Stress (physics) - Citizendia

Continuum mechanics

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Stress is a measure of the average amount of force exerted per unit area. Continuum mechanics is a branch of Mechanics that deals with the analysis of the Kinematics and mechanical behavior of materials modeled as a continuum e The law of conservation of mass/matter, also known as law of mass/matter conservation (or the Lomonosov - Lavoisier law says that the Mass of In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such In Physics, a force is whatever can cause an object with Mass to Accelerate. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. It is a measure of the intensity of the total internal forces acting within a body across imaginary internal surfaces, as a reaction to external applied forces and body forces. In Physics, a force is whatever can cause an object with Mass to Accelerate. A body force is a force that acts on the volume of a body and also can be defined as an external force acting throughout the mass of a body It was introduced into the theory of elasticity by Cauchy around 1822. Stress is a concept that is based on the concept of continuum. Continuum mechanics is a branch of Mechanics that deals with the analysis of the Kinematics and mechanical behavior of materials modeled as a continuum e In general, stress is expressed as

$\ \sigma = \frac{F}{A} \,$

where

$\ \sigma$ is the average stress, also called engineering or nominal stress, and

$\ F$ is the force acting over the area $\ A$ .

The SI unit for stress is the pascal (symbol Pa), which is a shorthand name for one newton (Force) per square metre (Unit Area). The unit for stress is the same as that of pressure, which is also a measure of Force per unit area. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface Engineering quantities are usually measured in megapascals (MPa) or gigapascals (GPa). For other meanings see Giga (disambiguation Giga- (symbol G is a prefix in the SI system of units denoting 109 In Imperial units, stress is expressed in pounds-force per square inch (psi) or kilopounds-force per square inch (ksi). Imperial units or the Imperial system is a collection of units first defined in the British Weights and Measures Act of 1824 The pound per square inch or more accurately pound-force per square inch (symbol psi or lbf/in² or lbf/in²) is a unit of

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. A strain gauge (alternatively strain gage) is a device used to measure the strain of an object The piezoresistive effect describes the changing Electrical resistance of a material due to applied Mechanical stress.

Stress as a tensor

Note: the Einstein summation convention of summing on repeated indices is used below. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational

In its full form, linear stress is a rank-two tensor quantity, and may be represented as a 3x3 matrix. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually A tensor may be seen as a linear vector operator - it takes a given vector and produces another vector as a result. In the case of the stress tensor, it takes the vector normal to any area element and yields the force (or "traction") acting on that area element. In matrix notation:

$F_i=\sum_{j=1}^3 \sigma_{ij} A_j$

where $\ A_i$ is the vector normal to a surface area element with a length equal to the area of the surface element, and $\ F_i$ is the force vector (or traction vector) acting on that element. Using index notation, we can eliminate the summation sign, since all sums will be the same over repeated indices. Index notation is used in Mathematics to refer to the elements of matrices or the components of a vector. Thus:

$F_i=\sigma_{ij} A_j \,$

Just as it is the case with a vector (which is actually a rank-one tensor), the matrix components of a tensor depend upon the particular coordinate system chosen. As with a vector, there are certain invariants associated with the stress tensor, whose value does not depend upon the coordinate system chosen (or the area element upon which the stress tensor operates). For a vector, there is only one invariant - the length. For a tensor, there are three - the eigenvalues of the stress tensor, which are called the principal stresses. It is important to note that the only physically significant parameters of the stress tensor are its invariants, since they are not dependent upon the choice of the coordinate system used to describe the tensor.

If we choose a particular surface area element, we may divide the force vector by the area (stress vector) and decompose it into two parts: a normal component acting normal to the stressed surface, and a shear component, acting parallel to the stressed surface. An axial stress is a normal stress produced when a force acts parallel to the major axis of a body, e. g. column. If the forces pull the body producing an elongation, it is termed tensile force. In Physics String Tension is the magnitude of the pulling force exerted by a string cable chain or similar object on another object If on the other hand the forces push the body reducing its length, it is termed compressive stresses. Bending stresses, e. This article is about structural behavior For other meanings see Bending (disambiguation. g. produced on a bent beam, are a combination of tensile and compressive stresses. Torsional stresses, e. In Solid mechanics, torsion is the twisting of an object due to an applied Torque. g. produced on twisted shafts, are shearing stresses.

