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Special strain–stress states in isotropic elasticity

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August 13, 2007 17:14



perform the following operations:

• determine complete strain states

• check the compatibility and balance equations (in the absence of body and inertial


• compute the spherical and deviatoric parts of the tensors and interpret the results in

terms of compression and shear

• determine the traction vectors acting on surfaces of an infinitesimal volume element

and interpret physically the conditions of positive definiteness of elastic moduli.




August 13, 2007 17:14

General principles in problems of elasticity


This chapter is devoted to the formulation of the complete elasticity problem. It begins with

the formulation of the regular problem of thermoelasticity. Displacement (Navier) and

stress (Beltrami–Michell) formulations are introduced, and the one-dimensional problem

of a spherical vessel under internal and external pressure is solved as an illustration.

The general principles applicable in linear elasticity are treated next. The superposition principle and the virtual work theorem are introduced, allowing the conditions for

the uniqueness of elastic solution to be established. The existence of the strain energy potential and the complementary energy potential is proven, and reciprocity theorems are

presented. Saint Venant torsion is considered in detail, and the more general Saint Venant

principle is introduced, together with Hoff’s counterexample and the von Mises–Sternberg



The complete system of equations of elasticity consists of the equations of kinematics

and dynamics, together with the linear elastic constitutive relations introduced in the

previous chapter. Solution of the complete system must be found in the form of three field


vector field of displacements, u ;

tensor field of small strains, ε ;

tensor field of stresses, σ .

Within domain


the following system of equations of linear thermoelasticity must be

• Kinematic equations




(∇ + ∇ T )u



• Constitutive equations of linear thermoelasticity:

σ = σ 0 + C : (εε − Aθ),


where σ 0 is a tensor of initial stresses, C is the tensor of elastic moduli, A is a tensor of

linear thermal expansion coefficients, and θ is a scalar field of temperature changes.




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4.1 The complete elasticity problem


The particular case when θ = 0 will be referred to as isothermal, and when σ 0 = 0 will

be referred to as the natural state.

• Equations of static equilibrium

div σ + f = 0,


where f is the vector of body forces.

A set of necessary and sufficient conditions must be imposed on the problem formulation in order to guarantee the existence and uniqueness of the solution. A problem

formulation that satisfies such conditions, following Hadamard, will be called a well-posed,

or regular linear thermoelastic problem.

The regular linear thermoelastic problem must be formulated by the above set of equations together with the provision of the following information:

• At every point within the domain :

• the tensor of elastic moduli C that is symmetric, positive definite, and bounded

• the tensor of linear thermal expansion coefficients A that is symmetric

• the tensor of initial stresses σ 0

• the vector of body forces f

• the scalar field of temperature changes θ.

• At every point on the domain boundary ∂ and for each direction e i , i = 1, 2, 3:

• either a component of the displacement vector,

ui = uB

i ,


where uB

i is a given function,

• or a component of the traction vector,

σ · n )i = tiB,


where tiB is a given function.

Let ∂ di denote the part of the boundary on which the boundary condition is imposed


in terms of displacements uB

i , and respectively let ∂ i denote the part of the boundary on

which the boundary condition is imposed in terms of tractions tiB. The complementarity

of the regions on which boundary conditions are imposed in terms of displacements and

tractions leads to the following partition of the boundary ∂ :













∀i = 1, 2, 3.


The above requirement of complementary partition means that at each boundary point

one of the factors must be prescribed in each of the terms in the expression for the

mechanical work of boundary traction, namely

u · σ · n ds =

ui σij nj ds


[u1 σ1j nj + u2 σ2j nj + u3 σ3j nj ] ds.



Regular, or well-posed problems form only a subset of all problems encountered in the

mechanics of linear thermoleastic solids. Examples of nonregular problems, also known



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General principles in problems of elasticity

as ill-posed problems, include nondestructive identification of defects, such as cracks or of

inclusions, or constitutive parameters, and tomographic reconstruction problems (Bonnet

and Constantinescu, 2005).

Some classical boundary conditions are

• Encastre:

u = 0,

that is,

ui = 0 ∀i.


• Traction-free surface:

σ ·n = 0

σij nj = 0



• Frictionless sliding contact:

Displacement normal to the boundary is prescribed, but lateral sliding is unrestrained,

so that shear tractions vanish,

u · n = ud,

t 1 · σ · n = 0,

and t 2 · σ · n = 0,


where t 1 and t 2 are two surface tangent vectors that are perpendicular to each other

and to the normal n .

