Cauchy stress tensor

In continuum mechanics, the Cauchy stress tensor, also called true stress tensor or simply stress tensor, completely defines the state of stress at a point inside a material in the deformed state, placement, or configuration. The second order tensor consists of nine components and relates a unit-length direction vector e to the traction vector T across a surface perpendicular to e:
The SI unit of both stress tensor and traction vector is the newton per square metre or pascal, corresponding to the stress scalar. The unit vector is dimensionless.
The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle for stress.
The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations: it is a central concept in the linear theory of elasticity. For large deformations, also called finite deformations, other measures of stress are required, such as the Piola–Kirchhoff stress tensor, the Biot stress tensor, and the Kirchhoff stress tensor.
According to the principle of conservation of linear momentum, if the continuum body is in static equilibrium it can be demonstrated that the components of the Cauchy stress tensor in every material point in the body satisfy the equilibrium equations. 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. However, in the presence of couple-stresses, i.e. moments per unit volume, the stress tensor is non-symmetric. This also is the case when the Knudsen number is close to one,, or the continuum is a non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as polymers.
There are certain invariants associated with the stress tensor, whose values do not depend upon the coordinate system chosen, or the area element upon which the stress tensor operates. These are the three eigenvalues of the stress tensor, which are called the principal stresses.

Euler–Cauchy stress principle – stress vector

The Euler–Cauchy stress principle states that upon any surface that divides the body, the action of one part of the body on the other is equivalent to the system of distributed forces and couples on the surface dividing the body, and it is represented by a field, called the traction vector, defined on the surface and assumed to depend continuously on the surface's normal unit vector.
To formulate the Euler–Cauchy stress principle, consider a surface passing through an internal material point dividing the continuous body into two segments, as seen in Figure 2.1a or 2.1b.
Following the classical dynamics of Newton and Euler, the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces and body forces. 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.
When the body is subjected to external surface forces or contact forces, following Euler's equations of motion, internal contact forces and moments are transmitted from point to point in the body, and from one segment to the other through the dividing surface, due to the mechanical contact of one portion of the continuum onto the other. On an element of area containing, with normal vector, the force distribution is equipollent to a contact force exerted at point and surface moment. In particular, the contact force is given by
where is the mean surface traction.
Cauchy's stress principle asserts that as tends to zero the ratio becomes and the couple stress vector vanishes. In specific fields of continuum mechanics the couple stress is assumed not to vanish; however, classical branches of continuum mechanics address non-polar materials which do not consider couple stresses and body moments.
The resultant vector is defined as the surface traction, also called stress vector, traction, or traction vector. given by at the point associated with a plane with a normal vector :
This equation means that the stress vector depends on its location in the body and the orientation of the plane on which it is acting.
This implies that the balancing action of internal contact forces generates a contact force density or Cauchy traction field that represents a distribution of internal contact forces throughout the volume of the body in a particular configuration of the body at a given time. It is not a vector field because it depends not only on the position of a particular material point, but also on the local orientation of the surface element as defined by its normal vector.
Depending on the orientation of the plane under consideration, the stress vector may not necessarily be perpendicular to that plane, i.e. parallel to, and can be resolved into two components :

one normal to the plane, called normal stress
:
and the other parallel to this plane, called the shear stress
:
Cauchy's postulate

According to the Cauchy Postulate, the stress vector remains unchanged for all surfaces passing through the point and having the same normal vector at, i.e., having a common tangent at. This means that the stress vector is a function of the normal vector only, and is not influenced by the curvature of the internal surfaces.

Cauchy's fundamental lemma

A consequence of Cauchy's postulate is Cauchy's Fundamental Lemma, also called the Cauchy reciprocal theorem, which states that the stress vectors acting on opposite sides of the same surface are equal in magnitude and opposite in direction. Cauchy's fundamental lemma is equivalent to Newton's third law of motion of action and reaction, and is expressed as

Cauchy's stress theorem—stress tensor

The state of stress at a point in the body is then defined by all the stress vectors T associated with all planes that pass through that point. However, according to Cauchy's fundamental theorem, also called Cauchy's stress theorem, merely by knowing the stress vectors on three mutually perpendicular planes, the stress vector on any other plane passing through that point can be found through coordinate transformation equations.
Cauchy's stress theorem states that there exists a second-order tensor field σ, called the Cauchy stress tensor, independent of n, such that T is a linear function of n:
This equation implies that the stress vector T at any point P in a continuum associated with a plane with normal unit vector n can be expressed as a function of the stress vectors on the planes perpendicular to the coordinate axes, i.e. in terms of the components σ_ij of the stress tensor σ.
To prove this expression, consider a tetrahedron with three faces oriented in the coordinate planes, and with an infinitesimal area dA oriented in an arbitrary direction specified by a normal unit vector n. The tetrahedron is formed by slicing the infinitesimal element along an arbitrary plane with unit normal n. The stress vector on this plane is denoted by T. The stress vectors acting on the faces of the tetrahedron are denoted as T, T, and T, and are by definition the components σ_ij of the stress tensor σ. This tetrahedron is sometimes called the Cauchy tetrahedron. The equilibrium of forces, i.e. Euler's first law of motion, gives:
where the right-hand-side represents the product of the mass enclosed by the tetrahedron and its acceleration: ρ is the density, a is the acceleration, and h is the height of the tetrahedron, considering the plane n as the base. The area of the faces of the tetrahedron perpendicular to the axes can be found by projecting dA into each face :
and then substituting into the equation to cancel out dA:
To consider the limiting case as the tetrahedron shrinks to a point, h must go to 0. As a result, the right-hand-side of the equation approaches 0, so
Assuming a material element with planes perpendicular to the coordinate axes of a Cartesian coordinate system, the stress vectors associated with each of the element planes, i.e. T, T, and T can be decomposed into a normal component and two shear components, i.e. components in the direction of the three coordinate axes. For the particular case of a surface with normal unit vector oriented in the direction of the x₁-axis, denote the normal stress by σ₁₁, and the two shear stresses as σ₁₂ and σ₁₃:
In index notation this is
The nine components σ_ij of the stress vectors are the components of a second-order Cartesian tensor called the Cauchy stress tensor, which can be used to completely define the state of stress at a point and is given by
where σ₁₁, σ₂₂, and σ₃₃ are normal stresses, and σ₁₂, σ₁₃, σ₂₁, σ₂₃, σ₃₁, and σ₃₂ are shear stresses. The first index i indicates that the stress acts on a plane normal to the X_i -axis, and the second index j denotes the direction in which the stress acts. A stress component is positive if it acts in the positive direction of the coordinate axes, and if the plane where it acts has an outward normal vector pointing in the positive coordinate direction.
Thus, using the components of the stress tensor
or, equivalently,
Alternatively, in matrix form we have
The Voigt notation representation of the Cauchy stress tensor takes advantage of the symmetry of the stress tensor to express the stress as a six-dimensional vector of the form:
The Voigt notation is used extensively in representing stress–strain relations in solid mechanics and for computational efficiency in numerical structural mechanics software.