Thermomechanical Models

Coupled thermal-mechanical constitutive models with temperature-dependent properties and thermal expansion.

void umat_elasticity_iso_T(const arma::vec&, const arma::vec&, arma::vec&, double&, arma::mat&, arma::mat&, arma::mat&, arma::mat&, const arma::mat&, const int&, const arma::vec&, const int&, arma::vec&, const double&, const double&, const double&, const double&, double&, double&, double&, double&, double&, double&, double&, const int&, const int&, const bool&, double&)

The elastic UMAT requires 2 constants:

props[0] : Young modulus

props[1] : Poisson ratio

props[2] : CTE

No statev is required for thermoelastic constitutive law

void umat_elasticity_ortho_T(const arma::vec&, const arma::vec&, arma::vec&, double&, arma::mat&, arma::mat&, arma::mat&, arma::mat&, const arma::mat&, const int&, const arma::vec&, const int&, arma::vec&, const double&, const double&, const double&, const double&, double&, double&, double&, double&, double&, double&, double&, const int&, const int&, const bool&, double&)

The elastic UMAT requires 2 constants:

props[0] : Young modulus

props[1] : Poisson ratio

props[2] : CTE

No statev is required for thermoelastic constitutive law

void umat_elasticity_trans_iso_T(const arma::vec&, const arma::vec&, arma::vec&, double&, arma::mat&, arma::mat&, arma::mat&, arma::mat&, const arma::mat&, const int&, const arma::vec&, const int&, arma::vec&, const double&, const double&, const double&, const double&, double&, double&, double&, double&, double&, double&, double&, const int&, const int&, const bool&, double&)

The elastic UMAT requires 2 constants:

props[0] : Young modulus

props[1] : Poisson ratio

props[2] : CTE

No statev is required for thermoelastic constitutive law

void umat_plasticity_iso_CCP_T(const arma::vec&, const arma::vec&, arma::vec&, double&, arma::mat&, arma::mat&, arma::mat&, arma::mat&, const arma::mat&, const int&, const arma::vec&, const int&, arma::vec&, const double&, const double&, const double&, const double&, double&, double&, double&, double&, double&, double&, double&, const int&, const int&, const bool&, double&)

The elastic-plastic UMAT with isotropic hardening requires 8 constants for a full thermomechanical coupling:

props[0] : density

props[1] : specific heat capacity

props[2] : Young modulus

props[3] : Poisson ratio

props[4] : CTE

props[5] : J2 equivalent yield stress limit : sigmaY

props[6] : hardening parameter k

props[7] : exponent m

The elastic-plastic UMAT with isotropic hardening requires 8 statev:

statev[0] : T_init : Initial temperature

statev[1] : Accumulative plastic parameter: p

statev[2] : Plastic strain 11: EP(0,0)

statev[3] : Plastic strain 22: EP(1,1)

statev[4] : Plastic strain 33: EP(2,2)

statev[5] : Plastic strain 12: EP(0,1) (*2)

statev[6] : Plastic strain 13: EP(0,2) (*2)

statev[7] : Plastic strain 23: EP(1,2) (*2)

void umat_plasticity_kin_iso_CCP_T(const arma::vec&, const arma::vec&, arma::vec&, double&, arma::mat&, arma::mat&, arma::mat&, arma::mat&, const arma::mat&, const int&, const arma::vec&, const int&, arma::vec&, const double&, const double&, const double&, const double&, double&, double&, double&, double&, double&, double&, double&, const int&, const int&, const bool&, double&)

The thermomechanical elastic-plastic UMAT with kinematic + isotropic hardening requires 9 constants:

props[0] : density

props[1] : specific heat capacity

props[2] : Young modulus

props[3] : Poisson ratio

props[4] : CTE

props[5] : J2 equivalent yield stress limit : sigmaY

props[6] : hardening parameter k

props[7] : exponent m

props[8] : linear kinematical hardening h

The elastic-plastic UMAT with kinematic + isotropic hardening requires 14 statev:

statev[0] : T_init : Initial temperature

statev[1] : Accumulative plastic parameter: p

statev[2] : Plastic strain 11: EP(0,0)

statev[3] : Plastic strain 22: EP(1,1)

statev[4] : Plastic strain 33: EP(2,2)

statev[5] : Plastic strain 12: EP(0,1) (*2)

statev[6] : Plastic strain 13: EP(0,2) (*2)

statev[7] : Plastic strain 23: EP(1,2) (*2)

statev[8] : Backstress 11: X(0,0)

statev[9] : Backstress 11: X(1,1)

statev[10] : Backstress 11: X(2,2)

statev[11] : Backstress 11: X(0,1)

statev[12] : Backstress 11: X(0,2)

statev[13] : Backstress 11: X(1,2)

void umat_sma_unified_T_T(const std::string &umat_name, const arma::vec &Etot, const arma::vec &DEtot, arma::vec &sigma, double &r, arma::mat &dSdE, arma::mat &dSdT, arma::mat &drdE, arma::mat &drdT, const arma::mat &DR, const int &nprops, const arma::vec &props, const int &nstatev, arma::vec &statev, const double &T, const double &DT, const double &Time, const double &DTime, double &Wm, double &Wm_r, double &Wm_ir, double &Wm_d, double &Wt, double &Wt_r, double &Wt_ir, const int &ndi, const int &nshr, const bool &start, double &tnew_dt)

Unified thermomechanical SMA model with coupled phase transformation and heat equation.

