where n is the number of Voigt units as shown in Figure. 1. Eqs. (2) and (3) can also be written in integral form: ³ c c c t ij m s s ij C J t s t dt s 0 2 ( ) ( ) H t t dt t t J t s t t C ij t m t t ij t c c c c ³ H H H H ( ) ( )exp exp 0 2 Figure. 1 Nonlinear Burger’s model A simple, stable integration operator for these equations is the central difference operator: t f f t t ' ' ' 2 1 , 2 2 1 f f f t t t ' ' where f is some function, tf is its value at the beginning of the increment, f' is the change in the function over the increment, and t' is the time increment. Jacobian matrix H V w' w' of the constitutive model, with V ' being the stress increments and H ' being the strain increments, can be derived by applying this to the above rate-dependent constitutive Eqs. (1)-(3). Applying the central difference method to the elastic strain component as depicted in Eq. (1), yields ij kk kk ij ij e ij e ij E E V G V X V V X H H ) 2 1 ( ) 2 1 ( 1 2 1 ' ' ' The stress and strain vectors for 3-D problem as defined in ABAQUS are xy zx yz zz yy xx V V V V V V V , , , , , xy zx yz zz yy xx H H H H H H H , , , , , E, X sC , sm tC , tm , H t 94
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