Is the Cauchy 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.”
Can stress tensor be non symmetric?
Although the theory generally predicts the stress to be non symmetric, the stress tensor can still be considered as symmetrical in the absence of external fields and when the inertia effects of internal rotations and couple stresses are neglected.
Why is a fluid stress tensor symmetric?
If the fluid particles have negligible angular momentum or if their angular momentum is not appreciably coupled to the external angular momentum, or if the equilibration time between the external and internal degrees of freedom is practically zero, the torque will be zero and the viscous stress tensor will be symmetric …
Is strain tensor symmetric?
Strain tensor ϵij is defined as a “symmetric” part of the displacement gradient, which is the first term in Eq. (2.12).
Which among these forces used in momentum equation is a tensor?
1. Which among these forces used in momentum equation is a tensor? Explanation: Viscous forces are tensors. The other forces given here (Gravitational, viscous and electromagnetic forces) are vectors.
Is stress a tensor quantity?
The restoring force per unit area is called stress. 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.
What is Cauchy stress formula?
The Kirchhoff stress tensor, τ is defined as: τ = det(F)σ. The Biot stress tensor, TB also called material stress tensor is defined as: TB = RtP, where R is the orthogonal tensor obtained during polar decomposition of F. The co-rotated Cauchy stress tensor σu, introduced by Green and Naghdi is defined as: σu = RtσR.
What is symmetry of stress tensor?
The symmetry of the stress tensor will be demonstrated in two ways. We argue that stress components located above and below the main diagonal represent torques that are equal but opposite. If the tensor is symmetric, then, those torques add up to zero.
What is meant by stress tensor?
The Stress Tensor Stress is defined as force per unit area. If we take a cube of material and subject it to an arbitrary load we can measure the stress on it in various directions (figure 4). These measurements will form a second rank tensor; the stress tensor.
What is stress strain tensor?
A simple example of a geophysically relevant tensor is stress. Stress, like pressure is defined as force per unit area. Pressure is isotropic, but if a material has finite strength, it can support different forces applied in different directions. These components form a second rank tensor; the stress tensor (Figure 1).
Why is the Cauchy stress tensor symmetric instead of nine?
At the same time, according to the principle of conservation of angular momentum, equilibrium requires that the summation of moments with respect to an arbitrary point is zero, which leads to the conclusion that the stress tensor is symmetric, thus having only six independent stress components, instead of the original nine.
How is the Cauchy stress vector related to surface forces?
Euler–Cauchy stress principle – stress vector. Thus, the total force applied to a body or to a portion of the body can be expressed as: Only surface forces will be discussed in this article as they are relevant to the Cauchy stress tensor.
How to calculate the Piola-Kirchhoff stress tensor?
First, determine the change in cross-sectional area from the reference to the current configuration. Next, if the Cauchy stress is to be taken as uniaxial, T = T11e1e1, calculate the first and second Piola–Kirchhoff stress tensors. Compare the 11-components of each of the three stress tensors.
Are there stress tensors based on the undeformed area?
Stress tensors based on the undeformed area can also be defined. They are known as the first and second Piola-Kirchhoff stress tensors. It is useful to be familiar with them not only because they appear in many works on continuum mechanics but also because one particular tensor may be more suitable in a particular problem.