Implementation of numerical integration schemes for the simulation of magnetic SMA constitutive response

Magnetic shape memory alloys (MSMAs), often also referred to as ferromagnetic shape memory alloys (FSMAs), exhibit a complex constitutive behavior that combines the nonlinear, dissipative and thermomechanically coupled features of classical shape memory response with strong magneto-mechanical coupling. The primary underlying mechanism causing the magnetic shape memory effect observed in the form of typical butterfly strain–magnetic field hysteresis curves is the magnetic-field-induced reorientation of martensitic variants. This phenomenon is exhibited by alloys that simultaneously feature high magnetic anisotropy and high twin boundary mobility in their ferromagnetic martensite phase. The associated magnetization hysteresis curves, however, suggest that, particularly in some loading regimes, the additional mechanisms of local magnetization rotation and magnetic domain wall motion strongly influence the macroscopic MSMA behavior [2].

In addition to experimental work following the first conclusive report of the magnetic shape memory effect by Ullakko et al [3], there has also been considerable research activity on the modeling of the MSMA response. For a detailed discussion of early model development (e.g. O'Handley [4], James and Wuttig [5], James and Hane [6], Likhachev and Ullakko [7], Hirsinger and Lexcellent [8], DeSimone [9], Kiefer and Lagoudas [1]) the reader is referred to [1, 10], the review article by Kiang and Tong [11] and the references therein. More recently, extended MSMA modeling concepts have been proposed. Kiefer and Lagoudas [2] presented a model that captures magneto-pseudo-elasticity and magneto-pseudo-plasticity in addition to the conventional magnetic shape memory effect. Seelecke et al [12] suggested a mesoscale free energy model. Modeling and simulation of shape memory micro-actuators was presented by Krevet and Kohl [13]. Stefanelli et al [14–16] suggested a mathematically well-founded model capable of capturing three-dimensional response. Finally, multi-scale approaches for the modeling and simulation of active materials have lately received much attention. In the context of MSMA modeling, we refer to the phase-field-based models suggested by Landis [17], Jin [18] and Nestler et al [19] and the numerical homogenization approach proposed by Conti, Lenz and Rumpf for the modeling of magneto-active composites containing MSMA particles [20].

The structure of this paper is as follows. In section 2, the key equations of the considered phenomenological MSMA model are reviewed. In section 3.2, we establish two fundamentally different numerical integration schemes for these constitutive equations. Simulation results obtained from implementations of the proposed algorithmic material models and their verification are presented in section 4. The paper concludes with a summary and an outlook on future work.

For the purpose of clarity and self-containment, the continuous formulation of the internal variable MSMA model is briefly summarized in this section. One of its central building blocks is the Gibbs free energy function, which characterizes the energy storage at a material point and the free space it occupies. This function is constructed as a weighted average of contributions from all possible martensitic variant/magnetic domain configurations. The motivation for this approach stems from crystallographically and magnetically compatible micro-scale configurations of coexisting twin bands and magnetic domains that have been observed experimentally in the ferromagnetic martensite phases of MSMAs [21, 22]. A two-dimensional schematic of such microstructure is displayed in figure 1. In the macroscopic modeling approach followed here, the microstructure is not geometrically fully resolved, but rather the effective influence of its evolution on the macroscopic response is taken into account through internal state variables.

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In order to limit complexity for the derivation of numerical integration algorithms following below, we consider a basic version of the MSMA model proposed in [1, 2] by assuming fixed local magnetization directions (θ_i = 0) and fixed domain volume fraction (α = const.). This reduced model has been shown to yield reasonable predictions of the magnetic-field-induced strain response for typical loading cases [1], while the magnetization response is less accurately captured. Under these assumptions, the particular form of the Gibbs free energy, cf equation (28) in [1], is given by

$$\begin{eqnarray} \displaystyle \fl g(\boldsymbol{\sigma },\boldsymbol{h},{\boldsymbol{\varepsilon }}^{r},\boldsymbol{m},\xi )&=&\displaystyle -\frac{1}{2}\boldsymbol{\sigma }:\mathcal{S}(\xi ):\boldsymbol{\sigma }-\boldsymbol{\sigma }:{\boldsymbol{\varepsilon }}^{r}\\ \displaystyle &&\displaystyle -~\frac{1}{2}\boldsymbol{h}\boldsymbol{\cdot }\boldsymbol{\mu }(\xi )\boldsymbol{\cdot }\boldsymbol{h}-\boldsymbol{h}\boldsymbol{\cdot }\boldsymbol{m}+{f}^{\xi }(\xi ). \end{eqnarray} \tag{ 1 }$$

Motivated by recent considerations concerning the formulation of thermodynamically consistent models in dissipative electromagneto-mechanics [23, 24], a slightly revised interpretation of this expression is offered here. The first and third terms in equation (1) are the standard energy storage contributions that are quadratic in the stress tensor σ and the magnetic field strength h and yield a linear mechanical and, respectively, magnetic response in the elastic regime. If the anisotropy of the martensitic variants is fully taken into account, the effective elastic compliance tensor (ξ): = (1 − ξ)^V₁ + ξ^V₂ = ^V₁ + ξΔ, with Δ: = ^V₂ − ^V₁, and the effective permeability tensor μ(ξ): = (1 − ξ)μ^V₁ + ξμ^V₂ = μ^V₁ + ξΔμ, with Δμ: = μ^V₂ − μ^V₁, depend on the variant volume fraction ξ³. The second and fourth terms enter via a Legendre–Fenchel transformation of the convex free energy and an assumed decomposition into elastic (reversible) and inelastic (remanent) contributions⁴. The term  − h⋅m is usually interpreted as the Zeeman or external field energy. Considering the previous argument and the local nature of (1), this interpretation must at least be handled with care in this context. Finally, f^ξ(ξ) is a scalar-valued hardening function, which accounts for additional energy storage due to the interaction of the variants during reorientation.

