Second-Order Damage Tensor Approach

• The second-order damage tensor approach is a framework that encodes anisotropic degradation using symmetric tensors representing both damage state and its thermodynamic conjugate.
• It encompasses finite-strain tensorial models, micromorphic regularizations, and higher-gradient formulations to control localization and ensure mesh-objectivity.
• The approach integrates spectral decomposition and machine-learning enhancements to accurately capture damage evolution in anisotropic materials while maintaining thermodynamic consistency.

Second-order damage tensor approach denotes a family of continuum damage formulations in which anisotropic degradation is encoded by a symmetric second-order tensor, while some higher-gradient variants additionally treat damage or displacement through second jets and the associated second-order stresses. Across the literature considered here, the approach appears in four closely connected forms: finite-strain tensorial damage models with thermodynamic driving forces of the same tensorial order, micromorphic regularizations that act on scalar projections of a damage tensor, micromechanics-based reconstructions in which fourth-order damage effects are represented through second-order tensors, and metric-independent geometric formulations in which second-order stresses act on second jets (Velden et al., 2024, Desmorat et al., 2017, Segev, 2014, Amiri-Hezaveh et al., 23 Jul 2025).

1. Terminological scope and defining ideas

The literature summarized here uses “second-order” in two distinct senses. In the constitutive damage literature, it refers to the order of the damage variable: the damage state is represented by a symmetric second-order tensor $\boldsymbol{D}$, and its thermodynamic conjugate $\boldsymbol{Y}$ is likewise a second-order tensor. In higher-gradient geometry, it refers to the order of differentiation: stresses act on second jets $J^2W$ and are energetically conjugate to second derivatives of a field (Velden et al., 2024, Segev, 2014).

Usage of “second-order”	Mathematical object	Typical role
Tensorial order	$\boldsymbol{D}$, $\boldsymbol{Y}$	Anisotropic damage state and conjugate force
Differential order	$J^2W$, second-order stress $S$	Higher-gradient virtual power, balance laws, boundary terms

Within the tensorial constitutive setting, the central internal variable is a symmetric second-order damage tensor $\boldsymbol{D}$. In the finite-strain anisotropic brittle model, the Helmholtz free energy is written as

$$\psi(C,D,\bar d,d,\nabla \bar d) = \psi_e(C,D)+\psi_d(\alpha_d)+\psi_h(D)+\psi_{\text{mic}}(d,\bar d,\nabla \bar d),$$

and the thermodynamic conjugate of the damage tensor is

$\boldsymbol{Y}:=-\frac{\partial \psi}{\partial D}.$

The reduced dissipation inequality becomes

$\boldsymbol{Y}$0

so both damage and its driving force are second-order tensors (Velden et al., 2024).

A closely related machine-learning formulation splits the damage state into an isotropic and an anisotropic part,

$\boldsymbol{Y}$1

with $\boldsymbol{Y}$2 structural tensors. This gives a tensorial encoding of orthotropic damage and of damage-induced anisotropy in initially isotropic materials (Amiri-Hezaveh et al., 23 Jul 2025).

A recurrent misconception is to treat a second-order damage tensor as merely a compact notational replacement for scalar damage. The cited work does not support that reduction. In these models, second-order tensors are introduced precisely to encode direction-dependent degradation, directional driving forces, and anisotropic evolution, rather than isotropic softening alone (Velden et al., 2024, Amiri-Hezaveh et al., 23 Jul 2025).

2. Finite-strain constitutive structure

The finite-strain formulations are written in the reference configuration with standard kinematics: $\boldsymbol{Y}$3 The balance of linear momentum in reference form is

$\boldsymbol{Y}$4

with $\boldsymbol{Y}$5 the second Piola–Kirchhoff stress. In the 2024 anisotropic brittle model there is no multiplicative decomposition of the deformation gradient into damage-related parts; damage acts through degradation functions applied to a standard hyperelastic energy defined in terms of $\boldsymbol{Y}$6 (Velden et al., 2024).

