Stress (mechanics)

In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are 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 shape change or structural failure.

However, models of continuum mechanics which explicitly express force as a variable generally fail to merge and describe deformation of matter and solid bodies, because the attributes of matter and solids are three dimensional. Classical models of continuum mechanics assume an average force and fail to properly incorporate "geometrical factors", which are important to describe stress distribution and accumulation of energy during the continuum.

The dimension of stress is that of pressure, and therefore the SI unit for stress is the pascal (symbol Pa), which is equivalent to one newton (force) per square meter (unit area), that is N/m². In Imperial units, stress is measured in pound-force per square inch, which is abbreviated as psi.

Introduction

"Stress" measures the average force per unit area of a surface within a deformable body on which internal forces act, specifically the intensity of the internal forces acting between particles of a deformable body across imaginary internal surfaces. These internal forces are produced between the particles in the body as a reaction to external forces. 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 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 $\backslash sigma\backslash ,\backslash !$, or intensity of internal forces, can be obtained by dividing the total normal force $F\_\backslash mathrm\; n\backslash ,\backslash !$ by the prism's cross-sectional area $A\backslash ,\backslash !$. The normal force is determined from the equilibrium of forces. 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 $\backslash sigma\backslash ,\backslash !$ is represented by a scalar called engineering stress or nominal stress that represents an average stress ($\backslash sigma\_\backslash mathrm\{avg\}\backslash ,\backslash !$) over the area, meaning that the stress in the cross-section is uniformly distributed. Thus, we have

:$\backslash sigma\_\backslash mathrm\{avg\}\; =\; \backslash frac\{F\_\backslash mathrm\; n\}\{A\}\backslash approx\backslash sigma\backslash ,\backslash !$.

In Figure 1.3, the normal stress is observed in two planes $m-m\backslash ,\backslash !$ and $n-n\backslash ,\backslash !$ of the axially loaded prismatic bar. The stress on plane $n-n\backslash ,\backslash !$, which is closer to the point of application of the load $F\backslash ,\backslash !$, varies more across the cross-section than that of plane $m-m\backslash ,\backslash !$. 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 $\backslash sigma\_\backslash mathrm\; \{avg\}\backslash ,\backslash !$. On the other hand, the variation of shear stress across the section of a prismatic bar cannot be assumed to be uniform.

A different type of stress occurs when transverse forces $F\_\backslash mathrm\backslash ,\backslash !$ act on the prismatic bar as shown in Figure 1.4. Considering the earlier cross-section, from static equilibrium, the internal force has a magnitude equal to $F\_\backslash mathrm\; s\backslash ,\backslash !$ opposite in direction and parallel to the cross-section. $F\_\backslash mathrm\; s\backslash ,\backslash !$ is called the shear force. Dividing the shear force $F\_\backslash mathrm\; s\backslash ,\backslash !$ by the cross-section's area $A\backslash ,\backslash !$ we obtain the shear stress. In this case the shear stress $\backslash tau\backslash ,\backslash !$ is a scalar quantity representing an average shear stress ($\backslash tau\_\backslash mathrm\{avg\}\backslash ,\backslash !$) in the section, i.e. the stress in the cross-section is uniformly distributed. In materials science and in engineering aspects, the average of the scalar shear force ($\backslash tau\_\backslash mathrm\{avg\}\backslash ,\backslash !$) is adequate for crystallized materials during brittle fracture and operates through the fractured cross-section or stress plane. However before any actual fracture occurs, it’s the strain or changes in potential energy from an applied load who is responsible for the fracture. But due to the brittleness and immediate fracture mechanism the force $F\_\backslash mathrm\; s\backslash ,\backslash !$ is generally used to define the property of the material instead of changes in potential energy.

:$\backslash tau\_\backslash mathrm\{avg\}=\; \backslash frac\{F\_\backslash mathrm\; s\}\{A\}\backslash approx\backslash tau\backslash ,\backslash !$

Stress modeling (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 $\backslash sigma\_\{ij\}\backslash ,\backslash !$ of a second-order tensor of type (0,2) known as the Cauchy stress tensor, $\backslash boldsymbol\backslash sigma\backslash ,\backslash !$:

:

\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 of stress distribution.

The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations where the differences in stress distribution in most cases can be neglected. 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 at 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.

