36 Drying of Wood

to the history of that point (« mem can deal with plasticity,creep, mechanosorption, etc.).The geometrical compatibility applies to thetotal strain field « tot . When solving the mechanicalproblems in terms of displacement, the total straintensor is deduced from the displacement field and thisgeometrical condition is automatically fulfilled withinthe domain. The stress field must satisfy the local mechanicalequilibrium and the boundary conditions.Finally, the complete formulation of the stress problemis given by Equation S1 through Equation S4.36.2.4.3 Memory EffectWhile describing the strain field, « mem , lies the entireproblem of developing a constitutive model forwood, which requires both theoretical and numericalwork. Comprehensive formulations are also verydifficult to characterize (Ranta-Maunus, 1975). Theproblem lies in the fact that the memory effect ofwood depends not only on the temperature and moisturecontent values but also on their variations in timeand on the history of their variations in time. This2 3A 0 0« sh ¼ H(~x) 4 0 B 0 5 (S1)0 0 CwithH(~x) ¼ 0 X(~x) X fsp if X (~x) $ X fspif X (~x) # X fsp8« tot« mec ¼ « elas þ « mem (S2)ij ¼ 1 2 (u i, j þ u j,i) over V>< s ij, j þ rf i ¼ 0 over Vs ij ¼ a ijkl (« totkl« 0 kl ) over V with «0 = « sh + « mem and 8i , G Di G Ti = Gs ij n j ¼ T ion G >:Tiu i ¼ D i ¼ 0 on G Di(S3)Remarks:. This static formulation requires that boundary and volumetric forces satisfy the global equilibrium.. G is the surface surrounding the domain V. G Di refers to the subdomain of G where the i componentof the displacement is known and G Ti to the subdomain of G where the i component of the tractionforce is known. In order to ensure the uniqueness of the solution, additional conditions are requiredon the boundary conditions: 8i, mes(G Di ) > 0. Otherwise, the solution is defined within a rigid bodymotion.. As wood is orthotropic, each behavior law involves nine independent terms. In fact, it is morecommon to define the inverse of a ijkl that, for the case of linear elasticity, leads to the generalizedHooke’s law:223« LL« RR« TT6 2« LR ¼ g LR¼74 2« LT ¼ g LT52« RT ¼ g RT641E L LR RLE R1E L LTE LE R TL RT 1E R0 0 0E T TR0 0 0E T0 0 0E T10 0 00 0G LR10 0 0 0 0G LT0 0 0 0 01G RT326475s LLs RRs TTs LRs LTs RT375(S4)ß 2006 by Taylor & Francis Group, LLC.