Inverse problems in elasticity
Marc BONNET, Andrei CONSTANTINESCU
Laboratoire de M
´
ecanique des Solides (UMR CNRS 7649), Ecole Polytechnique, 91128
Palaiseau cedex
E-mail: bonnet@lms.polytechnique.fr,
andrei.constantinescu@lms.polytechnique.fr
Abstract. This article is devoted to some inverse problems arising in the context of linear
elasticity, namely the identification of distributions of elastic moduli, model parameters, or
buried objects such as cracks. These inverse problems are considered mainly for three-
dimensional elastic media under equilibrium or dynamical conditions, and also for thin
elastic plates. The main goal is to overview some recent results, in an effort to bridge
the gap between studies of a mathematical nature and problems defined from engineering
practice. Accordingly, emphasis is given to formulations and solution techniques which are
well suited to general-purpose numerical methods for solving elasticity problems on complex
configurations, in particular the finite element method and the boundary element method. An
underlying thread of the discussion is the fact that useful tools for the formulation, analysis and
solution of inverse problems arising in linear elasticity, namely the reciprocity gap and the error
in constitutive equation, stem from variational and virtual work principles, i.e. fundamental
principles governing the mechanics of deformable solid continua. In addition, the virtual
work principle is shown to be instrumental for establishing computationally efficient formulae
for parameter or geometrical sensitivity, based on the adjoint solution method. Sensitivity
formulae are presented for various situations, especially in connection with contact mechanics,
cavity and crack shape perturbations, thus enriching the already extensive known repertoire of
such results. Finally, the concept of topological derivative and its implementation for the
identification of cavities or inclusions are expounded.
Inverse Problems 21:R1–R50 (2005)
Inverse problems in elasticity 2
1. Introduction
Elasticity theory describes the reversible deformation of solid bodies subjected to excitations
of various physical natures: mechanical, thermal, electromagnetical etc. Such excitations,
applied as distributions over the body (e.g. gravitation, Lorentz forces, thermal expansion)
or over the boundary (pressure, contact forces), generate strains (i.e. local deformations) and
stresses (i.e. local forces) in the material. Elasticity is a mechanical constitutive property of
materials whereby (i) a one-to-one relationship between instantaneous strains and stresses on
the current deformed configuration is assumed, and (ii) the material reverts to its initial state
if the sollicitation history is reversed.
Almost all natural or manufactured solid materials have a deformation range within
which their mechanical behaviour can be modelled by elasticity theory. For sufficiently small
strains, the elastic behaviour is considered as linear, i.e. strains and stresses are assumed to
be proportional to each other. A vast body of engineering experience shows that the theory
of linear elasticity allows an accurate modelling of many man-made or natural objects: civil
engineering structures, transportation vehicles, machines, the earth mantle (to list just a few),
and provides an essential tool for analysis and design. In addition to the basic theory for three-
dimensional solid media, specialized approches have been developed for cases featuring two
or more dissimilar length scales: composite media, slender structures (beams, plates, shells).
The theory of linearized elasticity has developed into one of the now classical areas of
mathemematical physics. Equilibrium problems are governed by elliptic partial differential
equations, similar to those of electrostatics but more complex in that physical quantities of
interest are described by tensor fields rather than vector fields. Closed-form solutions are
available only for simple geometries (usually corresponding to separable coordinate systems),
so that most real-life modelling studies are based on numerical solution methods. Dynamic
conditions give rise to hyperbolic partial differential equations.
The main types of inverse problems that arise in the context of linear elasticity, and more
generally of the mechanics of deformable solids, are similar to those encountered in other
areas of physics involving continuous media and distributed physical quantities, e.g. acous-
tics, electrostatics and electromagnetism. They are usually motivated by the desire or need
to overcome a lack of information concerning the properties of the system (a deformable
solid body or structure). Mathematical and numerical techniques for the reconstruction of
buried objects of a geometrical nature, such as cracks, cavities or inclusions, are the subject of
many investigations, e.g. [3, 7, 8, 15, 26, 31, 32, 44–46, 61, 76, 95, 96, 118, 120]. Mechanical
waves, such as ultrasonic or Lamb waves, are also frequently used in practical non-destructive
testing of structures, see e.g. [103–105, 112, 131] and the references provided therein.
