2D and 3D Rotation

The rotation matrix is used to rotate a vector or point in a coordinate system by a specific angle (in 2D)
or by a specific axis and angle (in 3D).

1. 2D Rotation Matrix

For a rotation by an angle θ (in radians or degrees) in a 2D coordinate system, the rotation matrix is:

cos θ − sin θ
R2D (θ) = [ ]
sin θ cos θ
​ ​ ​

If (x, y) is the point you want to rotate, the rotated point (x′ , y ′ ) is calculated as:

x′ cos θ − sin θ x
[ ′] = [ ][ ]
y sin θ cos θ y
​ ​ ​ ​

2. 3D Rotation Matrix

In 3D, a rotation is defined around a specific axis (x, y , or z ) by an angle θ . There are three basic rotation
matrices:

(a) Rotation about the X-axis:

1 0 0
RX (θ) = 0 cos θ
​ ​ ​ ​ − sin θ ​ ​

0 sin θ cos θ

(b) Rotation about the Y-axis:

cos θ 0 sin θ
RY (θ) =
​ 0 ​ 1​ 0 ​ ​

− sin θ 0 cos θ

(c) Rotation about the Z-axis:

cos θ − sin θ 0
RZ (θ) = sin θ
​ ​ ​ cos θ ​ 0 ​ ​

0 0 1

To rotate a point (x, y, z), you multiply it by the appropriate rotation matrix based on the axis of
rotation.

Combining Rotations in 3D

For a general rotation in 3D, you can combine rotations about all three axes:

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 1/14
 R3D (θx , θy , θz ) = RX (θx ) ⋅ RY (θy ) ⋅ RZ (θz )
​ ​ ​ ​ ​ ​ ​ ​ ​ ​

The order of multiplication matters because matrix multiplication is not commutative.

Let me know if you'd like an example of how to use these!

Stress and Strain Invariants

Stress and strain invariants are scalar quantities derived from the stress or strain tensor. These
invariants are independent of the coordinate system and provide a measure of the overall state of stress
or strain at a point. They are commonly used in continuum mechanics, material science, and
engineering analyses.

1. Stress Tensor

The Cauchy stress tensor in 3D is a 3 × 3 symmetric matrix:

σxx ​ σxy ​ σxz ​

σ = σyx ​ ​ ​ σyy ​ ​ σyz ​ ​ ​

σzx ​ σzy ​ σzz ​

Because it is symmetric (σij ​ = σji ), it contains 6 independent components.

​

Stress Invariants

The stress invariants are derived from the characteristic equation of the stress tensor:

det(σ − λI) = 0

This results in three principal stresses σ1 , σ2 , and σ3 , which lead to the three invariants:
​ ​ ​

1. First invariant (I1 ): Trace of the stress tensor (sum of principal stresses):
​

I1 = tr(σ) = σ1 + σ2 + σ3
​ ​ ​ ​

In terms of components:

I1 = σxx + σyy + σzz

​ ​ ​ ​

2. Second invariant (I2 ): Sum of principal minor determinants:

​

I 2 = σ1 σ2 + σ2 σ3 + σ3 σ1
​ ​ ​ ​ ​ ​ ​

In terms of components:
2 2 2
I2 = σxx σyy + σyy σzz + σzz σxx − σxy
​ ​ ​ − σyz − σzx ​ ​ ​ ​ ​ ​ ​

3. Third invariant (I3 ): Determinant of the stress tensor:

​

I3 = det(σ) = σ1 σ2 σ3
​ ​ ​ ​

In terms of components:

2
Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 2/14
 2
I3 = σxx (σyy σzz − σyz
​ ​) − σxy (σxy σzz − σyz σzx ) + σxz (σxy σyz − σyy σzx )
​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​

Strain Tensor

The strain tensor in 3D is also a symmetric 3 × 3 matrix, often represented as ϵ:

ϵxx ​ ϵxy ​ ϵxz ​

ϵ = ϵyx ​ ​ ϵyy ​
​ ϵyz ​
​ ​

ϵzx ​ ϵzy ​ ϵzz ​

Strain Invariants

The strain invariants are defined analogously to the stress invariants, based on the principal strains (
ϵ1 , ϵ2 , ϵ3 ):
​ ​ ​

