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Linear elasticity
Linear elasticity
is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity
Linear elasticity
models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of continuum mechanics. The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations (or strains) and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding. These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity
Linear elasticity
is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis.

Contents

1 Mathematical formulation

1.1 Direct tensor form 1.2 Cartesian coordinate
Cartesian coordinate
form 1.3 Cylindrical coordinate form 1.4 Spherical coordinate form

2 Isotropic homogeneous media

2.1 Elastostatics

2.1.1 Displacement formulation

2.1.1.1 The biharmonic equation

2.1.2 Stress formulation 2.1.3 Solutions for elastostatic cases

2.2 Elastodynamics – the wave equation

3 Anisotropic homogeneous media

3.1 Elastodynamics

3.1.1 Plane waves
Plane waves
and Christoffel equation

4 See also 5 References

Mathematical formulation[edit] Equations governing a linear elastic boundary value problem are based on three tensor partial differential equations for the balance of linear momentum and six infinitesimal strain-displacement relations. The system of differential equations is completed by a set of linear algebraic constitutive relations. Direct tensor form[edit] In direct tensor form that is independent of the choice of coordinate system, these governing equations are:[1]

Equation of motion, which is an expression of Newton's second law:

σ

+

F

= ρ

u

¨

displaystyle boldsymbol nabla cdot boldsymbol sigma +mathbf F =rho ddot mathbf u

Strain-displacement equations:

ε

=

1 2

[

u

+ (

u

)

T

]

displaystyle boldsymbol varepsilon = tfrac 1 2 left[ boldsymbol nabla mathbf u +( boldsymbol nabla mathbf u )^ T right],!

Constitutive equations. For elastic materials, Hooke's law
Hooke's law
represents the material behavior and relates the unknown stresses and strains. The general equation for Hooke's law
Hooke's law
is

σ

=

C

:

ε

,

displaystyle boldsymbol sigma = mathsf C : boldsymbol varepsilon ,

where

σ

displaystyle boldsymbol sigma

is the Cauchy stress tensor,

ε

displaystyle boldsymbol varepsilon

is the infinitesimal strain tensor,

u

displaystyle mathbf u

is the displacement vector,

C

displaystyle mathsf C

is the fourth-order stiffness tensor,

F

displaystyle mathbf F

is the body force per unit volume,

ρ

displaystyle rho

is the mass density,

displaystyle boldsymbol nabla

represents the nabla operator,

( ∙

)

T

displaystyle (bullet )^ T

represents a transpose,

( ∙ )

¨

displaystyle ddot (bullet )

represents the second derivative with respect to time, and

A

:

B

=

A

i j

B

i j

displaystyle mathbf A :mathbf B =A_ ij B_ ij

is the inner product of two second-order tensors (summation over repeated indices is implied). Cartesian coordinate
Cartesian coordinate
form[edit]

Note: the Einstein summation convention of summing on repeated indices is used below.

Expressed in terms of components with respect to a rectangular Cartesian coordinate
Cartesian coordinate
system, the governing equations of linear elasticity are:[1]

Equation of motion:

σ

j i , j

+

F

i

= ρ

t t

u

i

displaystyle sigma _ ji,j +F_ i =rho partial _ tt u_ i ,!

Engineering notation

σ

x

∂ x

+

τ

y x

∂ y

+

τ

z x

∂ z

+

F

x

= ρ

2

u

x

t

2

displaystyle frac partial sigma _ x partial x + frac partial tau _ yx partial y + frac partial tau _ zx partial z +F_ x =rho frac partial ^ 2 u_ x partial t^ 2 ,!

τ

x y

∂ x

+

σ

y

∂ y

+

τ

z y

∂ z

+

F

y

= ρ

2

u

y

t

2

displaystyle frac partial tau _ xy partial x + frac partial sigma _ y partial y + frac partial tau _ zy partial z +F_ y =rho frac partial ^ 2 u_ y partial t^ 2 ,!

τ

x z

∂ x

+

τ

y z

∂ y

+

σ

z

∂ z

+

F

z

= ρ

2

u

z

t

2

displaystyle frac partial tau _ xz partial x + frac partial tau _ yz partial y + frac partial sigma _ z partial z +F_ z =rho frac partial ^ 2 u_ z partial t^ 2 ,!

where the

( ∙ )

, j

displaystyle (bullet ) _ ,j

subscript is a shorthand for

( ∙ )

/

x

j

displaystyle partial (bullet ) /partial x_ j

and

t t

displaystyle partial _ tt

indicates

2

/

t

2

displaystyle partial ^ 2 /partial t^ 2

,

σ

i j

=

σ

j i

displaystyle sigma _ ij =sigma _ ji ,!

is the Cauchy stress tensor,

F

i

displaystyle F_ i ,!

are the body forces,

ρ

displaystyle rho ,!

is the mass density, and

u

i

displaystyle u_ i ,!

is the displacement.

These are 3 independent equations with 6 independent unknowns (stresses).

Strain-displacement equations:

ε

i j

=

1 2

(

u

j , i

+

u

i , j

)

displaystyle varepsilon _ ij = frac 1 2 (u_ j,i +u_ i,j ),!

Engineering notation

ϵ

x

=

u

x

∂ x

displaystyle epsilon _ x = frac partial u_ x partial x ,!

γ

x y

=

u

x

∂ y

+

u

y

∂ x

displaystyle gamma _ xy = frac partial u_ x partial y + frac partial u_ y partial x ,!

ϵ

y

=

u

y

∂ y

displaystyle epsilon _ y = frac partial u_ y partial y ,!

γ

y z

=

u

y

∂ z

+

u

z

∂ y

displaystyle gamma _ yz = frac partial u_ y partial z + frac partial u_ z partial y ,!

ϵ

z

=

u

z

∂ z

displaystyle epsilon _ z = frac partial u_ z partial z ,!

γ

z x

=

u

z

∂ x

+

u

x

∂ z

displaystyle gamma _ zx = frac partial u_ z partial x + frac partial u_ x partial z ,!

where

ε

i j

=

ε

j i

displaystyle varepsilon _ ij =varepsilon _ ji ,!

is the strain. These are 6 independent equations relating strains and displacements with 9 independent unknowns (strains and displacements).

Constitutive equations. The equation for Hooke's law
Hooke's law
is:

σ

i j

=

C

i j k l

ε

k l

displaystyle sigma _ ij =C_ ijkl ,varepsilon _ kl ,!

where

C

i j k l

displaystyle C_ ijkl

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[2]

C

i j k l

=

C

k l i j

=

C

j i k l

=

C

i j l k

displaystyle C_ ijkl =C_ klij =C_ jikl =C_ ijlk

.

