predicted intralaminar strength values shows a 3.7% underestimation.

Figure 9.

Comparison of the numerical and experimental results for the open-hole tensile strength of T800S/M21.

Figure 10 presents a comparison between the experimentally measured open-hole tensile strengths and the CDM predictions using material cards provided in Table 8 for four different CFRPs. The mean error recorded among the four CFRPs are 5.0%, with IM10/8552 showing a 7.0% underestimation, IM7/8552 showing an 8.9% overestimation, AS7/M21 showing a 3.5% overestimation, and AS4/8552 showing a 0.5% overestimation.

Figure 10.

Comparison of the numerical and experimental results for the open-hole tensile strengths of four CFRPs.

Table 8.

Material properties of various Hexcel® materials.

In conclusion, Figures 9 and 10 show that CDM predictions using material cards were based on Master Ply and equation (10) (where X_T was obtained through Master Ply concept) exhibit acceptable accuracy when compared to experimentally measured notched strengths, which suggested that the invariant-based model concept can effectively predict the intralaminar material properties of CFRPs.

Conclusions

In this work, an invariant-based methodology was proposed to reduce the number of experimental tests required for the characterisation of the material properties of CFRPs, enabling running finite element analyses without using fully experimentally measured material cards. Through this methodology, intralaminar strength and fracture toughness properties were estimated from Tsai’s modulus which allowed the determination of intralaminar properties using only the longitudinal Young’s modulus of either lamina or laminate.

The proposed invariant-based model was initially validated by comparing the predicted intralaminar properties with their experimental counterparts. Then, the predicted parameters were used to fill in the intralaminar strength and toughness properties in the material cards for various materials, allowing them to be used in finite element analysis of open-hole CFRP specimens. This validation involved comparing experimental measurements with finite element predictions of the open-hole tensile strengths for T800S/M21 and four different CFRPs. The CDM predictions showed reasonable agreement with the experimental data.

A comparison between the experimentally measured material card for T800S/M21 and the predicted material card by the invariant-based model showed that the model accurately predicted intralaminar material properties and could complete missing information in material cards. Furthermore, the accurate results obtained for Hexcel® CFRPs in tension suggested that the invariant-based model was a viable tool for predicting material properties and can serve as a basis for further refinement and application in composite material analysis.

Consequently, the invariant-based model has the potential to significantly advance the aerospace industry by reducing the need for extensive experimental testing, streamlining the material characterisation process, and simplifying composite laminate design, analysis, and optimisation, which can lead to lighter, stronger, and more efficient components while enhancing the reliability of composite material predictions.

Footnotes

Acknowledgments

The authors are thankful to Prof. Albertino Arteiro for providing constructive feedback and insightful suggestions on the research methodology. Additionally, the authors are thankful to Prof. Albertino Arteiro for providing constructive feedback and insightful suggestions on the research methodology.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the funding from the European Union's Horizon Europe research and innovation programme under grant agreement No. 101056682, through the project DIDEAROT - Digital Design strategies to certify and mAnufacture Robust cOmposite sTructures. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them.

ORCID iDs

Igor A. Rodrigues Lopes

Carolina Furtado

Pedro P. Camanho

Data Availability Statement

All the data employed in the current study is provided in the tables of this manuscript.

References

Diels

. Doxographi graeci: collegit recensuit prolegomenis indicibusque instruxit hermannus diels. Berlin, Germany: de Gruyter, 1929.

Arslan

. İlkçağ felsefe tarihi: sokrates öncesi yunan felsefesi. Istanbul: İstanbul Bilgi Üniversitesi Yayınlari, 2006.

Danzi

Campos

PJS

Arteiro

, et al. Longitudinal failure mechanisms and crack resistance curves of unidirectional thermoplastic composites. Eng Fract Mech 2023; 282: 109147.

Whitney

Nuismer

. Stress fracture criteria for laminated composites containing stress concentrations. J Compos Mater 1974; 8(3): 253–265.

