Nonlinear Waves in Rods and Beams of Power-Law Materials

Some novel traveling waves and special solutions to the 1D nonlinear dynamic equations of rod and beam of power-law materials are found in closed forms. The traveling solutions represent waves of high elevation that propagates without change of forms in time. These waves resemble the usual kink waves except that they do not possess bounded elevations. The special solutions satisfying certain boundary and initial conditions are presented to demonstrate the nonlinear behavior of the materials. This note demonstrates the apparent distinctions between linear elastic and nonlinear plastic waves.


Introduction
Free vibrations of rods and beams of power-law materials are considered.Analytic traveling wave solutions to the wave equations for power-law materials (see [1,2]) are obtained which represent kink waves of single elevation that propagates without change of forms in time.It is shown that, unlike the wave equations for linear materials, the nonlinear wave equations do not allow arbitrary traveling wave forms in an infinite rod or beam.The results demonstrate that the traveling fronts of the waves may sharpen or flatten as the wave speeds increase depending upon the power-law index  and the bulk modulus.For  > 1, the wave fronts sharpen, whereas for 0 <  < 1, the fronts flatten as the wave speeds increase.The solutions also demonstrate that the speeds of the nonlinear traveling waves depend not only on the material properties but also on the initial energy-level.It is well known that the speeds of waves for the linear elastic materials ( = 1, Hooke's law) depend only on the material properties in contrast to that of the waves in nonlinear materials.As far as we know these solutions are not available in literature, even though there are numerous research papers and books devoted to the discovery and study of traveling waves in elastic and plastic solids (see [3][4][5][6][7] for details).In the case of rods and beams of finite length, we also present some special solutions satisfying certain boundary and initial conditions.The closed formula solutions are expressed in terms of non-Euclidean sine functions (cf.[8]), which differ from the Euclidean sine functions corresponding to the waves in rods and beams of linear elastic materials.
The note is organized as follows.In Section 2, the powerlaw constitutive stress-strain equation is introduced.In Section 3, the potential energy and derivations of the wave equations of power-law materials are outlined.In Sections 4 and 5, closed-form solutions are derived.And, finally the results are summarized in Section 6.

Hollomon's Equation
It is well known that, in uniaxial state, the following powerlaw stress and strain relation is used for certain elastoplastic materials: where  is the axial stress,  is the axial strain, and  and  are engineering constants with values depending on the specific material.The materials satisfying (1) sometimes are also referred to as Ludwick or as Hollomon's materials in literature (cf.[1,2]).Many heat-treated metals are well-known power-law materials.For a given annealed metal or alloy,  and  depend on the heat treatment received by the metal or alloy.The values of  are typically between 0 and 1 for such metals.For a comprehensive list of experimental values of  and  of common annealed industrial metals, see, for example, [9].For some geological materials, such as certain rocks or ice, however, the values of  are greater than 1.In some biological tissues, experiments also indicate that the power-law index  satisfies 0 <  < 1 for bones such as tibia and femur, while  > 1 for cartilages such as common carotid artery and abdominal aorta (see, e.g., [10,11]).For a given value of 0 <  < 1, the stress-strain curve defined by (1) can result in a rapid increase in the yield stress for small strains or strain hardening.However, it can be the opposite for values of  > 1, for which large strains produce small stress or softening.For these reasons,  is called the strain-hardening or strain-softening exponent.Study of the mechanical properties of these heat-treated metals is very important in industries (see, e.g., [12], for stress analysis of beam columns made of Ludwick materials).If we allow  = 1, then (1) reduces to Hooke's law for linear elastic material and the constant , also called the bulk modulus, equals the corresponding Young's modulus .Power-law materials are a special case of a more general class of materials called Hencky plastics [13].Physically, the constitutive equation ( 1) describes the hardening or softening of materials showing an elastic-plastic transition.In the following, bold letters are used to denote vectors or matrices.A vector is considered as a single row matrix.The transpose of a matrix A is denoted by A  , and the inner product of two vectors u and v by uk  .The time derivative u/ is denoted by u .Let u(, , , ) = ((, , , ), V(, , , ), (, , , )) denote the displacement vector, the strain components, and   ,   ,   ,   ,   , and   the corresponding stress components.The following generalized power law can be derived from the Hencky total deformation theory [13]: where , where , , and ] are the material constants; see also Wei [14].Note that (3) is the three-dimensional version of (1).In the following two sections, wave equations of bars and beams made of the power-law elastoplastic materials are derived by (3) and the assumption of the Euler-Bernoulli beam theory.There are similar versions of generalized powerlaw stress-strain relations for strain-hardening or strainsoftening material in the literature and similar wave equations can be derived (see, e.g., [15][16][17][18][19][20]).

The Nonlinear Wave Equations
The potential energy for a power-law elastoplastic body occupying a three-dimension body  can by defined by where where  is the density, u = ( u , V , ẇ ) the velocity, f = (  ,   ,   ) the body force, and t = (  ,   ,   ) the surface force.See, for example, [21], for a standard definition of (u).
For completeness, the derivation of the wave equations of the power-law materials given in [14] is outlined here.It is well known that Hamilton's principle seeks an equilibrium state in time dependent mechanical systems (see, e.g., [21]).
Specifically, Hamilton's principle requires that we seek a displacement u so that, for any time interval [ 1 ,  2 ], u( 1 ) = u( 2 ) and u ( 1 ) = u ( 2 ), and for all displacement of the form u + k, where  is any real number, the first variation of the energy functional  satisfies The combination u() + k() is referred to as an admissible displacement for the mechanical system since it is required to satisfy some boundary conditions.It can be shown that if the displacement u satisfies (8) of Hamilton's principle, then it must also satisfy a differential wave equation under certain conditions.In particular, suppose that the cross-sectional area, denoted by , is a nonzero constant, and then for the rod, we have and for the corresponding Euler beam When  = 1, (9) reduces to the standard wave equation for the elastic bar and (10) to the standard wave equation for the elastic Euler beam The quantity   reduces to the second moment of inertia,   = ∫  || +1  reduces to  when  = 1 in the elastic beam theory, and the material constant  becomes Young's modulus  for linear elastic materials.In deriving the wave equations ( 9) and (10), we have made the assumption that the solutions  and V are continuously differentiable and their appropriate lower order derivatives are bounded or vanishing when || → ∞.By (8), we get Using integration by parts and interchange of the order of integration, with V( 1 ) = V( 2 ) = 0, and assuming that lim →±∞ |(, )/| −1 ((, )/) is bounded by a constant independent of  and lim →±∞ V(, ) = 0 uniformly in , we get the following: from (13).Since V,  1 , and  2 are arbitrary and  ̸ = 0, we then get ( 9) from ( 14).The corresponding beam equation ( 10) can be derived similarly which was reported in [14].
= (  ,   ,   ,   ,   ,   ) and  = ( x ,  y ,  z ,  xy ,  xz ,  yz ).The Lagrangian energy functional (u) equals the kinetic energy  minus the elastoplastic potential energy  plus the work  done by external force.It can be written as