The identification of distributed parameters [5, 12–14, 29, 39, 48, 49, 60, 70, 81–83, 116] (e.g.
elastic moduli, mass density, wave velocity) arises in connection with e.g. medical imaging
of tissues [11] or seismic exploration [98, 114, 123, 130, 136, 140]. The reconstruction of
residual stresses [9, 67, 106, 127] is a related topic with important engineering implications.
Models of complex engineering structures often feature local parameters that are not known
with sufficient accuracy, and therefore need to be corrected by exploiting experimental
Inverse problems in elasticity 3
information on the mechanical response of the structure. Model updating is often treated
as an inverse problem [17, 27, 41, 57, 100, 107, 126, 142], in particular because corrections
affect distributed parameters over a limited region of space which is not known beforehand.
Identification of sources or inaccessible boundary values (i.e. Cauchy problems in elasticity)
are also encountered [42, 50, 55, 94, 109–111]. Finally, the identification of homogeneous
constitutive properties is increasingly often made on the basis of measurements taken on
structures, for which simplifying assumptions such as constant states of strain or stress are
invalid, and inverse techniques are then developed for that purpose [51, 64, 69, 108, 137].
This article is devoted to some of the above-mentioned inverse problems, namely the
identification of (i) distributions of elastic moduli, (ii) model parameters, (iii) buried cracks
or other geometrical objects. These inverse problems will be considered mainly for three-
dimensional elastic media under equilibrium or dynamical conditions, and also for thin elastic
plates. The main goal is to overview some recent results, in an effort to bridge the gap
between studies of a mathematical nature and problems defined from engineering practice.
Accordingly, emphasis will be given to formulations and solution techniques which are well
suited to general-purpose numerical methods for solving elasticity problems on complex
configurations, in particular the finite element method [28, 125] and the boundary element
method [23, 40]. The important role of the variational principles of elasticity, which in
particular provide the foundations of the above-mentioned numerical solution methods, will
be highlighted throughout this article. Additionnally, investigations of a more mathematical
nature will also be reviewed.
The article is organised as follows. An overview of basic theory and equations of linear
elasticity under the small strain hypothesis, including the virtual work principle and varia-
tional formulations of equilibrium problems, is presented in Section 2. Then, Section 3 des-
cribes strategies for the identification of distributions of elastic moduli or cracks exploiting
the virtual work principle as an observation equation, with emphasis on the reciprocity gap
concept. The virtual work principle is also an effective tool for setting up parameter sensitivity
analyses and computing gradients of cost functions associated to identification problems, as
shown in Section 4. Then, Section 5 is devoted to cost functions and parameter identification
techniques based on the error in constitutive equation (ECE). Formulations for both three-
dimensional bodies and plates are presented, and the ability of the energy density function
associated to the ECE to outline the geometrical support of defects is discussed. Finally,
Section 6 is devoted to geometrical sensitivity tools, based on adjoint solutions, for defect
identification. A concise formulation for cavity shape sensitivity is followed by more recent
results concerning crack shape sensitivity and a presentation of the topological derivative
associated with wave scattering in the limit of vanishingly small objects.
A brief review of the typical orders of magnitude involved in elastic solid bodies will
close this introductory section. Values of elastic constitutive parameters for common isotropic
materials are given in table 1, where the Young modulus E and the Poisson ratio ν are
defined in terms of the basic experiment performed by applying a traction force F to both
extremities of a cylindrical bar. Under this experiment, the axial length stretches from −ℓ
0
to ℓ while the cross-section shrinks from S
0
to S, and one sets E = (F/S
0
)/(ℓ/ℓ
0
−1) and
Inverse problems in elasticity 4
E (GPa) ν ρ (kg/m
3
) σ
Y
(GPa) ε
Y
aluminium 71 0.34 2.6 150 – 400 .002 – .006
steel 210 0.29 7.8 200 – 1600 .002 – .007
titanium 105 0.34 4.5 700 – 900 .006 – .009
marble 26 0.3 2.8 10 .0004
glass 60 0.2 - 0.3 2.5 – 2.9 1200 .02
Table 1. Constitutive parameters of some common isotropic elastic materials: E Young
modulus, ν Poisson ratio, ρ mass density; σ
Y
and ε
Y
define the elastic limit, i.e. are the
stress and strain levels beyond which the material is no longer elastic.