1. First invariant (I1ϵ ): Trace of the strain tensor (volumetric strain):

​

I1ϵ = tr(ϵ) = ϵxx + ϵyy + ϵzz

​ ​ ​ ​

2. Second invariant (I2ϵ ): Sum of principal minor determinants:

​

I2ϵ = ϵ1 ϵ2 + ϵ2 ϵ3 + ϵ3 ϵ1
​ ​ ​ ​ ​ ​ ​

ϵ
3. Third invariant (I3 ): Determinant of the strain tensor:
​

I3ϵ = det(ϵ) ​

Physical Significance

1. First invariant (I1 ): Represents the mean or hydrostatic component of stress/strain.

​

2. Second invariant (I2 ): Related to the deviatoric stress/strain (shear component).

​

3. Third invariant (I3 ): Describes the volumetric changes in the material.

​

Would you like me to explain any specific application or provide an example calculation? 😊
Let's dive deeper into stress and strain invariants, their significance, derivation, and physical
interpretation in detail. This topic plays a crucial role in solid mechanics, particularly in material
behavior, yielding criteria, and failure theories.

Stress Tensor and Invariants

The stress tensor (σij ) represents the state of stress at a point in a material. It accounts for the normal
​

and shear stresses on a given plane.

Stress Tensor in 3D:

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 3/14
 σxx ​ σxy ​ σxz ​

σ = σxy ​
​
​ σyy ​
​ σyz ​
​ ​

σxz ​ σyz ​ σzz ​

σxx , σyy , σzz : Normal stresses along the x, y, z axes.

​ ​ ​

σxy , σyz , σzx : Shear stresses in the respective planes.

​ ​ ​

Principal Stresses and Principal Axes

The principal stresses (σ1 , σ2 , σ3 ) are the eigenvalues of the stress tensor.
​ ​ ​

The corresponding eigenvectors define the principal axes, where the shear stresses vanish.
The characteristic equation of the stress tensor is:

det(σ − λI) = 0

Here, λ are the eigenvalues (σ1 , σ2 , σ3 ). ​ ​ ​

Stress Invariants

Stress invariants are derived from the characteristic equation of the stress tensor. They are
independent of the coordinate system and are expressed in terms of principal stresses.

1. First Invariant (I1 ) – Trace ​

The first invariant is the trace of the stress tensor, representing the sum of the normal stresses or the
hydrostatic stress:

I1 = tr(σ) = σ1 + σ2 + σ3
​ ​ ​ ​

In terms of components:

I1 = σxx + σyy + σzz

​ ​ ​ ​

Physical Meaning: Measures the overall magnitude of normal stresses. For hydrostatic stress
states (σxx = σyy = σzz ), I1 dominates.
​ ​ ​ ​

2. Second Invariant (I2 ) – Deviatoric Stress Contribution

​

The second invariant is related to the sum of products of principal stresses (sum of minor determinants):

I 2 = σ1 σ2 + σ2 σ3 + σ3 σ1
​ ​ ​ ​ ​ ​ ​

In terms of components:
2 2 2
I2 = σxx σyy + σyy σzz + σzz σxx − σxy
​ ​ ​ − σyz − σzx ​ ​ ​ ​ ​ ​ ​

Physical Meaning: Correlates with the deviatoric stress (shear stress effects). It is used in yielding
criteria, such as the von Mises stress.

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 4/14
3. Third Invariant (I3 ) – Determinant ​

The third invariant is the determinant of the stress tensor or the product of principal stresses:

I3 = det(σ) = σ1 σ2 σ3
​ ​ ​ ​

In terms of components:
2
I3 = σxx (σyy σzz − σyz
​ ​ ) − σxy (σxy σzz − σyz σzx ) + σxz (σxy σyz − σyy σzx )
​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​

Physical Meaning: Indicates volumetric stress effects and is relevant for pressure-dominated
states.

Hydrostatic and Deviatoric Components

The stress tensor can be decomposed into two parts:

1. Hydrostatic Stress Tensor (σ hydro ): ​

Represents the mean or isotropic stress:

I1
σ hydro = I
​

3
​ ​

I is the identity matrix.

2. Deviatoric Stress Tensor (σ dev ): ​

Represents the shear or distortion stresses:

σ dev = σ − σ hydro
​ ​

Deviatoric stresses are responsible for material deformation.

