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How to Relate Stress and Strain
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We have considered deformation of continua and some of
the balance laws in them.
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These kinematics and mechanics apply to all the
continuous media. Then, where do the characteristics of
individual material come from?
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Properties unique to a certain material are determined by
the material’s internal constitution or physical make-up.
The quantitative expressions for such internal constitution
are called constitutive equations / laws / relations /
models.
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The role of constitutive relations is to relate stress (a
quantity needed for stating the linear momentum balance
law) and strain (a quantity used to describe motion of
continua).
Rules of Constitutive Theory
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In this lecture, we briefly review rules for formulating a
constitutive theory and then introduce linear elasticity.
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The constitutive relations are those between the internal
deformation quantified by strain and interfal forces
quantified by stress.
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The constitutive relations for a given material must give
meaningful response to all classes of deformations and
temperatures to which the body may be subjected.
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In case we need to develop a totally new constitutive
theory for a newly synthesized material for instance, we
need some guidelines by which a consitutive model can be
formulated.
Rules of Constitutive Theory
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Principle of Equipresence:
An independent variable assumed to be present in one
constitutive equation of a material should be assumed to
be present in all constitutive equations of the same
material unless its presence contradicts an assumed
symmetry of the material or contradicts of material
frame-indifference or some other fundamental principle.
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As one can see, this “principle” comes with many
conditions. Indeed, the development of a constitutive theory
depends highly on experience: researchers tend to avoid
complicating their equations with additional variables
unless experience (e.g., through experiments) indicates
that the variable are needed.
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Still, when applied, this principle can help revelaing the
coupling effects between different kinds of phenomena that
might otherwise be overlooked.
Rules of Constitutive Theory
Three fundamental postulates are assumed to be valid for any
constitutive theory of purely mechanical phenomena in a
continuous medium.
1. Principle of determinism for stress:
The stress in a body is determined by the history of the
motion of that body.
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This principle includes as special cases the Newtonian
fluid, which is history-independent, and linear elasticity, for
which there is only one hisotry, the reference state.
Viscoelasticity and plasticity are examples of more
complicated history-dependent theories.
Rules of Constitutive Theory
2. Principle of local action:
In determining the stress at a given particle, the motion
outside an arbitrary neighborhood of the particle may be
disregarded.
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Another seemingly obvious principle. However, there are
cases where we need to break this principle and consider
the responses from a finite neighborhood around a particle.
Such a theory is called “non-local”.
Rules of Constitutive Theory
3. Principle of material frame-indifference:
Constitutive equations must be invariant under changes of
frame of references. That is, two observers, even if in
relative motion (in the sense of the Galilean relativity) with
respect to each other, observe the same stress in a given
body.
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The assumed situation is that one observer is moving
(translation + rotation) relative to another. Distinguish the
chage of frame of references from coordinate
transformation, the latter of which is time-independent and
does not need to assume multiple observers.
Rules of Constitutive Theory
More on the material frame indifference
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Events (x,t):
x∗ = c(t) + Q(t)x
t∗ = t − a
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(1)
Vectors (v):
v∗ = Q(t)v
(2)
T∗ = Q(t)TQ(t)T
(3)
Rank 2 tensors (T):
Deformation gradient tensor (F):
F∗ = Q(t)F
(4)
Rules of Constitutive Theory
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Frame-indifference: Functions and fields whose values
are scalars, vectors, or tensors are called
frame-indifferent or objective if both the dependent and
the independent vector and tensor variables transform
according to Eq. (2) to (4), while the scalar variable are
unchanged (e.g., readings of a thermometer should be the
same for all the observers however they are moving).
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What does it mean for constitutive relations?
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If a constitutive equation is satisfied for a dynamical
process with a motion and a symmetric stress tensor, then
the constitutive equation must also be satisfied by the
motion and stress tensor that are transformed as in Eq. (1)
and (3),
Rules of Constitutive Theory
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The tedious and other-worldly discussion on seemingly
obvious statements sometimes pose practical challenges.
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Sometimes, the kinematics of a continuum is formulated in
terms of velocity rather than displacements. This point of
view makes a lot of sense when a body is deforming in a
time-dependent fashion. For instance, lithosphere that is
extending at a certain rate.
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In such a case, the deformation is better represented by
strain rate, not strain. If we assume a linear constitutive
relation between stress and strain, as in linear elasticity
we’ll see next, the constitutive relation will return stress
rate, not stress:
σ = Cε ⇒ σ˙ = Cε˙
Rules of Constitutive Theory
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Now the question is whether the stress rate is material
frame-indifferent. Predictably, the answer is no.
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To show this, we start with
σ ∗ = QσQT
(5)
By taking material time derivative, we get
T
˙
˙T
σ˙ ∗ = QσQ
˙ T + QσQ
+ Q(t)σ Q
Because of the last two terms in Eq. (6), σ˙ is not
frame-indifferent.
(6)
Rules of Constitutive Theory
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A slightly different definition of stress rate can be shown to
be frame-indifferent but there are more than one of such
stress rate forms.
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The most popular (due to easy computation) choice is
called Jaumann stress rate and defined as follows
σ
˚ = σ˙ + σW − Wσ
(7)
where W is the spin tensor, the skew-symmetric part of the
velocity gradient (dv/dx).
