连续介质力学的共轭应力应变对

1. First Piola-Kirchhoff Stress (P) and Deformation Gradient (F)

• First Piola-Kirchhoff Stress: Also known as the nominal stress, it relates the force in the current configuration to the area in the reference configuration.

• Deformation Gradient: Describes the local deformation of a material element.

• Conjugate Strain:F

• Usage: Convenient in formulations involving body forces and boundary conditions in the reference configuration.

2. Second Piola-Kirchhoff Stress (S) and Green-Lagrange Strain (E)

• Second Piola-Kirchhoff Stress: Related to the reference configuration, often used in theoretical and computational formulations.

• Green-Lagrange Strain: Measures the change in length and angles from the reference configuration.

• Conjugate Strain:



• Usage: Suitable for large deformations and used in finite element analysis.

3. Kirchhoff Stress (τ) and Logarithmic (Hencky) Strain (H)

• Kirchhoff Stress: Scaled version of the Cauchy stress, considering volume changes.

• Logarithmic Strain: Derived from the principal stretches, represents true strain.

• Conjugate Strain:





• Usage: Used in plasticity and viscoelasticity, representing true stress and strain measures.

4. Cauchy (True) Stress (σ) and Almansi Strain (e)

• Cauchy Stress: The true stress tensor, relating force to the current area.

• Almansi Strain: Measures the strain in the current configuration.

• Conjugate Strain:

  

• Usage: Suitable for formulations in the current configuration.

5. Mandel Stress (M) and  Green-Lagrange Strain (E)

• Mandel Stress: Used in plasticity, defined as M=F⋅S.

• Logarithmic Strain: Represents true strain, defined based on the right Cauchy-Green deformation tensor.

Green-Lagrange strain



• Usage: Useful in plasticity for finite deformations.

Comparison

1. Reference vs. Current Configuration:

• Reference Configuration: First Piola-Kirchhoff stress and second Piola-Kirchhoff stress are defined in the reference configuration, making them suitable for problems where the reference configuration is known and important.

• Current Configuration: Cauchy stress and Kirchhoff stress are defined in the current configuration, making them directly interpretable in terms of current forces and deformations.

2. Small vs. Large Deformations:

• Small Deformations: Cauchy stress and small strain measures (linear strain) are often used in small deformation problems.

• Large Deformations: Second Piola-Kirchhoff stress with Green-Lagrange strain, and Kirchhoff stress with logarithmic strain, are more suitable for large deformations.

3. Computational Use:

• Finite Element Analysis: Second Piola-Kirchhoff stress and Green-Lagrange strain are commonly used in finite element formulations due to their convenience in handling large deformations.

• Plasticity and Viscoelasticity: Mandel stress and logarithmic strain are often used in plasticity models to account for large, irreversible deformations.

Summary

• First Piola-Kirchhoff Stress (P) and Deformation Gradient (F)

• Second Piola-Kirchhoff Stress (S) and Green-Lagrange Strain (E)

• Kirchhoff Stress (τ) and Logarithmic Strain (H)

• Cauchy Stress (σ) and Almansi Strain (e)

• Mandel Stress (M) and Green-Lagrange Strain (E)

While both the second Piola-Kirchhoff (PK2) stress and Mandel stress are conjugate to the Green-Lagrange strain tensor, they serve different purposes and have different definitions. Here are the key differences:

Second Piola-Kirchhoff (PK2) Stress

1. Definition:

• The PK2 stress tensor (S) is defined in the reference (undeformed) configuration.

• It is given by:

  

• Here,σ is the Cauchy stress tensor,F is the deformation gradient, and J=det(F).

2. Use:

• PK2 stress is commonly used in theoretical and computational formulations involving the reference configuration.

• It is particularly useful for problems involving large deformations and is often used in finite element analysis.

3. Configuration:

• The PK2 stress relates forces in the deformed configuration to areas in the reference configuration.

Mandel Stress

1. Definition:

• The Mandel stress tensor (M) is a transformed version of the PK2 stress, incorporating the deformation gradient.

• It is given by:

  

• This relationship transforms the PK2 stress into a form that is useful for specific types of deformation analysis.

2. Use:

• Mandel stress is often used in plasticity and finite deformation theories.

• It provides a measure that is convenient for analyzing material behavior under plastic deformation and other complex loading conditions.

3. Configuration:

• The Mandel stress is still defined in terms of the reference configuration but incorporates the deformation gradient, making it more suitable for certain analytical and computational approaches.

Summary of Differences

• PK2 Stress:

• Defined purely in the reference configuration.

• Directly relates to forces and areas in the undeformed state.

• Used in general large deformation problems and finite element analysis.

• Mandel Stress:

• Incorporates the deformation gradient into the stress measure.

• Useful in plasticity and for specific deformation analyses.

• Provides a more complex but insightful view of stress that accounts for the deformation gradient.

Conjugate Strain

Despite these differences, both stress measures are conjugate to the Green-Lagrange strain tensor (E), which measures the change in length and angles from the reference configuration and is suitable for large deformations.

The Almansi strain tensor, also known as the Euler-Almansi strain tensor, is a measure of strain defined in the current (deformed) configuration of a material. It is particularly useful for describing large deformations. The Almansi strain tensor quantifies the deformation relative to the current configuration, making it suitable for problems involving large deformations.