Chapter 5
Constitutive Equations
We have studied stress and strain in the first part of this book. This is the introductory
chapter for the second part where stress and strain are related to constructing models for
tectonics. This chapter is the introduction to the stress-strain relationships of materials
used in the following chapters.
5.1
Stress-strain diagram
Tectonic deformation proceeds under high temperature and high pressure at depths. Laboratory
experiments have revealed the mechanical properties of rocks under those conditions [176, 247].
One popular type of apparatus loads a cylindrical rock sample with axial stresses, σ1 > σ2 = σ3 ,
to observe the stress-strain relation of the sample. The maximum stress, σ1 , is loaded along the
longitudinal axis of the sample. The load is laterally isotropic, σ2 = σ3 , which is called confining
pressure. The apparatus controls the longitudinal strain ε and confining pressure to measure the
change of longitudinal stress σ. Figure 5.1 shows the stress-strain diagrams of a hypothetical material
obtained using such apparatus. Given a confining pressure, the longitudinal strain ε and stress σ are
the input and output variables, respectively.
Basic Concepts
Deformation is called elastic if it is reversible. While applied strain is small, stress is proportional
to strain. However, the stress-strain curve becomes less steep beyond a limit called the proportional
limit (Fig. 5.1). Samples have their own limit beyond which deformation becomes irreversible.
Once the load under the limit is removed, the sample recovers its original, initial shape. The limit
is known as elastic limit or yield point, and the stress at which the sample yields is known as yield
stress, σY . If we use the convention that tension is positive stress, the yield stress is expressed as SY .
Proportional and elastic limits are indistinguishable for most rocks. Linear elasticity is the elastic
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126
CHAPTER 5. CONSTITUTIVE EQUATIONS
Figure 5.1: Stress-strain curve of a hypothetical material. Experimental apparatus controls the increase of longitudinal strain ε of a cylindrical sample to observe longitudinal stress σ at the ends
of the sample. For small strains, the sample behaves as an elastic material of which deformation is
reversible. However, the deformation become irreversible at the elastic limit of the sample, beyond
which brittle or plastic deformation occurs. Brittle failure of the sample abruptly relieves the stress
so that the breakdown of the sample is indicated by a vertical line in the stress-strain diagram.
behavior where stress and strain is proportional to each other1 . If a load over the yield stress is
applied, the sample may break or deform like a piece of plastic clay. They are brittle and plastic
deformations. Once the load is removed, stress and strain become smaller along the line parallel to
the line OA, remaining a permanent strain. Irreversible deformation over the yield point is known
as inelastic deformation.
Overburden causes a high confining pressure for a rock mass at depth so that broken pieces
slide on their interface rather, i.e., faulting is the important consequence of brittle deformation for
tectonics.
Simplification
Actual materials show a wide variation in behavior when subjected to loading. Their stress-strain
diagrams are affected by various factors including rock type, confining pressure, tempearture, strain
rate, chemical atmosphere, etc. Real materials respond in an extremely complicated fashion under
various loadings. Rocks tend to be plastic rather than brittle material for increasing temperature, as is
the case of a taffy. Deformation may dissipate kinetic energy to increase temperature that decreases
the strength of the deforming material. Increasing temperature may cause phase transition that may
affect the mechanical properties, and may cause an additional strain by thermal expansion. Thermal
1 Linear
elastic body hehaves equally for both compressional and tensile stresses. However, actual rocks do not. Rocks
have numerous cracks that decrease the rigidity of the rocks. Cracks open in tensile stresses and close in compression.
Laboratory experiments show that G in extension is in some cases tens of % smaller than that in compression [91]. The effect
of this asymmetry has been found to a mappable scale [178]. However, we neglect the nonlinearity in the following chapters.
5.1. STRESS-STRAIN DIAGRAM
127
Figure 5.2: Stress-strain curve and simple mechanical model for ideal materials. Displacement of
the mass, M, and force, F , depict the strain and stress, respectively. The springs represent elasticity.
Slidng on the rough surface indicates plasticity. Yield stress is represented by σY .
and mechanical phenomena are often coupled. Incremental deformation alters the internal structure
of a rock, i.e., the pile-up of microfractures, which in turn affects the strength of the rock against the
incremental deformation. A rock mass may consist of various types of rocks, each of which has its
own behavior.
Even a single crystal has its own complexity: its strength depends on direction with respect
to crystal axes. A material is called anistropic if some property depends on direction. Isotropic
materials show no such dependency. Although a rock mass is composed of various kinds of minerals
with anisotropy, most rocks have constituent minerals with random crysatalographic orientations,
vanishing anisotropy as a whole. Accordingly, details are often neglected but their average properties
are important for tectonics.
It is quite difficult to take into account all the observed phenomena under various conditions.
Instead, we need to define certain ideal materials such as ideal elastic solids. Such idealized materials
are useful in that they portray reasonably well over a definite range of loads and temperatures the
behavior of rocks. It is permitted to neglect the interaction of mechanical and thermal processes in
many situations.
