SYMPOSIUM: INSTABILITIES ACROSS THE SCALES Cairns, 14-17 September 2004 STRAIN LOCALIZATION & MATERIAL INSTABILITIES IN PLASTIC SOLIDS WITHIN A FULLY COUPLED THERMO-MECHANICAL CONTEXT Davide Bigoni Ahmed Benallal Università di Trento, Italy LMT – ENS de Cachan, France
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STRAIN LOCALIZATION & MATERIAL …bigoni/paper/queensland_bigoni...SYMPOSIUM: INSTABILITIES ACROSS THE SCALES Cairns, 14-17 September 2004 STRAIN LOCALIZATION & MATERIAL INSTABILITIES
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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
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 !