A damage mechanics based failure criterion for fiber reinforced polymers

Accepted Manuscript A damage mechanics based failure criterion for fiber reinforced polymers Elham Shahabi, Mohammad Reza Forouzan PII:

S0266-3538(16)30811-9

DOI:

10.1016/j.compscitech.2016.12.023

Reference:

CSTE 6613

To appear in:

Composites Science and Technology

Received Date: 25 July 2016 Revised Date:

17 November 2016

Accepted Date: 24 December 2016

Please cite this article as: Shahabi E, Forouzan MR, A damage mechanics based failure criterion for fiber reinforced polymers, Composites Science and Technology (2017), doi: 10.1016/ j.compscitech.2016.12.023. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A damage mechanics based failure criterion for fiber reinforced polymers Elham Shahabia, Mohammad Reza Forouzan b* Department of mechanical engineering, Isfahan University of Technology, Isfahan,

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a,b

8415683111, Iran.

*Corresponding author: Tel: +98-31-33915235, Fax: +98-31-33912628, E-mail address:

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[email protected] Abstract

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In this paper an energy-based failure envelope for unidirectional fiber reinforced polymers (FRP) has been developed. It is based on the stress and the damage energy release rate (DERR) components, which makes it suitable for being used as a damage surface in continuum damage mechanics (CDM), when a thermodynamically consistent evolution

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rule is intended to be used. The proposed failure surface is an interactive quadratic function of the DERR components accompanying with linear stress components in the form of an invariant. The mathematical formulation of the criterion is originated from the

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fully interactive Tsai-Wu criterion. The failure locus of the two criteria were compared

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with each other for several unidirectional FRP composites under several combined stress states. The two criteria were found to be in a reasonably good agreement for investigated composites with Eglass and carbon fibers, especially where the experimental data were available.

Keywords

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A. Polymer-matrix composites (PMCs); A. Glass fibers; C. Failure criterion; C. Damage mechanics; Thermodynamics of irreversible processes

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1. Introduction1 Fiber reinforced polymers (FRPs), due to their high strength-to-weight ratio, are very

attractive for weight reduction purposes in transportation industries. In particular, their

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use has become indispensable in aerospace vehicles where the weight reduction is a key parameter. The increasing use of composites in advanced industries requires an accurate

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modeling of their failure. However, due to their anisotropy and heterogeneity, the failure analysis in a structural composite element under combined stresses is quite complicated. Micro, meso and macro-mechanical approaches have been used for modeling of the failure in composites. Macro-mechanical approaches include failure criteria approach,

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fracture mechanics approach and continuum damage mechanics approach. Fracture mechanics based models have been successfully used for modeling of the delamination between adjacent layers with different angle plies. The intra-laminar damage, however,

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has been more effectively analyzed using either the failure criteria or the damage

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mechanics based models.

The failure criteria approach along with some kind of a degradation rule can be used to model strength reduction of structural elements under a critical loading [1]. The failure criteria can be classified into mode-dependent and mode-independent or interactive 1

Used abbreviations: FRP: Fiber Reinforced Polymers CDM: Continuum Damage Mechanics RVE: Representative Volume Element DERR: Damage Energy Release Rate

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criteria [2]. The best criteria among numerous proposed ones are those that better predict the material failure in the combined stress states. Many efforts have been devoted to

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compare different failure theories with each other and with experimental data [1, 3-5]. One of the most pervasive works was done by Hinton, Soden and Kaddour in three phases [6-8]. In the 2D state of stress [6], seventeen of the most popular failure theories were

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compared with each other in a number of predefined test cases. It was concluded that for the unidirectional lamina, the Tsai-Wu approach [9, 10] has described the available

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experimental results better than any other theory.

