4.1.1.6.1.18. pyfem.materials.SOVS module

Skorohod-Olevsky viscous sintering model for ceramic materials.

This module implements the Skorohod-Olevsky constitutive model for viscous sintering of ceramic powders. The model describes densification and deformation during sintering through viscous flow mechanisms, accounting for both densification-driven and stress-driven deformation.

class SkorohodOlevsky(props)[source]

Bases: BaseMaterial

Skorohod-Olevsky viscous sintering model for ceramics.

This class implements a constitutive model for viscous sintering of ceramic powders. The model describes the densification and creep deformation during sintering through viscous flow mechanisms.

The model is based on two key concepts:

  1. Densification (volumetric flow): The rate of densification is driven by sintering stress and hydrostatic stress:

    dρ/dt = (3ρ/2η_v) * (σ_sint - σ_m)

    where: - ρ: relative density (current density / theoretical density) - η_v: volumetric viscosity (depends on density and temperature) - σ_sint: sintering stress (capillary-driven stress from surface tension) - σ_m: mean (hydrostatic) stress = (σ₁₁ + σ₂₂ + σ₃₃)/3

  2. Deviatoric deformation (shape change): Viscous creep under deviatoric stress:

    ε̇_dev = s / (2η_s)

    where: - s: deviatoric stress tensor - η_s: shear viscosity

The viscosities are temperature and density dependent: - η_v = η₀ * f_v(ρ) * exp(Q/RT) - η_s = η₀ * f_s(ρ) * exp(Q/RT)

4.1.1.6.1.18. Required Properties

eta0float

Reference viscosity at reference temperature (Pa·s). Typical values: 1e10 - 1e14 Pa·s for ceramics at sintering temperatures.

Qfloat

Activation energy for viscous flow (J/mol). Typical values: 300000 - 700000 J/mol.

Rfloat, optional

Universal gas constant (J/(mol·K)). Default: 8.314

Tfloat

Temperature (K). For isothermal sintering analysis. Typical sintering temperatures: 1400-1800 K for ceramics.

rho0float

Initial relative density (green density). Typical values: 0.5 - 0.7 for powder compacts.

sigma_sintfloat

Sintering stress (Pa). Related to surface tension and particle size: σ_sint ≈ 3γ/r (γ: surface energy, r: particle radius). Typical values: 1e5 - 1e7 Pa.

n_volfloat, optional

Viscosity exponent for volumetric flow. Default: 2.0

n_shearfloat, optional

Viscosity exponent for shear flow. Default: 1.0

Examples

Material properties for alumina sintering at 1600 K:

<Material>

type = “SkorohodOlevsky”; eta0 = 1.0e12; # Pa·s - reference viscosity Q = 500000.0; # J/mol - activation energy T = 1600.0; # K - sintering temperature rho0 = 0.6; # Initial relative density sigma_sint = 1.0e6; # Pa - sintering stress n_vol = 2.0; # Volumetric viscosity exponent n_shear = 1.0; # Shear viscosity exponent R = 8.314; # J/(mol·K) - gas constant

</Material>

Notes

Physical Interpretation: - Relative density ρ increases from initial value (e.g., 0.6) towards 1.0 - Sintering stress drives densification even without external load - Applied compressive stress accelerates densification - Applied tensile stress can slow or reverse densification (swelling) - Shear deformation occurs through grain boundary sliding

The model captures: - Free sintering (densification without applied stress) - Pressure-assisted sintering (hot pressing) - Creep deformation under stress - Temperature-dependent sintering kinetics

References

  • Skorohod, V.V. (1972). “Rheological basis of the theory of sintering.” Naukova Dumka, Kiev.

  • Olevsky, E.A. (1998). “Theory of sintering: from discrete to continuum.” Materials Science and Engineering: R: Reports, 23(2), 41-100.

  • Olevsky, E.A. and Froyen, L. (2006). “Constitutive modeling of spark-plasma sintering of conductive materials.” Scripta Materialia, 55(12), 1175-1178.

getStress(kinematics) → Tuple[array, array][source]

Compute stress and tangent stiffness for the Skorohod-Olevsky model.

The algorithm integrates the viscous sintering equations: 1. Update relative density from volumetric viscous flow 2. Compute deviatoric viscous strain increment 3. Calculate elastic response with density-dependent moduli 4. Assemble consistent tangent matrix

Parameters:

kinematics (Kinematics) – Kinematics object containing strain information. Required attributes: - dstrain: strain increment (3 or 6 components)

Returns:

  • sigma (ndarray) – Stress tensor (3 or 6 components).

  • tang (ndarray) – Tangent stiffness matrix (3x3 or 6x6).

Notes

The densification rate follows: dρ/dt = (3ρ/2η_v) * (σ_sint - σ_m)

where η_v = η_ref * (1/ρ^n_vol - 1)

The deviatoric viscous strain rate follows: dε_dev/dt = s / (2η_s)

where η_s = η_ref * (1/ρ^n_shear - 1)

maximum_principal_stress(stress: array) → float[source]

Compute maximum principal stress (alternative to von Mises).

This is more appropriate for brittle materials like ceramics that fail primarily under tensile loading.

Parameters:

stress (ndarray) – Stress tensor in Voigt notation [σ11, σ22, σ33, σ23, σ13, σ12].

Returns:

Maximum principal stress (most tensile).

Return type:

float