김경종
444 lines
30 KiB
Markdown
444 lines
30 KiB
Markdown
<!-- source-page: 621 -->
|
||
|
||
To conclude the derivation, in the evaluation of $\partial \lambda / \partial^{t + \Delta t} e_k''$ we use the relation (6.228), the given material relationship of effective stress versus effective plastic strain, and the condition that the effective stress function must be zero.
|
||
|
||
Of course, we considered in the preceding discussion a very special but commonly used elastoplastic material assumption. We also considered the general three-dimensional stress state. However, the stress integrations and tangent stress-strain relationships for other stress and strain conditions can be derived directly using the above procedures. For example, in axisymmetric and plane strain conditions, the appropriate strain variables would simply be set to zero. For plane stress conditions, the stress throughout the thickness is assumed zero, and so on (see Fig. 4.5).
|
||
|
||
We have presented the solution procedure for von Mises plasticity with isotropic hardening, but the algorithm can also be developed directly for the conditions of kinematic hardening and combined isotropic-kinematic (i.e., mixed) hardening (see M. Kojić and K. J. Bathe [B, C], A. L. Eterovic and K. J. Bathe [A], and K. J. Bathe and F. J. Montáns [A]).
|
||
|
||
The salient feature of the plasticity model used here is that only a single state variable (the effective stress) needs to be solved from a governing equation (the effective stress function equation) to obtain the complete stress state. The solution procedure can of course also be employed for more complex plasticity models, in which a number of internal state variables or governing parameters define the stress state. In this case, we need to establish and solve the appropriate state variable equations in an analogous manner as for the effective stress function equation discussed above.
|
||
|
||
To illustrate the application of the solution procedure to another easily tractable material law, we consider in the following example the Drucker-Prager material model, which is widely used to characterize soil and rock structures.
|
||
|
||
EXAMPLE 6.26: Consider the Drucker-Prager material model for which the yield function at time $t + \Delta t$ is given by (see D. C. Drucker and W. Prager [A], C. S. Desai and H. J. Siriwardane [A]),
|
||
|
||
$$
|
||
{ } ^ { t + \Delta t } f _ { y } ^ { \mathrm{DP} } = \alpha { } ^ { t + \Delta t } I _ { 1 } + \sqrt { { } ^ { t + \Delta t } J _ { 2 } } - k \tag {a}
|
||
$$
|
||
|
||
where $^{t + \Delta t}I_1 = ^{t + \Delta t}\sigma_{ii}$ and $^{t + \Delta t}J_2 = \frac{1}{2} ^{t + \Delta t}S_{ij}^{t + \Delta t}S_{ij}$ and $\alpha, k$ are material property parameters, see Fig. E6.26. For example, if the cohesion $c$ and angle of friction $\theta$ of the material are measured in a triaxial compression test, we have
|
||
|
||
$$
|
||
\alpha = \frac {2 \sin \theta}{\sqrt {3} (3 - \sin \theta)}
|
||
$$
|
||
|
||
$$
|
||
k = \frac {6 c \cos \theta}{\sqrt {3} (3 - \sin \theta)}
|
||
$$
|
||
|
||
Consider the case of perfect plasticity, i.e., c and $\theta$ constant, and derive the relations for the stress integration.
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
√tJ₂
|
||
tfpDP = 0
|
||
-tI₁
|
||
</details>
|
||
|
||
Figure E6.26 Drucker-Prager yield function
|
||
|
||
<!-- source-page: 622 -->
|
||
|
||
Comparing the yield function of the Drucker-Prager material model with the von Mises yield function, we recognize that the mean stress is present in (a). Hence, volumetric plastic strains are present in the response when the Drucker-Prager material model is used.
|
||
|
||
The constitutive relation for the material is
|
||
|
||
$$
|
||
{ } ^ { t + \Delta t } S _ { i j } = \frac { 1 } { a _ { E } } ( { } ^ { t + \Delta t } e _ { i j } ^ { \prime } - { } ^ { t } e _ { i j } ^ { P \prime } - \Delta e _ { i j } ^ { P \prime } ) \tag {b}
|
||
$$
|
||
|
||
$$
|
||
{ } ^ { t + \Delta t } \sigma _ { m } = \frac { 1 } { a _ { m } } ( { } ^ { t + \Delta t } e _ { m } - { } ^ { t } e _ { m } ^ { P } - \Delta e _ { m } ^ { P } ) \tag {c}
|
||
$$
|
||
|
||
where ${}^t e_{ij}^{P'}$ and $\Delta e_{ij}^{P'}$ are the deviatoric plastic strains at time $t$ and their increments, ${}^t e_m^P$ and $\Delta e_m^P$ are the mean plastic strain at time $t$ and its increment, and $a_m = (1 - 2\nu)/E$ .
