Continuum mechanical analogous of Maxwell stress tensor

In summary, the Maxwell stress tensor in the static case can determine the total force acting on a system of charges within a bounded volume, as shown in equation (1). This is formally analogous to the stress tensor in continuum mechanics, which can determine the forces acting on a system of masses based on its components on a surface enclosing the system. In the 3D case, the stress tensor is equivalent to the Maxwell stress tensor.
  • #1
crick
43
4
Maxwell stress tensor ##\bar{\bar{\mathbf{T}}}## in the static case can be used to determine the total force ##\mathbf{f}## acting on a system of charges contanined in the volume bounded by ##S##

$$ \int_{S} \bar{\bar{\mathbf{T}}} \cdot \mathbf n \,\,d S=\mathbf{f}= \frac{d}{dt} \mathbf {{Q_{mech}}}\tag{1}$$

Where ## \mathbf {{Q_{mech}}}## is the (mechanical) momentum of the system of charges.

What theorem/relation is formally analogous to ##(1)## in continuum mechanics? I've read that also in continuum mechanic one can introduce a tensor such that the value of its components on a surface ##S## enclosing a system of masses determines the forces acting on the masses completely.

I could not find this analogy on Jackson or Griffiths, so what is the tensor that is similar to Maxwell stress tensor in mechanics? Is it the stress tensor? By which theorem does it determine the forces on a system of masses?
 
Physics news on Phys.org
  • #2
crick said:
Is it the stress tensor?
Partially, the stress tensor is the spatial-spatial part. The stress energy tensor also incorporates the momentum and energy densities.

Edit: So forget that. You were only asking about the 3D case, not the spacetime 4D case. It is just the stress tensor. The Maxwell stress tensor is the stress tensor of a continuum - the EM field.
 

1. What is the Continuum Mechanical Analogous of Maxwell Stress Tensor?

The Continuum Mechanical Analogous of Maxwell Stress Tensor is a mathematical concept used in the field of continuum mechanics to describe the stress distribution within a material. It is based on the principles of electromagnetism and is closely related to the Maxwell stress tensor used in electromagnetism.

2. How is the Continuum Mechanical Analogous of Maxwell Stress Tensor Calculated?

The Continuum Mechanical Analogous of Maxwell Stress Tensor is calculated using the constitutive equations of a material, which relate the stress and strain of a material. These equations take into account the material's properties such as elasticity, plasticity, and viscosity. The resulting stress tensor is a second-order tensor, which describes the stress at each point within the material.

3. What are the Applications of the Continuum Mechanical Analogous of Maxwell Stress Tensor?

The Continuum Mechanical Analogous of Maxwell Stress Tensor is widely used in the study of material behavior under different loading conditions. It is used in various engineering fields such as solid mechanics, fluid mechanics, and biomechanics. It is also used in the design and analysis of structures and materials, as well as in the development of new materials with specific properties.

4. How is the Continuum Mechanical Analogous of Maxwell Stress Tensor Different from the Maxwell Stress Tensor?

The Continuum Mechanical Analogous of Maxwell Stress Tensor is a generalization of the Maxwell stress tensor used in electromagnetism. While the Maxwell stress tensor only considers the force due to electromagnetic fields, the Continuum Mechanical Analogous of Maxwell Stress Tensor takes into account other forces such as gravitational, thermal, and mechanical forces. Additionally, the Continuum Mechanical Analogous of Maxwell Stress Tensor is applicable to all materials, while the Maxwell stress tensor is only applicable to materials with electromagnetic properties.

5. Are there any Limitations to the Continuum Mechanical Analogous of Maxwell Stress Tensor?

Like any mathematical model, the Continuum Mechanical Analogous of Maxwell Stress Tensor has its limitations. It is based on certain assumptions and simplifications, and its accuracy may be affected by factors such as material heterogeneity and complex loading conditions. Additionally, it may not accurately describe the behavior of materials at the microscopic level. Therefore, it is important to use the Continuum Mechanical Analogous of Maxwell Stress Tensor in conjunction with other experimental and theoretical methods to fully understand the behavior of materials under different conditions.

Similar threads

Dependence of the stress vector on surface orientation
    • Advanced Physics Homework Help
Replies
4
Views
1K
Electromagnetic stress tensor from pressure and tension
    • Electromagnetism
Replies
6
Views
1K
I Why Must Pure Force Depend on 4-Velocity?
    • Special and General Relativity
2
Replies
47
Views
3K
Hamiltonian for Charged Particles + EM-Field
    • Electromagnetism
Replies
1
Views
1K
Understanding the Maxwell Stress Tensor
    • Advanced Physics Homework Help
Replies
3
Views
3K
Maxwell Stress Tensor in the absence of a magnetic field
    • Electromagnetism
Replies
4
Views
2K
I Lorentz Force Equation: Coercion & Maxwell Stress Tensor
    • Special and General Relativity
Replies
11
Views
860
Maxwell Stress Tensor -> Force between magnets and perfect iron
    • Electromagnetism
Replies
4
Views
4K
I Divergence of first Piola-Kirchoff stress tensor
    • Classical Physics
Replies
1
Views
606
What Are the Fundamental Principles of Maxwell's Equations?
    • Electromagnetism
Replies
0
Views
2K

Hot Threads

Recent Insights

Back
Top