Stress (mechanics)

Continuum mechanics

Laws
Conservation of mass Conservation of momentum Conservation of energy Entropy inequality

Solid mechanics
Solids Stress · Deformation Compatibility Finite strain · Infinitesimal strain Elasticity (linear) · Plasticity Bending · Hooke's law Failure theory Fracture mechanics

Fluid mechanics
Fluids Fluid statics · Fluid dynamics Surface tension Navier–Stokes equations Viscosity: Newtonian, Non-Newtonian

Rheology
Viscoelasticity Smart fluids: Magnetorheological Electrorheological Ferrofluids Rheometry · Rheometer

Scientists
Bernoulli · Cauchy · Hooke Navier · Newton · Stokes

Figure 1.1 Stress in a loaded deformable material body assumed as a continuum.

Figure 1.2 Axial stress in a prismatic bar axially loaded

Figure 1.3 Normal stress in a prismatic (straight member of uniform cross-sectional area) bar. The stress or force distribution in the cross section of the bar is not necessarily uniform. However, an average normal stress $\sigma_\mathrm{avg}\,\!$ can be used

Figure 1.4 Shear stress in a prismatic bar. The stress or force distribution in the cross section of the bar is not necessarily uniform. However, an average shear stress $\tau_\mathrm{avg}\,\!$ is a reasonable approximation.^[1]

In continuum mechanics, stress is a measure of the intensity of the internal forces acting within a deformable body. Mathematically, it is a measure of the average force per unit area of a surface within a the body on which internal forces act. These internal forces are produced between the particles in the body as a reaction to external forces applied on the body. Because the loaded deformable body is assumed to behave as a continuum, these internal forces are distributed continuously within the volume of the material body, and result in deformation of the body's shape. Beyond certain limits of material strength, this can lead to a permanent change of shape or physical failure.

The SI unit for stress is the pascal (symbol Pa), which is equivalent to one newton (force) per square meter (unit area). The unit for stress is the same as that of pressure, which is also a measure of force per unit area.

1 Introduction
2 Forces in a continuum
3 Euler's laws of motion for a continuum
4 Euler-Cauchy stress principle
5 Equilibrium equations and symmetry of the stress tensor
6 Principal stresses and stress invariants
7 Maximum and minimum shear stresses
8 Stress deviator tensor
- 8.1 Invariants of the stress deviator tensor
9 Octahedral stresses
10 Analysis of stress
11 Stress transformation in plane stress and plane strain
12 Graphical representation of stress at a point
13 Graphical representation of the stress field
14 Alternative measures of stress
- 14.1 Piola-Kirchhoff stress tensor
  - 14.1.1 2^nd Piola-Kirchhoff stress tensor
15 See also
16 References
- 16.1 Bibliography

Introduction

In continuum mechanics, stress is a measure of the average force per unit area of a surface within a deformable body on which internal forces act. It is a measure of the intensity of the internal forces acting between particles of a deformable body across imaginary internal surfaces^[2]. These internal forces are produced between the particles in the body as a reaction to external forces applied on the body. External forces are either surface forces or body forces. Because the loaded deformable body is assumed to behave as a continuum, these internal forces are distributed continuously within the volume of the material body, i.e. the stress distribution in the body is expressed as a piecewise continuous function of space coordinates and time.

Normal and shear stresses

For the simple case of a body axially loaded, e.g. a prismatic bar subjected to tension or compression by a force passing through its centroid (Figures 1.2 and 1.3) the stress $\sigma\,\!$ , or intensity of internal forces, can be obtained by dividing the total normal force $F_\mathrm n\,\!$ , determined from the equilibrium of forces, by the cross-sectional area $A\,\!$ of the prism it is acting upon. The normal force can be a tensile force if acting outward from the plane, or compressive force if acting inward to the plane. In the case of a prismatic bar axially loaded, the stress $\sigma\,\!$ is represented by a scalar called engineering stress or nominal stress that represents an average stress ( $\sigma_\mathrm{avg}\,\!$ ) over the area, meaning that the stress in the cross section is uniformly distributed. Thus, we have

$\sigma_\mathrm{avg} = \frac{F_\mathrm n}{A}\approx\sigma\,\!$

A different type of stress is obtained when transverse forces $F_\mathrm\,\!$ are applied to the prismatic bar as shown in Figure 1.4. Considering the same cross-section as before, from static equilibrium the internal force has a magnitude equal to $F_\mathrm s\,\!$ and in opposite direction parallel to the cross-section. $F_\mathrm s\,\!$ is called the shear force. Dividing the shear force $F_\mathrm s\,\!$ by the area $A\,\!$ of the cross section we obtain the shear stress. In this case the shear stress $\tau\,\!$ is a scalar quantity representing an average shear stress ( $\tau_\mathrm{avg}\,\!$ ) in the section, i.e. the stress in the cross-section is uniformly distributed.

$\tau_\mathrm{avg}= \frac{F_\mathrm s}{A}\approx\tau\,\!$

In Figure 1.3, the normal stress is observed in two planes $m-m\,\!$ and $n-n\,\!$ of the axially loaded prismatic bar. The stress on plane $n-n\,\!$ , which is closer to the point of application of the load $F\,\!$ , varies more across the cross-section than that of plane $m-m\,\!$ . However, if the cross-sectional area of the bar is very small, i.e. the bar is slender, the variation of stress across the area is small and the normal stress can be approximated by $\sigma_\mathrm {avg}\,\!$ . On the other hand, the variation of shear stress across the section of a prismatic bar cannot be assumed to be uniform.

Stress modelling (Cauchy)

In general, stress is not uniformly distributed over the cross-section of a material body, and consequently the stress at a point in a given region is different from the average stress over the entire area. Therefore, it is necessary to define the stress not over a given area but at a specific point in the body (Figure 1.1). According to Cauchy, the stress at any point in an object assumed to behave as a continuum is completely defined by the nine components $\sigma_{ij}\,\!$ of a second-order tensor of type (0,2) known as the Cauchy stress tensor, $\boldsymbol\sigma\,\!$ :

$\boldsymbol{\sigma}= \left[{\begin{matrix} \sigma _{11} & \sigma _{12} & \sigma _{13} \\ \sigma _{21} & \sigma _{22} & \sigma _{23} \\ \sigma _{31} & \sigma _{32} & \sigma _{33} \\ \end{matrix}}\right] \equiv \left[{\begin{matrix} \sigma _{xx} & \sigma _{xy} & \sigma _{xz} \\ \sigma _{yx} & \sigma _{yy} & \sigma _{yz} \\ \sigma _{zx} & \sigma _{zy} & \sigma _{zz} \\ \end{matrix}}\right] \equiv \left[{\begin{matrix} \sigma _x & \tau _{xy} & \tau _{xz} \\ \tau _{yx} & \sigma _y & \tau _{yz} \\ \tau _{zx} & \tau _{zy} & \sigma _z \\ \end{matrix}}\right] \,\!$

The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr’s circle for stress.

The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations. For large deformations, also called finite deformations, other measures of stress, such as the first and second Piola-Kirchhoff stress tensors, the Biot stress tensor, and the Kirchhoff stress tensor, are required.

According to the principle of conservation of linear momentum, if a continuous body is in static equilibrium it can be demonstrated that the components of the Cauchy stress tensor in every material point in the body satisfy the equilibrium equations (Cauchy’s equations of motion for zero acceleration). At the same time, according to the principle of conservation of angular momentum, equilibrium requires that the summation of moments with respect to an arbitrary point is zero, which leads to the conclusion that the stress tensor is symmetric, thus having only six independent stress components instead of the original nine.

There are certain invariants associated with the stress tensor, whose values do not depend upon the coordinate system chosen or the area element upon which the stress tensor operates. These are the three eigenvalues of the stress tensor, which are called the principal stresses. 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). 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.

Stress analysis

The determination of the internal distribution of stresses, viz stress analysis, is required in engineering, for the study and design of structures, e.g. tunnels, dams, mechanical parts, and structural frames, among others, under prescribed or expected loads. To determine the distribution of stress in the structure it is necessary to solve a boundary-value problem by specifying the boundary conditions, i.e. displacements and/or forces on the boundary. Constitutive equations, such as Hooke’s Law for linear elastic materials, are used to describe the stress:strain relationship in these calculations. A boundary-value problem based on the theory of elasticity is applied to structures expected to deform elastically, with infinitesimal strains, under design loads. When the loads applied to the structure induce plastic deformations, the theory of plasticity is implemented.

The stress analysis can be simplified in cases where the physical dimensions and the distribution of loads allow the structure to be treated as one-dimensional or two-dimensional. For a two-dimensional analysis a plane stress or a plane strain condition can be assumed.

Alternatively, experimental determination of stresses can be carried out using the photoelastic method.

Approximate solutions for boundary-value problems can be obtained through the use of numerical methods such as the Finite Element Method, the Finite Difference Method, and the Boundary Element Method, which are implemented in computer programs. Analytical or closed-form solutions can be obtained for simple geometries, constitutive relations, and boundary conditions.

In designing structures, calculated stresses are restricted to be less than an specified allowable stress, also known as working or designed stress. Allowable stress is chosen as some fraction of the yield strength or of the ultimate strength of the material of which the structure is made. The ratio of the ultimate stress to the allowable stress is defined as the factor of safety. Laboratory tests are usually performed on material samples in order to determine the yield strength and the ultimate strength that the material can withstand before failure.

Engineering quantities are usually measured in megapascals (MPa) or gigapascals (GPa). In imperial units, stress is expressed in pounds-force per square inch (psi) or kilopounds-force per square inch (ksi).

