[[Image:Internal forces in continuum.svg|370px|right|thumb|Figure 2.1a Internal distribution of contact forces and couple stresses on a differential $dS\backslash ,\backslash !$ of the internal surface $S\backslash ,\backslash !$ in a continuum, as a result of the interaction between the two portions of the continuum separated by the surface]] [[Image:Internal forces in continuum 2.svg|370px|right|thumb|Figure 2.1b Internal distribution of contact forces and couple stresses on a differential $dS\backslash ,\backslash !$ of the internal surface $S\backslash ,\backslash !$ in a continuum, as a result of the interaction between the two portions of the continuum separated by the surface]] [[Image:stress vector.svg|370px|right|thumb|Figure 2.1c Stress vector on an internal surface S with normal vector n. Depending on the orientation of the plane under consideration, the stress vector may not necessarily be perpendicular to that plane, ''i.e.'' parallel to $\backslash mathbf\{n\}\backslash ,\backslash !$, and can be resolved into two components: one component normal to the plane, called ''normal stress'' $\backslash sigma\_\backslash mathrm\{n\}\; \backslash ,\backslash !$, and another component parallel to this plane, called the ''shearing stress'' $\backslash tau\; \backslash ,\backslash !$.]] The '''Euler–Cauchy stress principle''' is a law of the [[contunuum mechanics|mechanics of continuous bodies]] that 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 vector field '''T'''^('''n'''), called the stress vector, defined on the surface ''S'' and assumed to depend continuously on the surface's unit vector '''n'''.{{rp|p.66–96}} == Detailed formulation == 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''). The body is subjected to external surface forces '''F''' and body forces '''b'''. The internal contact forces transmitted from one segment to the other through the dividing plane, due to the action of one portion of the continuum onto the other, generate a force distribution on a small area Δ''S'', with a normal unit [[vector (geometry)|vector]] '''n''', on the dividing plane ''S''. The force distribution is equipollent to a contact force Δ'''F''' and a couple stress Δ'''M''', as shown in Figure 2.1a and 2.1b. Cauchy’s stress principle asserts{{rp|p.47–102}} that as Δ''S'' becomes very small and tends to zero the ratio Δ'''F'''/Δ''S'' becomes d'''F'''/d''S'' and the couple stress vector Δ'''M''' vanishes. In specific fields of continuum mechanics the couple stress is assumed not to vanish; however, classical branches of continuum mechanics address non-[[chemical polarity|polar]] materials which do not consider couple stresses and body moments. The resultant vector d'''F'''/d''S'' is defined as the ''stress vector'' or ''traction vector'' given by '''T'''^('''n''') = ''T''_i^('''n''') '''e'''_i at the point ''P'' associated with a plane with a normal vector '''n''': :$T^\{(\backslash mathbf\{n\})\}\_i=\; \backslash lim\_\{\backslash Delta\; S\; \backslash to\; 0\}\; \backslash frac\; \{\backslash Delta\; F\_i\}\{\backslash Delta\; S\}\; =\; \{dF\_i\; \backslash over\; dS\}.$ 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. Depending on the orientation of the plane under consideration, the stress vector may not necessarily be perpendicular to that plane, ''i.e.'' parallel to '''n''', and can be resolved into two components (Figure 2.1c): * one normal to the plane, called ''normal stress'' :$\backslash mathbf\{\backslash sigma\_\backslash mathrm\{n\}\}=\; \backslash lim\_\{\backslash Delta\; S\; \backslash to\; 0\}\; \backslash frac\; \{\backslash Delta\; F\_\backslash mathrm\; n\}\{\backslash Delta\; S\}\; =\; \backslash frac\{dF\_\backslash mathrm\; n\}\{dS\},$ :where d''F''_n is the normal component of the force d'''F''' to the differential area d''S'' * and the other parallel to this plane, called the ''shear stress'' :$\backslash mathbf\; \backslash tau=\; \backslash lim\_\{\backslash Delta\; S\; \backslash to\; 0\}\; \backslash frac\; \{\backslash Delta\; F\_\backslash mathrm\; s\}\{\backslash Delta\; S\}\; =\; \backslash frac\{dF\_\backslash mathrm\; s\}\{dS\},$ :where d''F''_s is the tangential component of the force d'''F''' to the differential surface area d''S''. The shear stress can be further decomposed into two mutually perpendicular