Thermodynamically consistent constitutive relations for the total infinitesimal strain tensor ε and the magnetic induction vector b follow from (1) through the standard Coleman and Noll procedure as

$$\begin{eqnarray} &&\begin{array}{@{}r@{}} \displaystyle \boldsymbol{\varepsilon }=-{\partial }_{\boldsymbol{\sigma }}g=\mathcal{S}:\boldsymbol{\sigma }+{\boldsymbol{\varepsilon }}^{r},\hspace{1em}\boldsymbol{b}=-{\partial }_{\boldsymbol{h}}g=\boldsymbol{\mu }\boldsymbol{\cdot }\boldsymbol{h}+\boldsymbol{m}. \end{array} \end{eqnarray} \tag{ 3 }$$

Note that these expressions are consistent with the additive decomposition assumption (cf the discussion in footnotes). Using relations (3), the reduced Clausius–Planck inequality may be expressed as

$$\begin{eqnarray} &&\mathcal{D}=-{\partial }_{\boldsymbol{I}}g\boldsymbol{\cdot }\dot {\boldsymbol{I}}=\boldsymbol{\sigma }:{\dot {\boldsymbol{\varepsilon }}}^{r}+\boldsymbol{h}\cdot \dot {\boldsymbol{m}}-{\partial }_{\xi }g\cdot \dot {\xi }\geq 0, \end{eqnarray} \tag{ 4 }$$

where I = [ε^r,m,ξ]^T denotes the vector of internal state variables⁵.

Motivated by the crystallographic considerations discussed above, it is further assumed, that the evolution of the reorientation strains and remanent magnetization is proportional to the evolution of the martensitic variant volume fraction, i.e.

$$\begin{eqnarray} &&{\dot {\boldsymbol{\varepsilon }}}^{r}=\boldsymbol{\Lambda }\dot {\xi }\tqs \text{and}\tqs \dot {\boldsymbol{m}}=\boldsymbol{r}\dot {\xi }, \end{eqnarray} \tag{ 5 }$$

where the evolution direction tensors Λ and r are defined as⁶

$$\begin{eqnarray} \displaystyle \boldsymbol{\Lambda }:={\varepsilon }^{r,\max }({\boldsymbol{e}}_{x}\otimes {\boldsymbol{e}}_{x}-{\boldsymbol{e}}_{y}\otimes {\boldsymbol{e}}_{y})\tqs \text{and}&&\displaystyle \\ \displaystyle \hspace{-3.5pc} \boldsymbol{r}:=(2 \alpha -1){\mu }_{0}{M}^{\mathrm{sat}}({\boldsymbol{e}}_{y}-{\boldsymbol{e}}_{x}).&&\displaystyle \end{eqnarray} \tag{ 6 }$$

The Cartesian base vectors e_i are assumed to coincide with crystallographic directions as shown in figure 1. For response predictions with the basic model and standard two-dimensional loading cases, α is a constant and takes the values of either one or zero, so that in an initial compressive stress-favored single variant state magnetic saturation is achieved in either positive or negative x direction, depending on the applied field direction. Using (5), the Clausius–Planck inequality (4) may further be reduced to

$$\begin{eqnarray} &&\mathcal{D}=(\boldsymbol{\sigma }:\boldsymbol{\Lambda }+\boldsymbol{h}\boldsymbol{\cdot }\boldsymbol{r}-{\partial }_{\xi }g)\dot {\xi }=:{\pi }^{\xi }\dot {\xi }\geq 0. \end{eqnarray} \tag{ 7 }$$

Based on (1), the driving force for variant reorientation takes the specific form

$$\begin{eqnarray} &&{\pi }^{\xi }(\boldsymbol{\sigma },\boldsymbol{h},\xi )=\boldsymbol{\sigma }:\boldsymbol{\Lambda }+\boldsymbol{h}\boldsymbol{\cdot }\boldsymbol{r}+\frac{1}{2}\boldsymbol{\sigma }:\Delta \mathcal{S}:\boldsymbol{\sigma }-{\partial }_{\xi }{f}^{\xi }. \end{eqnarray} \tag{ 8 }$$

Critical values for reorientation are determined by the reorientation yield function

$$\begin{eqnarray} &&{\Phi }^{\xi }(\boldsymbol{\sigma },\boldsymbol{h},\xi )=\vert {\pi }^{\xi } \vert -{Y}^{\xi }, \end{eqnarray} \tag{ 9 }$$

where Y^ξ > 0 is the material parameter quantifying the critical driving force. Because Y^ξ is directly related to the dissipation of the variant reorientation process via (7)–(9), there also exists an immediate correlation to the size of the magnetic-field-induced strain hysteresis. Note, further, that since in our approach the driving force (8) is an energetic measure, the reorientation process can be driven by stresses as well as magnetic fields.

