Jonah Patel
In physics and engineering, a constitutive equation is a relation between two or more physical quantities, approximating a material's response to external stimuli. Constitutive equations are important in fields such as fluid mechanics, solid-state physics, and structural analysis. Tensor invariants are scalar values calculated from tensors, unaffected by tensor rotations. They are essential in constitutive modeling as they provide a way to model material behavior accurately. By using tensor invariants, scientists and engineers can develop constitutive equations that accurately describe a material's response to external forces, such as stress and strain. This is particularly useful in the study of nonlinear materials, where the constitutive law linking stress and strain tensors is a simple relationship between two vectors in the principal space.
| Characteristics | Values |
|---|---|
| Definition | Tensor invariants are scalar values calculated from tensors that are unaffected by rotations of the tensor. |
| Importance in Constitutive Modeling | Tensor invariants are important in modeling material behavior. They are used to formulate closed-form expressions for the strain energy density or Helmholtz free energy of a nonlinear material with isotropic symmetry. |
| Examples | Principal invariants, mixed invariants, eigenvalues, the length of a vector, the norm of a tensor, and the trace of a tensor. |
| Applications | Constitutive equations for isotropic nonlinearly elastic materials, hydrodynamics, and turbulence. |
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What You'll Learn
- Tensor invariants are scalar values unaffected by tensor rotations, aiding material behaviour modelling
- The von Mises effective stress is an example of an invariant
- The length of a vector is an invariant
- The norm of any tensor is an invariant
- Mixed invariants between pairs of rank-two tensors can be defined and extracted
Tensor invariants are scalar values unaffected by tensor rotations, aiding material behaviour modelling
In physics and engineering, a constitutive equation is a relation between two or more physical quantities that is specific to a material or substance, approximating its response to external stimuli. Constitutive equations are used to solve physical problems, such as fluid mechanics, solid-state physics, and structural analysis.
Tensor invariants are scalar values calculated from tensors that are unaffected by tensor rotations. In other words, they remain constant despite changes in the tensor's orientation. This property is crucial in modelling material behaviour because it allows for the formulation of closed-form expressions for the strain energy density or Helmholtz free energy of a nonlinear material with isotropic symmetry.
For instance, the von Mises effective stress is an invariant because the double contraction with itself and the trace used to obtain the deviatoric part remains unchanged by rotations. The length of a vector is another example of a tensor invariant, as it does not alter with changes in the coordinate system.
In the context of constitutive modelling, tensor invariants are essential for understanding and predicting the behaviour of materials under various conditions. By leveraging the invariant properties of tensors, scientists and engineers can develop more accurate models that capture the intrinsic properties of materials. This is particularly valuable when dealing with complex materials that exhibit nonlinear behaviour or anisotropy.
Additionally, tensor invariants play a crucial role in the development of general invariant representations of constitutive equations for isotropic nonlinearly elastic materials. By constructing different sets of mutually orthogonal unit tensor bases, researchers can establish invariant relations and their geometrical interpretations in three-dimensional principal space. This facilitates a deeper understanding of the relationship between stress and strain tensors, ultimately improving our ability to model and predict material behaviour.
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The von Mises effective stress is an example of an invariant
In physics and engineering, a constitutive equation is a relation between two or more physical quantities, approximating a material's response to external stimuli. Constitutive equations are used to solve physical problems, such as fluid mechanics, solid-state physics, and structural analysis.
Tensor invariants are scalar values calculated from tensors that are unaffected by tensor rotations. They are important in constitutive modeling as they are used to model material behavior. The von Mises effective stress is an example of an invariant. It is a scalar representation of stress based on the second invariant of the deviatoric stress tensor. The von Mises stress equation was established by Richard Elder von Mises in 1913, with Heinrich Hencky providing a physical interpretation in 1924. The von Mises yield criterion is expressed as:
> {\\displaystyle k} is the yield stress of the material in pure shear. As shown later in this article, at the onset of yielding, the magnitude of the shear yield stress in pure shear is \\sqrt{3} times lower than the tensile yield stress in the case of simple tension. Thus, we have: ... {\\displaystyle \sigma \_y} is the tensile yield strength of the material. If we set the von Mises stress equal to the yield strength and combine the above equations, the von Mises yield criterion is written as: ... {\\displaystyle \sigma \_ {v} ^{2}=\\frac {1}{2}\\left [(\\sigma \_ {11}-\\sigma \_ {22})^ {2}+(\\sigma \_ {22}-\\sigma \_ {33})^ {2}+(\\sigma \_ {33}-\\sigma \_ {11})^ {2}+6\\left (\\sigma \_ {23} ^{2}+\\sigma \_ {31} ^{2}+\\sigma \_ {12} ^{2}\\right)]=\\frac {3}{2}s_ {ij}s_ {ij}}.
