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Encyclopedia > Stress (physics)
Continuum mechanics
Key topics
Conservation of mass
Conservation of momentum
Navier-Stokes equations
Classical mechanics
Stress · Strain · Tensor
Solid mechanics
Solids · Elasticity

Plasticity · Hooke's law
Rheology · Viscoelasticity Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... Image File history File links File links The following pages link to this file: Bernoullis equation ... The law of conservation of mass/matter, also known as law of mass/matter conservation (or the Lomonosov-Lavoisier law), states that the mass of a closed system of substances will remain constant, regardless of the processes acting inside the system. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations which describe the motion of fluid substances such as liquids and gases. ... Classical mechanics (also called Newtonian mechanics) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... This article is about the deformation of materials. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... Solid mechanics is the branch of physics and mathematics that concern the behavior of solid matter under external actions (e. ... For other uses, see Solid (disambiguation). ... Elasticity is a branch of physics which studies the properties of elastic materials. ... For other uses, see Plasticity. ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ... Rheology is the study of the deformation and flow of matter under the influence of an applied stress. ... Viscoelasticity, also known as anelasticity, describes materials that exhibit both viscous and elastic characteristics when undergoing plastic deformation. ...

Fluid mechanics
Fluids · Fluid statics
Fluid dynamics · Viscosity · Newtonian fluids
Non-Newtonian fluids
Surface tension
Scientists

Newton · Stokes · others Fluid mechanics is the subdiscipline of continuum mechanics that studies fluids, that is, liquids and gases. ... A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. ... Fluid statics (also called hydrostatics) is the science of fluids at rest, and is a sub-field within fluid mechanics. ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... Viscosity is a measure of the resistance of a fluid to deform under shear stress. ... A Newtonian fluid (named for Isaac Newton) is a fluid that flows like water—its shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. ... A non-Newtonian fluid is a fluid in which the viscosity changes with the applied strain rate. ... In physics, surface tension is an effect within the surface layer of a liquid that causes that layer to behave as an elastic sheet. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Sir George Gabriel Stokes, 1st Baronet (13 August 1819–1 February 1903) was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics (including the Navier-Stokes equations), optics, and mathematical physics (including Stokes theorem). ...

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Stress is a measure of force per unit area within a body. It is a body's internal distribution of force per area that reacts to external applied loads. Stress is often broken down into its shear and normal components as these have unique physical significance. In short, stress is to force as strain is to elongation. In physics, force is anything that can cause a massive body to accelerate. ... Area is a physical quantity expressing the size of a part of a surface. ... Shear stress is a stress state where the stress is parallel or tangential to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. ... Tensile stress (or tension) is the stress state leading to expansion (volume and/or length of a material tends to increase). ... This article is about the deformation of materials. ...


Solids, liquids and gases have stress fields. Static fluids support normal stress (hydrostatic pressure) but will flow under shear stress. Moving viscous fluids can support shear stress (dynamic pressure). Solids can support both shear and normal stress, with ductile materials failing under shear and brittle materials failing under normal stress. All materials have temperature dependent variations in stress related properties, and non-newtonian materials have rate-dependent variations. Hydrostatic pressure is the pressure exerted by a fluid due to its weight. ... Viscosity is a measure of the resistance of a fluid to deform under shear stress. ... Ductility is the physical property of being capable of sustaining large plastic deformations without fracture (in metals, such as being drawn into a wire). ... A material is brittle if it is subject to fracture when subjected to stress i. ... A non-Newtonian fluid is a fluid in which the viscosity changes with the applied strain rate. ...

Contents

Stress tensor

Stress is a second-order tensor with nine components, but can be fully described with six components due to symmetry in the absence of body moments. In N dimensions, the stress tensor mathbf sigma_{ij} is defined by: In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mechanics, a moment is a measure of the turning effect of a force about some point in space. ...

dF_i=sum_{j=1}^N sigma_{ji},dA_j

where the dFi are the components of the resultant force vector acting on a small area dA which can be represented by a vector dAj perpendicular to the area element, facing outwards and with length equal to the area of the element. In elementary mechanics, the subscripts are often denoted x, y, z rather than 1,2,3.

