A Large Strain Isotropic Elasticity Model Based on Molecular Dynamics Simulations of a Metallic Glass

Abstract

For an isotropic hyperelastic material, the free energy per unit reference volume, ψ [psi], may be expressed in terms of an isotropic function ψ = ¯ ψ(E) [psi = psi overscore (E)] of the logarithmic elastic strain E = ln V. We have conducted numerical experiments using molecular dynamics simulations of a metallic glass to develop the following simple specialized form of the free energy for circumstances in which one might encounter a large volumetric strain trE, but the shear strain √2|E0| [square root 2 pipe E subscript 0 pipe] (with E0 [E supscript 0] the deviatoric part of E) is small but not infinitesimal: ψ(E) = μ(trE) |E0|2 [psi (E)= mu (trE pipe E subscript 0 pipe superscript 2] + g(trE) , with μ(trE) = μr − (μr − μ0) exp„trE ǫr « [mu (trE) = mu subscript x - (mu subscript x - mu subscript 0) exp (trE divided by epsilon subscript x)], and g(trE) = κ0 (ǫc)2 »1 − „1 + trE ǫc «exp„− trE ǫc «– [g(trE) = kappa subscript 0 (epsilon subscript c) superscript 2 [1-(1 + trE divided by epsilon subscript c) exp (-trE divided by epsilon subscript c)]]. This free energy has five material constants — the two classical positive-valued shear and bulk moduli μ0 [mu subscript 0] and κ0 [kappa subscript 0] of the infinitesimal theory of elasticity, and three additional positive-valued material constants (μr, ǫr, ǫc) [(mu subscript r, epsilon subscript r, epsilon subscript c)], which are used to characterize the nonlinear response at large values of trE. In the large volumetric strain range −0.30 ≤ trE ≤ 0.15 but small shear strain range √2|E0| [square root 2 pipe E subscript 0 pipe < or about] 0.05 numerically explored in this paper, this simple five-constant model provides a very good description of the stress-strain results from our molecular dynamics simulations. D. L.