On the other hand, the CDM approach has been proved to be efficient in damage modeling of composites [11]. There is a large amount of literature on the CDM modeling of composites which differs from each other mainly in the definition of damage initiation

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and damage evolution rules. Various failure criteria such as the Hashin [12], Tsai [13] and Puck [14] have been used to describe the damage initiation [15-17]. Usually, the most challenging step in the development of an anisotropic progressive failure model is defining

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damage evolution rules for several damage parameters. The common solution to this

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problem is to use the standard bi-linear relationship [18, 19] or exponential damage evolution rules [20-22] based on the curve fitting to the maximum strain and fracture energies extracted from experimental data. Another approach for the problem is to adopt thermodynamically consistent evolution rules which are based on the maximization of damage dissipation energy [23-25]. In this approach, the damage evolution is described by a normality rule named flow rule which gives the growth of each damage parameter

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proportional to the gradient of a potential function with respect to the corresponding damage energy release rate (DERR) components. Therefore, it is required to describe the

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damage surface and potential functions in the DERR space rather than the stress or strain spaces. Although many efforts so far have been devoted to describe the damage threshold in terms of stress or strain components which have led to various failure criteria, the

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description of damage surface in terms of the DERR components has not received much attention. Only a few number of such theories exist [25, 26], none of which have exhibited

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the locus of suggested damage surfaces in the stress space.

In this paper an interactive failure surface based on the stress and the DERR components has been formulated, and its locus has been exhibited in the stress space. Contrary to the conventional failure theories which are based on the stress or strain components, the

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present criterion is based on the damage energy release rate (DERR) components. From one point of view, it can be used as a conservative failure criterion to predict the initiation of failure in the material. From another point of view, it can be considered as a damage

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surface in a CDM formulation if nonlinear behavior of the material is intended to be

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modeled. In the present work, the proposed failure surface has been considered as a failure criterion. It has been used to determine the failure locus for some unidirectional Eglass and carbon FRPs to be compared with the experimental data. In the cases that the experimental data were not available, the results were compared with the Tsai-Wu criterion, which is one of the most successful failure theories especially in predicting failure in unidirectional laminates [27].

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2. Continuum damage mechanics (CDM) formulation 2.1 Damage strain energy release rate

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In continuum damage mechanics, the damage parameter is introduced as a new state variable that describes the deterioration of the material due to the internal defects. The phenomenological description of anisotropic damage can be done by a second order

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damage parameter D ij [24]. The thermodynamic force associated with the damage

variable is also a second-order tensor, named damage energy release rate (DERR) tensor

Yij = − ρ

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as follows [23]:

∂ψ ∂Dij

(2)

where ψ is the free energy and ρ is the density of the material. The free energy of the

* ψ = 12 ε mn Emnpq ε pq

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damaged material can be defined as [28]:

(3)

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in which E* is the elastic modulus of the damaged material. Using the hypothesis of the equivalent strain energy, it is related to the undamaged modulus as [28]: (4)

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* −1 −1 E mnpq = M minj E ijkl M kplq

where M−1 is the damage effect tensor which relates the effective stress σ* in a fictitious undamaged configuration to the Cauchy stress σ in the damaged configuration as follows:

σ ij* = M imjnσ mn

(5)

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The definition of M−1 is not unique. Among several definitions, the one suggested by Cordebios and Sidroff has been used in here [29]: −1 M imjn = 12 [(δim − Dim )δ jn + (δ jn − D jn )δim ]

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(6)

where δ is the Kronecker delta and D is the damage parameter tensor which can be

assumed to be diagonal in the principal material coordinate of a unidirectional composite

0  0  D33 

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0  D11  Dij =  D22  sym.

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material [25]:

(7)

D11 , D22 and D33 represent the material failure in the fiber direction, transverse direction and through the thickness direction. Diagonal D implies that the material still remains

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orthotropic after damage. It is a reasonable assumption because the dominant modes of the damage reasonably fulfill this assumption [30]. Substituting equations (3,4) into equation (2) and doing some manipulation, the DERR

* ∂E mnpq

∂D ij

−1 ε pq = ε mn M mrns E rsuv

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Y ij = 12 ε mn

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tensor can be obtained as:

−1 ∂M upvq

∂D ij

ε pq

(8)