In the above description, little distinction is drawn between the "stress" and the "stress vector" since the body which is being stressed provides a particular coordinate system in which to discuss the effects of the stress. The distinction between "normal" and "shear" stresses is slightly different when considered independently of any coordinate system. The stress tensor yields a stress vector for a surface area element at any orientation, and this stress vector may be decomposed into normal and shear components. The normal part of the stress vector averaged over all orientations of the surface element yields an invariant value, and is known as the hydrostatic pressure. Fluid statics (also called hydrostatics) is the Science of Fluids at rest and is a sub-field within Fluid mechanics. Mathematically it is equal to the average value of the principal stresses (or, equivalently, the trace of the stress tensor divided by three). The normal stress tensor is then the product of the hydrostatic pressure and the unit tensor. Subtracting the normal stress tensor from the stress tensor gives what may be called the shear tensor. These two quantities are true tensors with physical significance, and their nature is independent of any coordinate system chosen to describe them. In fact, the extended Hooke's law is basically the statement that each of these two tensors is proportional to its strain tensor counterpart, and the two constants of proportionality (elastic moduli) are independent of each other. In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i Note that In rheology, the normal stress tensor is called extensional stress, and in acoustics is called longitudinal stress. Rheology is the study of the flow of matter mainly liquids but also soft solids or solids under conditions in which they flow rather than deform elastically Acoustics is the interdisciplinary science that deals with the study of Sound, Ultrasound and Infrasound (all mechanical waves in gases liquids and solids

Solids, liquids and gases have stress fields. Static fluids support normal stress but will flow under shear stress. Moving viscous fluids can support shear stress (dynamic pressure). Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. Solids can support both shear and normal stress, with ductile materials failing under shear and brittle materials failing under normal stress. Ductility is a mechanical property used to describe the extent to which materials can be deformed plastically or "stretched" into "wires" without All materials have temperature dependent variations in stress related properties, and non-newtonian materials have rate-dependent variations. A non-Newtonian fluid is a Fluid whose flow properties are not described by a single constant value of Viscosity.

Cauchy's stress principle

Cauchy's stress principle asserts that when a continuum body is acted on by forces, i. Continuum mechanics is a branch of Mechanics that deals with the analysis of the Kinematics and mechanical behavior of materials modeled as a continuum e e. surface forces and body forces, there are internal reactions (forces) throughout the body acting between the material points. A body force is a force that acts on the volume of a body and also can be defined as an external force acting throughout the mass of a body Based on this principle, Cauchy demonstrated that the state of stress at a point in a body is completely defined by the nine components $\ \sigma_{ij}$ of a second-order Cartesian tensor called the Cauchy stress tensor, given by

$\ \sigma_{ij}= \left[{\begin{matrix} \mathbf{T}^{(\mathbf{e}_1)} \\ \mathbf{T}^{(\mathbf{e}_2)} \\ \mathbf{T}^{(\mathbf{e}_3)} \\ \end{matrix}}\right] = \left[{\begin{matrix} \sigma _{11} & \sigma _{12} & \sigma _{13} \\ \sigma _{21} & \sigma _{22} & \sigma _{23} \\ \sigma _{31} & \sigma _{32} & \sigma _{33} \\ \end{matrix}}\right] \equiv \left[{\begin{matrix} \sigma _{xx} & \sigma _{xy} & \sigma _{xz} \\ \sigma _{yx} & \sigma _{yy} & \sigma _{yz} \\ \sigma _{zx} & \sigma _{zy} & \sigma _{zz} \\ \end{matrix}}\right] \equiv \left[{\begin{matrix} \sigma _x & \tau _{xy} & \tau _{xz} \\ \tau _{yx} & \sigma _y & \tau _{yz} \\ \tau _{zx} & \tau _{zy} & \sigma _z \\ \end{matrix}}\right]$

where

$\mathbf{T}^{(\mathbf{e}_1)}$ , $\mathbf{T}^{(\mathbf{e}_2)}$ , and $\mathbf{T}^{(\mathbf{e}_3)}$ are the stress vectors associated with the planes perpendicular to the coordinate axis,

$\ \sigma_{11}$ , $\ \sigma_{22}$ , and $\ \sigma_{33}$ are normal stresses, and

$\ \sigma_{12}$ , $\ \sigma_{13}$ , $\ \sigma_{21}$ , $\ \sigma_{23}$ , $\ \sigma_{31}$ , and $\ \sigma_{32}$ are shear stresses. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually

Relationship stress vector - stress tensor

The stress vector $\mathbf{T}^{(\mathbf{n})} \,$ at any point associated with a plane of normal vector $\ \mathbf{n}$ can be expressed as a function of the stress vectors on the planes perpendicular to the coordinate axes, i. e. components of the stress tensor $\ \sigma_{ij}$ . In tensor form this is:

$\ T_j^{(n)}= \sigma_{ij}n_i$

Transformation rule of the stress tensor

It can be shown that the stress tensor is a second order tensor; this is, under a change of the coordinate system, from an $\ x_i$ system to an $\ x^'_i$ system, the components $\ \sigma_{ij}$ in the initial system are transformed into the components $\ \sigma^'_{ij}$ in the new system according to the tensor transformation rule:

$\ \sigma^'_{ij}=a_{im}a_{jn}\sigma_{mn}$

where $a_{ij} \,$ is a rotation matrix. In Matrix theory, a rotation matrix is a real Square matrix whose Transpose is its inverse and whose Determinant is +1 In matrix form this is

$\ \left[{\begin{matrix} \sigma^'_{11} & \sigma^'_{12} & \sigma^'_{13} \\ \sigma^'_{21} & \sigma^'_{22} & \sigma^'_{23} \\ \sigma^'_{31} & \sigma^'_{32} & \sigma^'_{33} \\ \end{matrix}}\right]=\left[{\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{matrix}}\right]\left[{\begin{matrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \\ \end{matrix}}\right]\left[{\begin{matrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32} \\ a_{13} & a_{23} & a_{33} \\ \end{matrix}}\right]$

An easy visualization of this transformation for 2D and 3D stresses for simple rotations is Mohr's circle

Normal and shear stresses

The magnitude of the normal stress component, $\ \sigma_n$ , of any stress vector $\mathbf{T}^{(\mathbf{n})}$ acting on an arbitrary plane with normal vector $\ \mathbf{n}$ at a given point in terms of the component of the stress tensor $\ \sigma_{ij}$ is the dot product of the stress vector and the normal vector, thus

$\ \begin{align} \sigma_n &= \mathbf{T}^{(\mathbf{n})}\cdot \mathbf{n} \\ &=T^{(n)}_in_i \\ &=\sigma_{ij}n_in_j \end{align}$

The magnitude of the shear stress component, $\ \tau_n$ , can then be found using the Pythagorean theorem, thus

$\begin{align} \tau_n &=\sqrt{(T^{(n)})^2-\sigma_n^2} \\ &= \sqrt{T^{(n)}T^{(n)}-\sigma_n^2} \\ \end{align}$

Equilibrium equations and symmetry of the stress tensor

Figure 4. Mohr's circle is a graphical representation of any 2-D stress state proposed in 1892 by Christian Otto Mohr. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry 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,

$\ \sigma_{ji,j}+ F_i = 0$

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, i. In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T} e.

$\ \sigma_{ij}=\sigma_{ji}$

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, $K_{n}\rightarrow 1$ , e. The Knudsen number ( Kn) is a Dimensionless number defined as the Ratio of the molecular Mean free path length to a representative physical length g. Non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as polymers. A non-Newtonian fluid is a Fluid whose flow properties are not described by a single constant value of Viscosity. A polymer is a large Molecule ( Macromolecule) composed of repeating Structural units typically connected by Covalent Chemical bonds

Principal stresses and stress invariants

The components $\ \sigma_{ij}$ 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. In Mathematics and Theoretical physics, an invariant is a property of a system which remains unchanged under some transformation. 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 length of the vector is a physical quantity (a scalar) and is independent of the coordinate system chosen to represent the vector. Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end 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. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes Their direction vectors are the principal directions or eigenvectors. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes When the coordinate system is chosen to coincide with the eigenvectors of the stress tensor, the stress tensor is represented by a diagonal matrix:

$\sigma_{ij}= \begin{bmatrix} \sigma_1 & 0 & 0\\ 0 & \sigma_2 & 0\\ 0 & 0 & \sigma_3 \end{bmatrix}$

where $\ \sigma_1$ , $\ \sigma_2$ , and $\ \sigma_3$ , are the principal stresses. These principal stresses may be combined to form three other commonly used invariants, $\ I_1$ , $\ I_2$ , and $\ I_3$ , which are the first, second and third stress invariants, respectively. The first and third invariant are the trace and determinant respectively, of the stress tensor. Thus, we have

$\ \begin{align} I_1 &= \sigma_{1}+\sigma_{2}+\sigma_{3} \\ I_2 &= \sigma_{1}\sigma_{2}+\sigma_{2}\sigma_{3}+\sigma_{3}\sigma_{1} \\ I_3 &= \sigma_{1}\sigma_{2}\sigma_{3} \\ \end{align}$

Because of its simplicity, working and thinking in the principal coordinate system is often very useful when considering the state of the elastic medium at a particular point.