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). Specifying the boundary conditions, the boundary value problem is completely 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[edit] In cylindrical coordinates (

r , θ , z

displaystyle r,theta ,z

) the equations of motion are[1]

σ

r r

∂ r

+

1

r

σ

r θ

∂ θ

+

σ

r z

∂ z

+

1

r

(

σ

r r

σ

θ θ

) +

F

r

= ρ  

2

u

r

t

2

σ

r θ

∂ r

+

1

r

σ

θ θ

∂ θ

+

σ

θ z

∂ z

+

2

r

σ

r θ

+

F

θ

= ρ  

2

u

θ

t

2

σ

r z

∂ r

+

1

r

σ

θ z

∂ θ

+

σ

z z

∂ z

+

1

r

σ

r z

+

F

z

= ρ  

2

u

z

t

2

displaystyle begin aligned & frac partial sigma _ rr partial r + cfrac 1 r frac partial sigma _ rtheta partial theta + frac partial sigma _ rz partial z + cfrac 1 r (sigma _ rr -sigma _ theta theta )+F_ r =rho ~ frac partial ^ 2 u_ r partial t^ 2 \& frac partial sigma _ rtheta partial r + cfrac 1 r frac partial sigma _ theta theta partial theta + frac partial sigma _ theta z partial z + cfrac 2 r sigma _ rtheta +F_ theta =rho ~ frac partial ^ 2 u_ theta partial t^ 2 \& frac partial sigma _ rz partial r + cfrac 1 r frac partial sigma _ theta z partial theta + frac partial sigma _ zz partial z + cfrac 1 r sigma _ rz +F_ z =rho ~ frac partial ^ 2 u_ z partial t^ 2 end aligned

The strain-displacement relations are

ε

r r

=

u

r

∂ r

  ;    

ε

θ θ

=

1

r

(

u

θ

∂ θ

+

u

r

)

  ;    

ε

z z

=

u

z

∂ z

ε

r θ

=

1

2

(

1

r

u

r

∂ θ

+

u

θ

∂ r

u

θ

r

)

  ;    

ε

θ z

=

1

2

(

u

θ

∂ z

+

1

r

u

z

∂ θ

)

  ;    

ε

z r

=

1

2

(

u

r

∂ z

+

u

z

∂ r

)

displaystyle begin aligned varepsilon _ rr &= cfrac partial u_ r partial r ~;~~varepsilon _ theta theta = cfrac 1 r left( cfrac partial u_ theta partial theta +u_ r right)~;~~varepsilon _ zz = cfrac partial u_ z partial z \varepsilon _ rtheta &= cfrac 1 2 left( cfrac 1 r cfrac partial u_ r partial theta + cfrac partial u_ theta partial r - cfrac u_ theta r right)~;~~varepsilon _ theta z = cfrac 1 2 left( cfrac partial u_ theta partial z + cfrac 1 r cfrac partial u_ z partial theta right)~;~~varepsilon _ zr = cfrac 1 2 left( cfrac partial u_ r partial z + cfrac partial u_ z partial r right)end aligned

and the constitutive relations are the same as in Cartesian coordinates, except that the indices

1

displaystyle 1

,

2

displaystyle 2

,

3

displaystyle 3

now stand for

r

displaystyle r

,

θ

displaystyle theta

,

z

displaystyle z

, respectively. Spherical coordinate form[edit] In spherical coordinates (

r , θ , ϕ

displaystyle r,theta ,phi

) the equations of motion are[1]

σ

r r

∂ r

+

1

r

σ

r θ

∂ θ

+

1

r sin ⁡ θ

σ

r ϕ

∂ ϕ

+

1

r

( 2

σ

r r

σ

θ θ

σ

ϕ ϕ

+

σ

r θ

cot ⁡ θ ) +

F

r

= ρ  

2

u

r

t

2

σ

r θ

∂ r

+

1

r

σ

θ θ

∂ θ

+

1

r sin ⁡ θ

σ

θ ϕ

∂ ϕ

+

1

r

[ (

σ

θ θ

σ

ϕ ϕ

) cot ⁡ θ + 3

σ

r θ

] +

F

θ

= ρ  

2

u

θ

t

2

σ

r ϕ

∂ r

+

1

r

σ

θ ϕ

∂ θ

+

1

r sin ⁡ θ

σ

ϕ ϕ

∂ ϕ

+

1

r

( 2

σ

θ ϕ

cot ⁡ θ + 3

σ

r ϕ

) +

F

ϕ

= ρ  

2

u

ϕ

t

2

displaystyle begin aligned & frac partial sigma _ rr partial r + cfrac 1 r frac partial sigma _ rtheta partial theta + cfrac 1 rsin theta frac partial sigma _ rphi partial phi + cfrac 1 r (2sigma _ rr -sigma _ theta theta -sigma _ phi phi +sigma _ rtheta cot theta )+F_ r =rho ~ frac partial ^ 2 u_ r partial t^ 2 \& frac partial sigma _ rtheta partial r + cfrac 1 r frac partial sigma _ theta theta partial theta + cfrac 1 rsin theta frac partial sigma _ theta phi partial phi + cfrac 1 r [(sigma _ theta theta -sigma _ phi phi )cot theta +3sigma _ rtheta ]+F_ theta =rho ~ frac partial ^ 2 u_ theta partial t^ 2 \& frac partial sigma _ rphi partial r + cfrac 1 r frac partial sigma _ theta phi partial theta + cfrac 1 rsin theta frac partial sigma _ phi phi partial phi + cfrac 1 r (2sigma _ theta phi cot theta +3sigma _ rphi )+F_ phi =rho ~ frac partial ^ 2 u_ phi partial t^ 2 end aligned

Spherical coordinates (r, θ, φ) as commonly used in physics: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.

