Linear elasticity: Difference between revisions

Revision as of 16:45, 28 December 2024

Latest revision as of 18:13, 9 July 2025

(15 intermediate revisions by 9 users not shown)

Line 3:

{{Continuum mechanics|solid}}

'''Linear elasticity''' is a mathematical model as toof how solid objects [[deformation (physics)|deform]] and become internally [[stress (mechanics)|stressed]] by prescribed loading conditions. It is a simplification of the more general [[Finite strain theory|nonlinear theory of elasticity]] and a branch of [[continuum mechanics]].

The fundamental "linearizing" assumptions of linear elasticity are: [[Infinitesimal strain theory|infinitesimal strains]] or— meaning, "small" deformations (or strains)— and linear relationships between the components of stress and strain. In— additionhence the "linear" in its name. Linear elasticity is valid only for stress states that do not produce [[Yield (engineering)|yielding]]. Its assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in [[structural analysis]] and engineering design, often with the aid of [[finite element analysis]].

These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in [[structural analysis]] and engineering design, often with the aid of [[finite element analysis]].

==Mathematical formulation==

Line 44 ⟶ 42:

* [[Constitutive equations]]. The equation for Hooke's law is: <math display="block"> \sigma_{ij} = C_{ijkl} \, \varepsilon_{kl} </math> where <math>C_{ijkl}</math> is the stiffness tensor. These are 6 independent equations relating stresses and strains. The requirement of the symmetry of the stress and strain tensors lead to equality of many of the elastic constants, reducing the number of different elements to 21<ref>{{cite journal |last1=Belen'kii |last2= Salaev|date= 1988|title= Deformation effects in layer crystals|journal= Uspekhi Fizicheskikh Nauk|volume= 155|issue= 5|pages= 89–127|doi= 10.3367/UFNr.0155.198805c.0089|doi-access= free}}</ref> <math> C_{ijkl} = C_{klij} = C_{jikl} = C_{ijlk}</math>.

An elastostatic boundary value problem for an isotropic-homogeneous media is a system of 15 independent equations and equal number of unknowns (3 equilibrium equations, 6 strain-displacement equations, and 6 constitutive equations). SpecifyingBy specifying the boundary conditions, the boundary value problem is completelyfully defined. To solve the system two approaches can be taken according to boundary conditions of the boundary value problem: a '''displacement formulation''', and a '''stress formulation'''.

===Cylindrical coordinate form===

Line 62 ⟶ 60:

\varepsilon_{zr} = \cfrac{1}{2} \left(\cfrac{\partial u_r}{\partial z} + \cfrac{\partial u_z}{\partial r}\right)

\end{align}</math>

and the constitutive relations are the same as in Cartesian coordinates, except that the indices <math>1</math>,<math>2</math>,<math>3</math> now stand for <math>r</math>,<math>\theta</math>,<math>z</math>, respectively.

=== Spherical coordinate form ===

Line 83 ⟶ 81:

== (An)isotropic (in)homogeneous media ==

In [[Hooke's Lawlaw#Isotropic materials|isotropic]] media, the stiffness tensor gives the relationship between the stresses (resulting internal stresses) and the strains (resulting deformations). For an isotropic medium, the stiffness tensor has no preferred direction: an applied force will give the same displacements (relative to the direction of the force) no matter the direction in which the force is applied. In the isotropic case, the stiffness tensor may be written:{{citation needed|date=June 2012}} <math display="block"> C_{ijkl}

= K \, \delta_{ij}\, \delta_{kl}

+ \mu\, (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}- \tfrac{2}{3}\, \delta_{ij}\,\delta_{kl})

Line 89 ⟶ 87:

<math display="block"> \sigma_{ij} = K \delta_{ij} \varepsilon_{kk} + 2\mu \left(\varepsilon_{ij} - \tfrac{1}{3} \delta_{ij} \varepsilon_{kk}\right).</math>

This expression separates the stress into a scalar part on the left which may be associated with a scalar pressure, and a traceless part on the right which may be associated with shear forces. A simpler expression is:<ref>{{Cite book |lastlast1=Aki |firstfirst1=Keiiti |author-linklink1=Keiiti Aki |title=Quantitative seismology |last2=Richards |first2=Paul G. |author-link2=Paul G. richards |publisher=University Science Books |year=2002 |isbn=978-1-891389-63-4 |edition=2 |___location=Mill Valley, California}}</ref><ref>Continuum Mechanics for Engineers 2001 Mase, Eq. 5.12-2</ref>

<math display="block"> \sigma_{ij} = \lambda \delta_{ij} \varepsilon_{kk}+2\mu\varepsilon_{ij}</math>

where λ is [[Lamé's first parameter]]. Since the constitutive equation is simply a set of linear equations, the strain may be expressed as a function of the stresses as:<ref>{{cite book |title= Mechanics of Deformable Bodies |last=Sommerfeld|first=Arnold |author-link=Arnold Sommerfeld|year=1964 |publisher=Academic Press |___location=New York}}</ref>

Line 108 ⟶ 106:

====Displacement formulation====

In this case, the displacements are prescribed everywhere in the boundary. In this approach, the strains and stresses are eliminated from the formulation, leaving the displacements as the unknowns to be solved for in the governing equations.