The elastic free energy is specified as

$\boldsymbol{Y}$7

where $\boldsymbol{Y}$8 controls the degree of anisotropy, $\boldsymbol{Y}$9 is isotropic degradation, $J^2W$0 is anisotropic degradation, and $J^2W$1 is the undamaged hyperelastic energy. For the Neo-Hookean specialization,

$J^2W$2

and the stress follows from

$J^2W$3

The explicit degraded stress is

$J^2W$4

The damage functions are

$J^2W$5

so the second-order damage tensor enters both isotropically and directionally (Velden et al., 2024).

The same model introduces an additional kinematic damage hardening energy $J^2W$6 in the principal damage coordinates to keep the eigenvalues $J^2W$7 below $J^2W$8, and an isotropic hardening energy

$J^2W$9

with conjugate force

$\boldsymbol{D}$0

Damage onset is governed by

$\boldsymbol{D}$1

with

$\boldsymbol{D}$2

and associated evolution

$\boldsymbol{D}$3

together with Kuhn–Tucker conditions

$\boldsymbol{D}$4

Because $\boldsymbol{D}$5 depends on $\boldsymbol{D}$6 and $\boldsymbol{D}$7, the evolution law is fully tensorial (Velden et al., 2024).

A central constitutive requirement is the damage growth criterion: for any positive semi-definite increment $\boldsymbol{D}$8 and any $\boldsymbol{D}$9,

$\boldsymbol{Y}$0

equivalently

$\boldsymbol{Y}$1

The 2024 model shows that its chosen degradation structure satisfies this criterion, and also shows that a standard volumetric/isochoric split of the form

$\boldsymbol{Y}$2

will generally violate the criterion for the isochoric part at finite strains unless very specific forms are used (Velden et al., 2024).

3. Regularization, micromorphic extensions, and localization control

The 2024 finite-strain model regularizes the tensorial damage description through a Forest-type micromorphic framework. Besides the displacement field $\boldsymbol{Y}$3, it introduces a micromorphic tuple of scalar fields

$\boldsymbol{Y}$4

which are nonlocal counterparts of local scalar quantities

$\boldsymbol{Y}$5

themselves defined as projections of the damage tensor $\boldsymbol{Y}$6. The micromorphic balance equation is

$\boldsymbol{Y}$7

with $\boldsymbol{Y}$8 conjugate to $\boldsymbol{Y}$9 and $J^2W$0 conjugate to $J^2W$1 (Velden et al., 2024).

The micromorphic free-energy contribution is

$J^2W$2

with state laws

$J^2W$3

This is a quadratic penalty on $J^2W$4 plus a quadratic gradient term (Velden et al., 2024).

Three projection choices are emphasized. Model A performs full componentwise regularization: $J^2W$5 with

$J^2W$6

Model B regularizes principal invariants: $J^2W$7 Model C regularizes volumetric and deviatoric measures: $J^2W$8 The micromorphic contribution to the tensorial damage driving force is

$J^2W$9

which remains a second-order tensor (Velden et al., 2024).

The numerical role of this regularization is explicit. A local model with $S$0 and $S$1 exhibits strong mesh sensitivity, pathological localization into a single element row, and incorrect crack paths in the asymmetrically notched specimen. By contrast, all gradient-extended models A, B, and C show mesh-objective crack patterns and load–displacement curves. For the asymmetrically notched specimen, Models A and C produce nearly identical shear crack patterns and global responses. For the 3D rotor blade, using Model C with only two micromorphic degrees of freedom, five meshes ranging from $S$2 to $S$3 elements show very similar structural responses, and the coarsest mesh underestimates load capacity by only $S$4 (Velden et al., 2024).

This regularization strategy does not change the order of the damage variable itself: the underlying damage state remains second-order tensorial. The nonlocal fields are scalar projections chosen to control localization while preserving a tensorial thermodynamic driving force (Velden et al., 2024).

4. Metric-independent higher-gradient formulation

A separate, geometric line of work analyzes second-order stresses without assuming any metric. The body is modeled as an $S$5-dimensional smooth manifold $S$6, and a vector bundle $S$7 represents the field of interest. For mechanics $S$8 can represent virtual velocities or displacements; for damage mechanics, one may take $S$9 for scalar damage, $\boldsymbol{D}$0 for vector- or matrix-valued damage, or a more general bundle for internal variables (Segev, 2014).