Forces in a continuum

Continuum mechanics deals with deformable bodies, as opposed to rigid bodies. A solid is a deformable body that possesses shear strength, sc. a solid can support shear forces (forces parallel to the material surface on which they act). Fluids, on the other hand, do not sustain shear forces. For the study of the mechanical behavior of solids and fluids these are assumed to be continuous bodies, which means that the matter fills the entire region of space it occupies, despite the fact that matter is made of atoms, has voids, and is discrete. Therefore, when continuum mechanics refers to a point or particle in a continuous body it does not describe a point in the interatomic space or an atomic particle, rather an idealized part of the body occupying that point.

Following the classical dynamics of Newton and Euler, the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces $\mathbf F_C$ and body forces $\mathbf F_B$ .^[3] Thus, the total force $\mathcal F$ applied to a body or to a portion of the body can be expressed as:

$\mathcal F = \mathbf F_B + \mathbf F_C$

Surface forces or contact forces, expressed as force per unit area, can act either on the bounding surface of the body, as a result of mechanical contact with other bodies, or on imaginary internal surfaces that bound portions of the body, as a result of the mechanical interaction between the parts of the body to either side of the surface (Euler-Cauchy's stress principle). When a body is acted upon by external contact forces, internal contact forces are then transmitted from point to point inside the body to balance their action, according to Newton's second law of motion of conservation of linear momentum and angular momentum (for continuous bodies these laws are called the Euler's equations of motion). The internal contact forces are related to the body's deformation through constitutive equations. This article is concerned with the manner in which internal contact forces are mathematically described and how they relate to the motion of the body, independent of the body's material makeup.^[4]

The distribution of internal contact forces throughout the volume of the body is assumed to be continuous. Therefore, there exists a contact force density or Cauchy traction field ^[3] $\mathbf T(\mathbf n, \mathbf x, t)$ that represents this distribution in a particular configuration of the body at a given time $t\,\!$ . It is not a vector field because it depends not only on the position $\mathbf x$ of a particular material point, but also on the local orientation of the surface element as defined by its normal vector $\mathbf n$ ^[5].

Any differential area $dS\,\!$ with normal vector $\mathbf n$ of a given internal surface area $S\,\!$ , bounding a portion of the body, experiences a contact force $d\mathbf F_C\,\!$ arising from the contact between both portions of the body on each side of $S\,\!$ , and it is given by

$d\mathbf F_C= \mathbf T^{(\mathbf n)}\,dS$

where $\mathbf T^{(\mathbf n)}$ is the surface traction,^[6] also called stress vector,^[7] traction,^[8] or traction vector,^[9]. The stress vector is a frame-indifferent vector. The stress vector will be explained in more detail later (Euler-Cauchy's stress principle).

The total contact force on the particular internal surface $S\,\!$ is then expressed as the sum (surface integral) of the contact forces on all differential surfaces $dS\,\!$ :

$\mathbf F_C=\int_S \mathbf T^{(\mathbf n)}\,dS$

The concept of stress can then be thought as a measure of the intensity of the internal contact forces acting between particles of the body across imaginary internal surfaces.^[2] In other words, stress is a measure of the average quantity of force exerted per unit area of the surface on which these internal forces act. The intensity of contact forces is related, specifically in an inverse proportion, to the area of contact. For example, if we compare a force applied to a small area and a distributed load of the same resultant magnitude applied to a larger area, we find that the effects or intensities of these two forces are locally different because the stresses are not the same.

In continuum mechanics a body is considered stress-free if the only forces present are those inter-atomic forces (ionic, metallic, and van der Waals forces) required to hold the body together and to keep its shape in the absence of all external influences, including gravitational attraction.^[9]^[10] Stresses generated during manufacture of the body to a specific configuration are also excluded when considering stresses in a body. Therefore, the stresses considered in continuum mechanics are only those produced by deformation of the body, sc. only relative changes in stress are considered, not the absolute values of stress.

Body forces are forces originating from sources outside of the body^[11] that act on the volume (or mass) of the body. Saying that body forces are due to outside sources implies that the interaction between different parts of the body (internal forces) are manifested through the contact forces alone.^[6] These forces arise from the presence of the body in force fields, e.g. gravitational field (gravitational forces) or electromagnetic field (electromagnetic forces), or from inertial forces when bodies are in motion. As the mass of a continuous body is assumed to be continuously distributed, any force originating from the mass is also continuously distributed. Thus, body forces are specified by vector fields which are assumed to be continuous over the entire volume of the body,^[12] i.e. acting on every point in it. Body forces are represented by a body force density $\mathbf b(\mathbf x, t)$ (per unit of mass), which is a frame-indifferent vector field.

In the case of gravitational forces, the intensity of the force depends on, or is proportional to, the mass density $\mathbf \rho (\mathbf x, t)\,\!$ of the material, and it is specified in terms of force per unit mass ( $b_i\,\!$ ) or per unit volume ( $p_i\,\!$ ). These two specifications are related through the material density by the equation $\rho b_i = p_i\,\!$ . Similarly, the intensity of electromagnetic forces depends upon the strength (electric charge) of the electromagnetic field.

The total body force applied to a continuous body is expressed as

$\mathbf F_B=\int_V\mathbf b\,dm=\int_V \rho\mathbf b\,dV$

Body forces and contact forces acting on the body lead to corresponding moments of force (torques) relative to a given point. Thus, the total applied torque $\mathcal M$ about the origin is given by

$\mathcal M= \mathbf M_B + \mathbf M_C$

In certain situations, not commonly considered in the analysis of the mechanical behavior or materials, it becomes necessary to include two other types of forces: these are body moments and couple stresses^[13]^[14] (surface couples,^[11] contact torques^[12]). Body moments, or body couples, are moments per unit volume or per unit mass applied to the volume of the body. Couple stresses are moments per unit area applied on a surface. Both are important in the analysis of stress for a polarized dielectric solid under the action of an electric field, materials where the molecular structure is taken into consideration (e.g. bones), solids under the action of an external magnetic field, and the dislocation theory of metals.^[7]^[8]^[11]

Materials that exhibit body couples and couple stresses in addition to moments produced exclusively by forces are called polar materials.^[8]^[12] Non-polar materials are then those materials with only moments of forces. In the classical branches of continuum mechanics the development of the theory of stresses is based on non-polar materials.

Thus, the sum of all applied forces and torques (with respect to the origin of the coordinate system) in the body can be given by

$\mathcal F = \int_V \mathbf a\,dm = \int_S \mathbf T\,dS + \int_V \rho\mathbf b\,dV$

$\mathcal M = \int_S \mathbf r \times \mathbf T\,dS + \int_V \mathbf r \times \rho\mathbf b\,dV$

Euler's laws of motion for a continuum

The density of internal forces at every point in a deformable body are not necessarily equal, i.e. there is a distribution of stresses throughout the body. This variation of internal forces throughout the body is governed by Newton's second law of motion of conservation of linear momentum and angular momentum, which normally are applied to a mass particle but are extended in continuum mechanics to a body of continuously distributed mass. For continuous bodies these laws are called Euler’s equations of motion. If a body is represented as an assemblage of discrete particles, each governed by Newton’s laws of motion, then Euler’s equations can be derived from Newton’s laws. Euler’s equations can, however, be taken as axioms describing the laws of motion for extended bodies, independently of any particle structure.^[5]

Let the coordinate system $x_1, x_2, x_3\,\!$ be an inertial frame of reference. Let $\mathbf r$ be the position vector of a particle or point $\mathbf P$ in the continuous body with respect to the origin of the coordinate system, and $\mathbf v$ the velocity vector of point $\mathbf P$ .

Euler’s first axiom or law (law of balance of linear momentum or balance of forces) states that in an inertial frame the time rate of change of linear momentum $\mathbf p$ of an arbitrary portion of a continuous body is equal to the total applied force $\mathcal F$ acting on the considered portion, and it is expressed as

$\begin{align} \frac{d\mathbf p}{dt} &= \mathcal F \\ \frac{d}{dt}\int_V \rho\mathbf v\,dV&=\int_S \mathbf T\,dS + \int_V \rho\mathbf b\,dV \\ \end{align}$

Euler’s second axiom or law (law of balance of angular momentum or balance of torques) states that in an inertial frame the time rate of change of angular momentum $\mathbf L$ of an arbitrary portion of a continuous body is equal to the total applied torque $\mathcal M$ acting on the considered portion, and it is expressed as

$\begin{align} \frac{d\mathbf L}{dt} &= \mathcal M \\ \frac{d}{dt}\int_V \mathbf r\times\rho\mathbf v\,dV&=\int_S \mathbf r \times \mathbf T\,dS + \int_V \mathbf r \times \rho\mathbf b\,dV \\\end{align}$

The derivatives of $\mathbf p$ and $\mathbf L$ are material derivatives.

Euler-Cauchy stress principle

Figure 2.1a Internal distribution of contact forces and couple stresses on a differential $dS\,\!$ of the internal surface $S\,\!$ in a continuum, as a result of the interaction between the two portions of the continuum separated by the surface

Figure 2.1b Internal distribution of contact forces and couple stresses on a differential $dS\,\!$ of the internal surface $S\,\!$ in a continuum, as a result of the interaction between the two portions of the continuum separated by the surface

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 $\mathbf{n}\,\!$ , and can be resolved into two components: one component normal to the plane, called normal stress $\sigma_\mathrm{n} \,\!$ , and another component parallel to this plane, called the shearing stress $\tau \,\!$ .

