Mathematical analysis of a solution method for finite-strain holonomic plasticity of Cosserat materials

Abstract

This article deals with the mathematical derivation and the validation over benchmark examples of a numerical method for the solution of a finite-strain holonomic (rate-independent) Cosserat plasticity problem for materials, possibly with microstructure. Two improvements are made in contrast to earlier approaches: First, the micro-rotations are parameterized with the help of an Euler–Rodrigues formula related to quaternions. Secondly, as main result, a novel two-pass preconditioning scheme for searching the energy-minimizing solutions based on the limited memory Broyden–Fletcher–Goldstein–Shanno quasi-Newton method is proposed that consists of a predictor step and a corrector-iteration. After outlining the necessary adaptations to the model, numerical simulations compare the performance and efficiency of the new and the old algorithm. The proposed numerical model can be effectively employed for studying the mechanical response of complicated materials featuring large size effects.

Appendices

Appendix 1: List of symbols

$A\!:\!B$ :

Tensor product of A, B, below (6)

${{\varvec{u}}}\!\cdot \!{{\varvec{v}}}$ :

Inner product of ${{\varvec{u}}}$, ${{\varvec{v}}}\in {\mathbb {R}}^3$

${\text {sym}}(\sigma )$ :

Symmetric part of a tensor $\sigma$ (4)

${\text {skw}}(\sigma )$ :

Skew-symmetric part of $\sigma$ (4)

$\mathrm {tr}(\sigma )$ :

Trace of tensor $\sigma$

$\sigma ^{ T}$ :

Transpose of $\sigma$; $R^{ T}\!=\!R^{-1}$ for $R\!\in \!{\mathrm {SO}}(3)$

$\Vert \cdot \Vert$ :

Frobenius matrix norm (4)

$|\cdot |$ :

Euclidean vector norm in ${\mathbb {R}}^4$ (33)

$\varOmega \subset {\mathbb {R}}^3$ :

Reference domain, undeformed solid

(x, t):

Space and time coordinates

$\varphi$ :

Deformation vector of the solid (3)

$F\!=\!D\varphi$ :

Deformation tensor (3)

$F_{{\mathrm {e}}}$ :

Elasticity tensor (3)

$F_{{\mathrm {p}}}$ :

Plasticity tensor (3)

$R_{{\mathrm {\!e}}}$ :

Rotation tensor (1), (2), (3)

$U_{{\mathrm {\!e}}}$ :

(right) stretching tensor (3)

$K_{{\mathrm {\!e}}}$ :

(right) curvature tensor (39)

${\mathbb {I}}$ :

Identity tensor, $({\mathbb {I}})_{kl}=(\delta _{kl})_{kl}$ (10)

$\alpha$ :

Euler angle parameterization of $R_{{\mathrm {\!e}}}$ (1)

$\gamma$ :

Single-slip parametrization of $F_{{\mathrm {p}}}$ (10)

q :

Quaternion parameterization of $R_{{\mathrm {\!e}}}$ (2)

$q_D$ :

Dirichlet boundary values of q (8)

${\overline{q}}$ :

Conjugate of q (33)

${\mathscr {E}}$ :

Mechanical energy (13)

$h>0$ :

Discrete (fixed) time step (13)

$\gamma ^0$ :

Values of $\gamma$ at old time t (12)

$\kappa ^0$ :

Values of $\kappa$ at old time t (12)

$\kappa$ :

Dislocation density (12)

$V(\kappa )$ :

Dislocation energy (6)

$W_{{\mathrm {st}}}$ :

Stretching energy (4)

$W_{{\mathrm {c}}}$ :

Curvature energy (5)

X :

Back stress (dual variable to $F_{{\mathrm {p}}}$) (9),

$\xi$ :

Hardening (dual variable to $\kappa$) (9)

$f_{{\mathrm {ext}}}$ :

External volume forces (13)

$M_{{\mathrm {ext}}}$ :

External volume couples (13)

$\sigma _Y$ :

Yield stress (13)

$Q^*$ :

Dissipated energy (11)

${{\varvec{m}}}$ :

Slip vector (10)

${{\varvec{n}}}$ :

Slip normal (10)

$\rho >0$ :

Dislocation energy constant (13)

$g_{_D}$ :

Dirichlet boundary values of $\varphi$ (8)

$\varepsilon >0$ :

Regularization of $|\cdot |$, Remark 1

$\varLambda >0$ :

Lagrange parameter to $|q|^2=1$ (13)

$\lambda$, $\mu$:

Lamé parameters (4)

$\mu _c$ :

Cosserat couple modulus (4)

$L_c$ :

Internal length scale (5)

$\mu _2$ :

Parameter $\mu$ scaled by $L_c^2$ (5)

$\varepsilon _0$ :

Stop/precision parameter (19)

$d_1,d_2,d_3$ :

Spatial resolution (15), (24)

$\eta _1,\eta _2,\eta _3$ :

Points on the numerical mesh (14)

$N_{IJK}$ :

Discrete numerical weights (15), (24)

$\beta (t)$ :

Deformation parameter (29), (30).