The independence of the evolution directions on the stress state in (5), in combination with the dissipation expression (7), motivates the derivation of an associated evolution equation for the variant volume fraction based on the maximum dissipation principle:

$$\begin{eqnarray} &&\mathop{\mathop { \mathrm{minimize}}}\limits _{\mathrm{s.t.}\hspace{0.167em} {\pi }^{\xi }\in ~\mathbb{E}}-\mathcal{D}=-{\pi }^{\xi }\dot {\xi }, \end{eqnarray} \tag{ 10 }$$

where $\mathbb{E}:=\left\{{\pi }^{\xi }\in \mathbb{R}\mid {\Phi }^{\xi }({\pi }^{\xi })\leq 0\right\}$ defines the set of admissible driving forces. The constrained minimization problem (10) may be reformulated in terms of the dual problem:

$$\begin{eqnarray} &&\mathcal{L}({\pi }^{\xi },\lambda ):=-{\pi }^{\xi }\dot {\xi }+\lambda {\Phi }^{\xi }({\pi }^{\xi })\rightarrow \mathrm{stat.}, \end{eqnarray} \tag{ 11 }$$

where represents the Lagrange function and λ a Lagrange multiplier. The necessary Karush–Kuhn–Tucker optimality conditions associated with (11) are given by

$$\begin{eqnarray} \displaystyle \begin{array}{@{}rclrcl@{}} \displaystyle {\partial }_{{\pi }^{\xi }}\mathcal{L}&\displaystyle =&\displaystyle 0,\tqs &\displaystyle {\partial }_{\lambda }\mathcal{L}&\displaystyle \leq &\displaystyle 0,\\ \displaystyle \lambda &\displaystyle \geq &\displaystyle 0,\tqs &\displaystyle \lambda {\partial }_{\lambda }\mathcal{L}&\displaystyle =&\displaystyle 0. \end{array}&&\displaystyle \end{eqnarray} \tag{ 12 }$$

More specifically, this yields the associated evolution equation:

$$\begin{eqnarray} &&\dot {\xi }=\lambda {\partial }_{{\pi }^{\xi }}{\Phi }^{\xi }({\pi }^{\xi })=\mathrm{sign}({\pi }^{\xi })\lambda \end{eqnarray} \tag{ 13 }$$

and the Karush–Kuhn–Tucker conditions:

$$\begin{eqnarray} &&{\Phi }^{\xi }({\pi }^{\xi })\leq 0,\tqs \lambda \geq 0,\tqs \lambda {\Phi }^{\xi }({\pi }^{\xi })=0, \end{eqnarray} \tag{ 14 }$$

which conclude the set of constitutive equations.

In this section, algorithmic counterparts to the continuous internal variable MSMA model reviewed above shall be derived. In particular, two different algorithms will be formulated, namely a general return-type algorithm and a Fischer–Burmeister complementarity function-based algorithm. The former utilizes a classical predictor–corrector scheme to satisfy the Karush–Kuhn–Tucker conditions and is based on the integration of the tensor-valued evolution equation for the reorientation strains. The latter solves a complementary nonlinear unconstrained optimization problem via a standard Newton–Raphson scheme and is based on the implicit integration of the scalar-valued evolution equation for the martensitic variant volume fraction.

3.1. General return mapping algorithm

Here, we conceptually follow a general return mapping approach that is now well established for computational plasticity (see, e.g., Simo and Hughes [26]). Lagoudas et al successfully generalized and implemented such algorithms for the modeling of conventional shape memory response [27–29]. The intent of this section is to demonstrate that an analogous algorithmic format can be established for the considered MSMA model. The main difference is the fact that the coupled problem is magneto-mechanical in nature and involves the vector-valued magnetic field, in contrast to the previously considered thermomechanical problem involving the scalar-valued temperature as an additional independent state variable.

The derivation of the algorithmic material model begins with the integration of the evolution equations (5) in the time increment [t_n,t_n+1] according to

$$\begin{eqnarray} \begin{array}{@{}rc@{}} \displaystyle &\displaystyle {\boldsymbol{\varepsilon }}_{n+1}^{r}={\boldsymbol{\varepsilon }}_{n}^{r}+({\xi }_{n+1}-{\xi }_{n})\boldsymbol{\Lambda }\hspace{1em}\text{and}\\ \displaystyle &\displaystyle \hspace{-1.6pc} {\boldsymbol{m}}_{n+1}={\boldsymbol{m}}_{n}+({\xi }_{n+1}-{\xi }_{n})\boldsymbol{r}. \end{array} \end{eqnarray} \tag{ 15 }$$

Note again that the evolution directions have been assumed to be known constants. Closest point projection and convex cutting plane approaches [26, 27] practically coincide in this special case.

