Chapter 3

 

Constitutive Models MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzaiaeaa aaaaaaa8qacaWFtacaaa@3788@  Relations between Stress and Strain

 

 

 

3.8 Small strain viscoplasticity: creep and high strain rate deformation of crystalline solids

 

Viscoplastic constitutive equations are used to model the behavior of polycrystalline materials (metals and ceramics) that are subjected to stress at high temperatures (greater than half the melting point of the solid), and also to model the behavior of metals that are deformed at high rates of strain (greater than 100 per second). 

 

Viscoplasticity theory is a relatively simple extension of the rate independent plasticity model discussed in Section 3.6.  You may find it helpful to review this material before attempting to read this section.

 

 

3.8.1 Features of creep behavior

 

 Creep under uniaxial loading

 

1.      If a tensile specimen of a crystalline solid is subjected to a time independent stress, it will progressively increase in length.  A typical series of length-v-time curves is illustrated in the figure.

 

2.      The length-v-time plot has three stages: a transient period of primary creep, where the creep rate is high; a longer period of secondary creep, where the extension rate is constant; and finally a period of tertiary creep, where the creep rate again increases.   Most creep laws focus on modeling primary and secondary creep.  In fact, it is often sufficient to model only secondary creep.

 

3.      The rate of extension increases with stress.  A typical plot of secondary creep rate as a function of stress is shown in the figure.   There are usually three regimes of behavior: each regime can be fit (over a range of stress) by a power-law with the form ε ˙ A σ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaGaeyisISRaamyqaiabeo 8aZnaaCaaaleqabaGaamyBaaaaaaa@3771@ .  At low stresses, m12.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gacqGHijYUcaaIXaGaeyOeI0IaaG Omaiaac6cacaaI1aaaaa@36E0@ ; at intermediate stresses m2.57 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gacqGHijYUcaaIYaGaaiOlaiaaiw dacqGHsislcaaI3aaaaa@36E6@ , and at high stress m increases rapidly and can exceed 10-20.

 

4.      The rate of extension increases with temperature.   At a fixed stress, the temperature dependence of strain rate can be fit by an equation of the form ε ˙ = ε ˙ 0 exp(Q/kT) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaGaeyypa0JafqyTduMbai aadaWgaaWcbaGaaGimaaqabaGcciGGLbGaaiiEaiaacchacaGGOaGa eyOeI0Iaamyuaiaac+cacaWGRbGaamivaiaacMcaaaa@3E31@ , where Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgfaaaa@313E@  is an activation energy; k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgaaaa@3158@  is the Boltzmann constant, and T is temperature.  Like the stress exponent m,   the activation energy Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgfaaaa@313E@   can transition from one value to another as the temperature and stress level is varied.

 

5.      The various regimes for m and Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgfaaaa@313E@  are associated with different mechanisms of creep.  At low stress, creep occurs mostly by grain boundary sliding and diffusion.  At higher stresses it occurs as a result of thermally activated dislocation motion.   H.J. Frost and M.F. Ashby. “Deformation Mechanism Maps,” Pergamon Press, Elmsford, NY (1982) plot charts that are helpful to get a rough idea of which mechanism is likely to be active in a particular application.  A schematic of a deformation mechanism map is illustrated in the picture: the figure shows various regimes of plastic flow as a function of temperature (normalized by melting temperature), and stress (normalized by shear modulus)

 

6.      The creep behavior of a material is strongly sensitive to its microstructure (especially grain size and the size and distribution of precipitates) and composition.

 

 

  Creep under multi-axial loading

 

Under proportional multi-axial loading, creep shows all the same characteristics as rate independent plasticity: (i) plastic strains are volume preserving; (ii) creep rates are insensitive to hydrostatic pressure; (iii) the principal strain rates are parallel to the principal stresses; (iv) plastic flow obeys the Levy-Mises flow rule.   These features of behavior are discussed in more detail in Sect 3.6.1.  

 

 

 

3.8.2 Features of high-strain rate behavior

 

Stress-strain curves for metals have been measured for strain rates as high as 10 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIXaGaaGimamaaCaaaleqabaGaaG 4naaaaaaa@3529@  /sec.  The general form of the stress-strain curve is essentially identical to that measured at quasi-static strain rates (see Sect 3.6.1 for an example), but the flow stress increases with strain rate.  As an example, the experimental data of Klopp, Clifton and Shawki, Mechanics of Materials, 4, p. 375 (1985) for the high strain rate behavior of iron is reproduced in the figure.  The flow stress rises slowly with strain rate up to a strain rate of about 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIXaGaaGimamaaCaaaleqabaGaaG Onaaaaaaa@3528@ , and then begins to rise rapidly.

