Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 1, pp. 109-126, Warsaw 2009 TRAVELING WAVES IN A THIN LAYER COMPOSED OF NONLINEAR HYPERELASTIC ZAHORSKI’S MATERIAL Izabela Major Maciej Major Technical University of Częstochowa, Department of Civil Engineering, Częstochowa, Poland e-mail: admin@major.strefa.pl Thepropagationof nonlinear travellingwaves in theZahorski elastic lay- er is isnvestigated. By the averaging process over the width of the layer, we obtain a system of partial differential equations in one-dimensional space and time. A technique of phase planes is used to study the wave processes. Based on the phase trajectorymethod, we canmake an inter- pretation of conditions of propagation of the nonlinear travelling wave and can establish the existence conditions under which the phase plane contains physically acceptable solutions. Key words: travelling waves, phase plane, discontinuous surface, hyper- elastic materials 1. Introduction We can treat a layer as a one-dimensional elastic structure with one sca- lar variable representing transverse symmetric motion. However, the finished transverse dimensions carry the weight during propagation of elastic waves. In the simplest description, one scalar variable can be used to describe effects of finite transverse dimensions in the elastic layer that undergoes longitudinal and symmetrical transverse motion only. General equations describingmotion of an incompressible nonlinear elastic medium and symmetric lateral motion of the elastic layer are presented in Section 2. In this section, we derive equations of motion by averaging the equations of elasticity across the layer. The travelling waves are described in Section 3. We obtained a solution for the travelling wave propagating with speed V in the direction of the coordinate X1 depending on one parameter only. In Section 4, we use phase plane methods to classify different solutions 110 I. Major, M. Major for travelling waves that are possible. Some of the solutions to the differential equations do not correspond to physically acceptable waves propagating in the layer, and so additional restrictions must be imposed from the physical problem. We explore such restrictions in Section 5. We are able to establish conditions for the existence of physically acceptable solutions as representedby individual paths in the phase plane. Finally, in Section 6we present numerical analysis for travelling waves in the layer composed of the Zahorski material. 2. Basic equations Motion of a continuum is given by a set of functions (Truesdell and Toupin, 1960) xi =xi(Xα, t) i,α=1,2,3 (2.1) Weconsider an elastic layerwhichoccupies thematerial region X1 > 0 (Fig.1) in the direction of the axis X1. At the frontal area of the layer X1 = 0, the boundary conditions for deformations are given (Fu and Scott, 1989). We assume that motion described by equation (2.2) undergoes without imposing additional contact forces on the lateral planes of the layer X2 =±h (Coleman and Newman, 1990; Wright, 1981). Fig. 1. Motion of the layer; (a) main motion in the longitudinal direction, (b) secondarymotion in the transverse direction The symmetric motion of the considered travelling wave is described by the equations x1 =X1+u1(X1, t) x2 =X2+X2ε2(X1, t) x3 =X3 (2.2) The strain ε1, the gradient of the transversal strain κand speedof the particle of themedium ν1 and ν2 in both directions of the layer are equal, respectively ε1 =u1,1 κ= ε2,1 ν1 = ẋ1 = u̇1(X1, t) ν2 = ẋ2 =X2ε̇2(X1, t) (2.3) Traveling waves in a thin layer composed... 