|
||
|
||
The flow rule (6.213) gives
|
||
|
||
$$
|
||
\Delta e _ {i j} ^ {P} = \lambda \alpha \delta_ {i j} + \lambda \frac {^ {t + \Delta t} S _ {i j}}{2 \sqrt {^ {t + \Delta t} J _ {2}}}
|
||
$$
|
||
|
||
Hence $\Delta e_{m}^{P} = \frac{1}{3}\Delta e_{ii}^{P} = \lambda \alpha$ (d)
|
||
|
||
and $\Delta e_{ij}^{P,r} = \Delta e_{ij}^{P} - \Delta e_{m}^{P}\delta_{ij} = \lambda \frac{r + \Delta tS_{ij}}{2\sqrt{r + \Delta tJ_2}}$ (e)
|
||
|
||
Our objective is now to evaluate $\lambda$ . Since the material is nonhardening, this evaluation can be achieved analytically in terms of known quantities, and once $\lambda$ is known, the stresses for time $t + \Delta t$ can be evaluated directly.
|
||
|
||
Using the constitutive relation and the flow rule, we use (e) in (b), solve for $t^{+}\Delta t S_{ij}$ , and take the scalar product of both sides of the resulting equation to obtain
|
||
|
||
$$
|
||
\lambda = \sqrt {2} ^ {t + \Delta t} d - 2 a _ {E} \sqrt {^ {t + \Delta t} J _ {2}} \tag {f}
|
||
$$
|
||
|
||
where $^{t+\Delta t}d^{2}=^{t+\Delta t}e_{ij}^{\prime\prime}\cdot^{t+\Delta t}e_{ij}^{\prime\prime}$
|
||
|
||
and $^{t + \Delta t}e_{ij}^{\prime \prime} = ^{t + \Delta t}e_{ij}^{\prime} - ^{t}e_{ij}^{P'}$
|
||
|
||
We also use (d) in (c) to obtain
|
||
|
||
$$
|
||
{ } ^ { t + \Delta t } \sigma _ { m } = \frac { 1 } { a _ { m } } \left( { } ^ { t + \Delta t } e _ { m } ^ { \prime \prime } - \lambda \alpha \right) \tag {g}
|
||
$$
|
||
|
||
where $^{t + \Delta t}e_m'' = ^{t + \Delta t}e_m - ^t e_m^P$
|
||
|
||
Finally, we use the yield condition $t^{+\Delta t}f_y^{\mathrm{DP}} = 0$ and substitute for $t^{+\Delta t}I_1$ and $\sqrt{t^{+\Delta t}J_2}$ from (f) and (g) to obtain
|
||
|
||
$$
|
||
\lambda = \frac {\frac {3 \alpha}{a _ {m}} ^ {t + \Delta t} e _ {m} ^ {\prime \prime} + \frac {t + \Delta t d}{\sqrt {2} a _ {E}} - k}{3 \frac {\alpha^ {2}}{a _ {m}} + \frac {1}{2 a _ {E}}}
|
||
$$
|
||
|
||
With $\lambda$ known we can now evaluate directly the plastic strain increments from (b), (d), and (e) [where we use (f) to substitute in (e) for $\sqrt{t+\Delta t}J_{2}$ ]. The stresses at time $t + \Delta t$ are then obtained from (b) and (c).
|
||
|
||
<!-- source-page: 623 -->
|
||
|
||
# Thermoelastoplasticity and Creep
|
||
|
||
The effective-stress-function algorithm presented above was originally designed for the complex case of thermoelastoplasticity and creep (see K. J. Bathe, A. B. Chaudhary, E. N. Dvorkin, and M. Kojić [A]). In this case the relations presented earlier need to be generalized, and we obtain
|
||
|
||
$$
|
||
{ } ^ { t + \Delta t } \mathbf { S } = \frac { { } ^ { t + \Delta t } E } { 1 + { } ^ { t + \Delta t } \nu } ( { } ^ { t + \Delta t } \mathbf { e } ^ { t } - { } ^ { t + \Delta t } \mathbf { e } ^ { P } - { } ^ { t + \Delta t } \mathbf { e } ^ { C } ) \tag {6.253}
|
||
$$
|
||
|
||
$$
|
||
{ } ^ { t + \Delta t } \sigma _ { m } = \frac { { } ^ { t + \Delta t } E } { 1 - 2 ^ { t + \Delta t } \nu } ( { } ^ { t + \Delta t } e _ { m } - { } ^ { t + \Delta t } e ^ { T H } ) \tag {6.254}
|
||
$$
|
||
|
||
where the Young's modulus and Poisson's ratio are now considered temperature-dependent (which we model as time-dependent, with the temperature prescribed at each time step) and $t + \Delta t e^C$ , $t + \Delta t e^{TH}$ represent the creep and thermal strains, respectively. The thermal strain is calculated from
|
||
|
||
$$
|
||
{ } ^ { t + \Delta t } e ^ { T H } = { } ^ { t + \Delta t } \alpha _ { m } ( { } ^ { t + \Delta t } \theta - \theta _ { \text {ref} } ) \tag {6.255}
|
||
$$
|
||
|
||
where $t+\Delta t\alpha_{m}$ is the mean coefficient of thermal expansion and $\theta_{ref}$ is the reference temperature.