Appendix2: Use of quaternions in the Euler–Rodrigues formula

Following the classical notation in [56, 57], let

$$\begin{aligned} {\mathbb {H}}&:= \mathrm {span}_{\mathbb {R}}\{1,{{\varvec{i}}},{{\varvec{j}}},{{\varvec{k}}}\}\\&= \big \{q=q_0+q_1{{\varvec{i}}}+q_2{{\varvec{j}}}+q_3{{\varvec{k}}}\;\big |\;q_0,q_1,q_2,q_3\in {\mathbb {R}}\big \} \end{aligned}$$

denote the space of quaternions, where the quaternion imaginary units satisfy ${{\varvec{i}}}^2={{\varvec{j}}}^2={{\varvec{k}}}^2={{\varvec{i}}}{{\varvec{j}}}{{\varvec{k}}}=-1$. Let

$$\begin{aligned} {\mathbb {H}}_{{\mathrm {p}}}:=\{q=q_0+q_1{{\varvec{i}}}+q_2{{\varvec{j}}}+q_3{{\varvec{k}}}\in {\mathbb {H}}\;|\;q_0=0\} \end{aligned}$$

be the space of pure quaternions and

$$\begin{aligned} q=q_0+{\widehat{q}}:=q_0+q_1{{\varvec{i}}}+q_2{{\varvec{j}}}+q_3{{\varvec{k}}}. \end{aligned}$$

(31)

The set ${\mathbb {H}}$ is equipped with the multiplication (for $p,q\in {\mathbb {H}}$)

$$\begin{aligned} pq:=p_0q_0-{\widehat{p}}\cdot {\widehat{q}}+p_0{\widehat{q}}+q_0{\widehat{p}}+{\widehat{p}}\times {\widehat{q}}, \end{aligned}$$

(32)

where ${\widehat{p}}\cdot {\widehat{q}}:=p_1q_1+p_2q_2+p_3q_3$ specifies as above the inner product and ${\widehat{p}}\times {\widehat{q}}$ the vector product of ${\mathbb {R}}^3$, respectively. In general, $pq\not =qp$, so ${\mathbb {H}}$ is an associative, non-commutative algebra. Let ${\overline{q}}:=q_0-{\widehat{q}}$ be the conjugate of q and

$$\begin{aligned} |q|:=\big (q{\overline{q}}\big )^{1/2}=\big ({\overline{q}}q\big )^{1/2} =\big (q_0^2+q_1^2+q_2^2+q_3^2\big )^{1/2} \end{aligned}$$

(33)

be the modulus of q. By Formula (32), $q\in {\mathbb {H}}^*:={\mathbb {H}}{\setminus }\{0\}$ possesses the multiplicative inverse $q^{-1}=\frac{{\overline{q}}}{|q|^2}$. Let

$$\begin{aligned} {\mathrm {so}}(3):=\{\omega \in {\mathbb {R}}^{3\times 3}\;|\;\omega ^{ T}=-\omega \} \end{aligned}$$

be the Lie algebra of ${\mathrm {SO}}(3)$. The alternating skew tensor$\varepsilon :{\mathbb {H}}_{{\mathrm {p}}}\rightarrow {\mathrm {so}}(3)$ is defined by

$$\begin{aligned} \varepsilon ({\widehat{q}}):=\left( \begin{array}{*{20}c} 0 & -q_3 & q_2\\ q_3 & 0 & -q_1\\ -q_2 & q_1 & 0 \end{array}\right) . \end{aligned}$$

(34)

Evidently,

$$\begin{aligned} \varepsilon ({\widehat{q}})v={\widehat{q}}\times v\qquad \text{ for } v\in {\mathbb {R}}^3\simeq {\mathbb {H}}_{{\mathrm {p}}}. \end{aligned}$$

(35)

