Stress functions

Last updated January 19, 2022

In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces (&/or body forces that could be expressed as potentials) on the boundary are (using index notation) the equilibrium equation:

Beltrami stress functions

It can be shown ^[1] that a complete solution to the equilibrium equations may be written as

\sigma =\nabla \times \Phi \times \nabla

Using index notation:

\sigma _{ij}=\varepsilon _{ikm}\varepsilon _{jln}\Phi _{kl,mn}

Engineering notation
$\sigma _{x}={\frac {\partial ^{2}\Phi _{yy}}{\partial z\partial z}}+{\frac {\partial ^{2}\Phi _{zz}}{\partial y\partial y}}-2{\frac {\partial ^{2}\Phi _{yz}}{\partial y\partial z}}$		$\sigma _{xy}=-{\frac {\partial ^{2}\Phi _{xy}}{\partial z\partial z}}-{\frac {\partial ^{2}\Phi _{zz}}{\partial x\partial y}}+{\frac {\partial ^{2}\Phi _{yz}}{\partial x\partial z}}+{\frac {\partial ^{2}\Phi _{zx}}{\partial y\partial z}}$
$\sigma _{y}={\frac {\partial ^{2}\Phi _{xx}}{\partial z\partial z}}+{\frac {\partial ^{2}\Phi _{zz}}{\partial x\partial x}}-2{\frac {\partial ^{2}\Phi _{zx}}{\partial z\partial x}}$		$\sigma _{yz}=-{\frac {\partial ^{2}\Phi _{yz}}{\partial x\partial x}}-{\frac {\partial ^{2}\Phi _{xx}}{\partial y\partial z}}+{\frac {\partial ^{2}\Phi _{zx}}{\partial y\partial x}}+{\frac {\partial ^{2}\Phi _{xy}}{\partial z\partial x}}$
$\sigma _{z}={\frac {\partial ^{2}\Phi _{yy}}{\partial x\partial x}}+{\frac {\partial ^{2}\Phi _{xx}}{\partial y\partial y}}-2{\frac {\partial ^{2}\Phi _{xy}}{\partial x\partial y}}$		$\sigma _{zx}=-{\frac {\partial ^{2}\Phi _{zx}}{\partial y\partial y}}-{\frac {\partial ^{2}\Phi _{yy}}{\partial z\partial x}}+{\frac {\partial ^{2}\Phi _{xy}}{\partial z\partial y}}+{\frac {\partial ^{2}\Phi _{yz}}{\partial x\partial y}}$

where $\Phi _{mn}$ is an arbitrary second-rank tensor field that is at least twice differentiable, and is known as the Beltrami stress tensor.^[1] Its components are known as Beltrami stress functions. $\varepsilon$ is the Levi-Civita pseudotensor, with all values equal to zero except those in which the indices are not repeated. For a set of non-repeating indices the component value will be +1 for even permutations of the indices, and -1 for odd permutations. And $\nabla$ is the Nabla operator. For the Beltrami stress tensor to satisfy the Beltrami-Michell compatibility equations in addition to the equilibrium equations, it is further required that $\Phi _{mn}$ is at least four times continuously differentiable.

Maxwell stress functions

The Maxwell stress functions are defined by assuming that the Beltrami stress tensor $\Phi _{mn}$ is restricted to be of the form.^[2]

\Phi _{ij}={\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}}

The stress tensor which automatically obeys the equilibrium equation may now be written as:^[2]

$\sigma _{x}={\frac {\partial ^{2}B}{\partial z^{2}}}+{\frac {\partial ^{2}C}{\partial y^{2}}}$		$\sigma _{yz}=-{\frac {\partial ^{2}A}{\partial y\partial z}}$
$\sigma _{y}={\frac {\partial ^{2}C}{\partial x^{2}}}+{\frac {\partial ^{2}A}{\partial z^{2}}}$		$\sigma _{zx}=-{\frac {\partial ^{2}B}{\partial z\partial x}}$
$\sigma _{z}={\frac {\partial ^{2}A}{\partial y^{2}}}+{\frac {\partial ^{2}B}{\partial x^{2}}}$		$\sigma _{xy}=-{\frac {\partial ^{2}C}{\partial x\partial y}}$

The solution to the elastostatic problem now consists of finding the three stress functions which give a stress tensor which obeys the Beltrami–Michell compatibility equations for stress. Substituting the expressions for the stress into the Beltrami-Michell equations yields the expression of the elastostatic problem in terms of the stress functions:^[3]

\nabla ^{4}A+\nabla ^{4}B+\nabla ^{4}C=3\left({\frac {\partial ^{2}A}{\partial x^{2}}}+{\frac {\partial ^{2}B}{\partial y^{2}}}+{\frac {\partial ^{2}C}{\partial z^{2}}}\right)/(2-\nu ),

These must also yield a stress tensor which obeys the specified boundary conditions.

Airy stress function

The Airy stress function is a special case of the Maxwell stress functions, in which it is assumed that A=B=0 and C is a function of x and y only.^[2] This stress function can therefore be used only for two-dimensional problems. In the elasticity literature, the stress function $C$ is usually represented by $\varphi$ and the stresses are expressed as

\sigma _{x}={\frac {\partial ^{2}\varphi }{\partial y^{2}}}~;~~\sigma _{y}={\frac {\partial ^{2}\varphi }{\partial x^{2}}}~;~~\sigma _{xy}=-{\frac {\partial ^{2}\varphi }{\partial x\partial y}}-(f_{x}y+f_{y}x)

Where $f_{x}$ and $f_{y}$ are values of body forces in relevant direction.