 

 

 

3.8.3 Small-strain, viscoplastic constitutive equations

 

Viscoplastic constitutive equations are almost identical to the rate independent plastic equations in Section 3.6.  The main concepts in viscoplasticity are,

1.      Strain rate decomposition into elastic and plastic components;

2.      Elastic stress-strain law, which specifies the  elastic part of the strain rate in terms of stress rate;

3.      The plastic flow potential, which determines the magnitude of the plastic strain rate, given the stresses and the resistance of the material to flow.

4.       State Variables which characterize the resistance of the material to flow (analogous to yield stress)

5.      The plastic flow rule, which specifies the components of plastic strain rate under multiaxial loading. Recall that in rate independent plasticity the flow rule was expressed as the derivative of the yield surface with respect to stress.  In viscoplasticity, the flow rule involves the derivative of the plastic flow potential

6.      Hardening laws which specify the evolution of the state variables with plastic strain.

 

These are discussed in more detail below.

 

 Strain rate decomposition

 

We assume infinitesimal deformation, so shape changes are characterized by ε ij =( u i / x j + u j / x i )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaqadaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyA aaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaki abgUcaRiabgkGi2kaadwhadaWgaaWcbaGaamOAaaqabaGccaGGVaGa eyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaai aac+cacaaIYaaaaa@4880@ .

The strain rate is decomposed into elastic and plastic parts as

d ε ij dt = d ε ij e dt + d ε ij p dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacaWGKbGaeqyTdu2aa0baaS qaaiaadMgacaWGQbaabaaaaaGcbaGaamizaiaadshaaaGaeyypa0Za aSaaaeaacaWGKbGaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaam yzaaaaaOqaaiaadsgacaWG0baaaiabgUcaRmaalaaabaGaamizaiab ew7aLnaaDaaaleaacaWGPbGaamOAaaqaaiaadchaaaaakeaacaWGKb GaamiDaaaaaaa@4858@

 Elastic constitutive equations

 

The elastic strains are related to the stresses using the standard linear elastic stress-strain law.   The elastic strain rate follows as

d ε ˙ ij e dt = S ijkl d σ kl dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiqbew7aLzaacaWaa0 baaSqaaiaadMgacaWGQbaabaGaamyzaaaaaOqaaiaadsgacaWG0baa aiabg2da9iaadofadaWgaaWcbaGaamyAaiaadQgacaWGRbGaamiBaa qabaGcdaWcaaqaaiaadsgacqaHdpWCdaWgaaWcbaGaam4AaiaadYga aeqaaaGcbaGaamizaiaadshaaaaaaa@4478@

where S ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaaaaa@352A@  are the components of the elastic compliance tensor.  For the special case of an isotropic material with Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3132@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@

d ε ij e dt = 1+ν E d σ ij dt ν E d σ kk dt δ ij +α dΔT dt δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiabew7aLnaaDaaale aacaWGPbGaamOAaaqaaiaadwgaaaaakeaacaWGKbGaamiDaaaacqGH 9aqpdaWcaaqaaiaaigdacqGHRaWkcqaH9oGBaeaacaWGfbaaamaala aabaGaamizaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaaakeaa caWGKbGaamiDaaaacqGHsisldaWcaaqaaiabe27aUbqaaiaadweaaa WaaSaaaeaacaWGKbGaeq4Wdm3aaSbaaSqaaiaadUgacaWGRbaabeaa aOqaaiaadsgacaWG0baaaiabes7aKnaaBaaaleaacaWGPbGaamOAaa qabaGccqGHRaWkcqaHXoqydaWcaaqaaiaadsgacqqHuoarcaWGubaa baGaamizaiaadshaaaGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabe aaaaa@5D01@

 

 Plastic flow potential

 

The plastic flow potential specifies the magnitude of the plastic strain rate, as a function of stress and the resistance of the material to plastic flow.   It is very similar to the yield surface for a rate independent material.   The plastic flow potential is constructed as follows.

