Table of Contents
- 1 Is Cauchy stress symmetric?
- 2 What is symmetry equation of stress tensor?
- 3 Can a tensor be asymmetric?
- 4 Why is stress a tensor quantity?
- 5 Why is stress a second order tensor?
- 6 Why is stress tensor symmetric?
- 7 Why is the stress tensor symmetric?
- 8 What is the symmetry of the Cauchy stress tensor?
- 9 What is the shape of the stress tensor?
Is Cauchy stress symmetric?
According to conservation of linear momentum and angular momentum, one can derive that Cauchy stress tensor is symmetric and hence has only 6 independent components.
What is symmetry equation of stress tensor?
τij−τji=0, i.e., the stress tensor is symmetric at every point in space. (a) Effects of the same force acting at points above and below the x2-axis. (b) Tangential forces acting on all four faces. Figure 17.3.
Can stress tensor be asymmetric?
It was first shown by West that the stress tensor in the balance of linear momentum can be chosen to be symmetric. On a statistical mechanical basis an asymmetric stress tensor for materials with internal degrees of freedom is derived.
Can a tensor be asymmetric?
Asymmetric tensor fields are usually decomposed into a symmetric tensor field and a rotational vector field and then visualized simultaneously (but as two separate fields).
Why is stress a tensor quantity?
Stress has both magnitude and direction but it does not follow the vector law of addition thus, it is not a vector quantity. Instead, stress follows the coordinate transformation law of addition, and hence, stress is considered as a tensor quantity.
Is Cauchy stress True stress?
Stress Measures. The most fundamental and commonly used stress quantity is the Cauchy stress, also known as the true stress. It is defined by studying the forces acting on an infinitesimal area element in the deformed body.
Why is stress a second order tensor?
The stress state is a second order tensor since it is a quantity associated with two directions. As a result, stress components have 2 subscripts. A surface traction is a first order tensor (i.e. vector) since it a quantity associated with only one direction.
Why is stress tensor symmetric?
The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (Cauchy’s equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that the stress tensor is symmetric.”
What is symmetric antisymmetric tensor?
Antisymmetric and symmetric tensors A tensor A that is antisymmetric on indices and has the property that the contraction with a tensor B that is symmetric on indices and. is identically 0.
Why is the stress tensor symmetric?
What is the symmetry of the Cauchy stress tensor?
The symmetry of the Cauchy stress tensor is the result of applying conservation of angular momentum to an infinitesimal material element. This derivation assumes that there are no body moments. If there are body moments, the only way that angular momentum can be conserved is if the stress tensor is asymmetric.
What did Cauchy mean by stress matrix?
Cauchy introduced the idea of stress or traction vectors which are force vectors per unit area. He then introduced the stress as a symmetric matrix. The stress matrix is perhaps one of the early concepts that promoted the study of vectors and matrices.
What is the shape of the stress tensor?
Stress tensor is symmetric. Note a positive (tensile) component of stress acts on a face with a positive normal in a positive direction. Thus a stress acting on a negative normal face, in a negative direction is also positive.