T hi s pa per c ont ain s a co m p le te li s tin g of isotrop ic Ca rt es ia n tc nso rs of r an ks up to e ig ht with th e ir asso cia te d red uc tio n e q uat io n s fo r o bt ai nin g linea rl y ind e pe nd e nt se ts whe ne ve r th e red u c ti on is ca ll ed fo r . In p a rti c ul a r. th e li s tin g is co mp il e d onl y fo r isot ro pi c te n sors ass oc ia te d with t he ro ta tion gro up 0 +(3) of th e th ree ·dim e n s iona l und e r lyin g vec to r Sl)ac e. Based o n a n id e ntit y o rigi nal ly du e to C a pe lli ( 1887) , red uc tion equ a tio ns fo r te n so rs o f odd ra nk s be ginn in g a t r ank fiv e a nd e ve n ra nk s b e ginning a t ra nk e ig ht a r e s how n to b e n ont rivia l. S ign if ic a nce of t h e co mput a ti on a l re s ult in both pure and a ppli e d m a the matics is d isc u ssed .Key word s: Alge bra : a lt e rn a tin g t e nso r; Ca rt es ia n te n sor : C a pe lli 's id e ntit y; gr o up re prese nt a ti o n : in varia nt : is ut ro pi c te n sor: Kron ec kt-'r d e lt a : or! hogo ll a l ~r ou p: te nso r.

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The objective of this work is to represent general nonlinear viscoelasticity by a model based on a modified two-network theory. As a basic problem we examine Neubert and Saunders' data (1958) on the permanent set of crosslinked natural rubber samples after heating in a state of pure shear or simple extension. It is evident that the preferred configuration changes with time of heating. Our approach is to associate the preferred configuration with an internal measure of length in an initially isotropic material which becomes anisotropic after heating in a deformed state. Using Ericksen and Rivlin's work (1954) on anisotropic materials and a strain energy function which reduces for isotropic materials to Rivlin and Saunders' function (1951), we show an excellent consistency with the Neubert and Saunders' data. We also show consistency with some more recent data due to Djiauw and Gent (1973).

Experimental data on the volume changes accompanying simple tension of peroxide vulcanizates of natural gum rubber, as first reported by Penn in Trans. Soc. Rheol., 14, 509 (1970), are further interpreted within the framework of the theory of a hyperelastic material. Motivated by the formal connection between a hyperelastic material and a nonlinear viscoelastic fluid (Trans. Soc. Rheol., 9, 27 (1965)), and a mathematical result on the decomposition of the scalar potential of that fluid (Z. Angew. Math. Phys. 23, 780 (1972)), the interpretation of Penn's data leads to an explicit construction of a strain-energy function for rubbery materials: W(I1,I2,I3)=C1(I1−3)+C2(I2−3)+H̄(I1,I2)(I3−1)+18K(I3−1)2.

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