Homogenization

This module provides mean-field homogenization methods for predicting effective properties of composite materials.

Eshelby Tensor

The foundation of mean-field homogenization is the Eshelby tensor, which describes the strain field inside an ellipsoidal inclusion embedded in an infinite matrix.

Homogenization Schemes

Functions for computing effective properties using Mori-Tanaka, self-consistent, and other mean-field schemes.

class cylinder_multi : public phase_multi 

#include <cylinder_multi.hpp>

Cylindrical inclusion class for fiber-reinforced composite homogenization.

This class implements the mechanics of cylindrical (fiber) inclusions for mean-field homogenization of fiber-reinforced composites. Cylindrical inclusions are a limiting case of ellipsoids with one infinite axis.

Applications:

Unidirectional fiber composites
Long fiber reinforced polymers
Carbon fiber composites

Coordinate System:

The cylinder axis is aligned with the local x₃ direction
The cross-section is circular in the x₁-x₂ plane

Concentration Tensors:

The strain concentration relates macroscopic to local strain:

\[\boldsymbol{\varepsilon}^{fiber} = \mathbf{A} : \bar{\boldsymbol{\varepsilon}} \]

Local and global concentration tensors are related by rotation:

\[\mathbf{A} = \mathbf{R}^T : \mathbf{A}_{loc} : \mathbf{R} \]

See also

phase_multi for base class documentation

See also

ellipsoid_multi for general ellipsoidal inclusions

Public Functions

cylinder_multi(): Default constructor.

cylinder_multi(const arma::mat&, const arma::mat&, const arma::mat&, const arma::mat&, const arma::vec&, const arma::mat&, const arma::mat&, const arma::mat&, const arma::mat&)

Full constructor with all parameters.

Parameters:

A – Strain concentration tensor (global)
A_start – Initial strain concentration tensor
B – Stress concentration tensor (global)
B_start – Initial stress concentration tensor
A_in – Inelastic concentration vector
T_loc – Local interaction tensor
T – Global interaction tensor
A_loc – Local strain concentration tensor
B_loc – Local stress concentration tensor

cylinder_multi(const cylinder_multi&)

Copy constructor.

Parameters:: cm – cylinder_multi object to copy

~cylinder_multi(): Destructor.

virtual cylinder_multi &operator=(const cylinder_multi&)

Assignment operator.

Parameters:: cm – cylinder_multi object to assign from
Returns:: Reference to this object

Public Members

arma::mat T_loc

Interaction tensor in local coordinates (6×6)

Computed for cylinder geometry in the local frame where the cylinder axis is aligned with x₃.

arma::mat T

Interaction tensor in global coordinates (6×6)

Rotated from local to global using cylinder orientation

arma::mat A_loc

Strain concentration tensor in local coordinates (6×6)

Local frame concentration tensor before rotation to global frame

arma::mat B_loc

Stress concentration tensor in local coordinates (6×6)

Local frame concentration tensor before rotation to global frame

Friends

friend std::ostream &operator<<(std::ostream&, const cylinder_multi&)

Output stream operator for debugging.

Parameters:

os – Output stream
cm – cylinder_multi object to output

Returns:

Reference to output stream

class ellipsoid_multi : public phase_multi 

#include <ellipsoid_multi.hpp>

Ellipsoidal inclusion class for Eshelby-based homogenization.

This class implements the mechanics of ellipsoidal inclusions embedded in an infinite matrix, based on Eshelby’s classical solution (1957). It is used in mean-field homogenization methods such as:

Mori-Tanaka scheme
Self-consistent scheme
Differential scheme

Key Tensors:

The Eshelby tensor \( \mathbf{S} \) relates the constrained strain in an inclusion to its eigenstrain:

\[\boldsymbol{\varepsilon}^c = \mathbf{S} : \boldsymbol{\varepsilon}^* \]

Hill’s polarization tensor \( \mathbf{P} \) is related to the Eshelby tensor by:

\[\mathbf{P} = \mathbf{S} : \mathbf{L}_0^{-1} \]

The interaction tensor \( \mathbf{T} \) (Wu’s tensor) relates inclusion stress to the strain difference:

\[\mathbf{T} = \mathbf{L}_0 : (\mathbf{I} - \mathbf{S}) \]

See also

phase_multi for base class documentation

See also

Eshelby, J.D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. A 241, 376-396.