The Euler-Cauchy stress principle states that upon any surface (real or imaginary) that divides the body, the action of one part of the body on the other is equivalent (equipollent) to the system of distributed forces and couples on the surface dividing the body,^[15] and it is represented by a vector field $\mathbf{T}^{(\mathbf{n})}$ , called the stress vector, defined on the surface $S\,\!$ and assumed to depend continuously on the surfaces unit vectors $\mathbf n$ .^[8]^[12]

To explain this principle, we 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 (some authors use the cutting plane diagram and others use the diagram with the arbitrary volume inside the continuum enclosed by the surface $S\,\!$ ). The body is subjected to external surface forces $\mathbf{F}\,\!$ and body forces $\mathbf b\,\!$ . The internal contact forces being 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 $\Delta S\,\!$ , with a normal unit vector $\mathbf{n}\,\!$ , on the dividing plane $S\,\!$ . The force distribution is equipollent to a contact force $\Delta \mathbf F\,\!$ and a couple stress $\Delta \mathbf M\,\!$ , as shown in Figure 2.1a and 2.1b. Cauchy’s stress principle asserts^[9] that as $\Delta S\,\!$ becomes very small and tends to zero the ratio $\Delta \mathbf F / \Delta S\,\!$ becomes $d \mathbf F/dS\,\!$ and the couple stress vector $\Delta \mathbf M\,\!$ vanishes. In specific fields of continuum mechanics the couple stress is assumed not to vanish; however, as stated previously, in classical branches of continuum mechanics we deal with non-polar materials which do not consider couple stresses and body moments. The resultant vector $d \mathbf F/dS\,\!$ is defined as the stress vector or traction vector given by $\mathbf{T}^{(\mathbf{n})}=T_i^{(\mathbf{n})}\mathbf{e}_i\,\!$ at point $P\,\!$ associated with a plane with a normal vector $\mathbf{n}\,\!$ :

$T^{(\mathbf{n})}_i= \lim_{\Delta S \to 0} \frac {\Delta F_i}{\Delta S} = {dF_i \over dS}\,\!$

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 $\mathbf{n}\,\!$ , and can be resolved into two components:

one normal to the plane, called normal stress $\sigma_\mathrm{n} \,\!$

$\mathbf{\sigma_\mathrm{n}}= \lim_{\Delta S \to 0} \frac {\Delta F_\mathrm n}{\Delta S} = \frac{dF_\mathrm n}{dS}\,\!$

where $dF_\mathrm n\,\!$ is the normal component of the force $d \mathbf F\,\!$ to the differential area $dS\,\!$

and the other parallel to this plane, called the shearing stress $\tau \,\!$ .

$\mathbf \tau= \lim_{\Delta S \to 0} \frac {\Delta F_\mathrm s}{\Delta S} = \frac{dF_\mathrm s}{dS}\,\!$

where $dF_\mathrm s\,\!$ is the tangential component of the force $d \mathbf F\,\!$ to the differential surface area $dS\,\!$ . The shear stress can be further decomposed into two mutually perpendicular vectors.

Cauchy’s postulate

According to the Cauchy Postulate, the stress vector $\mathbf{T}^{(\mathbf{n})}$ remains unchanged for all surfaces passing through a point $\mathbf P$ and having the same normal vector $\mathbf n$ at $\mathbf P$ ,^[6]^[16] i.e. having a common tangent at $\mathbf P$ . This means that the stress vector is only a function of the normal vector $\mathbf n$ , and it 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^[6]^[10]^[11], also called the Cauchy reciprocal theorem^[17], which states that the stress vectors acting on opposite sides of the same surface are equal in magnitude and opposite in direction. Cauchy’s fundamental lemma is equivalent to Newton's third law of motion of action and reaction, and it is expressed as

$- \mathbf{T}^{(\mathbf{n})}= \mathbf{T}^{(- \mathbf{n})}\,\!$

Derivation of Cauchy’s Lemma
Derivation coming soon

Cauchy’s stress theorem – stress tensor

The state of stress at a point in the body is then defined by all the stress vectors $\mathbf{T}^{(\mathbf{n})}\,\!$ associated with all planes (infinite in number) that pass through that point ^[2]. However, according to Cauchy’s fundamental theorem ^[10], also called Cauchy’s stress theorem ^[11], 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 $\boldsymbol{\sigma}(\mathbf x,t)\,\!$ , called the Cauchy stress tensor, independent of $\mathbf n\,\!$ such that $\mathbf T\,\!$ is a linear function of $\mathbf n \,\!$ :

$\mathbf{T}^{(\mathbf n)}= \boldsymbol{\sigma}\cdot\mathbf n \quad \text{or} \quad T_j^{(n)}= \sigma_{ij}n_i\,\!$

This equation implies that the stress vector $\mathbf{T}^{(\mathbf n)}\,\!$ at any point $P\,\!$ in a continuum associated with a plane with normal vector $\mathbf{n}\,\!$ 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 $\sigma_{ij}\,\!$ of the stress tensor $\boldsymbol{\sigma}\,\!$ .

To prove this expression, consider a tetrahedron with three faces oriented in the coordinate planes, and with an infinitesimal area $dA\,\!$ oriented in an arbitrary direction specified by a normal vector $\mathbf{n}\,\!$ (Figure 2.2). The tetrahedron is formed by slicing the infinitesimal element along an arbitrary plane $\mathbf{n}\,\!$ . The stress vector on this plane is denoted by $\mathbf{T}^{(\mathbf{n})}\,\!$ . The stress vectors acting on the faces of the tetrahedron are denoted as $\mathbf{T}^{(\mathbf{e}_1)}\,\!$ , $\mathbf{T}^{(\mathbf{e}_2)}\,\!$ , and $\mathbf{T}^{(\mathbf{e}_3)}\,\!$ , and are by definition the components of the stress tensor $\sigma_{ij}\,\!$ . This tetrahedron is sometimes called the Cauchy tetrahedron. From equilibrium of forces, i.e. Euler’s first law of motion (Newton’s second law of motion), we have

$\mathbf{T}^{(\mathbf{n})}dA - \mathbf{T}^{(\mathbf{e}_1)}dA_1 - \mathbf{T}^{(\mathbf{e}_2)}dA_2 - \mathbf{T}^{(\mathbf{e}_3)}dA_3 = \rho \left( \frac{h}{3}dA \right) \mathbf{a}\,\!$

Figure 2.2. Stress vector acting on a plane with normal 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 of the equation represents the product of the mass enclosed by the tetrahedron and its acceleration: $\rho\,\!$ is the density, $\mathbf{a}\,\!$ is the acceleration, and $h\,\!$ is the height of the tetrahedron, considering the plane $\mathbf{n}\,\!$ as the base. The area of the faces of the tetrahedron perpendicular to the axes can be found by projecting $dA\,\!$ into each face (using the dot product):

$dA_1= \left(\mathbf{n} \cdot \mathbf{e}_1 \right)dA = n_1dA\,\!$

$dA_2= \left(\mathbf{n} \cdot \mathbf{e}_2 \right)dA = n_2dA\,\!$

$dA_3= \left(\mathbf{n} \cdot \mathbf{e}_3 \right)dA = n_3dA\,\!$

and then can be substituted into the equation to cancel out $dA\,\!$ :

$\mathbf{T}^{(\mathbf{n})} - \mathbf{T}^{(\mathbf{e}_1)}n_1 - \mathbf{T}^{(\mathbf{e}_2)}n_2 - \mathbf{T}^{(\mathbf{e}_3)}n_3 = \rho \left( \frac{h}{3} \right) \mathbf{a}\,\!$

To consider the limiting case as the tetrahedron shrinks to a point, $h$ must go to 0 (intuitively, plane $\mathbf{n}$ is translated along $\mathbf{n}$ towards $O$ ). As a result, the right-hand-side of the equation approaches 0, thus

$\mathbf{T}^{(\mathbf{n})} = \mathbf{T}^{(\mathbf{e}_1)}n_1 + \mathbf{T}^{(\mathbf{e}_2)}n_2 + \mathbf{T}^{(\mathbf{e}_3)}n_3\,\!$

Figure 2.3 Components of stress in three dimensions

Assuming a material element (Figure 2.3) with planes perpendicular to the coordinate axes of a Cartesian coordinate system, the stress vectors associated with each of the element planes, i.e. $\mathbf{T}^{(\mathbf{e}_1)}\,\!$ , $\mathbf{T}^{(\mathbf{e}_2)}\,\!$ , and $\mathbf{T}^{(\mathbf{e}_3)}\,\!$ can be decomposed into a normal component and two shear components, i.e. components in the direction of the three coordinate axes. For the particular case of a surface with normal unit vector oriented in the direction of the $x_1\,\!$ -axis, the normal stress is denoted by $\sigma_{11}\,\!$ , and the two shear stresses are denoted as $\sigma_{12}\,\!$ and $\sigma_{13}\,\!$ :

$\mathbf{T}^{(\mathbf{e}_1)}= T_1^{(\mathbf{e}_1)}\mathbf{e}_1 + T_2^{(\mathbf{e}_1)} \mathbf{e}_2 + T_3^{(\mathbf{e}_1)} \mathbf{e}_3 = \sigma_{11} \mathbf{e}_1 + \sigma_{12} \mathbf{e}_2 + \sigma_{13} \mathbf{e}_3\,\!$

$\mathbf{T}^{(\mathbf{e}_2)}= T_1^{(\mathbf{e}_2)}\mathbf{e}_1 + T_2^{(\mathbf{e}_2)} \mathbf{e}_2 + T_3^{(\mathbf{e}_2)} \mathbf{e}_3=\sigma_{21} \mathbf{e}_1 + \sigma_{22} \mathbf{e}_2 + \sigma_{23} \mathbf{e}_3\,\!$

$\mathbf{T}^{(\mathbf{e}_3)}= T_1^{(\mathbf{e}_3)}\mathbf{e}_1 + T_2^{(\mathbf{e}_3)} \mathbf{e}_2 + T_3^{(\mathbf{e}_3)} \mathbf{e}_3=\sigma_{31} \mathbf{e}_1 + \sigma_{32} \mathbf{e}_2 + \sigma_{33} \mathbf{e}_3\,\!$

In index notation this is

$\mathbf{T}^{(\mathbf{e}_i)}= T_j^{(\mathbf{e}_i)} \mathbf{e}_j = \sigma_{ij} \mathbf{e}_j\,\!$

The nine components $\sigma_{ij}\,\!$ of the stress vectors are the components of a second-order Cartesian tensor called the Cauchy stress tensor, which completely defines the state of stresses at a point and is given by

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

where

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

$\sigma_{12}\,\!$ , $\sigma_{13}\,\!$ , $\sigma_{21}\,\!$ , $\sigma_{23}\,\!$ , $\sigma_{31}\,\!$ , and $\sigma_{32}\,\!$ are shear stresses.