Camanho

Bessa

Catalanotti

, et al. Modeling the inelastic deformation and fracture of polymer composites - part ii: smeared crack model. Mech Mater 2013; 59: 36–49.

Maimí

Camanho

Mayugo

, et al. A continuum damage model for composite laminates: part i - constitutive model. Mech Mater 2007; 39: 897–908.

Maimí

Camanho

Mayugo

, et al. A continuum damage model for composite laminates: part II - computational implementation and validation. Mech Mater 2007; 39(10): 909–919.

Quintanas-Corominas

Maimí

Casoni

, et al. A 3D transversally isotropic constitutive model for advanced composites implemented in a high performance computing code. Eur J Mech Solid 2018; 71: 278–291.

Furtado

Catalanotti

Arteiro

, et al. Simulation of failure in laminated polymer composites: building-Block validation. Compos Struct 2019; 226: 111168.

10.

Cózar

Otero

Maimí

, et al. A three-dimensional plastic-damage model for polymer composite materials. Compos Appl Sci Manuf 2022; 163: 107198.

11.

Leone

. Deformation gradient tensor decomposition for representing matrix cracks in fiber-reinforced materials. Compos Appl Sci Manuf 2015; 76: 334–341.

12.

Rodrigues Lopes

Danzi

Arteiro

, et al. Modelling intralaminar damage mechanisms in fibre reinforced polymers at finite strains. Int J Solid Struct 2023; 282: 112449.

13.

Catalanotti

Xavier

Camanho

. Measurement of the compressive crack resistance curve of composites using the size effect law. Compos Appl Sci Manuf 2014; 56: 300–307.

14.

Tsai

Pagano

. Invariant properties of composite materials. Lancaster: Technomic Publishing Company, 1968.

15.

Grédiac

Fournier

Paris

, et al. Direct measurement of invariant mechanical parameters of composite plates. J Compos Mater 1999; 33(21): 2017–2036.

16.

Tsai

Melo

JDD

. An invariant-based theory of composites. Compos Sci Technol 2014; 100: 237–243.

17.

Arteiro

Sharma

Melo

, et al. A case for Tsai’s modulus, an invariant-based approach to stiffness. Compos Struct 2020; 252: 112683.

18.

Tsai

Melo

JDD

. Composite materials design and testing: Unlocking Mystery with Invariants. Stanford, CA: Stanford Aeronautics & Astronautics, 2015.

19.

Arteiro

Pereira

Bessa

, et al. A micro-mechanics perspective to the invariant-based approach to stiffness. Compos Sci Technol 2019; 176: 72–80.

20.

Tsai

Sihn

Melo

JDD

. Trace-based stiffness for a universal design of carbon-fiber reinforced composite structures. Compos Sci Technol 2015; 118: 23–30.

21.

Corporation

HexTow® IM10 carbon fiber product data sheet. 2023.

22.

Toray Composite Materials America

Toray T1100G carbon fiber product data sheet. 2018.

23.

Corporation

HexPly® M56 PrepReg system data sheet. 2017.

24.

Corporation

HexTow® IM8 carbon fiber product data sheet. 2023.

25.

Corporation

HexPly® M21 PrepReg system data sheet. 2020.

26.

Corporation

HexPly® M91 PrepReg system data sheet. 2020.

27.

Corporation

HexTow® IMA carbon fiber product data sheet. 2023.

28.

Toray Composite Materials America, Inc. 3960 PrepReg system data sheet. 2025.

29.

Camanho

Lambert

. A design methodology for mechanically fastened joints in laminated composite materials. Compos Sci Technol 2006; 66(15): 3004–3020.

30.

Erçin

. Stress gradient effects in laminated composites. Copyright - Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works; Analyte descriptor - NA 2013; 34: 385–400.

31.

Corporation

HexTow® IM5 carbon fiber product data sheet. 2023.