The strain tensor in spherical coordinates is

ε

r r

=

u

r

∂ r

ε

θ θ

=

1 r

(

u

θ

∂ θ

+

u

r

)

ε

ϕ ϕ

=

1

r sin ⁡ θ

(

u

ϕ

∂ ϕ

+

u

r

sin ⁡ θ +

u

θ

cos ⁡ θ

)

ε

r θ

=

1 2

(

1 r

u

r

∂ θ

+

u

θ

∂ r

u

θ

r

)

ε

θ ϕ

=

1

2 r

[

1

sin ⁡ θ

u

θ

∂ ϕ

+

(

u

ϕ

∂ θ

u

ϕ

cot ⁡ θ

)

]

ε

r ϕ

=

1 2

(

1

r sin ⁡ θ

u

r

∂ ϕ

+

u

ϕ

∂ r

u

ϕ

r

)

.

displaystyle begin aligned varepsilon _ rr &= frac partial u_ r partial r \varepsilon _ theta theta &= frac 1 r left( frac partial u_ theta partial theta +u_ r right)\varepsilon _ phi phi &= frac 1 rsin theta left( frac partial u_ phi partial phi +u_ r sin theta +u_ theta cos theta right)\varepsilon _ rtheta &= frac 1 2 left( frac 1 r frac partial u_ r partial theta + frac partial u_ theta partial r - frac u_ theta r right)\varepsilon _ theta phi &= frac 1 2r left[ frac 1 sin theta frac partial u_ theta partial phi +left( frac partial u_ phi partial theta -u_ phi cot theta right)right]\varepsilon _ rphi &= frac 1 2 left( frac 1 rsin theta frac partial u_ r partial phi + frac partial u_ phi partial r - frac u_ phi r right).end aligned

Isotropic homogeneous media[edit] In 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:

C

i j k l

= K

δ

i j

δ

k l

+ μ

(

δ

i k

δ

j l

+

δ

i l

δ

j k

2 3

δ

i j

δ

k l

)

displaystyle C_ ijkl =K,delta _ ij ,delta _ kl +mu ,(delta _ ik delta _ jl +delta _ il delta _ jk -textstyle frac 2 3 ,delta _ ij ,delta _ kl ),!

[citation needed]

where

δ

i j

displaystyle delta _ ij ,!

is the Kronecker delta, K  is the bulk modulus (or incompressibility), and

μ

displaystyle mu ,!

is the shear modulus (or rigidity), two elastic moduli. If the medium is inhomogeneous, the isotropic model is sensible if either the medium is piecewise-constant or weakly inhomogeneous; in the strongly inhomogeneous smooth model, anisotropy has to be accounted for. If the medium is homogeneous, then the elastic moduli will be independent of the position in the medium. The constitutive equation may now be written as:

σ

i j

= K

δ

i j

ε

k k

+ 2 μ (

ε

i j

1 3

δ

i j

ε

k k

) .

displaystyle sigma _ ij =Kdelta _ ij varepsilon _ kk +2mu (varepsilon _ ij -textstyle frac 1 3 delta _ ij varepsilon _ kk ).,!

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:[3]

σ

i j

= λ

δ

i j

ε

k k

+ 2 μ

ε

i j

displaystyle sigma _ ij =lambda delta _ ij varepsilon _ kk +2mu varepsilon _ ij ,!

[4]

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:[5]

ε

i j

=

1

9 K

δ

i j

σ

k k

+

1

2 μ

(

σ

i j

1 3

δ

i j

σ

k k

)

displaystyle varepsilon _ ij = frac 1 9K delta _ ij sigma _ kk + frac 1 2mu left(sigma _ ij -textstyle frac 1 3 delta _ ij sigma _ kk right),!

which is again, a scalar part on the left and a traceless shear part on the right. More simply:

ε

i j

=

1

2 μ

σ

i j

ν E

δ

i j

σ

k k

=

1 E

[ ( 1 + ν )

σ

i j

− ν

δ

i j

σ

k k

]

displaystyle varepsilon _ ij = frac 1 2mu sigma _ ij - frac nu E delta _ ij sigma _ kk = frac 1 E [(1+nu )sigma _ ij -nu delta _ ij sigma _ kk ],!

where ν is Poisson's ratio
Poisson's ratio
and E  is Young's modulus. Elastostatics[edit] Elastostatics is the study of linear elasticity under the conditions of equilibrium, in which all forces on the elastic body sum to zero, and the displacements are not a function of time. The equilibrium equations are then

σ

j i , j

+

F

i

= 0.

displaystyle sigma _ ji,j +F_ i =0.,!

Engineering notation (tau is shear stress)

σ

x

∂ x

+

τ

y x

∂ y

+

τ

z x

∂ z

+

F

x

= 0

displaystyle frac partial sigma _ x partial x + frac partial tau _ yx partial y + frac partial tau _ zx partial z +F_ x =0,!

τ

x y

∂ x

+

σ

y

∂ y

+

τ

z y

∂ z

+

F

y

= 0

displaystyle frac partial tau _ xy partial x + frac partial sigma _ y partial y + frac partial tau _ zy partial z +F_ y =0,!

τ

x z

∂ x

+

τ

y z

∂ y

+

σ

z

∂ z

+

F

z

= 0

displaystyle frac partial tau _ xz partial x + frac partial tau _ yz partial y + frac partial sigma _ z partial z +F_ z =0,!

This section will discuss only the isotropic homogeneous case. Displacement formulation[edit] 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 Law), eliminating the strains as unknowns:

σ

i j

= λ

δ

i j

ε

k k

+ 2 μ

ε

i j

= λ

δ

i j

u

k , k

+ μ

(

u

i , j

+

u

j , i

)

.

displaystyle begin aligned sigma _ ij &=lambda delta _ ij varepsilon _ kk +2mu varepsilon _ ij \&=lambda delta _ ij u_ k,k +mu left(u_ i,j +u_ j,i right).\end aligned ,!

Differentiating (assuming

λ

displaystyle lambda ,!

and

μ

displaystyle mu ,!

are spatially uniform) yields:

σ

i j , j

= λ

u

k , k i

+ μ

(

u

i , j j

+

u

j , i j

)

.

displaystyle sigma _ ij,j =lambda u_ k,ki +mu left(u_ i,jj +u_ j,ij right).,!

Substituting into the equilibrium equation yields:

λ

u

k , k i

+ μ

(

u

i , j j

+

u

j , i j

)

+

F

i

= 0

displaystyle lambda u_ k,ki +mu left(u_ i,jj +u_ j,ij right)+F_ i =0,!

or (replacing double (dummy) (=summation) indices k,k by j,j and interchanging indices, ij to, ji after the, by virtue of Schwarz' theorem)

μ

u

i , j j

+ ( μ + λ )

u

j , j i

+

F

i

= 0

displaystyle mu u_ i,jj +(mu +lambda )u_ j,ji +F_ i =0,!

where

λ

displaystyle lambda ,!

and

μ

displaystyle mu ,!

are Lamé parameters. In this way, the only unknowns left are the displacements, hence the name for this formulation. The governing equations obtained in this manner are called Navier-Cauchy equations or, alternatively, the elastostatic equations.

Derivation of Navier-Cauchy equations in Engineering notation

First, the

x

displaystyle x,!

-direction will be considered. Substituting the strain-displacement equations into the equilibrium equation in the

x

displaystyle x,!