The first index $i\,\!$ indicates that the stress acts on a plane normal to the $x_i\,\!$ axis, and the second index $j\,\!$ denotes the direction in which the stress acts. A stress component is positive if it acts in the positive direction of the coordinate axes, and if the plane where it acts has an outward normal vector pointing in the positive coordinate direction.

The Voigt notation representation of the Cauchy stress tensor takes advantage of the symmetry of the stress tensor to express the stress as a six-dimensional vector of the form:

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

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

Thus, using the components of the stress tensor

$\begin{align} \mathbf{T}^{(\mathbf{n})} &= \mathbf{T}^{(\mathbf{e}_1)}n_1 + \mathbf{T}^{(\mathbf{e}_2)}n_2 + \mathbf{T}^{(\mathbf{e}_3)}n_3 \\ & = \sum_{i=1}^3 \mathbf{T}^{(\mathbf{e}_i)}n_i \\ &= \left( \sigma_{ij}\mathbf{e}_j \right)n_i \\ &= \sigma_{ij}n_i\mathbf{e}_j \end{align}\,\!$

or, equivalently,

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

Alternatively, in matrix form we have

$\left[{\begin{matrix} T^{(n)}_1 & T^{(n)}_2 & T^{(n)}_3\end{matrix}}\right]=\left[{\begin{matrix} n_1 & n_2 & n_3 \end{matrix}}\right]\cdot \left[{\begin{matrix} \sigma _{11} & \sigma _{12} & \sigma _{13} \\ \sigma _{21} & \sigma _{22} & \sigma _{23} \\ \sigma _{31} & \sigma _{32} & \sigma _{33} \\ \end{matrix}}\right] \,\!$

Transformation rule of the stress tensor

It can be shown that the stress tensor is a contravariant second order tensor, which is a statement of how it transforms under a change of the coordinate system. From an $x_i\,\!$ 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 (Figure 2.4):

$\sigma^'_{ij}=a_{im}a_{jn}\sigma_{mn} \quad \text{or} \quad \boldsymbol{\sigma}' = \mathbf A \boldsymbol{\sigma} \mathbf A^T\,\!$

where $\mathbf A\,\!$ is the rotation matrix with components $a_{ij}\,\!$ . In matrix form this is

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

Figure 2.4 Transformation of the stress tensor

Expanding the matrix operation, and simplifying some terms by taking advantage of the symmetry of the stress tensor, gives:

$\sigma_{11}' = a_{11}^2\sigma_{11}+a_{12}^2\sigma_{22}+a_{13}^2\sigma_{33}+2a_{11}a_{12}\sigma_{12}+2a_{11}a_{13}\sigma_{13}+2a_{12}a_{13}\sigma_{23}\,\!$

$\sigma_{22}' = a_{21}^2\sigma_{11}+a_{22}^2\sigma_{22}+a_{23}^2\sigma_{33}+2a_{21}a_{22}\sigma_{12}+2a_{21}a_{23}\sigma_{13}+2a_{22}a_{23}\sigma_{23}\,\!$

$\sigma_{33}' = a_{31}^2\sigma_{11}+a_{32}^2\sigma_{22}+a_{33}^2\sigma_{33}+2a_{31}a_{32}\sigma_{12}+2a_{31}a_{33}\sigma_{13}+2a_{32}a_{33}\sigma_{23}\,\!$

$\begin{align} \sigma_{12}' = &a_{11}a_{21}\sigma_{11}+a_{12}a_{22}\sigma_{22}+a_{13}a_{23}\sigma_{33}\\ &+(a_{11}a_{22}+a_{12}a_{21})\sigma_{12}+(a_{12}a_{23}+a_{13}a_{22})\sigma_{23}+(a_{11}a_{23}+a_{13}a_{21})\sigma_{13} \end{align}\,\!$

$\begin{align} \sigma_{23}' = &a_{21}a_{31}\sigma_{11}+a_{22}a_{32}\sigma_{22}+a_{23}a_{33}\sigma_{33}\\ &+(a_{21}a_{32}+a_{22}a_{31})\sigma_{12}+(a_{22}a_{33}+a_{23}a_{32})\sigma_{23}+(a_{21}a_{33}+a_{23}a_{31})\sigma_{13}\end{align}\,\!$

$\begin{align} \sigma_{13}' = &a_{11}a_{31}\sigma_{11}+a_{12}a_{32}\sigma_{22}+a_{13}a_{33}\sigma_{33}\\ &+(a_{11}a_{32}+a_{12}a_{31})\sigma_{12}+(a_{12}a_{33}+a_{13}a_{32})\sigma_{23}+(a_{11}a_{33}+a_{13}a_{31})\sigma_{13}\end{align}\,\!$

A graphical representation of this transformation of stresses, for a two-dimensional (plane stress and plane strain) and a general three-dimensional state of stresses, is the Mohr's circle for stresses

Normal and shear stresses

The magnitude of the normal stress component, $\sigma_\mathrm{n}\,\!$ , 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

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

The magnitude of the shear stress component, $\tau_\mathrm{n}\,\!$ , acting in the plane formed by the two vectors $\mathbf{T}^{(\mathbf{n})}\,\!$ and $\mathbf n\,\!$ , can then be found using the Pythagorean theorem, thus

$\begin{align} \tau_\mathrm{n} &=\sqrt{ \left( T^{(n)} \right)^2-\sigma_\mathrm{n}^2} \\ &= \sqrt{T_i^{(n)}T_i^{(n)}-\sigma_\mathrm{n}^2} \\ \end{align} \,\!$

where $\left( T^{(n)} \right)^2 = T_i^{(n)}T_i^{(n)} = \left( \sigma_{ij}n_j \right) \left( \sigma_{ik}n_k \right)=\sigma_{ij}\sigma_{ik}n_jn_k\,\!$

Equilibrium equations and symmetry of the stress tensor

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,

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

For example, for a hydrostatic fluid in equilibrium conditions, the stress tensor takes on the form:

${\sigma_{ij}} = -p{\delta_{ij}}\$ ,

where $p$ is the hydrostatic pressure, and ${\delta_{ij}}\$ is the kronecker delta.

Derivation of equilibrium equations

Consider a continuum body (see Figure 4) occupying a volume $V\,\!$ , having a surface area $S\,\!$ , with defined traction or surface forces $T_i^{(n)}\,\!$ per unit area acting on every point of the body surface, and body forces $F_i\,\!$ per unit of volume on every point within the volume $V\,\!$ . Thus, if the body is in equilibrium the resultant force acting on the volume is zero, thus:

$\int_S T_i^{(n)}dS + \int_V F_i dV = 0\,\!$

By definition the stress vector is $T_i^{(n)} =\sigma_{ji}n_j\,\!$ , then

$\int_S \sigma_{ji}n_j\, dS + \int_V F_i\, dV = 0\,\!$

Using the Gauss's divergence theorem to convert a surface integral to a volume integral gives

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

$\int_V (\sigma_{ji,j} + F_i\,) dV = 0\,\!$

For an arbitrary volume the integral vanishes, and we have 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.e.

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

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,

$\begin{align} M_O &=\int_S (\mathbf{r}\times\mathbf{T})dS + \int_V (\mathbf{r}\times\mathbf{F})dV=0 \\ 0 &= \int_S\varepsilon_{ijk}x_jT_k^{(n)}dS + \int_V\varepsilon_{ijk}x_jF_k dV \\ \end{align}\,\!$

where $\mathbf{r}\,\!$ is the position vector and is expressed as

$\mathbf{r}=x_j\mathbf{e}_j\,\!$

Knowing that $T_k^{(n)} =\sigma_{mk}n_m\,\!$ and using Gauss's divergence theorem to change from a surface integral to a volume integral, we have

$\begin{align} 0 &= \int_S \varepsilon_{ijk}x_j\sigma_{mk}n_m\, dS + \int_V\varepsilon_{ijk}x_jF_k\, dV \\ &= \int_V (\varepsilon_{ijk}x_j\sigma_{mk})_{,m} dV + \int_V\varepsilon_{ijk}x_jF_k\, dV \\ &= \int_V (\varepsilon_{ijk}x_{j,m}\sigma_{mk}+\varepsilon_{ijk}x_j\sigma_{mk,m}) dV + \int_V\varepsilon_{ijk}x_jF_k\, dV \\ &= \int_V (\varepsilon_{ijk}x_{j,m}\sigma_{mk}) dV+ \int_V \varepsilon_{ijk}x_j(\sigma_{mk,m}+F_k)dV \\ \end{align} \,\!$

The second integral is zero as it contains the equilibrium equations. This leaves the first integral, where $x_{j,m}=\delta_{jm}\,\!$ , therefore

$\int_V (\varepsilon_{ijk}\sigma_{jk}) dV=0\,\!$

For an arbitrary volume V, we then have

$\varepsilon_{ijk}\sigma_{jk}=0\,\!$

which is satisfied at every point within the body. Expanding this equation we have

$\sigma_{12}=\sigma_{21}\,\!$ , $\sigma_{23}=\sigma_{32}\,\!$ , and $\sigma_{13}=\sigma_{31}\,\!$

or in general

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

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, $K_{n}\rightarrow 1\,\!$ , or the continuum is a Non-Newtonian fluid, which can lead to rotationally non-invariant fluids, such as polymers.

