International Journal of Civil Engineering and Technology (IJCIET) Volume 9, Issue 3, March 2018, pp. 282 291, Article ID: IJCIET_09_03_030 Available online at http://http://www.iaeme.com/ijciet/issues.asp?jtype=ijciet&vtype=9&itype=3 ISSN Print: 0976-6308 and ISSN Online: 0976-6316 IAEME Publication Scopus Indexed A STRAIN-BASED DAMAGE MECHANICS MODELING FOR WOVEN FABRIC COMPOSITES UNDER BIAXIAL FATIGUE LOADING Sajan Shakya and Kamal Bahadur Thapa Department of Civil Engineering, Pulchowk Campus, Institute of Engineering, Tribhuvan University, Nepal ABSTRACT A fatigue damage model for glass/epoxy woven fabric composites is developed in tension-tension fatigue loading. Glass-epoxy woven fabric composites during a tension-tension fatigue loading are highly affected by anisotropic nature of damage leading to the splitting failure modes. Damage is described through fourth-order tensor by using the second invariant of positive cones of strain tensor. The model is capable of capturing the stiffness degradation and inelastic deformations caused by fatigue induced distributed damage. The model, which is multi-axial, is checked against available experimental data, and found to be capable of predicting the fatigue behavior of glass fiber woven composites, under cyclic loading, very well. Key words: Glass-epoxy woven fabric composites, Damage, Fatigue, Anisotropic, Inelasticity, Biaxial loading Cite this Article: Sajan Shakya and Kamal Bahadur Thapa, A Strain-Based Damage Mechanics Modeling for Woven Fabric Composites Under Biaxial Fatigue Loading. International Journal of Civil Engineering and Technology, 9(3), 2018, pp. 282-291. http://www.iaeme.com/ijciet/issues.asp?jtype=ijciet&vtype=9&itype=3 1. INTRODUCTION Fiber-reinforced composites are increasingly used in numerous aerospace, automotive and construction industries because of their high specific strength and stiffness. Woven fabric composites are widely used in civil engineering, because they possess superior balanced properties in the fabric plane such as damage tolerance, dimensional stability, impact resistance, high strength and better performance during bi-axial stress state as compared to conventional unidirectional lamina. The ease of manufacturing and handling adds more to its beauty. The fatigue behavior of composite materials has been a topic of active research in the recent years. Widespread studies and researches on fatigue have concluded that as many as 90% of all the material failures are caused by fatigue (Abo-Elkhier, Hamada, & Bahei El- Deen, 2012). A number of experiments and researches on fatigue of fabric composite have http://www.iaeme.com/ijciet/index.asp 282 editor@iaeme.com
A Strain-Based Damage Mechanics Modeling for Woven Fabric Composites Under Biaxial Fatigue Loading occurred in the past. For homogeneous monolithic materials with isotropic material properties, fatigue failure is governed by a single crack that propagates in a direction perpendicular to the cyclic loading axis. However, Rim Ben Toumi (Toumi, Renard, Monin, & Nimdum, 2013) in his experiment, observed that in composite materials, the failure is characterized by multiple damage modes such as matrix cracking, fiber-matrix delamination and debonding, fiber breakage etc. Multitude of cleavage type distributed cracks, which are highly directional under tensile loading, were observed by Hansen (Hansen, 1999). In order to account for such progressive distributed damage reported by Hansen [3], Continuum Damage Mechanics theory has been considered to be a very useful tool for modeling fatigue behavior of woven fabric composite and has been followed by many researchers ( (Hansen, 1999) (Mao & Mahadevan, 2002) (Movaghghar & Ivanovich Lvov, 2012) (Wen & Yazdani, Anisotropic damage model for woven fabric composites during tension-tension fatigue, 2007) (Wen, Yazdani, Kim, & Abdelrahman, 2012) (HU & ZHANG, 2011)). The strain space approach is accepted to be very advantageous for numerical simulation and for complete stress-strain behavior ( (Thapa & Yazdani, A Strain Based Damage Mechanics Model for Plain Concrete, 2014) (Thapa & Yazdani, Combined damage and plasticity approach for modeling brittle materials with application to concrete, 2013) (Bhandari & Thapa, 2013)). The idea of using the changes in the material stiffness E as an indirect measure of crack growth, internal changes and energy dissipation due to damage process during fatigue has been widely used in past researches ( (Toumi, Renard, Monin, & Nimdum, 2013) (Hansen, 1999) (Mao & Mahadevan, 2002) (Bhandari & Thapa, 2013) (Subramanian, Reifsnider, & Stinchcomb, 1995) (Wu & Yao, 2010)). One