vectors. ===Cauchy’s postulate=== According to the ''Cauchy Postulate'', the stress vector '''T'''^('''n''') remains unchanged for all surfaces passing through the point ''P'' and having the same normal vector '''n''' at ''P'', i.e., having a common [[tangent]] at ''P''. This means that the stress vector is a function of the normal vector '''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'',{{rp|p.103–130}} 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 laws of motion|Newton's third law]] of motion of action and reaction, and is expressed as :$-\; \backslash mathbf\{T\}^\{(\backslash mathbf\{n\})\}=\; \backslash mathbf\{T\}^\{(-\; \backslash mathbf\{n\})\}.\backslash ,\backslash !$ ===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''': :$\backslash mathbf\{T\}^\{(\backslash mathbf\; n)\}=\; \backslash mathbf\; n\; \backslash cdot\backslash boldsymbol\{\backslash sigma\}\backslash quad\; \backslash text\{or\}\; \backslash quad\; T\_j^\{(n)\}=\; \backslash sigma\_\{ij\}n\_i.\backslash ,\backslash !$ 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 ''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 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 '''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'''^('''e'''₁), '''T'''^('''e'''₂), and '''T'''^('''e'''₃), 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 laws of motion|Euler’s first law of motion]] (Newton’s second law of motion), gives: :$\backslash mathbf\{T\}^\{(\backslash mathbf\{n\})\}\; \backslash ,\; dA\; -\; \backslash mathbf\{T\}^\{(\backslash mathbf\{e\}\_1)\}\; \backslash ,\; dA\_1\; -\; \backslash mathbf\{T\}^\{(\backslash mathbf\{e\}\_2)\}\; \backslash ,\; dA\_2\; -\; \backslash mathbf\{T\}^\{(\backslash mathbf\{e\}\_3)\}\; \backslash ,\; dA\_3\; =\; \backslash rho\; \backslash left(\; \backslash frac\{h\}\{3\}dA\; \backslash right)\; \backslash mathbf\{a\},\backslash ,\backslash !$ [[Image:Cauchy tetrahedron.svg|280px|right|thumb|Figure 2.2. Stress vector acting on a plane with normal unit vector '''n'''.
'''A note on the sign convention:''' The tetrahedron is formed by slicing a parallelepiped along an arbitrary plane '''n'''. So, the force acting on the plane '''n''' is the reaction exerted by the other half of the parallelepiped and has an opposite sign.]] 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): :$dA\_1=\; \backslash left(\backslash mathbf\{n\}\; \backslash cdot\; \backslash mathbf\{e\}\_1\; \backslash right)dA\; =\; n\_1\; \backslash ;\; dA,\backslash ,\backslash !$ :$dA\_2=\; \backslash left(\backslash mathbf\{n\}\; \backslash cdot\; \backslash mathbf\{e\}\_2\; \backslash right)dA\; =\; n\_2\; \backslash ;\; dA,\backslash ,\backslash !$ :$dA\_3=\; \backslash left(\backslash mathbf\{n\}\; \backslash cdot\; \backslash mathbf\{e\}\_3\; \backslash right)dA\; =\; n\_3\; \backslash ;\; dA,\backslash ,\backslash !$ and then substituting into the equation to cancel out d''A'': :$\backslash mathbf\{T\}^\{(\backslash mathbf\{n\})\}\; -\; \backslash mathbf\{T\}^\{(\backslash mathbf\{e\}\_1)\}n\_1\; -\; \backslash mathbf\{T\}^\{(\backslash mathbf\{e\}\_2)\}n\_2\; -\; \backslash mathbf\{T\}^\{(\backslash mathbf\{e\}\_3)\}n\_3\; =\; \backslash rho\; \backslash left(\; \backslash frac\{h\}\{3\}\; \backslash right)\; \backslash mathbf\{a\}.\backslash ,\backslash !$ 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 :$\backslash mathbf\{T\}^\{(\backslash mathbf\{n\})\}\; =\; \backslash mathbf\{T\}^\{(\backslash mathbf\{e\}\_1)\}\; n\_1\; +\; \backslash mathbf\{T\}^\{(\backslash mathbf\{e\}\_2)\}\; n\_2\; +\; \backslash mathbf\{T\}^\{(\backslash mathbf\{e\}\_3)\}\; n\_3.\backslash ,\backslash !$ ==See also====Equilibrium equations and symmetry of the stress tensor== [[Image:Equilibrium equation body.svg|200px|right|thumb|Figure 4. Continuum body in equilibrium]] When a body is in equilibrium the components of the stress tensor in every point of the body satisfy the equilibrium equations, :$\backslash sigma\_\{ji,j\}+\; F\_i\; =\; 0\; \backslash ,\backslash !$ For example, for a [[hydrostatic fluid]] in equilibrium conditions, the [[stress tensor]] takes on the form: :$\{\backslash sigma\_\{ij\}\}\; =\; -p\{\backslash delta\_\{ij\}\}\backslash$, where $p$ is the hydrostatic pressure, and $\{\backslash delta\_\{ij\}\}\backslash$ is the [[kronecker delta]]. :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of equilibrium equations |- |Consider a continuum body (see Figure 4) occupying a volume $V\backslash ,\backslash !