32.

Toray Composite Materials America, Inc. Toray T800H carbon fiber product data sheet. 2018.

33.

Corporation

HexPly® 8552 PrepReg system data sheet. 2020.

34.

Solvay® . Cycom®-977-3 data sheet, 2017.

35.

Corporation

HexTow® IM6 carbon fiber product data sheet. 2023.

36.

Marlett

Tomblin

. Hexcel 8552 im7 unidirectional prepreg 190 gsm & 35% rc qualification material property data report. National Center for Advanced Materials Performance, Wichita, Kansas Test Report CAM-RP-2009-015, Rev A 2011; 22: 1–238.

37.

Cytec Report . Cycom 5320, T40 uni-graphite 145gsm property data. 2015.

38.

Morimoto

Sugimoto

Katoh

, et al. Jaxa advanced composite materials mechanical properties database. JAXA Technical Report. 2015.

39.

Corporation

. HexPly 3501-6 high strength, damage-resistant. Structural epoxy matrix. Hexcel, 2020.

40.

AMADE . Thin ply program testing support. Girona, Spain: University of Girona, 2018.

41.

Toray Composite Materials America, Inc. Baseline carbon fiber with a 40-year production history. 2018.

42.

Toray Carbon Fibers America , Inc. T700S data sheet. 2021.

43.

Corporation

HexTow® AS4 carbon fiber product data sheet. 2023.

44.

Toray Composite Materials America, Inc . T700S data sheet. Toray Composite Materials America, Inc. 2024.

45.

Toray Composite Materials America , Inc. G-94M resin technical data sheet. 2021.

46.

Corporation

HexTow® AS7 carbon fiber product data sheet. 2023.

47.

Huchette

. Sur la complémentarité des approches expérimentales et numériques pour la modélisation des mécanismes d’endommagement des composites stratifiés. Université Paris. Composite Structures 2005; 6: 111386.

48.

Tomblin

Hooper

. Hexcel 8552 AS4 unidirectional prepreg at 190 gsm & 35% rc qualification material property data report. 2011.

49.

Toray Composite Materials America, Inc. 2510 PrepReg system data sheet. 2020.

50.

Caminero

García-Moreno

Rodríguez

. Damage resistance of carbon fibre reinforced epoxy laminates subjected to low velocity impact: effects of laminate thickness and ply-stacking sequence. Polym Test 2017; 63: 530–541.

51.

Zhang

. An efficient approach for predicting low-velocity impact force and damage in composite laminates. Compos Struct 2015; 130: 85–94.

52.

Shi

Pinna

Soutis

. Modelling impact damage in composite laminates: a simulation of intra- and inter-laminar cracking. Compos Struct 2014; 114: 10–19.

53.

Liao

Liu

. Finite element analysis of dynamic progressive failure of plastic composite laminates under low velocity impact. Compos Struct 2016; 159: 567–578.

54.

Tuo

, et al. Damage and failure mechanism of thin composite laminates under low-velocity impact and compression-after-impact loading conditions. Composites Part B Engineering 2019; 163: 642–654.

55.

Baluch

Falcó

Jiménez

, et al. An efficient numerical approach to the prediction of laminate tolerance to barely visible impact damage. Compos Struct 2019; 225: 111017.

56.

Caputo

De Luca

Sepe

. Numerical study of the structural behaviour of impacted composite laminates subjected to compression load. Composites Part B Engineering 2015; 79: 456–465.

57.

Tan

Falzon

Chiu

, et al. Predicting low velocity impact damage and compression-after-impact (cai) behaviour of composite laminates. Composites Part a Applied Science and Manufacturing 2015; 71: 212–226.

58.

Abir

Tay

Ridha

, et al. Modelling damage growth in composites subjected to impact and compression after impact. Compos Struct 2017; 168: 13–25.

59.