-direction we have

σ

x

= 2 μ

ε

x

+ λ (

ε

x

+

ε

y

+

ε

z

) = 2 μ

u

x

∂ x

+ λ

(

u

x

∂ x

+

u

y

∂ y

+

u

z

∂ z

)

displaystyle sigma _ x =2mu varepsilon _ x +lambda (varepsilon _ x +varepsilon _ y +varepsilon _ z )=2mu frac partial u_ x partial x +lambda left( frac partial u_ x partial x + frac partial u_ y partial y + frac partial u_ z partial z right),!

τ

x y

= μ

γ

x y

= μ

(

u

x

∂ y

+

u

y

∂ x

)

displaystyle tau _ xy =mu gamma _ xy =mu left( frac partial u_ x partial y + frac partial u_ y partial x right),!

τ

x z

= μ

γ

z x

= μ

(

u

z

∂ x

+

u

x

∂ z

)

displaystyle tau _ xz =mu gamma _ zx =mu left( frac partial u_ z partial x + frac partial u_ x partial z right),!

Then substituting these equations into the equilibrium equation in the

x

displaystyle x,!

-direction we have

σ

x

∂ x

+

τ

y x

∂ y

+

τ

z x

∂ z

+

F

x

= 0

displaystyle frac partial sigma _ x partial x + frac partial tau _ yx partial y + frac partial tau _ zx partial z +F_ x =0,!

∂ x

(

2 μ

u

x

∂ x

+ λ

(

u

x

∂ x

+

u

y

∂ y

+

u

z

∂ z

)

)

+ μ

∂ y

(

u

x

∂ y

+

u

y

∂ x

)

+ μ

∂ z

(

u

z

∂ x

+

u

x

∂ z

)

+

F

x

= 0

displaystyle frac partial partial x left(2mu frac partial u_ x partial x +lambda left( frac partial u_ x partial x + frac partial u_ y partial y + frac partial u_ z partial z right)right)+mu frac partial partial y left( frac partial u_ x partial y + frac partial u_ y partial x right)+mu frac partial partial z left( frac partial u_ z partial x + frac partial u_ x partial z right)+F_ x =0,!

Using the assumption that

μ

displaystyle mu

and

λ

displaystyle lambda

are constant we can rearrange and get:

(

λ + μ

)

∂ x

(

u

x

∂ x

+

u

y

∂ y

+

u

z

∂ z

)

+ μ

(

2

u

x

x

2

+

2

u

x

y

2

+

2

u

x

z

2

)

+

F

x

= 0

displaystyle left(lambda +mu right) frac partial partial x left( frac partial u_ x partial x + frac partial u_ y partial y + frac partial u_ z partial z right)+mu left( frac partial ^ 2 u_ x partial x^ 2 + frac partial ^ 2 u_ x partial y^ 2 + frac partial ^ 2 u_ x partial z^ 2 right)+F_ x =0,!

Following the same procedure for the

y

displaystyle y,!

-direction and

z

displaystyle z,!

-direction we have

(

λ + μ

)

∂ y

(

u

x

∂ x

+

u

y

∂ y

+

u

z

∂ z

)

+ μ

(

2

u

y

x

2

+

2

u

y

y

2

+

2

u

y

z

2

)

+

F

y

= 0

displaystyle left(lambda +mu right) frac partial partial y left( frac partial u_ x partial x + frac partial u_ y partial y + frac partial u_ z partial z right)+mu left( frac partial ^ 2 u_ y partial x^ 2 + frac partial ^ 2 u_ y partial y^ 2 + frac partial ^ 2 u_ y partial z^ 2 right)+F_ y =0,!

(

λ + μ

)

∂ z

(

u

x

∂ x

+

u

y

∂ y

+

u

z

∂ z

)

+ μ

(

2

u

z

x

2

+

2

u

z

y

2

+

2

u

z

z

2

)

+

F

z

= 0

displaystyle left(lambda +mu right) frac partial partial z left( frac partial u_ x partial x + frac partial u_ y partial y + frac partial u_ z partial z right)+mu left( frac partial ^ 2 u_ z partial x^ 2 + frac partial ^ 2 u_ z partial y^ 2 + frac partial ^ 2 u_ z partial z^ 2 right)+F_ z =0,!

These last 3 equations are the Navier-Cauchy equations, which can be also expressed in vector notation as

( λ + μ ) ∇ ( ∇ ⋅

u

) + μ

2

u

+

F

= 0

displaystyle (lambda +mu )nabla (nabla cdot mathbf u )+mu nabla ^ 2 mathbf u +mathbf F =0,!

Once the displacement field has been calculated, the displacements can be replaced into the strain-displacement equations to solve for strains, which later are used in the constitutive equations to solve for stresses. The biharmonic equation[edit] The elastostatic equation may be written:

(

α

2

β

2

)

u

j , i j

+

β

2

u

i , m m

= −

F

i

.

displaystyle (alpha ^ 2 -beta ^ 2 )u_ j,ij +beta ^ 2 u_ i,mm =-F_ i .,!

Taking the divergence of both sides of the elastostatic equation and assuming the body forces has zero divergence (homogeneous in domain) (

F

i , i

= 0

displaystyle F_ i,i =0,!

) we have

(

α

2

β

2

)

u

j , i i j

+

β

2

u

i , i m m

= 0.

displaystyle (alpha ^ 2 -beta ^ 2 )u_ j,iij +beta ^ 2 u_ i,imm =0.,!

Noting that summed indices need not match, and that the partial derivatives commute, the two differential terms are seen to be the same and we have:

α

2

u

j , i i j

= 0

displaystyle alpha ^ 2 u_ j,iij =0,!

from which we conclude that:

u

j , i i j

= 0.

displaystyle u_ j,iij =0.,!

Taking the Laplacian of both sides of the elastostatic equation, and assuming in addition

F

i , k k

= 0

displaystyle F_ i,kk =0,!

, we have

(

α

2

β

2

)

u

j , k k i j

+

β

2

u

i , k k m m

= 0.

displaystyle (alpha ^ 2 -beta ^ 2 )u_ j,kkij +beta ^ 2 u_ i,kkmm =0.,!

From the divergence equation, the first term on the left is zero (Note: again, the summed indices need not match) and we have:

β

2

u

i , k k m m

= 0

displaystyle beta ^ 2 u_ i,kkmm =0,!

from which we conclude that:

u

i , k k m m

= 0

displaystyle u_ i,kkmm =0,!

or, in coordinate free notation

4

u

= 0

displaystyle nabla ^ 4 mathbf u =0,!

which is just the biharmonic equation in

u

displaystyle mathbf u ,!