111 For assumedmotion (2.2), the left Cauchy-Green tensor and the deformation gradient have the form B=     ◦ ε 2 1 ◦ ε1X2κ 0 ◦ ε1X2κ (X2κ) 2+ ◦ ε 2 2 0 0 0 1     F=    ◦ ε1 0 0 X2κ ◦ ε2 0 0 0 1    (2.4) where ◦ ε1 =1+ε1 and ◦ ε2 =1+ε2. The invariants of the deformation B are I1 = I2 = ◦ ε 2 1+ ◦ ε 2 2+(X2κ) 2+1 I3 =1 (2.5) For an incompressible material, there is identity detF=1, then for the con- sidered material ◦ ε1 ◦ ε2 =1 (2.6) We assume that the layer is made of the Zahorski material characterised by the strain-energy function W(I1,I2)=µ[C1(I1−3)+C2(I2−3)+C3(I21 −9)] (2.7) where C1,C2 and C3 are elastic constants. According to Wesołowski (1972a,b) or Dai (2001), the Cauchy tensor has the form T=−qI+2µ[(C1+2C3I1)B−C2B−1] (2.8) where q is an arbitrary hydrostatic pressure. The first Piola-Kirchhoff stress tensor may be expressed by the Cauchy tensor T TR =TF −⊤ (2.9) and its non-zero components are given by TR11 =− ◦ ε2[q+2µC2(X2κ) 2]+2µ ◦ ε1 ( C1+2C3I1−C2 ◦ ε 4 2 ) TR12 =2µC2X2κ [ ◦ ε 2 1+ ◦ ε 2 2+(X2κ) 2 ] + qX2κ TR21 =2µX2κ(C1+2C3I1+C2) (2.10) TR22 =− ◦ ε1[q+2µC2(X2κ) 2]+2µ ◦ ε2 ( C1+2C3I1−C2 ◦ ε 4 1 ) TR33 =−q+2µ(C1+2C3I1−C2) 112 I. Major, M. Major For deformation gradient (2.4)2, the equations of motion are reduced to a system of equations for plane strain deformation TR11,1+TR12,2 = ρRu1,tt TR21,1+TR22,2 = ρRX2ε2,tt (2.11) TR33,3 =0 The boundary conditions at the lateral surfaces of the layer X2 = ±h, have the form TR12(X1,±h,X3)=TR22(X1,±h,X3)= 0 (2.12) We employ a procedure, which was described by Wright (1981), consisting in averaging equations of motion (2.11)1,2 along the cross-section A of the layer (see Fig.2). Fig. 2. Propagation of the travelling wave in the layer and cross-section A perpendicular to the axis X1 We assume that in the averaging process, boundary conditions (2.12) are satisfied. We multiply second equation of motion (2.11)2 by X2 and average both resulting equations and first equation (2.11)1 over thewidth of the layer, thus obtaining 1 2h h ∫ −h ∂TR11 ∂X1 dX2+ 1 2h h ∫ −h ∂TR12 ∂X2 dX2 = 1 2h h ∫ −h ρRü1 dX2 (2.13) 1 2h h ∫ −h ∂TR21 ∂X1 X2 dX2+ 1 2h h ∫ −h ∂TR22 ∂X2 X2 dX2 = 1 2h h ∫ −h ρRε̈2X 2 2 dX2 Traveling waves in a thin layer composed... 113 Taking into account boundary conditions (2.12), one obtains an averaged equ- ation of motion ∂ ∂X1 ( 1 2h h ∫ −h TR11 dX2 ) = ρRü1 (2.14) ∂ ∂X1 ( 1 2h h ∫ −h X2TR21 dX2 ) − 1 2h h ∫ −h TR22 dX2 = ρRε̈2 h2 3 Equations (2.14) are a consequence of the applied average process. Formotion (2.11), the cross-section of the layer remains plane, and the normal to the sur- face of cross-sections overlap the axis X1 (Fig.2). The analogical assumption was made in the paper by Braun and Kosiński (1999). In further analysis, we take the advantage of averaged equation (2.14)1 and equation (2.11)2. Substituting the components of Piola-Kirchhoff stress tensor (2.10) into (2.11)2 and integrating them with respect to X2, we obtain the equation of motion in the direction of the axis X2 q= µX22 ◦ ε1 { 2κ2(2C3 ◦ ε2−C2 ◦ ε1)+κ,1 [ C1+C2+2C3 ( ◦ ε 2 1+ ◦ ε 2 2+ 1 2 (X2κ) 2+1 )] + (2.15) +4C3κ ( ◦ ε1ε1,1+ ◦ ε2κ+ 1 2 X22κκ,1 ) − 1 2 ν−2o ε2,tt } + q1(X1, t) ◦ ε1 where q1 = (X1, t) is an arbitrary function and νo = √ µ/ρR is the speed of infinitesimal shear waves. We determine the function q1 = (X1, t) using boundary conditions (2.12) TR22 ∣ ∣ X2=±h =0 (seeMajor andMajor, 2007). The obtained equation depends on (X2) 2, then both boundary conditions are satisfied. Finally, for the Zahorski material, (2.15), has the form q= µ(X22−h2) ◦ ε1 { −2κ2C2 ◦ ε1+κ,1 [ C1+C2+2C3 ( ◦ ε 2 1+ ◦ ε 2 2+ 1 2 κ2(X22+h 2)+1 )] + +4C3κ [ ◦ ε1ε1,1+2 ◦ ε2κ+ 1 2 κκ,1(X 2 2+h 2) ] − 1 2 ν−2o ε2,tt } + (2.16) + 2µ ◦ ε2 ◦ ε1 { C1+2C3 [ ◦ ε 2 1+ ◦ ε 2 2+(hκ) 2+1 ] −C2 ◦ ε 2 1 [ ◦ ε 2 1+(hκ) 2 ]} 114 I. Major, M. Major Including (2.16), the left-hand side of equation (2.14)1 for the Zahorski mate- rial is ( 1 2h h ∫ −h TR11 dX2 ) ,1 =2µ [ ( ◦ ε1− ◦ ε 3 2) [ C1+2C3 ( ◦ ε 2 1+ ◦ ε 2 2+(hκ) 2+1 )] + (2.17) − 4 3 C3 ◦ ε1(hκ) 2+C2 ◦ ε2 ( ◦ ε 2 1− ◦ ε 2 2+ 2 3 (hκ)2 ) − ◦ε 2 2 h2 3 { 2C2κ 2◦ε1− 18 5 C3(hκ) 2κ,1+ + 1 2 ν−2o ε2,tt−κ,1 [ C1+2C3 ( ◦ ε 2 1+ ◦ ε 2 2+1 ) +C2 ] −4C3κ ( ◦ ε1ε1,1+2 ◦ ε2κ )} ] ,1 After transformation and differentiation, we obtain from (2.6) κ= ε2,1 =−ε1,1 ◦ ε −2 1 κ,1 =2ε 2 1,1 ◦ ε −3 1 −ε1,11 ◦ ε −2 1 (2.18) Including (2.6) and (2.18) in (2.17), we finally obtain an equation which con- tains the function ε1(X1, t) only [ ( ◦ ε1− ◦ ε −3 1 )[ C1+2C3 ( ◦ ε 2 1+ ◦ ε −2 1 +(hε1,1) 2◦ε −4 1 +1 )] − 4 3 C3 ◦ ε −3 1 (hε1,1) 2+ − h2 6 ν−2o ε2,tt ◦ ε −2 1 +C2 ( ◦ ε1− ◦ ε −3 1 + 2 3 (hε1,1) 2◦ε −5 1 ) − h2 3 { 2C2ε 2 1,1 ◦ ε −5 1 + (2.19) +4C3ε 2 1,1 ( ◦ ε −3 1 −2 ◦ ε −7 1 ) − ( 2ε21,1 ◦ ε −5 1 −ε1,11 ◦ ε −4 1 )[ C1+C2+ +2C3 ( ◦ ε 2 1+ ◦ ε −2 1 +1 )] − 18 5 C3(hε1,1) 2 ( 2ε21,1 ◦ ε −9 1 −ε1,11 ◦ ε −8 1 )} ] ,1 = 1 2 ν−2o u1,tt The above equation, (2.19), is the governing one-dimensional equation descri- bing nonlinear dynamics of the layers. 3. Traveling waves The phase ξ is defined by ξ=X1−Vt (3.1) where V is the speed of propagation of the travelling wave with a constant profile displaced along the axis X1. For the travelling wave with any profile, we express motion as a function of one parameter ξ only u1(X1, t)=u1(ξ) ε2(X1, t)= ε2(ξ) (3.2) Traveling waves in a thin layer composed... 115 Fig. 3. Propagation of the travelling wave with speed V Substituting (3.1) and integrating with respect to ξ, equation (2.19) for the travelling wave has the form (C1+C2+2C3) ( ◦ ε1− ◦ ε −3 1 ) +2C3 ( ◦ ε 3 1− ◦ ε −5 1 ) + h2 6 ν ◦ ε −2 1 [ ε1 ◦ ε1 ] ,ξξ + + h2 3 (C1+C2+2C3) ( 2ε21ξ ◦ ε −5 1 −ε1ξξ ◦ ε −4 1 ) + 2 3 C3h 2 ( ε21ξ ◦ ε −3 1 −ε1ξξ ◦ ε −2 1 + (3.3) +3ε21ξ ◦ ε −7 1 −ε1ξξ ◦ ε −6 1 ) + 3 5 C3h 4ε21ξ ( 4ε21ξ ◦ ε −9 1 −2ε1ξξ ◦ ε −8 1 ) = 1 2 νε1+d1 where ν =V 2/ν20 and d1 is a constant of integration. Multiplying (3.3) mutually by ε1,ξ, we integrate it once more to obtain 1 2 (C1+C2+2C3) ( ◦ ε 2 1+ ◦ ε −2 1 ) + 1 2 C3 ( ◦ ε 