Page 250 - The Combined Finite-Discrete Element Method

STRAIN SOFTENING BASED SMEARED FRACTURE MODEL 233

c
e
v v x
L L
Figure 7.2 Wave propagation in a strain-softening bar in tension.

Actual implementation of strain softening material models into finite element codes
has been associated with great difficulties regarding both sensitivity to mesh size and
mesh orientation. The problems associated with dynamic analysis are best illustrated by
a 1D strain softening bar, shown in Figure 7.2. The bar of length 2L with a unit cross-
section and mass ρ per unit length has the ends moving simultaneously outward with
constant opposite velocities, v, while the initial conditions at time t = 0 are given by zero
displacements and a zero displacement rate:

u(x, t = 0) =˙u(x, t = 0) = 0

In the case where the strains never exceed ε t (strain corresponding to peak stress), the
bar remains elastic and the differential equation of motion is hyperbolic:

2
2
∂ ∂ u ∂σ ∂ε ∂ u
(σ(ε)) − ρ = 0; or − ρ = 0; (7.1)
∂x ∂t 2 ∂ε ∂x ∂t 2
∂σ ∂u
After substituting E T = and ε = , this yields
∂ε ∂x
2
2
∂ u ∂ u
E T 2 − ρ 2 = 0 (7.2)
∂x ∂t
The solutions of (7.2) are the two waves travelling in opposite directions (Figure 7.2) at
velocity c and meeting at the centre of the bar, at which stage the strain at the midpoint
doubles. If this strain exceeds the strain corresponding to the peak stress, the strain-
softening region appears at the midpoint of the bar, the tangent modulus of elasticity
E T becomes negative (modulus E in Figure 7.1), and the differential equation of motion

within the strain-softening segment becomes elliptic. This ellipticity means that interaction
spreads immediately over finite distances, actually over the entire strain-softening region.
The response is in fact the same as if the bar was cut in two at midpoint. Thus, the wave
reflects as if the midpoint was a free end. In other words, the strain-softening happens
within an infinitesimal segment, and is confined to a single cross-sectional plane of the bar.
A similar problem arises in static analysis. A type of instability can again be investigated
on a 1D bar of length 2L with prescribed displacements ∆ at the bar ends, (Figure 7.3).
The governing equation of the problem is given by

2
∂ ∂σ ∂ε ∂ u ∂σ
(σ(ε)) = 0; or = 0; i.e. E T = 0; E T = (7.3)
∂x ∂ε ∂x ∂x 2 ∂ε
where for the softening region, the tangent modulus of elasticity E T becomes neg-
ative. With the assumption that only the segment around the midpoint represents a