Liu

Liao

Jia

, et al. Finite element analysis of dynamic progressive failure of carbon fiber composite laminates under low velocity impact. Compos Struct 2016; 149: 408–422.

60.

González

Maimí

Martín-Santos

, et al. Simulating drop-weight impact and compression after impact tests on composite laminates using conventional shell finite elements. Int J Solid Struct 2018; 144-145: 230–247.

61.

Long

Yao

Zhang

. Delamination prediction in composite laminates under low-velocity impact. Compos Struct 2015; 132: 290–298.

62.

Catalanotti

Arteiro

Hayati

, et al. Determination of the mode i crack resistance curve of polymer composites using the size-effect law. Eng Fract Mech 2013; 118: 49–65.

63.

Danzi

Tavares

Xavier

, et al. Effects of hybridization and ply thickness on the strength and toughness of composite laminates. J Compos Mater 2021; 55(30): 4601–4616.

64.

Furtado

Arteiro

Linde

, et al.

Is there a ply thickness effect on the mode i intralaminar fracture toughness of composite laminates?

Theor Appl Fract Mech 2020; 107: 102473.

65.

Dalli

Varandas

Scalici

, et al. Exploiting intrinsic size effect to overcome specimen scaling limitations in fracture toughness characterisation of composite materials. Eng Fract Mech 2023; 277: 108951.

66.

Tsai

Melo

JDD

. A unit circle failure criterion for carbon fiber reinforced polymer composites. Compos Sci Technol 2015; 123: 71–78.

67.

Soden

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

68.

ASTM International . ASTM D3518/D3518M-18 standard test method for In-Plane shear response of polymer matrix composite materials by tensile test of a ± 45° laminate. West Conshohocken, PA: ASTM International, 2018.

69.

ASTM International . ASTM D5379/D5379M-12 standard test method for shear properties of composite materials by the V-Notched beam method. West Conshohocken, PA: ASTM International, 2012.

70.

ASTM International . ASTM D7078/D7078M-12 standard test method for shear properties of composite materials by V-Notched rail shear method. West Conshohocken, PA: ASTM International, 2012.

71.

Choi

Jang

Shim

, et al. Determination of the in-plane shear modulus of unidirectional carbon fiber-reinforced plastics using digital image correlation and finite-element analysis. Compos Struct 2019; 229: 111392.

72.

Herman

Leon-Dufour

Peters

, et al. Evaluation of advanced approaches based on continuum damage mechanics for composite failure analysis: status and way forward. In: 25th international conference on composite, Porto, Portugal, July 19 - 22, 2022.

73.

Dalli

Catalanotti

Varandas

, et al. Mode i intralaminar fracture toughness of 2d woven carbon fibre reinforced composites: a comparison of stable and unstable crack propagation techniques. Eng Fract Mech 2019; 214: 427–448.

74.

Teixeira

Pinho

Robinson

. Thickness-dependence of the translaminar fracture toughness: experimental study using thin-ply composites. Compos Appl Sci Manuf 2016; 90: 33–44.

75.

Arteiro

. Structural mechanics of thin-ply laminated composites. Porto, Portugal: Faculdade de Engenharia da Universidade do Porto, 2016.

76.

Giddings

Bowen

Salo

, et al. Bistable composite laminates: effects of laminate composition on cured shape and response to thermal load. Compos Struct. 2010; 92(9): 2220–2225.

77.

Standard

ASTM.

ASTM D5766-D5766M-11: Standard test method for open-hole tensile strength of polymer matrix composite laminates. 2011.

Unveiling interrelations among material properties of polymer composite laminates

Abstract

Keywords

Introduction

A new invariant-based model for predicting intralaminar strength and toughness

Fiber-dominated strengths

Matrix-dominated strengths

Fiber-dominated toughness

Assessment of the estimated material cards using finite element models

Conclusions

Footnotes

Acknowledgments

Declaration of conflicting interests

Funding

ORCID iDs

Data Availability Statement

References