Cauchy stress tensor
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In
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mec ...
, the Cauchy stress tensor (symbol \boldsymbol\sigma, named after
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...
), 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 In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
consists of nine components \sigma_ and relates a unit-length direction vector e to the ''traction vector'' T(e) across an imaginary surface perpendicular to e: :\mathbf^ = \mathbf e \cdot\boldsymbol\quad \text \quad T_^= \sum_\sigma_e_i. The
SI base units The SI base units are the standard units of measurement defined by the International System of Units (SI) for the seven base quantities of what is now known as the International System of Quantities: they are notably a basic set from which all ...
of both stress tensor and traction vector are newton per square metre (N/m2) or pascal (Pa), corresponding to the stress scalar. The unit vector is
dimensionless Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...
. 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 Stress may refer to: Science and medicine * Stress (biology) Stress, whether physiological, biological or psychological, is an organism's response to a stressor, such as an environmental condition or change in life circumstances. When s ...
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 Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
, 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, K_\rightarrow 1, or the continuum is a non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as
polymers A polymer () is a substance or material that consists of very large molecules, or macromolecules, that are constituted by many repeating subunits derived from one or more species of monomers. Due to their broad spectrum of properties, b ...
. 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 In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
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 \mathbf^, called the traction vector, defined on the surface S and assumed to depend continuously on the surface's unit vector \mathbf n. To formulate the Euler–Cauchy stress principle, consider an imaginary surface S passing through an internal material point P 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 S). Following the classical dynamics of Newton and Euler, the motion of a material body is produced by the action of externally applied
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
s which are assumed to be of two kinds: surface forces \mathbf F and body forces \mathbf b.Smith & Truesdell p.97 Thus, the total force \mathcal F applied to a body or to a portion of the body can be expressed as: :\mathcal F = \mathbf b + \mathbf F 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'' \mathbf F, 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 S, 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 \Delta S containing P, with normal
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
\mathbf n, the force distribution is equipollent to a contact force \Delta \mathbf F exerted at point P and surface moment \Delta \mathbf M. In particular, the
contact force A contact force is any force that occurs because of two objects making contact with each other. Contact forces are very common and are responsible for most visible interactions between macroscopic The macroscopic scale is the length scale on ...
is given by :\Delta\mathbf F= \mathbf T^\,\Delta S where \mathbf T^ is the ''mean surface traction''. Cauchy's stress principle asserts that as \Delta S becomes very small and tends to zero the ratio \Delta \mathbf F/\Delta S becomes d\mathbf F/dS and the couple stress vector \Delta \mathbf M 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 d\mathbf F/dS is defined as the ''surface traction'', also called ''stress vector'', ''traction'', or ''traction vector''. given by \mathbf^=T_i^\mathbf_i at the point P associated with a plane with a normal vector \mathbf n: :T^_i= \lim_ \frac = . 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'' \mathbf T(\mathbf n, \mathbf x, t) 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 t. It is not a
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
because it depends not only on the position \mathbf x of a particular material point, but also on the local orientation of the surface element as defined by its normal vector \mathbf n.Lubliner Depending on the orientation of the plane under consideration, the stress vector may not necessarily be perpendicular to that plane, ''i.e.'' parallel to \mathbf n, and can be resolved into two components (Figure 2.1c): * one normal to the plane, called ''normal stress'' :\mathbf= \lim_ \frac = \frac, :where dF_\mathrm n is the normal component of the force d\mathbf F to the differential area dS * and the other parallel to this plane, called the ''
shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
'' :\mathbf \tau= \lim_ \frac = \frac, :where dF_\mathrm s is the tangential component of the force d\mathbf F to the differential surface area dS. The shear stress can be further decomposed into two mutually perpendicular vectors.


Cauchy's postulate

According to the ''Cauchy Postulate'', the stress vector \mathbf^ remains unchanged for all surfaces passing through the point P and having the same normal vector \mathbf n at P, i.e., having a common
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
at P. This means that the stress vector is a function of the normal vector \mathbf n 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 :- \mathbf^= \mathbf^.