such approach was used by Hansen (Hansen, 1999) in his fatigue damage model for woven fabric composite. Hansen (Hansen, 1999) performed experiments and observed damage initiation and growth in glass-reinforced woven composites both in static and fatigue environment. Three types of specimens were taken for experiment, namely: undamaged, barely visible impact damaged (BVID) and penetrated damage. Infrared thermography was used as non-destructive inspection technique for detecting damage initiation and growth. Hansen (Hansen, 1999) observed that fatigue produces continuously changing non-uniform stress fields due to impact induced local stress raiser and stress redistribution effects. Further three damage phases were noticed. In phase I, material property was degraded sharply and was caused by knee-effect, whereby glass-epoxy debonding and failures were observed in transverse fiber bundles resulting in microcracks. Saturation of these cracks indicated the end of phase I and beginning of phase II. In Phase II, friction between fibers and some delamination were observed that continued the damaging process keeping the microcrakcs still progressively distributed. Almost 92% of the fatigue life was governed by phase I and II. Similarly, phase III was characterized by localized damage, fiber breakage and failure of the specimen. Hansen (Hansen, 1999) has developed constitutive model for the fatigue damage for phase I and II using Continuum Damage Mechanics. The material compliance tensor was considered as an internal variable evolving with damage. However, the model was simple and reliable, it did not address anisotropic effects of cracking and permanent deformations due to fatigue environment. Chao (Wen & Yazdani, Anisotropic damage model for woven fabric composites during tension-tension fatigue, 2007) has developed anisotropic and inelastic fatigue damage model for glass-epoxy woven fabric composites during tension-tension fatigue based on damage mechanics theory which was capable of incorporating multitude of matrix cracking and interfacial debonding. The model was capable of capturing the elastic degradation process as http://www.iaeme.com/ijciet/index.asp 283 editor@iaeme.com
Sajan Shakya and Kamal Bahadur Thapa well as the permanent deformation due to damage and anisotropic characteristics also. The model predictions were compared with Hansen s experimental works. However, the formulation relied on stress based approach and subsequently the static stress-strain behavior has not been illustrated. Cyclic stress-strain behavior has also not been able to satisfy the developed S-N plot. The variation of inelastic strain accumulation with number of cycle has not been discussed. Further, Chao (Wen, Yazdani, Kim, & Abdelrahman, 2012) has developed generalized unified bounding surface approach guided by isotropic hardening/softening theories of plasticity and damage mechanics to predict the fatigue behavior of the woven composites under biaxial loadings, however, it still was stress space formulation. Thapa and Yazdani (Thapa & Yazdani, A Strain Based Damage Mechanics Model for Plain Concrete, 2014) has developed a damage model for structural concrete in strain space within the continuum thermodynamics framework. The equivalency of the stress and the strain space formulation was established. Rate independent behavior, infinitesimal deformations and isothermal conditions were assumed and Helmholtz Free Energy (HFE) was utilized as an energy potential to develop damage surface. Anisotropy caused by induced cracking was captured by using kinetic relations which was developed by adopting additive decomposition of the stiffness tensor. However, the model was capable of capturing the general mechanical behavior of concrete in both tension and compression, it lacks addressing the material non linearity under large confining pressure which actually can be captured using a plasticity type approach. It was concluded that strain based formulation involves iterative procedure which actually reduces processing time and also optimizes the data storage. Shakya and Thapa (Shakya & Thapa, 2018) developed a uniaxial fatigue damage model for glass/epoxy woven fabric composites in strain space using Continuum Damage Mechanics. This paper aims to perform the extension of the uniaxial strain space formulation of Shakya and Thapa (Shakya & Thapa, 2018) in biaxial fatigue damage modeling of glass/epoxy woven fabric composites so that the complete multi-axial stress-strain behavior of such material can also be illustrated. 