|
||
|
||
The relation (6.220) now becomes
|
||
|
||
$$
|
||
{ } ^ { t + \Delta t } \mathbf { S } = \frac { { } ^ { t + \Delta t } E } { 1 + { } ^ { t + \Delta t } \nu } ( { } ^ { t + \Delta t } \mathbf { e } ^ { \prime \prime } - \Delta \mathbf { e } ^ { P } - \Delta \mathbf { e } ^ { C } ) \tag {6.256}
|
||
$$
|
||
|
||
where the known strains are
|
||
|
||
$$
|
||
^ {t + \Delta t} \mathbf {e} ^ {\prime \prime} = ^ {t + \Delta t} \mathbf {e} ^ {\prime} - ^ {t} \mathbf {e} ^ {P} - ^ {t} \mathbf {e} ^ {C} \tag {6.257}
|
||
$$
|
||
|
||
Let us again consider the von Mises yield condition and isotropic hardening. The plastic strain increment $\Delta e^{P}$ is calculated in the same way as enumerated above except that now the effective stress–effective plastic strain curves are temperature-dependent (see Fig. 6.12). Hence, all relations derived are directly applicable, but the material constants are a function of temperature.
|
||
|
||
|
||
Figure 6.12 Effective stress–effective plastic strain curves at different temperatures (schematic representation)
|
||
|
||
The incremental creep strain $\Delta e^{c}$ is calculated quite analogously to the incremental plastic strain (see, for example, H. Kraus [A]). Using the $\alpha$ -method of time integration (see
|
||
|
||
<!-- source-page: 624 -->
|
||
|
||
M. D. Snyder and K. J. Bathe [A] and Section 9.6), we have
|
||
|
||
$$
|
||
\Delta \mathbf {e} ^ {C} = \Delta t ^ {\tau} \gamma^ {\tau} \mathbf {S} \tag {6.258}
|
||
$$
|
||
|
||
where $^{r}S = (1 - \alpha)^{r}S + \alpha^{r+\Delta t}S$ (6.259)
|
||
|
||
and $\alpha$ is the integration parameter $(0 \leq \alpha \leq 1)$ .
|
||
|
||
The function $\gamma$ is given by
|
||
|
||
$$
|
||
{ } ^ { \tau } \gamma = \frac { 3 } { 2 } \frac { { } ^ { \tau } \dot { \overline { { e } } } ^ { c } } { { } ^ { \tau } \overline { { \sigma } } } \tag {6.260}
|
||
$$
|
||
|
||
where the effective creep strain increment is
|
||
|
||
$$
|
||
\Delta \overline {{e}} ^ {C} = \sqrt {\frac {2}{3} \Delta \mathbf {e} ^ {C} \cdot \Delta \mathbf {e} ^ {C}} \tag {6.261}
|
||
$$
|
||
|
||
and the weighted effective stress is
|
||
|
||
$$
|
||
^ {r} \overline {{\sigma}} = (1 - \alpha) ^ {t} \overline {{\sigma}} + \alpha^ {t + \Delta t} \overline {{\sigma}} \tag {6.262}
|
||
$$
|
||
|
||
Since the material is assumed incompressible in creep, we have in uniaxial stress conditions $\overline{\sigma} = \sigma_{11}$ and $\overline{e}^C = e_{11}^C$ .
|
||
|
||
These relations are like those used for the calculation of the plastic strain increment, but in that case we performed the integration with $\alpha = 1$ (the Euler backward method).