$, having a surface area $S\backslash ,\backslash !$, with defined traction or surface forces $T\_i^\{(n)\}\backslash ,\backslash !$ per unit area acting on every point of the body surface, and body forces $F\_i\backslash ,\backslash !$ per unit of volume on every point within the volume $V\backslash ,\backslash !$. Thus, if the body is in [[Momentum#Relating_to_force_–_general_equations_of_motion|equilibrium]] the resultant force acting on the volume is zero, thus: :$\backslash int\_S\; T\_i^\{(n)\}dS\; +\; \backslash int\_V\; F\_i\; dV\; =\; 0\backslash ,\backslash !$ By definition the stress vector is $T\_i^\{(n)\}\; =\backslash sigma\_\{ji\}n\_j\backslash ,\backslash !$, then :$\backslash int\_S\; \backslash sigma\_\{ji\}n\_j\backslash ,\; dS\; +\; \backslash int\_V\; F\_i\backslash ,\; dV\; =\; 0\backslash ,\backslash !$ Using the [[Divergence theorem|Gauss's divergence theorem]] to convert a surface integral to a volume integral gives :$\backslash int\_V\; \backslash sigma\_\{ji,j\}\backslash ,\; dV\; +\; \backslash int\_V\; F\_i\backslash ,\; dV\; =\; 0\backslash ,\backslash !$ :$\backslash int\_V\; (\backslash sigma\_\{ji,j\}\; +\; F\_i\backslash ,)\; dV\; =\; 0\backslash ,\backslash !$ For an arbitrary volume the integral vanishes, and we have the ''equilibrium equations'' :$\backslash sigma\_\{ji,j\}\; +\; F\_i\; =\; 0\backslash ,\backslash !$ |} At the same time, 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 matrix|symmetric]], i.e. :$\backslash sigma\_\{ij\}=\backslash sigma\_\{ji\}\backslash ,\backslash !$ :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of symmetry of the stress tensor |- | Summing moments about point ''O'' (Figure 4) the resultant moment is zero as the body is in equilibrium. Thus, : where $\backslash mathbf\{r\}\backslash ,\backslash !$ is the position vector and is expressed as :$\backslash mathbf\{r\}=x\_j\backslash mathbf\{e\}\_j\backslash ,\backslash !$ Knowing that $T\_k^\{(n)\}\; =\backslash sigma\_\{mk\}n\_m\backslash ,\backslash !$ and using Gauss's divergence theorem to change from a surface integral to a volume integral, we have : The second integral is zero as it contains the equilibrium equations. This leaves the first integral, where $x\_\{j,m\}=\backslash delta\_\{jm\}\backslash ,\backslash !$, therefore :$\backslash int\_V\; (\backslash varepsilon\_\{ijk\}\backslash sigma\_\{jk\})\; dV=0\backslash ,\backslash !$ For an arbitrary volume V, we then have :$\backslash varepsilon\_\{ijk\}\backslash sigma\_\{jk\}=0\backslash ,\backslash !$ which is satisfied at every point within the body. Expanding this equation we have :$\backslash sigma\_\{12\}=\backslash sigma\_\{21\}\backslash ,\backslash !$, $\backslash sigma\_\{23\}=\backslash sigma\_\{32\}\backslash ,\backslash !$, and $\backslash sigma\_\{13\}=\backslash sigma\_\{31\}\backslash ,\backslash !$ or in general :$\backslash sigma\_\{ij\}=\backslash sigma\_\{ji\}\backslash ,\backslash !$ This proves that the stress tensor is symmetric |} 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, $K\_\{n\}\backslash rightarrow\; 1\backslash ,\backslash !$, or the continuum is a non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as [[polymers]]. ==References== Keith D. Hjelmstad (2005), [http://books.google.ca/books?id=LVTYjmcdvPwC&pg=PA103 "Fundamentals of Structural Mechanics"] (2nd edition). Prentice-Hall. ISBN 0-387-23330-X Teodor M. Atanackovic and Ardéshir Guran (2000), [http://books.google.ca/books?id=uQrBWdcGmjUC&pg=PA1 "Theory of Elasticity for Scientists and Engineers"]. Springer. ISBN 0-8176-4072-X Basar G. Thomas Mase and George E. Mase (1999), [http://books.google.ca/books?id=uI1ll0A8B_UC&pg=PA47 "Continuum Mechanics for Engineers"] (2nd edition). CRC Press. ISBN 0-8493-1855-6 {{harvnb|Truesdell|Topin|1960}} Peter Chadwick (1999), [http://books.google.ca/books?id=QSXIHQsus6UC&pg=PA95 "Continuum Mechanics: Concise Theory and Problems"]. Dover Publications, series "Books on Physics". ISBN 0-486-40180-4. pages Yuan-cheng Fung and Pin Tong (2001) [http://books.google.ca/books?id=hmyiIiiv4FUC&pg=PA66 "Classical and Computational Solid Mechanics"]. World Scientific. ISBN 981-02-4124-0