2. FORMULATION For an inelastic damaging process, with an assumption of small deformations which is valid for brittle materials and low frequency fatigues where thermal effects could be ignored, a constitutive relation between stress and strain tensors can be deduced from Thapa and Yazdani (Thapa & Yazdani, A Strain Based Damage Mechanics Model for Plain Concrete, 2014) utilizing fourth order material stiffness tensor as: ( ) ( ) (1) where, the stress and strain tensors are given by and respectively, denotes Helmholtz Free Energy, denotes the material stiffness tensor that depends on the state of microcrack growth and is the cumulative scalar fatigue damage parameter. Here, the tensor contraction operation is designated by : and the stress tensor corresponding to the inelastic damage is given by. Assuming N being the fatigue cycle number, taking rate form of Eq. (1) by differentiating with respect to N : ( ) ( ) ( ) (2) = ( ) http://www.iaeme.com/ijciet/index.asp 284 editor@iaeme.com
A Strain-Based Damage Mechanics Modeling for Woven Fabric Composites Under Biaxial Fatigue Loading where, denotes the stress increment when further damage in the material is prevented, is the stress relaxation rate due to further microcracking (elastic damage) and ( ) designates the stress tensor rate corresponding to the irreversible deformation due to microcracking. It is further assumed that, damage during fatigue loading degrades elastic properties and affects the stiffness tensor. The damage is recorded in the fourth order material stiffness tensor E. To account for induced anisotropy, the following additive decomposition of E is adopted: ( ) ( ) (3) where, is the stiffness tensor of the initial undamaged or uncracked material and ( ) is the overall stiffness degradation caused by damage during fatigue loadings. Further, the rate of stiffness tensor ( ) and that of the inelastic stress tensor are expressed as fluxes in thermodynamics state sense and are expressed in terms of evolutionary equations as, and (4) where, L and M are the fourth and second order response tensors that determine the direction of the elastic and inelastic damage processes. To progress further, specific forms of response tensor L and M must be specified. Since the damage is highly directional (i.e. anisotropic), the response tensor should be formulated to achieve anisotropy. For addressing anisotropy and formulating response tensor, let s decompose the strain tensor into its positive and negative cones. The positive and negative cones of the strain tensor holds the corresponding positive and negative eigenvalues of the system. Note that. Guided by the experimental results and observations for glass/epoxy woven fabric composite materials in tension by Hansen (Hansen, 1999), where majority of damage is shown to take place in the direction of applied strain and in tension regimes (i.e. the cleavage mode of cracking as shown in Fig.1), and further assuming that there is no coupling between cleavage type cracks in orthogonal direction, the following form of response tensors are proposed. Figure 1 Cleavage mode of cracking (5) ( ) (6) where, the symbol is the tensor product operation. Here, it should be noted that the material used in Hansen s work was assumed to be quasi-isotropic laminate, with the lay-up sequence of [(+45 0 #-45 0 )/(90 0 #0 0 )] s, made up of plain woven glass/epoxy prepreg fabrics. So, the strength values in different directions are considered equal initially. Here, α is the http://www.iaeme.com/ijciet/index.asp 285 editor@iaeme.com
Sajan Shakya and Kamal Bahadur Thapa material parameter and is introduced to incorporate the cumulative effect of damage caused by tensile deformations associated with biaxial direction (i.e. for biaxial loading condition as shown in Fig. 2) and is given by, Figure 2 Idealization of distributed cleavage mode of cracking during biaxial loading ( ) ( ) (7) where, is the traces of strain tensor i.e. sum of diagonals, λ is the maximum eigenvalue of applied strain tensor, is the Heaviside function of the minimum eigenvalue of ε and β is the material constant to be chosen based on the nature of brittle material. In this paper, an alternate and new damage evolution law is proposed using second invariant of the positive cone of the strain tensor, based on the fact that the damage in each cycle depends on elastic stiffness E, number of cycles N and second invariant of the positive cone of strain tensor. ( ) (8) where, A, B and m are material constants, N is the number of cycles and is the reference strain level. Differentiating with respect to number of cycles N, the increment of damage in one cycle is given as, ( ) (9) Substituting Eqs. (5), (6) and (9) into Eq. (4), yields ( ) (10) ( ) (11) ( ) Substituting Eqs. (3) and (4) into Eq. (2), we get: [ ( ) ] ( ) (12) ( ) 3. NUMERICAL SIMULATION The proposed model has five material parameters A, B, m, α and β. In order to check the validity of the model and to obtain the value of the parameters introduced, numerical simulation has been performed and the comparison is done with the experimental results by http://www.iaeme.com/ijciet/index.asp 286 editor@iaeme.com
