DOI 10.1515/jmbm-2012-0032 J Mech Behav Mater 2012; 21(1-2): 53–56 Short Communication George Petsos and Elias C. Aifantis* Some remarks on deformation, localization, and front propagation Keywords: gradient theory; propagation front; strain localization. *Corresponding author: Elias C. Aifantis, Laboratory of Mechanics and Materials, Aristotle University, Thessaloniki 54124, Greece, e-mail: mom@mom.gen.auth.gr Elias C. Aifantis: College of Engineering, Michigan Tech, Houghton, MI 49931, USA; and King Adbulaziz University, Jeddah 21589, Saudi Arabia George Petsos: Laboratory of Mechanics and Materials, Aristotle University, Thessaloniki 54124, Greece This note provides some analytical results concerning the localization of deformation in one-dimensional solids. It supplements earlier ones obtained by the second author and his coworkers in the past [1 –4]. These previous results were mainly concerned with stationary shear bands emerg- ing when the material enters the unstable (negative slope) regime of its (nonmonotonous) stress-strain relation. Such nonmonotonous constitutive equations were generalized with the Laplacian of strain to stabilize the solution of the nonlinear equation obtained by further using the equa- tion of stress equilibrium. The present results are derived within a more general framework based on the minimiza- tion of a gradient-dependent nonconvex free energy, but also taking into account viscosity and inertia. At the same time, they may also be viewed as being restricted by the assumption of the existence of free energy in far-from-ther- modynamic-equilibrium conditions and the validity of the variational principles used. This assumption was not used in the mechanical theory of fluid interfaces [5, 6], which was the motivation for the approach adopted in [1 –4]. We begin by defining kinetic ( T), free ( F), and dissipa- tion ( R) energies per unit length as follows [] { } 2 /2 /2 - /2 - /2 2 /2 - /2 1 ; ; 2 1 , 2 L L L NL L L L L u T dx F d f f x t R t ρ γ γ λ ∂ ⎛ ⎞ = = + ⎜ ⎟ ⎝ ⎠ ∂ ∂ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ∂ ∫ ∫ ∫ (1) where L denotes the specimen thickness in the x-direc- tion (- L/2 < x< L/2) loaded with a shear stress τ 0 in the perpendicular y-direction. The field variables u( x, t) and γ( x, t) = ∂ u/∂ x denote displacement and strain, respec- tively, whereas ρ and t denote mass density and time, respectively. The nonmonotonous function κ( γ) denotes the homogeneous part of the stress, whereas the homoge- neous part of the local free energy density functional f L [ γ] and its inhomogeneous gradient-dependent (nonlocal) counterpart f NL are given by [] () 2 2 0 0 - , . L NL f f l d x γ γ γ κγ γτγ ∂ ⎛ ⎞ = = ⎜ ⎟ ⎝ ⎠ ∂ ∫ (2) The quantity l is the gradient coefficient identified with an internal length, and λ denotes the viscosity co efficient (the Rayleigh dissipation constant). Following [7, 8], we postulate the variational prin- ciple , L u R u δ δ δ δ = ɺ where L = T- F is the Lagrangian of the system. As a result, we obtain the differential equation of dynamic equilibrium in its usual form, i.e., ρ∂ 2 u/∂ t 2 = ∂ τ/∂ x and the corresponding constitutive equation, i.e., τ( x, t) = κ( γ)-2 l 2 ∂ 2 γ/∂ x 2 + λ∂ γ/∂ t. By combin- ing the last two equations, we arrive at the following gov- erning partial differential equation for the displacement 2 2 4 3 2 2 2 4 2 -2 , u d u u u l t d x x tx κ ρ λ γ ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂∂ (3) and a similar one for the strain derived by differentiating Eq. (3) with respect to the spatial coordinate x, i.e., 2 2 2 2 4 3 2 2 2 2 4 2 -2 . d d l t d x d x x tx γ κ γ κ γ γ γ ρ λ γ γ ∂ ∂ ∂ ∂ ∂ ⎛ ⎞ = + + ⎜ ⎟ ⎝ ⎠ ∂ ∂ ∂ ∂ ∂∂ (4) Equations (3) and (4) are nonlinear wave equation with dissipation for the displacement u strain γ. The linear sound velocity υ s is given by () 0 s υ κ ρ =± ′ , where κ′ (0) denotes the Young modulus. To obtain the corresponding boundary conditions, we use the energy relation [7–9] ∂ E/∂ t = -2 R, where E= T+ F is the mechanical energy of the system. The result is 0 0 - /2 /2 - , - , - 0, 2 2 x L x L u L u L t t t t τ τ τ τ = = ∂ ∂ ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ∂ ∂ ⎣ ⎦ ⎣ ⎦ (5)