Using equation (15)₁, in combination with the time-discrete form of the consistency condition ${\dot {\Phi }}^{\xi }=0$, we obtain the nonlinear set of coupled algebraic equations:

$$\begin{eqnarray} \begin{array}{@{}rc@{}} \displaystyle &\displaystyle \hspace{-2.2pc} {\Phi }_{n+1}^{\xi }:={\Phi }^{\xi }({\boldsymbol{\sigma }}_{n+1},{\boldsymbol{h}}_{n+1},{\xi }_{n+1})=0,\\ \displaystyle &\displaystyle {\boldsymbol{R}}_{n+1}^{r}:=-{\boldsymbol{\varepsilon }}_{n+1}^{r}+{\boldsymbol{\varepsilon }}_{n}^{r}+({\xi }_{n+1}-{\xi }_{n})\boldsymbol{\Lambda }=\mathbf{0}. \end{array} \end{eqnarray} \tag{ 16 }$$

These equations can be solved in each time increment using the Newton–Raphson method, which requires the successive solution of the linearized relations:

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl \mathrm{Lin}~{\Phi }_{n+1}^{\xi (k)}:={\Phi }_{n+1}^{\xi (k)} + {\partial }_{\boldsymbol{\sigma }}{\Phi }_{n+1}^{\xi (k)}:\Delta {\boldsymbol{\sigma }}_{n+1}^{(k)} + {\partial }_{\xi }{\Phi }_{n+1}^{\xi (k)}\Delta {\xi }_{n+1}^{(k)}=0, \\ \displaystyle \fl \mathrm{Lin}~{\boldsymbol{R}}_{n+1}^{r(k)}:={\boldsymbol{R}}_{n+1}^{r(k)}-\Delta {\boldsymbol{\varepsilon }}_{n+1}^{r(k)}+\boldsymbol{\Lambda }\Delta {\xi }_{n+1}^{(k)}=\mathbf{0}, \end{array} \end{eqnarray} \tag{ 17 }$$

where Δ(⋅)^(k): = (⋅)^(k+1) − (⋅)^(k) denotes the increment of the respective quantity during the kth iteration. The algorithm is considered to be strain- and magnetic-field-driven, such that ε_n+1 and h_n+1 are prescribed and thus do not vary during the Newton–Raphson iteration at fixed time t_n+1.

On the other hand, the incremental version of the stress–strain relation (3)₁ is

$$\begin{eqnarray} \displaystyle \Delta {\boldsymbol{\varepsilon }}_{n+1}^{(k)}=\Delta {\mathcal{S}}_{n+1}^{(k)}:{\boldsymbol{\sigma }}_{n+1}^{(k)}+{\mathcal{S}}_{n+1}^{(k)}:\Delta {\boldsymbol{\sigma }}_{n+1}^{(k)}+\Delta {\boldsymbol{\varepsilon}}_{n+1}^{r(k)}=\mathbf{0}.&&\displaystyle \\ \displaystyle &&\displaystyle \end{eqnarray} \tag{ 18 }$$

Further recognizing that $\Delta {\mathcal{S}}_{n+1}^{(k)}=\Delta \mathcal{S}\Delta {\xi }_{n+1}^{(k)}$, with Δ: = ^V₂ − ^V₁, equation (18) can be solved for the increment of the reorientation strain, yielding

$$\begin{eqnarray} &&\Delta {\boldsymbol{\varepsilon }}_{n+1}^{r \hspace{0.167em} (k)}=-{\mathcal{S}}_{n+1}^{(k)}:\Delta {\boldsymbol{\sigma }}_{n+1}^{(k)}-\Delta \mathcal{S}:{\boldsymbol{\sigma }}_{n+1}^{(k)}\Delta {\xi }_{n+1}^{(k)}. \end{eqnarray} \tag{ 19 }$$

Then, eliminating $\Delta {\boldsymbol{\varepsilon }}_{n+1}^{r \hspace{0.167em} (k)}$ by equating (19) and (17)₂, the increment of the stress tensor is derived as

$$\begin{eqnarray} \displaystyle \fl \Delta {\boldsymbol{\sigma }}_{n+1}^{(k)}=\left\{\begin{array}{@{}ll@{}} \displaystyle {\mathcal{S}}_{n+1}^{(k)^{-1}}:[-{\boldsymbol{R}}_{n+1}^{r(k)}-\Delta {\xi }_{n+1}^{(k)}{\partial }_{\boldsymbol{\sigma }}{\Phi }_{n+1}^{\xi (k)}],&\displaystyle \quad \dot {\xi }\gt 0 \\ \displaystyle {\mathcal{S}}_{n+1}^{(k)^{-1}}:[-{\boldsymbol{R}}_{n+1}^{r(k)}+\Delta {\xi }_{n+1}^{(k)}{\partial }_{\boldsymbol{\sigma }}{\Phi }_{n+1}^{\xi (k)}],&\displaystyle \quad \dot {\xi }\lt 0 \end{array}\right.&&\displaystyle \\ \displaystyle &&\displaystyle \end{eqnarray} \tag{ 20 }$$

where we have recognized the relation ${\partial }_{\boldsymbol{\sigma }}{\Phi }_{n+1}^{\xi (k)}=\pm (\boldsymbol{\Lambda }+\Delta \mathcal{S}:{\boldsymbol{\sigma }}_{n+1}^{(k)})$ that holds through (8) and (9) and that the sign  ±  coincides with the respective sign of $\dot {\xi }$.

