constitutive modeling of anisotropic hyperelastic materials by polyconvex strain energy functions

6th European Solid Mechanics ConferenceESMC200628August–1September,2006Budapest,Hungary CONSTITUTIVE MODELING OF ANISOTROPIC HYPERELASTIC MATERIALS BY POLYCONVEX STRAIN ENERGY FUNCTIONSMikhail Itskov1,Alexander E.Ehret11Department of Continuum Mechanics,RWTH Aachen UniversityEilfschornsteinstr.18,52062Aachen,GermanyE-mail:itskov@km.rwth-aachen.deKeywords:Polyconvexity,Anisotropy,Hyperelasticity,Soft biological tissues,Calendered rubber Summary.Polyconvexity of a strain-energy function is a very important mathematical condition especially in the context of a boundary-value problem.In the present contribution,a class of polyconvex anisotropic strain-energy functions is proposed.They are given by a series represented by some convex scalar functions.Each term of this series a priori satisfies the condition of the energy and stress free natural state so that no additional restrictions have to be imposed.Special cases of the proposed hyperelastic model based on power and exponential function representations show very good agreement with experimental data.1INTRODUCTIONSoft biological tissues as well as many engineering materials,as for examplefiber-reinforced or calendered rubber-like elastomers,are characterized by strong anisotropy and are able to undergo large elastic deformations.In the constitutive modeling,these features should be taken into account in an adequate manner.Hyperelastic consti-tutive models for soft biological tissues as for example the well known Fung-elastic model[2]or the orthotropic St.Venant-Kirchhoffmodel(see,e.g.,[3])may be used to describe the material behavior.Unfortunately,these models are not elliptic and for this reason not polyconvex.Polyconvexity represents,however,a very important mathematical condition especially in the context of a boundary-value problem[1].Indeed,polyconvexity implies quasiconvexity which in turn ensures ellipticity or the so-called Legendre-Hadamard condition.It means that the acoustic tensor is positive definite so that the speed of displacement waves is always real for any direction of prop-agation.Furthermore,in combination with some continuity and growth requirements,quasiconvexity guarantees the existence of the global minimizer of the total elastic energy of the body[1].To benefit from these positive features we propose in the present contribution a class of polyconvex anisotropic strain energy functions.2ORTHOTROPIC AND TRANSVERSELY ISOTROPIC MATERIAL SYMMETRIESTo describe the material symmetry,one can make use of the so-called structural tensors defined byL i=l i⊗l i,i=1,2,3(1) for orthotropic and byL1=l1⊗l1,L2=L3=12(I−l1⊗l1)(2)for transversely isotropic material symmetry,where l i·l i=δi j denote a set of orthonormal base vectors in the principal material directions.3ANISOTROPIC POLYCONVEX STRAIN ENERGY FUNCTIONSA powerful tool to construct polyconvex functions is the following additive representation of the strain energy function W(F)W(F)=ˆW1(F)+ˆW2(adj F)+ˆW3(det F),adj F=F−1det F,(3) where F denotes the deformation gradient.Indeed,if the functionsˆW i(i=1,2,3)are convex with respect to F, adj F and det F,respectively,then the resulting strain energy function(3)is polyconvex[5].Taking into accountMikhail Itskov 1,Alexander E.Ehret 1this fact,we propose a strain energy function of the formW =14sr =1µr f r ˜I r +g r ˜Jr +h r III C ,III C =det C ,(4)where f r and g r are convex and monotonically increasing and h r are convex functions of their arguments.Theterms ˜Ir and ˜J r denote generalized invariants defined by ˜Ir =3 i =1w (r )i I i ,˜Jr =3 i =1w (r )i J i ,r =1,2,...,s ,(5)in terms of the classical orthotropic or transversely isotropic strain invariantsI i =tr(CL i ),J i =tr [(cof C )L i ],cof C =(adj C )T(6)and weight factors of the principal material directions w i ≥0,(i =1,2,3).Note that both the invariants I i and J i(6)are convex with respect to F and adj F ,respectively and III C is convex with respect to det F .The same holds forthe generalized invariants ˜Ir and ˜J r (5)as far as the weight factors w i are non-negative.It can further be shown that each term in the series (4)fulfills the condition of the energy and stress free natural state if the functions f r ,g r and h r satisfy the requirements [3,4]f r (1)=g r (1)=h r (1)=0,f ′r (1)=g ′r(1)=−12h ′r(1)=1.(7)4COMPARISON WITH EXPERIMENTAL DATAIn order to demonstrate a wide applicability of the proposed polyconvex model (4)we consider numerical exam-ples dealing with experimental data on di fferent materials.In the first example our model was fitted to uniaxial tension tests on calendered rubber sheets using power function representations for f r and g r .The second example deals with elongation tests on bovine pericardium,an important biomaterial for medical applications.Motivated by the typical exponential-like stress-strain behavior of soft biological tissues,an exponential representation was applied.In both cases,very accurate agreement with experiments has been obtained as seen in Figure 1.N o m i n a l s t r e s s t , M P aStretch λN o m i n a l s t r e s s t , M P aStretch λFigure 1:Experimental data for calendered rubber (left)and bovine pericardium (right)versus simulation by the polyconvex model (4)REFERENCES[1]Ball,J.M.(1977):Convexity conditions and existence theorems in non-linear elasticity.Arch.Rat.Mech.Anal.,63,337-403.[2]Fung,Y .C.(1993):Biomechanics.Mechanical Properties of Living Tissues.2nd Ed.,Springer,New York.[3]Itskov M.,Aksel,N.(2004):A class of orthotropic and transversely isotropic hyperelastic constitutive modelsbased on a polyconvex strain-energy function,Int.J.Solids Struct.41,3833-3848.[4]Itskov M.,Ehret,A.E.,Mavrilas D.(2005):A polyconvex anisotropic strain-energy function for soft collage-nous tissues,Biomech.Model.Mechanbiol.(in press)[5]Schr¨o der,J.,Ne ff,P.(2003):Invariant formulation of hyperelastic transverse isotropy based on polyconvexfree energy functions,Int.J.Solids Struct.40,401-445.。