1.      Define the plastic strain rate magnitude as ε ˙ e = 2 ε ˙ ij p ε ˙ ij p /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaaSbaaSqaaiaadwgaae qaaOGaeyypa0ZaaOaaaeaacaaIYaGafqyTduMbaiaadaqhaaWcbaGa amyAaiaadQgaaeaacaWGWbaaaOGafqyTduMbaiaadaqhaaWcbaGaam yAaiaadQgaaeaacaWGWbaaaOGaai4laiaaiodaaSqabaaaaa@3FF7@

2.      Let σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3434@  denote the stresses acting on the material, and let σ 1 , σ 2 , σ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaaabeaaki aacYcacqaHdpWCdaWgaaWcbaGaaGOmaaqabaGccaGGSaGaeq4Wdm3a aSbaaSqaaiaaiodaaeqaaaaa@39DD@  denote the principal stresses;

3.      Experiments show that the plastic strain rate is independent of hydrostatic pressure.  The strain rate must be a function of only the deviatoric stress components, defined as

S ij = σ ij σ kk δ ij /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiab gkHiTiabeo8aZnaaBaaaleaacaWGRbGaam4AaaqabaGccqaH0oazda WgaaWcbaGaamyAaiaadQgaaeqaaOGaai4laiaaiodaaaa@421D@

4.      Assume that the material is isotropic.  The strain rate can therefore only depend on the invariants of the deviatoric stress tensor.   The deviatoric stress has only two nonzero invariants. It is convenient to choose

σ e = 3 2 S ij S ij σ III =det(S) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGLbaabeaaki abg2da9maakaaabaWaaSaaaeaacaaIZaaabaGaaGOmaaaacaWGtbWa aSbaaSqaaiaadMgacaWGQbaabeaakiaadofadaWgaaWcbaGaamyAai aadQgaaeqaaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH dpWCdaWgaaWcbaGaamysaiaadMeacaWGjbaabeaakiabg2da9iGacs gacaGGLbGaaiiDaiaacIcacaWHtbGaaiykaaaa@6690@

In practice, only the first of these (the von Mises effective stress) is used in most flow potentials.

5.      The plastic flow potential can be represented graphically by plotting it as a function of the three principal stresses, exactly as the yield surface is shown graphically for a rate independent material.   An example is shown in the picture.  The lines show contours of constant plastic strain rate.

6.      For Drucker stability, the contours of constant strain rate must be convex, and the plastic strain rate must increase with strain rate in the direction shown in the figure.

 

7.      Just as the yield stress of a rate independent material can increase with plastic strain, the resistance of a viscoplastic material to plastic straining can also increase with strain.   The resistance to flow is characterized by one or more material state variables, which may evolve with plastic straining. 

 

The most general form for the flow potential of an isotropic material is thus

ε ˙ e =g( σ e , σ III ,State variables) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaaSbaaSqaaiaadwgaae qaaOGaeyypa0Jaam4zaiaacIcacqaHdpWCdaWgaaWcbaGaamyzaaqa baGccaGGSaGaeq4Wdm3aaSbaaSqaaiaadMeacaWGjbGaamysaaqaba GccaGGSaGaae4uaiaabshacaqGHbGaaeiDaiaabwgacaqGGaGaaeOD aiaabggacaqGYbGaaeyAaiaabggacaqGIbGaaeiBaiaabwgacaqGZb Gaaeykaaaa@4CAF@

where g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEgaaaa@3154@  must satisfy g(α σ e ,α σ III ,State vars)g( σ e , σ III ,State vars) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEgacaGGOaGaeqySdeMaeq4Wdm3aaS baaSqaaiaadwgaaeqaaOGaaiilaiabeg7aHjabeo8aZnaaBaaaleaa caWGjbGaamysaiaadMeaaeqaaOGaaiilaiaabofacaqG0bGaaeyyai aabshacaqGLbGaaeiiaiaabAhacaqGHbGaaeOCaiaabohacaqGPaGa eyyzImRaam4zaiaacIcacqaHdpWCdaWgaaWcbaGaamyzaaqabaGcca GGSaGaeq4Wdm3aaSbaaSqaaiaadMeacaWGjbGaamysaaqabaGccaGG SaGaae4uaiaabshacaqGHbGaaeiDaiaabwgacaqGGaGaaeODaiaabg gacaqGYbGaae4CaiaabMcaaaa@5D36@  for all α1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjabgwMiZkaaigdaaaa@3488@  (with state variables held fixed), and must also be a convex function of σ e , σ III MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGLbaabeaaki aacYcacqaHdpWCdaWgaaWcbaGaamysaiaadMeacaWGjbaabeaaaaa@3854@

 

Examples of flow potentials: Von-Mises flow potential with power-law rate sensitivity

 