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: :\mathbf^= \mathbf n \cdot\boldsymbol\quad \text \quad T_j^= \sigma_n_i. 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 In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
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: :\mathbf^ \, dA - \mathbf^ \, dA_1 - \mathbf^ \, dA_2 - \mathbf^ \, dA_3 = \rho \left( \fracdA \right) \mathbf, 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): :dA_1= \left(\mathbf \cdot \mathbf_1 \right)dA = n_1 \; dA, :dA_2= \left(\mathbf \cdot \mathbf_2 \right)dA = n_2 \; dA, :dA_3= \left(\mathbf \cdot \mathbf_3 \right)dA = n_3 \; dA, and then substituting into the equation to cancel out d''A'': :\mathbf^ - \mathbf^n_1 - \mathbf^n_2 - \mathbf^n_3 = \rho \left( \frac \right) \mathbf. 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 : \mathbf^ = \mathbf^ n_1 + \mathbf^ n_2 + \mathbf^ n_3. Assuming a material element (see figure at the top of the page) 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 In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
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: :\mathbf^= T_1^\mathbf_1 + T_2^ \mathbf_2 + T_3^ \mathbf_3 = \sigma_ \mathbf_1 + \sigma_ \mathbf_2 + \sigma_ \mathbf_3, :\mathbf^= T_1^\mathbf_1 + T_2^ \mathbf_2 + T_3^ \mathbf_3=\sigma_ \mathbf_1 + \sigma_ \mathbf_2 + \sigma_ \mathbf_3, :\mathbf^= T_1^\mathbf_1 + T_2^ \mathbf_2 + T_3^ \mathbf_3=\sigma_ \mathbf_1 + \sigma_ \mathbf_2 + \sigma_ \mathbf_3, In index notation this is :\mathbf^= T_j^ \mathbf_j = \sigma_ \mathbf_j. The nine components ''σij'' 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 :\boldsymbol= \sigma_ = \left
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= \left
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\equiv \left
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\equiv \left
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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 :\begin \mathbf^ &= \mathbf^n_1 + \mathbf^n_2 + \mathbf^n_3 \\ & = \sum_^3 \mathbf^n_i \\ &= \left( \sigma_\mathbf_j \right)n_i \\ &= \sigma_n_i\mathbf_j \end or, equivalently, :T_j^= \sigma_n_i. Alternatively, in matrix form we have :\left
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cdot \left
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The Voigt notation representation of the Cauchy stress tensor takes advantage of the
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
of the stress tensor to express the stress as a six-dimensional vector of the form: :\boldsymbol = \begin\sigma_1 & \sigma_2 & \sigma_3 & \sigma_4 & \sigma_5 & \sigma_6 \end^\textsf \equiv \begin\sigma_ & \sigma_ & \sigma_ & \sigma_ & \sigma_ & \sigma_ \end^\textsf. 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): :\sigma'_ = a_a_\sigma_ \quad \text \quad \boldsymbol' = \mathbf A \boldsymbol \mathbf A^\textsf, where A is a
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \t ...
with components ''aij''. In matrix form this is :\left
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\left
right Rights are law, legal, social, or ethics, ethical principles of freedom or Entitlement (fair division), entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal sy ...
left
right Rights are law, legal, social, or ethics, ethical principles of freedom or Entitlement (fair division), entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal sy ...
left
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Expanding the matrix operation, and simplifying terms using the symmetry of the stress tensor, gives :\begin \sigma_' = &a_^2\sigma_+a_^2\sigma_+a_^2\sigma_+2a_a_\sigma_+2a_a_\sigma_+2a_a_\sigma_, \\ \sigma_' = &a_^2\sigma_+a_^2\sigma_+a_^2\sigma_+2a_a_\sigma_+2a_a_\sigma_+2a_a_\sigma_, \\ \sigma_' = &a_^2\sigma_+a_^2\sigma_+a_^2\sigma_+2a_a_\sigma_+2a_a_\sigma_+2a_a_\sigma_, \\ \sigma_' = &a_a_\sigma_+a_a_\sigma_+a_a_\sigma_ \\ &+(a_a_+a_a_)\sigma_+(a_a_+a_a_)\sigma_+(a_a_+a_a_)\sigma_, \\ \sigma_' = &a_a_\sigma_+a_a_\sigma_+a_a_\sigma_ \\ &+(a_a_+a_a_)\sigma_+(a_a_+a_a_)\sigma_+(a_a_+a_a_)\sigma_, \\ \sigma_' = &a_a_\sigma_+a_a_\sigma_+a_a_\sigma_ \\ &+(a_a_+a_a_)\sigma_+(a_a_+a_a_)\sigma_+(a_a_+a_a_)\sigma_. \end 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 In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
of the stress vector and the normal unit vector: :\begin \sigma_\mathrm &= \mathbf^\cdot \mathbf \\ &=T^_i n_i \\ &=\sigma_n_i n_j. \end The magnitude of the shear stress component ''τ''n, acting orthogonal to the vector n, can then be found using the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
: :\begin \tau_\mathrm &=\sqrt \\ &= \sqrt, \end where :\left( T^ \right)^2 = T_i^ T_i^ = \left( \sigma_ n_j \right) \left(\sigma_ n_k \right) = \sigma_ \sigma_ n_j n_k.