Stress deviator tensor

The stress tensor $\ \sigma_{ij}$ 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, $\ p\delta_{ij}$ , which tends to change the volume of the stressed body; and
a deviatoric component called the stress deviator tensor, $\ s_{ij}$ , which tends to distort it.

$\ \sigma_{ij}= s_{ij} + p\delta_{ij}$

where $\ p$ is the mean stress given by

$\ p=\frac{\sigma_{kk}}{3}=\frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3}=\tfrac{1}{3}I_1$

The deviatoric stress tensor can be obtained by subtracting the hydrostatic stress tensor from the stress tensor:

$\begin{align} \ s_{ij} &= \sigma_{ij} - \frac{\sigma_{kk}}{3}\delta_{ij} \\ \left[{\begin{matrix} s_{11} & s_{12} & s_{13} \\ s_{21} & s_{22} & s_{23} \\ s_{31} & s_{32} & s_{33} \\ \end{matrix}}\right] &=\left[{\begin{matrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \\ \end{matrix}}\right]-\left[{\begin{matrix} p & 0 & 0 \\ 0 & p & 0 \\ 0 & 0 & p \\ \end{matrix}}\right] \\ &=\left[{\begin{matrix} \sigma_{11}-p & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22}-p & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33}-p \\ \end{matrix}}\right] \\ \end{align}$

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. In Mathematics, in the fields of Multilinear algebra and Representation theory, invariants of tensors are coefficients of the Characteristic polynomial It can be shown that the principal directions of the stress deviator tensor $\ s_{ij}$ are the same as the principal directions of the stress tensor $\ \sigma_{ij}$ . Thus, the characteristic equation is

$\ \left| s_{ij}- \lambda\delta_{ij} \right| = \lambda^3-J_1\lambda^2-J_2\lambda-J_3=0$

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. This deviatoric stress invariants can be expressed as a function of the components of $\ s_{ij}$ or its principal values $\ s_1$ , $\ s_2$ , and $\ s_3$ , or alternatively, as a function of $\ \sigma_{ij}$ or its principal values $\sigma_1 \,$ , $\sigma_2 \,$ , and $\sigma_3 \,$ . Thus,

$\begin{align} J_1 &= s_{kk}=0 \end{align}$

$\begin{align} J_2 &= \textstyle{\frac{1}{2}}s_{ij}s_{ji} \\ &= -s_1s_2 - s_2s_3 - s_3s_1 \\ &= \tfrac{1}{6}\left[(\sigma_{11} - \sigma_{22})^2 + (\sigma_{22} - \sigma_{33})^2 + (\sigma_{33} - \sigma_{11})^2 \right ] + \sigma_{12}^2 + \sigma_{23}^2 + \sigma_{31}^2 \\ &= \tfrac{1}{6}\left[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 \right ] \\ &= \tfrac{1}{3}I_1^2-I_2\\ J_3 &= \det(s_{ij}) \\ &= \tfrac{1}{3}s_{ij}s_{jk}s_{ki} \\ &= s_1s_2s_3 \\ &= \tfrac{2}{27}I_1^3 - \tfrac{1}{3}I_1 I_2 + I_3 \end{align}$

Because $s_{kk}=0 \,$ , 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 von Mises yield criterion suggests that the yielding of materials begins when the second deviatoric stress invariant \ J_2 reaches a critical The equivalent stress is defined as

$\sigma_e = \sqrt{3~J_2} = \sqrt{\cfrac{1}{2}~\left[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2\right]}$

Octahedral stresses

Figure 6. Octahedral stress planes

Considering the principal directions as the coordinate axes, a plane which normal vector makes equal angles with each of the principal axes, i. e. having direction cosines equal to $|1/\sqrt{3}|$ , 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 and octahedral shear stress, respectively. They are expressed as

$\ \sigma_{oct}=\tfrac{1}{3}(\sigma_1+\sigma_2+\sigma_3)$

$\ \begin{align} \tau_{oct}&=\tfrac{1}{3}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]^{1/2} \\ &=\sqrt{\tfrac{2}{3}J_2} \end{align}$

Analysis of stress

All real objects occupy a three-dimensional space. However, depending on the loading condition and viewpoint of the observer the same physical object can alternatively be assumed as one-dimensional or two-dimensional, thus simplifying the mathematical modelling of the object. Structural elements are used in Structural analysis to simplify the structure which is to be analysed

Uniaxial stress

If two of the dimensions of the object are very large or very small compared to the others, the object may be modelled as one-dimensional. In this case the stress tensor has only one component and is indistinguishable from a scalar. 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.