Principal stresses and stress invariants

At every point in a stressed body there are at least three planes, called principal planes, with normal vectors $\mathbf{n}\,\!$ , called principal directions, where the corresponding stress vector is perpendicular to the plane, i.e., parallel or in the same direction as the normal vector $\mathbf{n}\,\!$ , and where there are no normal shear stresses $\tau_\mathrm{n}\,\!$ . The three stresses normal to these principal planes are called principal stresses.

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. 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. Similarly, every second rank tensor (such as the stress and the strain tensors) has three independent invariant quantities associated with it. One set of such invariants are the principal stresses of the stress tensor, which are just the eigenvalues of the stress tensor. Their direction vectors are the principal directions or eigenvectors.

A stress vector parallel to the normal vector $\mathbf{n}\,\!$ is given by:

$\mathbf{T}^{(\mathbf{n})} = \lambda \mathbf{n}= \mathbf{\sigma}_\mathrm n \mathbf{n}\,\!$

where $\lambda\,\!$ is a constant of proportionality, and in this particular case corresponds to the magnitudes $\sigma_\mathrm{n}\,\!$ of the normal stress vectors or principal stresses.

Knowing that $T_i^{(n)}=\sigma_{ij}n_j\,\!$ and $n_i=\delta_{ij}n_j\,\!$ , we have

$\begin{align} T_i^{(n)} &= \lambda n_i \\ \sigma_{ij}n_j &=\lambda n_i \\ \sigma_{ij}n_j -\lambda n_i &=0 \\ \left(\sigma_{ij}- \lambda\delta_{ij} \right)n_j &=0 \\ \end{align}\,\!$

This is a homogeneous system, i.e. equal to zero, of three linear equations where $n_j\,\!$ are the unknowns. To obtain a nontrivial (non-zero) solution for $n_j\,\!$ , the determinant matrix of the coefficients must be equal to zero, i.e. the system is singular. Thus,

$\left|\sigma_{ij}- \lambda\delta_{ij} \right|=\begin{vmatrix} \sigma_{11} - \lambda & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} - \lambda & \sigma_{23} \\ \sigma_{31}& \sigma_{32} & \sigma_{33} - \lambda \\ \end{vmatrix}=0\,\!$

Expanding the determinant leads to the characteristic equation

$\left|\sigma_{ij}- \lambda\delta_{ij} \right| = -\lambda^3 + I_1\lambda^2 - I_2\lambda + I_3=0\,\!$

where

$\begin{align} I_1 &= \sigma_{11}+\sigma_{22}+\sigma_{33} \\ &= \sigma_{kk} \\ I_2 &= \begin{vmatrix} \sigma_{22} & \sigma_{23} \\ \sigma_{32} & \sigma_{33} \\ \end{vmatrix} + \begin{vmatrix} \sigma_{11} & \sigma_{13} \\ \sigma_{31} & \sigma_{33} \\ \end{vmatrix} + \begin{vmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22} \\ \end{vmatrix} \\ &= \sigma_{11}\sigma_{22}+\sigma_{22}\sigma_{33}+\sigma_{11}\sigma_{33}-\sigma_{12}^2-\sigma_{23}^2-\sigma_{13}^2 \\ &= \frac{1}{2}\left(\sigma_{ii}\sigma_{jj}-\sigma_{ij}\sigma_{ji}\right) \\ I_3 &= \det(\sigma_{ij}) \\ &= \sigma_{11}\sigma_{22}\sigma_{33}+2\sigma_{12}\sigma_{23}\sigma_{31}-\sigma_{12}^2\sigma_{33}-\sigma_{23}^2\sigma_{11}-\sigma_{13}^2\sigma_{22} \\ \end{align} \,\!$

The characteristic equation has three real roots $\lambda\,\!$ , i.e. not imaginary due to the symmetry of the stress tensor. The three roots $\lambda_1=\sigma_1\,\!$ , $\lambda_2=\sigma_2\,\!$ , and $\lambda_3=\sigma_3\,\!$ are the eigenvalues or principal stresses, and they are the roots of the Cayley–Hamilton theorem. The principal stresses are unique for a given stress tensor. Therefore, from the characteristic equation it is seen that the coefficients $I_1\,\!$ , $I_2\,\!$ and $I_3\,\!$ , called the first, second, and third stress invariants, respectively, have always the same value regardless of the orientation of the coordinate system chosen.

For each eigenvalue, there is a non-trivial solution for $n_j\,\!$ in the equation $\left(\sigma_{ij}- \lambda\delta_{ij} \right)n_j =0\,\!$ . These solutions are the principal directions or eigenvectors defining the plane where the principal stresses act. The principal stresses and principal directions characterize the stress at a point and are independent of the orientation of the coordinate system.

If we choose a coordinate system with axes oriented to the principal directions, then the normal stresses will be the principal stresses and the stress tensor is represented by a diagonal matrix:

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

The principal stresses may be combined to form the stress invariants, $I_1\,\!$ , $I_2\,\!$ , and $I_3\,\!$ .The first and third invariant are the trace and determinant respectively, of the stress tensor. Thus,

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

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.

Maximum and minimum shear stresses

The maximum shear stress or maximum principal shear stress is equal to one-half the difference between the largest and smallest principal stresses, and acts on the plane that bisects the angle between the directions of the largest and smallest principal stresses, i.e. the plane of the maximum shear stress is oriented $45^\circ$ from the principal stress planes. The maximum shear stress is expressed as

$\tau_\mathrm{max}=\frac{1}{2}\left|\sigma_\mathrm{max}-\sigma_\mathrm{min}\right|\,\!$

Assuming $\sigma_1\ge\sigma_2\ge\sigma_3\,\!$ then

$\tau_\mathrm{max}=\frac{1}{2}\left|\sigma_1-\sigma_3\right|\,\!$

The normal stress component acting on the plane for the maximum shear stress is non-zero and it is equal to

$\sigma_\mathrm{n}=\frac{1}{2}\left(\sigma_1+\sigma_3\right)\,\!$

Derivation of the maximum and minimum shear stresses ^[2]^[7]^[10]^[18]^[19]^[20]^[21]

The normal stress can be written in terms of principal stresses $(\sigma_1\ge\sigma_2\ge\sigma_3)\,\!$ as

$\begin{align} \sigma_\mathrm{n} &= \sigma_{ij}n_in_j \\ &=\sigma_1n_1^2 + \sigma_2n_2^2 + \sigma_3n_3^2\\ \end{align} \,\!$

Knowing that $\left( T^{(n)} \right)^2=\sigma_{ij}\sigma_{ik}n_jn_k$ , the shear stress in terms of principal stresses components is expressed as

$\begin{align} \tau_\mathrm{n}^2 &= \left( T^{(n)} \right)^2-\sigma_\mathrm{n}^2 \\ &=\sigma_1^2n_1^2+\sigma_2^2n_2^2+\sigma_3^2n_3^2-\left(\sigma_1n_1^2+\sigma_2n_2^2+\sigma_3n_3^2\right)^2 \\ &=(\sigma_1^2-\sigma_2^2)n_1^2+(\sigma_2^2-\sigma_3^2)n_2^2+\sigma_3^2-\left[\left(\sigma_1-\sigma_3\right)n_1^2+\left(\sigma_2-\sigma_2\right)n_2^2+\sigma_3\right]^2 \\ &= (\sigma_1-\sigma_2)^2n_1^2n_2^2+(\sigma_2-\sigma_3)^2n_2^2n_3^2+(\sigma_1-\sigma_3)^2n_1^2n_3^2 \\ \end{align} \,\!$

The maximum shear stress at a point in a continuum body is determined by maximizing $\tau_\mathrm{n}^2\,\!$ subject to the condition that

$n_in_i=n_1^2+n_2^2+n_3^2=1\,\!$

This is a constrained maximization problem, which can be solved using the Lagrangian multiplier technique to convert the problem into an unconstrained optimization problem. Thus, the stationary values (maximum and minimum values)of $\tau_\mathrm{n}^2\,\!$ occur where the gradient of $\tau_\mathrm{n}^2\,\!$ is parallel to the gradient of $F\,\!$ .

The Lagrangian function for this problem can be written as

$\begin{align} F\left(n_1,n_2,n_3,\lambda\right) &= \tau^2 + \lambda \left(g\left(n_1,n_2,n_3\right) - 1 \right) \\ &= \sigma_1^2n_1^2+\sigma_2^2n_2^2+\sigma_3^2n_3^2-\left(\sigma_1n_1^2+\sigma_2n_2^2+\sigma_3n_3^2\right)^2+\lambda\left(n_1^2+n_2^2+n_3^2-1\right)\\ \end{align} \,\!$

where $\lambda\,\!$ is the Lagrangian multiplier (which is different from the $\lambda\,\!$ use to denote eigenvalues).