. Stress formulation[edit] In this case, the surface tractions are prescribed everywhere on the surface boundary. In this approach, the strains and displacements are eliminated leaving the stresses as the unknowns to be solved for in the governing equations. Once the stress field is found, the strains are then found using the constitutive equations. There are six independent components of the stress tensor which need to be determined, yet in the displacement formulation, there are only three components of the displacement vector which need to be determined. This means that there are some constraints which must be placed upon the stress tensor, to reduce the number of degrees of freedom to three. Using the constitutive equations, these constraints are derived directly from corresponding constraints which must hold for the strain tensor, which also has six independent components. The constraints on the strain tensor are derivable directly from the definition of the strain tensor as a function of the displacement vector field, which means that these constraints introduce no new concepts or information. It is the constraints on the strain tensor that are most easily understood. If the elastic medium is visualized as a set of infinitesimal cubes in the unstrained state, then after the medium is strained, an arbitrary strain tensor must yield a situation in which the distorted cubes still fit together without overlapping. In other words, for a given strain, there must exist a continuous vector field (the displacement) from which that strain tensor can be derived. The constraints on the strain tensor that are required to assure that this is the case were discovered by Saint Venant, and are called the "Saint Venant compatibility equations". These are 81 equations, 6 of which are independent non-trivial equations, which relate the different strain components. These are expressed in index notation as:

ε

i j , k m

+

ε

k m , i j

ε

i k , j m

ε

j m , i k

= 0.

displaystyle varepsilon _ ij,km +varepsilon _ km,ij -varepsilon _ ik,jm -varepsilon _ jm,ik =0.,!

Engineering notation

2

ϵ

x

y

2

+

2

ϵ

y

x

2

= 2

2

ϵ

x y

∂ x ∂ y

displaystyle frac partial ^ 2 epsilon _ x partial y^ 2 + frac partial ^ 2 epsilon _ y partial x^ 2 =2 frac partial ^ 2 epsilon _ xy partial xpartial y ,!

2

ϵ

y

z

2

+

2

ϵ

z

y

2

= 2

2

ϵ

y z

∂ y ∂ z

displaystyle frac partial ^ 2 epsilon _ y partial z^ 2 + frac partial ^ 2 epsilon _ z partial y^ 2 =2 frac partial ^ 2 epsilon _ yz partial ypartial z ,!

2

ϵ

x

z

2

+

2

ϵ

z

x

2

= 2

2

ϵ

z x

∂ z ∂ x

displaystyle frac partial ^ 2 epsilon _ x partial z^ 2 + frac partial ^ 2 epsilon _ z partial x^ 2 =2 frac partial ^ 2 epsilon _ zx partial zpartial x ,!

2

ϵ

x

∂ y ∂ z

=

∂ x

(

ϵ

y z

∂ x

+

ϵ

z x

∂ y

+

ϵ

x y

∂ z

)

displaystyle frac partial ^ 2 epsilon _ x partial ypartial z = frac partial partial x left(- frac partial epsilon _ yz partial x + frac partial epsilon _ zx partial y + frac partial epsilon _ xy partial z right),!

2

ϵ

y

∂ z ∂ x

=

∂ y

(

ϵ

y z

∂ x

ϵ

z x

∂ y

+

ϵ

x y

∂ z

)

displaystyle frac partial ^ 2 epsilon _ y partial zpartial x = frac partial partial y left( frac partial epsilon _ yz partial x - frac partial epsilon _ zx partial y + frac partial epsilon _ xy partial z right),!

2

ϵ

z

∂ x ∂ y

=

∂ z

(

ϵ

y z

∂ x

+

ϵ

z x

∂ y

ϵ

x y

∂ z

)

displaystyle frac partial ^ 2 epsilon _ z partial xpartial y = frac partial partial z left( frac partial epsilon _ yz partial x + frac partial epsilon _ zx partial y - frac partial epsilon _ xy partial z right),!

The strains in this equation are then expressed in terms of the stresses using the constitutive equations, which yields the corresponding constraints on the stress tensor. These constraints on the stress tensor are known as the Beltrami-Michell equations of compatibility:

σ

i j , k k

+

1

1 + ν

σ

k k , i j

+

F

i , j

+

F

j , i

+

ν

1 − ν

δ

i , j

F

k , k

= 0.

displaystyle sigma _ ij,kk + frac 1 1+nu sigma _ kk,ij +F_ i,j +F_ j,i + frac nu 1-nu delta _ i,j F_ k,k =0.,!

In the special situation where the body force is homogeneous, the above equations reduce to

( 1 + ν )

σ

i j , k k

+

σ

k k , i j

= 0.

displaystyle (1+nu )sigma _ ij,kk +sigma _ kk,ij =0.,!

[6]

A necessary, but insufficient, condition for compatibility under this situation is

4

σ

=

0

displaystyle boldsymbol nabla ^ 4 boldsymbol sigma = boldsymbol 0

or

σ

i j , k k ℓ ℓ

= 0

displaystyle sigma _ ij,kkell ell =0

.[1] These constraints, along with the equilibrium equation (or equation of motion for elastodynamics) allow the calculation of the stress tensor field. Once the stress field has been calculated from these equations, the strains can be obtained from the constitutive equations, and the displacement field from the strain-displacement equations. An alternative solution technique is to express the stress tensor in terms of stress functions which automatically yield a solution to the equilibrium equation. The stress functions then obey a single differential equation which corresponds to the compatibility equations. Solutions for elastostatic cases[edit]

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

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 (later Lord Kelvin) in 1848 (Thomson 1848). This solution is the analog of Coulomb's law
Coulomb's law
in electrostatics. A derivation is given in Landau & Lifshitz.[7]:§8 Defining

a = 1 − 2 ν

displaystyle a=1-2nu ,!

b = 2 ( 1 − ν ) = a + 1

displaystyle b=2(1-nu )=a+1,!

where

ν

displaystyle nu ,!

is Poisson's ratio, the solution may be expressed as

u

i

=

G

i k

F

k

displaystyle u_ i =G_ ik F_ k ,!

where

F

k

displaystyle F_ k ,!

is the force vector being applied at the point, and

G

i k

displaystyle G_ ik ,!

is a tensor Green's function which may be written in Cartesian coordinates as:

G

i k

=

1

4 π μ r

[

(

1 −

1

2 b

)

δ

i k

+

1

2 b

x

i

x

k

r

2

]

displaystyle G_ ik = frac 1 4pi mu r left[left(1- frac 1 2b right)delta _ ik + frac 1 2b frac x_ i x_ k r^ 2 right],!