Figure 1  Stress tensor

The components mathbf {it sigma_{ij}} of the stress tensor depend on the orientation of the plane that passes through the point under consideration, i.e on the viewpoint of the observer. This would lead to the ridiculous conclusion that the stress on a structure, and hence its proximity to failure, depends on the viewpoint of the observer. However, every tensor, including stress, has invariants that do not depend on the choice of viewpoint. The length of a first-order tensor, i.e a vector, is a simple example. The existence of invariants means that the components seen by one observer are related, via the tensor transformation relations, to those seen by any other observer. The transformation relations for a second-order tensor like stress are different from those of a first-order tensor, which is why it is misleading to speak of the stress 'vector'. Mohr's circle method is a graphical method for performing stress (or strain) transformations. Stress tensor Illustrated with Sodipodi for Windows source: me Duk 02:00, 23 Oct 2004 (UTC) File links The following pages link to this file: Stress (physics) Categories: GFDL images ... Stress tensor Illustrated with Sodipodi for Windows source: me Duk 02:00, 23 Oct 2004 (UTC) File links The following pages link to this file: Stress (physics) Categories: GFDL images ... In physics, invariants are usually quantities conserved (unchanged) by the symmetries of the physical system. ... Look up length, width, breadth in Wiktionary, the free dictionary. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... A tensor is a generalization of the concepts of vectors and matrices. ... Stress is a measure of force per unit area within a body. ...


When the stress tensor is needed to fully describe the state of stress in a body, it is useful to break the concept up into smaller parts that have physical significance. In a 1-dimensional system, such as a uniaxially loaded bar, stress is simply equal to the applied force divided by the cross-sectional area of the bar (see also pressure). The 2-D or 3-D cases are more complex. In three dimensions, the internal force acting on a small area dA of a plane that passes through a point P can be resolved into three components: one normal to the plane and two parallel to the plane (see Figure 1). The normal component divided by dA gives the normal stress (usually denoted σ), and the parallel components divided by the area dA give shear stresses (denoted τ or τ in elementary textbooks). If the area dA is finite then, strictly, these are average stresses. In the limit, when dA approaches zero, the stresses become stresses at the point P. In general, stress varies from point to point and so is a tensor field. This article is about pressure in the physical sciences. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... 2-dimensional renderings (ie. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ... Tensile stress (or tension) is the stress state leading to expansion (volume and/or length of a material tends to increase). ... Shear stress is a stress state where the stress is parallel or tangential to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ...


Stress in one-dimensional bodies

All real objects occupy three-dimensional space. However, if two dimensions are very large or very small compared to the others, the object may be modelled as one-dimensional. This simplifies the mathematical modelling of the object. One-dimensional objects include a piece of wire loaded at the ends and viewed from the side, and a metal sheet loaded on the face and viewed up close and through the cross section. Structural elements are used in structural analysis to simplify the structure which is to be analysed. ...


For one-dimensional objects, the stress tensor has only one component and is indistinguishable from a scalar. The simplest definition of stress, σ = F/A, where A is the initial cross-sectional area prior to the application of the load, is called engineering stress or nominal stress. However, when any material is stretched, its cross-sectional area may change by an amount that depends on the Poisson's ratio of the material. Engineering stress neglects this change in area. The stress axis on a stress-strain graph is often engineering stress, even though the sample may undergo a substantial change in cross-sectional area during testing. Figure 1: Rectangular specimen subject to compression, with Poissons ratio circa 0. ... A stress-strain curve is a graph derived from measuring load (stress - σ) versus extension (strain - ε) for a sample of a material. ...


True stress is an alternative definition in which the initial area is replaced by the current area. In engineering applications, the initial area is always known, and so calculations using nominal stress are generally easier. For small deformation, such as in practical material usage, the reduction in cross-sectional area is small and the distinction between nominal and true stress is insignificant; so the change of cross-sectional area could be assumed to be a constant value. This is not so for the large deformations typical of elastomers and plastic materials when the change in cross-sectional areas can be significant. The term elastomer is often used interchangeably with the term rubber, and is preferred when referring to vulcanisates. ... For other uses, see Plasticity. ...


In one dimension, conversion between true stress and nominal (engineering) stress is given by

σtrue = (1 + εe)(σe),

where εe is nominal (engineering) strain, and σe is nominal (engineering) stress. The relationship between true strain and engineering strain is given by This article is about the deformation of materials. ...

εtrue = ln(1 + εe).

In uniaxial tension, true stress is then greater than nominal stress. The converse holds in compression.