Strain Tensor and Invariants

The strain tensor (ϵij ) represents the deformation state in a material.
​

Strain Tensor in 3D:

ϵxx ​ ϵxy ​ ϵxz ​

ϵ = ϵxy ​ ​ ϵyy ​ ​ ϵyz ​ ​ ​

ϵxz ​ ϵyz ​ ϵzz ​

ϵxx , ϵyy , ϵzz : Normal strains.

​ ​ ​

ϵxy , ϵyz , ϵzx : Shear strains.

​ ​ ​

Strain Invariants

The strain invariants are similar in form to stress invariants and are defined based on principal strains (
ϵ1 , ϵ2 , ϵ3 ):
​ ​ ​

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 5/14
1. First Invariant (I1ϵ ) – Trace
​

I1ϵ = tr(ϵ) = ϵ1 + ϵ2 + ϵ3
​ ​ ​ ​

In terms of components:

I1ϵ = ϵxx + ϵyy + ϵzz

​ ​ ​ ​

Physical Meaning: Represents the volumetric strain (change in volume).

2. Second Invariant (I2ϵ ) ​

I2ϵ = ϵ1 ϵ2 + ϵ2 ϵ3 + ϵ3 ϵ1
​ ​ ​ ​ ​ ​ ​

In terms of components:

I2ϵ = ϵxx ϵyy + ϵyy ϵzz + ϵzz ϵxx − ϵ2xy − ϵ2yz − ϵ2zx
​ ​ ​ ​ ​ ​ ​ ​ ​

Physical Meaning: Describes distortional effects in strain.

3. Third Invariant (I3ϵ ) ​

I3ϵ = det(ϵ) = ϵ1 ϵ2 ϵ3
​ ​ ​ ​

Physical Meaning: Indicates volume changes caused by deformation.

Applications of Invariants

1. Material Behavior and Failure:

Yield criteria like von Mises and Tresca use stress invariants to predict yielding.
Drucker-Prager and Mohr-Coulomb criteria incorporate hydrostatic stress (I1 ) for pressure- ​

sensitive materials.
2. Finite Element Analysis:
Stress invariants help simplify complex stress states into scalar measures for visualization and
computation.
3. Plasticity and Damage Models:
Deviatoric invariants (I2 ) are crucial for understanding plastic deformation.
​

4. Hydrostatic and Deviatoric Decomposition:

Engineers use hydrostatic and deviatoric components to separate volumetric and shear
effects in structural materials.

Let me know if you need any additional clarifications, derivations, or practical examples! 😊
In the context of material failure criteria, particularly the von Mises yield criterion, J2 plays a central ​

role. Let's break it down step by step to explain what J invariants are and their significance in material
failure.

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 6/14
Deviatoric Stress and the J -Invariants

To describe yielding and failure of materials under stress, it is common to separate the stress tensor into
two components:

1. Hydrostatic Stress Component (σ hydro ): Represents the mean or volumetric stress, associated
​

with dilation (volume change).

I1
σ hydro = I, where I1 = tr(σ).
​

3
​ ​ ​

2. Deviatoric Stress Component (σ dev ): Represents the distortional (shear) stress, responsible for
​

shape change (no volume change).

σ dev = σ − σ hydro ​ ​

The deviatoric stress tensor is:

′
σxx ​ σxy ​ σxz ​

′
σ dev ​ = σxy ​
​
​ σyy ​
​ σyz , ​
​ ​

′
σxz ​ σyz ​ σzz ​

where the deviatoric normal stresses are:

′ I1 ′ I1 ′ I1
= σxx − , = σyy − , = σzz − .
​ ​ ​

σxx σyy σzz

3 3 3
​ ​ ​ ​ ​ ​ ​ ​ ​

J -Invariants
The J -invariants are derived from the deviatoric stress tensor σ dev and are coordinate-system ​

independent. They describe distortional (shear) effects in a material:

1. First Deviatoric Invariant (J1 ): ​

J1 = tr(σ dev ) ​ ​

For the deviatoric stress tensor, J1 ​ = 0, because the trace of the deviatoric stress tensor is
always zero.
2. Second Deviatoric Invariant (J2 ): ​

1
J2 = σ dev : σ dev
2
​ ​ ​ ​

In component form:

1
J2 = ∑ σij′ σij′
2
​ ​ ​ ​

i,j

Physical Meaning: Represents the magnitude of the distortional (shear) stress. It is

proportional to the square of the von Mises equivalent stress.
3. Third Deviatoric Invariant (J3 ): ​

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 7/14
 J3 = det(σ dev )
​ ​

Physical Meaning: Related to the shape of the deviatoric stress state. It is often associated
with the Lode angle, which measures the relative influence of the principal deviatoric stresses.