Creep is often modeled using the a flow potential of the form

g( σ e ,{ σ 0 (i) })= i=1 N ε ˙ 0 (i) exp( Q i /kT) ( σ e σ 0 (i) ) m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4zaiaacIcacqaHdpWCdaWgaaWcba GaamyzaaqabaGccaGGSaWaaiWaaeaacqaHdpWCdaqhaaWcbaGaaGim aaqaaiaacIcacaWGPbGaaiykaaaaaOGaay5Eaiaaw2haaiaacMcacq GH9aqpdaaeWbqaaiqbew7aLzaacaWaa0baaSqaaiaaicdaaeaacaGG OaGaamyAaiaacMcaaaGcciGGLbGaaiiEaiaacchacaGGOaGaeyOeI0 IaamyuamaaBaaaleaacaWGPbaabeaakiaac+cacaWGRbGaamivaiaa cMcadaqadaqaamaalaaabaGaeq4Wdm3aaSbaaSqaaiaadwgaaeqaaa GcbaGaeq4Wdm3aa0baaSqaaiaaicdaaeaacaGGOaGaamyAaiaacMca aaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaWGTbWaaSbaaWqaai aadMgaaeqaaaaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOtaaqd cqGHris5aaaa@5EED@

where ε ˙ 0 (i) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaa0baaSqaaiaaicdaae aacaGGOaGaamyAaiaacMcaaaaaaa@3546@ , Q i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgfadaWgaaWcbaGaamyAaaqabaaaaa@3258@  and m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gadaWgaaWcbaGaamyAaaqabaaaaa@3294@ , i=1..N are material properties ( Q i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgfadaWgaaWcbaGaamyAaaqabaaaaa@3278@  are the activation energies for the various mechanisms that contribute to creep); k is the Boltzmann constant and T is temperature.  The model is most often used with N=1, but more terms are required to fit material behavior over a wide range of temperatures and strain rates. The potential has several state variables, σ 0 (i) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaDaaaleaacaaIWaaabaGaai ikaiaadMgacaGGPaaaaaaa@3559@ .   To model steady state creep, you can take σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaaa a@3311@  to be constant; to model transient creep, σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaaa a@3311@  must increase with strain.  An example of an evolution law for σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaaa a@3311@  is given below. 

 

High strain rate deformation is also modeled using a power law Mises flow potential: the following form is sometimes used:

g( σ e , σ 0 )={ 00< σ e / σ 0 <1 ε ˙ 0 (1) [ ( σ e / σ 0 ) m 1 1 ]1< σ e / σ 0 <α ε ˙ 0 (2) [ ( σ e / σ 0 ) m 2 1 ]α< σ e / σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4zaiaacIcacqaHdpWCdaWgaaWcba GaamyzaaqabaGccaGGSaGaeq4Wdm3aa0baaSqaaiaaicdaaeaaaaGc caGGPaGaeyypa0ZaaiqaaeaafaqabeWabaaabaGaaGimaiaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaicdacqGH8aapcqaHdpWCdaWgaaWcbaGaamyzaa qabaGccaGGVaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaOGaeyipaWJa aGymaaqaaiqbew7aLzaacaWaa0baaSqaaiaaicdaaeaacaGGOaGaaG ymaiaacMcaaaGcdaWadaqaamaabmaabaGaeq4Wdm3aaSbaaSqaaiaa dwgaaeqaaOGaai4laiabeo8aZnaaBaaaleaacaaIWaaabeaaaOGaay jkaiaawMcaamaaCaaaleqabaGaamyBamaaBaaameaacaaIXaaabeaa aaGccqGHsislcaaIXaaacaGLBbGaayzxaaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaigdacqGH8aapcqaHdpWCdaWgaaWcbaGaamyzaa qabaGccaGGVaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaOGaeyipaWJa eqySdegabaGafqyTduMbaiaadaqhaaWcbaGaaGimaaqaaiaacIcaca aIYaGaaiykaaaakmaadmaabaWaaeWaaeaacqaHdpWCdaWgaaWcbaGa amyzaaqabaGccaGGVaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaaGcca GLOaGaayzkaaWaaWbaaSqabeaacaWGTbWaaSbaaWqaaiaaikdaaeqa aaaakiabgkHiTiaaigdaaiaawUfacaGLDbaacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlabeg7aHjabgYda8iabeo8aZnaaBaaaleaacaWGLbaabeaa kiaac+cacqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaaGccaGL7baaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVdaa@233F@