Example:  A steel bolt of diameter 5 mm has a cross-sectional area of 19.6 mm2. A load of 50 N induces a stress (force distributed over the cross section) of σ = 50/19.6 = 2.55 MPa (N/mm2). This can be thought of as each square millimeter of the bolt supporting 2.55 N of the total load. In another bolt with half the diameter, and hence a quarter the cross-sectional area, carrying the same 50 N load, the stress will be quadrupled (10.2 MPa). For other uses, see Steel (disambiguation). ... Screws come in a variety of shapes and sizes for different purposes. ... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ... A millimetre (American spelling: millimeter), symbol mm is an SI unit of length that is equal to one thousandth of a metre. ... Area is a physical quantity expressing the size of a part of a surface. ... For other uses, see Newton (disambiguation). ... The megapascal, symbol MPa is an SI unit of pressure. ...


The ultimate tensile strength is a property of a material and is usually determined experimentally from a uniaxial tensile test. It allows the calculation of the load that would cause fracture. The compressive strength is a similar property for compressive loads. The yield strength is the value of stress causing plastic deformation. Tensile strength isthe measures the force required to pull something such as rope, wire, or a structural beam to the point where it breaks. ... Tensile stress (or tension) is the stress state leading to expansion (volume and/or length of a material tends to increase). ... For fractures in geologic formations, see Rock fracture. ... Compressive strength is the capacity of a material to withstand axially directed pushing forces. ... Yield strength, or the yield point, is defined in engineering and materials science as the stress at which a material begins to plastically deform. ... For other uses, see Plasticity. ...


Stress in two-dimensional bodies

Cracks in rock resulting from stress.
Cracks in rock resulting from stress.

All real objects occupy 3-dimensional space. However, if one dimension is very large or very small compared to the others, the object may be modelled as two-dimensional. This simplifies the mathematical modelling of the object. Two-dimensional objects include a piece of wire loaded on the sides and viewed up close and through the cross-section and a metal sheet loaded in-plane and viewed face-on. Image File history File linksMetadata Size of this preview: 800 × 600 pixelsFull resolution (1600 × 1200 pixel, file size: 949 KB, MIME type: image/jpeg) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File linksMetadata Size of this preview: 800 × 600 pixelsFull resolution (1600 × 1200 pixel, file size: 949 KB, MIME type: image/jpeg) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... For fractures in geologic formations, see Rock fracture. ...


Notice that the same physical, three-dimensional object can be modelled as one-dimensional, two-dimensional or even three-dimensional, depending on the loading and viewpoint of the observer.


Plane stress

Plane stress is a two-dimensional state of stress (Figure 2). This 2-D state models well the state of stresses in a flat, thin plate loaded in the plane of the plate. Figure 2 shows the stresses on the x- and y-faces of a differential element. Not shown in the figure are the stresses in the opposite faces and the external forces acting on the material. Since moment equilibrium of the differential element shows that the shear stresses on the perpendicular faces are equal, the 2-D state of stresses is characterized by three independent stress components (σx, σy, τxy). Note that forces perpendicular to the plane can be abbreviated. For example, σx is an abbreviation for σxx. This notation is described further below. In real engineering components, stress (and strain) are 3-D tensors, however when one of the dimensions of the material is much smaller than the other two, it can be neglected and the resulting state of stress becomes bidimensional[1]. This state is known as plane stress because the normal... A standard definition of mechanical equilibrium is: A system is in mechanical equilibrium when the sum of the forces, and torque, on each particle of the system is zero. ...

Figure 2  Stresses normal and tangent to faces
Figure 2  Stresses normal and tangent to faces

See also plane strain. Stress in 2D fig 3 File links The following pages link to this file: Stress (physics) Categories: GFDL images ... Stress in 2D fig 3 File links The following pages link to this file: Stress (physics) Categories: GFDL images ... In real engineering components, stress (and strain) are 3-D tensors but in prismatic structures such as a long metal billet, the length of the structure is much greater than the other two dimensions. ...


Principal stresses in 2-D

Augustin Louis Cauchy was the first to demonstrate that at a given point, it is always possible to locate two orthogonal planes in which the shear stress vanishes. These planes in which the normal forces are acting are called the principal planes, while the normal stresses on these planes are the principal stresses. They are the eigenvalues of the stress tensor and are orthogonal because the stress tensor is symmetric (as per the spectral theorem). Eigenvalues are invariants with respect to choice of basis and are the roots of the Cayley–Hamilton theorem (although the term 'the' invariants usually means (I1,I2,I3)). Mohr's circle is a graphical method of extracting the principal stresses in a 2-dimensional stress state. The maximum and minimum principal stresses are the maximum and minimum possible values of the normal stresses. The maximum principal stress controls brittle fracture. Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... Fig. ... In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ... In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every square matrix over the real or complex field, satisfies its own characteristic equation. ...