It may be also compactly written as:

G

i k

=

1

4 π μ

[

δ

i k

r

1

2 b

2

r

x

i

x

k

]

displaystyle G_ ik = frac 1 4pi mu left[ frac delta _ ik r - frac 1 2b frac partial ^ 2 r partial x_ i partial x_ k right],!

and it may be explicitly written as:

G

i k

=

1

4 π μ r

[

1 −

1

2 b

+

1

2 b

x

2

r

2

1

2 b

x y

r

2

1

2 b

x z

r

2

1

2 b

y x

r

2

1 −

1

2 b

+

1

2 b

y

2

r

2

1

2 b

y z

r

2

1

2 b

z x

r

2

1

2 b

z y

r

2

1 −

1

2 b

+

1

2 b

z

2

r

2

]

displaystyle G_ ik = frac 1 4pi mu r begin bmatrix 1- frac 1 2b + frac 1 2b frac x^ 2 r^ 2 & frac 1 2b frac xy r^ 2 & frac 1 2b frac xz r^ 2 \ frac 1 2b frac yx r^ 2 &1- frac 1 2b + frac 1 2b frac y^ 2 r^ 2 & frac 1 2b frac yz r^ 2 \ frac 1 2b frac zx r^ 2 & frac 1 2b frac zy r^ 2 &1- frac 1 2b + frac 1 2b frac z^ 2 r^ 2 end bmatrix ,!

In cylindrical coordinates (

ρ , ϕ , z

displaystyle rho ,phi ,z,!

) it may be written as:

G

i k

=

1

4 π μ r

[

1 −

1

2 b

z

2

r

2

0

1

2 b

ρ z

r

2

0

1 −

1

2 b

0

1

2 b

z ρ

r

2

0

1 −

1

2 b

ρ

2

r

2

]

displaystyle G_ ik = frac 1 4pi mu r begin bmatrix 1- frac 1 2b frac z^ 2 r^ 2 &0& frac 1 2b frac rho z r^ 2 \0&1- frac 1 2b &0\ frac 1 2b frac zrho r^ 2 &0&1- frac 1 2b frac rho ^ 2 r^ 2 end bmatrix ,!

where r is total distance to point. It is particularly helpful to write the displacement in cylindrical coordinates for a point force

F

z

displaystyle F_ z ,!

directed along the z-axis. Defining

ρ

^

displaystyle hat mathbf rho ,!

and

z

^

displaystyle hat mathbf z ,!

as unit vectors in the

ρ

displaystyle rho ,!

and

z

displaystyle z,!

directions respectively yields:

u

=

F

z

4 π μ r

[

1

4 ( 1 − ν )

ρ z

r

2

ρ

^

+

(

1 −

1

4 ( 1 − ν )

ρ

2

r

2

)

z

^

]

displaystyle mathbf u = frac F_ z 4pi mu r left[ frac 1 4(1-nu ) , frac rho z r^ 2 hat mathbf rho +left(1- frac 1 4(1-nu ) , frac rho ^ 2 r^ 2 right) hat mathbf z right],!

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.

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[8] for the normal force and Cerruti for the tangential force and a derivation is given in Landau & Lifshitz.[7]:§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
Cartesian coordinates
as [note: a=(1-2ν) and b=2(1-ν), ν== Poissons ratio]:

G

i k

=

1

4 π μ

[

b r

+

x

2

r

3

a

x

2

r ( r + z

)

2

a z

r ( r + z )

x y

r

3

a x y

r ( r + z

)

2

x z

r

3

a x

r ( r + z )

y x

r

3

a y x

r ( r + z

)

2

b r

+

y

2

r

3

a

y

2

r ( r + z

)

2

a z

r ( r + z )

y z

r

3

a y

r ( r + z )

z x

r

3

+

a x

r ( r + z )

z y

r

3

+

a y

r ( r + z )

b r

+

z

2

r

3

]

displaystyle G_ ik = frac 1 4pi mu begin bmatrix frac b r + frac x^ 2 r^ 3 - frac ax^ 2 r(r+z)^ 2 - frac az r(r+z) & frac xy r^ 3 - frac axy r(r+z)^ 2 & frac xz r^ 3 - frac ax r(r+z) \ frac yx r^ 3 - frac ayx r(r+z)^ 2 & frac b r + frac y^ 2 r^ 3 - frac ay^ 2 r(r+z)^ 2 - frac az r(r+z) & frac yz r^ 3 - frac ay r(r+z) \ frac zx r^ 3 + frac ax r(r+z) & frac zy r^ 3 + frac ay r(r+z) & frac b r + frac z^ 2 r^ 3 end bmatrix ,!

Other solutions:

Point force inside an infinite isotropic half-space.[9] Point force on a surface of an isotropic half-space.[6] Contact of two elastic bodies: the Hertz solution (see Matlab code).[10] See also the page on Contact mechanics.

Elastodynamics – the wave equation[edit]

This section needs expansion with: more principles, a brief explanation to each type of wave. You can help by adding to it. (September 2010)

Elastodynamics is the study of elastic waves and involves linear elasticity with variation in time. An elastic wave is a type of mechanical wave that propagates in elastic or viscoelastic materials. The elasticity of the material provides the restoring force of the wave. When they occur in the Earth
Earth
as the result of an earthquake or other disturbance, elastic waves are usually called seismic waves. The wave equation of elastodynamics is simply the equilibrium equation of elastostatics with an additional inertial term:

σ

j i , j

+

F

i

= ρ

u ¨

i

= ρ

t t

u

i

.

displaystyle sigma _ ji,j +F_ i =rho , ddot u _ i =rho ,partial _ tt u_ i .,!

If the material is isotropic and homogeneous (i.e. the stiffness tensor is constant throughout the material), the elastodynamic wave equation has the form:

μ

u

i , j j

+ ( μ + λ )

u

j , i j

+

F

i

= ρ

t t

u

i

o r

μ

2

u

+ ( μ + λ ) ∇ ( ∇ ⋅

u

) +

F

= ρ

2

u

t

2

.

displaystyle mu u_ i,jj +(mu +lambda )u_ j,ij +F_ i =rho partial _ tt u_ i quad mathrm or quad mu nabla ^ 2 mathbf u +(mu +lambda )nabla (nabla cdot mathbf u )+mathbf F =rho frac partial ^ 2 mathbf u partial t^ 2 .,!