where ε ˙ 0 (2) = ε ˙ 0 (1) ( α m 1 1)/( α m 2 1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaa0baaSqaaiaaicdaae aacaGGOaGaaGOmaiaacMcaaaGccqGH9aqpcuaH1oqzgaGaamaaDaaa leaacaaIWaaabaGaaiikaiaaigdacaGGPaaaaOGaaiikaiabeg7aHn aaCaaaleqabaGaamyBamaaBaaameaacaaIXaaabeaaaaGccqGHsisl caaIXaGaaiykaiaac+cacaGGOaGaeqySde2aaWbaaSqabeaacaWGTb WaaSbaaWqaaiaaikdaaeqaaaaakiabgkHiTiaaigdacaGGPaaaaa@490F@  and ε ˙ 0 (1) ,α, m 1 , m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaa0baaSqaaiaaicdaae aacaGGOaGaaGymaiaacMcaaaGccaGGSaGaeqySdeMaaiilaiaad2ga daWgaaWcbaGaaGymaaqabaGccaGGSaGaamyBamaaBaaaleaacaaIYa aabeaaaaa@3CA9@  are material properties, while σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaaa a@3311@  is a strain, strain rate and temperature dependent state variable, which represents the quasi-static yield stress of the material and evolves with deformation as described below.  In this equation, m 2 < m 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gadaWgaaWcbaGaaGOmaaqabaGccq GH8aapcaWGTbWaaSbaaSqaaiaaigdaaeqaaaaa@3549@  so as to model the transition in strain rate sensitivity at high strain rates; while α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3227@  controls the point at which the transition occurs.

 

 Plastic flow rule

 

The plastic flow rule specifies the components of plastic strain rate resulting from a multiaxial state of stress.   It is constructed so that:

1.      The plastic strain rate satisfies the Levy-Mises plastic flow rule

2.      The viscoplastic stress-strain law satisfies the Drucker stability criterion

3.      The flow rule predicts a plastic strain magnitude consistent with the flow potential.

 

Both (1) and (2) are satisfied by

ε ˙ ij p = 3 2 g( σ e , σ III ,State vars) [ ( g/ σ kl )( g/ σ kl ) ] g σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaa0baaSqaaiaadMgaca WGQbaabaGaamiCaaaakiabg2da9maakaaabaWaaSaaaeaacaaIZaaa baGaaGOmaaaaaSqabaGcdaWcaaqaaiaadEgacaGGOaGaeq4Wdm3aaS baaSqaaiaadwgaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaWGjbGa amysaiaadMeaaeqaaOGaaiilaiaabofacaqG0bGaaeyyaiaabshaca qGLbGaaeiiaiaabAhacaqGHbGaaeOCaiaabohacaqGPaaabaWaamWa aeaadaqadaqaaiabgkGi2kaadEgacaGGVaGaeyOaIyRaeq4Wdm3aaS baaSqaaiaadUgacaWGSbaabeaaaOGaayjkaiaawMcaamaabmaabaGa eyOaIyRaam4zaiaac+cacqGHciITcqaHdpWCdaWgaaWcbaGaam4Aai aadYgaaeqaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaaaamaalaaa baGaeyOaIyRaam4zaaqaaiabgkGi2kabeo8aZnaaBaaaleaacaWGPb GaamOAaaqabaaaaaaa@68EC@

If g depends on stress only through the Von-Mises effective stress, this expression can be simplified to

ε ˙ ij p =g( σ e ,State vars) 3 S ij 2 σ e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaa0baaSqaaiaadMgaca WGQbaabaGaamiCaaaakiabg2da9iaadEgacaGGOaGaeq4Wdm3aaSba aSqaaiaadwgaaeqaaOGaaiilaiaabofacaqG0bGaaeyyaiaabshaca qGLbGaaeiiaiaabAhacaqGHbGaaeOCaiaabohacaqGPaWaaSaaaeaa caaIZaGaam4uamaaBaaaleaacaWGPbGaamOAaaqabaaakeaacaaIYa Gaeq4Wdm3aaSbaaSqaaiaadwgaaeqaaaaaaaa@4C46@

For the particular case of the Power-law von Mises flow potential this gives

ε ˙ ij p = n=1 N ε ˙ 0 (n) exp( Q n /kT) ( σ e σ 0 (n) ) m n 3 S ij 2 σ e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGafqyTduMbaiaadaqhaaWcbaGaamyAai aadQgaaeaacaWGWbaaaOGaeyypa0ZaaabCaeaacuaH1oqzgaGaamaa DaaaleaacaaIWaaabaGaaiikaiaad6gacaGGPaaaaOGaciyzaiaacI hacaGGWbGaaiikaiabgkHiTiaadgfadaWgaaWcbaGaamOBaaqabaGc caGGVaGaam4AaiaadsfacaGGPaWaaeWaaeaadaWcaaqaaiabeo8aZn aaBaaaleaacaWGLbaabeaaaOqaaiabeo8aZnaaDaaaleaacaaIWaaa baGaaiikaiaad6gacaGGPaaaaaaaaOGaayjkaiaawMcaamaaCaaale qabaGaamyBamaaBaaameaacaWGUbaabeaaaaaaleaacaWGUbGaeyyp a0JaaGymaaqaaiaad6eaa0GaeyyeIuoakmaalaaabaGaaG4maiaado fadaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaaGOmaiabeo8aZnaa BaaaleaacaWGLbaabeaaaaaaaa@5E12@