In polar coordinates the expressions are:

\sigma _{rr}={\frac {1}{r}}{\frac {\partial \varphi }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}\varphi }{\partial \theta ^{2}}}~;~~\sigma _{\theta \theta }={\frac {\partial ^{2}\varphi }{\partial r^{2}}}~;~~\sigma _{r\theta }=\sigma _{\theta r}=-{\frac {\partial }{\partial r}}\left({\frac {1}{r}}{\frac {\partial \varphi }{\partial \theta }}\right)

Morera stress functions

The Morera stress functions are defined by assuming that the Beltrami stress tensor $\Phi _{mn}$ tensor is restricted to be of the form ^[2]

\Phi _{ij}={\begin{bmatrix}0&C&B\\C&0&A\\B&A&0\end{bmatrix}}

The solution to the elastostatic problem now consists of finding the three stress functions which give a stress tensor which obeys the Beltrami-Michell compatibility equations. Substituting the expressions for the stress into the Beltrami-Michell equations yields the expression of the elastostatic problem in terms of the stress functions:^[4]

$\sigma _{x}=-2{\frac {\partial ^{2}A}{\partial y\partial z}}$		$\sigma _{yz}=-{\frac {\partial ^{2}A}{\partial x^{2}}}+{\frac {\partial ^{2}B}{\partial y\partial x}}+{\frac {\partial ^{2}C}{\partial z\partial x}}$
$\sigma _{y}=-2{\frac {\partial ^{2}B}{\partial z\partial x}}$		$\sigma _{zx}=-{\frac {\partial ^{2}B}{\partial y^{2}}}+{\frac {\partial ^{2}C}{\partial z\partial y}}+{\frac {\partial ^{2}A}{\partial x\partial y}}$
$\sigma _{z}=-2{\frac {\partial ^{2}C}{\partial x\partial y}}$		$\sigma _{xy}=-{\frac {\partial ^{2}C}{\partial z^{2}}}+{\frac {\partial ^{2}A}{\partial x\partial z}}+{\frac {\partial ^{2}B}{\partial y\partial z}}$

Prandtl stress function

The Prandtl stress function is a special case of the Morera stress functions, in which it is assumed that A=B=0 and C is a function of x and y only.^[4]

Notes

1 2 Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Elsevier Science & Technology Books. p. 363. ISBN 978-0-12-605811-6.
1 2 3 4 Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 364
↑ Knops (1958) p327
1 2 Sadd, M. H. (2005) Elasticity: Theory, Applications, and Numerics, Elsevier, p. 365

Related Research Articles

Laplaces equation Second order partial differential equation

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as

Navier–Stokes equations Equations describing the motion of viscous fluid substances

In physics, the Navier–Stokes equations are certain partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of the gradient of a scalar always being equal to zero.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols $,, or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δ f (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (p) .$

In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller than any relevant dimension of the body; so that its geometry and the constitutive properties of the material at each point of space can be assumed to be unchanged by the deformation.

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas.

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group $O(p, q)$ . As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold. The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols.

In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami.

In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of thin structures that react elastically to external forces.

The Newman–Penrose (NP) formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— $in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.$

The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as its application and formulation for different families of fluids.

The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.

In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions. The typical thickness to width ratio of a plate structure is less than 0.1. A plate theory takes advantage of this disparity in length scale to reduce the full three-dimensional solid mechanics problem to a two-dimensional problem. The aim of plate theory is to calculate the deformation and stresses in a plate subjected to loads.

Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load.

The Uflyand-Mindlin theory of vibrating plates is an extension of Kirchhoff–Love plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1948 by Yakov Solomonovich Uflyand (1916-1991) and in 1951 by Raymond Mindlin with Mindlin making reference to Uflyand's work. Hence, this theory has to be referred to us Uflyand-Mindlin plate theory, as is done in the handbook by Elishakoff, and in papers by Andronov, Elishakoff, Hache and Challamel, Loktev, Rossikhin and Shitikova and Wojnar. In 1994, Elishakoff suggested to neglect the fourth-order time derivative in Uflyand-Mindlin equations. A similar, but not identical, theory in static setting, had been proposed earlier by Eric Reissner in 1945. Both theories are intended for thick plates in which the normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface. The Uflyand-Mindlin theory is used to calculate the deformations and stresses in a plate whose thickness is of the order of one tenth the planar dimensions while the Kirchhoff–Love theory is applicable to thinner plates.

In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference. Many constitutive equations are designed in the form of a relation between a stress-rate and a strain-rate. The mechanical response of a material should not depend on the frame of reference. In other words, material constitutive equations should be frame-indifferent (objective). If the stress and strain measures are material quantities then objectivity is automatically satisfied. However, if the quantities are spatial, then the objectivity of the stress-rate is not guaranteed even if the strain-rate is objective.

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

References

Sadd, Martin H. (2005). Elasticity - Theory, applications and numerics. New York: Elsevier Butterworth-Heinemann. ISBN 0-12-605811-3. OCLC 162576656.
Knops, R. J. (1958). "On the Variation of Poisson's Ratio in the Solution of Elastic Problems". The Quarterly Journal of Mechanics and Applied Mathematics. Oxford University Press. 11 (3): 326–350. doi:10.1093/qjmam/11.3.326.