Certain invariants are 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

Stress analysis is the determination of the internal distribution of stresses in a structure. It is needed in engineering for the study and design of structures such as tunnels, dams, mechanical parts, and structural frames, under prescribed or expected loads. To determine the distribution of stress in a structure, the engineer needs to solve a boundary-value problem by specifying the boundary conditions. These are displacements and forces on the boundary of the structure.

Constitutive equations, such as Hooke’s Law for linear elastic materials, describe the stress-strain relationship in these calculations.

When a structure is expected to deform elastically (and resume its original shape), a boundary-value problem based on the theory of elasticity is applied, with infinitesimal strains, under design loads.

When the applied loads permanently deform the structure, the theory of plasticity applies.

Stress analysis is simplified when the physical dimensions and the distribution of loads allow the structure to be treated as one- or two-dimensional. For a two-dimensional analysis a plane stress or a plane strain condition can be assumed. Alternatively, stresses can be experimentally determined.

Computer-based approximations for boundary-value problems can be obtained through numerical methods such as the Finite Element Method, the Finite Difference Method, and the Boundary Element Method. Analytical or closed-form solutions can be obtained for simple geometries, constitutive relations, and boundary conditions.

Theoretical background

Continuum mechanics deals with deformable bodies, as opposed to rigid bodies. The stresses considered in continuum mechanics are only those produced by deformation of the body, sc. relative changes are considered rather than absolute values. 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. Stresses generated during manufacture of the body to a specific configuration are also excluded.

Following classical Newtonian and Eulerian dynamics, 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 and body forces.

Surface forces, or contact forces, 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 bind 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 external contact forces act on a body, internal contact forces pass 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. These laws are called Euler's equations of motion for continuous bodies. The internal contact forces are related to the body's deformation through constitutive equations. This article provides mathematical descriptions of internal contact forces and how they relate to the body's motion, independent of the body's material makeup.

Stress can be thought as a measure of the internal contact forces' intensity acting between particles of the body across imaginary internal surfaces. that act on its volume (or mass). This implies that the internal forces manifest through the contact forces alone. These forces arise from the presence of the body in force fields, (e.g., a gravitational field). 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 assumed to be continuous over the body's volume.

The density of internal forces at every point in a deformable body is not necessarily even, i.e. there is a distribution of stresses. This variation of internal forces is governed by the laws of conservation of linear and angular momentum, which normally apply to a mass particle but extend in continuum mechanics to a body of continuously distributed mass. 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.

Euler–Cauchy stress principle

The Euler–Cauchy stress principle states that upon any surface (real or imaginary) that divides the body, the action of one part of the body on the other is equivalent (equipollent) to the system of distributed forces and couples on the surface dividing the body, and it is represented by a vector field T⁽ⁿ⁾, called the stress vector, defined on the surface S and assumed to depend continuously on the surface's unit vector n.

To explain this principle, consider an imaginary surface S passing through an internal material point P dividing the continuous body into two segments, as seen in Figure 2.1a or 2.1b (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 F and body forces b. The internal contact forces transmitted from one segment to the other through the dividing plane, due to the action of one portion of the continuum onto the other, generate a force distribution on a small area ΔS, with a normal unit vector n, on the dividing plane S. The force distribution is equipollent to a contact force ΔF and a couple stress ΔM, as shown in Figure 2.1a and 2.1b. Cauchy’s stress principle asserts i.e., having a common tangent at P. This means that the stress vector is a function of the normal vector n only, and is not influenced by the curvature of the internal surfaces.

Cauchy’s fundamental lemma

A consequence of Cauchy’s postulate is Cauchy’s Fundamental Lemma, which states that the stress vectors acting on opposite sides of the same surface are equal in magnitude and opposite in direction. Cauchy’s fundamental lemma is equivalent to Newton's third law of motion of action and reaction, and is expressed as

:$-\; \backslash mathbf\{T\}^\{(\backslash mathbf\{n\})\}=\; \backslash mathbf\{T\}^\{(-\; \backslash mathbf\{n\})\}.\backslash ,\backslash !$

Cauchy’s stress theorem—stress tensor

The state of stress at a point in the body is then defined by all the stress vectors T⁽ⁿ⁾ associated with all planes (infinite in number) that pass through that point. The principal normal stresses can then be used to calculate the von Mises stress and ultimately the safety factor and margin of safety.

