SYMPOSIUM: INSTABILITIES ACROSS THE SCALESCairns, 14-17 September 2004

STRAIN LOCALIZATION & MATERIAL INSTABILITIES IN PLASTIC SOLIDS WITHIN A FULLY

COUPLED THERMO-MECHANICAL CONTEXT

Davide Bigoni Ahmed BenallalUniversità di Trento, Italy LMT – ENS de Cachan, France

MOTIVATION

Sequence of thermal images at a crack tip in a C300 steel sample loaded at 3 m/s (from Ravichandran, Rosakis & co-workers 2000-2002)

Thermal image of a shear band:temperature distribution is highly nonuniform

TEMPERATURE AFFECTS SHEAR BAND NUCLEATION & GROWTH

UNDER ISOTHERMAL CONDITIONS …

emergence of a stationary discontinuity(Hill 1966)

vanishing speed of an acceleration wave(Mandel 1964)

unbounded growth of a small perturbation in the form of a planar wave (Rice 1976)

… ALL THESE CRITERIA ARE EQUIVALENT

WHAT HAPPENS IN A FULLY COUPLED THERMOMECHANICAL CONTEXT ?

ANSWER IS NOT TRIVIAL SINCE DISCONTINUITY CONDITIONS ON TEMPERATURE AND HEAT FLUX ARE INVOLVED

ANALOGY BETWEEN COUPLED THERMOPLASTICITY & FLUID-SATURATED POROPLASTICITY

Strain localization for fluid saturated poroplastic solids has been addressed by

Rice (1975)Vardoulakis (1985)Loret & Harireche (1992)Benallal & Comi (2003)

… BUT THE PROBLEM IS STILL OPEN …

THE CONSTITUTIVE FRAMEWORK

STRESS, TEMPERATURE, THERMODYNAMICAL FORCES …

RATE CONSTITUTIVE EQUATIONS WITH ISOTHERMAL QUANTITIES

TEMPERATURE, HEAT FLUX & DISSIPATION

RATE CONSTITUTIVE EQUATIONS WITH ADIABATIC QUANTITIES

PROPAGATING DISCONTINUITY JUMPS

STATIONARY DISCONTINUITY JUMPS

TWO ACOUSTIC TENSORS

TWO-STRAIN LOCALIZATION CONDITIONS “IN THE RICE SENSE”

SINGULARITY OF ACOUSTIC TENSORS

ACCELERATION WAVES

PERTURBATIONS IN THERMOMECHANICAL PROBLEMS

… AND THE ADIABATIC ACOUSTIC TENSOR ?

AN EXAMPLE: THERMOELASTOPLASTICITY OF INCONEL 718

STRAIN LOCALIZATION

SOME CONCLUSIONS

for isothermal elastoplasticity:strain localization;stationarity of acceleration waves;unbounded growth of perturbations;coincide with the singularity of the acoustic tensor.

This is NOT true when thermal-effects and thermomechanical couplings are considered.

Propagation (& stationarity!) of acceleration waves is governed by the isothermal acoustic tensor, while strain localization & unbounded growth are governed by either isothermal or adiabatic acoustic tensor

CAN WE SAY MORE? WHAT IS THE FORM IN WHICH THE INSTABILITY MANIFESTS ITSELF ?

… A POSSIBILITY WOULD BE TO USE …

THE IDEA (Bigoni & Capuani, JMPS 2002):• dipole (self-equilibrated load) perturbing a pre-stressed infinite medium.

σ1

F

Fσ1

Material instabilities become evident!

THE DIFFICULTY:find the Green’s function for a pre-stressed infinite medium. → always possible following JR Willis (1992)

A BENEFIT:a boundary element technique can be set up (but integral equations are also needed and may be not easy to find…)

PERTURBATION & SHEAR BANDS IN STATICS

PERTURBATIONS & SHEAR BANDS IN DYNAMICS

RESULTS FOR MOONEY-RIVLIN

STRAIN LOCALIZATION IS USUALLY CONSIDERED NOT POSSIBLE FOR MOONEY-RIVLIN MATERIALtherefore:OUR APPROACH CAN REVEAL FEATURES WHICH MAY REMAIN UNDETECTED USING STANDARD METHODS

WAVE PATTERNS TEND TO DEGENERATE TO FAMILIES OF PLANE WAVES WHEN THE ELLIPTIC BOUNDARY IS APPROACHED

“SHADOWING EFFECT”AT INCREASING FREQUENCY

RESULTS FOR DEFORMATION THEORY OF PLASTICITY

QUASI-STATIC APPROXIMATION

STRONG DEPENDENCE ON THE SCALE PARAMETER RELATED TO THE CIRCULAR FREQUENCY Ω:

aΩ/c

CONCLUSIONS

FIND A GOOD STUDENT WHO WORKS OUT THE GREEN’S FUNCTION FOR INCREMENTAL THERMOELASTIC-PLASTIC DEFORMATIONS !

(GOOD LUCK!)