By direct inspection, it is straightforward to verify that for every $q\in S^3$

$$\begin{aligned} R_{{\mathrm {\!e}}}(q)v:=qv{\overline{q}}\qquad \text{ for } v\in {\mathbb {R}}^3\simeq {\mathbb {H}}_{{\mathrm {p}}}\end{aligned}$$

(36)

defines a rotation in ${\mathrm {SO}}(3)$. Using (32), this leads to

$$\begin{aligned} R_{{\mathrm {\!e}}}(q)\,=\,(2q_0^2-|q|^2){\mathbb {I}}+2{\widehat{q}}\otimes {\widehat{q}}+2q_0\varepsilon ({\widehat{q}}). \end{aligned}$$

(37)

Plugging in the above definitions, this coincides with Formula (2).

The mapping $R_{{\mathrm {\!e}}}$ thus introduced has the properties

$$\begin{aligned} R_{{\mathrm {\!e}}}(1)={\mathbb {I}},\qquad R_{{\mathrm {\!e}}}({\overline{q}})=R_{{\mathrm {\!e}}}(q)^{ T},\qquad R_{{\mathrm {\!e}}}(pq)=R_{{\mathrm {\!e}}}(p)R_{{\mathrm {\!e}}}(q) \end{aligned}$$

and is therefore an algebra-homomorphism. It is a double cover of ${\mathrm {SO}}(3)$, especially it is non-unique, since

$$\begin{aligned} R_{{\mathrm {\!e}}}(q)=R_{{\mathrm {\!e}}}(-q)\qquad \text{ for } q\in S^3. \end{aligned}$$

(38)

In comparison, the parameterization (1) breaks down for $\alpha _2=\frac{\pi }{2}$, in which case $\alpha _1$ and $\alpha _3$ denote a rotation around the same axis. In summary, both (2) and (1) set up rivaling charts on the manifold ${\mathrm {SO}}(3)$ which have certain disadvantages when used globally.

Eular angles may be more direct to interpret and require one parameter less than quaternions. Nevertheless, the quaternion description is preferable here, as is discussed in Sect. 4.

Formula (2) can be used to interpolate between rotations and allows to introduce a distance in ${\mathrm {SO}}(3)$, see, e.g. [58]. This is a prerequisite to studying surface energies between grains or particles of different orientations [59].

For $x\in {\mathbb {R}}^3$ and a quaternion field $q=q(x)$, the m-th material curvature vector or Darboux vector is given by

$$\begin{aligned} K_{{\mathrm {\!e}}}^m(q):=2{\overline{q}}\partial _{m}q\in {\mathbb {H}}_{{\mathrm {p}}},\qquad 1\le m\le 3. \end{aligned}$$

(39)

The following lemma computes the derivatives of $R_{{\mathrm {\!e}}}(q)$ and $K_{{\mathrm {\!e}}}(q)$ in ${\mathbb {H}}$ with $|q|=1$.

Lemma 1

(Lie Derivatives of $R_{{\mathrm {\!e}}}$ and $K_{{\mathrm {\!e}}}^m$) Let$q=q(x):{\mathbb {R}}^3\rightarrow S^3$and$1\le l,m\le 3$. Then

$$\begin{aligned} \partial _{l}R_{{\mathrm {\!e}}}(q)&= R_{{\mathrm {\!e}}}(q)\varepsilon (K_{{\mathrm {\!e}}}^l(q)), \end{aligned}$$

(40)

$$\begin{aligned} \partial _{l}K_{{\mathrm {\!e}}}^m(q)&= 2{\overline{q}}\big [\partial _{l}\partial _{m}q-\partial _{l}q{\overline{q}}\partial _{m}q\big ]. \end{aligned}$$

(41)

Proof

An elementary proof of (40) can be found in [60], Chapter 11. The following proof is a modification of an argument in [61]. Let $v\in {\mathbb {R}}^3\simeq {\mathbb {H}}_{{\mathrm {p}}}$ and let $w\in {\mathbb {R}}^3$ denote various changing vectors. Then it holds