The second jet space $\boldsymbol{D}$1 encodes the value, first derivatives, and second derivatives of a section at a point. A second-order stress at $\boldsymbol{D}$2 is a linear map

$\boldsymbol{D}$3

and a stress field is a smooth section

$\boldsymbol{D}$4

For a region $\boldsymbol{D}$5 and a virtual field $\boldsymbol{D}$6, the virtual power is

$\boldsymbol{D}$7

In this framework, second-order stresses are precisely the objects conjugate to second gradients, and the construction does not require a Riemannian metric (Segev, 2014).

Direct analysis on $\boldsymbol{D}$8 is limited, so second-order stresses are represented by non-holonomic stresses on the iterated jet bundle $\boldsymbol{D}$9. There is a canonical inclusion

$$\psi(C,D,\bar d,d,\nabla \bar d) = \psi_e(C,D)+\psi_d(\alpha_d)+\psi_h(D)+\psi_{\text{mic}}(d,\bar d,\nabla \bar d),$$0

and every second-order stress

$$\psi(C,D,\bar d,d,\nabla \bar d) = \psi_e(C,D)+\psi_d(\alpha_d)+\psi_h(D)+\psi_{\text{mic}}(d,\bar d,\nabla \bar d),$$1

can be represented as

$$\psi(C,D,\bar d,d,\nabla \bar d) = \psi_e(C,D)+\psi_d(\alpha_d)+\psi_h(D)+\psi_{\text{mic}}(d,\bar d,\nabla \bar d),$$2

for some

$$\psi(C,D,\bar d,d,\nabla \bar d) = \psi_e(C,D)+\psi_d(\alpha_d)+\psi_h(D)+\psi_{\text{mic}}(d,\bar d,\nabla \bar d),$$3

The representation is non-unique, but it allows second-order problems to be analyzed by iterating the first-order stress machinery (Segev, 2014).

The divergence operator is defined first on first-order stresses and then applied twice. For a non-holonomic stress $$\psi(C,D,\bar d,d,\nabla \bar d) = \psi_e(C,D)+\psi_d(\alpha_d)+\psi_h(D)+\psi_{\text{mic}}(d,\bar d,\nabla \bar d),$$4, one obtains

$$\psi(C,D,\bar d,d,\nabla \bar d) = \psi_e(C,D)+\psi_d(\alpha_d)+\psi_h(D)+\psi_{\text{mic}}(d,\bar d,\nabla \bar d),$$5

where

$$\psi(C,D,\bar d,d,\nabla \bar d) = \psi_e(C,D)+\psi_d(\alpha_d)+\psi_h(D)+\psi_{\text{mic}}(d,\bar d,\nabla \bar d),$$6

On the boundary, $$\psi(C,D,\bar d,d,\nabla \bar d) = \psi_e(C,D)+\psi_d(\alpha_d)+\psi_h(D)+\psi_{\text{mic}}(d,\bar d,\nabla \bar d),$$7 can itself be decomposed, producing surface forces, higher-order boundary interactions, and, on piecewise smooth boundaries, edge or corner interactions (Segev, 2014).

For second-gradient damage models this gives a coordinate-free route from energetic dependence on $$\psi(C,D,\bar d,d,\nabla \bar d) = \psi_e(C,D)+\psi_d(\alpha_d)+\psi_h(D)+\psi_{\text{mic}}(d,\bar d,\nabla \bar d),$$8 or $$\psi(C,D,\bar d,d,\nabla \bar d) = \psi_e(C,D)+\psi_d(\alpha_d)+\psi_h(D)+\psi_{\text{mic}}(d,\bar d,\nabla \bar d),$$9 to balance laws and higher-order natural boundary conditions. A plausible implication is that the phrase “second-order damage” may denote either a tensorial internal variable or a second-gradient energetic structure, and the two notions are compatible rather than mutually exclusive (Segev, 2014).

