Regularization and perturbation technique to solve plasticity problems

Abstract

This paper investigates new procedures to solve plasticity problems by using the asymptotic numerical method (ANM). As the elastic-plastic behavior involves two unilateral conditions, we replace these two conditions by regular functions depending upon the stress field and its time derivative which permits one to take into account elastic-plastic transition and elastic unloading. Several applications in structural plasticity problems are presented to assess the ability of the proposed algorithm.

Appendix

In the appendix, one deduces of the power series the recurrence formula (38), that gives the constitutive relation at any order. One can find in [2, 24] the general method to calculate the coefficients of the series. This technique is applied to the regularized plasticity model (36) where, for simplicity, the analysis is limited to isotropic materials. Thus the first Eq. 36 is written as:

$$\frac{1+\nu}{E}\;\sigma-\frac{\nu}{E}\;\textrm{tr}(\sigma)I=\varepsilon-\varepsilon^p $$

(43)

Furthermore, two additional variables F and D are introduced to simplify the function G(f,σ _e ), that is split in three elementary equations:

$$F=f^2 $$

(44)

$$D=\frac{F\sigma_e}{2\;\mu}+\eta_1\left(\frac{3}{2}+\frac{h}{2\;\mu}(1+f)\right) $$

(45)

$$G\;D=\eta_1 $$

(46)

When applied to the simple Eqs. 44, 45, 46, the perturbation technique yields the k coefficient of the Taylor series of G. It is written as a function of the coefficients of the series of f and σ _e :

$$ F_k=\sum\limits_{i=0}^{k}{f_i\;f_{k-i}} $$

(47)

$$D_k=\frac{1}{2\;\mu}\sum\limits_{i=0}^{k}{F_i\;\sigma_{e\;k-i}}+\eta_1\frac{h}{2\;\mu}f_k $$

(48)

$$G_k=-\frac{1}{D_0}\sum\limits_{i=0}^{k-1}{G_i\;D_{k-i}} $$

(49)

Next, the same rules are applied to the modified constitutive law (36), where the first equation has been written in the form Eq. 43. This leads to:

$$\frac{1+\nu}{E}\sigma_k-\frac{\nu}{E}\;\textrm{tr}(\sigma_k)I=\varepsilon_k-\varepsilon_k^p $$

(50)

$$ \varepsilon^{p}_k=\lambda_k\;n_0+\overline{\varepsilon}_k^{p\;nl} \quad \quad \overline{\varepsilon}_k^{p\;nl}=\frac{1}{k}\sum\limits_{i=1}^{k-1}{(k-i)\lambda_{k-i}n_i} $$

(51)

$$\lambda_k=\frac{\dot{\varepsilon}_c}{k}G_0\;H_{k-1}+\lambda_k^{nl} \quad \quad \lambda_k^{nl}=\frac{\dot{\varepsilon}_c}{k}\sum\limits_{i=1}^{k-1}{H_{k-i-1}G_i} $$

(52)

$$\begin{array}{rcl} H_{k-1}&=&\frac{H_0}{2H_0-\xi_0}\xi_{k-1}+H_{k-1}^{nl} \\ \\ H_{k-1}^{nl}&=&\frac{-1}{2H_0-\xi_0}\sum\limits_{i=1}^{k-2}{H_i(H_{k-i-1}-\xi_{k-i-1})} \end{array}$$

(53)

$$\xi_{k-1}=\frac{k}{\dot{\varepsilon}_c} n_0:\varepsilon_k+\xi_{k-1}^{nl}\quad \quad \!\! \xi_{k-1}^{nl}=\frac{1}{\dot{\varepsilon}_c}\sum\limits_{i=1}^{k-1}{(k-i\,)\varepsilon_{k-i}:n_i} $$

(54)

$$n_k=\frac{3}{2}\frac{\sigma_k^d}{q_0}+n_k^{nl} \quad \quad n_k^{nl}=\frac{-1}{q_0}\sum\limits_{i=0}^{k-1}{n_i\;q_{k-i}} $$

