Viscoplasticity
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Viscoplasticity is a model for rate-dependent plasticity. Rate dependent plasticity is important for (high-speed) transient plasticity calculations. It should be used in combination with a plasticity law. Viscoplasticity influences the stresses via the plastic strains.
Viscoplastic solids are those which exhibit permanent deformations after the application of loads (like plastic solids) but which continue to undergo a creep flow as a function of time under the influence of the applied load (equilibrium is impossible).
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[edit] Background
Development of mathematical modeling of viscoplasticity started in 1910 with the representation of primary creep by Andrew’s law. Followed by Norton’s law in 1929 which links the rate of secondary creep to the stress. And in 1934, it followed by Odqvist’s generalization of Norton’s law to the multiaxial case.
The first IUTAM Symposium “Creep in Structures” organized by Hoff took place in 1960. It became the starting point of a great development in viscoplasticity with the works of Hoff, Rabotnov, Perzyna, Hult, and Lemaitre for the isotropic hardening laws, and those of Kratochvil, Malinini and Khadjinsky, Ponter and Leckie, and Chaboche for the kinematic hardening laws.
[edit] Domain of validity and use
The theory of viscoplasticity describes the flow of matter by creep, which in contrast to plasticity, depends on time. For metals and alloys, it corresponds to mechanism linked to the movement of dislocations in grains climb, deviation, polygonization-with superposed effects of inter-crystalline gliding. The mechanism begins to arise as soon as the temperature is greater than approximately one third of the absolute melting temperature. In fact, certain alloys exhibit viscoplasticity at room temperature (300K). Time effect must be taken into consideration as well. For polymers,wood, bitumen,etc. the theory of viscoplasticity must be used as soon as the load has passed the limit of elasticity or viscoelasticity.
[edit] Phenomenological aspects
[edit] Hardening test
The hardening curves of a viscoplastic material are not significantly different from those of plastic material. Nevertheless, three essential differences are apparent: at the same strain, the higher the rate of strain (or stress) is, the higher will be the stress. A change in the rate of strain during the test results in an immediate change in the stress – strain curve. Also, the concept of a plastic yield limit is no longer strictly applicable.
The hypothesis of portioning the partitioning the strains by decoupling is still applicable in most cases (where the strains are small):
- ε = εe + εp
where:
- εe is the linear elastic strain and
- εp is the viscoplastic strain.
[edit] Creep test
The classical creep curve represents the evolution of strain as a function of time in a material subjected to uniaxial stress at a constant temperature . This curve shows three phases or periods of behavior: a ‘primary’ creep phase 0 ≤ ε ≤ ε1 during which hardening of the material leads to a decrease in the rate of flow which is initially very high. A ‘secondary’ creep phase ε1 ≤ ε ≤ ε2during which the rate of flow is almost constant. Finally a ‘tertiary’ creep phase ε2 ≤ ε ≤ εR in which the usual increase in the strain rate up to the fracture strain.
[edit] Relaxation test
Relaxation tests demonstrate the decrease in the stress which results from maintaining a volume element in uniaxial loading at constant strain. In fact these tests characterize the viscosity and can be used to determine the relation which exists between the stress and the rate of viscoplasticity strain. In terms of rates:
ε˙= εe˙ + εp˙ where, with the linear elasticity εe˙ = -σ˙/E. Thus, each point on the relaxation curve σ(t) gives the stress and rate of viscoplastic strain εp˙ = -σ˙/E
[edit] Influence of temperature
The viscosity phenomena, in metals or polymers, are highly influenced by temperature. The stress which generate a given viscoplastic strain in a given time diminishes as a function of temperature.
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The most commonly used visco plastic solids are, Perfectly viscoplastic solid,Elastic perfectly viscoplastic solid,Elastoviscoplastic hardening solid.
[edit] Perfectly viscoplastic solid
The rate of permanent strain (as for viscous fluids) is a function of the stress:σ(ε ̇) Norton model σ=λε ̇^(1/N)
Applications: rough schematic representation of metals and alloys at temperatures higher than one third of their absolute melting point (in K).
[edit] Elastic perfectly viscoplastic solid
The elasticity is no longer considered negligible but the rate of plastic strain is only a function of the stress . There is no influence of hardening. |σ|<σ_s→ε=ε_e=σ⁄E |σ|≥σ_s→ε=ε_e+ε_p ε ̇=σ ̇⁄E+f(σ)
[edit] Elastoviscoplastic hardening solid
This is the most complex schematic representation because the stress depends on the plastic strain rate and on the plastic strain itself or on some other hardening variable. |σ|<σ_s→ε=ε_e=σ⁄E
|σ|≥σ_s→ε=ε_e+ε_p
σ=Eε_e=f(ε_p,ε ̇_p)
Applications : metals and alloys at medium to high temperatures, wood under high loads.
[edit] See also
[edit] References
- J.Lemaitre and J.L.Chaboche "Mechanics of solid materials"
