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Chapter 3
Constitutive
Models  Relations between Stress and Strain
3.13 Constitutive models for
contacting surfaces and interfaces in solids
Many
practical problems involve two contacting surfaces that roll or slide against
one another: examples include machine elements such as gears and bearings;
machining and metal forming processes; or slip along a geological fault. In addition, models of deformation and
failure in materials must often account for the nucleation and growth of
cracks in the solid. In these
applications, constitutive equations must be used to specify the forces
transmitted across the interface or contacting surface as a function of their
relative motion.
The
simplest and most familiar such constitutive law is Coulomb friction, which
relates the normal and tangential tractions acting across a contacting
surface. More complex constitutive
laws are also available, which can model very complex interactions between
surfaces. In this section, we outline
two general classes of interface law: (i) Cohesive zone models, which are
used to model interfaces in materials or adhesion between very clean (often
nanoscale) surfaces; and (ii) Models that are intended to model contact and
friction between two sliding surfaces.
3.13.1
Cohesive zone models of interfaces
Cohesive
zone models are usually used to model the nucleation and propagation of
cracks along an interface between two solids, and to model adhesion between
two contacting surfaces. The figure
illustrates the problem to be solved.
We assume that
 Two
solids meet at a surface S.
In the
undeformed configuration, the interface is free of traction, and there is no
overlap or separation between the solids along S
When
the solid is loaded, forces are transmitted across the interface, while the
two solids may separate, slide, or overlap at the interface. The notion that two solids may
interpenetrate can be disturbing at first sight. However, the surface S where the solids meet does not represent a plane of atoms it merely characterizes the equilibrium
separation between the two solids when the interface is stress free. If the two solids overlap, this means that
the atomic or material planes just adjacent to the interface move closer
together.
We
shall assume that the relative displacement of the two solids across S is small compared with any
characteristic dimension of the solid; and also that the two contacting
solids themselves experience small shape changes.
A
“cohesive zone law” relates the relative motion of the two solids adjacent to
S to the tractions transmitted
across S. A large number of such constitutive
equations have been developed, but there are two general classes: (i)
reversible force-displacement laws, in which the traction is simply a
function of the relative distance between the two surfaces, and independent
of the history of loading. These are
often used to model nucleation and growth of a crack on an interface that is
subjected to monotonically increasing loading, where irreversibility plays no
role; and are also used to model interaction between surfaces of nanoscale
structures, whose dimension can be comparable to the distance of action of
long-range interatomic forces. (ii)
Irreversible force-displacement laws, which model failure processes that lead
to the creation of new free surface in the solid. These could include separation of atomic
planes due to cleavage, or more complex processes such as rupture by void
nucleation and coalescence, or fatigue.
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Kinematics: The relative motion of the two solids is
characterized as follows:
1. Let n denote
a unit vector normal to the interface.
The sense of n is arbitrary
(i.e. it can point up or down, as you prefer). Once n
has been chosen, however, we designate the two material surfaces adjacent to by and ,
with n the outward normal to .
2. Introduce two mutually perpendicular unit vectors that are tangent to the interface.
3. Let and denote the displacement of two material
points that are just adjacent to a point on S in the undeformed solid.
4. Let denote the relative displacement of two
initially coincident points. To specify constitutive equations, it is
convenient to characterize the relative displacement using the three scalar
Cartesian components of in the basis . If the interface is isotropic (i.e. its
response is independent of the direction of the relative tangential
displacement between the surfaces), the behavior of the interface depends
only on and .
Kinetics:
The forces acting between the two surfaces are characterized as follows:
1. Two points that are initially coincident in the
undeformed interface are assumed to exert equal and opposite tractions on one
another. Since the relative
displacements of and are assumed to be small, and both solids are
assumed to experience small shape changes, there is no need to distinguish
between forces acting on the deformed and undeformed solids. Let and denote the force per unit area acting on and ,
respectively.
2. Since ,
the tractions can be characterized by the three scalar components of in the basis .
The
constitutive equations for the interface must relate to .
Constitutive
equations representing reversible separation between interfaces are the
simplest cohesive zone laws. For these
models, the tractions are a function only of the relative displacement of the
material planes adjacent to the interface, and are independent of the history
or rate of loading. This means that the traction-displacement relation for
the interface is reversible. The interface
will heal if the two surfaces are brought back into contact after separation.
The
constitutive equations relating to for a reversible interface are constructed
as follows:
1. The traction-displacement relation is most
conveniently characterized by a scalar inter-planar potential by setting
The value of represents
the work done per unit area in separating the interface by .
2. A number of different functions are used to
approximate . Here, we will just give one example (a
simplified version of a potential suggested by Xu and Needleman, (1995), J
Mech. Phys. Solids 42 p. 1397)
Here,
,
while are material properties. Their physical
significance is discussed below.