$$\begin{aligned} \varepsilon (K_{{\mathrm {\!e}}}^l(q))v&= \varepsilon (2{\overline{q}}\partial _{l}q)v&\text{ by } \text{ Eq. } \text{(A.9) }\\&= 2{\overline{q}}\partial _{l}q\times v&\text{ by } \text{ Eq. } \text{(A.5) }\\&= 2{\overline{q}}\partial _{l}q v&\text{ by } \text{ Eq. } \text{(A.2) }\\&= 2\widehat{{\overline{q}}\partial _{l}qv}&\text{ by } \text{ Eq. } \text{(A.1) }\\&= {\overline{q}}\partial _{l}qv-\overline{{\overline{q}}\partial _{l}q v}&\text{ since } w-{\overline{w}} =2{\widehat{w}}\\&= {\overline{q}}\partial _{l}qv+v\partial _{l}{\overline{q}}q&\text{ since } {\overline{v}}=-v\\&= {\overline{q}}(\partial _{l}qv{\overline{q}}+qv\partial _{l}{\overline{q}})q\quad&\text{ since } {\overline{q}}q=|q|^2=1\\&= {\overline{q}}(\partial _{l}(qv{\overline{q}}))q&\text{ since } \partial _{l}v=0\\&= {\overline{q}}(\partial _{l}R_{{\mathrm {\!e}}}(q)v)q&\text{ by } \text{ Eq. } \text{(A.6) }\\&= R_{{\mathrm {\!e}}}(q)^{ T}\partial _{l}R_{{\mathrm {\!e}}}(q)v&\text{ since } (R_{{\mathrm {\!e}}}(q)w)^{ T}={\overline{q}}wq. \end{aligned}$$

As this is true for every $v\in {\mathbb {R}}^3\simeq {\mathbb {H}}_{{\mathrm {p}}}$, this shows

$$\begin{aligned} \varepsilon (K_{{\mathrm {\!e}}}^l(q))=R_{{\mathrm {\!e}}}(q)^{ T}\partial _{l}R_{{\mathrm {\!e}}}(q). \end{aligned}$$

Multiplication with $R_{{\mathrm {\!e}}}(q)$ from the left yields (40).

In order to show (41), multiplying (39) with q from the left yields

$$\begin{aligned} 2\partial _{m}q=qK_{{\mathrm {\!e}}}^m(q). \end{aligned}$$

Consequently,

$$\begin{aligned} 2\partial _{l}\partial _{m}q=\partial _{l}qK_{{\mathrm {\!e}}}^m(q)+q\partial _{l}K_{{\mathrm {\!e}}}^m(q) \end{aligned}$$

or equivalently

$$\begin{aligned} q\partial _{l}K_{{\mathrm {\!e}}}^m(q)=2\partial _{l}\partial _{m}q-\partial _{l}qK_{{\mathrm {\!e}}}^m(q). \end{aligned}$$

Multiplication of this identity with ${\overline{q}}$ from the left leads to

$$\begin{aligned} \partial _{l}K_{{\mathrm {\!e}}}^m(q)=2{\overline{q}}\partial _{l}\partial _{m}q-{\overline{q}}\partial _{l}qK_{{\mathrm {\!e}}}^m(q). \end{aligned}$$

With (39), this shows (41). $\square$

Applying the results of Lemma 1 to $W_{{\mathrm {c}}}$, it holds by Eqns. (40) and (34),

$$\begin{aligned} W_{{\mathrm {c}}}(q)&= \mu _2\sum _{l=1}^3||\partial _{l}R_{{\mathrm {\!e}}}(q)||^2=\mu _2\sum _{l=1}^3||R_{{\mathrm {\!e}}}(q) \varepsilon (K_{{\mathrm {\!e}}}^l(q))||^2 \\&=\mu _2\sum _{l=1}^3||\varepsilon (K_{{\mathrm {\!e}}}^l(q))||^2 \\&= 2\mu _2\sum _{l=1}^3\Big [({K_{{\mathrm {\!e}}}}_1^l(q))^2+({K_{{\mathrm {\!e}}}}_2^l(q))^2 +({K_{{\mathrm {\!e}}}}_3^l(q))^2\Big ] \\&=2\mu _2\sum _{l=1}^3|K_{{\mathrm {\!e}}}^l(q)|^2. \end{aligned}$$

(42)

For the first derivative, using (39) and (41), this results in

$$\begin{aligned} \partial _{m}W_{{\mathrm {c}}}(q)&= 4\mu _2\sum _{l=1}^3\widehat{\partial _{m}K_{{\mathrm {\!e}}}^l(q)}\cdot \widehat{K_{{\mathrm {\!e}}}^l(q)} \\&= 16\mu _2\sum _{l=1}^3\big [{\overline{q}}(\partial _{m}\partial _{l}q-\partial _{m}q{\overline{q}}\partial _{l}q)\big ]\cdot \big [ {\overline{q}}\partial _{l}q\big ]. \end{aligned}$$

(43)