Chapter 3

 

Constitutive Models $–$ 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 $\dot{\epsilon }\approx A{\sigma }^m$.  At low stresses, $m\approx 1-2.5$; at intermediate stresses $m\approx 2.5-7$, 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 $\dot{\epsilon }={\dot{\epsilon }}_0\exp (-Q/kT)$, where $Q$ is an activation energy; $k$ is the Boltzmann constant, and T is temperature.  Like the stress exponent m,   the activation energy $Q$  can transition from one value to another as the temperature and stress level is varied.

 

5.      The various regimes for m and $Q$ 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 ${10}^7$ /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 ${10}^6$, 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 ${\epsilon }_{ij}=\left(\partial u_i/\partial x_j+\partial u_j/\partial x_i\right)/2$.

The strain rate is decomposed into elastic and plastic parts as

$\frac{d{\epsilon }_{ij}^{}}{dt}=\frac{d{\epsilon }_{ij}^e}{dt}+\frac{d{\epsilon }_{ij}^p}{dt}$

 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

$\frac{d{\dot{\epsilon }}_{ij}^e}{dt}=S_{ijkl}\frac{d{\sigma }_{kl}}{dt}$

where $S_{ijkl}$ are the components of the elastic compliance tensor.  For the special case of an isotropic material with Young’s modulus $E$ and Poisson’s ratio $\nu$

$\frac{d{\epsilon }_{ij}^e}{dt}=\frac{1+\nu }{E}\frac{d{\sigma }_{ij}}{dt}-\frac{\nu }{E}\frac{d{\sigma }_{kk}}{dt}{\delta }_{ij}+\alpha \frac{d\Delta T}{dt}{\delta }_{ij}$

 

 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 ${\dot{\epsilon }}_e=\sqrt{2{\dot{\epsilon }}_{ij}^p{\dot{\epsilon }}_{ij}^p/3}$

2.      Let ${\sigma }_{ij}$ denote the stresses acting on the material, and let ${\sigma }_1,{\sigma }_2,{\sigma }_3$ 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

$S_{ij}={\sigma }_{ij}-{\sigma }_{kk}{\delta }_{ij}/3$

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

${\sigma }_e=\sqrt{\frac{3}{2}{S}_{ij}{S}_{ij}}{\sigma }_{III}=\det (S)$

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

${\dot{\epsilon }}_e=g({\sigma }_e,{\sigma }_{III},\text{State variables)}$

where $g$ must satisfy $g(\alpha {\sigma }_e,\alpha {\sigma }_{III},\text{State vars)}\ge g({\sigma }_e,{\sigma }_{III},\text{State vars)}$ for all $\alpha \ge 1$ (with state variables held fixed), and must also be a convex function of ${\sigma }_e,{\sigma }_{III}$

 

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

$g({\sigma }_e,\left\{{\sigma }_0^{(i)}\right\})={\displaystyle \sum _{i=1}^N{\dot{\epsilon }}_0^{(i)}\exp (-Q_i/kT){\left(\frac{{\sigma }_e}{{\sigma }_0^{(i)}}\right)}^{m_i}}$

where ${\dot{\epsilon }}_0^{(i)}$, $Q_i$ and $m_i$, i=1..N are material properties ( $Q_i$ 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, ${\sigma }_0^{(i)}$.   To model steady state creep, you can take ${\sigma }_0$ to be constant; to model transient creep, ${\sigma }_0$ must increase with strain.  An example of an evolution law for ${\sigma }_0$ is given below. 

 

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

$g({\sigma }_e,{\sigma }_0^{})=\{\begin{array}{c}00<{\sigma }_e/{\sigma }_0<1\\ {\dot{\epsilon }}_0^{(1)}\left[{\left({\sigma }_e/{\sigma }_0\right)}^{m_1}-1\right]1<{\sigma }_e/{\sigma }_0<\alpha \\ {\dot{\epsilon }}_0^{(2)}\left[{\left({\sigma }_e/{\sigma }_0\right)}^{m_2}-1\right]\alpha <{\sigma }_e/{\sigma }_0\end{array}$

where ${\dot{\epsilon }}_0^{(2)}={\dot{\epsilon }}_0^{(1)}({\alpha }^{m_1}-1)/({\alpha }^{m_2}-1)$ and ${\dot{\epsilon }}_0^{(1)},\alpha ,m_1,m_2$ are material properties, while ${\sigma }_0$ 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, $m_2<m_1$ so as to model the transition in strain rate sensitivity at high strain rates; while $\alpha$ 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