The extreme values of these functions are

$\frac{\partial F}{\partial n_1}=0 \qquad \frac{\partial F}{\partial n_2}=0 \qquad \frac{\partial F}{\partial n_3}=0 \,\!$

thence

$\frac{\partial F}{\partial n_1} = n_1\sigma_1^2-2n_1\sigma_1\left(\sigma_1 n_1^2+\sigma_2 n_2^2+\sigma_3 n_3^2\right)+\lambda n_1 = 0\,\!$

$\frac{\partial F}{\partial n_2} = n_2\sigma_2^2-2n_2\sigma_2\left(\sigma_1 n_1^2+\sigma_2 n_2^2+\sigma_3 n_3^2\right)+\lambda n_2 = 0\,\!$

$\frac{\partial F}{\partial n_3} = n_3\sigma_3^2-2n_3\sigma_3\left(\sigma_1 n_1^2+\sigma_2 n_2^2+\sigma_3 n_3^2\right)+\lambda n_3 = 0\,\!$

These three equations together with the condition $n_in_i=1\,\!$ may be solved for $\lambda, n_1, n_2,\,\!$ and $n_3\,\!$

By multiplying the first three equations by $n_1,\,n_2,\,\!$ and $n_3\,\!$ , respectively, and knowing that $\sigma_\mathrm{n} = \sigma_{ij}n_in_j=\sigma_1n_1^2 + \sigma_2n_2^2 +\sigma_3n_3^2\,\!$ we obtain

$n_1^2\sigma_1^2-2\sigma_1n_1^2\sigma_\mathrm{n}+n_1^2\lambda=0\,\!$

$n_2^2\sigma_2^2-2\sigma_2n_2^2\sigma_\mathrm{n}+n_2^2\lambda=0\,\!$

$n_3^2\sigma_3^2-2\sigma_1n_3^2\sigma_\mathrm{n}+n_3^2\lambda=0\,\!$

Adding these three equations we get

$\begin{align} \left[n_1^2\sigma_1^2+n_2^2\sigma_2^2+n_3^2\sigma_3^2\right]-2\left(\sigma_1n_1^2+\sigma_2n_2^2+\sigma_3n_3^2\right)\sigma_\mathrm{n}+\lambda\left(n_1^2+n_2^2+n_3^2\right)&=0 \\ \left[\tau_\mathrm{n}^2+\left(\sigma_1n_1^2+\sigma_2n_2^2+\sigma_3n_3^2\right)^2\right]-2\sigma_\mathrm{n}^2+\lambda&=0 \\ \left[\tau_\mathrm{n}^2+\sigma_\mathrm{n}^2\right]-2\sigma_\mathrm{n}^2+\lambda &=0 \\ \lambda &= \sigma_\mathrm{n}^2-\tau_\mathrm{n}^2 \\ \end{align}\,\!$

this result can be substituted into each of the first three equations to obtain

$\begin{align} \frac{\partial F}{\partial n_1} = n_1\sigma_1^2-2n_1\sigma_1\left(\sigma_1 n_1^2+\sigma_2 n_2^2+\sigma_3 n_3^2\right)+\left(\sigma_\mathrm{n}^2-\tau_\mathrm{n}^2\right) n_1 &= 0 \\ n_1\sigma_1^2-2n_1\sigma_1\sigma_\mathrm{n}+\left(\sigma_\mathrm{n}^2-\tau_\mathrm{n}^2\right) n_1 &= 0 \\ \left(\sigma_1^2-2\sigma_1\sigma_\mathrm{n}+\sigma_\mathrm{n}^2-\tau_\mathrm{n}^2\right) n_1 &= 0 \\ \end{align}\,\!$

Doing the same for the other two equations we have

$\frac{\partial F}{\partial n_2}=\left(\sigma_2^2-2\sigma_2\sigma_\mathrm{n}+\sigma_\mathrm{n}^2-\tau_\mathrm{n}^2\right) n_2 = 0\,\!$

$\frac{\partial F}{\partial n_3}= \left(\sigma_3^2-2\sigma_3\sigma_\mathrm{n}+\sigma_\mathrm{n}^2-\tau_\mathrm{n}^2\right) n_3= 0\,\!$

A first approach to solve these last three equations is to consider the trivial solution $n_i=0\,\!$ . However this options does not fulfill the constrain $n_in_i=1\,\!$ .

Considering the solution where $n_1=n_2=0\,\!$ and $n_3 \neq 0\,\!$ , it is determine from the condition $n_in_i=1\,\!$ that $n_3=\pm1\,\!$ , then from the original equation for $\tau_\mathrm{n}^2\,\!$ it is seen that $\tau_\mathrm{n}=0\,\!$ . The other two possible values for $\tau_\mathrm{n}\,\!$ can be obtained similarly by assuming

$n_1=n_3=0\,\!$ and $n_2 \neq 0\,\!$

$n_2=n_3=0\,\!$ and $n_1 \neq 0\,\!$

Thus, one set of solutions for these four equations is:

$\begin{align} n_1&=0,\,\,n_2&=0,\,\,n_3&=\pm1,\,\,\tau_\mathrm{n}&=0 \\ n_1&=0,\,\,n_2&=\pm1,\,\,n_3&=0,\,\,\tau_\mathrm{n}&=0 \\ n_1&=\pm1,\,\,n_2&=0,\,\,n_3&=0,\,\,\tau_\mathrm{n}&=0 \\ \end{align}\,\!$

These correspond to minimum values for $\tau_\mathrm{n}\,\!$ and verifies that there are no shear stresses on planes normal to the principal directions of stress, as shown previously.

A second set of solutions is obtained by assuming $n_1=0,\, n_2\neq0\,\!$ and $n_3 \neq 0\,\!$ . Thus we have

$\frac{\partial F}{\partial n_2}= \sigma_2^2-2\sigma_2\sigma_\mathrm{n}+\sigma_\mathrm{n}^2-\tau_\mathrm{n}^2 = 0 \,\!$

$\frac{\partial F}{\partial n_3}=\sigma_3^2-2\sigma_3\sigma_\mathrm{n}+\sigma_\mathrm{n}^2-\tau_\mathrm{n}^2 = 0 \,\!$

To find the values for $n_2\,\!$ and $n_3\,\!$ we first add these two equations

$\begin{align} \sigma_2^2-\sigma_3^2-2\sigma_2\sigma_n+2\sigma_2\sigma_n&=0 \\ \sigma_2^2-\sigma_3^2-2\sigma_n\left(\sigma_2-\sigma_3\right)&=0 \\ \sigma_2+\sigma_3&=2\sigma_n \\ \end{align}\,\!$

Knowing that for $n_1=0\,\!$

$\sigma_\mathrm{n} = \sigma_1n_1^2+\sigma_2n_2^2 + \sigma_3n_3^2=\sigma_2n_2^2 + \sigma_3n_3^2\,\!$

and

$n_1^2+n_2^2+n_3^2=n_2^2+n_3^2=1 \,\!$

we have

$\begin{align} \sigma_2+\sigma_3&=2\sigma_n \\ \sigma_2+\sigma_3&=2\left(\sigma_2n_2^2 + \sigma_3n_3^2\right) \\ \sigma_2+\sigma_3&=2\left(\sigma_2n_2^2 + \sigma_3\left(1-n_2^2\right)\right)&=0 \\ \end{align}\,\!$

and solving for $n_2\,\!$ we have

$n_2=\pm\frac{1}{\sqrt 2}\,\!$

Then solving for $n_3\,\!$ we have

$n_3=\sqrt{1-n_2^2}=\pm\frac{1}{\sqrt 2}\,\!$

and

$\begin{align} \tau_\mathrm{n}^2&=(\sigma_2-\sigma_3)^2n_2^2n_3^2 \\ \tau_\mathrm{n}&=\frac{\sigma_2-\sigma_3}{2}\end{align}\,\!$

The other two possible values for $\tau_\mathrm{n}\,\!$ can be obtained similarly by assuming

$n_2=0,\, n_1\neq0\,\!$ and $n_3 \neq 0\,\!$

$n_3=0,\, n_1\neq0\,\!$ and $n_2 \neq 0\,\!$

Therefore the second set of solutions for $\frac{\partial F}{\partial n_1}=0\,\!$ , representing a maximum for $\tau_\mathrm{n}\,\!$ is

$n_1=0,\,\,n_2=\pm\frac{1}{\sqrt 2},\,\,n_3=\pm\frac{1}{\sqrt 2},\,\,\tau_\mathrm{n}=\pm\frac{\sigma_2-\sigma_3}{2}\,\!$

$n_1=\pm\frac{1}{\sqrt 2},\,\,n_2=0,\,\,n_3=\pm\frac{1}{\sqrt 2},\,\,\tau_\mathrm{n}=\pm\frac{\sigma_1-\sigma_3}{2}\,\!$

$n_1=\pm\frac{1}{\sqrt 2},\,\,n_2=\pm\frac{1}{\sqrt 2},\,\,n_3=0,\,\,\tau_\mathrm{n}=\pm\frac{\sigma_2-\sigma_3}{2}\,\!$

Therefore, assuming $\sigma_1\ge\sigma_2\ge\sigma_3\,\!$ , the maximum shear stress is expressed by

$\tau_\mathrm{max}=\frac{1}{2}\left|\sigma_1-\sigma_3\right|=\frac{1}{2}\left|\sigma_\mathrm{max}-\sigma_\mathrm{min}\right|\,\!$

and it can be stated as being equal to one-half the difference between the largest and smallest principal stresses, acting on the plane that bisects the angle between the directions of the largest and smallest principal stresses.

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. 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. These 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 equivalent stress is defined as

$\sigma_\mathrm e = \sqrt{3~J_2} = \sqrt{\tfrac{1}{2}~\left[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2 \right]} \,\!$

Octahedral stresses

Figure 6. Octahedral stress planes

Considering the principal directions as the coordinate axes, a plane whose 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 $\sigma_\mathrm{oct}\,\!$ and octahedral shear stress $\tau_\mathrm{oct}\,\!$ , respectively.

