Introduction
Stress tensors are physical and mathematical objects that quantify the internal forces on a given body. They are commonly used across a multitude of disciplines, such as engineering, physics, and mathematics. Stress tensors are most commonly represented as an array of nine elements, an upper triangular matrix.
Definition
In general, stress tensors can be defined by a one-to-one mapping between the external forces acting on the given body and the internal forces exerted by the body itself. This can be represented as a three-dimensional array of nine elements
[σ11, σ12, σ13,
σ21, σ22, σ23,
σ31, σ32, σ33],
where each element symbolizes a component of the internal forces within the body. The tensor elements σ11, σ12, and σ13 represent the force components in the x direction; σ21, σ22, and σ23 represent the force components in the y direction; and σ31, σ32, and σ33 represent the force components in the z direction.
Stress Components
There are three primary stress components: normal stress, shear stress, and hydrostatic stress. Normal stress is the force applied perpendicular to the surface of a body or element. It is typically in the form of compression or tension, which can be represented by the element σ11 in a stress tensor. Shear stress is the force applied parallel to the surface of a body or element. This is represented by the elements σ12, σ13, σ21, and σ23 within the stress tensor. Finally, hydrostatic stress is the pressure that is applied uniformly over the surface of the body. This is represented by the elements σ31, σ32, and σ33.
Applications
Stress tensors are used in a variety of applications across multiple disciplines. In engineering, they are commonly used to calculate the distribution of stress in a structure. In physics, they are used to describe the distribution of force within a body in static equilibrium. In mathematics, they are used to describe the curvature of a manifold in three-dimensional space. They are also used in geology to describe the deformation of rocks.
Conclusion
Stress tensors are mathematical and physical objects that represent the internal forces within a body. They are represented as a three-dimensional array of nine elements, an upper triangular matrix. There are three primary components of stress: normal stress, shear stress, and hydrostatic stress. Stress tensors are used in a variety of applications across multiple disciplines, such as engineering, physics, and mathematics.