:$\backslash sigma\_\{1\},\backslash sigma\_\{2\}=\; \backslash frac\{\backslash sigma\_\{x\}\; +\; \backslash sigma\_\{y\}\}\{2\}\; \backslash pm\; \backslash sqrt\{\backslash left\; (\backslash frac\{\backslash sigma\_\{x\}\; -\; \backslash sigma\_\{y\}\}\{2\}\backslash right)^2\; +\; \backslash tau\_\{xy\}^2\}\backslash ,\backslash !$

Using just the part of the equation under the square root is equal to the maximum and minimum shear stress for plus and minus. This is shown as:

:$\backslash tau\_\{max\},\backslash tau\_\{min\}=\; \backslash pm\; \backslash sqrt\{\backslash left\; (\backslash frac\{\backslash sigma\_\{x\}\; -\; \backslash sigma\_\{y\}\}\{2\}\backslash right)^2\; +\; \backslash tau\_\{xy\}^2\}\backslash ,\backslash !$

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 $45^\backslash circ$ from the principal stress planes. The maximum shear stress is expressed as

:$\backslash tau\_\backslash mathrm\{max\}=\backslash frac\{1\}\{2\}\backslash left|\backslash sigma\_\backslash mathrm\{max\}-\backslash sigma\_\backslash mathrm\{min\}\backslash right|\backslash ,\backslash !$

Assuming $\backslash sigma\_1\backslash ge\backslash sigma\_2\backslash ge\backslash sigma\_3\backslash ,\backslash !$ then

:$\backslash tau\_\backslash mathrm\{max\}=\backslash frac\{1\}\{2\}\backslash left|\backslash sigma\_1-\backslash sigma\_3\backslash right|\backslash ,\backslash !$

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

$\backslash sigma\_\backslash mathrm\{n\}=\backslash frac\{1\}\{2\}\backslash left(\backslash sigma\_1+\backslash sigma\_3\backslash right)\backslash ,\backslash !$

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of the maximum and minimum shear stresses |- |The normal stress can be written in terms of principal stresses $(\backslash sigma\_1\backslash ge\backslash sigma\_2\backslash ge\backslash sigma\_3)\backslash ,\backslash !$ as

:

Knowing that $\backslash left(\; T^\{(n)\}\; \backslash right)^2=\backslash sigma\_\{ij\}\backslash sigma\_\{ik\}n\_jn\_k$, the shear stress in terms of principal stresses components is expressed as

:

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

:$n\_in\_i=n\_1^2+n\_2^2+n\_3^2=1\backslash ,\backslash !$

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 $\backslash tau\_\backslash mathrm\{n\}^2\backslash ,\backslash !$ occur where the gradient of $\backslash tau\_\backslash mathrm\{n\}^2\backslash ,\backslash !$ is parallel to the gradient of $F\backslash ,\backslash !$.

The Lagrangian function for this problem can be written as

:

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

The extreme values of these functions are

:$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; n\_1\}=0\; \backslash qquad\; \backslash frac\{\backslash partial\; F\}\{\backslash partial\; n\_2\}=0\; \backslash qquad\; \backslash frac\{\backslash partial\; F\}\{\backslash partial\; n\_3\}=0\; \backslash ,\backslash !$

thence

:$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; n\_1\}\; =\; n\_1\backslash sigma\_1^2-2n\_1\backslash sigma\_1\backslash left(\backslash sigma\_1\; n\_1^2+\backslash sigma\_2\; n\_2^2+\backslash sigma\_3\; n\_3^2\backslash right)+\backslash lambda\; n\_1\; =\; 0\backslash ,\backslash !$ :$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; n\_2\}\; =\; n\_2\backslash sigma\_2^2-2n\_2\backslash sigma\_2\backslash left(\backslash sigma\_1\; n\_1^2+\backslash sigma\_2\; n\_2^2+\backslash sigma\_3\; n\_3^2\backslash right)+\backslash lambda\; n\_2\; =\; 0\backslash ,\backslash !$ :$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; n\_3\}\; =\; n\_3\backslash sigma\_3^2-2n\_3\backslash sigma\_3\backslash left(\backslash sigma\_1\; n\_1^2+\backslash sigma\_2\; n\_2^2+\backslash sigma\_3\; n\_3^2\backslash right)+\backslash lambda\; n\_3\; =\; 0\backslash ,\backslash !$

These three equations together with the condition $n\_in\_i=1\backslash ,\backslash !$ may be solved for $\backslash lambda,\; n\_1,\; n\_2,\backslash ,\backslash !$ and $n\_3\backslash ,\backslash !$