This function implements the fully coupled thermomechanical version of the unified phenomenological SMA model. Compared to the mechanical-only version (umat_sma_unified_T), this adds thermal coupling through the heat equation, requiring density and specific heat capacity as additional material parameters, and producing thermal tangent operators as output.

Elastic Symmetry and Criteria Selection:

The elastic behavior and transformation criteria are determined by the umat_name parameter:

umat_name	Elasticity	Criteria	Number of props
SMADI	Isotropic	Drucker	31
SMADC	Cubic	Drucker	33
SMAAI	Isotropic	Anisotropic Drucker (DFA)	38
SMAAC	Cubic	Anisotropic Drucker (DFA)	40

Material Parameters (props):

SMADI (Isotropic elasticity, Drucker criteria) — 31 props:

Index	Symbol	Description	Units
0	\( \rho \)	Density	kg/m^3
1	\( c_{pA} \)	Specific heat capacity of Austenite	J/(kg.K)
2	\( c_{pM} \)	Specific heat capacity of Martensite	J/(kg.K)
3	flagT	Temperature extrapolation (0=linear, 1=smooth)	-
4	\( E_A \)	Young’s modulus of Austenite	MPa
5	\( E_M \)	Young’s modulus of Martensite	MPa
6	\( \nu_A \)	Poisson’s ratio of Austenite	-
7	\( \nu_M \)	Poisson’s ratio of Martensite	-
8	\( \alpha_A \)	CTE of Austenite	1/K
9	\( \alpha_M \)	CTE of Martensite	1/K
10	\( H_{min} \)	Minimal transformation strain magnitude	-
11	\( H_{max} \)	Maximal transformation strain magnitude	-
12	\( k_1 \)	Exponential evolution parameter for \( H^{cur} \)	-
13	\( \sigma_{crit} \)	Critical stress for \( H^{cur} \) evolution	MPa
14	\( C_A \)	Clausius-Clapeyron slope (M \( \rightarrow \) A)	MPa/K
15	\( C_M \)	Clausius-Clapeyron slope (A \( \rightarrow \) M)	MPa/K
16	\( M_{s0} \)	Martensite start temperature at zero stress	K
17	\( M_{f0} \)	Martensite finish temperature at zero stress	K
18	\( A_{s0} \)	Austenite start temperature at zero stress	K
19	\( A_{f0} \)	Austenite finish temperature at zero stress	K
20	\( n_1 \)	Martensite start smooth exponent	-
21	\( n_2 \)	Martensite finish smooth exponent	-
22	\( n_3 \)	Austenite start smooth exponent	-
23	\( n_4 \)	Austenite finish smooth exponent	-
24	\( \sigma_{caliber} \)	Calibration stress for \( C_A \) / \( C_M \)	MPa
25	\( b \)	Tension-compression asymmetry parameter (Prager)	-
26	\( n \)	Tension-compression asymmetry exponent (Prager)	-
27	\( c_\lambda \)	Penalty function exponent start point	-
28	\( p_{0\lambda} \)	Penalty function limit value	-
29	\( n_\lambda \)	Penalty function power law exponent	-
30	\( \alpha_\lambda \)	Penalty function power law parameter	-

SMADC (Cubic elasticity, Drucker criteria) — 33 props:

Index	Symbol	Description	Units
0	\( \rho \)	Density	kg/m^3
1	\( c_{pA} \)	Specific heat capacity of Austenite	J/(kg.K)
2	\( c_{pM} \)	Specific heat capacity of Martensite	J/(kg.K)
3	flagT	Temperature extrapolation (0=linear, 1=smooth)	-
4	\( E_A \)	Young’s modulus of Austenite ([100] direction)	MPa
5	\( E_M \)	Young’s modulus of Martensite ([100] direction)	MPa
6	\( \nu_A \)	Poisson’s ratio of Austenite	-
7	\( \nu_M \)	Poisson’s ratio of Martensite	-
8	\( G_A \)	Shear modulus of Austenite	MPa
9	\( G_M \)	Shear modulus of Martensite	MPa
10–32		Same as SMADI props[8–30] (common SMA parameters)

SMAAI (Isotropic elasticity, anisotropic Drucker criteria) — 38 props:

