ADVANCED NUMERICAL MODELING OF CRACKING PROCESSES IN REINFORCED
CONCRETE
Rena C. Yu
Co-workers: G. Ruiz, J.R. Carmona and X.X. Zhang
Department of Applied Mechanics and Engineering Projects,UCLM, Spain
Outline
The Objective
Finite element methodology
Validation against experimental results on plain and reinforced concrete elements
Conclusions
The Objective
Feeding measured materialparameters to model explicitly
Distributed micro-cracking Multiple macro cracks Deterioration of the interface
Finite Element Methodology
Cohesive approach to fractureCohesive elements for the matrix material fractureInterface elements with frictional contact effects for the failure between concrete and steel rebar.
Modified explicit dynamic relaxation method as an alternative solver
The standard dynamic relaxation methodThe modified algorithm
Explicit cohesive approach to fracture
The crack topology
Cohesive surface
222nS +=
Effective opening displacement:
Decomposition of the opening displacement
Effective traction:
222nS ttt +=
A general cohesive law
Damage = /Gc
Cohesive Elements in 3D
3D cohesive element (Ortiz & Pandolfi, 1999)
How to handle the topology changes ?
(Pandolfi & Ortiz, 2002)
Fragmentation algorithm (1)
Fragmentation algorithm (2)
Dynamic fracture in plain and reinforced concrete elements
Complex fracture in plain concrete
Dynamic splitting test modeled by Ruiz, Ortiz & Pandolfi (2000)
Dynamic mixed-mode fracture in plain conc.
Ruiz, Ortiz & Pandolfi (2001)
Dynamic fracture in RC beams
Yu, Zhang & Ruiz (2007)
What happens in static regime?
Difficulties Encountered
Concrete as a quasi-brittle material is non-linear.
The modeling of static multi-cracking processes in concrete has been hindered due to its high non-linearity, traditional implicit solvers often fail to give convergent solutions in those circumstances.
We look for an alternative solver which is robust and compatible with the explicit framework we have.
Dynamic Relaxation Algorithm
Governing equation for a static problem for a certain load step n
Transformed dynamic system equation
extnn
inn FxF =)(
extnn
innnn FxFxCxM =++ )(
...
Dynamic Relaxation Algorithm
Obtain the solution by recourse to the Newmark scheme
The explicit time integration scheme is obtained by setting =0 and 0.5, the system is solved in two steps
])2/1[(1....
2.
1+
+ +++=k
n
k
n
k
nkn
kn xxtxtxx
])1[(1.....1. ++
++=k
n
k
n
k
n
k
n xxtxx
k
n
k
nkn
kn xtxtxx
..2
.1 )2/(++=+
k
n
k
n
k
npred xtxx...1.
)1( +=+
Dynamic Relaxation Algorithm
Update internal forces vector, obtain the accelerations and velocities (corrector step)
(=0.5 gives the well-known central difference scheme).Mass and damping matrices are chosen to be diagonal to
preserve the explicit form of the time-stepping integrator.
+=
++
+ 1.11
1..)()(
k
npredkn
inn
extn
k
n xCxFFtCMx
1...1. ++
+=k
n
k
npred
k
n xtxx
MC =
DR Parameters
M, and t are arbitrary and are selected to produce the fastest and most stable convergence to the steady-state, or the static solution of the physical problem.
DR parameters M, t
Courant-Friedricks-Lewy stability condition
One effective way of estimating the fictitious lumped mass matrix is to determine the mass of each element such that the time for the elastic wave to travel through each single element is the same,which is computed as
xmacrtt /2=
eeecr cht /=
DR parameters M, t
The highest frequency of vibration associated with an element
where
For a given time step, the fictitious mass density can be estimated as
t is serving as an iteration counter, set to 1,
eexema hc /2=
eec /)2(2 +=
2)/)(2( ee ht+
DR parameters C=M
The lowest participating mode of vibration is critically damped
Rayleigh quotient
where K is a diagonal estimate of the tangent stiffness matrix at the current iteration.
nmi 2=
MxxKxx TTnmi /2 =
Convergence criteria
Total kinetic energy ( steady state ?)
tol
k
keK
xM