4 1+ ◦ ε −4 1 ) + h2 12 νε21,ξ ◦ ε −4 1 + −h 2 6 (C1+C2+2C3)ε 2 1,ξ ◦ ε −4 1 − 1 3 C3h 2ε21,ξ ( ◦ ε −2 1 + ◦ ε −6 1 ) + (3.4) − 3 10 C3h 4ε41,ξ ◦ ε −8 1 = 1 4 νε21+d1ε1+d2 where d2 is another constant of integration. This equation gives a solution for the travelling wave propagatingwith the speed V in the X1 direction and depending on one parameter ξ (3.1) only. If the constant ν = ρV 2/µ and the constants of integration d1 and d2 are known, we could find the solution. For the constant C3 =0,weobtain the equation ofmotion for theMooney- Rivlin material (compareMajor andMajor, 2007). 116 I. Major, M. Major 4. Phase plane analysis of propagation of the travelling wave in the layer By constructing phase portraits of the solution in the (ε1,ε1,ξ) plane, we can made an interpretation of the conditions of propagation of nonlinear travelling waves (Dai, 2001; Major andMajor, 2006). Fig. 4. Deformation of the layer during propagation of the travelling wave For the considered travelling wave we can make an approximation of equ- ation (3.4) for a small slope angle surface of the layer to the axis X1 (see Fig.4) tanα≈α= lim ∆X1→0 h2−h1 ∆X1 = lim ∆X1→0 ∆ε2 ∆X1 h= dε2 dX1 h= ε2,1h (4.1) according to (2.3)2 ε2,1 = κ, then tanα≈ κh. Assumed that there is a little modification to the surface of the layer slope to the axis X1, we have |κh|≪ 1 or |hε1,1 ◦ ε −2 1 |≪ 1 (4.2) The last equality follows from (2.18)1. First, we introduce the dimensionless variables b1 = 2C1 ν−2C1 b2 = 2C2 ν−2C1 b3 = 2C3 ν−2C1 D1 = 2d1 ν−2C1 D2 = 2d2 ν−2C1 (4.3) Multiplying equations of motion (3.4) by 4/(ν−2C1) and substituting (4.3), we obtain an approximate form Traveling waves in a thin layer composed... 117 (b1+ b2+2b3) ( ◦ ε 2 1+ ◦ ε −2 1 ) +b3 ( ◦ ε 4 1+ ◦ ε −4 1 ) + h2 3 ε21,ξ ◦ ε −4 1 (1− b2)+ (4.4) −2 3 b3h 2ε21,ξ ( ◦ ε −2 1 + ◦ ε −4 1 + ◦ ε −6 1 ) − 3 5 b3h 4ε41,ξ ◦ ε −8 1 = ε 2 1(1+ b1)+2D1ε1+2D2 According to (4.2)2, the expression h 4ε4 1,ξ ◦ ε −8 1 is infinitesimal superior in rank compared to the rest of term of equation (4.4). Now we introduce the following transformation ζ = √ 3 h ξ (4.5) Apart fromthe scaling factor, ζ is just the current configuration coordinate X1 in terms of the phase ξ, and (4.4) takes the form ε21,ζ =F(ε1,D2) (4.6) where F(ε1,D2)= (4.7) = ◦ ε 6 1 [ 2D1ε1+2D2+ε 2 1(1+ b1) ] − (b1+ b2+2b3) ( ◦ ε 8 1+ ◦ ε 4 1 ) + b3 ( ◦ ε 10 1 + ◦ ε 2 1 ) (1− b2) ◦ ε 2 1−2b3 ( ◦ ε 4 1+ ◦ ε 2 1+1 ) D2 is an argument of F in above equations (4.6) and (4.7) because different curves in the phase plane correspond to different values of D2. More precisely, the parameters b1, b2, b3 and D1 uniquely determine a portrait, and then D2 determines the curves in that portrait. We introduce a denotation y= ε1,ζ = √ F(ε1,D2) (4.8) whose first derivative with respect to ζ is equal y,ζ = 1 2 F ′(ε1,D2) then tanβ= dy dε1 = F ′ 2y (4.9) where derivatives of F(ε1,D2) with respect to ε1 are denoted by prime. This system shows immediately that equilibria in the phase plane satisfy y = 0, F ′(ε1,D2)= 0. This indicates a specific character of the nonlinear system, which have one or several equilibrium positions, and depends on the function F(ε1,D2) (Dai, 2001). 