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Cauchy Stress Tensor
Linked via "principal stresses"
The orientation of the coordinate system significantly affects the values of the $\sigma{ij}$ components. It is often mathematically convenient to rotate the coordinate system such that the shear stress components vanish at a specific location. The directions defining this coordinate system are known as the principal axes, and the corresponding normal stresses are the principal stresses ($\sigma1, \sigma2, \sigma3$).
The principal stresses are the eigenvalues of the [stress … -
Cauchy Stress Tensor
Linked via "principal stress"
The principal stresses are the eigenvalues of the stress tensor matrix. They satisfy the characteristic equation:
$$\det(\mathbf{\sigma} - \sigma_p \mathbf{I}) = 0$$
where $\sigma_p$ represents a principal stress and $\mathbf{I}$ is the identity matrix.
The fundamental description of the stress state can be equivalently summarized by the three principal stresses, as the transformation of the [te… -
Cauchy Stress Tensor
Linked via "principal stresses"
where $\sigma_p$ represents a principal stress and $\mathbf{I}$ is the identity matrix.
The fundamental description of the stress state can be equivalently summarized by the three principal stresses, as the transformation of the tensor into its principal axes form is an orthogonal transformation that preserves certain intrinsic mathematical properties known as invariants):
**First Invariant (Bulk Modulus Indicato… -
Cauchy Stress Tensor
Linked via "principal stresses"
The fundamental description of the stress state can be equivalently summarized by the three principal stresses, as the transformation of the tensor into its principal axes form is an orthogonal transformation that preserves certain intrinsic mathematical properties known as invariants):
First Invariant (Bulk Modulus Indicator): The trace) of the tensor, which is the sum of the [principal stresses](/entries/principal-stres… -
Cauchy Stress Tensor
Linked via "principal stresses"
The deviatoric stress tensor ($\mathbf{s}$) captures the components responsible for changing the shape (distortion) of the material element:
$$\mathbf{s} = \mathbf{\sigma} - \sigma_h \mathbf{I}$$
The trace) of the deviatoric tensor is identically zero ($\text{Tr}(\mathbf{s}) = 0$). The principal values of $\mathbf{s}$ are directly related to the differences between the principal stresses ($\sigma1, \sigma2, \sigma_3$).
Relation to Other Stress Measures