5. Micromechanical and harmonic foundations

A micromechanics-based route starts from a crack density function $\boldsymbol{Y}:=-\frac{\partial \psi}{\partial D}.$0, expanded as

$\boldsymbol{Y}:=-\frac{\partial \psi}{\partial D}.$1

where $\boldsymbol{Y}:=-\frac{\partial \psi}{\partial D}.$2 are totally symmetric traceless harmonic tensors. In many homogenization schemes for an initially isotropic elastic matrix with a dilute distribution of penny-shaped cracks, the effective damage affecting elasticity depends on the scalar, second-order, and fourth-order crack density tensors. The resulting fourth-order damage tensor can be written

$\boldsymbol{Y}:=-\frac{\partial \psi}{\partial D}.$3

so a classical micromechanical description is naturally fourth-order (Desmorat et al., 2017).

The key reduction mechanism is the harmonic product

$\boldsymbol{Y}:=-\frac{\partial \psi}{\partial D}.$4

defined for harmonic tensors. For second-order harmonic tensors $\boldsymbol{Y}:=-\frac{\partial \psi}{\partial D}.$5 and $\boldsymbol{Y}:=-\frac{\partial \psi}{\partial D}.$6,

$\boldsymbol{Y}:=-\frac{\partial \psi}{\partial D}.$7

Sylvester’s theorem and the harmonic factorization theorem imply that fourth-order harmonic tensors can be factorized into lower-order harmonic factors. In 2D the fourth-order harmonic part is an exact harmonic square,

$\boldsymbol{Y}:=-\frac{\partial \psi}{\partial D}.$8

and for practical 3D crack density measurements on thin or thick walled structures the fourth-order harmonic part can be represented as a harmonic square over the set of mechanically accessible directions

$\boldsymbol{Y}:=-\frac{\partial \psi}{\partial D}.$9

Then, for all $\boldsymbol{Y}$00,

$\boldsymbol{Y}$01

with $\boldsymbol{Y}$02 scalar and $\boldsymbol{Y}$03, $\boldsymbol{Y}$04 second-order deviators (Desmorat et al., 2017).

This permits a second-order reparametrization of the fourth-order damage tensor: $\boldsymbol{Y}$05 The associated Gibbs free enthalpy density is taken as

$\boldsymbol{Y}$06

which yields the effective compliance

$\boldsymbol{Y}$07

A phenomenological variant introduces a second-order tensor $\boldsymbol{Y}$08 with

$\boldsymbol{Y}$09

and chooses

$\boldsymbol{Y}$10

with

$\boldsymbol{Y}$11

This gives

$\boldsymbol{Y}$12

and, in the illustrating discrete-element example on micro-concrete with $\boldsymbol{Y}$13 MPa, $\boldsymbol{Y}$14, about $\boldsymbol{Y}$15 particles, and more than $\boldsymbol{Y}$16 broken beams at the end of loading, the low-damage response is consistent with a hydrostatic sensitivity $\boldsymbol{Y}$17 (Desmorat et al., 2017).

The 2016 harmonic reconstruction results sharpen the algebraic side of this picture. They show explicit equivariant reconstruction formulas for transverse isotropic and orthotropic fourth-order harmonic tensors using second-order covariants, and for trigonal and tetragonal classes up to a cubic fourth-order covariant remainder. For instance, a transversely isotropic fourth-order harmonic tensor satisfies

$\boldsymbol{Y}$18

At the same time, there is no globally defined equivariant section $\boldsymbol{Y}$19 on the full space $\boldsymbol{Y}$20, and cubic symmetry is an obstruction to pure second-order reconstruction. This clarifies both the mathematical strength and the limitation of second-order reductions of fourth-order damage descriptions (Olive et al., 2016).