 

 Hardening rule

 

The hardening rule specifies the evolution of state variables with plastic straining.   Many different forms of hardening rule are used (including kinematic hardening laws such as those discussed in 3.6.5).   A simple example of an isotropic hardening law which is often used to model transient creep is

σ 0 (i) = Y i ( 1+ ε e (i) ε 0 (i) ) 1/ n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aa0baaSqaaiaaicdaaeaaca GGOaGaamyAaiaacMcaaaGccqGH9aqpcaWGzbWaaSbaaSqaaiaadMga aeqaaOWaaeWaaeaacaaIXaGaey4kaSYaaSaaaeaacqaH1oqzdaqhaa WcbaGaamyzaaqaaiaacIcacaWGPbGaaiykaaaaaOqaaiabew7aLnaa DaaaleaacaaIWaaabaGaaiikaiaadMgacaGGPaaaaaaaaOGaayjkai aawMcaamaaCaaaleqabaGaaGymaiaac+cacaWGUbWaaSbaaWqaaiaa dMgaaeqaaaaaaaa@49BF@

where ε e (i) = ε ˙ 0 (i) exp( Q i /kT) ( σ e / σ 0 (i) ) m i dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTdu2aa0baaSqaaiaadwgaaeaaca GGOaGaamyAaiaacMcaaaGccqGH9aqpdaWdbaqaaiqbew7aLzaacaWa a0baaSqaaiaaicdaaeaacaGGOaGaamyAaiaacMcaaaGcciGGLbGaai iEaiaacchacaGGOaGaeyOeI0IaamyuamaaBaaaleaacaWGPbaabeaa kiaac+cacaWGRbGaamivaiaacMcacaGGOaGaeq4Wdm3aaSbaaSqaai aadwgaaeqaaOGaai4laiabeo8aZnaaDaaaleaacaaIWaaabaGaaiik aiaadMgacaGGPaaaaOGaaiykamaaCaaaleqabaGaamyBamaaBaaame aacaWGPbaabeaaaaGccaWGKbGaamiDaaWcbeqab0Gaey4kIipaaaa@558E@  is the accumulated strain associated with each mechanism of creep, and Y i , ε 0 (i) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamywamaaBaaaleaacaWGPbaabeaaki aacYcacaaMc8UaaGPaVlaaykW7cqaH1oqzdaqhaaWcbaGaaGimaaqa aiaacIcacaWGPbGaaiykaaaaaaa@3CF7@  and n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gadaWgaaWcbaGaamyAaaqabaaaaa@3275@  are material constants.  The law is usually used only with N=1.

 

Similar hardening laws are used in constitutive equations for high strain rate deformation; but in this case the flow strength is made temperature dependent.  The following formula is sometimes used

σ 0 =Y[1β(T T 0 )] ( 1+ ε e ε 0 ) 1/n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaO Gaeyypa0JaamywaiaacUfacaaIXaGaeyOeI0IaeqOSdiMaaiikaiaa dsfacqGHsislcaWGubWaaSbaaSqaaiaaicdaaeqaaOGaaiykaiaac2 fadaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiabew7aLnaaDaaaleaa caWGLbaabaaaaaGcbaGaeqyTdu2aa0baaSqaaiaaicdaaeaaaaaaaa GccaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaai4laiaad6gaaaaa aa@4A9B@

where ε e = 3 ε ˙ ij p ε ˙ ij p /2 dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyTdu2aaSbaaSqaaiaadwgaaeqaaO Gaeyypa0Zaa8qaaeaadaGcaaqaaiaaiodacuaH1oqzgaGaamaaDaaa leaacaWGPbGaamOAaaqaaiaadchaaaGccuaH1oqzgaGaamaaDaaale aacaWGPbGaamOAaaqaaiaadchaaaGccaGGVaGaaGOmaaWcbeaakiaa dsgacaWG0baaleqabeqdcqGHRiI8aaaa@445C@  is the total accumulated strain, T is temperature, and Y,n, T 0 ,β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMfacaGGSaGaamOBaiaacYcacaWGub WaaSbaaSqaaiaaicdaaeqaaOGaaiilaiabek7aIbaa@37D3@  are material properties.  More sophisticated hardening laws make the flow stress a function of strain rate MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  see Clifton “High strain rate behavior of metals.” Applied Mechanics Reviews 43 (5) (1990) S9-S22 for more details.