Von Mises Yield Criterion and J2 ​

The von Mises yield criterion is based solely on J2 , as it assumes yielding depends only on distortional
​

(shear) energy, independent of hydrostatic stress. The criterion states:

3J2 = σeq ​ ​ ​

Here:

σeq : Von Mises equivalent stress (used to compare complex stress states to uniaxial tests).
​

In terms of principal stresses:

1
σeq = [(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 ]
2
​ ​ ​ ​ ​ ​ ​ ​ ​

Alternatively, J2 in terms of stress tensor components can be written as:

​

1
J2 = [(σxx − σyy )2 + (σyy − σzz )2 + (σzz − σxx )2 ] + σxy
2 2
+ σyz 2
+ σzx
6
​ ​ ​ ​ ​ ​ ​ ​ ​ ​

Significance of J2 : ​

1. Shear-Dominated Yielding: Von Mises criterion assumes yielding occurs due to shear stresses
only. J2 quantifies the energy stored in distortional deformation.
​

2. Material Independence: It works well for ductile materials (metals) because their failure is
primarily shear-driven.
3. Hydrostatic Independence: Since J2 excludes hydrostatic stress (I1 ), it is ideal for describing
​ ​

yielding that is not affected by pressure.

Applications of J2 : ​

1. Von Mises Stress Calculation: Used extensively in engineering to check material failure under
complex loading.
2. Finite Element Analysis (FEA): J2 is a key input for plasticity models and stress visualization.
​

3. Ductile Failure Prediction: For materials where shear governs failure (e.g., metals), J2 -based ​

criteria provide accurate predictions.

Let me know if you'd like an example of von Mises stress calculation or a deeper dive into another
aspect! 😊
The I invariants (stress invariants) and J invariants (deviatoric stress invariants) are both used to
describe the state of stress in a material. They are closely related but serve different purposes in
material analysis. Below is a detailed explanation of each set of invariants and their relationships.

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 8/14
1. Stress Tensor Invariants (I )

The I invariants are calculated from the stress tensor σ , which includes both normal and shear
components. They are derived from the characteristic equation of the stress tensor and represent
quantities that remain constant regardless of the coordinate system.

Definition of I -Invariants

1. First Stress Invariant (I1 ): The trace of the stress tensor, representing the sum of the normal
​

stresses:

I1 = tr(σ) = σx + σy + σz = σ1 + σ2 + σ3
​ ​ ​ ​ ​ ​ ​

Physical Meaning: Represents the mean or hydrostatic stress.

2. Second Stress Invariant (I2 ): Related to the sum of products of the principal stresses:
​

I 2 = σ1 σ2 + σ2 σ3 + σ3 σ1
​ ​ ​ ​ ​ ​ ​

In terms of components:
2 2 2
I2 = σx σy + σy σz + σz σx − τxy
​ ​ − τyz
​ − τzx ​ ​ ​ ​ ​ ​ ​

Physical Meaning: Correlates with both normal and shear stress effects.
3. Third Stress Invariant (I3 ): The determinant of the stress tensor or the product of the principal
​

stresses:

I 3 = σ1 σ2 σ3
​ ​ ​ ​

In terms of components:

I3 = det(σ) ​

Physical Meaning: Represents volumetric stress effects.

2. Deviatoric Stress Tensor and J -Invariants

The deviatoric stress tensor (σ dev ) describes the stress state excluding hydrostatic stress. The J
​

invariants are derived from this tensor and are directly linked to distortional or shear effects.

Definition of J -Invariants

1. First Deviatoric Invariant (J1 ): ​

J1 = tr(σ dev ) ​ ​

For the deviatoric stress tensor, J1 ​ = 0, because the trace of the deviatoric stress tensor is
zero by definition (hydrostatic stress is removed).
2. Second Deviatoric Invariant (J2 ): ​

1
J2 = σ dev : σ dev
2
​ ​ ​ ​

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 9/14
 In component form:

1
J2 = [(σx − σy )2 + (σy − σz )2 + (σz − σx )2 ] + τxy
2 2
+ τyz 2
+ τzx
6
​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​

Physical Meaning: Quantifies the magnitude of shear stress.