When a structural element is elongated or compressed, its cross-sectional area changes by an amount that depends on the Poisson's ratio of the material. Poisson's ratio ( ν) named after Simeon Poisson, is the ratio of the relative contraction strain, or transverse strain (normal to In engineering applications, structural members experience small deformations and the reduction in cross-sectional area is very small and can be neglected, i. e. , the cross-sectional area is assumed constant during deformation. For this case, the stress is called engineering stress or nominal stress. In some other cases, e. g. , elastomers and plastic materials, the change in cross-sectional area is significant, and the stress must be calculated assuming the current cross-sectional area instead of the initial cross-sectional area. An elastomer is a Polymer with the property of Elasticity. The term which is derived from elastic polymer, is often used interchangeably with the term This is termed true stress and is expressed as

$\sigma_\mathrm{true} = (1 + \varepsilon_e)(\sigma_e) \,$ ,

where

$\ \varepsilon_e$ is the nominal (engineering) strain, and

$\ \sigma_e$ is nominal (engineering) stress.

The relationship between true strain and engineering strain is given by

$\varepsilon_\mathrm{true} = \ln(1 + \varepsilon_e) \,$ .

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

Plane stress

A state of plane stress exist when one of the principal stresses is zero, stresses with respect to the thin surface are zero. This usually occurs in structural elements where one dimension is very small compared to the other two, i. e. the element is flat or thin, and the stresses are negligible with respect to the smaller dimension as they are not able to develop within the material and are small compared to the in-plane stresses. Therefore, the face of the element is not acted by loads and the structural element can be analyzed as two-dimensional, e. g. thin-walled structures such as plates subject to in-plane loading or thin cylinders subject to pressure loading. The stress tensor can then be approximated by:

$\ \sigma_{ij} = \begin{bmatrix} \sigma_{11} & \sigma_{12} & 0 \\ \sigma_{21} & \sigma_{22} & 0 \\ 0 & 0 & 0\end{bmatrix}$ .

The corresponding strain tensor is:

$\ \varepsilon_{ij} = \begin{bmatrix} \varepsilon_{11} & \varepsilon_{12} & 0 \\ \varepsilon_{21} & \varepsilon_{22} & 0 \\ 0 & 0 & \varepsilon_{33}\end{bmatrix}$

in which the non-zero $\ \varepsilon_{33}$ term arises from the Poisson effect. This strain term can be temporarily removed from the stress analysis to leave only the in-plane terms, effectively reducing the analysis to two dimensions.

Plane strain

If one dimension is very large compared to the others, the principal strain in the direction of the longest dimension is constrained and can be assumed as zero, yielding a plane strain condition. In this case, though all principal stresses are non-zero, the principal stress in the direction of the longest dimension can be disregarded for calculations. Thus, allowing a two dimensional analysis of stresses, e. g. a dam analyzed at a cross section loaded by the reservoir. A dam is a barrier that divides waters. Dams generally serve the primary purpose of retaining water while other structures such as Floodgates, Levees

Mohr's circle for stresses

Mohr's circle is a graphical representation of any 2-D stress state and was named for Christian Otto Mohr. Mohr's circle is a graphical representation of any 2-D stress state proposed in 1892 by Christian Otto Mohr. Christian Otto Mohr ( October 8, 1835 &ndash October 2, 1918) was a German Civil engineer, one of the most celebrated Mohr's circle may also be applied to three-dimensional stress. In this case, the diagram has three circles, two within a third.