Knowing that the stress tensor of point O (Figure 6) in the principal axes is

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

the stress vector on an octahedral plane is then given by:

$\begin{align} \mathbf{T}_\mathrm{oct}^{(\mathbf{n})}&= \sigma_{ij}n_i\mathbf{e}_j \\ &=\sigma_1n_1\mathbf{e}_1+\sigma_2n_2\mathbf{e}_2+\sigma_3n_3\mathbf{e}_3\\ &=\tfrac{1}{\sqrt{3}}(\sigma_1\mathbf{e}_1+\sigma_2\mathbf{e}_2+\sigma_3\mathbf{e}_3) \end{align} \,\!$

The normal component of the stress vector at point O associated with the octahedral plane is

$\begin{align} \sigma_\mathrm{oct} &= T^{(n)}_in_i \\ &=\sigma_{ij}n_in_j \\ &=\sigma_1n_1n_1+\sigma_2n_2n_2+\sigma_3n_3n_3 \\ &=\tfrac{1}{3}(\sigma_1+\sigma_2+\sigma_3)=\tfrac{1}{3}I_1 \end{align} \,\!$

which is the mean normal stress or hydrostatic stress. This value is the same in all eight octahedral planes. The shear stress on the octahedral plane is then

$\begin{align} \tau_\mathrm{oct} &=\sqrt{T_i^{(n)}T_i^{(n)}-\sigma_\mathrm{n}^2} \\ &=\left[\tfrac{1}{3}(\sigma_1^2+\sigma_2^2+\sigma_3^2)-\tfrac{1}{9}(\sigma_1+\sigma_2+\sigma_3)^2\right]^{1/2} \\ &=\tfrac{1}{3}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]^{1/2} = \tfrac{1}{3}\sqrt{2I_1^2-6I_2} = \sqrt{\tfrac{2}{3}J_2} \end{align} \,\!$

Analysis of stress

The analysis of stress within a body implies the determination at each point of the body of the magnitudes of the nine stress components. In other words, it is the determination of the internal distribution of stresses. A stress analysis is required in engineering, e.g., civil engineering and mechanical engineering, for the study and design of structures, e.g., tunnels, dams, mechanical parts, and structural frames among others, under prescribed or expected loads.

To determine the distribution of stress in the structure it is necessary to solve a boundary-value problem by specifying the boundary conditions, i.e. displacements and/or forces on the boundary. Constitutive equations, such as e.g. Hooke's Law for linear elastic materials, are used to describe the stress:strain relationship in these calculations. A boundary-value problem based on the theory of elasticity is applied to structures expected to deform elastically, i.e. infinitesimal strains, under design loads. When the loads applied to the structure induce plastic deformations, the theory of plasticity is implemented.

Approximate solutions for boundary-value problems can be obtained through the use numerical methods such as the Finite Element Method, the Finite Difference Method, and the Boundary Element Method, which are implemented in computer programs. Analytical or close-form solutions can be obtained for simple geometries, constitutive relations, and boundary conditions.

Alternatively, experimental determination of stresses can be carried out using the photoelastic method.

In design of structures, calculated stresses are restricted to be less than an specified allowable stress, also known as working or designed stress, that is chosen as some fraction of the yield strength or of the ultimate strength of the material which the structure is made of. The ratio of the ultimate stress to the allowable stress is defined as the factor of safety. Laboratory test are usually performed on material samples in order to determine the yield strength and the ultimate strength that the material can withstand before failure.

All real objects occupy a three-dimensional space. The stress analysis can be simplified in cases where the physical dimensions and the loading conditions allows the structure to be assumed as one-dimensional or two-dimensional. For a two-dimensional analysis a plane stress or a plane strain condition can be assumed.

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 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. 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. This is termed true stress and is expressed as

$\sigma_\mathrm{true} = (1 + \varepsilon_\mathrm e)(\sigma_\mathrm e)\,\!$ ,

where

$\varepsilon_\mathrm e\,\!$ is the nominal (engineering) strain, and

$\sigma_\mathrm e\,\!$ is nominal (engineering) stress.

The relationship between true strain and engineering strain is given by

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

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

Figure 7.1 Plane stress state in a continuum.

Plane stress

A state of plane stress exists when one of the three principal $\left(\sigma_1, \sigma_2, \sigma_3 \right)\,\!$ , stresses is 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. In this case, 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 other three non-zero components remain constant over the thickness of the plate. The stress tensor can then be approximated by:

$\sigma_{ij} = \begin{bmatrix} \sigma_{11} & \sigma_{12} & 0 \\ \sigma_{21} & \sigma_{22} & 0 \\ 0 & 0 & 0 \end{bmatrix} \equiv \begin{bmatrix} \sigma_{x} & \tau_{xy} & 0 \\ \tau_{yx} & \sigma_{y} & 0 \\ 0 & 0 & 0 \end{bmatrix}\,\!$ .

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's 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.

Figure 7.2 Plane strain state in a continuum.

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 (Figure 7.2). 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.

Stress transformation in plane stress and plane strain

Consider a point $P\,\!$ in a continuum under a state of plane stress, or plane strain, with stress components $(\sigma_x, \sigma_y, \tau_{xy})\,\!$ and all other stress components equal to zero (Figure 7.1, Figure 8.1). From static equilibrium of an infinitesimal material element at $P\,\!$ (Figure 8.2), the normal stress $\sigma_\mathrm{n}\,\!$ and the shear stress $\tau_\mathrm{n}\,\!$ on any plane perpendicular to the $x\,\!$ - $y\,\!$ plane passing through $P\,\!$ with a unit vector $\mathbf n\,\!$ making an angle of $\theta\,\!$ with the horizontal, i.e. $\cos \theta\,\!$ is the direction cosine in the $x\,\!$ direction, is given by:

$\sigma_\mathrm{n} = \frac{1}{2} ( \sigma_x + \sigma_y ) + \frac{1}{2} ( \sigma_x - \sigma_y )\cos 2\theta + \tau_{xy} \sin 2\theta\,\!$

$\tau_\mathrm{n} = -\frac{1}{2}(\sigma_x - \sigma_y )\sin 2\theta + \tau_{xy}\cos 2\theta\,\!$

These equations indicate that in a plane stress or plane strain condition, one can determine the stress components at a point on all directions, i.e. as a function of $\theta\,\!$ , if one knows the stress components $(\sigma_x, \sigma_y, \tau_{xy})\,\!$ on any two perpendicular directions at that point. It is important to remember that we are considering a unit area of the infinitesimal element in the direction parallel to the $y\,\!$ - $z\,\!$ plane.

Figure 8.1 - Stress transformation at a point in a continuum under plane stress conditions.

Figure 8.2 - Stress components at a plane passing through a point in a continuum under plane stress conditions.

The principal directions (Figure 8.3), i.e. orientation of the planes where the shear stress components are zero, can be obtained by making the previous equation for the shear stress $\tau_\mathrm{n}\,\!$ equal to zero. Thus we have:

$\tau_\mathrm{n} = -\frac{1}{2}(\sigma_x - \sigma_y )\sin 2\theta + \tau_{xy}\cos 2\theta=0\,\!$

and we obtain

$\tan 2 \theta_\mathrm{p} = \frac{2 \tau_{xy}}{\sigma_x - \sigma_y}\,\!$

This equation defines two values $\theta_\mathrm{p}\,\!$ which are $90^\circ\,\!$ apart (Figure 8.3). The same result can be obtained by finding the angle $\theta\,\!$ which makes the normal stress $\sigma_\mathrm{n}\,\!$ a maximum, i.e. $\frac{d\sigma_\mathrm{n}}{d\theta}=0\,\!$

The principal stresses $\sigma_1\,\!$ and $\sigma_2\,\!$ , or minimum and maximum normal stresses $\sigma_\mathrm{max}\,\!$ and $\sigma_\mathrm{min}\,\!$ , respectively, can then be obtained by replacing both values of $\theta_\mathrm{p}\,\!$ into the previous equation for $\sigma_\mathrm{n}\,\!$ . This can be achieved by rearranging the equations for $\sigma_\mathrm{n}\,\!$ and $\tau_\mathrm{n}\,\!$ , first transposing the first term in the first equation and squaring both sides of each of the equations then adding them. Thus we have

$\begin{align} \left[ \sigma_\mathrm{n} - \tfrac{1}{2} ( \sigma_x + \sigma_y )\right]^2 + \tau_\mathrm{n}^2 &= \left[\tfrac{1}{2}(\sigma_x - \sigma_y)\right]^2 + \tau_{xy}^2 \\ (\sigma_\mathrm{n} - \sigma_\mathrm{avg})^2 + \tau_\mathrm{n}^2 &= R^2 \end{align}\,\!$

where

$R = \sqrt{\left[\tfrac{1}{2}(\sigma_x - \sigma_y)\right]^2 + \tau_{xy}^2} \quad \text{and} \quad \sigma_\mathrm{avg} = \tfrac{1}{2} ( \sigma_x + \sigma_y )\,\!$

which is the equation of a circle of radius $R\,\!$ centered at a point with coordinates $[\sigma_\mathrm{avg}, 0]\,\!$ , called Mohr's circle. But knowing that for the principal stresses the shear stress $\tau_\mathrm{n} = 0\,\!$ , then we obtain from this equation:

$\sigma_1 =\sigma_\mathrm{max} = \tfrac{1}{2}(\sigma_x + \sigma_y) + \sqrt{\left[\tfrac{1}{2}(\sigma_x - \sigma_y)\right]^2 + \tau_{xy}^2}\,\!$

$\sigma_2 =\sigma_\mathrm{min} = \tfrac{1}{2}(\sigma_x + \sigma_y) - \sqrt{\left[\tfrac{1}{2}(\sigma_x - \sigma_y)\right]^2 + \tau_{xy}^2}\,\!$

Figure 8.3 - Transformation of stresses in two dimensions, showing the planes of action of principal stresses, and maximum and minimum shear stresses.