The von Mises stress is directly related to the deviatoric strain energy term, which is derived from Hooke's Law. The deviatoric stress is represented by σ', and the von Mises yield criterion helps to predict ductile materials' behavior under complex 3D loading conditions. It is used to check yield conditions in various engineering disciplines, including aerospace, civil, and marine engineering.
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The length of a vector is an invariant
Constitutive equations are used in physics and engineering to model the response of materials to external stimuli, such as applied fields or forces. These equations involve constitutive relations, which are functions of tensors. Tensors are mathematical objects that describe physical quantities and their transformations.
Tensor invariants are scalar values derived from tensors that remain unchanged under certain transformations, such as rotations. In the context of constitutive modeling, tensor invariants play a crucial role in understanding and predicting the behaviour of materials under various conditions.
The length of a vector is an example of a tensor invariant. It represents the magnitude or size of the vector in a given space. Regardless of how the coordinate system is chosen or transformed, the length of the vector remains the same. This property is known as being "invariant under coordinate transformation" or "coordinate invariant."
Mathematically, if we consider a vector v in a given space, its length or magnitude is calculated using the Euclidean norm:
||v|| = sqrt(v . v)
Where "sqrt" denotes the square root, and "v . v" represents the dot product of the vector with itself, resulting in a scalar value. This scalar value, representing the length of the vector, is a tensor invariant.
The invariance of the vector's length implies that the geometric properties of the vector, such as its magnitude or direction, remain unchanged despite changes in the coordinate system. This concept is essential in fields like mechanics, physics, and engineering, where vector calculations and transformations are prevalent. By utilizing tensor invariants, scientists and engineers can make predictions about the behaviour of materials and develop accurate models that are independent of specific coordinate choices.
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The norm of any tensor is an invariant
Tensor invariants are scalar values calculated from tensors that remain unchanged by tensor rotations. They are important in constitutive modelling as they are used in modelling material behaviour. Constitutive equations are used to approximate a material's response to external stimuli, such as applied fields or forces. Constitutive modelling is used in various fields, including physics and engineering.
Contractions are used as tensor norms because they remain unchanged under tensor transformations. For example, the double contraction of a tensor with itself is an invariant, and it can be used to calculate the norm of a tensor. This is demonstrated in the equation for the von Mises effective stress, where the double contraction and the trace are used to obtain the deviatoric part of the tensor.
Additionally, mixed invariants between pairs of rank two tensors can be defined and determined using algorithms such as the Faddeev-LeVerrier algorithm. The invariants of rank three, four, and higher-order tensors can also be determined. These invariants are independent of rotations of the coordinate system and are commonly used in formulating expressions for strain energy density or Helmholtz free energy in nonlinear materials with isotropic symmetry.
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Mixed invariants between pairs of rank-two tensors can be defined and extracted
In physics and engineering, a constitutive equation is a relation between two or more physical quantities that is specific to a material or substance. Constitutive equations are used to solve physical problems, such as fluid mechanics, solid-state physics, and structural analysis.
Tensor invariants are scalar values calculated from tensors that have the special property of being unaffected by rotations of the tensor(s). They are important in constitutive modeling as they can be used to model material behavior. For example, the von Mises effective stress is an invariant because the double contraction with itself and the trace (used to get the deviatoric part) is an invariant.
$$
\boldsymbol{a}\underline{\boldsymbol{v}}_i = \lambda_i \underline{\boldsymbol{v}}_i
$$
Where $\boldsymbol{a}$ is a tensor, $\underline{\boldsymbol{v}}_i$ is an eigenvector, and $\lambda_i$ is the corresponding eigenvalue. The eigenvectors of a symmetric tensor are orthogonal, and the number of independent components (degrees of freedom) of a symmetric tensor is equal to 6.
In addition to the principal invariants, there are also main invariants that are functions of the principal invariants. These are given by:
$\begin{aligned
J_{1}&=\lambda _{1}+\lambda _{2}+\lambda _{3}=I_{1}\\
J_{2}&=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=I_{1}^{2}-2I_{2}\\
J_{3}&=\lambda _{1}^{3}+\lambda _{2}^{3}+\lambda _{3}^{3}=I_{1}^{3}-3I_{1}I_{2}+3I_{3}
\end{aligned}
Where $I_1$, $I_2$, and $I_3$ are the principal invariants. These main invariants are important in engineering applications, especially in the study of nonlinear materials with isotropic symmetry.
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