Y11 , Y22 and Y33 represent the rate of energy released due to the damage growth in the three principal directions of the orthotropic material. The greater the amount of Yij , the greater the potential of the material for the Dij growth. This physical interpretation makes the Yij tensor a good candidate to be employed instead of the stress or strain tensors as

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the building block in describing the failure envelope. The physical interpretation of a failure criterion based on the Yij components is that the material fails when a specified

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amount of energy would be released due to its failure. 2.2 damage evolution rule

Damage evolution is an irreversible process that dissipates the energy according to the

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following relation:

(9)

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Π =Y ij D& ij ≥ 0

According to the maximum dissipation principal, among all the admissible states that fulfill the above inequality, the one takes place that maximizes the dissipated energy Π . During the damage process the stress state should remain over the damage surface i.e. g = 0 .

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Maximizing the dissipated energy under the constraint g = 0 leads to the following expression using the Lagrange multiplier method [23]: ∂g D& ij = λ& ∂Y ij

(10)

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in which g is the damage surface and λ& is the Lagrange multiplier which is determined

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using the constraint g = 0 . Using the evolution equation (10) requires the definition of the damage surface g in terms of the conjugate forces σ, Y i.e. g = g (σ, Y) . The damage surface represents the threshold of damage, thus, a failure surface can be used as a damage surface in continuum damage modeling and/or as a conservative failure criterion along with some kind of a degradation rule for modeling of an abrupt damage in composites.

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3. Invariant energy-based failure criteria Interactive failure theories are essentially the development of the successful yielding

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theories of ductile materials in 1950's, to account for the material anisotropy and the difference in tensile and compressive strengths observed in FRP composites. From the

most popular interactive failure theories are the well-known Tsai-Hill, the Hoffman and

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the Tsai-Wu criteria which are quadratic failure theories and are generally based on the stress components. Similar to the Tsai-Wu theory, an interactive failure criterion can be

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proposed based on the DERR components accompanying with linear terms of stress as follows: H i σ i + J ijY iY j = 1

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where H and J are second and forth order tensors which are related to the basic

stresses.

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strengths of the material as is explained in the following for two-dimensional state of

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Assuming the damage parameter tensor to be diagonal, the DERR tensor Y is also obtained to be diagonal. Furthermore, the shear strengths of an orthotropic material in its

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principal material coordinate should be independent of their sign, thus the expanded form of the above equation becomes:

H 1σ1 + H 2σ 2 + H 3σ 3 + H 4 σ 4 + H 5 σ 5 + H 6 σ 6 + J 11Y 12 + J 22Y 22 + J 33Y 32 + 2J 12Y 1Y 2 + 2J 13Y 1Y 3 + 2J 23Y 2Y 3 = 1

(12)

which is reduced to the following expression in a 2D state of stress: H 1σ 1 + H 2σ 2 + H 4 σ 4 + J 11Y 1 2 + J 22Y 22 + 2 J 12Y 1Y 2 = 1

(13)

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The linear terms of the above equation with normal stresses allow for the distinction between the tensile and compressive strengths. The linear term with shear stress is used

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for adjusting the in-plane shear strength of the material. The five principal strength data, i.e. F1t , F1c , F2t , F2c , Fs , should be used to calculate the

corresponding ultimate stains at failure. The ultimate strain data are used in equation (8)

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to compute the five corresponding ultimate values of the DERR components, i.e.

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Y 1t ,Y 1c ,Y 2t ,Y 2c ,Y 1s =Y 2s . Inserting these five values in equation (13) leads to a set of five linear equations with five unknowns, H 1 , H 2 , H 4 , J 11 , J 22 .

