MATERIAL DAMPING AND CRYSTAL PLASTICITY Pratheek Shanthraj MODEL: Consider a rod (figure 1) having length L, cross-­‐sectional area A, mass M and Young’s modulus E, subjected to an impact load f ext . The macroscopic damping of the elastic response of the rod due to microscopic slip occurring on crystallographic planes is characterized in this study. The microscopic plasticity is accounted for by using a 1D crystal plasticity constitutive formulation. €
fext LENGTH, L Figure 1. 1D CONSTITUTIVE FORMULATION FOR DYNAMIC CRYSTAL PLASTICITY: The crystal plasticity framework used in this study is based on the formulation developed by Asaro and Rice [1] and Zikry [2] adapted here to 1D and small strains. Here, the total strain-­‐rate is additively decomposed into an elastic and plastic component as: ε˙ = ε˙e + ε˙ p (1) The elastic strain-­‐rate is given by Hooke’s law €
σ˙
ε˙e =
E (2)
and the plastic strain-­‐rate is defined in terms of the crystallographic slip rates as: €
ε˙ p = ∑ γ˙ (α ) cosθ α cos φα
α
(3)
€
where, θ α is the angle made by the slip plane normal with the loading axis, φα is the angle made by the slip direction with the loading axis and γ˙(α ) is the slip rate on the slip system α . €
€
For a rate-­‐dependent inelastic formulation, the slip-­‐rates are functions of the resolved shear and reference stresses. The power €
law €
1
−1
⎡τ (α ) ⎤⎡ τ (α ) ⎤ m
⎥
⎥⎢
γ˙ (α ) = γ˙ ref ⎢
⎣ τ ref ⎦⎢⎣ τ ref ⎥⎦ (4)
is used here. The reference shear-­‐strain, γ˙ref , corresponds to a reference shear stress, τ ref . The rate €
sensitivity parameter, m, is material dependent. The rate-­‐
independent limit is approached as m approaches zero. The resolved shear stress, τ (α ) , on slip system α , is given in terms € of the stress, σ , as €
τ (α ) = σ cosθ α cos φα (5) €
€
Substituting equations 2-­‐5 into equation 1, we obtain €
€
1
€
⎡ σ ⎤⎡ σ ⎤ m −1
σ˙
ε˙ = + γ˙ ref ⎢ ⎥⎢ ⎥ ∑ cos θ α cos φα
E
⎣τ ref ⎦⎣ τ ref ⎦
α
(6)
Where ∑ cosθ α cos φα is a constant and is known as the Taylor factor. Equation 6 α
€
can be expressed in terms of velocity, v, displacement, u, and internal force, f int to obtain 1
⎡ f ⎤⎡ f ⎤ m −1
€
L
˙
v = u˙ = f int
+ γ˙ref L ⎢ int ⎥⎢ int ⎥ ∑ cosθ α cos φα
EA
⎣ Aτ ref ⎦⎣ Aτ ref ⎦
α
(7)
Furthermore, accounting for dynamics by including the inertia force we obtain €
Mv˙ = f ext − f int NUMERICAL METHOD: €
The above ODEs can be expressed as the following non-­‐linear non-­‐linear system (8) 1
⎡
⎤
−1
m
⎡
⎤
⎡
⎤
f
EA
f
⎢
⎥
f˙int =
v − γ˙ref L ⎢ int ⎥ ⎢ int ⎥ ∑ cos θ α cos φα ⎥
⎢
L
⎣ Aτ ref ⎦ ⎣ Aτ ref ⎦
α
⎢⎣
⎥⎦
1
v˙ = [ f ext − f int ]
M
€
u˙ = v (9) €
subjected to an initial velocity and f ext = 0 , corresponding to an impact load, with the rod initially at rest. The €
above system is the formulation for a single element (Figure 1). This system of ODEs, and a system corresponding to a discretization into 10 finite elements are solved on MATLAB using the material properties in Table 1. €
The solution obtained, u, is fitted to the response, utot, of a spring-­‐damper system (Figure 3), which is governed by the following system of equations: M˙u˙tot + c1u˙ tot + c 2 u˙ tot − u˙ spring = 0
M˙u˙tot + c1u˙ tot + kuspring = 0
and the spring constant and damping coefficients are obtained using least squares parameter estimation. €
K C2 C 1 Figure 2. (
)
Property Length, L Cross section area, A Young’s modulus, E Mass, M Reference shear stress, τ ref Reference shear-­‐strain, γ˙ref Taylor factor, ∑ cosθ α cos φα α€
€
Table 1. Value 10 mm 1 mm2 100 GPa 2.7x10-­‐5 Kg 200 MPa 1x10-­‐3 s-­1 0.5 €
RESULTS: Equations 9, is solved for using rate sensitivity parameters of 0.05 to 1 to corresponding to a range of material behaviors from rate-­‐independent to viscoelastic for the single element formulation. A plot of the displacement is shown in Figure 3. We can see that the displacement is damped exponentially indicating a viscous type damping on the macroscopic scale. The damping is dependent on the rate sensitivity parameter, where the vibration at higher rate sensitivity parameters are dampened faster. We can also see that there is a positive bias of the displacement, which is due to the permanent plastic deformation associated with an applied tensile impact load. Further, the damping profiles are fitting to an exponential decaying function (using the method of least-­‐squares) and is shown in Figure 4. The natural frequency of the rod is given by EA
ω0 =
= 6.0858 × 10 5 rad s LM
The damping ratios are calculated from the corresponding decay constant and the natural frequency and is listed in Table 2. €
Figure 2. Figure 3. Rate sensitivity parameter, m 1 0.2 0.1 0.05 Table 2. Damping ratio, ξ 0.0250 0.0082 0.0049 €
0.0025 Further the finite element discretization of the system (equation 9) is solved for a material rate sensitivity of 1. A plot of the displacement is shown in Figure 5. Figure 5. Using a least squares parameter estimation, the values obtained for the stiffness, k, is 2.14 x 105 Nm-­‐1 and the damping coefficients, C1 and C2, are 0.01 Pas and 35 Pas respectively. We can see that the spring-­‐damper model captures the permanent creep deformation and the exponential decaying of the vibrations. Furthermore, due to comparative low value of C1 with respect to C2, a simple spring-­‐damper system in series may be sufficient to capture the dynamics of microscale plasticity. REFERENCES [1] Asaro RJ, Rice JR. J Mech Phys Solids 1977;25:309. [2] Zikry MA. Comput. Struct 1994;50:337.