 

 

 

3.8.4 Representative values of parameters for viscoplastic models of creeping solids

 

Fitting material parameters to test data is conceptually straightforward: the flow potential has been constructed so that for a uniaxial tension test with σ 11 =σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCaaa@36C0@ , all other stress components zero, the uniaxial plastic strain rate is

ε ˙ 11 p = n=1 N ε ˙ 0 (n) exp( Q n /kT) ( σ σ 0 (n) ) m n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGafqyTduMbaiaadaqhaaWcbaGaaGymai aaigdaaeaacaWGWbaaaOGaeyypa0ZaaabCaeaacuaH1oqzgaGaamaa DaaaleaacaaIWaaabaGaaiikaiaad6gacaGGPaaaaOGaciyzaiaacI hacaGGWbGaaiikaiabgkHiTiaadgfadaWgaaWcbaGaamOBaaqabaGc caGGVaGaam4AaiaadsfacaGGPaWaaeWaaeaadaWcaaqaaiabeo8aZb qaaiabeo8aZnaaDaaaleaacaaIWaaabaGaaiikaiaad6gacaGGPaaa aaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaamyBamaaBaaameaaca WGUbaabeaaaaaaleaacaWGUbGaeyypa0JaaGymaaqaaiaad6eaa0Ga eyyeIuoaaaa@5534@

so the properties can be fit directly to the results of a series of uniaxial tensile tests conducted at different temperatures and applied stresses. To model steady-state creep  σ 0 (n) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaDaaaleaacaaIWaaabaGaai ikaiaad6gacaGGPaaaaaaa@355E@  can be taken to be constant.  One, or two terms in the series is usually sufficient to fit material behavior over a reasonable range of temperature and stress. 

 

Creep rates are very sensitive to the microstructure and composition of a material, so for accurate predictions you will need to find data for the particular material you plan to use.  Frost and Ashby “Deformation Mechanism Maps,” Pergamon Press, 1982 provide approximate expressions for creep rates of a wide range of materials, as well as references to experimental data.  As a rough guide, approximate values for a 1-term fit to creep data for polycrystalline Al alloys subjected to stresses in the range 5-60MPa are listed in the table. 

 

Approximate creep parameters for polycrystalline Al alloys

ε ˙ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaaSbaaSqaaiaaicdaae qaaaaa@32FE@  (sec-1)

σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaaa a@3311@  (MPa)

m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gaaaa@315A@

Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgfaaaa@313E@  (J)

1.3× 10 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaGGUaGaaG4maiabgEna0kaaig dacaaIWaWaaWbaaSqabeaacaaI4aaaaaaa@370D@

20

4

2.3× 10 19 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacaGGUaGaaG4maiabgEna0kaaig dacaaIWaWaaWbaaSqabeaacqGHsislcaaIXaGaaGyoaaaaaaa@38B7@

 

 

3.8.5 Representative values of parameters for viscoplastic models of high strain rate deformation

 

The material parameters in constitutive models for high strain rate deformation can also be fit to the results of a uniaxial tension or compression test.  For the model described in Section 3.7.3, the steady-state uniaxial strain rate as a function of stress is

ε ˙ ={ 00<σ/ σ 0 <1 ε ˙ 0 (1) [ ( σ/ σ 0 ) m 1 1 ]1<σ/ σ 0 <α ε ˙ 0 (2) [ ( σ/ σ 0 ) m 2 1 ]α<σ/ σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGafqyTduMbaiaacqGH9aqpdaGabaqaau aabeqadeaaaeaacaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaiabgY da8iabeo8aZjaac+cacqaHdpWCdaWgaaWcbaGaaGimaaqabaGccqGH 8aapcaaIXaaabaGafqyTduMbaiaadaqhaaWcbaGaaGimaaqaaiaacI cacaaIXaGaaiykaaaakmaadmaabaWaaeWaaeaacqaHdpWCcaGGVaGa eq4Wdm3aaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzkaaWaaWbaaS qabeaacaWGTbWaaSbaaWqaaiaaigdaaeqaaaaakiabgkHiTiaaigda aiaawUfacaGLDbaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGym aiabgYda8iabeo8aZjaac+cacqaHdpWCdaWgaaWcbaGaaGimaaqaba GccqGH8aapcqaHXoqyaeaacuaH1oqzgaGaamaaDaaaleaacaaIWaaa baGaaiikaiaaikdacaGGPaaaaOWaamWaaeaadaqadaqaaiabeo8aZj aac+cacqaHdpWCdaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaa daahaaWcbeqaaiaad2gadaWgaaadbaGaaGOmaaqabaaaaOGaeyOeI0 IaaGymaaGaay5waiaaw2faaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqyS deMaeyipaWJaeq4WdmNaai4laiabeo8aZnaaBaaaleaacaaIWaaabe aaaaaakiaawUhaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 oaaa@16C3@