|
||
|
||
The evaluation of the scalar function $\tau \gamma$ is based on a creep law. A typical uniaxial creep law used in practice is
|
||
|
||
$$
|
||
^ \prime e ^ {C} = a _ {0} ^ {\prime} \sigma^ {a _ {1}} t ^ {a _ {2}} e ^ {- a _ {3} / (t \theta + 2 7 3. 1 6)} \tag {6.263}
|
||
$$
|
||
|
||
where $e^{C}$ and $\sigma$ are the creep strain and the stress, respectively, $\theta$ is the temperature in degrees Celsius, and $a_{0}$ , $a_{1}$ , $a_{2}$ , $a_{3}$ are constants. The generalization of (6.263) to multiaxial conditions is achieved by substitution of the effective stress and effective creep strain for the uniaxial variables,
|
||
|
||
$$
|
||
{ } ^ { t } \overline { { e } } ^ { C } = a _ { 0 } { } ^ { t } \overline { { \sigma } } ^ { a _ { 1 } } t ^ { a _ { 2 } } e ^ { - a _ { 3 } / ( t \theta + 2 7 3 . 1 6 ) } \tag {6.264}
|
||
$$
|
||
|
||
This equation and other creep laws are of the form
|
||
|
||
$$
|
||
{ } ^ { t } \overline { { e } } ^ { C } = f _ { 1 } ( { } ^ { t } \overline { { \sigma } } ) f _ { 2 } ( t ) f _ { 3 } ( { } ^ { t } \theta ) \tag {6.265}
|
||
$$
|
||
|
||
and we use
|
||
|
||
$$
|
||
\Delta \overline {{e}} ^ {C} = \Delta t f _ {1} (\tau \overline {{\sigma}}) \dot {f} _ {2} (\tau) f _ {3} (\tau \theta) \tag {6.266}
|
||
$$
|
||
|
||
where $\tau\theta=(1-\alpha)^{t}\theta+\alpha^{t+\Delta t}\theta$ (6.267)
|
||
|
||
and $\tau = t + \alpha \Delta t$ . The incremental creep law in (6.266) is based on experimental evidence and corresponds to a time differentiation of the function $f_{2}$ only.
|
||
|
||
Using equations (6.266) and (6.262), the function $\gamma$ can be determined for a given value of $\overline{\sigma}$ , and the creep strain increment can be calculated. This approach corresponds to the so-called time hardening procedure. Physical observations, however, show that the use of the strain hardening procedure gives better results for variable stress conditions. In the strain hardening method, the creep strain rate is expressed in terms of the effective creep strain $\overline{e}^{C}$ instead of the time $\tau$ . This is achieved by evaluating from (6.265) and (6.266) the pseudotime $\tau_{p}$ ,
|
||
|
||
$$
|
||
^ {\prime} \bar {e} ^ {C} + f _ {1} \left(^ {\prime} \bar {\sigma}\right) f _ {3} \left(^ {\prime} \theta\right) \left[ \alpha \Delta t \dot {f} _ {2} \left(\tau_ {p}\right) - f _ {2} \left(\tau_ {p}\right) \right] = 0 \tag {6.268}
|
||
$$
|
||
|
||
<!-- source-page: 625 -->
|
||
|
||
In general, this equation needs to be solved numerically for $\tau_{p}$ . The creep strain increment $\Delta\overline{e}^{c}$ can then be computed from (6.266) with $\tau$ replaced by $\tau_{p}$ . Figure 6.13 illustrates schematically the difference between the assumptions of time and strain hardening in creep strain calculations.
|
||
|
||
|
||
|
||
<details>
|
||
<summary>line</summary>
|
||
|
||
| Time Point | Creep strain (σ₁) | Creep strain (σ₂) | Stress (σ₁) | Stress (σ₂) |
|
||
|------------|-------------------|-------------------|-------------|-------------|
|
||
| tₐ | σ₁ | σ₂ | σ₁ | σ₂ |
|
||
| tₐ_c | σ₂ | σ₂ | σ₁ | σ₂ |
|
||
</details>
|
||
|
||
Figure 6.13 Creep strain in time hardening and strain hardening assumptions. In the time hardening assumption, curve A-B defines the incremental creep strain from time $t_{a}$ onward. In the strain hardening assumption, curve A'-B' defines the incremental creep strain from time $t_{a}$ onward.
|
||
|
||
We should note that for cyclic loading conditions, additional considerations are necessary to account for the reversal of stress (see H. Kraus [A]).
|
||
|
||
The key observation regarding the above computation is that the inelastic strain increments are a function of the unknown effective stress $t^{+\Delta t}\overline{\sigma}$ only. To solve for this stress value, we use the effective stress function for the problem.