118 I. Major, M. Major Equation (4.9)2 describes a straight line tangent to the trajectory at the function of the phase coordinates (ε1,y). The phase points are called ordina- ry or regular points if the tangent is determinate, however if the tangent is indeterminate, i.e. dy/dε1 = yζ/ε1,ζ → 0 the points are called singular points or equilibrium points. The phase point is the equilibrium point if y= ε1,ζ =0 ⇒ F(ε1,D2)= 0 yζ =0 ⇒ F ′(ε1,D2)= 0 (4.10) Discharging necessary equilibriumcondition (4.10) after substituting (4.7), we have ◦ ε 6 1[2D1ε1+2D2+ε 2 1(1+ b1)]− (b1+ b2+2b3) ( ◦ ε 8 1+ ◦ ε 4 1 ) − b3 ( ◦ ε 10 1 + ◦ ε 2 1 ) =0 (4.11) 6 ◦ ε 5 1[2D1ε1+2D2+ε 2 1(1+b1)]+2 ◦ ε 6 1[D1+ε1(1+ b1)]+ −4(b1+ b2+2b3) ( 2 ◦ ε 7 1+ ◦ ε 3 1 ) −2b3 ( 5 ◦ ε 9 1+ ◦ ε1 ) =0 Eliminating D2 and simplifying, we obtain a polynomial equation [ ◦ ε 5 1[D1+ε1(1+ b1)]+2b3 ( 1− ◦ε 8 1 ) +(b1+ b2+2b3) ( ◦ ε 2 1− ◦ ε 6 1 )] ◦ ε 2 1 =0 (4.12) The character of each equilibrium can be found by linearisation of (4.8) and (4.9). If ε1 = ε1e, y =0 is a solution to (4.12), according with (4.10) we have F(ε1e,D2)=F ′(ε1e,D2)= 0 (4.13) then close to the equilibrium point y=Y ε= ε1e+Λ (4.14) where Y and Λ are small perturbations. Substituting (4.14) into (4.8), we have Y =(ε1e+Λ)ζ (4.15) which entails that Λζ =Y (4.16) Similarly, substituting (4.13) into (4.9), we obtain Yζ = 1 2 F ′[(ε1e+Λ),D2] = 1 2 F ′(ε1e,D2)+ 1 2 F ′′(ε1e,D2)(ε1e+Λ−ε1e) (4.17) Traveling waves in a thin layer composed... 119 According to (4.13) 0.5F ′(ε1e,D2)= 0, then Yζ = 1 2 F ′′(ε1e,D2)Λ (4.18) where D2 is the parameter value representing the equilibriumpoint (compare Dai, 2001). From the analysis described by Dai (2001) and Osiński (1980) it follows that if F ′′(ε1e,D2) < 0 the singular point (equilibrium point) at the phase plane is a center. Such a point sets a stable state of equilibrium. However, if F ′′(ε1e,D2)> 0, the singular point is a saddle and the state of equilibrium is unstable. In the degenerate case, in which F ′′(ε1e,D2)= 0, we obtain a cusp point (see Fig.5). We obtain the curve in the phase plane directly by taking square roots of F(ε1,D2). Fig. 5. Graphs of the functions F (a) and phase trajectory y in the phase plane (b) (ε1c – denotes the center point, ε1s – saddle point and ε1n – cusp point) The foregoing discussion indicates connection between the location and nature of equilibria as well as the form of graphs of F(ε1,D2). The real curves in the phase plane are described by equation (4.7) y=± √ F(ε1,D2). 5. Discussion about physically acceptable solutions The phase portrait method allows one to find solutions to differential sys- tem (4.6). However, not all curves in the phase plane are interesting for the 120 I. Major, M. Major physical problem at hand. Our main task consists in characterising such por- traits, whose values of b1, b2, b3 and D1, represent physically meaningful behaviour. With some approximation, we can assume that in the case of compression or tension of a thin rubber layer, the physically acceptable value ε1 is in the interval from −0.5 to 0.5. According to Theorem 1 from the paper by Dai (2001, p.104), in order that there be a physically