A Strain-Based Damage Mechanics Modeling for Woven Fabric Composites Under Biaxial Fatigue Loading Hansen (Hansen, 1999). Due to scarcity of experimental data in the literature for the measurement of these parameters in the numerical simulation, judgments were used to obtain the numerical results. On the basis of the experiment carried out by Hansen (Hansen, 1999), the following constants were used for numerical simulation. A = 0.03, B = -0.74, m = 1.4, α = 0.0035 and β = 0.015. These parameters were estimated by comparing predicted results and experimental results over a range of applied stresses. The initial strain tensor was taken ε + = [0.007 0 0] corresponding to the applied far-field stress tensor of σ + = [155 0 0] MPa. The reference strain was fixed at 0.0156. In the case of BVID specimen, the initial uniform impact damage was considered to be 2.5% prior to fatigue. The theoretical static stress-strain curves for both uniaxial and biaxial monotonic loading conditions have been illustrated in Fig.3. The variation of both the longitudinal and transverse strains with respect to the applied farfield stress have also been illustrated in this figure. The comparison with experimental data from Hansen (Hansen, 1999) for uniaxial loading condition is done. The strength of woven fabric composite is found to be reduced in biaxial loading condition because the curve shows that the failure stress and strain drastically reduces in biaxial loading condition as compared uniaxial loading condition. Similarly, Fig.4 clearly shows that the proposed model captures the initial damage due to the knee effect quite well. The model is useful in prediction of Phase I and Phase II but could not predict Phase III. The predicted S-N curve is compared with the experimental S-N curve by Hansen (Hansen, 1999) for BVID specimen. The effect of material parameter m on predicted S-N curve is also shown in Fig.5. It can been seen that the trend of predicted and experimental S-N curves matches each other which is quite satisfactory. Figure 3 Uniaxial and biaxial static monotonic stress-strain curve http://www.iaeme.com/ijciet/index.asp 287 editor@iaeme.com
Sajan Shakya and Kamal Bahadur Thapa Figure 4 Longitudinal stiffness reduction with number of cycles Figure 5 Comparison of experimental S-N curve by Hansen [3] with predicted S-N curves The fatigue stress-strain curves are shown in Fig.6 and 7 for uniaxial loading condition. Fig.6 illustrates the elastic degradation as well as a permanent deformation due to inelastic behavior. However, Fig.7 illustrates the nature when the composite is treated as perfectly elastic with only reversible deformations, though the damage is progressively accumulating as evidenced by degradation of longitudinal stiffness. This behavior corresponds to an idealized case whereby crack faces close perfectly upon unloading and is achieved in the model by letting zero value to α. But in most of the heterogeneous materials like glass-fiber composites, permanent deformations take place which has been replicated in Fig.6. The inelastic damage accumulation i.e. the cumulative permanent deformation under fatigue loading is also illustrated in Fig.8. http://www.iaeme.com/ijciet/index.asp 288 editor@iaeme.com
A Strain-Based Damage Mechanics Modeling for Woven Fabric Composites Under Biaxial Fatigue Loading Figure 6 Cyclic stress-strain behavior of woven composite during inelastic damage accumulation (N f = 48993, ε f = 0.0156, σ f = 155MPa) Figure 7 Cyclic stress-strain behavior for elastic damaging process (N f = 129355, ε f = 0.0156, σ f = 155MPa) Figure 8 Inelastic damage accumulation http://www.iaeme.com/ijciet/index.asp 289 editor@iaeme.com