|
||
|
||
Let us substitute into (6.256) for $\Delta \mathbf{e}^P$ from (6.225) and for $\Delta \mathbf{e}^C$ from (6.258). Since all material properties are now a function of temperature, we obtain
|
||
|
||
$$
|
||
^ {t + \Delta t} \mathbf {S} = \frac {1}{^ {t + \Delta t} a _ {E} + \alpha \Delta t ^ {\tau} \gamma + \lambda} \left[ ^ {t + \Delta t} \mathbf {e} ^ {\prime \prime} - (1 - \alpha) \Delta t ^ {\tau} \gamma^ {\prime} \mathbf {S} \right] \tag {6.269}
|
||
$$
|
||
|
||
where $t+\Delta t a_{E}=\frac{1+t+\Delta t \nu}{t+\Delta t E}$ (6.270)
|
||
|
||
Taking the scalar product of both sides in (6.269), we find that the unknown effective stress satisfies
|
||
|
||
$$
|
||
a ^ {2} {} ^ {t + \Delta t} \overline {{{\sigma}}} ^ {2} + b ^ {\tau} \gamma - c ^ {2} {} ^ {\tau} \gamma^ {2} - d ^ {2} = 0 \tag {6.271}
|
||
$$
|
||
|
||
<!-- source-page: 626 -->
|
||
|
||
where
|
||
|
||
$$
|
||
a = ^ {t + \Delta t} a _ {E} + \alpha \Delta t ^ {\tau} \gamma + \lambda
|
||
$$
|
||
|
||
$$
|
||
b = 3 (1 - \alpha) \Delta t ^ {t + \Delta t} \mathbf {e} ^ {\prime \prime} \cdot {} ^ {t} \mathbf {S} \tag {6.272}
|
||
$$
|
||
|
||
$$
|
||
c = (1 - \alpha) \Delta t ^ {\prime} \overline {{{\sigma}}}
|
||
$$
|
||
|
||
$$
|
||
d ^ {2} = \frac {3}{2} ^ {t + \Delta t} \mathbf {e} ^ {n} \cdot {} ^ {t + \Delta t} \mathbf {e} ^ {n}
|
||
$$
|
||
|
||
The coefficients b, c, and d are constants that depend only on known values, whereas the coefficient a is a function of $t^{+\Delta t}\overline{\sigma}$ . Since $t^{+\Delta t}\overline{\sigma}$ is the variable to be solved for, we define the effective stress function,
|
||
|
||
$$
|
||
f (\overline {{{\sigma}}} ^ {*}) = a ^ {2} (\overline {{{\sigma}}} ^ {*}) ^ {2} + b ^ {\tau} \gamma - c ^ {2} {} ^ {\tau} \gamma^ {2} - d ^ {2} \tag {6.273}
|
||
$$
|
||
|
||
The function $f(\overline{\sigma}^{*})$ is zero at $^{t+\Delta t}\overline{\sigma}$ (see Fig. 6.14). Hence, the value of $^{t+\Delta t}\overline{\sigma}$ can, in general, be evaluated by any numerical iterative scheme that calculates the zero of a function, for example, a stable and efficient bisection technique. Once $^{t+\Delta t}\overline{\sigma}$ is known, we can use the above equations to evaluate the inelastic strains and stresses at time $t + \Delta t$ .
|
||
|
||
|
||
|
||
<details>
|
||
<summary>line</summary>
|
||
| σ̄* | f(σ̄*) |
|
||
| ------ | ----- |
|
||
| t+Δt/σ̄ | 0 |
|
||
</details>
|
||
|
||
Figure 6.14 Effective stress function, schematically shown, with zero value at $t^{+ \Delta t} \overline{\sigma}$
|
||
|
||
This solution scheme for the thermoplastic and creep inelastic response is clearly an extension of the method presented for the isothermal plastic strain calculations. Therefore, the considerations given earlier regarding the accuracy of the solution scheme are applicable here also. However, for the creep strain calculations the $\alpha$ -method with $0 \leq \alpha \leq 1$ is used in the incremental relations. In practice, stability considerations usually require that $\alpha \geq \frac{1}{2}$ (the $\alpha$ -integration scheme is unconditionally stable in linear analysis provided $\alpha \geq \frac{1}{2}$ ), and frequently we use $\alpha = 1$ . We consider the $\alpha$ -integration scheme in some detail in Section 9.6.
|
||
|
||
In the foregoing presentation we discussed how the stresses at time $t + \Delta t$ are calculated but did not present the evaluation of the tangent stress-strain relationship. This evaluation requires the derivative of the stresses with respect to the strains [see (6.244)], and we refer to the remarks given at the end of the next section in which we consider viscoplastic strains.
|
||
|
||
# Viscoplasticity
|
||
|
||
The plasticity model considered above does not model time effects that physically occur in the material. Such effects may be important and a viscoplastic material model may be more appropriate to characterize the material response. Let us consider a quite widely used
|
||
|
||
<!-- source-page: 627 -->
|
||
|
||
viscoplastic model proposed by P. Perzyna [A]. The model uses the concepts of the von Mises plasticity model but introduces time-rate effects. An important aspect of the theory of viscoplasticity is that there is no yield condition but instead the rate of the inelastic response is determined by the instantaneous difference between the effective stress and the “material” effective stress.