acceptable solution we must obtain for the func- tion F(ε1,D2) a center point in the region of the physically acceptable va- lue ε1. Supposing that this point exists for ε1 = ε1c (then F(ε1c,D2)= 0 and F ′(ε1c,D2)= 0), we can find D1 and D2 as functions of ε1c, which determine this center D1 = (b1+ b2+2b3) ( ◦ ε 6 1c− ◦ ε 2 1c ) +2b3 ( ◦ ε 8 1c−1 ) ◦ ε 5 1c −ε1c(1+ b1) (5.1) D2 = ε1c(b1+ b2+2b3) ( ◦ ε 2 1c− ◦ ε 6 1c ) +2b3ε1c ( 1− ◦ε 8 1c ) ◦ ε 5 1c + + (b1+ b2+2b3) ( ◦ ε 6 1c+ ◦ ε 2 1c ) + b3 ( ◦ ε 8 1c+1 ) 2 ◦ ε 4 1c + 1 2 ε21c(1+ b1) where ◦ ε1c =1+ε1c. After substituting (5.1) into (4.12), we obtain (ε1−ε1c) ◦ ε 2 1 { ◦ ε 5 1(1+ b1)− (b1+ b2+2b3) ◦ ε 2 1 ◦ ε 3 1c ( ◦ ε 3 1 ◦ ε 3 1c+3(1+A)+B ) + (5.2) −2b3 ◦ ε 5 1c [ ◦ ε 5 1 ◦ ε 5 1c(3(1+A)+B)+5(1+W)+10(A+B)+Z ]} =0 where A= ε1+ε1c W = ε 3 1+ε 2 1ε1c+ε1ε 2 1c+ε 3 1c B= ε21+ε1ε1c+ε 2 1c Z = ε 4 1+ε 3 1ε1c+ε 2 1ε 2 1c+ε1ε 3 1c+ε 4 1c (5.3) Traveling waves in a thin layer composed... 121 The expression in the square brackets of equation (5.2) follows determina- tion of other equilibrium points ◦ ε 5 1(1+ b1)− (b1+ b2+2b3) ◦ ε 2 1 ◦ ε 3 1c [ ◦ ε 3 1 ◦ ε 3 1c+3(1+A)+B ] + (5.4) − 2b3 ◦ ε 5 1c [ ◦ ε 5 1 ◦ ε 5 1c(3(1+A)+B)+5(1+C)+10(A+B)+D ] =0 Finally, by computing F ′′(ε1c,D2) and substituting (5.1), we find that ε1c will be a center if 1 2 F ′′(ε1c,D2)= (1+ b1) ◦ ε 6 1c− (b1+ b2+2b3) ( 3 ◦ ε 2 1c+ ◦ ε 6 1c ) −2b3 ( 3 ◦ ε 8 1c+5 ) (1− b2) ◦ ε 2 1c−2b3 ( ◦ ε 4 1c+ ◦ ε 2 1c+1 ) <0 (5.5) In order to obtain physically acceptable solutions, we must have ν > 2C1. Equation (5.5) is not satisfied if ν < 2C1. It results from the paper of Dai (2001) that there is a second point except for the point of stable state of equilibrium. It is a point of unstable state of equilibrium – the saddle point. Since ν > 2C1, we see from (4.3) that b1 > 0, b2 > 0 and b3 > 0. The difference in the signs ofmarks of terms in expression (5.4) suggests that there is a positive root, which we assume to be equal ε1 = ε1s. Equation (5.2) takes the form (ε1−ε1c)(ε1−ε1s) ◦ ε 2 1 ◦ ε 2 1c { (1−b2−2b3) ◦ ε 2 1 ◦ ε 2 1c 3(1+A1)+B1 [ 8E+3E2+3(ε21A1+ε 2 1cA2+ε 2 1sA)+ +6(1+F)+AF +ε21sB+ε 2 1ε 2 1c+3(ε1ε1c+ε1ε1s+ε1cε1s) ] + −2b3 [ ◦ ε 2 1 ◦ ε 2 1c(3(1+A)+B)(3(1+A2)+B2) ] + (5.6) + 2b3 ◦ ε 2 1s(3(1+A1)+B1) [ 20E+10E2+7(ε21A1+ε 2 1cA2+ε 2 1sA)+ +2EF +2G+15+12F +ε1ε1cB+ε1ε1sB2+ε1cε1sB1 ] } =0 122 I. Major, M. Major where we have eliminated b1 using the fact that ε1 = ε1s is a root b1= (1−b2−2b3) ◦ ε 3 1s ◦ ε 3 1c 3(1+A1)+B1 −b2−2b3 ◦ ε 3 1s ◦ ε 3 1c−2b3 ( 1+ 5(1+W1)+10(A1+B1)+Z1 ◦ ε 2 1s ◦ ε 2 1c[3(1+A1)+B1] ) (5.7) where ◦ ε1s =1+ε1s and we used the following variables A1 = ε1s+ε1c A2 = ε1s+ε1 B1 = ε 2 1s+ε1sε1c+ε 2 1c B2 = ε 2 1s+ε1sε1+ε 2 1 E = ε1+ε1s+ε1c F = ε1ε1sε1c G= ε31+ε 3 1s+ε 3 1c W1 = ε 3 1s+ε 2 1sε1c+ε1sε 2 1c+ε 3 1c Z1 = ε 4 1s+ε 3 1sε1c+ε 2 1sε 2 1c+ε1sε 3 1c+ε 4 1c (5.8) however A,B,W ,Z are according to (5.3). Substituting b1 (see (5.7)) into (5.5), we obtain the following expressions for ε1c and ε1s, respectively 1 2 F ′′(ε1c,D2)= ε1c−ε1s (1− b2) ◦ ε 2 1c−2b3 ( ◦ ε 4 1c+ ◦ ε 2 1c+1 ) · · ( (1− b2−2b3) ◦ ε 5 1c(6+4ε1c+ε 2 1c+2ε1cε1s+8ε1s+3ε 2 1s) 3(1+ε1s+ε1c)+ε 2 1s+ε1sε1c+ε 2 1c + +6b3 ◦ ε 5 