By multiplying the first three equations by $n\_1,\backslash ,n\_2,\backslash ,\backslash !$ and $n\_3\backslash ,\backslash !$, respectively, and knowing that $\backslash sigma\_\backslash mathrm\{n\}\; =\; \backslash sigma\_\{ij\}n\_in\_j=\backslash sigma\_1n\_1^2\; +\; \backslash sigma\_2n\_2^2\; +\backslash sigma\_3n\_3^2\backslash ,\backslash !$ we obtain

:$n\_1^2\backslash sigma\_1^2-2\backslash sigma\_1n\_1^2\backslash sigma\_\backslash mathrm\{n\}+n\_1^2\backslash lambda=0\backslash ,\backslash !$ :$n\_2^2\backslash sigma\_2^2-2\backslash sigma\_2n\_2^2\backslash sigma\_\backslash mathrm\{n\}+n\_2^2\backslash lambda=0\backslash ,\backslash !$ :$n\_3^2\backslash sigma\_3^2-2\backslash sigma\_1n\_3^2\backslash sigma\_\backslash mathrm\{n\}+n\_3^2\backslash lambda=0\backslash ,\backslash !$

Adding these three equations we get

:

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

:

Doing the same for the other two equations we have :$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; n\_2\}=\backslash left(\backslash sigma\_2^2-2\backslash sigma\_2\backslash sigma\_\backslash mathrm\{n\}+\backslash sigma\_\backslash mathrm\{n\}^2-\backslash tau\_\backslash mathrm\{n\}^2\backslash right)\; n\_2\; =\; 0\backslash ,\backslash !$

:$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; n\_3\}=\; \backslash left(\backslash sigma\_3^2-2\backslash sigma\_3\backslash sigma\_\backslash mathrm\{n\}+\backslash sigma\_\backslash mathrm\{n\}^2-\backslash tau\_\backslash mathrm\{n\}^2\backslash right)\; n\_3=\; 0\backslash ,\backslash !$

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

Considering the solution where $n\_1=n\_2=0\backslash ,\backslash !$ and $n\_3\; \backslash neq\; 0\backslash ,\backslash !$, it is determine from the condition $n\_in\_i=1\backslash ,\backslash !$ that $n\_3=\backslash pm1\backslash ,\backslash !$, then from the original equation for $\backslash tau\_\backslash mathrm\{n\}^2\backslash ,\backslash !$ it is seen that $\backslash tau\_\backslash mathrm\{n\}=0\backslash ,\backslash !$. The other two possible values for $\backslash tau\_\backslash mathrm\{n\}\backslash ,\backslash !$ can be obtained similarly by assuming

:$n\_1=n\_3=0\backslash ,\backslash !$ and $n\_2\; \backslash neq\; 0\backslash ,\backslash !$ :$n\_2=n\_3=0\backslash ,\backslash !$ and $n\_1\; \backslash neq\; 0\backslash ,\backslash !$

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

:

These correspond to minimum values for $\backslash tau\_\backslash mathrm\{n\}\backslash ,\backslash !$ 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 $n\_1=0,\backslash ,\; n\_2\backslash neq0\backslash ,\backslash !$ and $n\_3\; \backslash neq\; 0\backslash ,\backslash !$. Thus we have

:$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; n\_2\}=\; \backslash sigma\_2^2-2\backslash sigma\_2\backslash sigma\_\backslash mathrm\{n\}+\backslash sigma\_\backslash mathrm\{n\}^2-\backslash tau\_\backslash mathrm\{n\}^2\; =\; 0\; \backslash ,\backslash !$

:$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; n\_3\}=\backslash sigma\_3^2-2\backslash sigma\_3\backslash sigma\_\backslash mathrm\{n\}+\backslash sigma\_\backslash mathrm\{n\}^2-\backslash tau\_\backslash mathrm\{n\}^2\; =\; 0\; \backslash ,\backslash !$

To find the values for $n\_2\backslash ,\backslash !$ and $n\_3\backslash ,\backslash !$ we first add these two equations

:

Knowing that for $n\_1=0\backslash ,\backslash !$

:$\backslash sigma\_\backslash mathrm\{n\}\; =\; \backslash sigma\_1n\_1^2+\backslash sigma\_2n\_2^2\; +\; \backslash sigma\_3n\_3^2=\backslash sigma\_2n\_2^2\; +\; \backslash sigma\_3n\_3^2\backslash ,\backslash !$

and

:$n\_1^2+n\_2^2+n\_3^2=n\_2^2+n\_3^2=1\; \backslash ,\backslash !$

we have

:

and solving for $n\_2\backslash ,\backslash !$ we have

:$n\_2=\backslash pm\backslash frac\{1\}\{\backslash sqrt\; 2\}\backslash ,\backslash !$

