JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 4, pp. 777-787, Warsaw 2005 VELOCITY OF ACCELERATION WAVE PROPAGATING IN HYPERELASTIC ZAHORSKI AND MOONEY-RIVLIN MATERIALS Maciej Major Department of Civil Engineering, Technical University of Częstochowa e-mail: admin@major.strefa.pl This paper studies homogeneous static deformation of a incompressible body. It shows a comparative analysis of a wave process in hyperelastic materialswhichhave linear (Mooney-Rivlinmaterial) andnonlinear (Za- horski material) dependences on invariants of the deformation tensor. The numerical analysis clearly demonstrates fundamental quantitative differences in the process of propagation of the accelerationwave. These differences are the consequence of calculation of elastic potentials which has been assumed in the study. Key words: discontinuous surface, acceleration waves, hyperelastic materials. 1. Introduction The general principles governing the mechanics of continuum, in particu- lar its motion, are essential to determine body behaviour being influenced by external forces. Consequently, what becomes especially important, is the ana- lysis of propagation of a disturbancebeingmodeled as a ”moving discontinuity surface” in the continuum. Anaccelerationwave isa surfaceofdiscontinuity.Muchworkhasbeendone on the subject of acceleration waves in hyperelastic materials, for example by Wright (1973),Truesdell andNoll (1965),Varley (1965),Chen(1968) orJeffrey (1982). Studies on the acceleration wave propagation have been the object of interest in a range of scientific domains which make use of mathematical models of the continuum in their deliberations. For many years, there have been lots of researches to recognize the process of wave propagation as well as 778 M.Major accompanying transport of energy andmomentum. In the case of acceleration waves, these processes demonstrate great complexity. Their intensities change depending on series of factors. It gives a meaningful sense to the analysis of propagation of a weak discontinuity wave in the continuum. There has been a progress in this domain since newmeasurementmethods have been applied. In the experimental research,measurements ofwave propa- gation velocity allows one to determinematerial constantsmore precisely. The methods being permanently improved, give new possibilities to experimental analysis within both compressible and incompressible materials. 2. Basic dependences Motion of a three-dimensional continuum is represented by a set of func- tions xi =xi(Xα, t) i,α=1,2,3 (2.1) The coordinates xi describe the current position at a time t of a material point in terms of its position Xα in the reference configuration. The deformation gradient and the particle velocity have the components xiα = ∂xi ∂Xα ẋi = ∂xi ∂t (2.2) The left Cauchy-Green tensor is defined by Bik =xiαx k βg αβ (2.3) Its principal invariants I1, I2 and I3 are the deformation invariants. Incompressible, isotropic elasticmaterial is characterized (i) by the internal constraint of incompressibility, I3 =1 and (ii) by a strain-energy function W =W(I1,I2) (2.4) The internal energy W per unit undeformed volume is expressed as function of the two free deformation invariants. In this case, the first Piola-Kirchhoff stress tensor has the components TRi α = ∂W ∂xiα +pXαi = ∂W ∂Ik ∂Ik ∂xiα +pXαi (2.5) where p is a hydrostatic reaction stress. Velocity of acceleration wave propagating... 