To compute increments of the volume fraction, relation (20) is substituted into (17)₁. After some straightforward manipulations we find

$$\begin{eqnarray} \displaystyle \fl \Delta {\xi }_{n+1}^{(k)}=\left\{\begin{array}{@{}ll@{}} \displaystyle \frac{{\Phi }_{n+1}^{\xi (k)} - {\partial }_{\boldsymbol{\sigma }}{\Phi }_{n+1}^{\xi (k)}:{\mathcal{S}}_{n+1}^{(k)^{-1}}:{\boldsymbol{R}}_{n+1}^{r(k)}}{{\partial }_{\boldsymbol{\sigma }}{\Phi }_{n+1}^{\xi (k)}:{\mathcal{S}}_{n+1}^{(k)-1}:{\partial }_{\boldsymbol{\sigma }}{\Phi }_{n+1}^{\xi (k)} - {\partial }_{\xi }{\Phi }_{n+1}^{\xi (k)}},&\displaystyle \,\, \dot {\xi }\gt 0 \\ \displaystyle -\frac{{\Phi }_{n+1}^{\xi (k)} - {\partial }_{\boldsymbol{\sigma }}{\Phi }_{n+1}^{\xi (k)}:{\mathcal{S}}_{n+1}^{(k)^{-1}}:{\boldsymbol{R}}_{n+1}^{r(k)}}{{\partial }_{\boldsymbol{\sigma }}{\Phi }_{n+1}^{\xi (k)}:{\mathcal{S}}_{n+1}^{(k)-1}:{\partial }_{\boldsymbol{\sigma }}{\Phi }_{n+1}^{\xi (k)} + {\partial }_{\xi }{\Phi }_{n+1}^{\xi (k)}},&\displaystyle \,\, \dot {\xi }\lt 0 . \end{array}\right.&&\displaystyle \\ \displaystyle &&\displaystyle \end{eqnarray} \tag{ 21 }$$

The general return algorithm is summarized in table 1. In this general notation, the algorithmic model is nearly identical to the respective algorithm for conventional shape memory behavior presented by Lagoudas et al in [27, 28]. The difference, of course, is contained in the details of the specific expressions, such as the driving force (19) and its partial derivatives. It must also be pointed out that special coding effort is required to ensure that ξ_n+1, and strictly speaking also its increments, may only range between zero and one. For details of different approaches to solving this problem, the reader is again referred to the discussions in [27, 28] or [30].

Table 1.  General return mapping algorithm for magnetic SMA response.

(i) Initialization: set k = 0, ${\xi }_{n+1}^{(0)}={\xi }_{n}$, $\qquad {\boldsymbol{\varepsilon}}_{n+1}^{r(0)}={\boldsymbol{\varepsilon}}_{n}^{r}$, ${\boldsymbol{m}}_{n+1}^{(0)}={\boldsymbol{m}}_{n}$

(ii) Trial step:$\hspace{0.167em} \hspace{0.167em} \hspace{0.167em} {\boldsymbol{\sigma}}_{n+1}^{(k)}={\mathcal{S}}_{n+1}^{(k)^{-1}}:[{\boldsymbol{\varepsilon}}_{n+1}-{\boldsymbol{\varepsilon}}_{n+1}^{r(k)}]$

$\qquad \qquad \qquad {\boldsymbol{b}}_{n+1}^{(k)}={\mu }_{0}{\boldsymbol{h}}_{n+1}+{\boldsymbol{m}}_{n+1}^{(k)}$

$\qquad \qquad \qquad {\Phi }_{n+1}^{(k)}=\Phi [{\boldsymbol{\sigma}}_{n+1}^{(k)},{\boldsymbol{h}}_{n+1},{\xi }_{n+1}^{(k)}]$ (see equations (8) and (9))

$\qquad \qquad \qquad {\boldsymbol{R}}_{n+1}^{r(k)}=-{\boldsymbol{\varepsilon}}_{n+1}^{r(k)}+{\boldsymbol{\varepsilon}}_{n}^{r}+\boldsymbol{\Lambda},[{\xi }_{n+1}^{(k)}-{\xi }_{n}]$

(iii) Reorientation and convergence check:

IF $\vert {\Phi }_{n+1}^{(k)}\vert \leq \mathrm{tol}$ AND $\mid \mid {\boldsymbol{R}}_{n+1}^{r(k)}\parallel \leq \mathrm{tol}$

THEN store quantities at tn+1 and EXIT to global iteration

ELSE continue to 4.

(iv) Reorientation correction step:

Compute the increments $\Delta {\xi }_{n+1}^{(k)}$, $\Delta {\boldsymbol{\sigma}}_{n+1}^{(k)}$ and $\Delta {\boldsymbol{\varepsilon}}_{n+1}^{r(k)}$ via equations (19)–(21).

