FLAC3D Theory and Background • Constitutive Models

Maxwell Model

The classical notion of Newtonian viscosity is that the rate of strain is proportional to stress. Stress-strain relations can be developed for viscous flow in a way similar to elastic deformation. The derivation of the equations in three dimensions can be found, for example, in Jaeger (1969).

Viscoelastic materials exhibit both viscous and elastic behaviors. One such material is the Maxwell material, which can be represented in one dimension by a spring (with elastic constant \(k\)) in series with dashpot (of viscous constant \(\eta\)). The incremental force/displacement law for this material can be written as

(1)\[\dot u = {\dot F \over k} + {F \over \eta}\]

where \(\dot u\) is the velocity and \(F\) is the force. Denoting the new value of force by \(F'\) and the old value by \(F^\circ\), over a timestep of \(\Delta t\), we can rewrite Equation (1) as

(2)\[{\Delta u\over\Delta t}={F'-F^\circ\over k\Delta t} +{F'+F^\circ\over2\eta}\]

This is a central difference equation, because the velocity is calculated at the midpoint between the instances when \(F'\) and \(F^\circ\) are defined. Solving for \(F'\)

(3)\[F'=(F^\circ C_1+k\Delta u)\,C_2\]

where:

\[C_1=1-{k\Delta t\over2\eta}\]

\[C_2={1\over{1+{k\Delta t\over2\eta}}}\]

An equation identical to Equation (3) can be written for the relation between deviatoric stresses and strain increments:

(4)\[\sigma_{ij}^d=(\sigma_{ij}^{d^\circ}C_1+2G \ \Delta \epsilon_{ij}^d)\,C_2\]

where:

\[\Delta \epsilon_{ij}^d=\Delta \epsilon_{ij}-{1\over3} \ \Delta \epsilon_{ij}\,\delta_{ij}\]

\[\sigma_{ij}^{d^\circ}=\sigma_{ij}^\circ-{1\over3} \ \sigma_{ij}^\circ \,\delta_{ij}\]

\[C_1=1-{G\Delta t\over2\eta}\]

\[C_2={1\over{1+{G\Delta t\over2\eta}}}\]

Here, \(\Delta \epsilon_{ij}\) are the components of the “input” strain-increment tensor, \(\sigma_{ij}^\circ\) are the components of the previous stress tensor, and \(G\) is the shear modulus. For the volumetric component of stress and strain, we assume that there are no viscous effects (elastic relations apply):

(5)\[\sigma^{\rm iso}={1\over3} \ \sigma_{kk}^\circ+K\Delta \epsilon_{kk}\]

where \(K\) is the bulk modulus. The final stress tensor is given by the sum of the deviatoric and isotropic parts:

(6)\[\sigma_{ij}=\sigma_{ij}^d+\sigma^{\rm iso}\delta_{ij}\]

The material properties required for this model are shear and bulk moduli (for the elastic behavior) and viscosity. Under an applied shear stress, the material flows continuously, but it behaves elastically under an applied isotropic stress.

maxwell Model Properties

Use the following keywords with the zone property command to set these properties of the Maxwell (classical viscoplastic) model.

maxwell

bulk f

bulk modulus, K

poisson f

Poisson’s ratio, v

shear f

shear modulus, G

young f

Young’s modulus, E

viscosity f

dynamic viscosity, η

Notes:

• Only one of the two options is required to define the elasticity: bulk modulus K and shear modulus G, or Young’s modulus E and Poisson’s ratio v.
• The creep behavior is triggered by deviatoric stress, while the volumetric behavior does not consider creep.