The two dimensional Cauchy stress tensor is defined as:

sigma_{ij}= left[{begin{matrix} {sigma _x } & {tau _{xy}}  {tau _{xy}} & {sigma _y }  end{matrix}}right]

Then principal stresses σ12 are equal to:

sigma _1 = frac {sigma _x + sigma _ y}{2} + sqrt{ left( frac {sigma _x - sigma _ y}{2} right)^2 + {tau _{xy}}^2 }
sigma _2 = frac {sigma _x + sigma _ y}{2} - sqrt{ left( frac {sigma _x - sigma _ y}{2} right)^2 + {tau _{xy}}^2 }

Those formulas have geometrical interpretation in the form of Mohr Circle presented in section below.


Mohr's circle

A graphical representation of any 2-D stress state was proposed by Christian Otto Mohr in 1882. Consider the state of stress at a point P in a body (Figure 2). The Mohr's circle may be constructed as follows. Year 1882 (MDCCCLXXXII) was a common year starting on Sunday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Friday of the 12-day slower Julian calendar). ...

  1. Draw two perpendicular axes with the horizontal axis representing normal stress, while the vertical axis the shear stress.
  2. Plot the state of stress on the x-plane as the point A, whose abscissa (x value) is the magnitude of the normal stress, σx (tension is positive), and whose ordinate (y value) is the shear stress (clockwise shear is positive).
  3. Mark the magnitude of the normal stress σy on the horizontal axis (tension being positive).
  4. Mark the midpoint of the two normal stresses, O (Figure 3).
  5. Draw the circle with radius OA, centered at O (Figure 4).
  6. A point on the Mohr's circle represents the state of stresses on a particular plane at the point P. Of special interest are

the points where the circle crosses the horizontal axis, for they represent the magnitudes of the principal stresses (Figure 5). Abscissa means the x coordinate on an (x, y) graph; the input of a mathematical function against which the output is plotted. ... The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... Ordinate means the y coordinate on an (x, y) graph; the plotted output of a mathematical function. ...

Figure 3  Mohr's circle, stage 1
Figure 4  Mohr's circle, stage 2
Figure 4  Mohr's circle, stage 2
Figure 5  Mohr's circle, stage 3


Construction of Mohrs circle stage 1. ... Construction of Mohrs circle stage 1. ... Stage 2 in construction of Mohrs circle. ... Stage 2 in construction of Mohrs circle. ... Stage 3 in construction of Mohrs circle. ... Stage 3 in construction of Mohrs circle. ...


Mohr's circle may also be applied to three-dimensional stress. In this case, the diagram has three circles, two within a third.


Engineers use Mohr's circle to find the planes of maximum normal and shear stresses, as well as the stresses on known weak planes. For example, if the material is brittle, the engineer might use Mohr's circle to find the maximum component of normal stress (tension or compression); and for ductile materials, the engineer might look for the maximum shear stress. A material is brittle if it is subject to fracture when subjected to stress i. ... Ductility is the physical property of being capable of sustaining large plastic deformations without fracture (in metals, such as being drawn into a wire). ...


Stress in three dimensional bodies

Cauchy's principle

Cauchy enunciated the principle that, within a body, the forces that an enclosed volume imposes on the remainder of the material must be in equilibrium with the forces upon it from the remainder of the body.


This intuition provides a route to characterizing and calculating complicated patterns of stress. To be exact, the stress at a point may be determined by considering a small element of the body that has an area ΔA, over which a force ΔF acts. By making the element infinitesimally small, the stress vector σ is defined as the limit:

 sigma = lim_{Delta A to 0} frac {Delta F} {Delta A} = {dF over dA}

Being a tensor, the stress has two directional components: one for force and one for plane orientation; in three dimensions these can be two forces within the plane of the area A, the shear components, and one force perpendicular to A, the normal component. Therefore the shear stress can be further decomposed into two orthogonal force components within the plane. This gives rise to three total stress components acting on this plane. For example in a plane orthogonal to the x axis, there can be a normal force applied in the x direction and a combination of y and z in plane force components. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...