Then solving for $n\_3\backslash ,\backslash !$ we have :$n\_3=\backslash sqrt\{1-n\_2^2\}=\backslash pm\backslash frac\{1\}\{\backslash sqrt\; 2\}\backslash ,\backslash !$

and

:

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

:$n\_2=0,\backslash ,\; n\_1\backslash neq0\backslash ,\backslash !$ and $n\_3\; \backslash neq\; 0\backslash ,\backslash !$ :$n\_3=0,\backslash ,\; n\_1\backslash neq0\backslash ,\backslash !$ and $n\_2\; \backslash neq\; 0\backslash ,\backslash !$

Therefore the second set of solutions for $\backslash frac\{\backslash partial\; F\}\{\backslash partial\; n\_1\}=0\backslash ,\backslash !$, representing a maximum for $\backslash tau\_\backslash mathrm\{n\}\backslash ,\backslash !$ is

:$n\_1=0,\backslash ,\backslash ,n\_2=\backslash pm\backslash frac\{1\}\{\backslash sqrt\; 2\},\backslash ,\backslash ,n\_3=\backslash pm\backslash frac\{1\}\{\backslash sqrt\; 2\},\backslash ,\backslash ,\backslash tau\_\backslash mathrm\{n\}=\backslash pm\backslash frac\{\backslash sigma\_2-\backslash sigma\_3\}\{2\}\backslash ,\backslash !$

:$n\_1=\backslash pm\backslash frac\{1\}\{\backslash sqrt\; 2\},\backslash ,\backslash ,n\_2=0,\backslash ,\backslash ,n\_3=\backslash pm\backslash frac\{1\}\{\backslash sqrt\; 2\},\backslash ,\backslash ,\backslash tau\_\backslash mathrm\{n\}=\backslash pm\backslash frac\{\backslash sigma\_1-\backslash sigma\_3\}\{2\}\backslash ,\backslash !$

:$n\_1=\backslash pm\backslash frac\{1\}\{\backslash sqrt\; 2\},\backslash ,\backslash ,n\_2=\backslash pm\backslash frac\{1\}\{\backslash sqrt\; 2\},\backslash ,\backslash ,n\_3=0,\backslash ,\backslash ,\backslash tau\_\backslash mathrm\{n\}=\backslash pm\backslash frac\{\backslash sigma\_2-\backslash sigma\_3\}\{2\}\backslash ,\backslash !$

Therefore, assuming $\backslash sigma\_1\backslash ge\backslash sigma\_2\backslash ge\backslash sigma\_3\backslash ,\backslash !$, the maximum shear stress is expressed by

:$\backslash tau\_\backslash mathrm\{max\}=\backslash frac\{1\}\{2\}\backslash left|\backslash sigma\_1-\backslash sigma\_3\backslash right|=\backslash frac\{1\}\{2\}\backslash left|\backslash sigma\_\backslash mathrm\{max\}-\backslash sigma\_\backslash mathrm\{min\}\backslash right|\backslash ,\backslash !$

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 $\backslash sigma\_\{ij\}\backslash ,\backslash !$ 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, $p\backslash delta\_\{ij\}\backslash ,\backslash !$, which tends to change the volume of the stressed body; and # a deviatoric component called the stress deviator tensor, $s\_\{ij\}\backslash ,\backslash !$, which tends to distort it. So:

:$\backslash sigma\_\{ij\}=\; s\_\{ij\}\; +\; p\backslash delta\_\{ij\},\backslash ,$

where $p\backslash ,\backslash !$ is the mean stress given by

:$p=\backslash frac\{\backslash sigma\_\{kk\}\}\{3\}=\backslash frac\{\backslash sigma\_\{11\}+\backslash sigma\_\{22\}+\backslash sigma\_\{33\}\}\{3\}=\backslash tfrac\{1\}\{3\}I\_1.\backslash ,$

Note that convention in solid mechanics differs slightly from what is listed above. In solid mechanics, pressure is generally defined as negative one-third the trace of the stress tensor.