The coefficient J 12 shows the interaction between Y 1 and Y 2 . It should be measured by a test in which both Y 1 and Y 2 are nonzero. This requires a combined stress state such as

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biaxial or off-axis tension tests. Such experimental data are not usually reported in the data sheets available for conventional FRPs. However, from a mathematical point of view there are upper and lower bounds for J 12 as explained in the Appendix. Although J 12 is

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expected to be a material dependent parameter, for the present time the mean value of

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the admissible interval, i.e. J 12 = 0 , is assigned to it. 4. Results and discussion The proposed failure criterion is compared with the Tsai-Wu criterion and the experimental data [31] available for unidirectional Eglass/LY556Epoxy and T300/BSL914C FRP composites. The proposed failure criterion is then compared with the Tsai-Wu criterion, for unidirectional Eglass/Polyester FRP composite. The material properties are

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listed in Table 1. The ultimate values of the DERR components have been calculated and used to compute the model constants H, J as are listed in Table 2.

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Figs. 1a and 1b show the proposed failure criterion in comparison with the Tsai-Wu criterion and experimental data in the bi-axial and normal-shear stress states, for

Eglass/LY556Epoxy lamina. It can be seen in Fig. 1a that the proposed failure criterion is in

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a very close agreement with the Tsai-Wu criterion in the 1st, 2nd and 4th quadrants.

However, it differs from it in the 3rd quadrant where the experimental data are not

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available. In Fig. 1b, both the proposed criterion and the Tsai-Wu criterion properly predict the interaction between the shear and the transverse stresses. Figs. 2a and 2b show the proposed failure criterion for a T300/BSL914C lamina. It can be seen in Fig. 2a that the proposed failure criterion coincides with the Tsai-Wu criterion in

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the 1st and 2nd quadrants with tensile transverse stresses. In the 3rd quadrant however, the proposed criterion is more conservative than the Tsai-Wu criterion, while it is more extended than the Tsai-Wu criterion in the 4th quadrant. Unfortunately, no experimental

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data exist for biaxial states of stress in Fig. 2a. Thus, it is not possible to judge between the

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two failure theories where they deviate from each other. However, since the Tsai's theory predicts excessive strength in the 3rd quadrant compared to other failure theories studied by Kaddour et al. [6], the more conservative predictions of the proposed criterion in this quadrant seems to be more reliable for design purposes. In Fig. 2b, the proposed criterion has predicted an enhancement in shear strength under the application of moderate longitudinal tensile loads which is not captured by the Tsai-Wu criterion.

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To further investigate the characteristics of the proposed criterion and reveals the effect of different terms of equation (13) on the failure surface geometry it is depicted in more

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details for another type of Glass-FRP composite, Eglass/Polyester, in Fig. 3. Since the experimental data are not available for this material in bi-axial and normal-shear stress

states it is compared to the Tsai-Wu criterion which has been ranked highly for predicting

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the strength of unidirectional lamina under combined loads [27]. Furthermore, the two criteria were shown to be in a very close agreement with each other for the previously

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mentioned Glass-FRP composite material, Eglass/LY556Epoxy, in Fig. 1.

Figs. 3a, 3c and 3d show a reasonably good agreement between the two criteria in bi-axial and normal-shear stress states for a pultruded Eglass/Polyester unidirectional composite material. To illustrate the effect of linear terms in equation (13) on the failure envelope

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geometry, the two criteria are also plotted in the bi-axial stress states without the contribution of their linear terms which results in equal tensile and compressive strengths as illustrated in Fig. 3b. It can be seen that the failure envelope of the proposed criterion

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without its linear terms is parallelogram-shaped compared to the quasi-elliptical shape of

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the Tsai-Wu criterion. This is due to the contribution of the Poisson's ratio in the formulation of the DERR components which are based on the strain components (equation (8)). Thus, the parallelogram shape of the energy-based theory resembles that of the maximum strain theory in the σ1 − σ 2 plane. Moreover, to have an idea about the effect of the interaction parameter J12 on the failure surface geometry, the criterion has been demonstrated using non-zero interaction terms,

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J 12 = ±0.1J 12* , J 12 = ± J 12* and J 12 = ±10J 12* in Fig. 4, with J 12* = J 11J 22 as a norm for the

interaction term as defined in the Appendix. It can be seen in Figs. 4a, 4b and 4c that

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J 12 = ±0.1J 12* and J 12 = J 12* do not change the geometry of the failure envelope

considerably from that of Fig. 3a with J 12 = 0 . However, J 12 = − J 12* and J 12 = ±10J 12*

results in either convex or open envelopes in the stress space as can be seen in Fig. 4d, 4e

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and 4f, which are not acceptable.