The considerations above can be generalized to three dimensions. However, this is very complicated, since each shear loading produces shear stresses in one orientation and normal stresses in other orientations, and vice versa. Often, only certain components of stress will be important, depending on the material in question.


The von Mises stress is derived from the distortion energy theory and is a simple way to combine stresses in three dimensions to calculate failure criteria of ductile materials. In this way, the strength of material in a 3-D state of stress can be compared to a test sample that was loaded in one dimension. Von Mises stress, , or simply Mises stress, is a scalar function of the components of the stress tensor that gives an appreciation of the overall magnitude of the tensor. ...


The stress tensor in 3-D

(see also viscosity and Hooke's law for development of the stress tensor in viscous and elastic materials respectively) Viscosity is a measure of the resistance of a fluid to deform under shear stress. ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ...


Because the behavior of a body does not depend on the coordinate systems used to measure it, stress can be described by a tensor. In the absence of body moments, the stress tensor is symmetric and can always be resolved into the sum of two symmetric tensors: This article does not cite any references or sources. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mechanics, a moment is a measure of the turning effect of a force about some point in space. ...

  • a mean or hydrostatic stress tensor, involving only pure tension and compression; and
  • a shear or deviatoric stress tensor, involving only shear stress.

In the case of a fluid, Pascal's law shows that the hydrostatic stress is the same in all directions, at least to a first approximation, so can be captured by the scalar quantity pressure. Thus, in the case of a solid, the hydrostatic (or isostatic) pressure p is defined as one third of the trace of the tensor, i.e., the mean of the diagonal terms. A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. ... In the physical sciences, Pascals law or Pascals principle states that for all points at the same absolute height in a connected body of an incompressible fluid at rest, the fluid pressure is the same, even if additional pressure is applied on the fluid at some place. ... In physics, a scalar is a simple physical quantity that does not depend on direction, and therefore does not depend on the choice of a coordinate system. ... This article is about pressure in the physical sciences. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...


For viscos fluids the chauchy stress tensor σ is defined as:


σij = − pδij + λ * ekkδij + 2ηeij


If the fluid is incompressible it follows that:

p = frac{mathrm{tr}(T)}{3} = frac{sigma_{11} + sigma_{22} + sigma_{33}}{3}

If the fluid is compressible the assumtion above is true, if the viscosity of compression μD vanishs:  mu_D = lambda^*+frac{2}{3}eta


Principal stresses in 3-D

The three dimensional Cauchy stress tensor is defined as:

sigma_{ij}= left[{begin{matrix} {sigma _x } & {tau _{xy}} & {tau _{xz}}  {tau _{yx}} & {sigma _y } & {tau _{yz}}  {tau _{zx}} & {tau _{zy}} & {sigma _z }  end{matrix}}right]

In equilibrium, τyx = τxy, τzx = τxz, and τzy = τyz, so the matrix is effectually symmetric. If not in equilibrium, other methods - not outlined here - must be used to make it symmetric before calculations can begin.


To calculate the principal stresses σ12 and σ3 the three invariants of the Cauchy stress tensor must be calculated:

{ I _ 1 = sigma _x + sigma _ y + sigma _ z }!
 I _ 2 = sigma _x sigma _ y + sigma _ y sigma _ z + sigma _ z sigma _ x - {tau _{xy}}^2 - {tau _{yz}}^2 - {tau _{xz}}^2
 I _ 3 = sigma _x sigma _ y sigma _ z + 2tau _{xy}tau _{yz}tau _{xz} - sigma _x {tau _{yz}}^2 - sigma _ y {tau _{xz}}^2 - sigma _z {tau _{xy}}^2

Then the characteristic equation of 3-D principal stresses is expressed as:

{ sigma ^ 3 - I _ 1 sigma ^2 + I _ 2 sigma - I _ 3 = 0}!

The three roots of this equation are principal stresses σ12 and σ3. When they are found it can be shown that the three invariants can be expressed in terms of principal stresses:

{ I _ 1 = sigma _ 1 + sigma _ 2 + sigma _ 3 }!
{ I _ 2 = sigma _ 1 sigma _ 2 + sigma _ 2 sigma _ 3 + sigma _ 3 sigma _ 1 }!
{ I _ 3 = sigma _ 1 sigma _ 2 sigma _ 3 }!

Generalized notation

In the generalized stress tensor notation, the tensor components are written σij, where i and j are in {1;2;3}.