(v) Update internal state variables:

$\qquad \qquad \qquad {\xi }_{n+1}^{(k+1)}={\xi }_{n+1}^{(k)}+\Delta {\xi }_{n+1}^{(k)}$

$\qquad \qquad \qquad {\boldsymbol{\varepsilon}}_{n+1}^{r(k+1)}={\boldsymbol{\varepsilon}}_{n+1}^{r(k)}+\Delta {\boldsymbol{\varepsilon}}_{n+1}^{r(k)}$

$\qquad \qquad \qquad {\boldsymbol{m}}_{n+1}^{(k+1)}={\boldsymbol{m}}_{n}+({\xi }_{n+1}^{(k+1)}-{\xi }_{n})\boldsymbol{r}$

Set k = k + 1 and return to 2.

In addition to current stresses and magnetic inductions, the algorithmic material model also provides the current tangent to be used, for example, in the iterative solution of nonlinear global boundary value problems via the finite element method. Components of the tangent appear in the differential form of the constitutive relations as

$$\begin{eqnarray} \displaystyle \begin{array}{@{}rcl@{}} \displaystyle d{\boldsymbol{\sigma}}_{n+1}&\displaystyle =&\displaystyle {\mathbb{C}}_{n+1}^{e r}:d{\boldsymbol{\varepsilon}}_{n+1}+{\mathbb{h} }_{n+1}\cdot d{\boldsymbol{h}}_{n+1} \\ \displaystyle -d{\boldsymbol{b}}_{n+1}&\displaystyle =&\displaystyle {\mathbb{h} }_{n+1}^{T}:d{\boldsymbol{\varepsilon}}_{n+1}+{\boldsymbol{\beta }}_{n+1}\cdot d{\boldsymbol{h}}_{n+1}. \end{array} &&\displaystyle \end{eqnarray} \tag{ 22 }$$

It is observed that in the reorientation regime, where h_n+1 ≠ 0, (22) has the expected piezo-magnetic structure, even though the respective standard coupling term was not introduced in the free energy function. Note also that treating the negative magnetic induction as a dependent state variable allows working with a symmetric tangent [23].

In summary, the current stresses, magnetic induction and consistent tangent matrix are updated by the algorithmic material model according to the relations

$$\begin{eqnarray} &&\displaystyle \begin{array}{@{}rcl@{}} \displaystyle \left[\begin{array}{@{}c@{}} \displaystyle {\boldsymbol{\sigma}}_{n+1}\\ \displaystyle -{\boldsymbol{b}}_{n+1} \end{array}\right]&\displaystyle =&\displaystyle \left[\begin{array}{@{}c@{}} \displaystyle {\mathcal{S}}_{n+1}^{-1}:({\boldsymbol{\varepsilon}}_{n+1}-{\boldsymbol{\varepsilon}}_{n+1}^{r})\\ \displaystyle -{\mu }_{0}{\boldsymbol{h}}_{n+1}-{\boldsymbol{m}}_{n+1} \end{array}\right], \\ \displaystyle {\mathbb{C}}_{n+1}&\displaystyle :=&\displaystyle \left[\begin{array}{@{}cc@{}} \displaystyle {\mathbb{C}}_{n+1}^{e r}&\displaystyle {\mathbb{h} }_{n+1}\\ \displaystyle {\mathbb{h} }_{n+1}^{T}&\displaystyle {\boldsymbol{\beta }}_{n+1} \end{array}\right]. \end{array}&&\displaystyle \end{eqnarray} \tag{ 23 }$$

Following standard procedure (see [27]), the consistent algorithmic tangent matrix C_n+1, for forward reorientation $(\dot {\xi }\gt 0)$, can be shown to consist of the submatrices

$$\begin{eqnarray} \displaystyle \begin{array}{@{}rcl@{}} \displaystyle {\mathbb{C}}_{n+1}^{\mathrm{er}}&\displaystyle :=&\displaystyle {\mathcal{S}}_{n+1}^{-1}-{\mathbb{A}}_{n+1}\otimes {\mathbb{A}}_{n+1},\\ \displaystyle {\mathbb{h} }_{n+1}&\displaystyle :=&\displaystyle -{\mathbb{A}}_{n+1}\otimes {\mathbb{b} }_{n+1},\\ \displaystyle {\boldsymbol{\beta }}_{n+1}&\displaystyle :=&\displaystyle -({\mu }_{0}\mathbf{1}+{\mathbb{b} }_{n+1}\otimes {\mathbb{b} }_{n+1}), \end{array}&&\displaystyle \end{eqnarray} \tag{ 24 }$$

with the abbreviations

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl {\mathbb{A}}_{n+1}:=\frac{{\mathcal{S}}_{n+1}^{-1}:{\partial }_{\boldsymbol{\sigma}}{\Phi }_{n+1}^{\xi }}{c}\hspace{0.167em} \tqs {\mathbb{b} }_{n+1}:=\frac{{\partial }_{\boldsymbol{h}}{\Phi }_{n+1}^{\xi }}{c}, \\ \displaystyle c:=\sqrt{{\partial }_{\boldsymbol{\sigma}}{\Phi }_{n+1}^{\xi }:{\mathcal{S}}_{n+1}^{-1}:{\partial }_{\boldsymbol{\sigma}}{\Phi }_{n+1}^{\xi }-{\partial }_{\xi }{\Phi }_{n+1}^{\xi }}. \end{array} \end{eqnarray} \tag{ 25 }$$

For the reverse reorientation process, similar expressions follow in an analogous manner.