Note

All tensors are computed in the local coordinate system of the ellipsoid and can be rotated to global coordinates using the ellipsoid orientation.

Public Functions

ellipsoid_multi(): Default constructor.

ellipsoid_multi(const arma::mat&, const arma::mat&, const arma::mat&, const arma::mat&, const arma::vec&, const arma::mat&, const arma::mat&, const arma::mat&, const arma::mat&, const arma::mat&, const arma::mat&)

Full constructor with all parameters.

Parameters:

A – Strain concentration tensor
A_start – Initial strain concentration tensor
B – Stress concentration tensor
B_start – Initial stress concentration tensor
A_in – Inelastic concentration vector
S_loc – Local Eshelby tensor
P_loc – Local polarization tensor
T_loc – Local interaction tensor
T – Global interaction tensor
T_in_loc – Local inelastic interaction tensor
T_in – Global inelastic interaction tensor

ellipsoid_multi(const ellipsoid_multi&)

Copy constructor.

Parameters:: em – ellipsoid_multi object to copy

~ellipsoid_multi(): Destructor.

virtual void fillS_loc(const arma::mat&, const ellipsoid&)

Compute the Eshelby tensor in local coordinates.

Computes \( \mathbf{S} \) based on the matrix stiffness and ellipsoid geometry using numerical integration over the unit sphere.

Parameters:

L0 – Matrix (reference medium) stiffness tensor (6×6)
ell – Ellipsoid geometry (aspect ratios and orientation)

virtual void fillP_loc(const arma::mat&, const ellipsoid&)

Compute Hill’s polarization tensor in local coordinates.

Computes \( \mathbf{P} = \mathbf{S} : \mathbf{L}_0^{-1} \)

Parameters:

L0 – Matrix stiffness tensor (6×6)
ell – Ellipsoid geometry

virtual void fillT(const arma::mat&, const arma::mat&, const ellipsoid&)

Compute the interaction tensor.

Computes the interaction tensor \( \mathbf{T} \) that relates the stress in the inclusion to the strain mismatch with the matrix.

Parameters:

L0 – Matrix stiffness tensor (6×6)
Lt – Inclusion tangent stiffness tensor (6×6)
ell – Ellipsoid geometry

virtual void fillT_iso(const arma::mat&, const arma::mat&, const ellipsoid&)

Compute the interaction tensor for isotropic matrix.

Optimized computation when the matrix is isotropic, using analytical expressions for the Eshelby tensor.

Parameters:

L0 – Isotropic matrix stiffness tensor (6×6)
Lt – Inclusion tangent stiffness tensor (6×6)
ell – Ellipsoid geometry

virtual void fillT_mec_in(const arma::mat&, const arma::mat&, const ellipsoid&)

Compute the inelastic interaction tensor.

Computes the tensor relating eigenstrain/transformation strain to the resulting stress perturbation.

Parameters:

L0 – Matrix stiffness tensor (6×6)
Lt – Inclusion tangent stiffness tensor (6×6)
ell – Ellipsoid geometry

virtual ellipsoid_multi &operator=(const ellipsoid_multi&)

Assignment operator.

Parameters:: em – ellipsoid_multi object to assign from
Returns:: Reference to this object

Public Members

arma::mat S_loc

Eshelby tensor in local coordinates (6×6 in Voigt notation)

The Eshelby tensor depends only on the ellipsoid aspect ratios and the matrix Poisson ratio (for isotropic matrix).

arma::mat P_loc

Hill’s polarization tensor in local coordinates (6×6 in Voigt notation)

Related to Eshelby tensor: \( \mathbf{P} = \mathbf{S} : \mathbf{L}_0^{-1} \)

arma::mat T_loc

Interaction tensor (Wu’s tensor) in local coordinates (6×6)

