2.1 by the skew symmetric part of the velocity

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 e ective 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 e ective plastic strain rate is de ned on the basis of the conventionale ective plastic strain rate and the gradient of the conventional e ective 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 con guration 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 e ective 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 con guration. The consistancy condition andhigher order boundary condition expressed asQ? e ? i;i = 0; t = ini (2.8)Kirchho stress measures are de ned as&ij = Jij ; &e= Je; q = JQ; i = Ji (2.9)Where J is de ned as determinant of the metric tensor. The incremental formulation of principleof virtual power, in updated lagrangian framework(where deformaed con guration 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 con guration.Fleck and Hutchinson 11 de ned the plastic potential in rate-independent theory as(EP ) = SEP0c?EP??dEP?(2.11)Where c is an e ective stress which is work-conjugate to the e ective 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 de ned as( _EP ;EP ) = S_EP0c? _EP?;EP ?d _EP?(2.12)4Where c is work-conjugate to the e ective 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 e ective stress and i is higher order stress de nd asq = c_EP_ P (2.14)i = c_EPl2?_ P;i (2.15)By substituting the these expressions in equation (2.4), the e ective stress is given as following2c = q2 + l?2? ii (2.16)when excluding the material length scale by setting l? = 0, the e ective stress, c , reduces to VonMises stress and the e ective plastic strain rate, _EP , equals the conventional e ective plastic strainrate, _ P . The behaviour of the viscous material is modeled by power law for the e ective 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 sti ness tensor which is given byRijkl = E1 + ?12(ikjl + iljk) + 1 ? 2ijkl? (2.19)E ective 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 e ective 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.