3. Third Deviatoric Invariant (J3 ): ​

J3 = det(σ dev )
​ ​

Physical Meaning: Represents distortional stress effects related to the relative magnitudes of
the principal deviatoric stresses. It is often used to calculate the Lode angle, which
distinguishes different shear-dominant stress states.

Relationship Between I and J Invariants

The I -invariants and J -invariants are connected because J -invariants are derived from the deviatoric
component of the stress tensor. The relationship is primarily defined by the decomposition of the stress
tensor into hydrostatic and deviatoric parts:

σ = σ hydro + σ dev ​ ​

where:

σ hydro = I31 I
​
​

σ dev = σ − σ hydro
​ ​

Key Relationships:

1. First Invariant Relationship (I1 and J1 ): ​ ​

The trace of the stress tensor equals the trace of the hydrostatic stress tensor:

I1 = 3P ,
​ J1 = 0 ​

I1
Here, P = 3
is the hydrostatic pressure.
​

2. Second Invariant Relationship (I2 and J2 ): ​ ​

The second deviatoric invariant is related to the second stress invariant:

1
J2 = I2 − I12
3
​ ​ ​ ​

This relationship removes the hydrostatic stress effects from I2 . ​

3. Third Invariant Relationship (I3 and J3 ): ​ ​

The third deviatoric invariant is related to the third stress invariant:

1 2
J3 = I3 − I1 I2 + I13
3 27
​ ​ ​ ​ ​ ​ ​

This relationship isolates the volumetric stress contributions from I3 . ​

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 10/14
Physical Interpretation of the Relationships

I1 represents the hydrostatic stress, while J2 represents the shear stress.

​ ​

J2 and J3 are essential for describing yielding and distortion in materials because they exclude
​

hydrostatic stress, which does not contribute to plastic deformation.

The von Mises yield criterion uses J2 because it assumes that yielding is driven by distortional
​

energy (shear effects), not volumetric changes.

Summary Table of Relationships

Invariant Expression Physical Meaning

I1 ​ σ1 + σ2 + σ3
​ ​ ​
Hydrostatic stress (mean normal stress).

J1 ​
0 Trace of deviatoric stress tensor (always zero).

I2 ​
Sum of minor principal determinants Combines normal and shear stresses.
1 2
J2 ​ I2 −​ I
3 1
​ ​
Magnitude of deviatoric stress (shear stress effects).

I3 ​ Determinant of σ Volumetric stress effects.

J3 ​ I3 − 13 I1 I2 +
​ ​ ​ ​
2 3
​I
27 1
​
Distortional effects, related to Lode angle.

Practical Use of I and J Invariants in Material Failure

Von Mises Criterion: Relies on J2 (distortional energy). ​

3J2 = σeq
​ ​ ​

Hydrostatic Stress Effects: Materials like soils and polymers use I1 (pressure sensitivity) in failure
​

criteria (e.g., Drucker-Prager).

Lode Angle and J3 : Differentiates between shear-dominated stress states (e.g., uniaxial tension vs.
​

pure shear).

Let me know if you'd like examples or derivations! 😊

Yes, besides the commonly used I -invariants (stress invariants) and J -invariants (deviatoric stress
invariants), there are other types of invariants used in the context of stress-strain relationships and
material behavior analysis. These include strain invariants, energy-based invariants, and others
tailored to specific material models or failure theories. Let’s explore the broader landscape:

1. Strain Invariants
Similar to the stress invariants, the strain tensor (ε) can also be analyzed using invariants. These are
useful in describing material deformation independently of the coordinate system.

Principal Strains

Strain invariants are derived from the principal strains ε1 , ε2 , ε3 or the strain tensor itself.
​ ​ ​

Strain Invariants (E ):

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 11/14
 1. First Strain Invariant (E1 ): ​

E1 = tr(ε) = ε1 + ε2 + ε3
​ ​ ​ ​

Physical Meaning: Volumetric strain or the relative change in volume of the material.
2. Second Strain Invariant (E2 ): ​

E2 = ε1 ε2 + ε2 ε3 + ε3 ε1
​ ​ ​ ​ ​ ​

Physical Meaning: Combines normal and shear deformation effects.