(caution: subscript notation in this section is different from the rest of the article - the order of subscripts is reversed)


The first step is to number the sides of the cube. When the lines are parallel to a vector base (vec{e_1},vec{e_2},vec{e_3}), then:

  • the sides perpendicular to vec{e_j} are called j and -j; and
  • from the center of the cube, vec{e_j} points toward the j side, while the -j side is at the opposite.

Figure 6  Cube face nomenclature Wikipedia does not have an article with this exact name. ...


σij is then the component along the i axis that applies on the j side of the cube. (Or in books in the English language, σij is the stress on the i face acting in the j direction -- the transpose of the subscript notation herein. But transposing the subscript notation produces the same stress tensor, since a symmetric matrix is equal to its transpose.)


Figure 7  Stress tensor components Wikipedia does not have an article with this exact name. ...


This generalized notation allows an easy writing of equations of the continuum mechanics, such as the generalized Hooke's law: Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ...

sigma_{ij} = sum_{kl} C_{ijkl} cdot varepsilon_{kl}

The correspondence with the former notation is thus:

x 1
y 2
z 3
σxx σ11
τxy σ12
τxz σ13
...

Why is Newtonian stress a symmetric tensor?

The fact that the Newtonian stress is a symmetric tensor follows from some simple considerations. The force on a small volume element will be the sum of all the stress forces over the surface area of that element. Suppose we have a volume element in the form of a long bar with a triangular cross section, where the triangle is a right triangle. We can neglect the forces on the ends of the bar, because they are small compared to the faces of the bar. Let vec{A} be the vector area of one face of the bar, vec{B} be the area of the other, and vec{C} be the area of the "hypotenuse face" of the bar. It can be seen that In geometry, for a finite planar surface of scalar area , the vector area is defined as a vector whose magnitude is and whose direction is perpendicular to the plane, as determined by the right-hand screw rule on the rim. ...

vec{C}=-vec{A}-vec{B}

Let's say sigma(vec{A}) is the force on area vec{A} and likewise for the other faces. Since the stress is by definition the force per unit area, it is clear that

sigma(kvec{A})=ksigma(vec{A})

The total force on the volume element will be:

vec{F}=sigma(vec{A})+sigma(vec{B})-sigma(vec{A}+vec{B})

Let's suppose that the volume element contains mass, at a constant density. The important point is that if we make the volume smaller, say by halving all lengths, the area will decrease by a factor of four, while the volume will decrease by a factor of eight. As the size of the volume element goes to zero, the ratio of area to volume will become infinite. The total stress force on the element is proportional to its area, and so as the volume of the element goes to zero, the force/mass (i.e. acceleration) will also become infinite, unless the total force is zero. In other words:

sigma(vec{A}+vec{B})=sigma(vec{A})+sigma(vec{B})

This, along with the second equation above, proves that the σ function is a linear vector operator (i.e. a tensor). By an entirely analogous argument, we can show that the total torque on the volume element (due to stress forces) must be zero, and that it follows from this restriction that the stress tensor must be symmetric. Torque applied via an adjustable end wrench Relationship between force, torque, and momentum vectors in a rotating system In physics, torque (or often called a moment) can informally be thought of as rotational force or angular force which causes a change in rotational motion. ...


However, there are two fundamental ways in which this mode of thinking can be misleading. First, when applying this argument in tandem with the underlying assumption from continuum mechanics that the Knudsen number is strictly less than one, then in the limit K_{n}rightarrow 1, the symmetry assumptions in the stress tensor may break down. This is the case of Non-Newtonian fluid, and can lead to rotationally non-invariant fluids, such as polymers. The other case is when the system is operating on a purely finite scale, such as is the case in mechanics where Finite deformation tensors are used. Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... The Knudsen number (Kn) is the ratio of the molecular mean free path length to a representative physical length scale. ... A non-Newtonian fluid is a fluid in which the viscosity changes with the applied strain rate. ... A polymer is a long, repeating chain of atoms, formed through the linkage of many molecules called monomers. ... In continuum mechanics, finite deformation tensors are used when the deformation of a body is sufficiently large to invalidate the assumptions inherent in small strain theory. ...