\( \mathbf{T}_{loc} = \mathbf{L}_0 : (\mathbf{I} - \mathbf{S}) \)

arma::mat T

Interaction tensor in global coordinates (6×6)

Rotated from local to global using ellipsoid orientation

arma::mat T_in_loc

Inelastic interaction tensor in local coordinates (6×6)

Used for eigenstrain/transformation strain contributions

arma::mat T_in: Inelastic interaction tensor in global coordinates (6×6)

Public Static Attributes

static int mp: Number of integration points in polar direction for numerical integration.

static int np: Number of integration points in azimuthal direction.

static arma::vec x: Gauss points for polar integration.

static arma::vec wx: Gauss weights for polar integration.

static arma::vec y: Gauss points for azimuthal integration.

static arma::vec wy: Gauss weights for azimuthal integration.

Friends

friend std::ostream &operator<<(std::ostream&, const ellipsoid_multi&)

Output stream operator for debugging.

Parameters:

os – Output stream
em – ellipsoid_multi object to output

Returns:

Reference to output stream

class layer_multi : public phase_multi 

#include <layer_multi.hpp>

Layer class for laminated composite homogenization.

This class implements the mechanics of layered composites for mean-field homogenization. It is used for materials with a stratified microstructure where layers are stacked in a specific direction.

Applications:

Laminated composites (e.g., carbon fiber laminates)
Layered geological media
Multilayer coatings
Sandwich structures

Coordinate System:

The stacking direction is aligned with the normal direction (n)
The in-plane directions are tangential (t)

Stress/Strain Decomposition:

For layered media, it is convenient to decompose stress and strain into normal (n) and tangential (t) components:

Normal stress components: \( \sigma_{nn}, \sigma_{nt} \)
Tangential strain components: \( \varepsilon_{tt} \)

The continuity conditions at layer interfaces impose:

Continuous normal stress: \( \sigma_n^{(1)} = \sigma_n^{(2)} \)
Continuous tangential strain: \( \varepsilon_t^{(1)} = \varepsilon_t^{(2)} \)

See also

phase_multi for base class documentation

Public Functions

layer_multi(): Default constructor.

layer_multi(const arma::mat&, const arma::mat&, const arma::mat&, const arma::mat&, const arma::vec&, const arma::mat&, const arma::mat&, const arma::mat&, const arma::mat&, const arma::vec&, const arma::vec&)

Full constructor with all parameters.

Parameters:

A – Strain concentration tensor
A_start – Initial strain concentration tensor
B – Stress concentration tensor
B_start – Initial stress concentration tensor
A_in – Inelastic concentration vector
Dnn – Normal-normal stiffness block
Dnt – Normal-tangential stiffness block
dXn – Deformation gradient derivative (normal)
dXt – Deformation gradient derivative (tangential)
sigma_hat – Interface stress variable
dzdx1 – Field gradient through thickness

layer_multi(const layer_multi&)

Copy constructor.

Parameters:: lm – layer_multi object to copy

~layer_multi(): Destructor.

virtual layer_multi &operator=(const layer_multi&)

Assignment operator.

Parameters:: lm – layer_multi object to assign from
Returns:: Reference to this object

Public Members

arma::mat Dnn

Normal-normal block of the tangent modulus (3×3)

Submatrix of the stiffness tensor relating normal stress to normal strain: \( \mathbf{D}_{nn} = \frac{\partial \boldsymbol{\sigma}_n}{\partial \boldsymbol{\varepsilon}_n} \)

arma::mat Dnt

Normal-tangential block of the tangent modulus (3×3)

Submatrix coupling normal stress to tangential strain: \( \mathbf{D}_{nt} = \frac{\partial \boldsymbol{\sigma}_n}{\partial \boldsymbol{\varepsilon}_t} \)

arma::mat dXn

Derivative of deformation gradient with respect to normal position (3×3)

Used in incremental formulations for geometric nonlinearity

arma::mat dXt

Derivative of deformation gradient with respect to tangential position (3×3)

Used in incremental formulations for geometric nonlinearity

arma::vec sigma_hat

Stress-like quantity for interface conditions (6×1)

Intermediate variable used in the layer homogenization algorithm

arma::vec dzdx1

Derivative of local field with respect to layer position (6×1)

Used for computing gradients through the layer thickness

Friends

friend std::ostream &operator<<(std::ostream&, const layer_multi&)

Output stream operator for debugging.

