Cauchy stress tensor

In continuum mechanics, the Cauchy stress tensor formula_1, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components formula_2 that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector n to the traction vector T(n) across an imaginary surface perpendicular to n:
where,
The SI units of both stress tensor and stress vector are N/m2, 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 (Cauchy's equations of motion for zero acceleration). 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, formula_5, 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 (real or imaginary) that divides the body, the action of one part of the body on the other is equivalent (equipollent) to the system of distributed forces and couples on the surface dividing the body", and it is represented by a field formula_6, called the traction vector, defined on the surface formula_7 and assumed to depend continuously on the surface's unit vector formula_8.
To formulate the Euler–Cauchy stress principle, consider an imaginary surface formula_7 passing through an internal material point formula_10 dividing the continuous body into two segments, as seen in Figure 2.1a or 2.1b (one may use either the cutting plane diagram or the diagram with the arbitrary volume inside the continuum enclosed by the surface formula_7).
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 formula_12 and body forces formula_13. Thus, the total force formula_14 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" formula_12, 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 formula_7, due to the mechanical contact of one portion of the continuum onto the other (Figure 2.1a and 2.1b). On an element of area formula_18 containing formula_10, with normal vector formula_8, the force distribution is equipollent to a contact force formula_21 exerted at point P and surface moment formula_22. In particular, the contact force is given by
where formula_24 is the "mean surface traction".
Cauchy's stress principle asserts that as formula_18 becomes very small and tends to zero the ratio formula_26 becomes formula_27 and the couple stress vector formula_22 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 formula_27 is defined as the "surface traction", also called "stress vector", "traction", or "traction vector". given by formula_30 at the point formula_10 associated with a plane with a normal vector formula_8:
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" formula_34 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 formula_35. It is not a vector field because it depends not only on the position formula_36 of a particular material point, but also on the local orientation of the surface element as defined by its normal vector formula_8.
Depending on the orientation of the plane under consideration, the stress vector may not necessarily be perpendicular to that plane, "i.e." parallel to formula_8, and can be resolved into two components (Figure 2.1c):
Cauchy’s postulate.
According to the "Cauchy Postulate", the stress vector formula_6 remains unchanged for all surfaces passing through the point formula_10 and having the same normal vector formula_8 at formula_10, i.e., having a common tangent at formula_10. This means that the stress vector is a function of the normal vector formula_8 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(n) associated with all planes (infinite in number) 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(x, t), 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(n) 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 "σij" 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 d"A" oriented in an arbitrary direction specified by a normal unit vector n (Figure 2.2). 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(n). The stress vectors acting on the faces of the tetrahedron are denoted as T(e1), T(e2), and T(e3), and are by definition the components "σij" 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 (Newton's second 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 d"A" into each face (using the dot product):
and then substituting into the equation to cancel out d"A":
To consider the limiting case as the tetrahedron shrinks to a point, "h" must go to 0 (intuitively, the plane n is translated along n toward "O"). As a result, the right-hand-side of the equation approaches 0, so
Assuming a material element (Figure 2.3) 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(e1), T(e2), and T(e3) 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 "x"1-axis, denote the normal stress by "σ"11, and the two shear stresses as "σ"12 and "σ"13:
In index notation this is
The nine components "σij" of the stress vectors are the components of a second-order Cartesian tensor called the "Cauchy stress tensor", which completely defines the state of stress at a point and is given by
where "σ"11, "σ"22, and "σ"33 are normal stresses, and "σ"12, "σ"13, "σ"21, "σ"23, "σ"31, and "σ"32 are shear stresses. The first index "i" indicates that the stress acts on a plane normal to the "Xi" -axis, and the second index "j" denotes the direction in which the stress acts (For example, σ12 implies that the stress is acting on the plane that is normal to the 1st axis i.e.;"X"1 and acts along the 2nd axis i.e.;"X"2). 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.
Transformation rule of the stress tensor.
It can be shown that the stress tensor is a contravariant second order tensor, which is a statement of how it transforms under a change of the coordinate system. From an "xi"-system to an " xi' "-system, the components "σij" in the initial system are transformed into the components "σij' " in the new system according to the tensor transformation rule (Figure 2.4):
where A is a rotation matrix with components "aij". In matrix form this is
Expanding the matrix operation, and simplifying terms using the symmetry of the stress tensor, gives
The Mohr circle for stress is a graphical representation of this transformation of stresses.
Normal and shear stresses.
The magnitude of the normal stress component "σ"n of any stress vector T(n) acting on an arbitrary plane with normal unit vector n at a given point, in terms of the components "σij" of the stress tensor σ, is the dot product of the stress vector and the normal unit vector:
The magnitude of the shear stress component "τ"n, acting orthogonal to the vector n, can then be found using the Pythagorean theorem:
where
Balance laws – Cauchy's equations of motion.
Cauchy's first law of motion.
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.
For example, for a hydrostatic fluid in equilibrium conditions, the stress tensor takes on the form:
where formula_78 is the hydrostatic pressure, and formula_79 is the kronecker delta.
Cauchy's second law of motion.
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, formula_5, or the continuum is a non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as polymers.
Principal stresses and stress invariants.
At every point in a stressed body there are at least three planes, called "principal planes", with normal vectors formula_82, called "principal directions", where the corresponding stress vector is perpendicular to the plane, i.e., parallel or in the same direction as the normal vector formula_82, and where there are no normal shear stresses formula_84. The three stresses normal to these principal planes are called "principal stresses".