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5. Conclusion

An energy based failure criterion based on the DERR and stress components in the form of an invariant has been developed. The tensorial nature of the formulation makes it independent of the choice of the coordinate system. The DERR components represent the

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potential of the material for the damage growth. The physical interpretation of a failure criterion based on the DERR components is that the material begins to fail when a specified amount of energy would be released due to its failure. The predictions of the

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proposed failure envelope were shown to be in a close agreement with the available experimental data for Eglass/LY556Epoxy and T300/BSL914C unidirectional laminates. The

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failure surface has also been demonstrated for Eglass/Polyester unidirectional composite in comparison with the Tsai-Wu criterion. It was shown that the proposed surface was in a reasonably good agreement with the Tsai-Wu criterion, which is one of the most successful failure theories in predicting failure in unidirectional laminates. Finally, the failure locus was drawn using various J 12 coefficients. It was shown that the failure envelopes were not so sensitive to the J 12 coefficient inside its admissible domain. So the

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assumption J 12 = 0 can be used when it is not easy to perform combined stress experiments. However, having sufficient experimental data available, nonzero J 12

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coefficient can be adjusted to provide the best match with experimental data. Acknowledgements

This research did not receive any specific grant from funding agencies in the public,

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commercial, or not-for-profit sectors. Appendix

J 11Y 12 + J 22Y 22 + 2J 12Y 1Y 2 = 1

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The proposed criterion without its linear terms reduces to

(A.1)

which is a quadratic curve (conic section) in the Y 1 −Y 2 space with its discriminant as: (A.2)

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Discriminant = J 11J 22 − J 122

Depending on the magnitude of the discriminant, the quadratic curves have different shapes. A positive discriminant represents an ellipse, a negative discriminant represents a

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hyperbola and zero discriminant leads to parallel lines. In order to avoid an open failure surface in the Y 1 −Y 2 space, its discriminant should be positive which leads to the

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following upper and lower bounds for the interactive coefficient:

− J 11J 22 < J 12 < J 11J 22

(A.3)

Defining J * = J 11J 22 the upper and lower bounds of the J 12 is defined as: −J * < J 12 < J *

(A.4)

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Note that the assumption J 12 = 0 is always admissible because it ensures a closed

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elliptical failure surface in the Y 1 −Y 2 space.

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Figure captions Fig.1 Comparison of the present criterion with the Tsai-Wu criterion and experimental

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data available for an Eglass/LY556Epoxy unidirectional laminate.in: (a) bi-axial stress state, (b) normal-shear stress state

Fig.2 Comparison of the present criterion with the Tsai-Wu criterion and experimental

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data available for a T300/BSL914C unidirectional laminate.in: (a) bi-axial stress state, (b) normal-shear stress state

Eglass/Polyester composite in:

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Fig.3 Comparison of the present criterion with the Tsai-Wu criterion for an

(a) bi-axial stress state, (b) bi-axial stress state without the contribution of the linear stress terms, (c) and (d) normal-shear stress states

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Fig.4 Comparison of the present criterion with the Tsai-Wu criterion for Eglass/Polyester composite laminate using:

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J 12 = −10J 12*

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(a) J 12 = 0.1J 12* , (b) J 12 = − 0.1J 12* , (c) J 12 = J 12* , (d) J 12 = −1J 12* , (e) J 12 = 10J 12* , (f)

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Table 1. Material properties Eglass/

T300/

Eglass/

LY556Epoxy *

BSL914C*

Polyester

E1

45.6

Transverse modulus (GPa)

E2

16.2

In-plane shear modulus (GPa)

G12

5.83

Major Poisson's ratio

υ12

0.278

138

29.6

11

10.0

5.5

4.1

0.28

0.29

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Longitudinal modulus (GPa)

