Hysteretic equivalent continuum model of nanocomposites with interfacial stick-slip

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1 Hysteretic equivalent continuum model of nanocomposites with interfacial stick-slip Giovanni Formica 1, Michela Talò, Walter Lacarbonara 1 Dipartimento di Architettura, Università degli Studi Roma Tre formica@uniroma3.it Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma La Sapienza michela.talo@uniroma1.it, walter.lacarbonara@uniroma1.it Keywords: Eshelby-Mori-Tanaka homogenization, interfacial stick-slip, damping capacity. SUMMARY. The high dissipation properties exhibited by nanocomposite materials based on carbon nanotubes have recently drawn considerable attention in various fields including structural engineering for their potential applications. In this work an inelastic equivalent continuum theory is presented for nanocomposites combining the Eshelby-Mori-Tanaka approach (equivalent inclusion method and average stress theorem) with the concept of Mura of inhomogeneous inclusions having inelastic eigenstrains. The inelastic eigenstrains, which are here introduced to account for the stick-slip between carbon nanotubes and the matrix, are described by an evolution law based on a micromechanical adjustment of the von Mises yield criterion. The main constitutive parameters affecting the hysteretic behavior of the nanocomposites are identified through parametric analyses, and subsequent optimization studies are carried out to determine optimal material parameter combinations yielding maximum damping capacity. 1 INTRODUTION arbon nanotube (NT) composites exhibit interesting nonlinear mechanical features under cyclic loading such as hysteretic phenomena due to debonding [1] or stick-slip [] between the NTs and the surrounding hosting matrix. Although several independent experiments showed the high damping capacity of NT composites [3, ], only a few works dealt with nonlinear models capable of either describing the damping behavior of such nanostructured materials or characterizing the NT/matrix interface and the dissipation mechanism therein localized. In the last decade, some relevant works, e.g. [5], proposed elasto-plastic multi-scale models to describe the mechanical response of NT nanocomposites by adopting an imperfect interface to account for the slippage of the NTs with respect to the matrix. Other models have attempted to incorporate dissipative mechanical features of the nanocomposites, including the multi-scale, mass-spring model proposed by Fraternali et al. [6] for foam materials integrated by carpets of unidirectional NTs. The present work addresses an inelastic continuum theory, which can provide effective tools to either predict or characterize the energy dissipation due to the shear stick-slip between NTs and matrix. The developed nonlinear constitutive theory features an incremental formulation obtained as a combination of the Eshelby equivalent inclusion method [7], the Mori-Tanaka homogenization method [8, 9], and the treatment of inhomogeneous inclusions with inelastic eigenstrains proposed by Mura [1]. The introduced inelastic eigenstrains have the meaningful role of modeling the frictional sliding at the NT/matrix interface, and allow to define the elasto-plastic constitutive law through a micromechanical adjustment of the von Mises criterion. 1

2 The model has the virtue of accounting for micro-scale characteristics such as the NTs interface constitutive properties, in addition to NTs geometry, orientation, volume fraction as well as the macroscopic elastic properties. Moreover, the model has been conceived in such a way to be suitably implemented in standard finite element codes devoted to computational elasto-plasticity using time integrations. Numerical results obtained for polymer- and epoxy-based nanocomposites show the possibility of achieving optimal material performance. THE NONLINEAR MODEL Within the context of equivalent continuum micromechanics, the carbon nanotube composite is assumed to be a two-phase perfectly aligned unidirectional composite material made of a linearly elastic hosting matrix B M embedding aligned NTs (inclusions) B (see Figure 1). The volume fractions of the NTs and the matrix are defined in terms of the volume of the NTs, denoted by V, and the volume of the entire nanocomposite V ; thus φ := V /V and φ M = 1 φ characterize the volume concentration of NTs and matrix, respectively. 1 B 3 BM Figure 1: Two-phase unidirectional composite featuring the interfacial sliding of the aligned carbon nanotubes B embedded in the hosting matrix B M..1 Incremental onstitutive Equations The governing equations of the model are obtained starting from the incremental form of the Mori-Tanaka approach: Ė = φ M Ė M + φ Ė, (1) Ṫ = φ M Ṫ M + φ Ṫ () where the equivalent incremental strain and stress tensors, Ė and Ṫ, are given in terms of the equivalent continuum average strain and stress rates (ĖM, Ṫ M) in B M and (Ė, Ṫ ) in B, respectively. The incremental constitutive laws for the two phases are expressed as Ṫ M = L M : Ė M, Ṫ = L : (Ė ĖP ), (3) where (L, L M ) are the elastic tensors in B M and B, respectively. The incremental plastic strain Ė P introduced in (3) for the NT phase plays the role of describing the shear-stick slip at the NT/matrix interface. The interfacial frictional sliding between the NTs and the surrounding matrix causes a local inelastic behavior at the NT/matrix interface which can be effectively described as an elasto-plastic behavior localized in the NTs. According to the Eshelby equivalent inclusion