(55)

$$f_k=\frac{1}{\sigma_{e\;0}}\left(q_k-\sigma_{e\;k}(1+f_0)-\sum\limits_{i=1}^{k-1}{f_i \sigma_{e k-i}}\right) $$

(56)

$$q_k=\frac{3}{2q_0}\sum\limits_{i=0}^{k}{\sigma^d_i:\sigma^d_{k-i}}-\frac{1}{2q_0}\sum\limits_{i=1}^{k-1}{q_i:q_{k-i}} $$

(57)

$$\sigma_{e\;k}=h\;\lambda_k $$

(58)

In most of these recurrence formula, the right hand side has been split in two parts. The first part corresponds to the tangent behavior, for instance λ _k n ₀ in Eq. 51. The second part, denoted by the mark nl, includes the contribution of the lower order terms and accounts for the non linear effects.

In the specific case of elasto-plasticity, one can distinguish two groups of equations. The first group contains Eqs. 50, 51, 52, 53 and 54 and corresponds to the non linear relation between the stress rate and the strain rate. These five equations will be combined to get the recurrence formula (38), that relates σ _k and ε _k . The second group of equations (47), (48), (49), (55), (56), (57) and (58) defines the yield function f _k , the normal n _k , the equivalent stress q _k and the effective stress σ _{e k} . These variables f, q and σ _e appears in the first group, but only via the orders lower than k. As a consequence, the second group of equations can be implemented after the first group.

It is convenient to eliminate the plastic strain, the plastic multiplier, H _k − 1 and ξ _k − 1 to recover the stress-strain relation (38), that can be introduced in a finite element framework, see “Finite element applications”. This condensation process is easy, because the perturbation technique leads to linear equations.

First one eliminates ξ _k − 1 from Eq. 54, that is substituted in Eq. 53. Next one drops H _k − 1 and λ _k from Eqs. 52 and 51. This yields the following relation between the plastic strain $\varepsilon_k^p$ and the strain ε _k :

$$\varepsilon^{p}_k=\frac{G_0\;H_0}{2H_0-\xi_0}(n_0 \otimes n_0)\varepsilon_k+\varepsilon_k^{p\;nl} $$

(59)

where

$$\left\{ \begin{array}{ll} \varepsilon_k^{p\;nl}=\overline{\varepsilon}_k^{p\;nl}+\overline{\lambda}_{k-1}^{nl}\;n_0\\[4pt] \overline{\lambda}_{k-1}^{nl}=\lambda_{k-1}^{nl}+\dot{\varepsilon}_c\frac{G_0}{k}\overline{H}_{k-1}^{nl}\\[4pt] \overline{H}_{k-1}^{nl}=H_{k-1}^{nl}+\frac{H_0}{(2H_0-\xi_0)}\xi_{k-1}^{nl} \end{array} \right. $$

(60)

Last, one substitutes Eq. 59 into Eq. 50 to obtain:

$$\frac{1+\nu}{E}\sigma_k\!-\!\frac{\nu}{E}\;\textrm{tr}(\sigma_k)I\!=\!\varepsilon_k-\frac{G_0\;H_0}{2H_0-\xi_0}(n_0 \otimes n_0)\varepsilon_k\!-\!\varepsilon_k^{p\;nl} $$

(61)

In the case of an isotropic material, Eq. 61 can be inverted analytically, by considering the spherical part of the tensor Eq. 61:

$$\frac{1-2\nu}{E}\textrm{tr}(\sigma_k)=\textrm{tr}(\varepsilon_k) $$

(62)

By combining Eqs. 61 and 62, one obtains the constitutive law at order k in the form Eq. 38.

$$\begin{array}{l}\sigma_k=\frac{E}{1+\nu}\left[ \varepsilon_k+\frac{\nu}{1-2\nu}\textrm{tr}(\varepsilon_k)I\vphantom{\frac{G_0\;H_0}{2H_0-\xi_0}}\right. \\ \quad\qquad\qquad \left.-\frac{G_0\;H_0}{2H_0-\xi_0}(n_0 \otimes n_0)\varepsilon_k-\varepsilon_k^{p\;nl}\right] \end{array}$$

(63)