${\dot{\epsilon }}_{ij}^p=\sqrt{\frac{3}{2}}\frac{g({\sigma }_e,{\sigma }_{III},\text{State vars)}}{\left[\left(\partial g/\partial {\sigma }_{kl}\right)\left(\partial g/\partial {\sigma }_{kl}\right)\right]}\frac{\partial g}{\partial {\sigma }_{ij}}$

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

${\dot{\epsilon }}_{ij}^p=g({\sigma }_e,\text{State vars)}\frac{3S_{ij}}{2{\sigma }_e}$

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

${\dot{\epsilon }}_{ij}^p={\displaystyle \sum _{n=1}^N{\dot{\epsilon }}_0^{(n)}\exp (-Q_n/kT){\left(\frac{{\sigma }_e}{{\sigma }_0^{(n)}}\right)}^{m_n}}\frac{3S_{ij}}{2{\sigma }_e}$

 

 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

${\sigma }_0^{(i)}=Y_i{\left(1+\frac{{\epsilon }_e^{(i)}}{{\epsilon }_0^{(i)}}\right)}^{1/n_i}$

where ${\epsilon }_e^{(i)}={\displaystyle \int {\dot{\epsilon }}_0^{(i)}\exp (-Q_i/kT){({\sigma }_e/{\sigma }_0^{(i)})}^{m_i}dt}$ is the accumulated strain associated with each mechanism of creep, and $Y_i,{\epsilon }_0^{(i)}$ and $n_i$ 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

${\sigma }_0=Y[1-\beta (T-T_0)]{\left(1+\frac{{\epsilon }_e^{}}{{\epsilon }_0^{}}\right)}^{1/n}$

where ${\epsilon }_e={\displaystyle \int \sqrt{3{\dot{\epsilon }}_{ij}^p{\dot{\epsilon }}_{ij}^p/2}dt}$ is the total accumulated strain, T is temperature, and $Y,n,T_0,\beta$ are material properties.  More sophisticated hardening laws make the flow stress a function of strain rate $–$ 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 ${\sigma }_{11}=\sigma$, all other stress components zero, the uniaxial plastic strain rate is

${\dot{\epsilon }}_{11}^p={\displaystyle \sum _{n=1}^N{\dot{\epsilon }}_0^{(n)}\exp (-Q_n/kT){\left(\frac{\sigma }{{\sigma }_0^{(n)}}\right)}^{m_n}}$

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  ${\sigma }_0^{(n)}$ 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. 

 

 

 

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

$\dot{\epsilon }=\{\begin{array}{c}00<\sigma /{\sigma }_0<1\\ {\dot{\epsilon }}_0^{(1)}\left[{\left(\sigma /{\sigma }_0\right)}^{m_1}-1\right]1<\sigma /{\sigma }_0<\alpha \\ {\dot{\epsilon }}_0^{(2)}\left[{\left(\sigma /{\sigma }_0\right)}^{m_2}-1\right]\alpha <\sigma /{\sigma }_0\end{array}$

The material constants $m_1,m_2,\alpha ,{\dot{\epsilon }}_0^{(1)}$, and the flow stress ${\sigma }_0$ can be determined from a series of uniaxial tension tests conducted at different temperatures and levels of applied stress; while ${\dot{\epsilon }}_0^{(2)}$ can be found from ${\dot{\epsilon }}_0^{(2)}={\dot{\epsilon }}_0^{(1)}({\alpha }^{m_1}-1)/({\alpha }^{m_2}-1)$. If strain hardening can be neglected, ${\sigma }_0$ is a temperature dependent constant, which could be approximated crudely as ${\sigma }_0=Y\left(1-\beta (T-T_0)\right)$, where $Y,\beta ,T_0$ 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 $\beta$ was estimated by assuming that the solid loses all strength at the melting point of Al (approximately 650C).