779 Using the propagation condition for the acceleration wave in the reference configuration (Wesołowski, 1974) (Aαi β k NαNβ −ρRgikU 2)Ak +CNαX α i =0 (2.6) and Nα = 1 J xiαni dS dSR = U u xiαni (2.7) and identity Xαix i β ≡ δ α β (2.8) we obtain the equation (Aαi β k NαNβ −ρRδikU 2)Ak+C U u ni =0 (2.9) Multiplying (2.9) by ni, we have Aαi β k NαNβA kni−ρRδikU 2Akni+C U u =0 (2.10) According to (Wesołowski, 1974), the first order function of the material and coordinates of the vector N normal to the discontinuity surface in the refe- rence configuration is the acoustical tensor Qik =A α i β k NαNβ (2.11) Then (2.10) takes the form Qikn iAk−ρRA knk+C U u =0 (2.12) where C =−Qikn iAk u U (2.13) because in incompressible bodies there are only transverse waves (Aknk =0). Substituting (2.13) into (2.10), we obtain (Qik−Qrkn rni−ρRδikU 2)Ak =0 (2.14) Taking a notation ∗ Qik =Qik−Qrkn rni (2.15) for the reduced acoustical tensor, we obtain the propagation condition second- order discontinuity surface in the reference configuration ( ∗ Qik−ρRδikU 2)Ak =0 (2.16) 780 M.Major 3. Homogeneous deformations in isotropic materials We consider homogenous static deformation of an incompressible body in the coincide Cartesian coordinate systems {xi} and {Xα}, represented by x1 =λX1 x2 =λX2 x3 = 1 λ2 X3 (3.1) where λ= const. For medium deformation (3.1) being considered, the deformation gradient does not depend on Xα coordinate and time t. For a homogenous material, the material functions are constant in time and space. The deformation gradient, its converse, and the left and right Cauchy- Green deformation tensor, are [xiα] =     λ 0 0 0 λ 0 0 0 1 λ2     [ Xαi] =      1 λ 0 0 0 1 λ 0 0 0 λ2      (3.2) [Bij] = [Cαβ] =     λ2 0 0 0 λ2 0 0 0 1 λ4     and the deformation invariants I1 =2λ 2+ 1 λ4 I2 =λ 4+ 2 λ2 I3 =1 (3.3) We have to define the vector n normal to the discontinuity surface in the current configuration in order to work out the coordinates of the reduced acoustical tensor. We assume that the discontinuity surface in the reference configuration propagates in the X3 direction (Fig.1). The normal vector in the reference configuration takes the form N = [cosα,sinα,0] (3.4) The unit vector in the current configuration (Wesołowski, 1974) nk = JX α kNα dSR dS (3.5) Velocity of acceleration wave propagating... 781 Fig. 1. The surface elements dSR and dS and the discontinuity surface ΣR and Σ in the reference and current configuration According to (Fig.1), for the deformation described by (3.1), we have dSR = dLR ·dX3 = √ (dX1)2+(dX2)2 ·dX3 (3.6) dS = dL ·dx3 = √ λ2(dX1)2+λ2(dX2)2 · dX3 λ2 Substituting (3.6) into (3.5), we obtain a formula for the coordinates of the vector n in the current configuration nk =X α kNαλ (3.7) The analysis will be continued for a special isotropic elastic material charac- terized by the constitutive equation (Zahorski, 1962) W(I1,I2)=σρR =C1(I1−3)+C2(I2−3)+C3(I 2 1 −9) (3.8) where C1, C2 and C3 are constants. The special case C3 =0 corresponds to the Mooney-Rivlin material. For the analyzed material σi = ∂σ ∂Ii σik = ∂2σ ∂Ii∂Ik i,k=1,2 (3.9) we obtain the properties σ1 = 1 ρR (C1+2C3I1) σ2 = C2 ρR σ12 =σ21 =σ22 =0 σ11 = 2C3 ρR (3.10) 782 M.Major According to (Wesołowski, 1974), the coordinates of the first order functions of the material tensor can be calculated from the equation Aαi β k = ρR ∂2σ ∂xiα∂x k β = ρR { σ1 ∂2I1 ∂xiα∂x k β +σ11 ∂I1 ∂xkβ ∂I1 ∂xiα +σ2 ∂2I2 ∂xiα∂x k β } (3.11) where ∂I1 ∂xiα =2xi α ∂I2 ∂xiα =2(I1xi α−Birx r α) ∂2I1 ∂xiα∂x k β =2gikg αβ (3.12) ∂2I2 ∂xiα∂x k β =2[2xi αxk β −gikxr αxrβ −xi βxk α+(I1gik−Bik)g αβ] Substituting (3.12) into (3.11), we obtain the first order function of the material tensor for the material characterized by strain-energy function (3.8) Aαi β k = ρR { 2σ1gikg αβ + (3.13) +2σ2[2xi αxk β −gikC αβ −xi βxk α+(I1gik−Bik)g αβ]+4σ11xi αxk β } According to (2.11) and including (3.13), we have the acoustical tensor Qik = ρR { 2σ1gik+2σ2(2xi αxk β −gikC αβ −xi βxk α)NαNβ + (3.14) +2σ2(I1gik−Bik)+4σ11xi αxk βNαNβ } Substituting the deformation gradient and deformation tensor described by expressions (3.2) into (3.14) and including the vector N from (3.4), we obtain components of the acoustical tensor Q11 =2 [ C1+C2 ( λ2cos2α+ 1 λ4 ) +2C3 ( 1 λ4 +λ2(2+cos2α) )] Q12 =Q21 =(C2+2C3)λ 2 sin2α Q22 =2 [ C1+C2 ( λ2 sin2α+ 1 λ4 ) +2C3 ( 1 λ4 +λ2(2+sin2α) )] (3.15) Q33 =2 [ C1+C2λ 2+2C3 ( 2λ2+ 1 λ4 )] Q13 =Q23 =Q31 =Q32 =0 Velocity of acceleration wave propagating... 783 Fordeformation (3.1) beingconsidered, thevector N = [cosα,sinα] according to (3.7) passes into the normal vector n in the current configuration, and this vector is the same like the vector N in the reference configuration n=N = [cosα,sinα] (3.16) The coordinates vector n allows one to simplify expression (2.15), thus ∗ Qik =Qik−Q1kn1ni−Q2kn2ni i,k=1,2,3 (3.17) Substituting (3.15) and (3.16) into the above equation, we obtain components of the tensor ∗ Qik ∗ Q11 =2 [ C1+C2 1 λ4 +2C3 ( 2λ2+ 1 λ4 )] sin2α ∗ Q12 = ∗ Q21 =− [ C1+C2 1 λ4 +2C3 ( 2λ2+ 1 λ4 )] sin2α ∗ Q22 =2 [ C1+C2 1 λ4 +2C3 ( 2λ2+ 1 λ4 )] cos2α (3.18) ∗ Q33 =Q33 =2 [ C1+C2λ 2+2C3 ( 2λ2+ 1 λ4 )] ∗ Q13 = ∗ Q23 = ∗ Q31 = ∗ Q32 =0 The reduced acoustical tensor matrix [ ∗ Qik] could be simplified [ ∗ Qik] =      ∗ Q11 ∗ Q12 0 ∗ Q21 ∗ Q22 0 0 0 ∗ Q33      (3.19) The expression ρRU 2 is an eigenvalue of the ∗ Qik tensor. From the charac- teristic equation, we determine the propagation velocity of the discontinuity surface in the reference configuration (U1,2) 2 = 1 2ρR [ ∗ Q11+ ∗ Q22± √ ( ∗ Q11− ∗ Q22) 2+4 ∗ Q12 ∗ Q21 ] (U3) 2 = ∗ Q33 ρR (3.20) 784 M.Major Having substituted the components ∗ Qik, see (3.18), into expressions (3.20), we have U1 = √ 2 ρR [ C1+C2 1 λ4 +2C3 ( 2λ2+ 1 λ4 )] U2 =0 (3.21) U3 = √ 2 ρR [ C1+C2λ2+2C3 ( 2λ2+ 1 λ4 )] The eingenvector D(1) for the velocity U1 satisfies the equation      ∗ Q11− (U1) 2ρR ∗ Q12 0 ∗ Q21 ∗ Q22− (U1) 2ρR 0 0 0 ∗ Q33− (U1) 2ρR          D (1) 1 D (1) 2 D (1) 3     =    0 0 0    (3.22) from the above equation we directly have D (1) 1 D (1) 2 =− ∗ Q12 ∗ Q11− (U1) 2ρR =− ∗ Q22− (U1) 2ρR ∗ Q21 (3.23) Substituting (3.18) into (3.23), we obtain D (1) 1 D (1) 2 =− sinα cosα (3.24) Fig. 2. The unit vector D(1) for the acceleration wave propagating with the velocity U1 in spatial coordinates For the vector n normal to the discontinuity surface, the unit vector in the wave amplitude direction D(1) for the propagation velocity U1 has the coordinates D (1) = [D (1) 1 ,D (1) 2 ,D (1) 3 ] = [−sinα,cosα,0] (3.25) Velocity of acceleration wave propagating... 