The elastodynamic wave equation can also be expressed as

(

δ

k l

t t

A

k l

[ ∇ ] )

u

l

=

1 ρ

F

k

displaystyle (delta _ kl partial _ tt -A_ kl [nabla ]),u_ l = frac 1 rho F_ k ,!

where

A

k l

[ ∇ ] =

1 ρ

i

C

i k l j

j

displaystyle A_ kl [nabla ]= frac 1 rho ,partial _ i ,C_ iklj ,partial _ j ,!

is the acoustic differential operator, and

δ

k l

displaystyle delta _ kl ,!

is Kronecker delta. In isotropic media, the stiffness tensor has the form

C

i j k l

= K

δ

i j

δ

k l

+ μ

(

δ

i k

δ

j l

+

δ

i l

δ

j k

2 3

δ

i j

δ

k l

)

displaystyle C_ ijkl =K,delta _ ij ,delta _ kl +mu ,(delta _ ik delta _ jl +delta _ il delta _ jk - frac 2 3 ,delta _ ij ,delta _ kl ),!

where

K

displaystyle K,!

is the bulk modulus (or incompressibility), and

μ

displaystyle mu ,!

is the shear modulus (or rigidity), two elastic moduli. If the material is homogeneous (i.e. the stiffness tensor is constant throughout the material), the acoustic operator becomes:

A

i j

[ ∇ ] =

α

2

i

j

+

β

2

(

m

m

δ

i j

i

j

)

displaystyle A_ ij [nabla ]=alpha ^ 2 partial _ i partial _ j +beta ^ 2 (partial _ m partial _ m delta _ ij -partial _ i partial _ j ),!

For plane waves, the above differential operator becomes the acoustic algebraic operator:

A

i j

[

k

] =

α

2

k

i

k

j

+

β

2

(

k

m

k

m

δ

i j

k

i

k

j

)

displaystyle A_ ij [mathbf k ]=alpha ^ 2 k_ i k_ j +beta ^ 2 (k_ m k_ m delta _ ij -k_ i k_ j ),!

where

α

2

=

(

K +

4 3

μ

)

/

ρ

β

2

= μ

/

ρ

displaystyle alpha ^ 2 =left(K+ frac 4 3 mu right)/rho qquad beta ^ 2 =mu /rho ,!

are the eigenvalues of

A [

k

^

]

displaystyle A[ hat mathbf k ],!

with eigenvectors

u

^

displaystyle hat mathbf u ,!

parallel and orthogonal to the propagation direction

k

^

displaystyle hat mathbf k ,!

, respectively. The associated waves are called longitudinal and shear elastic waves. In the seismological literature, the corresponding plane waves are called P-waves and S-waves (see Seismic wave). Anisotropic homogeneous media[edit] Main article: Hooke's law For anisotropic media, the stiffness tensor

C

i j k l

displaystyle C_ ijkl ,!

is more complicated. The symmetry of the stress tensor

σ

i j

displaystyle sigma _ ij ,!

means that there are at most 6 different elements of stress. Similarly, there are at most 6 different elements of the strain tensor

ε

i j

displaystyle varepsilon _ ij ,!

. Hence the fourth-order stiffness tensor

C

i j k l

displaystyle C_ ijkl ,!

may be written as a matrix

C

α β

displaystyle C_ alpha beta ,!

(a tensor of second order). Voigt notation
Voigt notation
is the standard mapping for tensor indices,

i j

=

α

=

11

22

33

23 , 32

13 , 31

12 , 21

1

2

3

4

5

6

displaystyle begin matrix ij&=\Downarrow &\alpha &=end matrix begin matrix 11&22&33&23,32&13,31&12,21\Downarrow &Downarrow &Downarrow &Downarrow &Downarrow &Downarrow &\1&2&3&4&5&6end matrix ,!

With this notation, one can write the elasticity matrix for any linearly elastic medium as:

C

i j k l

C

α β

=

[

C

11

C

12

C

13

C

14

C

15

C

16

C

12

C

22

C

23

C

24

C

25

C

26

C

13

C

23

C

33

C

34

C

35

C

36

C

14

C

24

C

34

C

44

C

45

C

46

C

15

C

25

C

35

C

45

C

55

C

56

C

16

C

26

C

36

C

46

C

56

C

66

]

.

displaystyle C_ ijkl Rightarrow C_ alpha beta = begin bmatrix C_ 11 &C_ 12 &C_ 13 &C_ 14 &C_ 15 &C_ 16 \C_ 12 &C_ 22 &C_ 23 &C_ 24 &C_ 25 &C_ 26 \C_ 13 &C_ 23 &C_ 33 &C_ 34 &C_ 35 &C_ 36 \C_ 14 &C_ 24 &C_ 34 &C_ 44 &C_ 45 &C_ 46 \C_ 15 &C_ 25 &C_ 35 &C_ 45 &C_ 55 &C_ 56 \C_ 16 &C_ 26 &C_ 36 &C_ 46 &C_ 56 &C_ 66 end bmatrix .,!

As shown, the matrix

C

α β

displaystyle C_ alpha beta ,!

is symmetric, this is a result of the existence of a strain energy density function which satisfies

σ

i j

=

∂ W

ε

i j

displaystyle sigma _ ij = frac partial W partial varepsilon _ ij

. Hence, there are at most 21 different elements of

C

α β

displaystyle C_ alpha beta ,!

. The isotropic special case has 2 independent elements:

C

α β

=

[

K + 4 μ  

/

3

K − 2 μ  

/

3

K − 2 μ  

/

3

0

0

0

K − 2 μ  

/

3

K + 4 μ  

/

3

K − 2 μ  

/

3

0

0

0

K − 2 μ  

/

3

K − 2 μ  

/

3

K + 4 μ  

/

3

0

0

0

0

0

0

μ  

0

0

0

0

0

0

μ  

0

0

0

0

0

0

μ  

]

.

displaystyle C_ alpha beta = begin bmatrix K+4mu /3&K-2mu /3&K-2mu /3&0&0&0\K-2mu /3&K+4mu /3&K-2mu /3&0&0&0\K-2mu /3&K-2mu /3&K+4mu /3&0&0&0\0&0&0&mu &0&0\0&0&0&0&mu &0\0&0&0&0&0&mu end bmatrix .,!

The simplest anisotropic case, that of cubic symmetry has 3 independent elements:

C

α β

=

[

C

11

C

12

C

12

0

0

0

C

12

C

11

C

12

0

0

0

C

12

C

12

C

11

0

0

0

0

0

0

C

44

0

0

0

0

0

0

C

44

0

0

0

0

0

0

C

44

]

.

displaystyle C_ alpha beta = begin bmatrix C_ 11 &C_ 12 &C_ 12 &0&0&0\C_ 12 &C_ 11 &C_ 12 &0&0&0\C_ 12 &C_ 12 &C_ 11 &0&0&0\0&0&0&C_ 44 &0&0\0&0&0&0&C_ 44 &0\0&0&0&0&0&C_ 44 end bmatrix .,!

