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, may be expressed in terms of an isotropic function ψ =ψ(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 tr E, but the shear strain √ 2|E 0 | (with E 0 the deviatoric part of E) is small but not infinitesimal:, withThis free energy has five material constants -the two classical positive-valued shear and bulk moduli µ 0 and κ 0 of the infinitesimal theory of elasticity, and three additional positive-valued material constants (µr, ǫr, ǫc), which are used to characterize the nonlinear response at large values of tr E. In the large volumetric strain range −0.30 ≤ tr E ≤ 0.15 but small shear strain range √ 2|E 0 | 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.

The Exponentiated Hencky-Logarithmic Strain Energy. Part I: Constitutive Issues and Rank-One Convexity

“…For these reasons the quadratic Hencky model is used in theoretical investigations and in physical applications [149,8,39,40,77,192,86,88,87,83]. We observe also a renewed interest in this class of isotropic slightly compressible hyperelastic solids originally proposed by Hencky [100,102,101,103,98]. The strain energy W H is also often used in commercial FEM-codes.…”

“…This decomposition problem was studied later in the papers [7,114]. Gearing and Anand [89] (see also [98,68]) recently proposed an energy of the form 17…”

“…The strain measure (natural strain) then has been extensively used over the years to report experimental true-stress-true-strain data. More recently, in [98] a modified Hencky energy is proposed which is motivated by in depth molecular dynamics simulations for a metalic glass 7 . Hill [108,107] (see also [53,150,151]) has discussed the advantage of the logarithmic strain measure 8 in setting up a class of constitutive inequalities, based on a family of measures of finite strain and their corresponding conjugate stresses, for both elastic and elasto-plastic solids.…”

Iron-based bulk metallic glasses

“…For instance, the fracture energy (toughness) of Fe-based BMG alloys was correlated with changes in the G/K ratio as well as the Poisson's ratio by Lewandowski et al 205 Through a broad survey of the literature, it was shown 209,210 model also predicts the importance of elastic constant ratio on the toughness of metallic glasses. 211 The ductility of conventional crystalline metals has been shown to increase by the introduction of thin metallic glassy ribbons into them. 212 Therefore, researchers have utilised this concept to increase the ductility of BMG alloys and have achieved some amount of success.…”

The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity, planar polyconvexity and existence of minimizers

“…In the first part [36] we have summarized the well-known unique features of the quadratic Hencky strain energy W H based exclusively on the natural strain tensor log U . The Hencky model is definitely one of the most widely used strain energies in the small elastic strain regime [21,23,22,24,8,9,20]. In [36], however, we also pointed out that the quadratic Hencky energy has some serious shortcomings.…”