I'm hoping that someone can share the physical significance (i.e. type of motion) resulting when the (spatial) velcity gradient L is constant. I'm not certain if L constant wrt to only position or also time. Gurtin (The Mechanics & Thermodynamics of Continua, pp. 107-08) uses this assumption to develop the motion equation given below.
I think (?!) that the resulting motion can include rigid body and uniform stretch but not sure. I'm also wondering if the exponential term, additive decomposition of L into spin & stretch rate, & multiplicative decomposition of F into F = RU = vR suggests a Lie group/algebra relationship for this motion.
Thanks,
John
EQ'N 14.5 with L constant & to = 0
(The Mechanics
& Thermodynamics of Continua, p. 108)
G(X, t) =
xo + e^(Lt) Fo(X-Xo) where
X =
position in reference configuration
x = position in current
configuration
to = 0
Fo=
F(to) = deformation gradient
Possible Answers
I have received two answers from chatting with colleagues:
1) Constant L assumes constant strain throughout the body. For example, a 3-noded triangle in finite element analysis (fea).
2) Constant L assumes constant acceleration which is a common fea assumption & reasonable with small time steps.
I'm comfortable with these explanations so wanted to share.
Thanks, John.