• Prescribed normal and shear tractions:

σ · n = −p n + q1t 1 + q2t 2

n · σ · n = −p

and t 1 · σ · n = q1


and t 2 · σ · n = q2 .



The regular linear thermoelastic problem can be solved in the displacement formulation,

that is, by assuming that the vector displacement field is the principal unknown variable.

This technique is associated with the names of Lame´ and Clapeyron. Substituting equation

(4.1) into (4.2) and then into (4.3), and using the symmetry of the elastic stiffness tensor

C, one obtains the displacement equations of equilibrium:

σ 0 + C : (grad u − Aθ)) + f = 0 .

div (σ


In the displacement formulation the following solution procedure may be adopted:

An admissible displacement field satisfying the displacement boundary conditions

(4.4) is assumed and substituted into kinematic equations (4.1) and linear thermoelastic

constitutive equations (4.2). It is then verified whether the stresses obtained in this way

satisfy the static equilibrium equations (4.3) and the tractions satisfy the boundary conditions (4.5). If not, an improved trial admissible displacement field is selected and the

procedure is repeated.

Conversely, once a displacement vector field u is found that satisfies the displacement

equations of equilibrium (4.14), with the strain defined by the kinematic equations (4.1),

and stress defined by the linear thermoelastic constitutive equation (4.2), then the stress

equation of equilibrium (4.3) is satisfied.

In the isothermal case, θ = 0, and in the absence of initial stresses σ 0 (i.e., in the natural

state), equation (4.14) reduces to

C : grad u ) + f = 0 .

div (C




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4.3 Stress formulation


Equation (4.15) for isotropic material reduces to the Navier equation,

µ u + (λ + µ)grad div u + f = 0 ,


or, equivalently, in terms of engineering elastic constants,


2(1 + ν)

grad div u +

f = 0.

1 − 2ν





u = grad div u − curl curl u ,


equation (4.16) can be rewritten as

(λ + 2µ)grad div u − µcurl curl u + f = 0 .


The application to the above equation of the operators div and curl respectively leads to

the following results:

(λ + 2µ) div u = −div f ,


µ curl u = −curl f .


If the body force field is such that div f and curl f both vanish, then both div u and curl u

are harmonic fields. Furthermore, changing the order of operators yields

div u = 0


curl u = 0.

Hence, by equation (4.18), u is a vector field that is harmonic; that is,

u = 0.


Thus, if the body force field f is divergence-free and curl-free, then the displacement field

is biharmonic.

From the kinematic equations (4.1), it follows that

ε = 0;


that is, the strain tensor is also biharmonic.


In the stress formulation, the tensor stress field is used as the principal unknown variable.

This technique is associated with the names of Beltrami and Michell. The strains are

determined from the stresses using the compliance form of the linear thermoelasticity

constitutive equations (4.2), namely,

σ − σ 0 ).

ε = Aθ + C−1 : (σ


As demonstrated in the discussion of kinematics, in order for a given tensor strain field to

correspond to a compatible displacement field, the strain compatibility equation must be


inc ε = −(curl (curl ε T )T = 0 .




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General principles in problems of elasticity

Substitution of (4.24) into the above equation leads to the stress equation of compatibility:

Aθ + C−1 : (σ

σ − σ 0 )] = 0 .

inc [A


This equation ensures that a unique vector displacement field u can be constructed. Provided the resulting displacements satisfy boundary conditions on parts ∂ di of the boundary, a complete solution of the linear thermoelastic problem is obtained.

The stress formulation solution procedure may be adopted as follows. First, a statically

admissible tensor stress field is selected, that is, a field that satisfies stress equilibrium

conditions (4.3) and traction boundary on parts ∂ ti of the boundary. Strains are then

determined in terms of stresses using equation (4.24) and substituted into the compatibility

equation (4.25). If these equations are not satisfied, another trial admissible stress field

must be selected. If compatibility is verified, displacements can be obtained by back

integration, and displacement boundary conditions on parts ∂ di of the boundary enforced.

In the isothermal case, θ = 0, and in the absence of initial stress σ 0 , equation (4.26)

reduces to

C−1 : σ ) = 0 .

inc (C


For isotropic linearly elastic material the compliance form of the constitutive linear elastic

equations is




σ − (tr σ ) 1 .