3. Third Strain Invariant (E3 ): ​

E3 = ε1 ε2 ε3
​ ​ ​ ​

Physical Meaning: Represents the total deformation, including all volumetric and shear
components.

Deviatoric Strain Invariants (e):

These are derived from the deviatoric strain tensor (εdev ), which removes the volumetric strain (E1 ):
​ ​

1. e1 = tr(εdev ) = 0
​ ​

1
2. e2 = 2 εdev : εdev
​ ​ ​ ​

3. e3 = det(εdev )
​ ​

2. Energy-Based Invariants
Energy-based invariants are crucial in modeling material behavior, particularly for nonlinear elasticity,
plasticity, and failure criteria.

Strain Energy Density (SED)

The strain energy density (W ) measures the amount of energy stored in a material due to deformation.
It can be decomposed into volumetric and distortional parts:

1. Volumetric Strain Energy:

2
1
= f (I1 ) = K ( )
I1 ​

Wvol
2 3
​ ​ ​ ​

Related to bulk modulus (K ) and hydrostatic deformation.

2. Distortional Strain Energy:

Wdev = f (J2 ) ​

Related to shear modulus (G) and distortional deformation.

3. Combined Invariants for Advanced Material Models

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 12/14
In advanced material models, combinations of invariants (stress, strain, and energy-based) are used to
describe complex behaviors such as anisotropy, pressure sensitivity, and time-dependent behavior.

Examples:

1. Drucker-Prager Criterion (for pressure-sensitive materials like soils and rocks):

Uses both I1 and J2 :
​ ​

F = αI1 + ​ J2 − k = 0
​ ​

2. Modified Mohr-Coulomb Criterion:

Includes I1 and the Lode angle (θ ), which depends on J2 and J3 .
​ ​ ​

3. Hyperelastic Models:
Invariants of the Green-Lagrange strain tensor (E ) or Cauchy-Green deformation tensor (
C ) are used:
I1 = tr(C)
​

I2 = 12 (tr(C)2 − tr(C 2 ))
​ ​

I3 = det(C)
​

4. Polar Decomposition and Rotation-Based Invariants

In cases of finite deformation, the deformation gradient F is decomposed into stretch (U ) and rotation
(R) components using polar decomposition:

F =R⋅U

Invariants are derived from the stretch tensor (U ):

1. I1 ​ = tr(U 2 )
2. I2 ​ = tr(U 2 ) − tr(U )2
3. I3 ​ = det(U )

5. Plasticity and Damage-Related Invariants

In elasto-plastic and damage models, invariants are used to track deformation history and damage
evolution:

1. Accumulated Plastic Strain (p):

Scalar invariant representing the total plastic deformation:

t
2 pl pl
p=∫ ε̇ : ε̇ dt
3
​ ​ ​

2. Damage Variables (D ):
Scalar invariant tracking material degradation.

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 13/14
6. Higher-Order Tensor Invariants
For anisotropic materials, higher-order invariants are used:

Fabric Tensors: Used in modeling materials with preferred orientations (e.g., composites, granular
materials).
Anisotropic Invariants: Depend on material symmetry and can involve the stress and strain
tensors in combination with structural tensors.

Summary
Type Invariants Physical Meaning
Stress
I1 , I2 , I3 Hydrostatic stress, combined stress effects.
Invariants
​ ​ ​

Deviatoric
J2 , J3 Shear stress and distortional effects.
Stress
​ ​

Strain
E1 , E2 , E3 ; Deviatoric: e2 , e3 Volumetric strain, shear strain.
Invariants
​ ​ ​ ​

Volumetric (Wvol ) and distortional (Wdev )

Energy-Based Bulk and shear energy contributions.
​ ​

energy.
Plastic strain invariants (p), damage Cumulative plastic deformation and damage
Plasticity
variables (D ). tracking.

Advanced Pressure-sensitivity, anisotropy, and complex

Lode angle (θ ), combinations of I1 , J2 , J3 .
Models behavior.
​ ​ ​

These invariants are central to material modeling in engineering, particularly in finite element analysis
(FEA), constitutive modeling, and failure prediction. If you need detailed examples of how these
invariants are calculated or used, let me know! 😊

Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 14/14