785 Making analogical deliberation for the velocity U3, we find that the unit vector D(3) does not depend on the angle α. Then we have D (3) = [D (3) 1 ,D (3) 2 ,D (3) 3 ] = [0,0,±1] (3.26) Fig. 3. The unit vector D(3) for the acceleration wave propagating with the velocity U3 in spatial coordinates 4. Numerical results The subject of numerical analysis are the expressions for velocity of pro- pagation of the acceleration wave in a hyperelastic material assumed in the calculations. The analysis is based on formulas represented by the velocity as function of the preliminary deformation (extension) λ. The density of rubber assumed in the analysis is ρ=1190kg/m3. C1,C2 and C3 constants correspond to one kind of rubber represented by (Zahorski, 1962) C1 =6.278 ·10 4Pa C2 =8.829 ·10 3Pa C3 =6.867 ·10 3Pa The velocity distributions U1 and U3 of the acceleration wave presented in the Fig.4 point out considerable both quantitative and qualitative differences between the Zahorski andMooneyRivlinmaterials. They result fromdifferent strain-energy functions of hyperelastic materials assumed in the calculations. This difference is based on the linear (MooneyRivlin material) and nonlinear (Zahorski material) dependence on the Cauchy-Green deformation invariants. The preliminary strain of rubber in the interval inwhich both compression andtensionarepossible (λ∈< 0.5,2>) for thevelocity U1 and U3 inZahorski material, increase the velocity of the compression and tension strain. 786 M.Major Fig. 4. The propagation velocity of the acceleration wave in the Zahorski material (U1Z and U3Z) and theMooney-Rivlin material (U1M and U3M) for λ∈< 0.5,2> For the Mooney-Rivlin material, this increase occurs for compression for thevelocity U1 and for tension for thevelocity U3, however thedecrease occurs for the velocity U1 in the case of tension strain, and for the velocity U3 in the case of compression strain. Such dynamical behaviour of the incompressible medium is different than in the case of behaviour of theMurnaghanmaterial (Major, 2001). Consisten- tly, the increase of propagation velocity of the acceleration wave and tension cause a decrease in this velocity. References 1. Chen P.J., 1968, The growth of accelerationwaves in arbitrary form in homo- geneouslydeformed elastic materials,Arch. Rat. Mech. Anal., 30, 1, 81-89 2. Jeffrey A., 1982, Acceleration wave propagation in hyperelastic rods of va- riable cross-section,Wave Motion, 4, 173-180 3. Major M., 2001, Fala przyspieszenia we wstępnie odkształconym materiale Murnaghana, Zeszyty Naukowe Politechniki Śląskiej, 1514, Budownictwo, 93, Gliwice, 305-314 4. TruesdellC.,NollW., 1965,TheNon-Linear Field Theories ofMechanics, Springer-Verlag, Berlin 5. Varley E., 1965, Acceleration fronts in viscoelastic materials, Arch. Rat. Mech. Anal., 19, 3, 215-225 Velocity of acceleration wave propagating... 787 6. WesołowskiZ., 1974,Zagadnienia dynamiczne nieliniowej teorii sprężystości, PWN,Warszawa 7. WrightT.W., 1973,Accelerationwaves in simple elasticmaterials,Arch. Rat. Mech. Anal., 50, 4, 237-277 8. Zahorski S., 1962, Doświadczalne badania niektórych własności mechanicz- nych gumy,Rozprawy Inżynierskie, 10, 1, 191-207 Prędkość fali przyspieszenia propagującej w hipersprężystym materiale Zahorskiego i Mooneya-Rivlina Streszczenie W pracy rozważana jest jednokrotna deformacja statyczna ciała nieściśliwego. Przedstawiona została analiza porównawcza procesu falowego zachodzącegowmate- riałach hipersprężystych, które rózni liniowa (materiałMooneya-Rivlina) i nieliniowa (materiał Zahorskiego) zależność od niezmienników tensora odkształcenia Cauchy- -Greena. Przeprowadzona analiza numerycznawyraźnie wykazała istotne różnice ilo- ściowewprocesiepropagacji fali przyspieszenia.Różnice te sąnastępstwemprzyjętych do obliczeń potencjałów sprężystych. Manuscript received May 30, 2005; accepted for print July 4, 2005