Theories of tectonics utilize the phenomenological models of idealized materials. The stressstrain diagrams of common, idealized materials are shown in Fig. 5.2. A linear elastic body shows
a linear stress-strain relation. A rigid-perfectly-plastic body has no elastic behavior. The adverb
“perfectly” carries the property that stress is constant at σY for incremental strain. This model may
represent not only plastic behavior but also a frictional sliding of faults.
CHAPTER 5. CONSTITUTIVE EQUATIONS
128
5.2
Representation theorem
A constitutive equation indicates a mathematical relationship among the statical, kinematic, and
thermal variables, which will describe the behavior of the material when subjected to applied mechanical or thermal loads. The equation relates stress S with strain F for statical equillibria or with L
for kinematic problems. The stress caused by the deformation F or rate of deformation L is calculated
by the equation.
Whatever the material in question is, its constitutive equation must follow the principle of material frame-indefference if the stress in the material is caused by purely mechanical phenomena
[132, 244]. Namely, constitutive equations must be invariant under changes in the frame of reference. Namely, two observers, even if in relative motion with respect to each other, observe the same
stress in the body. This is also called the principle of material objectivity. The principle provides a
tight constraint to mathematical formulation of the constitutive equations.
The change of reference is given by
x = o(t) + Q(t) · x,
(5.1)
where o(t) is a vector and Q(t) is an orthogonal tensor representing the translation and rotation of
the coordinate system, respectively. Since the stress tensor S defines a linear transformation (Eq.
(3.10)) between the two vectors N and t(N ), the tensorial representations of the stress for the two
reference systems are related to each other by the equation S = Q · S · QT (Exercise 5.1). However,
the deformation gradient F is transformed in a different way. To show this, suppose that the frames
coincide at time t0 so that ξ = ξ at t = t0 . Note that the spatial coordinates x and x are dependent
on time t, but the material coordinates ξ and ξ are not. From Eq. (5.1) we obtain
dx = Q(t) · dx = Q(t) · F · dξ = Q(t) · F · dξ.
(5.2)
In this case, dξ = dξ , so that dx = F dξ = F dξ. Combining Eq. (5.2), we have
F − Q(t) · F · dξ = 0
for any ξ. Therefore, we obtain
F = Q(t) · F,
(5.3)
showing that the deformation gradient tensor transforms like a vector (Eq. (C.11)).
Now let S = F F be a constitutive equation, where F ( ) represents a tensor-valued function of
a second-order tensor. Stress tensor S is transformed by the equation S = Q · S · QT , and Eq. (5.3)
transforms the deformation gradient. The function F indicates the material property of a subject
material, so that the function does not depend on the choice of reference frames. Therefore, the
function also applies to the primed tensors, that is, S = F F . Accordingly, we have
Q · F F · QT = F Q · F .
5.3. EXERCISES
129
We are able to constrain the equation further for isotropic materials. The rotation of reference
frames should not affect the function F for such materials, so that we have F F = F Q · F . As
the deformation gradient can always be decomposed as F = R · U = V · R, we have
F F = F R · U = F (Q · R) · U .
In this equation, (Q · R) is an orthogonal tensor, therefore, the function has only the argument U or
V. The constitutive equation of isotropic material is, therefore,
S=F V .
(5.4)
If an isotropic material is extended, tensile stress may be induced in the extensional direction,
that is, the principal axes of V and S may be parallel to each other (Exercise 5.2). Therefore, the
function F does not vary principal axes, so that the Cayley-Hamilton theorem applies to this case.
Here, Eq. (C.46) works as the recursive relation to generate Vn from Vn−1 , Vn−2 and Vn−3 , so that Vn
can be expressed by the linear combination of V2 , V and I. Namely,
V3 = VI V2 − VII V + VIII I,
V4 = VI V3 − VII V2 + VIII V = (VI2 − VII )V2 + (VIII − VI VII )V + VI VIII I
and so on, where VI , VII and VIII are the basic invariants of V. The function F is assumed not to
change eigenvectors, so that Eq. (C.27) holds for V.
The Taylor series of V have terms of Vn , which can be rewritten with V2 , V and I. Thus
S = F (V) = φ0 I + φ1 V + φ2 V2 ,
(5.5)
where the coefficients φ1 , φ2 and φ3 are the scalar-valued functions of the basic invariants2 . The
concrete forms of φ1 , φ2 and φ3 depend on F. We have seen that isotropic materials have the simple
constitutive equation (Eq. (5.5)). This is known as the representation theorem.
5.3
5.1
Exercises
Demonstrate that the stress tensor is transformed as S = Q · S · QT .
5.2 Consider what if an anisotropic material undergoes uniaxial compression. Are the principal
strain axes parallel to the principal stress axes?
2 Equation (5.5) seems to be a quadratic equation of
invariants.
V, but the coefficients depend on the higher-order terms of the