1c[3(1+ε1s+ε1c)+ε 2 1s+ε1sε1c+ε 2 1c]+ + 2b3(15+25ε1c+15ε 2 1c+3ε 3 1c+20ε1cε1s+6ε 2 1cε1s) ◦ ε 2 1s[3(1+ε1s+ε1c)+ε 2 1s+ε1sε1c+ε 2 1c] + + 2b3(4ε1cε 2 1s+20ε1s+10ε 2 1s+2ε 3 1s) ◦ ε 2 1s[3(1+ε1s+ε1c)+ε 2 1s+ε1sε1c+ε 2 1c] ) (5.9) 1 2 F ′′(ε1s,D2)= ε1s−ε1c (1−b2) ◦ ε 2 1s−2b3 ( ◦ ε 4 1s+ ◦ ε 2 1s+1 ) · · ( (1− b2−2b3) ◦ ε 5 1s(6+4ε1s+ε 2 1s+2ε1sε1c+8ε1c+3ε 2 1c) 3(1+ε1c+ε1s)+ε 2 1c+ε1cε1s+ε 2 1s + +6b3 ◦ ε 5 1s[3(1+ε1c+ε1s)+ε 2 1c+ε1cε1s+ε 2 1s]+ Traveling waves in a thin layer composed... 123 + 2b3(15+25ε1s+15ε 2 1s+3ε 3 1s+20ε1sε1c+6ε 2 1sε1c) ◦ ε 2 1c[3(1+ε1c+ε1s)+ε 2 1c+ε1cε1s+ε 2 1s] + + 2b3(4ε1sε 2 1c+20ε1c+10ε 2 1c+2ε 3 1c) ◦ ε 2 1c[3(1+ε1c+ε1s)+ε 2 1c+ε1cε1s+ε 2 1s] ) According to conclusions featured at condition (4.18), we can see that if 1 2 F ′′(ε1c,D2)< 0 or 1 2 F ′′(ε1s,D2)> 0 (5.10) we obtain a center point or saddle point in the phase plane, respectively. 6. Numerical analysis The numerical analysis is based on equation (4.6) obtained for the Zahorski material and carried out for the function F(ε1,D2) (see (4.7)). The constant D1 depends on ε1, and according to (4.12) we have D1(ε1)= (b1+ b2+2b3) ( ◦ ε 6 1− ◦ ε 2 1 ) +2b3 ( ◦ ε 8 1−1 ) ◦ ε 5 1 −ε1(1+ b1) (6.1) Analogously, the constant D2 (which depends on ε1 too), according to (4.10)1 is equal D2(ε1)= ε1(b1+ b2+2b3) ( ◦ ε 2 1− ◦ ε 6 1 ) +2b3ε1 ( 1− ◦ε 8 1 ) ◦ ε 5 1 + (6.2) + (b1+ b2+2b3) ( ◦ ε 6 1+ ◦ ε 2 1 ) + b3 ( ◦ ε 8 1+1 ) 2 ◦ ε 4 1 + 1 2 ε21(1+ b1) Then, we can determine the constants D1 and D2 from (6.1) and (6.2), re- spectively, for a chosen value of ε1 (in this paper ε1 =0.5). In Fig.6 there are three graphs of the functions Fi(ε1) ≡ Fi(ε1,D2), i = 1,2,3, for constant D2 calculated according to (6.2) and for ε1 = 0.5. The functions y(ε1) denotes yi(ε1)≡± √ Fi(ε1) for i=1,2,3. In the analysis we assumed the rubber density ρ = 1190kg/m3 and the shear modulus µ=1.432 ·105N/m2. The constants C1, C2 and C3 are cha- racteristic for the kind of rubber described by Zahorski (1962) and take the 124 I. Major, M. Major Fig. 6. Graphs for the rubber OKA-1made of the Zahorski material (µ=1.46kG/cm2, ρ=1190kg/m3) for the speed V =25m/s and constants b1 =0.203, b2 =0.029, b3 =0.022 and D1 =−0.125 (according to (6.1) for ε1 =0.5); (a) distribution of functions: for F1(ε1) the constant D2 =0.342 (according to (6.2) for ε1 =0.5), for F2(ε1) – D2 =0.348 and for F3(ε1) – D2 =0.335, respectively, (b) phase trajectory values C1 =4.299·104N/m2,C2 =0.604·104N/m2 and C3 =0.47·104N/m2. The constants b1, b2 and b3 were calculated according to (4.3)1,2,3. Figure 6b shows phase trajectories in the coordinate system (ε1,ζ = y,ε1) for the functions Fi(ε1), i=1,2,3 found fromFig.6a for the Zahorski mate- rial. The constants D1 and D2 for ε1 = 0.5 are calculated according to (6.1) and (6.2). InFig.6, the constant D1 =−0.125 and the constant D2 calculated from (6.2) is 0.342. The constants D2 = 0.348 and D2 = 0.335 have been established arbitrarily, but here it is fixed at D2 =0.342. The center point is obtained for ε1 ∼= 0.07, and the graph is contained within the interval ε1 = 〈−0.36;0.5〉 of physically acceptable solutions (see Section 5). We find that propagation of the travelling wave in the thin layer composed of Zahorski’s material is possible for compression and tension. The solution has a periodic character for closed curves in an area limited by the solid line shown in Fig.6b, and can be a solitary wave for solutions represented by a homoclinic orbit (see the solid line in Fig.6b). Traveling waves in a thin layer composed... 