Mohr's circle is used to find the principal stresses, maximum shear stresses, and principal 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. Ductility is a mechanical property used to describe the extent to which materials can be deformed plastically or "stretched" into "wires" without

Piola-Kirchhoff stress tensor

In the case of finite deformations, the Piola-Kirchhoff stress tensors are used to express the stress relative to the reference configuration. This is in contrast to the Cauchy stress tensor which expresses the stress relative to the present configuration. Stress is a measure of the average amount of Force exerted per unit Area. For infinitesimal deformations or rotations, the Cauchy and Piola-Kirchoff tensors are identical. These tensors take their names from Gabrio Piola and Gustav Kirchhoff. Gabrio Piola (1794–1850 was an Italian physicist The Piola-Kirchhoff stress tensor bears his name Gustav Robert Kirchhoff ( March 12, 1824 &ndash October 17, 1887) was a German Physicist who contributed to the fundamental

1^st Piola-Kirchhoff stress tensor

Whereas the Cauchy stress tensor, $\ \sigma_{ij}$ , relates forces in the present configuration to areas in the present configuration, the 1^st Piola-Kirchhoff stress tensor, $\ K_{Lj}$ relates forces in the present configuration with areas in the reference ("material") configuration. Stress is a measure of the average amount of Force exerted per unit Area. $\ K_{Lj}$ is given by

$K_{Lj}=J X_{L,i} \sigma_{ij} \!$

where $\ J$ is the Jacobian, and $\ X_{L,i}$ is the inverse of the deformation gradient. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant.

Because it relates different coordinate systems, the 1^st Piola-Kirchhoff stress is a two-point tensor. Two-point tensor s or double vector s are Tensor -like quantities which transform as vectors with respect to each of their indices and are used in Continuum mechanics In general, it is not symmetric. The 1^st Piola-Kirchhoff stress is the 3D generalization of the 1D concept of engineering stress. Stress is a measure of the average amount of Force exerted per unit Area.

If the material rotates without a change in stress state (rigid rotation), the components of the 1^st Piola-Kirchhoff stress tensor will vary with material orientation.

The 1^st Piola-Kirchhoff stress is energy conjugate to the deformation gradient.

2^nd Piola-Kirchhoff stress tensor

Whereas the 1^st Piola-Kirchhoff stress relates forces in the current configuration to areas in the reference configuration, the 2^nd Piola-Kirchhoff stress tensor $\ S_{IJ}$ relates forces in the reference configuration to areas in the reference configuration. The force in the reference configuration is obtained via a mapping that preserves the relative relationship between the force direction and the area normal in the current configuration.

$S_{IJ}=J X_{I,k} X_{J,l} \sigma_{kl} \!$

This tensor is symmetric.

If the material rotates without a change in stress state (rigid rotation), the components of the 2^nd Piola-Kirchhoff stress tensor will remain constant, irrespective of material orientation.

The 2^nd Piola-Kirchhoff stress tensor is energy conjugate to the Green-Lagrange strain.

Voigt notation

The principle of conservation of angular momentum (in the absence of body couples) requires that the Cauchy stress tensor be symmetric, i. e. , $\ \sigma_{ij} = \sigma_{ji}$ . The Voigt notation representation of the Cauchy stress tensor takes advantage of this symmetry to express the stress as a 6-dimensional vector of the form

$\boldsymbol{\sigma} = \begin{bmatrix}\sigma_1 & \sigma_2 & \sigma_3 & \sigma_4 & \sigma_5 & \sigma_6 \end{bmatrix}^T \equiv \begin{bmatrix}\sigma_{11} & \sigma_{22} & \sigma_{33} & \sigma_{23} & \sigma_{31} & \sigma_{12} \end{bmatrix}^T$

The Voigt notation is used extensively in representing stress-strain relations in solid mechanics and for computational efficiency in numerical structural mechanics software.

Books

Dieter, G. This article is about structural behavior For other meanings see Bending (disambiguation. Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions Residual stresses are stresses that remain after the original cause of the stresses (external forces heat gradient has been removed Shot peening is a Process used to produce a Compressive residual stress layer and modify mechanical properties of Metals It entails impacting The stress-energy tensor (sometimes stress-energy-momentum tensor is a Tensor quantity in Physics that describes the Density and Flux During testing of a material sample the stress–strain curve is a graphical representation of the relationship between stress, derived from measuring the load applied on the A stress concentration (often called stress raisers or stress risers) is a location in an object where stress is concentrated The von Mises yield criterion suggests that the yielding of materials begins when the second deviatoric stress invariant \ J_2 reaches a critical The yield strength or yield point of a Material is defined in Engineering and Materials science as the stress at which a material A yield surface is a five-dimensional surface in the six-dimensional space of stresses. 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.

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Contents

Stress as a tensor

Cauchy's stress principle