3.2. Fischer–Burmeister complementarity function-based algorithm

The main purpose of this alternative approach to integrating the MSMA constitutive equations is to reduce the complexity, and thus increase the efficiency, of the algorithmic treatment. This is achieved by (i) eliminating the need for a predictor–corrector-type scheme, (ii) automatically constraining the range of the variant volume fraction and (iii) utilizing a scalar-valued, rather than a tensorial, evolution equation. Aspects (i) and (ii) can be realized by employing Fischer–Burmeister complementarity functions [31]. These have successfully been used by Schmidt–Baldassari in the context of rate-independent multi-surface plasticity [32] and by Bartel et al for martensitic phase transformation modeling [33–35]. Recall that (iii) is possible for this particular model due to the assumed independence of the evolution directions on the stress state.

The starting point of the algorithmic model derivation is again the (implicit) integration of the evolution equations for the internal state variables, in this case (13), i.e.

$$\begin{eqnarray} &&{\xi }_{n+1}={\xi }_{n}+\mathrm{sign}({\pi }_{n+1}^{\xi })\hspace{0.167em} {\gamma }_{n+1}\hspace{0.167em} \end{eqnarray} \tag{ 26 }$$

where γ_n+1: = Δtλ_n+1 is the reorientation increment. Updates of the reorientation strains and the remanent magnetization are now computed in a post-processing step by means of equation (15)₂, after a converged solution for ξ_n+1 has been obtained.

The basic idea of the Fischer–Burmeister approach is to convert the inequality-constrained optimization problem (11) into a dual unconstrained problem in terms of the complementarity function:

$$\begin{eqnarray} &&\sqrt{({\Phi }_{n+1}^{\xi })^{2}+{\gamma }_{n+1}^{2}}+{\Phi }_{n+1}^{\xi }-{\gamma }_{n+1}=0 \hspace{0.167em} \end{eqnarray} \tag{ 27 }$$

which a priori satisfies the Karush–Kuhn–Tucker conditions (14). It is then straightforward to implement additional constraints, e.g. to restrict the variant volume fraction to the range 0 ≤ ξ_n+1 ≤ 1. To this end, we consider the enhanced Gibbs free energy function:

$$\begin{eqnarray} &&{g}_{n+1}^{\mathrm{enh}}:={g}_{n+1}+{\Gamma }_{n+1}{r}_{n+1}+{\bar {\Gamma }}_{n+1}{\bar {r}}_{n+1}, \end{eqnarray} \tag{ 28 }$$

with the additional constraints given by r_n+1: =− ξ_n+1 ≤ 0, ${\bar {r}}_{n+1}:={\xi }_{n+1}-1\leq 0$ and Γ_n+1, ${\bar {\Gamma }}_{n+1}$ acting as Lagrange multipliers. This results in the enhanced variant reorientation driving force expression:

$$\begin{eqnarray} \displaystyle \fl {\pi }_{n+1}^{\xi ,\mathrm{enh}}&:=&\displaystyle {\boldsymbol{\sigma}}_{n+1}:\boldsymbol{\Lambda}+{\boldsymbol{h}}_{n+1}\boldsymbol{\cdot }\boldsymbol{r}-{\partial }_{\xi }{g}_{n+1}^{\mathrm{enh}}\\ \displaystyle &=&\displaystyle {\boldsymbol{\sigma}}_{n+1}:\boldsymbol{\Lambda}+{\boldsymbol{h}}_{n+1}\boldsymbol{\cdot }\boldsymbol{r}-{\partial }_{\xi }{f}_{n+1}^{\xi }+{\Gamma }_{n+1}-{\bar {\Gamma }}_{n+1}. \end{eqnarray} \tag{ 29 }$$

The time-discrete optimization problem then takes the specific form:

$$\begin{eqnarray} \displaystyle \fl {\boldsymbol{R}}_{n+1}({\boldsymbol{f}}_{n+1})&&\displaystyle \\ &&\displaystyle :=\left[\begin{array}{@{}c@{}} \displaystyle {\xi }_{n+1}-{\xi }_{n}-\mathrm{sign}({\pi }_{n+1}^{\xi ,\mathrm{enh}})\hspace{0.167em} {\gamma }_{n+1} \\ \displaystyle \sqrt{({\Phi }_{n+1}^{\xi ,\mathrm{enh}})^{2}+{\gamma }_{n+1}^{2}}+{\Phi }_{n+1}^{\xi ,\mathrm{enh}}-{\gamma }_{n+1} \\ \displaystyle \sqrt{{r}_{n+1}^{2}+{\Gamma }_{n+1}^{2}}+{r}_{n+1}-{\Gamma }_{n+1} \\ \displaystyle \sqrt{{\bar {r}}_{n+1}^{2}+{\bar {\Gamma }}_{n+1}^{2}}+{\bar {r}}_{n+1}-{\bar {\Gamma }}_{n+1} \end{array}\right]=\mathbf{0},&&\displaystyle \end{eqnarray} \tag{ 30 }$$

with ${\boldsymbol{f}}_{n+1}:=[{\xi }_{n+1},{\gamma }_{n+1},{\Gamma }_{n+1},{\bar {\Gamma }}_{n+1}]^{T}$. This nonlinear relation may be solved via the Newton–Raphson method, which results in the standard update relation:

$$\begin{eqnarray} &&{\boldsymbol{f}}_{n+1}^{(k+1)}={\boldsymbol{f}}_{n+1}^{(k)}-[{d}_{\boldsymbol{f}}{\boldsymbol{R}}_{n+1}^{(k)}]^{-1}{\boldsymbol{R}}_{n+1}^{(k)}. \end{eqnarray} \tag{ 31 }$$

