Evolution law for the plastic eigenstrain

As shown in the literature, a significant enhancement of damping capacity in CNT composites is obtained by dispersing CNTs aligned along a preferential direction. Moreover, in several applications (e.g., [11]; [29]), the CNTs are directly fabricated as aligned forests immersed in a hosting material. Thus, the CNTs can be assumed to be cylindrical inhomogeneous inclusions aligned, for example, along the 3-direction (see Fig. 3.2) that leads to a specific form of the Eshelby tensor S [64].

Note that, as mentioned, Eq. (3.5) states that limit conditions, enforced on the interfacial stress discontinuity, are automatically satisfied by limit conditions imposed on the stress within the inclusions. Thus, the activation of the interfacial stick-slip mechanism can be regulated through the CNT shear stresses (T_C23, T_C13), whence the stick-slip mechanism is described by the associated incremental plastic shear strains ˙γ₁₃^P and ˙γ₂₃^P, corresponding to the only nontrivial components of ˙E^P. These shear strain rates turn to be the same (i.e.,

˙γ₁₃^P = ˙γ₂₃^P = ˙γ^P) as a result of the relevant symmetry.

The presented model for the elastoplastic-like behavior of the CNT inclusions can be framed within the unified theory of viscoplasticity of [14]. A simplification of such a general theory is adopted, by considering a perfect elastoplastic behavior, and a constitutive restric-tion in order to ensure rate independence. The evolurestric-tion law for the effective plastic strain

Figure 3.2: Representative volume element of a unidirectional CNT composite. Slippage of a carbon nanotube in the hosting matrix and the interfacial shear stress distribution.

rate ˙γ^Pis then defined by the following functions and rules:

˙γ^P=







0 if T_C: ˙E < 0,

˙¯

γ^PT¯^VM S_o

^m

sgn(TC23) if TC: ˙E ≥ 0, (3.21)

where the power function is regulated by the constitutive parameters S_oand m, whence ˙¯γ^Pis a constitutive function to be prescribed in order to ensure the rate independence of the model while ¯T^VMis a suitable measure of the effective stress taken in its ratio to the threshold stress So. The inequalities on the right-hand sides of the equation define the loading/unloading conditions. This equation, combined with the constitutive law of Eq. (3.15), leads to an equivalent nonlinear model which can be easily implemented in standard finite element codes.

Moreover, it benefits from low computational costs, since there is no need to implement return mapping algorithms (see e.g. [77]). Indeed, definitions of either yielding functions or elastic domains are not explicitly used.

The main parameters introduced in the power function are the parameter Sowhich rep-resents the interfacial shear strength (ISS) between the two constituents, thus regulating the value of the CNT shear stress for which slippage at the CNT-matrix interface takes place, and the parameter m which determines the smoothness of the nonlinear stress-strain curves describing the behavior of the CNT phase.

Finally, in Eq. (3.21), the definition of ¯T^VMis consistent with the mechanical meaning of So; in fact, it represents the effective shear stress for the onset of inelastic strain at the CNT in-terface, derived by a micromechanical adjustment of the von Mises criterion. To this end, the octahedral (von Mises) shear stress function is applied to the interfacial stress discontinuity T :=¯ JT K:

T¯^VM=

3

2T¯^dev: ¯T^dev^1/2

(3.22) where ¯T^dev is the deviatoric part of the interfacial stress discontinuity ¯T . The condition given by Eq. (3.5) allows to use the inclusion stress T_C instead of the stress discontinuity ¯T and, in particular, to prove that both expressions for the stress deviatoric part and the von Mises stress are equivalent except for the non-dimensional scalar quantity cµ, i.e.,

T¯^dev = cµT^dev_C , T¯^VM= cµT_C^VM (3.23)

where c_µ is given by

cµ:= (1 − µM/µC) , (3.24)

with µ_M and µ_C being the elastic shear moduli of the matrix and CNT phases, respectively.

Therefore, the plastic evolution law given by Eq. (3.21) can be recast in terms of the stress TC only, as

Rate independence in the one-dimensional problem

The rate-independent nature of the elastoplastic model is guaranteed by a suitable definition of the function ˙¯γ^P in Eq. (3.25). As [14] suggested, a general viscoplastic model can be modified to become rate independent by zeroing the overstress. In the specific case, this is equivalently obtained by imposing ˙T_C^VM → 0 as T_C^VM → S_o.

To this end, the investigation is limited to a one-dimensional problem, where all stress and strain tensors have as nontrivial components those corresponding to the nontrivial com-ponents of ˙E^P.

Consequently, the plastic consistency condition ˙T_C^VM → 0 reduces to ˙TC 23 → 0, while the incremental constitutive law, given by Eq. (3.17), can be expressed as

T˙_C23 = L^E_C ˙γ − L^P_C˙γ^P. (3.26)

Substituting Eq. (3.25) into Eq. (3.26) enforces the rate independence assuming

˙¯

γ^P=L^E_C ˙γ

L^P_C . (3.27)

Thus, the evolution law (3.25) can be rewritten as

˙γ^P=

The incremental constitutive laws can also be cast in terms of an elastoplastic tangent mod-ulus for the CNT phase

T˙C23 = L^E_C

while for the composite material the tangent modulus can be derived from T˙₂₃=

Considering general three-dimensional states of stress and strain, the assumptions made on the rate independence entail, through the plastic flow rule (3.28), the positive definiteness of the dissipation power:

P_C^D:= TC : ˙E^P= (TC13+ TC23) ˙γ^p≥ 0 . (3.31)

These conditions guarantee thermodynamical consistency of the inelastic model for the inclu-sion phase. Such a condition can be verified for the one-dimeninclu-sional problem here proposed by simply substituting the plastic flow rule (3.28) into the dissipation power definition (3.31).

As a consequence of condition (3.20) – which relates the stress description in the composite to that in the inclusion – it can be proved that the thermodynamical consistency of the inclusion material model implies the thermodynamical consistency of the composite material model. Indeed, due to the isotropic nature of the elastic tensors, we can integrate Eq. (3.20) over time and rewrite the result as

T = λ_Mtr(E) I + 2 µ_ME + φ_C (c_λtr(T_C) I + 2 c_µT_C) (3.32) where

c_λ:= λ_Cµ_M/µ_C− λ_M 3 λC+ 2 µC

, (3.33)

λM and λC are the first Lam´e parameters of the matrix and CNTs, respectively.

Therefore, also the composite material is thermodynamically consistent because its dissi-pation power

P_C^D= 2[µM(E13+ E23) + φCcµ(TC13+ TC23)] ˙γ^p (3.34) turns out to be positive when the inclusion dissipation power (3.31) is positive, due to the stiffer properties of the inclusion material with respect to the matrix material (i.e, cµ ≥ 0) and of the positive definiteness of the power exhibited by an elastic stress response of a purely matrix material. In particular, for the one-dimensional problem whereby E13= E23= γ23/2 and T_C13= T_C23, the dissipation power becomes P_C^D= 2(µ_Mγ₂₃+ 2φ_Cc_µT_C23) ˙γ^p.

3.3 Numerical shear tests and measures of damping