First, the strain-displacement equations are substituted into the constitutive equations (Hooke's Lawlaw), eliminating the strains as unknowns:

<math display="block">\sigma_{ij} = \lambda \delta_{ij} \varepsilon_{kk}+2\mu\varepsilon_{ij}

= \lambda\delta_{ij}u_{k,k}+\mu\left(u_{i,j}+u_{j,i}\right).

Line 192 ⟶ 190:

===== Thomson's solution - point force in an infinite isotropic medium =====

TheThomson's solution or Kelvin's solution is the most important solution of the Navier–Cauchy or elastostatic equation is for that of a force acting at a point in an infinite isotropic medium. This solution was found by [[William Thomson, 1st Baron Kelvin|William Thomson]] (later Lord Kelvin) in 1848 (Thomson 1848). This solution is the analog of [[Coulomb's law]] in [[electrostatics]]. A derivation is given in Landau & Lifshitz.<ref name="LL">{{cite book |title=Theory of Elasticity |edition=3rd|last=Landau |first=L.D. |author-link=Lev Landau |author2=Lifshitz, E. M. |author-link2=Evgeny Lifshitz |year=1986 |publisher=Butterworth Heinemann |___location=Oxford, England |isbn=0-7506-2633-X}}</ref>{{rp|§8}} Defining

<math display="block">a = 1-2\nu</math>

<math display="block">b = 2(1-\nu) = a+1</math>

Line 229 ⟶ 227:

It can be seen that there is a component of the displacement in the direction of the force, which diminishes, as is the case for the potential in electrostatics, as 1/''r'' for large ''r''. There is also an additional ρ-directed component.

======Frequency ___domain Green's function======

===== Boussinesq–Cerruti solution - point force at the origin of an infinite isotropic half-space =====

Another useful solution is that of a point force acting on the surface of an infinite half-space. It was derived by Boussinesq<ref>{{cite book |title=Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques |last=Boussinesq |first=Joseph |author-link=Joseph Boussinesq |year=1885 |publisher=Gauthier-Villars |___location=Paris, France |url=http://name.umdl.umich.edu/ABV5032.0001.001 |archive-date=2024-09-03 |access-date=2007-12-19 |archive-url=https://web.archive.org/web/20240903234441/https://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV5032.0001.001 |url-status=live }}</ref> for the normal force and Cerruti for the tangential force and a derivation is given in Landau & Lifshitz.<ref name="LL" />{{rp|§8}} In this case, the solution is again written as a Green's tensor which goes to zero at infinity, and the component of the stress tensor normal to the surface vanishes. This solution may be written in Cartesian coordinates as [recall: <math>a=(1-2\nu)</math> and <math>b=2(1-\nu)</math>, <math>\nu</math> = Poisson's ratio]:

Rewrite the Navier-Cauchy equations in component form<ref>{{cite web |last=Bouchbinder |first=Eran |title= Linear Elasticity I (Non‑Equilibrium Continuum Physics)|url=https://www.weizmann.ac.il/chembiophys/bouchbinder/sites/chemphys.bouchbinder/files/uploads/Courses/2021/TAs/TA4-Linear_elasticity-I.pdf |website=Weizmann Institute of Science |publisher=Department of Chemical and Biological Physics |date=5 May 2021 |access-date=20 May 2025}}</ref>

<math display="block">G_{ik} = \frac{1}{4\pi\mu} \begin{bmatrix}

<math display="block">(\lambda + \mu)\partial_i \partial_j u_j +\mu\partial_j\partial_j u_i =-F_i</math>

Convert this to frequency ___domain, where derivative <math> \partial_i</math> maps to <math>\sqrt{-1}q_i</math>, where <math>q</math> is the wave vector

<math display="block">(\lambda + \mu)q_i q_j u_j +\mu|q|^2u_i =F_i</math>

Spatial frequency ___domain force to displacement Green's function is the inverse of the above

<math>G_{ij}(q) = \frac{1}{\mu}\bigg[\frac{\delta_{ij}}{|q|^2} -\frac{1}{b}\frac{q_iq_j}{|q|^4}\bigg]</math>

The stress to strain Green's function <math>\Gamma</math> is<ref>{{cite journal | last1=Moulinec | first1=H. | last2=Suquet | first2=P. | title=A fast numerical method for computing the linear and nonlinear mechanical properties of composites | journal=Comptes Rendus de l'Académie des Sciences, Série II | volume=318 | pages=1417–1423 | year=1994 | url=https://lma-software-craft.cnrs.fr/wp-content/uploads/2020/11/CRAS_Moulinec_Suquet_1994.pdf | access-date=2025-05-17}}</ref>