ν = −(
p
S/S
0
−1)/(ℓ/ℓ
0
−1). Linear elasticity is usually valid when strains are small (typical
magnitudes are 10
−3
or less), and is also restricted to stress levels below a certain threshold
σ
Y
beyond which irreversible constitutive properties, e.g. plasticity, set in (see Table 1).
Strains and displacements can be measured directly, e.g. using strain gages, whereas
stresses can only be measured indirectly. Classical strain gages are reliable up to 10
−6
but
offer only a “pointwise” measurement, in practice over an area of a few square millimeters.
Modern technology based on laser interferometry or image correlation techniques [21] are
reliable up to 10
−5
but allow measurements of practically continuous fields over extended
areas. Such experimental techniques yield rich experimental data and are therefore well-suited
to identification problems. The importance of the latter techniques is increasing as inversion
techniques specifically exploiting availability of field quantities (either on the boundary or
over part of the domain itself) become accessible.
2. Review of governing equations
2.1. Fundamental field equations for three-dimensional elasticity
The deformation of an elastic body, occupying in its undeformed state the region Ω ⊂ R
3
bounded by the surface S, is usually described in terms of a vector displacement field u(x, t)
(x ∈ Ω) which is such that the deformation process moves a small material element lying
at x to its new position x + u(x, t). The linearized elasticity theory [74] is established on
the assumption of small strains, namely |∇u(x, t)| ≪ 1. In that case, the changes in metric
induced by the deformation are described by the linearized strain tensor ε(x, t), defined as a
differential operator on u by:
ε[u](x, t) = (∇u(x, t) + ∇u
T
(x, t))/2 (1)
This equation is often referred to as the compatibility equation for small deformations. The
strain ε(x, t) is a symmetric second-order tensor.
The material is characterized by two constitutive parameters: its mass density distribution
ρ(x), associated with the kinetic energy
T (u) =
1
2
Z
Ω
ρ|
˙
u|
2
dV
Inverse problems in elasticity 5
(where the dot denotes time differentiation) and the fourth-order tensor of elastic moduli
C(x), hereafter referred to as the elasticity tensor, associated with the elastic strain energy
E(u) =
1
2
Z
Ω
ε[u]:C :ε[u] dV (2)
The elasticity tensor C defines a positive definite quadratic form over the 6-dimensional space
of symmetric second-order tensors. Therefore, C has the following symmetries:
C
ijkℓ
= C
kℓij
= C
jikℓ
(1 ≤ i, j, k, ℓ ≤ 3) (3)
and hence has at most 21 independent coefficients. In the simplest situation of isotropic
elasticity, C depends on only two independent moduli. For instance, C can be expressed in
terms of the Lam
´
e coefficients (λ, µ):
C
ijkℓ
= λδ
ij
δ
kℓ
+ µ(δ
ik
δ
jℓ
+ δ
jk
δ
iℓ
) (4)
Other commonly used elastic parameters are the Young modulus E and the Poisson ratio ν,
which are related to the Lam
´
e constants by
E =
µ(3λ + 2µ)
λ + µ
ν =
λ
2(λ + µ)
(5)
The stress tensor σ describes internal forces: the traction vector
p[n](x, t) = σ(x, t).n(x) (6)
is such that p[n] dS is the elementary force applied on a infinitesimal surface patch dS
of unit normal n located at x ∈ Ω. The fundamental balance equation of the dynamics of
deformable bodies (an extension of Newton’s second law to a small material element) is then:
div σ(x, t) + f(x, t) − ρ(x)
¨
u(x, t) = 0 (7)
where f (x, t) is a given distribution of body forces. The constitutive assumption of linearized
elasticity, adopted in this article, postulates that the stress tensor σ(x, t) depends linearly on
the linearized strain tensor, i.e.:
σ(x, t) = C(x):ε[u](x, t) (8)
On combining the three field equations (1), (7) and (8) and eliminating ε and σ, the
displacement field is found to be governed by the partial differential equation
[A
C
u](x, t) + f (x, t) − ρ(x)
¨
u(x, t) = 0 (9)
with the elasticity operator A
C
defined by:
A
C
u := div (C : ε[u]) = div (C : ∇u) (10)
(where the last equality stems from the constitutive symmetries (3)). Equation (9) is the
analog for linear elasticity of the hyperbolic linear wave equation. Besides, a well-posed
elastodynamic problem features initial conditions
u(x, 0) = u
0
(x)
˙
u(x, 0) = v
0
(x) (x in Ω) (11)