Fischer–Burmeister functions are known to be numerically sensitive, in particular due to the fact that their derivatives are non-smooth. However, their application to the problem at hand leads to a well-behaved and robust algorithmic scheme. We also want to remark that an analogous algorithm could have been constructed on the basis of the evolution equation for the reorientation strains (15), by simply replacing the first entry of the residual vector (30)₁ accordingly. Such an approach is needed in cases where the evolution directions depend on the stress state, which is, in fact, the more common case.

For the second algorithm, the tangent moduli are approximated numerically. This is achieved by evaluating the constitutive relations at a slightly perturbed strain and magnetic field state and relating increments in the resulting stresses and magnetic induction to the magnitude of the perturbation (see, e.g., Miehe [36]).

For verification purposes, numerical response simulations shall now be compared to the corresponding closed-form analytical expressions previously reported in [1] for typical loading cases. To this end, the proposed integration schemes are tested by means of a constitutive driver algorithm, following the same conceptual idea proposed in [37, 23].

In particular, we consider cyclic magnetic loading h = h_ye_y, with |h_y| ≤ h_max, under constant uniaxial stress bias σ = σ_xxe_x⊗e_x, with σ_xx = σ ≤ 0. In this most common loading case, reorientation occurs from a stress-favored single variant configuration to a magnetic-field-favored variant state. For the proposed deformation and magnetic-field-driven algorithmic material models, which compute current stresses according to ${\boldsymbol{\sigma}}_{n+1}=\hat {\boldsymbol{\sigma}}({\boldsymbol{\varepsilon}}_{n+1},{\boldsymbol{h}}_{n+1};{\boldsymbol{I}}_{n})$, the constitutive driver algorithm must iteratively determine current strain components in such a manner that the one-dimensional stress state is approximated with desired accuracy. More specifically, this is achieved by solving the nonlinear relation

$$\begin{eqnarray} &&{\boldsymbol{R}}_{n+1}^{\mathrm{dr}}:=({\sigma }_{x x}^{n+1}-\sigma ){\boldsymbol{e}}_{x}\otimes {\boldsymbol{e}}_{x}+{\bar {\boldsymbol{\sigma}}}_{n+1}=\mathbf{0} \end{eqnarray} \tag{ 32 }$$

by means of the Newton–Raphson method, which leads to an update relation for the total strains analogous to (31), i.e.

$$\begin{eqnarray} &&{\boldsymbol{\varepsilon}}_{n+1}^{(k+1)}={\boldsymbol{\varepsilon}}_{n+1}^{(k)}-[{\mathbb{C}}_{n+1}^{(k)}]^{-1}:{\boldsymbol{R}}_{n+1}^{\mathrm{dr}\hspace{0.167em} (k)}. \end{eqnarray} \tag{ 33 }$$

Simulation results are displayed in figure 2 for three different compressive stress levels. These computations have been carried out for the trigonometric hardening case with the model parameters given in [1] (page 4320, table 2)⁷. It was observed that, for reasonable load step size and convergence tolerance, numerical results obtained with both algorithmic material models practically coincide with analytical solutions.

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It has been demonstrated that the internal variable MSMA model previously proposed by Kiefer and Lagoudas [1, 2] is well suited for numerical implementation. Two alternative algorithms for the integration of the coupled, nonlinear and inelastic constitutive equations have been presented. The first represents a classical predictor–corrector general return mapping scheme, the second a Newton–Raphson-type solution scheme based on an unconstrained nonlinear optimization problem in terms of Fischer–Burmeister complementarity functions. The greater numerical efficiency of the second algorithm is explained by the elimination of the need for a predictor–corrector-type scheme, the automatic range restriction of the variant volume fraction and the scalar nature of the underlying evolution equation. Both algorithmic models were verified by comparison to closed-form analytical solutions for experimentally relevant loading cases.

Current work is concerned with the extension of the proposed integration algorithms for the more general versions of the MSMA model [2]. In a first step, the case of finite magnetocrystalline anisotropy is considered, so that local magnetization vectors are free to rotate away from easy axes. This generalization is particularly important for the prediction of more accurate magnetization curves. In a second step, magnetic domain wall motion will also be allowed, since this mechanism becomes especially relevant at low stress levels and general magnetic loading cases. Finally, an extension of the proposed MSMA model to three dimensions is highly desirable. While the modeling concept is principally generalizable in this sense, this last step is expected to be associated with some effort. Another important on-going effort is the implementation of the proposed algorithmic material models into finite element codes that numerically solve the associated fully coupled magneto-mechanical field equations (see, e.g., [23, 24, 30]).