<math>\Gamma_{khij} = \frac{1}{4\mu |q|^2}(\delta_{ki}q_hq_j+\delta_{hi}q_kq_j+\delta_{kj}q_hq_i+\delta_{hj}q_kq_i) -\frac{\lambda+\mu}{\mu(\lambda+2\mu)}\frac{q_iq_jq_kq_h}{|q|^4}</math>

where <math>\epsilon_{kh} = \Gamma_{khij}\sigma_{ij}</math>

===== Boussinesq–Cerruti solution - point force at the origin of an infinite isotropic half-space =====

Another useful solution is that of a point force acting on the surface of an infinite half-space.<ref name="tribonet" /> It was derived by Boussinesq<ref>{{cite book |title=Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques |last=Boussinesq |first=Joseph |author-link=Joseph Boussinesq |year=1885 |publisher=Gauthier-Villars |___location=Paris, France |url=http://name.umdl.umich.edu/ABV5032.0001.001 |archive-date=2024-09-03 |access-date=2007-12-19 |archive-url=https://web.archive.org/web/20240903234441/https://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV5032.0001.001 |url-status=live }}</ref> for the normal force and Cerruti for the tangential force and a derivation is given in Landau & Lifshitz.<ref name="LL" />{{rp|§8}} In this case, the solution is again written as a Green's tensor which goes to zero at infinity, and the component of the stress tensor normal to the surface vanishes. This solution may be written in Cartesian coordinates as [recall: <math>a=(1-2\nu)</math> and <math>b=2(1-\nu)</math>, <math>\nu</math> = Poisson's ratio]:

<math display="block">G_{ik} = \frac{1}{4\pi\mu} \begin{bmatrixr}

\frac{b}{r}+\frac{x^2}{r^3}-\frac{ax^2}{r(r+z)^2}-\frac{az}{r(r+z)} &

\begin{bmatrix}

\frac{xy}{r^3}-\frac{axy}{r(r+z)^2}&

\frac{xzb r + z}{r^3 + z}- + \frac{ax(2 r (\nu r + z) + z^2) x^2}{r^2 (r + z)^2}\\ &

\frac{(2 r (\nu r + z) + z^2) x y}{r^2 (r + z)^2} &

\frac{xyx z}{r^32} - \frac{axya x}{r(r + z)^2}& \\

\frac{yx}{(2 r^3} -(\frac{ayxnu r + z) + z^2) y x}{r^2 (r + z)^2} &

\frac{b}{ r} +\frac{y^2 z}{r^3 + z}- + \frac{ay^(2}{ r (\nu r + z) + z^2) y^2}-\frac{az}{r^2 (r + z)^2} &

\frac{yzy z}{r^32} - \frac{aya y}{r(r + z)} \\

\frac{zxz x}{r^32}- + \frac{axa x}{r(r + z)} &

\frac{zyz y}{r^32}- + \frac{aya y}{r(r + z)} &

\frac{b}{r} + \frac{z^2}{r^32}

\end{bmatrix}

</math>

===== Other solutions =====

* Point force inside an infinite isotropic half-space.<ref>{{cite journal |last=Mindlin |first= R. D.|author-link=Raymond D. Mindlin |year=1936|title=Force at a point in the interior of a semi-infinite solid |journal=Physics |volume=7| issue= 5| pages=195–202 |url= http://www.dtic.mil/get-tr-doc/pdf?AD=AD0012375|archive-url= https://web.archive.org/web/20170923074956/http://www.dtic.mil/get-tr-doc/pdf?AD=AD0012375|url-status= dead|archive-date= September 23, 2017|doi=10.1063/1.1745385 |bibcode = 1936Physi...7..195M|url-access=subscription}}</ref>

* Point force on a surface of an isotropic half-space.<ref name="tribonet" />

* Contact of two elastic bodies: the Hertz solution (see [http://www.tribonet.org/cmdownloads/hertz-contact-calculator/ Matlab code]).<ref>{{cite journal |last=Hertz |first= Heinrich|author-link=Heinrich Hertz |year=1882 |title=Contact between solid elastic bodies |journal=Journal für die reine und angewandte Mathematik|volume=92}}</ref> See also the page on [[Contact mechanics]].

Line 275 ⟶ 290:

is the ''acoustic differential operator'', and <math> \delta_{kl}</math> is [[Kronecker delta]].

In [[Hooke's Lawlaw#Isotropic materials|isotropic]] media, the stiffness tensor has the form

<math display="block"> C_{ijkl}

= K \, \delta_{ij}\, \delta_{kl}