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Showing σ
˚ is frame-indifferent (also called objective) is
straightforward. How to get an objective stress rate is a
separate question and to answer it takes a whole new set
of mathematical machinery. So, we skip that part in this
course.
Linear Elasticity
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Hooke’s law for a 1 dimensional mass-spring system:
F = −kx
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If no damping force acts on it, the system is conservative,
meaning by definition that there is a potential function U(x)
such that F = −∇U.
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In this 1D example, integration to get U is straightforward
and U = 12 kx 2 .
Linear Elasticity
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A material is called ideally elastic when a body formed of
the material recovers its original form completely upon
removal of the forces causing the deformation, and there is
a one-to-one relationship between the state of stress and
the state of strain, for a given temperature.
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The one-to-one relationship precludes behaviors like creep
at constant load or stress relaxation at constant strain.
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The classical elastic constitutive equations, often called the
generalized Hooke’s law, are nine equations expressing
the stress components as linear homonenous (i.e., all the
terms are of the same power) functions of the nine strain
components:
σij = Cijkl εkl
(8)
Linear Elasticity
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The rank 4 tensor, Cijkl , has 81(= 34 ) components.
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However, recall that stress and strain tensor are
symmetric: i.e., σij = σji and εkl = εlk .
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Thus,
Cijkl = Cjikl and Cijkl = Cijlk .
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We further consider the case in which the material is
elastically isotropic, i.e., there are no preferred directions
in the material. Then, the elastic constants (Cijkl ) must be
the same at a given particle for all possible choices of
rectangular Cartesian coordinates in which stress and
strain components are evaluated.
Linear Elasticity
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The most general rank 4 tensor that satisfy all of the above
symmetry and isotropy conditions is
Cijkl = λδij δkl + µ(δik δjl + δil δjk )
(9)
(see Malvern Sec. 6.1 and 6.2 for further details.)
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The constitutive relation becomes
σij = λδij δkl + µ(δik δjl + δil δjk ) εkl
(10)
Finally, after some simplification, we reach the isotropic
generalized Hooke’s law:
σij = λεkk δij + 2µεij ,
where λ and µ are called Lamé’s constants.
(11)
Linear Elasticity
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By setting i = j in Eq. (11), we find
σii = (3λ + 2µ)εii
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By substituting εkk = σkk /(3λ + 2µ) (from Eq. (12)) into Eq.
(11), we obtain
εij = −
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(12)
1
λ
σkk δij +
σij
2µ(3λ + 2µ)
2µ
(13)
Recall the definitions of Young’s modulus and Poisson’s
relation:
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Hooke’s law: σxx = Eεxx , etc, where E is the Young’s
modulus.
Poisson’s relation: e.g., εyy = εzz = −νεxx = − Eν σxx , where
ν is the Poisson’s ratio.
Linear Elasticity
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From these relations, we derive the following for 3D:
1
[σxx − ν(σyy + σzz )]
E
1
= [σyy − ν(σzz + σxx )]
E
1
= [σzz − ν(σxx + σyy )]
E
1
=
σxy , etc.
2µ
εxx =
εyy
εzz
εxy
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(14)
The above set of equations can be generalized to the
following indicial notation:
ν
1+ν
εij = − σkk δij +
σij
E
E
(15)
Linear Elasticity
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It is also convenient to decompose the stress and strain
tensor into deviatoric and volumetric parts:
1
sij = σij − σkk δij
3
1
ij = εij − εkk δij
3
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(16)
Then, the whole relationship can be expressed by the two
equations:
sij = 2µij and p = −Ke,
(17)
where p = −σkk /3 is the pressure, e = εkk si the volume
strain, and K is the bulk modulus, related to Lamé’s
constants by the relation
2
K =λ+ µ
3
(18)
Linear Elasticity
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In the case of isotropic elasticity, all the elastic constants
(λ, µ, E, ν, K ) can be expressed in terms of just two
independent constants.
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µ, the shear modulus, is often denoted as G.
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Poisson’s ratio: Which is better for wine bottle, cork or
rubber?
Negative Poisson’s ratio?
http://www.youtube.com/watch?v=HJ1Ck6FIqwU
Linear Elasticity: Plane Stress
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Stress state in which σ3j = 0 with j = 1 . . . 3.
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In terms of principal stress, σ1 6= 0, σ2 6= 0 and σ3 = 0.
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From Eq. (15),
1
(σ1 − νσ2 )
E
1
ε2 = (σ2 − νσ1 )
E
ν
ε3 = − (σ1 + σ2 )
E
ε1 =
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Let’s assume that the lithosphere is in a lithostatic
condition: i.e., diviatroic stresses are zero and
σ1 = σ2 = σ3 = ρgh.
(19)
Linear Elasticity: Plane Stress
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Furthermore, deviatoric stresses are in the plane stress
condition and the two horizontal principal stresses are
equal in magnitude.
∆σ1 = ∆σ2 6= 0 but ∆σ3 = 0.
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Then, we get
ε1 = ε2 =
ε3 = −
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1−ν
∆σ1
E
2ν
∆σ1 .
E
(20)
Let’s think about the arguments given by Turcotte and
Schubert for the plane stress condition in the lithosphere.