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

:

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 $s\_\{ij\}\backslash ,\backslash !$ are the same as the principal directions of the stress tensor $\backslash sigma\_\{ij\}\backslash ,\backslash !$. Thus, the characteristic equation is

:$\backslash left|\; s\_\{ij\}-\; \backslash lambda\backslash delta\_\{ij\}\; \backslash right|\; =\; \backslash lambda^3-J\_1\backslash lambda^2-J\_2\backslash lambda-J\_3=0,\backslash ,$

where $J\_1\backslash ,\backslash !$, $J\_2\backslash ,\backslash !$ and $J\_3\backslash ,\backslash !$ 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 $s\_\{ij\}\backslash ,\backslash !$ or its principal values $s\_1\backslash ,\backslash !$, $s\_2\backslash ,\backslash !$, and $s\_3\backslash ,\backslash !$, or alternatively, as a function of $\backslash sigma\_\{ij\}\backslash ,\backslash !$ or its principal values $\backslash sigma\_1\backslash ,\backslash !$, $\backslash sigma\_2\backslash ,\backslash !$, and $\backslash sigma\_3\backslash ,\backslash !$ . Thus,

:

Because $s\_\{kk\}=0\backslash ,\backslash !$, 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

:$\backslash sigma\_\backslash mathrm\; e\; =\; \backslash sqrt\{3~J\_2\}\; =\; \backslash sqrt\{\backslash tfrac\{1\}\{2\}~\backslash left[(\backslash sigma\_1-\backslash sigma\_2)^2\; +\; (\backslash sigma\_2-\backslash sigma\_3)^2\; +\; (\backslash sigma\_3-\backslash sigma\_1)^2\; \backslash right]\}\; \backslash ,.$

Octahedral stresses

Considering the principal directions as the coordinate axes, a plane whose normal vector makes equal angles with each of the principal axes (i.e. having direction cosines equal to $|1/\backslash sqrt\{3\}|\backslash ,\backslash !$) 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 $\backslash sigma\_\backslash mathrm\{oct\}\backslash ,\backslash !$ and octahedral shear stress $\backslash tau\_\backslash mathrm\{oct\}\backslash ,\backslash !$, respectively.

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

:

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

:

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

:

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

:

Alternative measures of stress

Other useful stress measures 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 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.

Whereas the Cauchy stress tensor, $\backslash boldsymbol\{\backslash sigma\}$ 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. Describing the stress, strain and deformation 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, $\backslash boldsymbol\{P\}$ 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, $\backslash boldsymbol\{P\}$ relates forces in the present configuration with areas in the reference ("material") configuration. :$\backslash boldsymbol\{P\}\; =\; J~\backslash boldsymbol\{\backslash sigma\}\backslash cdot\backslash boldsymbol\{F\}^\{-T\}$ where $\backslash boldsymbol\{F\}$ is the deformation gradient and $J=\; \backslash det\backslash boldsymbol\{F\}$ is the Jacobian determinant.

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

:$P\_\{iL\}\; =\; J~\backslash sigma\_\{ik\}~F^\{-1\}\_\{Lk\}\; =\; J~\backslash sigma\_\{ik\}~\backslash cfrac\{\backslash partial\; X\_L\}\{\backslash partial\; x\_k\}~\backslash ,\backslash !$

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

In index notation with respect to an orthonormal basis, :$S\_\{IL\}=J~F^\{-1\}\_\{Ik\}~F^\{-1\}\_\{Lm\}~\backslash sigma\_\{km\}\; =\; J~\backslash cfrac\{\backslash partial\; X\_I\}\{\backslash partial\; x\_k\}~\backslash cfrac\{\backslash partial\; X\_L\}\{\backslash partial\; x\_m\}~\backslash sigma\_\{km\}\; \backslash !\backslash ,\backslash !$

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 remain constant, irrespective of material orientation.

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

See also

References

Bibliography

Further reading

Dieter, G. E. (3 ed.). (1989). Mechanical Metallurgy. New York: McGraw-Hill. ISBN 0-07-100406-8.

Landau, L.D. and E.M.Lifshitz. (1959). Theory of Elasticity.

Love, A. E. H. (4 ed.). (1944). Treatise on the Mathematical Theory of Elasticity. New York: Dover Publications. ISBN 0-486-60174-9.

Category:Classical mechanics Category:Tensors Category:Materials science Category:Elasticity (physics) Category:Plasticity Category:Solid mechanics Category:Mechanics