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: : \sigma_+ F_i = 0 , where \sigma_ = \sum_j \partial_j \sigma_ For example, for a hydrostatic fluid in equilibrium conditions, the stress tensor takes on the form: : = -p , where p is the hydrostatic pressure, and \ is the
kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
. :


Cauchy's second law of motion

According to the principle of
conservation of angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
, 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: :\sigma_=\sigma_ : 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, K_\rightarrow 1, or the continuum is a non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as
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.


Principal stresses and stress invariants

At every point in a stressed body there are at least three planes, called ''principal planes'', with normal vectors \mathbf, 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 \mathbf, and where there are no normal shear stresses \tau_\mathrm. The three stresses normal to these principal planes are called ''principal stresses''. The components \sigma_ 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
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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 In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...
. A stress vector parallel to the normal unit vector \mathbf is given by: :\mathbf^ = \lambda \mathbf= \mathbf_\mathrm n \mathbf where \lambda is a constant of proportionality, and in this particular case corresponds to the magnitudes \sigma_\mathrm of the normal stress vectors or principal stresses. Knowing that T_i^=\sigma_n_j and n_i=\delta_n_j, we have :\begin T_i^ &= \lambda n_i \\ \sigma_n_j &= \lambda n_i \\ \sigma_n_j - \lambda n_i &= 0 \\ \left(\sigma_ - \lambda\delta_\right)n_j &=0 \\ \end This is a homogeneous system, i.e. equal to zero, of three linear equations where n_j are the unknowns. To obtain a nontrivial (non-zero) solution for n_j, the determinant matrix of the coefficients must be equal to zero, i.e. the system is singular. Thus, :\left, \sigma_ - \lambda\delta_\ = \begin \sigma_ - \lambda & \sigma_ & \sigma_ \\ \sigma_ & \sigma_ - \lambda & \sigma_ \\ \sigma_ & \sigma_ & \sigma_ - \lambda \\ \end = 0 Expanding the determinant leads to the ''characteristic equation'' :\left, \sigma_- \lambda\delta_\ = -\lambda^3 + I_1\lambda^2 - I_2\lambda + I_3 = 0 where :\begin I_1 &= \sigma_ + \sigma_ + \sigma_ \\ &= \sigma_ = \text(\boldsymbol) \\ pt I_2 &= \begin \sigma_ & \sigma_ \\ \sigma_ & \sigma_ \\ \end + \begin \sigma_ & \sigma_ \\ \sigma_ & \sigma_ \\ \end + \begin \sigma_ & \sigma_ \\ \sigma_ & \sigma_ \\ \end \\ &= \sigma_\sigma_ + \sigma_\sigma_ + \sigma_\sigma_ - \sigma_^2 - \sigma_^2 - \sigma_^2 \\ &= \frac\left(\sigma_\sigma_ - \sigma_\sigma_\right) = \frac\left \left( \text(\boldsymbol) \right)^ - \text\left(\boldsymbol^\right) \right\\ pt I_3 &= \det(\sigma_) = \det(\boldsymbol)\\ &= \sigma_\sigma_\sigma_ + 2\sigma_\sigma_\sigma_ - \sigma_^2\sigma_ - \sigma_^2\sigma_ - \sigma_^2\sigma_ \\ \end The characteristic equation has three real roots \lambda_i, i.e. not imaginary due to the symmetry of the stress tensor. The \sigma_1 = \max \left( \lambda_1,\lambda_2,\lambda_3 \right), \sigma_3 = \min \left(\lambda_1, \lambda_2, \lambda_3\right) and \sigma_2 = I_1 - \sigma_1 - \sigma_3, are the principal stresses, functions of the eigenvalues \lambda_i. The eigenvalues are the roots of the
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. The principal stresses are unique for a given stress tensor. Therefore, from the characteristic equation, the coefficients I_1, I_2 and I_3, 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 n_j in the equation \left(\sigma_ - \lambda\delta_ \right)n_j = 0. These solutions are the principal directions or
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s 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: :\sigma_ = \begin \sigma_1 & 0 & 0\\ 0 & \sigma_2 & 0\\ 0 & 0 & \sigma_3 \end The principal stresses can be combined to form the stress invariants, I_1, I_2, and I_3. The first and third invariant are the trace and determinant respectively, of the stress tensor. Thus, :\begin I_1 &= \sigma_ + \sigma_ + \sigma_ \\ I_2 &= \sigma_\sigma_ + \sigma_\sigma_ + \sigma_\sigma_ \\ I_3 &= \sigma_\sigma_\sigma_ \\ \end 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. :\sigma_, \sigma_ = \frac \pm \sqrt Using just the part of the equation under the
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is equal to the maximum and minimum shear stress for plus and minus. This is shown as: :\tau_\max,\tau_\min = \pm \sqrt