6. Spectral representation and algorithmic tangents

Because second-order damage models are frequently written in principal directions, spectral decomposition is central to implementation. For a symmetric second-order tensor $\boldsymbol{Y}$21, the invariants are

$\boldsymbol{Y}$22

and the eigenvalues are roots of

$\boldsymbol{Y}$23

Introducing the deviatoric tensor

$\boldsymbol{Y}$24

with

$\boldsymbol{Y}$25

the eigenvalues can be written in closed trigonometric form, and the tensor admits the spectral representation

$\boldsymbol{Y}$26

A key identity is

$\boldsymbol{Y}$27

which yields eigenprojectors without computing eigenvectors explicitly (Panteghini, 2023).

The same work gives derivatives needed for consistent Newton tangents: $\boldsymbol{Y}$28 and closed-form expressions for $\boldsymbol{Y}$29. It also distinguishes three multiplicity cases: all eigenvalues distinct, two equal, and triple eigenvalue. In the two-eigenvalue case,

$\boldsymbol{Y}$30

for the repeated eigenspace, while in the triple-eigenvalue case

$\boldsymbol{Y}$31

For isotropic tensor functions $\boldsymbol{Y}$32 with shared eigenbasis,

$\boldsymbol{Y}$33

which provides the consistent tangent operator in spectral form (Panteghini, 2023).

These formulas transfer directly to second-order damage tensors. A plausible implication is that robust treatment of coincident damage eigenvalues is not a secondary numerical detail but part of the constitutive definition, because isotropic damage states and transverse-isotropic states are precisely multiplicity cases of a symmetric second-order tensor (Panteghini, 2023).

7. Physics-augmented machine-learning generalizations

A recent extension formulates anisotropic damage mechanics as a data-driven constitutive model based explicitly on the second-order damage tensor approach for both compressible and incompressible materials. The free energy is expressed as an isotropic function of the right Cauchy–Green tensor together with structural tensors that encode virgin anisotropy or damage-induced anisotropy. For compressible materials, the generic free energy is

$\boldsymbol{Y}$34

with generalized structural tensors

$\boldsymbol{Y}$35

and invariants

$\boldsymbol{Y}$36

The stress is obtained from

$\boldsymbol{Y}$37

Polyconvexity is enforced by composing convex, non-decreasing ICNNs with polyconvex invariants, and normality is enforced by an energy shift and additional stress-correction terms so that

$\boldsymbol{Y}$38

(Amiri-Hezaveh et al., 23 Jul 2025).

Damage attenuation is modeled by a family of convex, decreasing functions

$\boldsymbol{Y}$39

with fixed exponents $\boldsymbol{Y}$40. The scalar conjugate forces are

$\boldsymbol{Y}$41

and at tensor level

$\boldsymbol{Y}$42

Damage evolution is driven by a potential

$\boldsymbol{Y}$43

with

$\boldsymbol{Y}$44

and evolution equations

$\boldsymbol{Y}$45

$\boldsymbol{Y}$46

subject to Kuhn–Tucker conditions. The derivatives $\boldsymbol{Y}$47 are represented by MLPs composed with softplus and exponential to enforce non-negativity (Amiri-Hezaveh et al., 23 Jul 2025).

The same framework introduces a new anisotropic generic format for initially isotropic materials: $\boldsymbol{Y}$48 so anisotropy is induced solely through direction-dependent damage variables $\boldsymbol{Y}$49. To reduce training cost, a two-stage decoupled scheme is used: Stage I identifies elastic energies from unloading branches while treating attenuation values cycle-wise as constants, and Stage II calibrates damage-related parameters on full loading cycles. A short Stage III can then fine-tune all parameters jointly (Amiri-Hezaveh et al., 23 Jul 2025).

The numerical evidence reported for this data-driven extension is specific. Benchmarks include incompressible isotropic, transversely isotropic, and compressible orthotropic materials, and the framework is also validated against experimental data on double-network hydrogels with anisotropic Mullins-type damage. In those experiments, an isotropic damage model fits the equi-biaxial response reasonably but fails to capture directional differences in unequal-biaxial and planar tests, especially in the $\boldsymbol{Y}$50-direction stress, whereas the anisotropic second-order damage tensor model captures all three loading modes and principal stress components accurately while maintaining non-negative energy and positive dissipation (Amiri-Hezaveh et al., 23 Jul 2025).