2.1 A Viscoplastic version of Fleck and Hutchinsons TheoryIn this section viscoplastic formulation for isotropic materials which allows for nite strains of therate-independent strain gradient plasticity theory presented by Fleck and Hutchinson 11 was out-lined. The kinematical basis is based on the rate-independent nite strain generalization of the straingradient theory by Niordson and Redanz 12 formulated within an updated Lagrangian framework.2.1.1 Material ModelLet ui denote the displacement vector and u_ i is the velocity led. With Lij = u_ i;j denoting thevelocity gradient,the material spin ij , is given by the skew symmetric part of the velocity gradientasij = 12(Lij ? Lji) (2.1)The strain rate is the symmetric part of the velocity gradient, which is decomposed into elastic andplastic part_ ij = 12(Lij + Lji) = _ Eij + _ Pij (2.2)The direction of plastic strain rate is given by mij = 32Sij~e; where Sij = ij ? 13 ijkk denotes thestress deviator and e =¼32SijSij is von mises eective stress with ij being the Cauchy stresstensor and ij the Kronecker delta function. The plastic strain rate components can be expressedas product of its magnitude, _ P =¼23 _ Pij _ Pij and its direction_ Pij = mij _ P (2.3)A nonlocal measure of the eective plastic strain rate is dened on the basis of the conventionaleective plastic strain rate and the gradient of the conventional eective plastic strain rate throughthe incremental relation_EP2 = _ P2 + l2?_ P;i _ P;i (2.4)where l2?is a material length parameter. Following Fleck and Hutchinson11, assuming that theplastic strain gradients contribute to the internal work, the principle of virtual power in total form3in the deformed conguration may be formulated asSV?ij _ Eij + Q _ P + i _ P;j?dV = SS?Tiu_ i + t_P ?dS (2.5)where Q is a generalized eective stress which is work-conjugate to the plastic strain rate magnitude,_ P and i is a higher order stress which is work-conjugate to the gradient of the plastic strain ratemagnitude, _ P;j . V and S are current volume and surface respectively. Ti and ti denotes the surfacetraction and higher order surface traction. The formulation can also be expressed asSV?ij _ Eij + (Q? e) _ P + i _ P;j?dV = SS?Tiu_ i + t_P ?dS (2.6)The strong form of the eld equations is found by requiring the principle of virtual power to holdfor all admissible variations in u_ i and _P . The classical force balance law and boundary conditionsare obtained asij;j = 0; Ti = ijnj (2.7)Where nj is the surface unit normal in the deformed conguration. The consistancy condition andhigher order boundary condition expressed asQ? e ? i;i = 0; t = ini (2.8)Kirchho stress measures are dened as&ij = Jij ; &e= Je; q = JQ; i = Ji (2.9)Where J is dened as determinant of the metric tensor. The incremental formulation of principleof virtual power, in updated lagrangian framework(where deformaed conguration is taken as areference) can be expressed as(Niordson and Redanz12)SV0?S&ij_ij ? ij(2_ik_kj ? LkjLki) + (q_ ? _ &e) _ P + -i _ P0;i?dV0 = SS0? _T 0i u_ i + _ t0 _ P ?dS0 (2.10)WhereS&ij = &_ij ? &kjik ? &ikjk, is Jaumann rate of the kircho stress, and-i = _i ? Likk isconvected rate of higher order kircho stress and subscript “0” refers to the reference conguration.Fleck and Hutchinson 11 dened the plastic potential in rate-independent theory as(EP ) = SEP0c?EP??dEP?(2.11)Where c is an eective stress which is work-conjugate to the eective plastic strain, EP and thefunction c?EP?) denotes the uniaxial tensile stress versus plastic strain curve of the material. Avisco plastic potential for rate-dependent version dened as( _EP ;EP ) = S_EP0c? _EP?;EP ?d _EP?(2.12)4Where c is work-conjugate to the eective plastic strain rate, _EP and the function c? _EP?;EP ?denotes the uniaxial tensile stress versus plastic strain rate curve.Taking the variation of the potentialby use of equation (2.4) gives = c _EP = c? _ P_EP _ P +l2?_ P;i_EP _ P;i ? = q _ P + i _ P;i (2.13)where q is generalized eective stress and i is higher order stress dend asq = c_EP_ P (2.14)i = c_EPl2?_ P;i (2.15)By substituting the these expressions in equation (2.4), the eective stress is given as following2c = q2 + l?2? ii (2.16)when excluding the material length scale by setting l? = 0, the eective stress, c , reduces to VonMises stress and the eective plastic strain rate, _EP , equals the conventional eective plastic strainrate, _ P . The behaviour of the viscous material is modeled by power law for the eective plasticstrain rate_EP = _ 0? cg(EP )?(1~m)(2.17)Where _ 0 is a reference strain rate and m is strain rate hardening exponent.The incremental elastic constitutive equation for viscoplastic material can be expressed asS&ijt = Rijkl(kl ?mklP ) (2.18)Where t is the time step and Rijkl is elastic stiness tensor which is given byRijkl = E1 + ?12(ikjl + iljk) + 1 ? 2ijkl? (2.19)Eective stress and higher order stress increments can be derived from equation (2.14) and (2.15)using equation (2.17) and expressed as followsq_t = c_EP?(m? 1) _ P_EP _EP +_ P ? + ?_EP_ 0?mdgdEP _ Pt (2.20)-it = l2?? c_EP?(m? 1)_ P;i_EP _EP +_ P;i ? + ?_EP_ 0?mdgdEP _ P;it? (2.21)Where the change in eective plastic strain rate is given as _EP = _ P_EP _ P + l2?_ P;i_EP _ P;i . The valuesof _ P and _ P;i are taken from previous increment, such that unknowns are ij ;_ P and _ P;i only.