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 45^\circ from the principal stress planes. The maximum shear stress is expressed as : \tau_\max = \frac\left, \sigma_\max - \sigma_\min\ Assuming \sigma_1 \ge \sigma_2 \ge \sigma_3 then : \tau_\max = \frac\left, \sigma_1 - \sigma_3\ 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 : \sigma_\text = \frac\left(\sigma_1 + \sigma_3\right) :


Stress deviator tensor

The stress tensor \sigma_ can be expressed as the sum of two other stress tensors: # a ''mean hydrostatic stress tensor'' or ''volumetric stress tensor'' or ''mean normal stress tensor'', \pi\delta_, which tends to change the volume of the stressed body; and # a deviatoric component called the ''stress deviator tensor'', s_, which tends to distort it. So :\sigma_ = s_ + \pi\delta_,\, where \pi is the mean stress given by :\pi = \frac = \frac = \fracI_1.\,
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(p) 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. :p = \zeta\, \nabla\cdot\vec - \pi = \zeta\,\frac - \pi = \sum_k\zeta\,\frac - \pi, where \zeta is a proportionality constant (viz. the Volume viscosity), \nabla\cdot is the divergence operator, x_k is the ''k'':th Cartesian coordinate, \vec is the
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and u_k is the ''k'':th Cartesian component of \vec. The deviatoric stress tensor can be obtained by subtracting the hydrostatic stress tensor from the Cauchy stress tensor: :\begin s_ &= \sigma_ - \frac\delta_,\,\\ \left
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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 s_ are the same as the principal directions of the stress tensor \sigma_. Thus, the characteristic equation is :\left, s_ - \lambda\delta_\ = \lambda^3 - J_1\lambda^2 - J_2\lambda - J_3 = 0, where J_1, J_2 and J_3 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 s_ or its principal values s_1, s_2, and s_3, or alternatively, as a function of \sigma_ or its principal values \sigma_1, \sigma_2, and \sigma_3. Thus, :\begin J_1 &= s_=0, \\ pt J_2 &= \frac s_s_ = \frac\operatorname\left(\boldsymbol^2\right) \\ &= \frac\left(s_1^2 + s_2^2 + s_3^2\right) \\ &= \frac\left \sigma_ - \sigma_)^2 + (\sigma_ - \sigma_)^2 + (\sigma_ - \sigma_)^2 \right + \sigma_^2 + \sigma_^2 + \sigma_^2 \\ &= \frac\left \sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 \right\\ &= \fracI_1^2 - I_2 = \frac\left operatorname\left(\boldsymbol^2\right) - \frac\operatorname(\boldsymbol)^2\right \\ pt J_3 &= \det(s_) \\ &= \fracs_s_s_ = \frac \text\left(\boldsymbol^3\right) \\ &= \frac\left(s_1^3 + s_2^3 + s_3^3\right) \\ &= s_1 s_2 s_3 \\ &= \fracI_1^3 - \fracI_1 I_2 + I_3 = \frac\left text(\boldsymbol^3) - \operatorname\left(\boldsymbol^2\right) \operatorname(\boldsymbol) + \frac\operatorname(\boldsymbol)^3\right\, \end Because s_ = 0, 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 :\sigma_\text = \sqrt = \sqrt\,.


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 , 1/\sqrt, ) 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'' \sigma_\text and ''octahedral shear stress'' \tau_\text, 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'', s_ = \frac I. Knowing that the stress tensor of point O (Figure 6) in the principal axes is :\sigma_ = \begin \sigma_1 & 0 & 0\\ 0 & \sigma_2 & 0\\ 0 & 0 & \sigma_3 \end the stress vector on an octahedral plane is then given by: :\begin \mathbf_\text^ &= \sigma_ n_i\mathbf_j \\ &= \sigma_1 n_1\mathbf_1 + \sigma_2 n_2\mathbf_2 + \sigma_3 n_3\mathbf_3\\ &= \frac(\sigma_1\mathbf_1 + \sigma_2\mathbf_2 + \sigma_3\mathbf_3) \end The normal component of the stress vector at point O associated with the octahedral plane is :\begin \sigma_\text &= T^_i n_i \\ &= \sigma_n_i n_j \\ &= \sigma_1 n_1 n_1 + \sigma_2 n_2 n_2 + \sigma_3 n_3 n_3 \\ &= \frac(\sigma_1 + \sigma_2 + \sigma_3) = \fracI_1 \end 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 :\begin \tau_\text &= \sqrt \\ &= \left frac\left(\sigma_1^2 + \sigma_2^2 + \sigma_3^2\right) - \frac(\sigma_1 + \sigma_2 + \sigma_3)^2\right\frac \\ &= \frac\left \sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2\right\frac = \frac\sqrt = \sqrt \end


See also

* Cauchy momentum equation * Critical plane analysis * Stress–energy tensor


Notes


References

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{{Authority control Tensor physical quantities Solid mechanics Continuum mechanics Structural analysis