|
||
|
||
The implementation of the model is representative of that of viscoplastic models and can be achieved using the methods already presented.
|
||
|
||
Let us consider the Perzyna model without temperature effects. In this case the model postulates that the increments in strain are at any time
|
||
|
||
$$
|
||
d e _ {i j} = d e _ {i j} ^ {E} + d e _ {i j} ^ {V P} \tag {6.274}
|
||
$$
|
||
|
||
where the superscripts E and VP denote elastic and viscoplastic strain increments. The elastic strain increments are calculated as usual, and the viscoplastic strain increments at time t are
|
||
|
||
$$
|
||
d e _ {i j} ^ {v p} = \left\{ \begin{array}{c c} \beta \phi (^ {t} \overline {{\sigma}}) \frac {3}{2 ^ {t} \overline {{\sigma}}} ^ {t} S _ {i j} d t & \text { if } ^ {t} \overline {{\sigma}} > ^ {t} \overline {{\sigma}} _ {0} \\ 0 & \text { if } ^ {t} \overline {{\sigma}} \leq {} ^ {t} \overline {{\sigma}} _ {0} \end{array} \right. \tag {6.275}
|
||
$$
|
||
|
||
In this relation $\beta$ is a material constant, $\overline{\sigma}$ is the current effective stress, and
|
||
|
||
$$
|
||
\phi (^ {t} \overline {{\sigma}}) = \left(\frac {^ {t} \overline {{\sigma}} - ^ {t} \overline {{\sigma}} _ {0}}{^ {t} \overline {{\sigma}} _ {0}}\right) ^ {N} \tag {6.276}
|
||
$$
|
||
|
||
where $\overline{\sigma}_{0}$ is the material effective stress, that is, the effective stress corresponding to the accumulated effective viscoplastic strain $\overline{e}^{VP}$ (see Fig. 6.15), and N is another material constant. The relations for calculating deviatoric stresses, the effective stress, and the effective inelastic strain were given earlier [see (6.216), (6.227), and (6.224)]. We note that the expression for the viscoplastic strains in (6.275) is of the form of the expression for the creep strains [see (6.258)] and the plastic strains [see (6.225)] because the underlying physical phenomena of all these strain components are similar (but different time scales are applicable). A consequence is that the viscoplastic strains also correspond to an incompressible response [as do the plastic and creep strain components in (6.225) and (6.258)].
|
||
|
||
The model requires the elastic constants $E$ (Young's modulus) and $\nu$ (Poisson's ratio), the material constants $\beta$ , $N$ , and the curve in Fig. 6.15. We note that $\beta$ (with unit 1/time) and $N$ determine the rate behavior of the material. That is, viscoplastic strains are accumulated as long as the effective stress is larger than the material effective stress, and the rate of such accumulation is determined by $\beta$ and $N$ .
|
||
|
||
|
||
|
||
<details>
|
||
<summary>line</summary>
|
||
| Accumulated effective viscoplastic strain | Material effective stress |
|
||
| ----------------------------------------- | ------------------------- |
|
||
| tσ̄₀ | tσ̄₀ |
|
||
| t̄ₑ^VP | t̄ₑ^VP |
|
||
</details>
|
||
|
||
Figure 6.15 Schematic representation of material effective stress versus accumulated effective viscoplastic strain
|
||
|
||
<!-- source-page: 628 -->
|
||
|
||
Let us now consider the calculation of the stresses at time $t + \Delta t$ . We proceed as in the case of plasticity and creep. Assuming that the total strains corresponding to time $t + \Delta t$ and all stress and strain variables corresponding to time t are known, we have (as in the case of plasticity and creep)
|
||
|
||
$$
|
||
{ } ^ { t + \Delta t } \sigma _ { m } = \frac { E } { 1 - 2 \nu } { } ^ { t + \Delta t } e _ { m } \tag {6.277}
|
||
$$
|
||
|
||
$$
|
||
{ } ^ { t + \Delta t } \mathbf { S } = \frac { E } { 1 + \nu } ( { } ^ { t + \Delta t } \mathbf { e } ^ { n } - \Delta \mathbf { e } ^ { V P } ) \tag {6.278}
|
||
$$
|
||
|
||
$$
|
||
{ } ^ { t + \Delta t } \mathbf { e } ^ { \prime \prime } = { } ^ { t + \Delta t } \mathbf { e } ^ { t } - { } ^ { t } \mathbf { e } ^ { V P } \tag {6.279}
|
||
$$
|
||
|
||
where the variables are as in (6.214) to (6.221) but we consider viscoplastic strains instead of plastic strains.