Same as SMADI (props[0–30]) followed by 7 DFA parameters:

Index	Symbol	Description
31	\( F_{dfa} \)	F parameter of DFA criteria
32	\( G_{dfa} \)	G parameter of DFA criteria
33	\( H_{dfa} \)	H parameter of DFA criteria
34	\( L_{dfa} \)	L parameter of DFA criteria
35	\( M_{dfa} \)	M parameter of DFA criteria
36	\( N_{dfa} \)	N parameter of DFA criteria
37	\( K_{dfa} \)	K parameter of DFA criteria

SMAAC (Cubic elasticity, anisotropic Drucker criteria) — 40 props:

Same as SMADC (props[0–32]) followed by 7 DFA parameters (props[33–39]).

State Variables (statev) — 17 variables:

Index	Symbol	Description	Units
0	\( T_{init} \)	Initial/reference temperature	K
1	\( \xi \)	Martensitic volume fraction	-
2	\( \varepsilon^{tr}_{11} \)	Transformation strain component 11	-
3	\( \varepsilon^{tr}_{22} \)	Transformation strain component 22	-
4	\( \varepsilon^{tr}_{33} \)	Transformation strain component 33	-
5	\( \gamma^{tr}_{12} \)	Transformation strain component 12 (engineering)	-
6	\( \gamma^{tr}_{13} \)	Transformation strain component 13 (engineering)	-
7	\( \gamma^{tr}_{23} \)	Transformation strain component 23 (engineering)	-
8	\( \xi_F \)	Forward martensitic volume fraction	-
9	\( \xi_R \)	Reverse martensitic volume fraction	-
10	\( \rho \Delta s_0 \)	Entropy difference between phases (M - A)	MPa/K
11	\( \rho \Delta E_0 \)	Internal energy difference between phases (M - A)	MPa
12	\( D \)	Stress dependence parameter for transformation limits	-
13	\( a_1 \)	Forward hardening parameter	MPa
14	\( a_2 \)	Reverse hardening parameter	MPa
15	\( a_3 \)	Equilibrium hardening parameter	MPa
16	\( Y_{0t} \)	Initial transformation critical value	MPa

References:

• Chatziathanasiou, D. (2016). Ph.D. Thesis — Phenomenological constitutive modeling of shape memory alloys.

• Chemisky, Y., Chatziathanasiou, D., Kumar, P., & Lagoudas, D. C. (2014). “A constitutive model for cyclic actuation of high-temperature shape memory alloys.” Mechanics of Materials, 68, 120-136.

See also

umat_sma_unified_T() for the mechanical-only version (no thermal coupling)

See also

L_iso() for isotropic stiffness tensor (SMADI, SMAAI)

See also

L_cubic() for cubic stiffness tensor (SMADC, SMAAC)

Note

Elastic convention: isotropic uses “Enu”, cubic uses “EnuG” for L_iso / L_cubic

Note

The flagT parameter controls temperature extrapolation: 0 for linear, 1 for smooth

Note

Legacy aliases SMAUT and SMANI map to SMADI and SMAAI respectively

Parameters:

• umat_name – Model variant (SMADI, SMADC, SMAAI, SMAAC)

• Etot – Total strain tensor at beginning of increment (Voigt notation: 6x1)

• DEtot – Strain increment tensor (Voigt notation: 6x1)

• sigma – Cauchy stress tensor (Voigt notation: 6x1) [output]

• r – Heat source rate [output]

• dSdE – Mechanical tangent modulus \( \partial \sigma / \partial \varepsilon \) (6x6) [output]

• dSdT – Thermal stress tangent \( \partial \sigma / \partial T \) (6x1) [output]

• drdE – Thermal-mechanical tangent \( \partial r / \partial \varepsilon \) (1x6) [output]

• drdT – Thermal tangent \( \partial r / \partial T \) (scalar) [output]

• DR – Rotation increment matrix (3x3) for objective integration

• nprops – Number of material properties

• props – Material properties vector (layout depends on umat_name)

• nstatev – Number of state variables (17)

• statev – State variables vector [input/output]

• T – Temperature at beginning of increment

• DT – Temperature increment

• Time – Time at beginning of increment

• DTime – Time increment

• Wm – Total mechanical work [output]

• Wm_r – Recoverable (elastic) mechanical work [output]

• Wm_ir – Irrecoverable mechanical work stored in transformation [output]

• Wm_d – Dissipated mechanical work (hysteresis) [output]

• Wt – Total thermal work [output]

• Wt_r – Recoverable thermal work [output]

• Wt_ir – Irrecoverable thermal work [output]

• ndi – Number of direct stress components (typically 3)

• nshr – Number of shear stress components (typically 3)

• start – Flag indicating first increment (true) or continuation (false)

• tnew_dt – Suggested new time step ratio for adaptive time stepping [output]