The material constants m 1 , m 2 ,α, ε ˙ 0 (1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gadaWgaaWcbaGaaGymaaqabaGcca GGSaGaamyBamaaBaaaleaacaaIYaaabeaakiaacYcacqaHXoqycaGG SaGafqyTduMbaiaadaqhaaWcbaGaaGimaaqaaiaacIcacaaIXaGaai ykaaaaaaa@3C89@ , and the flow stress σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaaa a@3311@  can be determined from a series of uniaxial tension tests conducted at different temperatures and levels of applied stress; while ε ˙ 0 (2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaa0baaSqaaiaaicdaae aacaGGOaGaaGOmaiaacMcaaaaaaa@3514@  can be found from ε ˙ 0 (2) = ε ˙ 0 (1) ( α m 1 1)/( α m 2 1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaa0baaSqaaiaaicdaae aacaGGOaGaaGOmaiaacMcaaaGccqGH9aqpcuaH1oqzgaGaamaaDaaa leaacaaIWaaabaGaaiikaiaaigdacaGGPaaaaOGaaiikaiabeg7aHn aaCaaaleqabaGaamyBamaaBaaameaacaaIXaaabeaaaaGccqGHsisl caaIXaGaaiykaiaac+cacaGGOaGaeqySde2aaWbaaSqabeaacaWGTb WaaSbaaWqaaiaaikdaaeqaaaaakiabgkHiTiaaigdacaGGPaaaaa@490F@ . If strain hardening can be neglected, σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaaa a@3311@  is a temperature dependent constant, which could be approximated crudely as σ 0 =Y( 1β(T T 0 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaki abg2da9iaadMfadaqadaqaaiaaigdacqGHsislcqaHYoGycaGGOaGa amivaiabgkHiTiaadsfadaWgaaWcbaGaaGimaaqabaGccaGGPaaaca GLOaGaayzkaaaaaa@3EB9@ , where Y,β, T 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMfacaGGSaGaeqOSdiMaaiilaiaads fadaWgaaWcbaGaaGimaaqabaaaaa@3606@  are constants and T is temperature.  Viscoplastic properties of materials are very strongly dependent on their composition and microstructure, so for accurate predictions you will need to find data for the actual material you intend to use.  Clifton, R.J. “High strain rate behavior of metals.” Applied Mechanics Reviews 43 (5) (1990) S9-S22 describes several experimental techniques for testing material at high strain rates, and contains references to experimental data.  As a rough guide, parameter values for 1100-0 Al alloy (fit to data in Clifton’s paper cited earlier) are listed in the table below.   The value of β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIbaa@3209@  was estimated by assuming that the solid loses all strength at the melting point of Al (approximately 650C).

 

Approximate constitutive parameters for high strain rate behavior of Al alloy

ε ˙ 0 (1) ( s 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaa0baaSqaaiaaicdaae aacaGGOaGaaGymaiaacMcaaaGccaaMc8UaaGPaVlaaykW7caGGOaGa ci4CamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaacMcaaaa@3DEF@

ε ˙ 0 (2) ( s 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaa0baaSqaaiaaicdaae aacaGGOaGaaGOmaiaacMcaaaGccaaMc8UaaGPaVlaaykW7caGGOaGa ci4CamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaacMcaaaa@3DF0@

α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3207@

m 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gadaWgaaWcbaGaaGymaaqabaaaaa@3241@

m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gadaWgaaWcbaGaaGOmaaqabaaaaa@3242@

Y MNm 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMfacaaMc8UaaGPaVlaaykW7caqGnb GaaeOtaiaab2gadaahaaWcbeqaaiabgkHiTiaaikdaaaaaaa@3A4E@

β( K 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIjaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaiikaiaadUeadaahaaWcbeqa aiabgkHiTiaaigdaaaGccaGGPaaaaa@4269@

T 0 (K) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaGcca aMc8UaaGPaVlaaykW7caGGOaGaam4saiaacMcaaaa@38FB@

100

2.4 10 7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacaGGUaGaaGinaiaaykW7caaMc8 UaaGPaVlaaigdacaaIWaWaaWbaaSqabeaacaaI3aaaaaaa@3998@

1.6

15

0.1

50

0.00157

298