3 method, Ė P can be then treated as an incremental eigenstrain field in the inclusions, so that the induced elasto-plastic response can be averaged out throughout the equivalent homogenized body of the composite. Thus, employing the Eshelby equivalent inclusion method [7] yields L : (Ė ĖP ) = L M : (Ė ĖP Ė ) () where Ė represents a fictitious incremental eigenstrain which allows to express the incremental stress Ṫ acting in the NT inclusions in terms of the elastic tensor L M of the matrix. By making use of the Mori-Tanaka average stress and strain theorem [9], the incremental strains of the two phases can be expressed as a combination of the equivalent incremental strain filed Ė and incremental perturbation strain fields. These incremental perturbation strains are given in terms of increments of the eigenstrains. Thus, by treating the plastic strain increment ĖP as an eigenstrain increment [], a total incremental eigenstrain can be defined as Ė = Ė + ĖP. onsequently the incremental strains of the two phases are give by Ė M = Ė φ S : E, Ė = Ė + φ M S : E (5) where S is the Eshelby tensor. By introducing the incremental strain definitions given by (5) into the stress equivalence condition (), the total incremental eigenstrain can be obtained as Ė = H : (L : Ė P L : Ė), (6) where L := L L M is the jump in the material elastic tensors of the two phases and the tensor H is defined as: H := (L M + φ M L : S) 1. (7) Therefore, by using the constitutive equations (3) and the expressions (5), and by further exploiting (6), the incremental equivalent stress given by () becomes where Ṫ = L 1 : Ė φ L : Ė P, (8) L 1 := L M : (I + φ H : L ), L := L M : H : L. (9) The resulting equivalent elastic tensor L 1 turns out to be the equivalent elastic tensor of a purely elastic two-phase composite [8, 9] obtained by the standard Eshelby-Mori-Tanaka approach. Subsequently, the equivalent constitutive law for the inclusions in terms of average incremental strains can be obtained as Ṫ = L 1 : Ė L : Ė P (1) where L 1 := L φ M L : S : H : L, L := L φ M L : S : H : L. (11) 3

4 . Evolution law for the inelastic eigenstrain As a consequence of the Eshelby equivalent inclusion method (), an incremental stress discontinuity at the NT/matrix interface can be defined as T := L : (Ė ĖP ) = (I L M : L -1 ) : Ṫ. (1) The stress discontinuity turns out to be explicitly related to the incremental NT stress. This implies that all limit conditions enforced on the interface become limit conditions for the stress states in the NT inclusions. In particular, due to the nature of the interfacial slippage and the symmetry of the cylindrical body representing the NTs (see Figure 1), the incremental plastic shear strains γ P 13 = γ P 3 = γ P are assumed as the only nontrivial components of ĖP. Hence, the activation of the interfacial energy dissipation can be regulated by the NT shear stress components (T 13, T 3 ), related to the corresponding incremental plastic shear strains. The evolution of the plastic eigenstrain γ P is described by a constitutive law belonging to the unified theory of viscoplasticity of haboche [11]. The rate independence of the resulting elastoplastic model is enforced by introducing suitable conditions. onsidering a perfect elasto-plastic behavior of the NT phase, the evolution law for the plastic strain rate γ P can be formulated as an adaptation of such a theory, by the following functions and rules: if T : Ė <, γ P = γ P ( T VM S o ) m sgn(t3 ) if T : Ė, where the inequalities on the right-hand sides of the equation define the loading/unloading conditions. The power function in (13) is controlled by the key parameters S o and m, which represent the NT/matrix interfacial shear strength and the smoothness of the nonlinear stress-strain curves, respectively. S o controls the NT shear stress value that activates the stick-slip and m regulates the transition between the elastic and the plastic phase. The function T VM is a micromechanical adjustment of the von Mises shear stress function applied to the interfacial stress discontinuity defined in (1). T VM represents a measure of the effective stress taken in its ratio to the threshold stress S o, T VM := ( 3 T dev : T dev) 1/ where T dev is the deviatoric part of the interfacial stress discontinuity T. Finally, the constitutive function γ P is introduced to ensure the rate independence. As known, a general viscoplastic model can be properly reduced to a rate-independent model by forcing the overstress to vanish [11]. In the present case, this is simply achieved by T VM as T VM S o. 3 DAMPING APAITY A first estimate of the damping capacity can be predicted by computing the cyclic hysteretic response of the unidirectional NT-composites. To this end, a set of numerical uniform straindriven tests is carried out by assuming a uniform state of equivalent shear deformation whereby the shear strains between the NT axis and any pair of orthogonal directions are equal. The performed tests consist of cyclic loading/unloading paths with the following assigned sinusoidal law: (13) (1) γ 3 (t)/ = γ / sin (πt/t ), (15)