The case of transverse isotropy, also called polar anisotropy, (with a single axis (the 3-axis) of symmetry) has 5 independent elements:

C

α β

=

[

C

11

C

11

− 2

C

66

C

13

0

0

0

C

11

− 2

C

66

C

11

C

13

0

0

0

C

13

C

13

C

33

0

0

0

0

0

0

C

44

0

0

0

0

0

0

C

44

0

0

0

0

0

0

C

66

]

.

displaystyle C_ alpha beta = begin bmatrix C_ 11 &C_ 11 -2C_ 66 &C_ 13 &0&0&0\C_ 11 -2C_ 66 &C_ 11 &C_ 13 &0&0&0\C_ 13 &C_ 13 &C_ 33 &0&0&0\0&0&0&C_ 44 &0&0\0&0&0&0&C_ 44 &0\0&0&0&0&0&C_ 66 end bmatrix .,!

When the transverse isotropy is weak (i.e. close to isotropy), an alternative parametrization utilizing Thomsen parameters, is convenient for the formulas for wave speeds. The case of orthotropy (the symmetry of a brick) has 9 independent elements:

C

α β

=

[

C

11

C

12

C

13

0

0

0

C

12

C

22

C

23

0

0

0

C

13

C

23

C

33

0

0

0

0

0

0

C

44

0

0

0

0

0

0

C

55

0

0

0

0

0

0

C

66

]

.

displaystyle C_ alpha beta = begin bmatrix C_ 11 &C_ 12 &C_ 13 &0&0&0\C_ 12 &C_ 22 &C_ 23 &0&0&0\C_ 13 &C_ 23 &C_ 33 &0&0&0\0&0&0&C_ 44 &0&0\0&0&0&0&C_ 55 &0\0&0&0&0&0&C_ 66 end bmatrix .,!

Elastodynamics[edit] The elastodynamic wave equation for anisotropic media can be expressed as

(

δ

k l

t t

A

k l

[ ∇ ] )

u

l

=

1 ρ

F

k

displaystyle (delta _ kl partial _ tt -A_ kl [nabla ]),u_ l = frac 1 rho F_ k ,!

where

A

k l

[ ∇ ] =

1 ρ

i

C

i k l j

j

displaystyle A_ kl [nabla ]= frac 1 rho ,partial _ i ,C_ iklj ,partial _ j ,!

is the acoustic differential operator, and

δ

k l

displaystyle delta _ kl ,!

is Kronecker delta. Plane waves
Plane waves
and Christoffel equation[edit] A plane wave has the form

u

[

x

,

t ] = U [

k

x

− ω

t ]

u

^

displaystyle mathbf u [mathbf x ,,t]=U[mathbf k cdot mathbf x -omega ,t], hat mathbf u ,!

with

u

^

displaystyle hat mathbf u ,!

of unit length. It is a solution of the wave equation with zero forcing, if and only if

ω

2

displaystyle omega ^ 2 ,!

and

u

^

displaystyle hat mathbf u ,!

constitute an eigenvalue/eigenvector pair of the acoustic algebraic operator

A

k l

[

k

] =

1 ρ

k

i

C

i k l j

k

j

.

displaystyle A_ kl [mathbf k ]= frac 1 rho ,k_ i ,C_ iklj ,k_ j .,!

This propagation condition (also known as the Christoffel equation) may be written as

A [

k

^

]

u

^

=

c

2

u

^

displaystyle A[ hat mathbf k ], hat mathbf u =c^ 2 , hat mathbf u ,!

where

k

^

=

k

/

k

k

displaystyle hat mathbf k =mathbf k / sqrt mathbf k cdot mathbf k ,!

denotes propagation direction and

c = ω

/

k

k

displaystyle c=omega / sqrt mathbf k cdot mathbf k ,!

is phase velocity. See also[edit]

Continuum mechanics

Laws

Conservations

Energy Mass Momentum

Inequalities

Clausius–Duhem (entropy)

Solid mechanics

Stress Deformation Compatibility Finite strain Infinitesimal strain Elasticity (linear) Plasticity Bending Hooke's law Material failure theory Fracture mechanics

Contact mechanics (frictional)

Fluid mechanics

Fluids

Statics · Dynamics Archimedes' principle · Bernoulli's principle Navier–Stokes equations Poiseuille equation · Pascal's law Viscosity (Newtonian · non-Newtonian) Buoyancy · Mixing · Pressure

Liquids

Surface tension Capillary action

Gases

Atmosphere Boyle's law Charles's law Gay-Lussac's law Combined gas law

Plasma

Rheology

Viscoelasticity Rheometry Rheometer

Smart fluids

Magnetorheological Electrorheological Ferrofluids

Scientists

Bernoulli Boyle Cauchy Charles Euler Gay-Lussac Hooke Pascal Newton Navier Stokes

v t e

Castigliano's method Clapeyron's theorem (elasticity) Contact mechanics Deformation Elasticity (physics) GRADELA Hooke's law Infinitesimal strain
Infinitesimal strain
theory Michell solution Plasticity (physics) Signorini problem Spring system Stress (mechanics) Stress functions

References[edit]

^ a b c d e Slaughter, W. S., (2002), The linearized theory of elasticity, Birkhauser. ^ Belen'kii; Salaev (1988). "Deformation effects in layer crystals". Uspekhi Fizicheskikh Nauk. 155: 89. doi:10.3367/UFNr.0155.198805c.0089.  ^ Aki, Keiiti; Richards, Paul G. (2002). Quantitative Seismology (2 ed.). Sausalito, California: University Science Books.  ^ Continuum Mechanics for Engineers 2001 Mase, Eq. 5.12-2 ^ Sommerfeld, Arnold (1964). Mechanics of Deformable Bodies. New York: Academic Press.  ^ a b tribonet (2017-02-16). "Elastic Deformation". Tribology. Retrieved 2017-02-16.  ^ a b Landau, L.D.; Lifshitz, E. M. (1986). Theory of Elasticity (3rd ed.). Oxford, England: Butterworth Heinemann. ISBN 0-7506-2633-X.  ^ Boussinesq, Joseph (1885). Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques. Paris, France: Gauthier-Villars.  ^ Mindlin, R. D. (1936). " Force
Force
at a point in the interior of a semi-infinite solid". Physics. 7 (5): 195–202. Bibcode:1936Physi...7..195M. doi:10.1063/1.1745385.  ^ Hertz, Heinrich (1882). "Contact between solid elastic bodies". Journal für die reine und angewandte Mathematik. 92. 

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