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Symbol

Property

F1t

1280

1500

490

Longitudinal compressive strength (MPa)

F1c

800

900

435

Transverse tensile strength (MPa)

F2t

40

27

60

F2c

145

200

90

73

80

30

Longitudinal tensile strength (MPa)

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Transverse compressive strength (MPa) In-plane shear strength (MPa)

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Stiffness and strength properties from [32]

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*

Fs

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Table 2. Model data

BSL914C

Polyester

Y 1t

3.59e7

1.63e7

8.11e6

Y 1c

1.40e7

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Damage energy release rate

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corresponding to F1t (Pa)

Damage energy release rate

Eglass/

LY556Epoxy Damage energy release rate

corresponding to F1c (Pa)

T300/

RI PT

Symbol Eglass/

Quantity

5.87e6

6.39e6

Y 2t

9.88e4

6.63e4

3.36e5

Y 2c

1.30e6

3.64e6

7.57e5

Y 1s =Y 2s

4.57e5

5.82e5

1.5e5

Model constant 1 (Pa)-2

J11

1.62e-15

8.25e-15

1.90e-14

Model constant 2 (Pa)-2

J 22

2.69e-12

6.34e-13

3.37e-12

Model constant 4 (Pa)-1

H1

-8.51e-10

-7.95e-10

-5.12e-10

Model constant 5 (Pa)-1

H2

2.43e-8

3.69e-8

1.03e-8

Model constant 6 (Pa)-1

H4

6.0e-9

9.78e-9

3.08e-8

corresponding to F2t (Pa)

TE D

Damage energy release rate corresponding to F2c (Pa)

Damage energy release rate

AC C

EP

corresponding to Fs (Pa)

ACCEPTED MANUSCRIPT

Proposed criterion Tsai-Wu criterion Experiment [31]

Proposed criterion Tsai-Wu criterion Experiment [31]

1

(c)

1.4 1.2

-1

1

/

/

0

0.8

-3

0.6

-4

0.4

-5

0.2

-6

SC

-2

(c)

RI PT

2

0 -0.5

0.5

AC C

EP

TE D

/

1.5

-4

M AN U

-1.5

-2

/

0

ACCEPTED MANUSCRIPT

Proposed criterion Tsai-Wu criterion Experiment [31]

Proposed criterion Tsai-Wu criterion Experiment [31] 1.8

(a)

1.6

0

(b)

1.4

-2

1.2 1

-4

0.8 0.6 0.4

-8

0.2 0 -1

0

1

AC C

EP

TE D

-2

-0.8

-0.3

M AN U

-10

SC

-6

RI PT

2

0.2

0.7

1.2

ACCEPTED MANUSCRIPT

Proposed criterion Tsai-Wu criterion Strength data

Proposed criterion Tsai-Wu criterion Strength data

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

-2

-2.5

-2.5 -0.5

0.5

1.5

-1.5

-0.5

M AN U

-1.5

(b)

1.5

SC

1.5

2

(a)

RI PT

2

0.5

1.5

1.4

1.4

(c)

1.2

(d)

1.2 1

1

0.8

0.6 0.4 0.2 0 -0.5

0.5

AC C

EP

-1.5

TE D

0.8

1.5

0.6 0.4 0.2 0 -2

-1

0

1

ACCEPTED MANUSCRIPT Proposed criterion Tsai-Wu criterion Strength data

Proposed criterion Tsai-Wu criterion Strength data

2

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

-2

-2.5

RI PT

1

-2.5 -1.5

-0.5

0.5

1.5

2

-1.5

2

-0.5

0.5

1.5

M AN U

1.5

(b)

1.5

SC

1.5

2

(a)

(c)

(d)

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5 -2 -2.5 -0.5

2

(e)

1.5 1

0.5

-2 -2.5

1.5

EP

-1.5

TE D

-1.5

-1.5

-0.5

0.5

1.5

2

(f)

1.5 1 0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

-2

AC C

0.5

-2.5

-2.5 -1.5

-0.5

0.5

1.5

-1.5

-0.5

0.5

1.5