|
||
|
||
The viscoplastic strain increment is given by [compare (6.258)]
|
||
|
||
$$
|
||
\Delta \mathbf {e} ^ {V P} = \Delta t ^ {\tau} \gamma^ {\tau} \mathbf {S} \tag {6.280}
|
||
$$
|
||
|
||
where using the $\alpha$ -method of integration,
|
||
|
||
$$
|
||
{ } ^ { \tau } \mathbf { S } = ( 1 - \alpha ) { } ^ { \prime } \mathbf { S } + \alpha { } ^ { t + \Delta t } \mathbf { S } \tag {6.281}
|
||
$$
|
||
|
||
and the scalar $\tau\gamma$ is given by
|
||
|
||
$$
|
||
{ } ^ { \tau } \gamma = \left\{ \begin{array} { l l } \beta { } ^ { \tau } \phi \frac { 3 } { 2 { } ^ { \tau } \overline { { \sigma } } } & \text { if } { } ^ { \tau } \overline { { \sigma } } > { } ^ { \tau } \overline { { \sigma } } _ { 0 } \\ 0 & \text { if } { } ^ { \tau } \overline { { \sigma } } \leq { } ^ { \tau } \overline { { \sigma } } _ { 0 } \end{array} \right. \tag {6.282}
|
||
$$
|
||
|
||
We note that $\gamma$ depends on $\overline{\sigma}$ and $\overline{\sigma}_{0}$ , where $\overline{\sigma}_{0}$ depends on the accumulated effective viscoplastic strain (see Fig. 6.15). Hence, as in the analysis of creep response, the above relations represent a one-parameter system of equations in the effective stress $\tau+\Delta\overline{\sigma}$ , which is obtained as the zero of the effective stress function,
|
||
|
||
$$
|
||
f (\overline {{{\sigma}}} ^ {*}) = a ^ {2} (\overline {{{\sigma}}} ^ {*}) ^ {2} + b ^ {\tau} \gamma - c ^ {2} {} ^ {\tau} \gamma^ {2} - d ^ {2} \tag {6.283}
|
||
$$
|
||
|
||
where $a = a_{E} + \alpha \Delta t^{\tau}\gamma$
|
||
|
||
$$
|
||
b = 3 (1 - \alpha) \Delta t ^ {t + \Delta t} \mathbf {e} ^ {\prime \prime} \cdot {} ^ {t} \mathbf {S} \tag {6.284}
|
||
$$
|
||
|
||
$$
|
||
c = (1 - \alpha) \Delta t ^ {\prime} \overline {{\sigma}}
|
||
$$
|
||
|
||
$$
|
||
d ^ {2} = \frac {3}{2} ^ {t + \Delta t} \mathbf {e} ^ {n} \cdot {} ^ {t + \Delta t} \mathbf {e} ^ {n}
|
||
$$
|
||
|
||
$$
|
||
a _ {E} = \frac {1 + \nu}{E}
|
||
$$
|
||
|
||
This function is obtained as in the case of thermoplasticity and creep [see (6.271) and (6.272)] but neglecting all temperature dependency and the plastic parameter $\lambda$ (since the viscoplastic strains are calculated by the procedure used for the creep strains). Of course, a dependence on temperature for the material properties could be included directly. Indeed, an important consideration in the use of viscoplastic models is that various functional dependencies can be directly, and with relative ease, included in the calculations. The basic reason for this ease in use is that there is no explicit yield condition. Instead, the solution is obtained by integrating the inelastic strains until the effective stress is equal to the material effective stress (given by the effective viscoplastic strain). This integration is
|
||
|
||
<!-- source-page: 629 -->
|
||
|
||
efficiently performed with the $\alpha$ -method because with $\alpha \geq \frac{1}{2}$ the integration can proceed with relatively large time steps (see Section 9.6).
|
||
|
||
We considered in the above discussions of thermoelastoplasticity and creep and viscoplasticity only the evaluation of the stresses corresponding to given total strains. The consistent tangent constitutive matrices would be calculated as discussed for the case of plasticity but, in general, can be obtained only in analytical form provided tractable functional relationships for the inelastic strains are used. If the analytical derivation is not possible, a numerical evaluation of the tangent constitutive relation can be achieved by use of a finite difference scheme to calculate the required differentiations in (6.208) (see, for example, M. Kojić and K. J. Bathe [B]).
|
||
|
||
# 6.6.4 Large Strain Elastoplasticity
|
||
|
||
The formulations for inelastic response discussed in the previous section have been presented for small or large displacement response with small strains. Additional considerations are important when large strains are modeled.