125 For the constant C3 = 0, we can receive graphs for the Mooney-Rivlin material.Thesolutionsdifferquantitatively (compareMajor andMajor, 2007). References 1. Braun M., Kosiński S., 1999, Evolution behavior of transverse shocks in a nonlinear elastic layer, International Series of Numerical Mathematics, 129, 119-128 2. ColemanB.D., NewmanD.C., 1990,Onwaves in slender elastic rods,Arch. Rational. Mech. Anal., 109, 39-61 3. Dai H.-H., 2001, Nonlinear dispersive waves in a circular rod composed of a Mooney-Rivlinmaterial,Nonlinear elasticity: Theory and applications,London Mathematical Society Lecture Note Series, 283, 392-432 4. FuY.B., ScottN.H., 1989,Accelerationwaves and shockwaves in transver- sely isotropic elastic non-conductors, Int. J. Engng. Sci., 22, 11, 1379-1396 5. Major I., Major M., 2006, Phase plane analysis for nonlinear travelling wa- ves in incompressible hyperelastic circular rod, Vibrations in Physical System, XXII, 243-249 6. Major I., Major M., 2007, Nonlinear travelling waves in a thin layer com- posed of the Mooney-Rivlin material, Journal of Theoretical and Applied Me- chanics, 45, 2, 259-275 7. Osiński Z., 1980,Teoria drgań, PWN,Warszawa 8. TruesdellC., ToupinR.A., 1960,The classical field theories,Handbuch der Physik, III/1, Springer-Verlag, Berlin 9. Wesołowski Z., 1972a, Fala akustycznaw cylindrze odkształconymw sposób skończony,Rozprawy Inżynierskie, 20, 4, 613-628 10. Wesołowski Z., 1972b,Wprowadzenie do nieliniowej teorii sprężystości,Wy- dawnictwoUczelniane Politechniki Poznańskiej, Poznań 11. WrightT.W., 1981,Nonlinearwaves in rods,Proc. of the IUTAMSymposium on Finite Elasticity, D.E. Carlson and R.T. Shields (edit.), 423-443,Martinus Nijhoff The Hague 12. Zahorski S., 1962, Doświadczalne badania niektórych własności mechanicz- nych gumy,Rozprawy Inżynierskie, 10, 1, 193-207 126 I. Major, M. Major Fale biegnące w cienkiej warstwie wykonanej z nieliniowego hipersprężystego materiału Zahorskiego Streszczenie W pracy omówiono zagadnienia dotyczące propagacji nieliniowej fali biegnącej w cienkiej sprężystej warstwie wykonanej z materiału Zahorskiego.Metoda polegają- ca na uśrednieniu równań ruchuwprzekrojupoprzecznymwarstwyprzy założeniu, że uśrednione wielkości spełniają równania ruchu i warunki brzegowe pozwala na przy- bliżone rozwiązanie zagadnienia propagacji fali biegnącej w warstwie hipersprężystej zastosowano. Otrzymane w ten sposób równania zastosowano do opisu procesów fa- lowych dla rozpatrywanychw pracy fal biegnących. Do analizy procesów falowych użyta została technika płaszczyzny fazowej. W oparciu o metodę trajektorii fazowej zinterpretowano warunki propagacji nieli- niowej fali oraz ustalono warunki istnienia fizycznie akceptowalnych rozwiązań. Manuscript received March 25, 2008; accepted for print May 9, 2008