Sajan Shakya and Kamal Bahadur Thapa In Fig. 9, theoretical cyclic stress-strain curve for biaxial fatigue loading is illustrated. The elastic degradation as well as a permanent deformation due to inelastic behavior have been replicated. The stress tensor is fixed at 155MPa, after which unloading is done. 1747 cycles is observed as a fatigue life to reach the fatigue failure strain of 0.0156. This fatigue life is much less than that of uniaxial loading condition. The reason behind such less fatigue life might be that the orthogonal microcracks during biaxial loading condition accelerates and activates each other. The inelastic damage accumulation under biaxial fatigue loading is presented in Fig. 10. At the fatigue cycle of 1747, the inelastic strain is found to be nearly 0.0012 as also seen in Fig. 9. Figure 9 Cyclic stress-strain behavior for biaxial loading (N f = 1747, ε f = 0.0156, σ f = 155MPa) Figure 10 Inelastic damage accumulation during biaxial loading 4. CONCLUSIONS An anisotropic-inelastic fatigue damage model is established for glass/epoxy woven fabric composite in tension-tension fatigue utilizing Continuum Damage Mechanics formulation in strain-space. The distributed progressive weakening of the composite due to the formation of micro-cracks and micro-voids is reflected through the reduction of fourth-order material stiffness tensor. The damage mechanics formulation is cast within the general framework of the internal variable theory of thermodynamics considering rate-independent behavior, http://www.iaeme.com/ijciet/index.asp 290 editor@iaeme.com
A Strain-Based Damage Mechanics Modeling for Woven Fabric Composites Under Biaxial Fatigue Loading infinitesimal deformations and Helmholtz Free Energy (HFE) as an energy potential. The proposed damage evolution law consists five damage parameters. For the numerical simulation, the parameters, A, B, m, α and β are determined from the sensitivity study and the values are selected such that they best fit the experimental data. This model considers the effects of multi-axial loading condition too. So it has a potential to be used in modeling of retrofitting of real structural components like bridge pier, beam, column etc. REFERENCES [1] M. Abo-Elkhier, A. Hamada and A. Bahei El-Deen, "Preediction of fatigue life of glass fiber reinforced polyster composites using model testing," International Journal of Fatigue, pp. 1-8, 2012. [2] R. B. Toumi, J. Renard, M. Monin and P. Nimdum, "Fatigue damage modeling of continuous E-glass fibre/epoxy composite," 5th Fatigue Design Conference, Fatigue Design 2013, vol. 66, pp. 723-736, 2013. [3] U. Hansen, "Damage Developement in Woven Fabric Composites during Tension- Tension Fatigue," Journal of Composite Materials, vol. 33, no. 7, pp. 614-639, 1999. [4] H. Mao and S. Mahadevan, "Fatigue damage modeling of composite materials," Composite Structures, vol. 58, pp. 405-410, 2002. [5] A. Movaghghar and G. Ivanovich Lvov, "Theoretical and Experimental Study of Fatigue Strength of Plain Woven Glass/Epoxy Composite," Journal of Mechanical Engineering, vol. 58, no. 3, pp. 175-182, 2012. [6] C. Wen and S. Yazdani, "Anisotropic damage model for woven fabric composites during tension-tension fatigue," Composite Structures, vol. 82, no. 1, pp. 127-131, 2007. [7] C. Wen, S. Yazdani, Y. J. Kim and M. Abdelrahman, "Bounding Surface Approach to the Modeling of Anisotropic Fatigue Damage in Woven Fabric Composites," Open Journal of Composite Materials, vol. 2, pp. 125-132, 2012. [8] Z.-g. HU and Y. ZHANG, "Continuum damage mechanics based modeling progressive failure of woven-fabric composite laminate under low velocity impact," Journal of Zhejiang University, vol. 11, no. 3, pp. 151-164, 2011. [9] K. B. Thapa and S. Yazdani, "A Strain Based Damage Mechanics Model for Plain Concrete," International Journal of Civil Engineering Reseacrh, vol. 5, no. 1, pp. 27-40, 2014. [10] K. B. Thapa and S. Yazdani, "Combined damage and plasticity approach for modeling brittle materials with application to concrete," International Journal of Civil and Structural Engineeering, vol. 3, no. 3, pp. 513-525, 2013. [11] D. Bhandari and K. B. Thapa, "Constitutive Modeling of Concrete Confined by FRP Composite Jackets utilizing Damage Mechanics Theory," in ResearchGate, India, 2013. [12] S. Subramanian, K. Reifsnider and W. Stinchcomb, "A cummulative damage model to predict the fatigue life of composite laminates including the effect of a fiber-matrix interphase," International Journal of Fatigue, vol. 17, no. 5, pp. 343-351, 1995. [13] F. Wu and W. Yao, "A fatigue damage model of composite materials," International Journal of Fatigue, vol. 32, pp. 134-138, 2010. [14] S. Shakya and K. B. Thapa, "A Strain Based Fatigue Damage Model for Woven Fabric Composites," in IOE Graduate Conference, Kathmandu, 2018. [15] Satyam Kumar, Dr. Uday Krishna Ravella, Atul Kumar Shrivastava, Dr.Midathoda Anil and Dr. S. K. Kumar Swamy A Review on Human Cervical Fatigue Measurement Technologies and Data Analysis Methods. International Journal of Mechanical Engineering and Technology, 8(7), 2017, pp. 1474 1484 http://www.iaeme.com/ijciet/index.asp 291 editor@iaeme.com