Equilibrium conditions

The state of stress as defined by the stress tensor is an equilibrium state if the following conditions are satisfied:

 frac {partial {sigma_{11}}} {partial {x_{1}}} + frac {partial {sigma_{21}}} {partial {x_{2}}} + frac {partial {sigma_{31}}} {partial {x_{3}}} = f_{1}
 frac {partial {sigma_{12}}} {partial {x_{1}}} + frac {partial {sigma_{22}}} {partial {x_{2}}} + frac {partial {sigma_{32}}} {partial {x_{3}}} = f_{2}
 frac {partial {sigma_{13}}} {partial {x_{1}}} + frac {partial {sigma_{23}}} {partial {x_{2}}} + frac {partial {sigma_{33}}} {partial {x_{3}}} = f_{3}

σij are the components of the tensor, and f 1 , f 2 , and f 3 are the body forces (force per unit volume).


These equations can be compactly written using Einstein notation in which repeated indices are summed. Defining partial_i as partial/partial x_i the equilibrium conditions are written: This article or section does not adequately cite its references or sources. ...

partial_jsigma_{ji}=f_i

The equilibrium conditions may be derived from the condition that the net force on an infinitesimal volume element must be zero. Consider an infinitesimal cube aligned with the x1, x2, and x3 axes, with one corner at xi and the opposite corner at xi + dxi and having each face of area dA. Consider just the faces of the cube which are perpendicular to the x1 axis. The area vector for the near face is [ − dA,0,0] and for the far face it is [dA,0,0]. The net stress force on these two opposite faces is

dF_i=sigma_{1i}([x_1+dx_1,x_2,x_3]),dA-sigma_{1i}([x_1,x_2,x_3]),dA approx partial_1sigma_{1i},dV

A similar calculation can be carried out for the other pairs of faces. The sum of all the stress forces on the infinitesimal cube will then be

dF_i=partial_jsigma_{ji},dV

Since the net force on the cube must be zero, it follows that this stress force must be balanced by the force per unit volume fi on the cube (e.g., due to gravitation, electromagnetic forces, etc.) which yields the equilibrium conditions written above.


Equilibrium also requires that the resultant moment on the cube of material must be zero. Taking the moment of the forces above about any suitable point, it follows that, for equilibrium in the absence of body moments In mechanics, a moment is a measure of the turning effect of a force about some point in space. ...

sigma_{ij}= sigma_{ji},.

The stress tensor is then symmetric and the subscripts can be written in either order.


Stress in big bending deformation

For big deformations in of the body, the stress in the cross-section is calculated using an extended version of this formula. First the following assumptions must be made: Image File history File links Big_bending_asymptote_stress. ...

  1. Assumption of flat sections - before and after deformation the considered section of body remains flat (i.e. is not swirled).
  2. Shear and normal stresses in this section that are perpendicular to the normal vector of cross section have no influence on normal stresses that are parallel to this section.

The big bending is usually assumed when bending radius ρ is smaller than ten section heights h:

ρ < 10h

With those assumptions the stress in big bending is calculated as:

 sigma = frac {F} {A} + frac {M} {rho A} + {frac {M} {{I_x}'}}y{frac {rho}{rho +y}}

where

F is the normal force
A is the section area
M is the bending moment
ρ is the local bending radius (the radius of bending at the current section)
Ix' is the area moment of inertia along the x axis, at the y place (see Steiner's theorem)
y is the position along y axis on the section area in which the stress σ is calculated

When bending radius ρ approaches infinity and y is zero, the original formula is back: In physics, force is anything that can cause a massive body to accelerate. ... Area is a physical quantity expressing the size of a part of a surface. ... A bending moment in physics is an example of an internal force that is induced in a restrained structural element when external forces are applied. ... The second moment of area, also known as the second moment of inertia and the area moment of inertia, is a property of a shape that is used to predict its resistance to bending and deflection. ... The parallel axes rule can be used to determine the moment of inertia of a rigid object about any axis, given the moment of inertia of the object about the parallel axis through the objects center of mass and the perpendicular distance between the axes. ...

sigma = {F over A} pm frac {My}{I} .

Stress measurement

As with force, stress cannot be measured directly but is usually inferred from measurements of strain and knowledge of elastic properties of the material. Devices capable of measuring stress indirectly in this way are strain gauges and piezoresistors. Look up strain in Wiktionary, the free dictionary. ... Typical foil strain gauge. ... Piezoresistors are resistors made from a piezoresistive material and are usually used for measurement of mechanical stress. ...