Parameters:

os – Output stream
lm – layer_multi object to output

Returns:

Reference to output stream

class phase_multi

#include <phase_multi.hpp>

Base class for phases in multiphase homogenization schemes.

The phase_multi class serves as the base class for all phase types used in mean-field homogenization methods. It stores the concentration tensors that relate macroscopic strain/stress to local phase-level quantities.

Concentration Tensor Relationships:

The strain concentration tensor \( \mathbf{A} \) relates macroscopic strain to local strain:

\[\boldsymbol{\varepsilon}^{(r)} = \mathbf{A}^{(r)} : \bar{\boldsymbol{\varepsilon}} \]

The stress concentration tensor \( \mathbf{B} \) relates macroscopic stress to local stress:

\[\boldsymbol{\sigma}^{(r)} = \mathbf{B}^{(r)} : \bar{\boldsymbol{\sigma}} \]

For inelastic behavior, an additional concentration vector handles eigenstrain contributions:

\[\boldsymbol{\varepsilon}^{(r)} = \mathbf{A}^{(r)} : \bar{\boldsymbol{\varepsilon}} + \mathbf{A}_{in}^{(r)} \]

Derived Classes:

ellipsoid_multi: Ellipsoidal inclusions (Eshelby-based methods)
cylinder_multi: Cylindrical inclusions
layer_multi: Layered composites

Subclassed by cylinder_multi, ellipsoid_multi, layer_multi

Public Functions

phase_multi()

Default constructor.

Initializes concentration tensors to zero matrices/vectors of appropriate size

phase_multi(const arma::mat &A_init, const arma::mat &A_start_init, const arma::mat &B_init, const arma::mat &B_start_init, const arma::vec &A_in_init)

Constructor with parameters.

Parameters:

A_init – Initial strain concentration tensor (6×6)
A_start_init – Initial strain concentration tensor for start state (6×6)
B_init – Initial stress concentration tensor (6×6)
B_start_init – Initial stress concentration tensor for start state (6×6)
A_in_init – Initial inelastic concentration vector (6×1)

phase_multi(const phase_multi &pm)

Copy constructor.

Parameters:: pm – phase_multi object to copy

virtual ~phase_multi(): Virtual destructor.

void to_start()

Reset current tensors to start-of-increment values.

Sets A = A_start and B = B_start for iterative schemes

void set_start()

Store current tensors as start-of-increment values.

Sets A_start = A and B_start = B after convergence

virtual phase_multi &operator=(const phase_multi &pm)

Assignment operator.

Parameters:: pm – phase_multi object to assign from
Returns:: Reference to this object

Public Members

arma::mat A

Strain concentration tensor (6×6 matrix in Voigt notation)

Relates macroscopic strain to local phase strain: \( \boldsymbol{\varepsilon}^{local} = \mathbf{A} : \bar{\boldsymbol{\varepsilon}} \)

arma::mat A_start

Strain concentration tensor at the start of increment.

Stored for incremental schemes and convergence checks

arma::mat B

Stress concentration tensor (6×6 matrix in Voigt notation)

Relates macroscopic stress to local phase stress: \( \boldsymbol{\sigma}^{local} = \mathbf{B} : \bar{\boldsymbol{\sigma}} \)

arma::mat B_start

Stress concentration tensor at the start of increment.

Stored for incremental schemes and convergence checks

arma::vec A_in

Inelastic strain concentration vector (6×1 in Voigt notation)

Accounts for eigenstrain/transformation strain contributions: \( \boldsymbol{\varepsilon}^{local} = \mathbf{A} : \bar{\boldsymbol{\varepsilon}} + \mathbf{A}_{in} \)

Friends

friend std::ostream &operator<<(std::ostream &os, const phase_multi &pm)

Output stream operator for debugging.

Parameters:

os – Output stream
pm – phase_multi object to output

Returns:

Reference to output stream