5 where the shear strain amplitude is expressed by the constant value γ / := k 3 t, with t being the period. The polymers employed for the linearly elastic hosting matrix are epoxy resin and PEEK (poly-ether-ether-ketone), which show good mechanical properties as well as good chemical resistance and stability at high temperatures. The Young modulus E is chosen to be 5 GPa for epoxy resin and. GPa for PEEK, respectively, while the Poisson ratio ν is equal to. for both polymers. The chosen NTs are the same for the two composites and consist of single walled NTs with Young s modulus E = 97 GPa and Poisson ratio ν =.8. NTs play a major role in defining the mechanical and dissipative properties of the nanocomposite [, 1] especially because all the parameters involved in the model are referred to their features. In agreement with the literature [13, 1], the following reference parameter values are assumed: S o = 1 MPa for the interfacial shear strength; m = 1 for the evolution law s exponent; φ = 1% for the NTs volume fraction; and k 3 = 1 5 /s for the prescribed shear strain rate. The loading/unloading cycles and the hysteretic loops for the two nanocomposite materials are obtained in Figure, where the equivalent shear stress T 3 and the NT stress T 3 are plotted versus the prescribed shear strain γ 3 / over one cycle, together with the first loading curve. [MPa] T (a) NT/PEEK T 3 [MPa] (b) 3 x / x / ( c) NT/EPOXY x / T 3 [MPa] (d) / 3 x Figure : Nonlinear stress-strain curves obtained for (a)-(b) NT/PEEK nanocomposite and (c)-(d) NT/Epoxy nanocomposite under a cyclic shear test. The nonlinear elasto-plastic response of the NT/Epoxy and the NT/PEEK composites is due to the perfectly elasto-plastic behavior of the NT phase, as Figure b,d shows. The specific damping capacity of the nanocomposite can be directly measured by the dissipated energy per cycle W D (computed as the area of the hysteresis loop), taken in its ratio to the maximum stored energy W E (computed as the area under the first loading curve). The specific damping capacity per radiant represents the so-called loss factor Λ, which for low levels of damping is related to the damping ratio ξ by Λ = ξ = W D π W E. (16) 5