Useful relationships between stress and strain can be obtained from the above equation

by applying trace and divergence operators, respectively:

1 − 2ν

(tr σ )




div ε = − grad σ +

div σ .



tr ε =



To obtain the compatibility equation of stress for isotropic material, expression (4.28) is

substituted into the strain compatibility equation in the form

ε + grad grad (tr ε ) − (∇ + ∇ T )εε = 0 .


In conjunction with the stress equilibrium equation (4.3), the following result is obtained:




grad grad (tr σ ) −

(tr σ ) 1 + (∇ + ∇ T )ff = 0 .




The above equation is known as the Beltrami–Michell equation.

Taking the trace of the above equation, it is found that

(tr σ ) = −


div f .



Substituting this result back into the Beltrami–Michell equation, it is found that




grad grad (tr σ ) +

div f 1 + (∇ + ∇ T )ff = 0.






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4.4 Example: spherical shell under pressure




Figure 4.1. The upper half of a spherical reservoir, a spherical

shell under pressure.




In the absence of body forces, f = 0 , the stress compatibility equation takes the simple




grad grad (tr σ ) = 0.



If the volume forces are constant, both div f and grad f vanish, and the following results

are obtained. It is found from equation (4.33) that

(tr σ ) = 0.


Taking the Laplacian of the Beltrami–Michell equation and using the commutativity of

differential operators, it is established that the tensor field of stresses is biharmonic:

σ = 0.


Equations (4.22), (4.23), and (4.37) demonstrate that for an isothermal natural state

under the action of a constant body force, the elastic fields of displacements, strains, and

stresses are all biharmonic. Therefore a close relationship exists between the family of

solutions of linear thermoelastic problems and the solutions of the biharmonic equation.

The Mathematica package supplied with this book provides an efficient means of

evaluating differential operators of arbitrary tensor fields in various orthogonal coordinate

systems, and thus of verifying biharmonicity of these fields and their suitability as elastic

solutions. Some tools are provided to establish equivalence between differential forms, as

well as methods of writing analytical expression for the general solutions of the biharmonic

equation in different coordinate systems.


Let us consider the problem of a spherical reservoir under pressure (see Figure 4.1). The

reservoir is a spherical shell with internal and external radii of ri and re , respectively,

made from a linear elastic material with moduli (λ, µ). We shall seek to compute the



August 13, 2007 17:14


General principles in problems of elasticity

complete solution to this problem, assuming that the internal and the external pressure

fields are uniform and equal to p i and p e , respectively. Gravity and thermal effects are


In order to solve the problem we first set the coordinate system to spherical and then start

to search for a general form of solution under the assumption of spherical symmetry,

with a displacement

u (r, θ, ϕ) = ur (r)eer .

Strains and stresses are computed next.

SetCoordinates[Spherical[r, t, p]]

CoordinatesToCartesian[{r, t, p}]

u[r_, t_, p_] := {ur[r], 0, 0}

eps = Strain[ u [r, t, p]]

sig = Lambda Tr[eps ] IdentityMatrix[3 ] + 2 Mu eps

The equation to be satisfied is the stress field equilibrium. Note that only the divergence

component along e r is nonzero. This leads to the following equation:

(λ + 2µ)


1 ∂ 2

r ur (r)

r2 ∂r

= 0.

The general solution of this equation is computed using DSolve. The option GeneratedParameters sets the name of the constants of integration.

(divsig = Simplify[Div[ sig ]] ) // MatrixForm

solur = DSolve[ divsig[[1]] == 0, ur, r ,

GeneratedParameters -> CR][[1, 1]]

uu = u [r, t, p] /. solur

eps = Strain[ uu ]

sig = \[Lambda] Tr[eps ] IdentityMatrix[3 ] + 2 \ [Mu] eps

The general form of displacement, strain, and stress fields satisfying the balance equations in spherical coordinates with spherical symmetry is






ε (r, θ, ϕ) = c1 − 2 3 e r ⊗ e r + c1 + 3 (eeθ ⊗ e θ + e ϕ ⊗ e ϕ )





σ (r, θ, ϕ) = (3λ + 2µ)c1 − 4µ 3 e r ⊗ e r + (3λ + 2µ)c1 + 2µ 3 (eeθ ⊗ e θ + e ϕ ⊗ e ϕ ) ,



u (r, θ, ϕ) = c1 r +

where the constants c1 and c2 will be determined from the boundary conditions.

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