The components formula_2 of the stress tensor depend on the orientation of the coordinate system at the point under consideration. However, the stress tensor itself is a physical quantity and as such, it is independent of the coordinate system chosen to represent it. There are certain invariants associated with every tensor which are also independent of the coordinate system. For example, a vector is a simple tensor of rank one. In three dimensions, it has three components. The value of these components will depend on the coordinate system chosen to represent the vector, but the magnitude of the vector is a physical quantity (a scalar) and is independent of the Cartesian coordinate system chosen to represent the vector (so long as it is normal). Similarly, every second rank tensor (such as the stress and the strain tensors) has three independent invariant quantities associated with it. One set of such invariants are the principal stresses of the stress tensor, which are just the eigenvalues of the stress tensor. Their direction vectors are the principal directions or eigenvectors.
A stress vector parallel to the normal unit vector formula_82 is given by:
where formula_88 is a constant of proportionality, and in this particular case corresponds to the magnitudes formula_89 of the normal stress vectors or principal stresses.
Knowing that formula_90 and formula_91, we have
This is a homogeneous system, i.e. equal to zero, of three linear equations where formula_93 are the unknowns. To obtain a nontrivial (non-zero) solution for formula_93, the determinant matrix of the coefficients must be equal to zero, i.e. the system is singular. Thus,
Expanding the determinant leads to the "characteristic equation"
where
The characteristic equation has three real roots formula_98, i.e. not imaginary due to the symmetry of the stress tensor. The formula_99, formula_100 and formula_101, are the principal stresses, functions of the eigenvalues formula_98. The eigenvalues are the roots of the characteristic polynomial. The principal stresses are unique for a given stress tensor. Therefore, from the characteristic equation, the coefficients formula_103, formula_104 and formula_105, called the first, second, and third "stress invariants", respectively, always have the same value regardless of the coordinate system's orientation.
For each eigenvalue, there is a non-trivial solution for formula_93 in the equation formula_107. These solutions are the principal directions or eigenvectors defining the plane where the principal stresses act. The principal stresses and principal directions characterize the stress at a point and are independent of the orientation.
A coordinate system with axes oriented to the principal directions implies that the normal stresses are the principal stresses and the stress tensor is represented by a diagonal matrix:
The principal stresses can be combined to form the stress invariants, formula_103, formula_104, and formula_105. The first and third invariant are the trace and determinant respectively, of the stress tensor. Thus,
Because of its simplicity, the principal coordinate system is often useful when considering the state of the elastic medium at a particular point. Principal stresses are often expressed in the following equation for evaluating stresses in the x and y directions or axial and bending stresses on a part. The principal normal stresses can then be used to calculate the von Mises stress and ultimately the safety factor and margin of safety.
Using just the part of the equation under the square root is equal to the maximum and minimum shear stress for plus and minus. This is shown as:
Maximum and minimum shear stresses.
The maximum shear stress or maximum principal shear stress is equal to one-half the difference between the largest and smallest principal stresses, and acts on the plane that bisects the angle between the directions of the largest and smallest principal stresses, i.e. the plane of the maximum shear stress is oriented formula_115 from the principal stress planes. The maximum shear stress is expressed as
Assuming formula_117 then
When the stress tensor is non zero the normal stress component acting on the plane for the maximum shear stress is non-zero and it is equal to
Stress deviator tensor.
The stress tensor formula_2 can be expressed as the sum of two other stress tensors:
So
where formula_124 is the mean stress given by
Pressure (formula_78) is generally defined as negative one-third the trace of the stress tensor minus any stress the divergence of the velocity contributes with, i.e.
where formula_88 is a proportionality constant, formula_129 is the divergence operator, formula_130 is the "k":th Cartesian coordinate, formula_131 is the velocity and formula_132 is the "k":th Cartesian component of formula_131.
The deviatoric stress tensor can be obtained by subtracting the hydrostatic stress tensor from the Cauchy stress tensor:
Invariants of the stress deviator tensor.
As it is a second order tensor, the stress deviator tensor also has a set of invariants, which can be obtained using the same procedure used to calculate the invariants of the stress tensor. It can be shown that the principal directions of the stress deviator tensor formula_122 are the same as the principal directions of the stress tensor formula_2. Thus, the characteristic equation is
where formula_138, formula_139 and formula_140 are the first, second, and third "deviatoric stress invariants", respectively. Their values are the same (invariant) regardless of the orientation of the coordinate system chosen. These deviatoric stress invariants can be expressed as a function of the components of formula_122 or its principal values formula_142, formula_143, and formula_144, or alternatively, as a function of formula_2 or its principal values formula_146, formula_147, and formula_148. Thus,
Because formula_150, the stress deviator tensor is in a state of pure shear.
A quantity called the equivalent stress or von Mises stress is commonly used in solid mechanics. The equivalent stress is defined as
Octahedral stresses.
Considering the principal directions as the coordinate axes, a plane whose normal vector makes equal angles with each of the principal axes (i.e. having direction cosines equal to formula_152) is called an "octahedral plane". There are a total of eight octahedral planes (Figure 6). The normal and shear components of the stress tensor on these planes are called "octahedral normal stress" formula_153 and "octahedral shear stress" formula_154, respectively. Octahedral plane passing through the origin is known as the "π-plane" ("π" not to be confused with "mean stress" denoted by "π" in above section) "." On the "π-plane", formula_155.
Knowing that the stress tensor of point O (Figure 6) in the principal axes is
the stress vector on an octahedral plane is then given by:
The normal component of the stress vector at point O associated with the octahedral plane is
which is the mean normal stress or hydrostatic stress. This value is the same in all eight octahedral planes.
The shear stress on the octahedral plane is then