|
||
|
||
The extension of the infinitesimal small strain theory of plasticity to large strains can be achieved by a number of alternative formulations (see, for example, A. E. Green and P.M. Naghdi [A], E.H. Lee [A], J. Lubliner [A], J.C. Simo [A], G.Weber and L. Anand [A], A.L. Eterovic and K.J. Bathe [A], D.N. Kim, F.J. Montáns, and K.J. Bathe [A] and M.A. Caminero, F. J. Montáns, and K.J. Bathe [A]). However, our purpose here is to merely introduce the basic considerations, and hence we briefly discuss only one formulation, namely, a total strain formulation based on Cauchy stresses and logarithmic strains.
|
||
|
||
The basic considerations in any formulation relate to the choice of adequate stress and strain measures, the characterization of the elastic behavior, and the proper characterization of plastic flow.
|
||
|
||
An effective large strain procedure should surely reduce to the formulations presented in the previous section when the strains are small. However, an important feature of the materially-nonlinear-only and large displacement-small strain analysis procedures presented is that these formulations are total strain and not rate-type formulations. That is, the equations of equilibrium are written for time $t + \Delta t$ and the total strain for that time is calculated. Hence, numerical integration is used only in the calculation of the inelastic strain from time t to time $t + \Delta t$ . In contrast, using a rate-type formulation, rates of stress, strain, and rotational effects are integrated, which leads to additional numerical errors and, for an accurate solution, requires significantly smaller solution steps than are needed in a total strain formulation.
|
||
|
||
For large deformation analysis, rate-type formulations are frequently based on the Jaumann stress rate–velocity strain description (see Example 6.24). In addition to the numerical integration errors, such a hypoelastic stress-strain description also leads to non-conservative and therefore nonphysical response predictions in purely elastic cyclic motions (see M. Kojić and K. J. Bathe [A]). This nonphysical behavior may be judged to be small, but is not due to numerical integration error.
|
||
|
||
A natural approach based on micromechanical observations for large strain elastoplasticity is based on the hyperelastic material description with the product decomposition of the deformation gradient into elastic and plastic parts (see E. H. Lee [A], J. R. Rice [A], and R. J. Asaro [A]). Such an approach also lends itself to a total formulation that is a natural extension of the infinitesimal strain formulation discussed in the previous section.
|
||
|
||
<!-- source-page: 630 -->
|
||
|
||
One important feature of the large strain elastoplastic analysis is that the uniaxial stress-strain law used to characterize the response is given by the Cauchy stress-logarithmic strain relationship (see Fig. 6.16). The yield condition, flow rule, and hardening rule are used for the Cauchy stresses.
|
||
|
||
|
||
|
||
<details>
|
||
<summary>line</summary>
|
||
|
||
| Logarithmic strain | Cauchy stress (Force / Current area) |
|
||
| ------------------ | ------------------------------------ |
|
||
| 0 | 0 |
|
||
| >0.5 | σyv |
|
||
</details>
|
||
|
||
Figure 6.16 Large strain elastoplastic one-dimensional response model
|
||
|
||
Using the multiplicative decomposition of the deformation gradient (to characterize the large strain elastoplastic deformation of a body), we have
|
||
|
||
$$
|
||
\mathbf {X} = \mathbf {X} ^ {E} \mathbf {X} ^ {P} \tag {6.285}
|
||
$$
|
||
|
||
where $X^{E}$ and $X^{P}$ represent, respectively, the elastic and plastic deformation gradients. The relation (6.285) is assumed to hold throughout the response, but for ease of writing we do not include the left superscripts and subscripts (until we present the actual computational procedure); hence for example, we have $X \equiv \oint X$ .
|
||
|
||
Relation (6.285) is a key equation in our large strain elastoplasticity formulation. At time $t$ the relation (6.285) reads $\mathbf{X} = \mathbf{X}^E \mathbf{X}^P$ , or we may—as mentioned following (6.29) and used in Example 6.9—also write $\mathbf{X} = \mathbf{X}^E \mathbf{X}^P$ for conceptual understanding. Hence, the approach used is conceptually based on a relaxed hypothetical configuration (corresponding to $\tau$ ) which for each particle is obtained by unloading the material from the current configuration to a state of zero stress in such a way that no inelastic process takes place. The plastic deformation gradient $\mathbf{X}^P$ corresponds to the deformation from the original to this hypothetical configuration. The elastic deformations and therefore stresses are measured using $\mathbf{X}^E$ , which is thought of as a deformation gradient measured from the relaxed configuration $\tau$ .
|
||
|
||
Let us, as in Section 6.6.3, characterize the material with the von Mises yield condition and isotropic hardening. Hence, our only aim in the following discussion is to extend the plasticity formulation of Section 6.6.3 to large strains.
|
||
|
||
As in small strain plasticity, we assume that the plastic deformation is incompressible; hence,
|
||
|
||
$$
|
||
\det \mathbf {X} ^ {P} = 1 \tag {6.286}
|
||
$$
|