The principle of the minimum of the dissipation potential for nonisothermal processes states that

see [3]. In Equation (1),

are the rates of the internal variables of the specific material model,

**q** is the heat flux vector, and

is the thermodynamic dissipation, which is given by the second law of thermodynamics. This is

with the Helmholtz free energy *ψ* and the (absolute) temperature *θ*. We introduce two kinds of internal variables, namely, a vector of volume fractions of the crystallographic phases denoted by **λ**=(*λ*_{i}), *λ*_{0} representing the austenite and

representing the specific martensitic variants, where

is alloy dependent, and a set of three Euler angles

**α**, which indicates the average or mean orientation direction of the polycrystalline arrangement.

Thus, the rate of the internal variables is

We denote by

the so-called dissipation potential introduced in [15] comprising the material-dependent dissipative terms. We decompose the dissipation potential into a transformational and a thermal part as

Moreover, we set

and follow [3] to set

where

*r*_{λ} and

*r*_{α} are the dissipation parameters for phase transformation and reorientation, respectively, and

*μ* is the reciprocal of the heat conductivity [3].

The Lagrangean

in Equation (1) is not minimized with respect to the rates of the strains. Hence, we drop the term

and find

The last two terms in Equation (3), -*γλ* and

are introduced to include the constraints of positivity and mass conservation, respectively, by means of the Lagrange parameter

*β* and the Kuhn-Tucker parameters

**γ**=(

*γ*_{i}). For more details, we refer to [4, 5].

The stationarity condition of

with respect to the rates of the volume fractions reads, due to the nondifferentiability of

at

the differential inclusion

where **β**=β**1** and **1** denotes the

-dimensional vector with value 1 in all components. The stationarity condition of

with respect to the rates of the Euler angles gives

Finally, stationarity of

with respect to

**q** yields

which is nothing else but Fourier’s law of heat conduction.

It turns out to be convenient to invert Equations (4) and (5), which amounts to performing a Legendre transformation of the dissipation potentials

and

For this purpose, let us introduce an active set (of variants)

as

to account for the constraint of positivity and furthermore **p**^{λ}=-*∂ψ*/*∂***λ** and **p**^{α}=-*∂ψ*/*∂***α** as thermodynamically conjugated driving forces. Inserting this into Equation (4), we find

Following [4, 5], we introduce the active deviator as

with

as number of active phases. This allows to define yield functions Φ

_{λ}≤0 and Φ

_{α}≤0, which indicate whether phase transformation and/or reorientation take place, respectively. They are given as

and

where

. We take the formula given in [3] to derive the heat conduction equation. Using our approach for

it is given as

where *κ* denotes the heat capacity. Structural heating

is defined via

For the Helmholtz free energy, we take the approach from [4, 5], now including temperature-dependent parts, as

where **Q**=**Q**(*α*) denotes a rotation matrix, and the effective transformation strain

the effective stiffness tensor

and the effective caloric part of the energy

are given by

According to [16], the temperature-dependent caloric part of the energy is given as

where we have omitted terms that are identical for all phases and therefore are not relevant for our formulation.

We are now able to write the final system of evolution equations as

with

and

as consistency parameters. The Kuhn-Tucker conditions

close the system of equations, in combination with the consistency condition

which updates the active set. The heat conduction equation reads

with **b**=(*b*_{A}, *b*_{M}, …, *b*_{M}), where *b*_{A} and *b*_{M} denote the entropic constants for austenite and martensite, respectively.

To evaluate the material model, it is necessary to derive the driving forces. For the volume fractions, they are

see also [4, 5]. The rotation matrix is given in terms of Euler angles as

with **α**={*φ*, *ν*, *ω*} and *φ*, *ω*∈[0, 2*π*], *ν*∈[0, *π*]. Now, we can calculate the corresponding driving forces as

Whereas the first term in Equation (25) reads

the entries of the second part *∂***Q**/*∂α*=(*∂***Q**/*∂φ*, *∂***Q**/*∂ν*, *∂***Q**/*∂ω*) are

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