Linear elasticity

Mathematical formulation

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

In direct tensor form that is independent of the choice of coordinate system, these governing equations are:

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

∇ ⋅ σ + F = ρ u ¨

Strain-displacement equations:

ε = 1 2 [ ∇ u + ( ∇ u ) T ]

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

σ = C : ε ,

where σ is the Cauchy stress tensor, ε is the infinitesimal strain tensor, u is the displacement vector, C is the fourth-order stiffness tensor, F is the body force per unit volume, ρ is the mass density, ∇ represents the nabla operator, ( ∙ ) T represents a transpose, ( ∙ ) ¨ represents the second derivative with respect to time, and A : B = A i j B i j is the inner product of two second-order tensors (summation over repeated indices is implied).

Cartesian coordinate form

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 system, the governing equations of linear elasticity are:

Equation of motion:

σ j i , j + F i = ρ ∂ t t u i where the ( ∙ ) , j subscript is a shorthand for ∂ ( ∙ ) / ∂ x j and ∂ t t indicates ∂ 2 / ∂ t 2 , σ i j = σ j i is the Cauchy stress tensor, F i are the body forces, ρ is the mass density, and 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 ) where ε i j = ε j i 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 is:

σ i j = C i j k l ε k l where C i j k l 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 C i j k l = C k l i j = C j i k l = C i j l k .

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

In cylindrical coordinates ( r , θ , z ) the equations of motion are

∂ 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

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 )

and the constitutive relations are the same as in Cartesian coordinates, except that the indices 1 , 2 , 3 now stand for r , θ , z , respectively.

Spherical coordinate form

In spherical coordinates ( r , θ , ϕ ) the equations of motion are

∂ 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

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 ) .

Isotropic homogeneous media

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 )

where δ i j is the Kronecker delta, K  is the bulk modulus (or incompressibility), and μ 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 ) .

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:

σ i j = λ δ i j ε k k + 2 μ ε i j

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:

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 )

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 ]

where ν is Poisson's ratio and E  is Young's modulus.

Elastostatics

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.

This section will discuss only the isotropic homogeneous case.

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 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 ) .

Differentiating yields:

σ i j , j = λ u k , k i + μ ( u i , j j + u j , i j ) .

Substituting into the equilibrium equation yields:

λ u k , k i + μ ( u i , j j + u j , i j ) + 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

where λ and μ 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.

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

The elastostatic equation may be written:

( α 2 − β 2 ) u j , i j + β 2 u i , m m = − 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 ) we have

( α 2 − β 2 ) u j , i i j + β 2 u i , i m m = 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

from which we conclude that:

u j , i i j = 0.

Taking the Laplacian of both sides of the elastostatic equation, and assuming in addition F i , k k = 0 , we have

( α 2 − β 2 ) u j , k k i j + β 2 u i , k k m m = 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

from which we conclude that:

u i , k k m m = 0

or, in coordinate free notation ∇ 4 u = 0 which is just the biharmonic equation in u .

Stress formulation

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.

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.

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.

A necessary, but insufficient, condition for compatibility under this situation is ∇ 4 σ = 0 or σ i j , k k ℓ ℓ = 0 .

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.

Other solutions:

Point force inside an infinite isotropic half-space.

Point force on a surface of an isotropic half-space.

Contact of two elastic bodies: the Hertz solution (see Matlab code). See also the page on Contact mechanics.

Elastodynamics – the wave equation

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 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 .

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 .

The elastodynamic wave equation can also be expressed as

( δ k l ∂ t t − A k l [ ∇ ] ) u l = 1 ρ F k

where

A k l [ ∇ ] = 1 ρ ∂ i C i k l j ∂ j

is the acoustic differential operator, and δ k l 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 )

where K is the bulk modulus (or incompressibility), and μ 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 )

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 )

where

α 2 = ( K + 4 3 μ ) / ρ β 2 = μ / ρ

are the eigenvalues of A [ k ^ ] with eigenvectors u ^ parallel and orthogonal to the propagation direction 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

For anisotropic media, the stiffness tensor C i j k l is more complicated. The symmetry of the stress tensor σ i j 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 . Hence the fourth-order stiffness tensor C i j k l may be written as a matrix C α β (a tensor of second order). 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

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 ] .

As shown, the matrix C α β is symmetric, this is a result of the existence of a strain energy density function which satisfies σ i j = ∂ W ∂ ε i j . Hence, there are at most 21 different elements of C α β .

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   ] .

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 ] .

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 ] .

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 ] .

Elastodynamics

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

where

A k l [ ∇ ] = 1 ρ ∂ i C i k l j ∂ j

is the acoustic differential operator, and δ k l is Kronecker delta.

Plane waves and Christoffel equation

A plane wave has the form

u [ x , t ] = U [ k ⋅ x − ω t ] u ^

with u ^ of unit length. It is a solution of the wave equation with zero forcing, if and only if ω 2 and 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 .

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

A [ k ^ ] u ^ = c 2 u ^

where k ^ = k / k ⋅ k denotes propagation direction and c = ω / k ⋅ k is phase velocity.