When $\tau_{xy}=0\,\!$ the infinitesimal element is oriented in the direction of the principal planes, thus the stresses acting on the rectangular element are principal stresses: $\sigma_x = \sigma_1\,\!$ and $\sigma_y = \sigma_2\,\!$ . Then the normal stress $\sigma_\mathrm{n}\,\!$ and shear stress $\tau_\mathrm{n}\,\!$ as a function of the principal stresses can be determined by making $\tau_{xy}=0\,\!$ . Thus we have

$\sigma_\mathrm{n} = \frac{1}{2} ( \sigma_1 + \sigma_2 ) + \frac{1}{2} ( \sigma_1 - \sigma_2 )\cos 2\theta\,\!$

$\tau_\mathrm{n} = -\frac{1}{2}(\sigma_1 - \sigma_2 )\sin 2\theta\,\!$

Then the maximum shear stress $\tau_\mathrm{max}\,\!$ occurs when $\sin 2\theta = 1\,\!$ , i.e. $\theta = 45^\circ\,\!$ (Figure 8.3):

$\tau_\mathrm{max} = \frac{1}{2}(\sigma_1 - \sigma_2 )\,\!$

Then the minimum shear stress $\tau_\mathrm{min}\,\!$ occurs when $\sin 2\theta = -1\,\!$ , i.e. $\theta = 135^\circ\,\!$ (Figure 8.3):

$\tau_\mathrm{min} = -\frac{1}{2}(\sigma_1 - \sigma_2 )\,\!$

Graphical representation of stress at a point

Mohr's circle, Lame's stress ellipsoid (together with the stress director surface), and Cauchy's stress quadric are two-dimensional graphical representations of the state of stress at a point. They allow for the graphical determination of the magnitude of the stress tensor at a given point for all planes passing through that point. Mohr's circle is the most common graphical method.

Mohr's circle

Mohr's circle, named after Christian Otto Mohr, is the locus of points that represent the state of stress on individual planes at all their orientations. The abscissa, $\sigma_\mathrm{n}\,\!$ , and ordinate, $\tau_\mathrm{n}\,\!$ , of each point on the circle are the normal stress and shear stress components, respectively, acting on a particular cut plane with a unit vector $\mathbf n\,\!$ with components $\left(n_1, n_2, n_3 \right)\,\!$ .

Lame's stress ellipsoid

The surface of the ellipsoid represents the locus of the endpoints of all stress vectors acting on all planes passing through a given point in the continuum body. In other words, the endpoints of all stress vectors at a given point in the continuum body lie on the stress ellipsoid surface, i.e., the radius-vector from the center of the ellipsoid, located at the material point in consideration, to a point on the surface of the ellipsoid is equal to the stress vector on some plane passing through the point. In two dimensions, the surface is represented by an ellipse (Figure coming).

Cauchy's stress quadric

The Cauchy's stress quadric, also called the stress surface, is a surface of the second order that traces the variation of the normal stress vector $\sigma_\mathrm n \,\!$ as the orientation of the planes passing through a given point is changed.

Graphical representation of the stress field

The complete state of stress in a body at a particular deformed configuration, i.e., at a particular time during the motion of the body, implies knowing the six independent components of the stress tensor $(\sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{12}, \sigma_{23}, \sigma_{13})\,\!$ , or the three principal stresses $(\sigma_1, \sigma_2, \sigma_3)\,\!$ , at each material point in the body at that time. However, numerical analysis and analytical methods allow only for the calculation of the stress tensor at a certain number of discrete material points. To graphically represent in two dimensions this partial picture of the stress field different sets of contour lines can be used^[19]:

Isobars are curves along which the principal stress, e.g., $\sigma_1\,\!$ is constant.
Isochromatics are curves along which the maximum shear stress is constant. This curves are directly determined using photoelasticity methods.
Isopachs are curves along which the mean normal stress is constant
Isostatics or stress trajectories are a system of curves which are at each material point tangent to the principal axes of stress.
Isoclinics are curves on which the principal axes make a constant angle with a given fixed reference direction. These curves can also be obtained directly by photoelasticity methods.
Slip lines are curves on which the shear stress is a maximum.

Alternative measures of stress

The Cauchy stress tensor is not the only measure of stress that is used in practice. Other measures of stress include the first and second Piola-Kirchhoff stress tensors, the Biot stress tensor, and the Kirchhoff stress tensor.

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. For infinitesimal deformations or rotations, the Cauchy and Piola-Kirchhoff tensors are identical. These tensors take their names from Gabrio Piola and Gustav Kirchhoff.

Whereas the Cauchy stress tensor, $\boldsymbol{\sigma}$ relates stresses in the current configuration, the deformation gradient and strain tensors are described by relating the motion to the reference configuration; thus not all tensors describing the state of the material are in either the reference or current configuration. Having the stress, strain and deformation all described either in the reference or current configuration would make it easier to define constitutive models (for example, the Cauchy Stress tensor is variant to a pure rotation, while the deformation strain tensor is invariant; thus creating problems in defining a constitutive model that relates a varying tensor, in terms of an invariant one during pure rotation; as by definition constitutive models have to be invariant to pure rotations). The 1^st Piola-Kirchhoff stress tensor, $\boldsymbol{P}$ is one possible solution to this problem. It defines a family of tensors, which describe the configuration of the body in either the current or the reference state.

The 1^st Piola-Kirchhoff stress tensor, $\boldsymbol{P}$ relates forces in the present configuration with areas in the reference ("material") configuration.

$\boldsymbol{P} = J~\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T}$

where $\boldsymbol{F}$ is the deformation gradient and $J= \det\boldsymbol{F}$ is the Jacobian determinant.

In terms of components with respect to an orthonormal basis, the first Piola-Kirchhoff stress is given by

$P_{iL} = J~\sigma_{ik}~F^{-1}_{Lk} = J~\sigma_{ik}~\cfrac{\partial X_L}{\partial x_k}~\,\!$

Because it relates different coordinate systems, the 1^st Piola-Kirchhoff stress is a two-point tensor. In general, it is not symmetric. The 1^st Piola-Kirchhoff stress is the 3D generalization of the 1D concept of engineering stress.

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 $\boldsymbol{S}$ 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.

$\boldsymbol{S} = J~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{F}^{-T} ~.$

In index notation with respect to an orthonormal basis,

$S_{IL}=J~F^{-1}_{Ik}~F^{-1}_{Lm}~\sigma_{km} = J~\cfrac{\partial X_I}{\partial x_k}~\cfrac{\partial X_L}{\partial x_m}~\sigma_{km} \!\,\!$

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 finite strain tensor.

References

↑ Walter D. Pilkey, Orrin H. Pilkey (1974). Mechanics of solids. p. 292. http://books.google.com/books?id=d7I8AAAAIAAJ&q=average+shear+stress+approximation&dq=average+shear+stress+approximation&ei=FdBkS837NJPyNMLQzNgB&cd=2.
↑ ^2.0 ^2.1 ^2.2 ^2.3 Chen
↑ ^3.0 ^3.1 Smith & Truesdell p.97
↑ Slaughter
↑ ^5.0 ^5.1 Lubliner
↑ ^6.0 ^6.1 ^6.2 ^6.3 Liu
↑ ^7.0 ^7.1 ^7.2 Wu
↑ ^8.0 ^8.1 ^8.2 ^8.3 Fung
↑ ^9.0 ^9.1 ^9.2 Mase
↑ ^10.0 ^10.1 ^10.2 ^10.3 Atanackovic
↑ ^11.0 ^11.1 ^11.2 ^11.3 ^11.4 Irgens
↑ ^12.0 ^12.1 ^12.2 ^12.3 Chadwick
↑ Maxwell pointed out that nonvanishing body moments exist in a magnet in a magnetic field and in a dielectric material in an electric field with different planes of polarization. Fung p.76.
↑ Couple stresses and body couples were first explored by Voigt and Cosserat, and later reintroduced by Mindlin in 1960 on his work for Bell Labs on pure quartz crystals. Richards p.55.
↑ Truesdell and Topin 1960
↑ Basar
↑ Hjelmstad
↑ Chatterjee
↑ ^19.0 ^19.1 Jaeger
↑ Ameen
↑ Prager

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Stress (mechanics)

Contents

Introduction

Normal and shear stresses

Stress modelling (Cauchy)

Stress analysis

Forces in a continuum

Euler's laws of motion for a continuum

Euler-Cauchy stress principle

Cauchy’s postulate

Cauchy’s fundamental lemma

Cauchy’s stress theorem – stress tensor

Transformation rule of the stress tensor

Normal and shear stresses

Equilibrium equations and symmetry of the stress tensor

Principal stresses and stress invariants

Maximum and minimum shear stresses

Stress deviator tensor

Invariants of the stress deviator tensor

Octahedral stresses

Analysis of stress

Uniaxial stress

Plane stress

Plane strain

Stress transformation in plane stress and plane strain

Graphical representation of stress at a point

Mohr's circle

Lame's stress ellipsoid

Cauchy's stress quadric

Graphical representation of the stress field

Alternative measures of stress

Piola-Kirchhoff stress tensor

2nd Piola-Kirchhoff stress tensor

See also

References

Bibliography

2^nd Piola-Kirchhoff stress tensor