Units

The SI unit for stress is the pascal (symbol Pa), the same as that of pressure. Since the pascal is so small (1 N/m2), engineering quantities are usually measured in megapascals (MPa) or gigapascals (GPa) In US Customary units, stress is expressed in pounds-force per square inch (psi) or kilopounds-force per square inch (ksi). See also pressure. Look up si, Si, SI in Wiktionary, the free dictionary. ... The pascal (symbol: Pa) is the SI derived unit of pressure or stress (also: Youngs modulus and tensile strength). ... mega- (symbol M) is an SI prefix in the SI system of units denoting a factor of 106, i. ... Look up giga- in Wiktionary, the free dictionary. ... A pressure gauge reading in PSI (red scale) and kPa (black scale) The pound-force per square inch (symbol: lbf/in²) is a non-SI unit of pressure based on avoirdupois units. ... This article is about pressure in the physical sciences. ...


Residual stress

Residual stresses are stresses that remain after the original cause of the stresses has been removed. Residual stresses occur for a variety of reasons, including inelastic deformations and heat treatment. Heat from welding may cause localized expansion. When the finished weldment cools, some areas cool and contract more than others, leaving residual stresses. Castings may also have large residual stresses due to uneven cooling. Residual stresses are stresses that remain after the original cause of the stresses (external forces, heat gradient) has been removed. ...


While uncontrolled residual stresses are undesirable, many designs rely on controlled ones. For example, toughened glass and prestressed concrete rely on residual stress to prevent brittle failure. Similarly, a gradient in martensite formation leaves residual stress in some swords with particularly hard edges (notably the katana), which can prevent the opening of edge cracks. In certain types of gun barrels made with two telescoping tubes forced together, the inner tube is compressed while the outer tube stretches, preventing cracks from opening in the rifling when the gun is fired. These tubes are often heated or dunked in liquid nitrogen to aid assembly. Architectural glass has been used in buildings since the 11th century. ... Traditional reinforced concrete is based on the use of steel reinforcement bars, rebar, inside poured concrete. ... A material is brittle if it is subject to fracture when subjected to stress i. ... Martensite in AISI 4140 steel 0. ... Look up hardness in Wiktionary, the free dictionary. ... This article or section does not cite any references or sources. ... Rifling of a Canon de 75 modèle 1897 A 35 caliber Remington, with a microgroove rifled barrel with a right hand twist. ... General Name, Symbol, Number nitrogen, N, 7 Chemical series nonmetals Group, Period, Block 15, 2, p Appearance colorless gas Standard atomic weight 14. ...


In manufacturing, press fits are the most common intentional use of residual stress. Automotive wheel studs, for example, are pressed into holes on the wheel hub. The holes are smaller than the studs, requiring force to drive the studs into place. The residual stresses fasten the parts together. Nails are another example. A pile of nails. ...


See also

3-D elasticity is one of three methods of structural analysis. ... // Linear elasticity The linear theory of elasticity models the macroscopic mechanical properties of solids assuming small deformations. ... This article needs to be cleaned up to conform to a higher standard of quality. ... This article is about the deformation of materials. ... The strain tensor, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation: the diagonal coefficients εii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms εij = 1/2 γij (i... This article is in need of attention from an expert on the subject. ... A stress-strain curve is a graph derived from measuring load (stress - σ) versus extension (strain - ε) for a sample of a material. ... A stress concentration is a phenomenon encounterered in mechanical engineering where an object under load has higher than average local stresses due to its shape. ... Von Mises stress, , or simply Mises stress, is a scalar function of the components of the stress tensor that gives an appreciation of the overall magnitude of the tensor. ... Yield strength, or the yield point, is defined in engineering as the amount of strain that a material can undergo before moving from elastic deformation into plastic deformation. ... Yield surface is described in three dimensional space of principal stresses (), and encompasses the elastic region of material behavior. ...

Books

  • Dieter, G. E. (3 ed.). (1989). Mechanical Metallurgy. New York: McGraw-Hill. ISBN 0-07-100406-8.
  • Love, A. E. H. (4 ed.). (1944). Treatise on the Mathematical Theory of Elasticity. New York: Dover Publications. ISBN 0-486-60174-9.
  • Marsden, J. E., & Hughes, T. J. R. (1994). Mathematical Foundations of Elasticity. New York: Dover Publications. ISBN 0-486-67865-2.
  • L.D.Landau and E.M.Lifshitz. (1959). Theory of Elasticity.

External links

  • ESDU Stress Analysis Methods
  • True stress and true strain

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