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One important thing in the continuum mechanics is the choice of the stress and the strain. This short note focuses on summary of the different types of stresses and strains and explains why we need many of them. The mathematical definitions of these stresses and strains can be found in many textbooks!
One important thing in the continuum mechanics is the choice of the stress and the strain. This short note focuses on summary of the different types of stresses and strains and explains why we need many of them. The mathematical definitions of these stresses and strains can be found in many textbooks!
1. Tensile example
1. Tensile example
In this example, two different types of stress and strain are considered. If the stress and strain based on the original length and cross section, we have the engineering stress and engineering strain. If the stress and strain based on the current length and cross section, we have logarithmic strain and true stress.
In this example, two different types of stress and strain are considered. If the stress and strain based on the original length and cross section, we have the engineering stress and engineering strain. If the stress and strain based on the current length and cross section, we have logarithmic strain and true stress.
If the elongation is infinitesimal or very small compared the initial length, there is no difference between those stresses and strains. In this case, it is more convenient to formulate the equilibrium equations in the undeformed geometry. However, if the elongation is large, the geometric nonlinearity (changes of length and cross section) should be considered.
If the elongation is infinitesimal or very small compared the initial length, there is no difference between those stresses and strains. In this case, it is more convenient to formulate the equilibrium equations in the undeformed geometry. However, if the elongation is large, the geometric nonlinearity (changes of length and cross section) should be considered.
2. Stress measures
2. Stress measures
Cauchy stress (True stress) is a current force acting on an area of the deformed body. Cauchy stress tensor is symmetric with 6 stress components describing the stress in a Cartesian coordinate system. Cauchy stress is commonly used for the infinitesimal deformation. Why not for large deformation? Because it is more convenient to formulate the motion (or equilibrium) equations when the "deformed area" does not change!
Cauchy stress (True stress) is a current force acting on an area of the deformed body. Cauchy stress tensor is symmetric with 6 stress components describing the stress in a Cartesian coordinate system. Cauchy stress is commonly used for the infinitesimal deformation. Why not for large deformation? Because it is more convenient to formulate the motion (or equilibrium) equations when the "deformed area" does not change!
For large deformation (finite deformation), the 1st Piola-Kirchhoff stress could be used. 1st Piola-Kirchhoff stress tensor is unsymmetric and defined as a current force acting on an area of the reference body. Besides, 2nd Piola-Kirchhoff stress is symmetric and defined as a reference force acting on an area of the reference body. Why do we need Piola-Kirchhoff stress?
For large deformation (finite deformation), the 1st Piola-Kirchhoff stress could be used. 1st Piola-Kirchhoff stress tensor is unsymmetric and defined as a current force acting on an area of the reference body. Besides, 2nd Piola-Kirchhoff stress is symmetric and defined as a reference force acting on an area of the reference body. Why do we need Piola-Kirchhoff stress?
Let consider a beam in a reference state (undeformed) and in a current state (deformed). If we want to know the tension of the beam along x-direction, we need to know the Piola-Kirchhoff stress as it tracks the tension along the axis of material, not in a fixed Cartesian space as Cauchy stress.
Let consider a beam in a reference state (undeformed) and in a current state (deformed). If we want to know the tension of the beam along x-direction, we need to know the Piola-Kirchhoff stress as it tracks the tension along the axis of material, not in a fixed Cartesian space as Cauchy stress.
Another stress measure is the Kirchhoff stress which is Cauchy stress scaled with volume change (Jacobian = determinant of deformation gradient). It gives some convenience for some numerical development.
Another stress measure is the Kirchhoff stress which is Cauchy stress scaled with volume change (Jacobian = determinant of deformation gradient). It gives some convenience for some numerical development.
3. Strain measures
3. Strain measures
The strain measures must be selected corresponding to the stress measures to give accurate strain energy density. This requirement refers as energetic conjugates. Some conjugate stress-strain pairs are:
The strain measures must be selected corresponding to the stress measures to give accurate strain energy density. This requirement refers as energetic conjugates. Some conjugate stress-strain pairs are:
Cauchy (True) stress/ Kirchhoff stress - Engineering (True) Strain
Cauchy (True) stress/ Kirchhoff stress - Engineering (True) Strain
1st Piola-Kirchhoff stress - Deformation Gradient
1st Piola-Kirchhoff stress - Deformation Gradient
2nd Piola-Kirchhoff stress - Green-Lagrange strain
2nd Piola-Kirchhoff stress - Green-Lagrange strain
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