On energy-based coupled elastoplastic damage theories: Constitutive modeling and computational aspects

A new thermodynamically consistent continuum model for hardening plasticity coupled with damage

“…The form (17) of the generalised elastic relation recovers the one used by many authors (Simo and Ju, 1987;Ju, 1989). The coupled effect of damage on elastic stiffness and on the hardening moduli is apparent.…”

Computational mechanics of fatigue and life predictions for composite materials and structures

“…The solution to these equations gives the asymptotic expansion of the strain field in RVE as (10) where is the elastic strain in macro domain and is a damage-induced macroscopic strain; is the so-called elastic strain concentration function [25] given by (11) where is Kronecker delta; is termed as the local distribution function of damage-induced strain, which can be obtained by solving a linear boundary value problem in with Y-periodic boundary conditions, i.e. (12) where is a Y-periodic third rank tensor with symmetry , and .…”

“…The evolution of the nonlocal phase static damage at a given time can be expressed as (25) where ; the operator denotes the positive part, i.e. ; the phase deformation history parameter is determined from the evolution of the phase damage equivalent strain, denoted by (26) where represents the threshold value of damage equivalent strain prior to the initiation of phase damage; is defined as the square root of the phase damage energy release rate [39] (27) Since is assumed to be a positive definite fourth order tensor it follows that and consequently, must hold due to the energy dissipation inequality in (24).…”

“…Without loss of generality this can be accomplished by reformulating (25) and (26) as (28) (29) where is termed here as a nonlocal "pseudo-damage" parameter. By setting the so-called "gauge function" [29] as (30) the rate form of the static phase damage evolution law (29) turns into (31) where can be expressed as an arctangent form damage evolution proposed in [20]: (32) where are material constants; denotes the threshold of the strain history parameter beyond which the damage will develop very quickly.…”

Numerical Simulation of Brittle Damage in Concrete Specimens

“…-scalar value representing the local damage parameter (ranging from 0 for the virgin material to 1, which represents the failure or zero stress) p о -material specific mass £ j -strain tensor У о И -effective stress tensor У ф 0 -denotes the initial (undamaged) elastic stored energŷ°j kl -fourth order tensor of the elastic stiffness of the undamaged material ijkl -fourth order tensor of the elastic stiffness of the damaged material a -parameter taking into consideration the shear resistance f t -material yield stress in tension f c -material yield stress in compression [ 5,6 ] , a n d f ou rt h -ord er t en sors [7], h ave b een p rop osed . H ow ever, th ese th eori es sh ow ei t h er a d i scon ti n u ou s stress-strai n resp on se w h en t h e u n i l at eral con d i ti on t ak es p l ace o r a n u n accep t ab l e n on -sym m et ri c el asti c b eh avi or f or som e l oad i n g con d i ti on s [8] .…”