6 Thus, applying (16), the obtained damping ratio is found to be ξ = 6.8 % for the NT/PEEK nanocomposite, and ξ = 6.98 % for the NT/Epoxy nanocomposite. Previous sensitivity studies [] showed that the damping capacity of NT nanocomposites can be maximized by varying the constitutive parameters values of the model (φ, S o, m, γ /) in physically allowed ranges, which are chosen to be: 5 S o 5 MPa, 1 m 5 and 1 6 γ / 1 with the NT volume fraction φ assuming the following values [5, 1, 15, ]%. Optimal combination of these key parameters for each prescribed NT volume fraction can be reached by carrying out an optimization study based on the differential evolution algorithm. This method represents a population-based optimization strategy and a reliable function optimizer which provides at low computational costs the tuned combination of the parameters for a maximized damping capacity. The obtained optimal sets of the material parameters are given in Tables 1 and for the NT/PEEK and NT/Epoxy composites, respectively, while Figure 3 shows the resulting curves of the computed damping ratio ξ versus the prescribed strain amplitude γ /. φ [%] S o [MPa] m [ ] γ / 1 3 ξ [%] Table 1: Maximized damping ratio ξ for the NT/PEEK composite. φ [%] S o [MPa] m [ ] γ / 1 3 ξ [%] Table : Maximized damping ratio ξ for the NT/Epoxy composite. The damping ratio reaches the value of ξ = 7% for both NT/PEEK and NT/Epoxy nanocomposites with a NT volume fraction equal to 1%. This represents an interesting result since an increment of the order of 35 7% can be found comparing the damping ratio of the neat epoxy (ξ =.1.%) with that of the NT/Epoxy composite (ξ = 7%). ONLUSIONS The proposed theory delivers an efficient elasto-plastic model to describe the nanohysteresis and investigate damping capacity in unidirectional NT nanocomposites arising from the slippage of the NT carpets within the hosting matrix. The theoretical results are compared with experimental data available from the literature indicating that the damping capacity of NT/polymer composites is significantly improved with respect to that of neat polymer resins. Furthermore, the strain amplitude is shown to be a key parameter of the model, as highlighted by the strain-controlled cyclic tests which show the localization of a maximum damping ratio in the neighborhood of given strain amplitudes. The differential evolution algorithm has been employed to seek optimal combinations of these constitutive parameters leading to global maxima of the damping ratio ξ for the 6

7 (a) NT/PEEK 1 (b) NT/EPOXY / x 1-3 / x 1-3 Figure 3: Damping ratio versus the strain amplitude curves for the NT/PEEK (a) and NT/Epoxy (b) nanocomposites, obtained for optimal combinations of the main parameters of the model. NT/Expoxy and NT/PEEK nanocomposites with different NT volume fractions. The results confirm the possibility of tailoring the mechanical ant dissipative properties of NT-composites by suitable functionalization and fabrication processes. AKNOWLEDGEMENTS This work was partially supported by the Specialized International ollaborative Program (Grant No. UD98GD) under the Agency for Defense Development (ADD), Republic of Korea. The PRIN Grant No. 1BFXRHS- is gratefully acknolwedged in the form of a PhD fellowship awarded to Michela Talò. References [1] Formica G., Lacarbonara W., Debonding model of carbon nanotubes in a nanostructured composite, ompos. Struct., 96, (1). [] Formica G., Talò M., Lacarbonara W., Nonlinear modeling of carbon nanotube composites dissipation due to interfacial stick-slip, Int. J. Plasticity, accepted for publication (13). [3] Rajoria H., Jalili N., Passive vibration damping enhancement using carbon nanotube-epoxy reinforced composites, ompos. Sci. Technol., 65, (5). [] Suhr J., Koratkar N. A., Energy dissipation in carbon nanotube composites: a review, J. Mater. Sci.,, (8). [5] Barai P., Weng G., A theory of plasticity for carbon nanotube reinforced composites, Int. J. Plasticity, 7, (11). [6] Fraternali F., Blesgen T., Amendola A., Daraio., Multiscale mass-spring models of carbon nanotube foams, J. Mech. Phys. Solids, 159, 89-1 (11). [7] Eshelby J.D., The determination of the elastic field of an ellipsoidal inclusion, and related problems, P. Roy. Soc. A-Math. Phy., 1, (1957). 7

8 [8] Benveniste Y., A new approach to the application of Mori Tanaka s theory in composite materials, Mech. Mater. 6, (1987). [9] Mori T., Tanaka K., Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall. Mater., 1, (1973). [1] Mura T., Micromechanics of defects in solids, Martinus Nijhoff, The Netherlands (1987). [11] haboche J. L., A review of some plasticity and viscoplasticity constitutive theories, Int. J. Plasticity,, (8). [1] Zhou X., Shin E., Wang K.W., Bakis.E., Interfacial damping characteristics of carbon nanotube-based composites, ompos. Sci. Technol., 6, 5-37 (). [13] Ogasawara T., Tsuda T., Takeda N., Stress-strain behavior of multi-walled carbon nanotube/peek composites, ompos. Sci. Technol., 71, (11). [1] Tsuda T., Ogasawara T., Deng F., Takeda N., Direct measurements of interfacial shear strength of multi-walled carbon nanotube/peek composite using a nano-pullout method, ompos. Sci. Technol., 71, (11). 8