3. Formulas relating to can be calculated by differentiating the
potential. The result is
where . The traction-displacement relations are
plotted in the figure. Under purely
normal tensile loading, the interface has work of separation ,
and the noremal traction reaches a value of at an interface separation . Under purely shear loading, the tangential
traction has a maximum value at a tangential shear displacement .
Constitutive
equations modeling irreversible separation between interfaces. Most
interfaces do not heal when brought back into contact after separation. In applications where interfaces are
subjected to cyclic loading, more
complicated constitutive equations must be used to account for this
irreversible behavior. Again, a very
large number of such constitutive equations have been developed: we will illustrate their general features using a
model adapted from Ortiz and Pandolfi, Int J. Numerical Methods in
Engineering, 44 1276 (1999) as a
representative example.
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The
behavior of the interface can be illustrated using its response to a purely
normal tensile traction, shown in the traction-separation law on the
right. The interface initially
responds elastically, with a constant stiffness ,
so that . As
long as the interface is reversible and
undamaged. If the displacement
exceeds ,
the interface begins to accumulate irreversible damage, which causes the
stress to drop. At the same time, the
damage reduces the stiffness of the interface, so that during unloading the
traction-displacement relation remains linear, but with a reduced slope. Note
that the total work of separation for the interface under tensile loading is .
The constitutive equation is constructed as follows:
- The material
surfaces and are completely prevented from
overlapping, by enforcing the constraint
- The magnitude of the
relative displacement between and is quantified by a scalar parameter ,
where is a material property, which controls
the relative stiffness and strength of the interface to normal and shear
loading.
- Similarly, the
magnitude of the traction can be quantified by an effective stress
- The tractions acting
between and are related to the relative
displacement by an elastic potential
by setting
Here, is a scalar parameter that quantifies the
irreversible damage accumulated by the interface.
- A linear
traction-displacement relation is constructed by making a quadratic function of ,
as follows
Here, is a material property that corresponds to
the slope of the traction-displacement relation for the undamaged
interface. It follows that the
tractions are related to the displacements by
- The constitutive law
is completed by devising an appropriate equation governing the evolution
of D. remains constant if the traction on the
interface is less than its current strength; if the interface is
unloaded, or if D reaches
1. Otherwise, D must evolve so that the
strength of the interface decreases linearly from its initial value to zero as the effective displacement increases from to .
This requires
Representative
values for properties of cohesive zones.
The
two constitutive laws contain the following parameters:
- The reversible
interface can be conveniently characterized by its strength, ;
the total work of tensile separation ,
and the parameter that controls the ratio of shear to
normal strength.
- The irreversible
interface can be characterized by its strength, ;
the total work of tensile separation ,
the parameter and the displacement at the instant of maximum stress.
It
is difficult to give precise values for these material properties. This is partly because the constitutive
equations are used to model a variety of physical processes that lead to
failure, and partly because there is no simple way to measure the values of
the parameters. The following
guidelines are usually followed:
- If the cohesive zone
is used to model atomic-scale cleavage in a brittle elastic material, or
adhesion between two elastic solids, then is set equal to the fracture toughness
of the interface (typically ), and the peak strength of the
material where E is the Young’s modulus.
Available data suggests that interfaces are stronger in shear
than in tension so is usually taken to be slightly less
than 1 ( ).
Computations are usually not strongly sensitive to the shape of
the cohesive zone, so can be taken to be approximately 1 in
the irreversible model.
- If the cohesive zone
is intended to model both the plastic zone and the failure process at
the tip of a crack in an otherwise elastic solid, then is set equal to the fracture toughness
of the solid (fracture toughness values are tabulated in Section 9.3.6),
while is taken to be roughly three times the
yield stress of the solid in uniaxial tension (yield stress values are
tabulated in Section 3.6.9).
Again, ,
while in the irreversible model.
- Cohesive zones are
sometimes used to model material separation at the tip of a crack in a
plastic solid, together with an elastic-plastic constitutive equation
for the two solids adjacent to the cohesive interface. In this case it is not usually clear
what process the cohesive zone represents. Experience shows that if the strength of the cohesive zone is taken to be too
high (greater than approximately three times the yield stress of the
plastic material) the crack will never propagate. If the strength of the cohesive zone
is less than the yield stress, there is no plasticity. Consequently . It is difficult to interpret the
meaning of in these models, but fortunately
simulations tend to be relatively insensitive to . A value for is usually estimated by choosing a
sensible characteristic length (between and
) for or and setting for the reversible model; or for the irreversible version.
3.13.2
Models of contact and friction between surfaces
Experimental
observations of friction between contacting surfaces
A
friction experiment is conceptually very simple: two surfaces are pressed
into contact by a controlled normal pressure p, and the specimens are loaded so as to induce a state of
uniform shear traction acting between the contacting surfaces. The experiment seeks to answer the
following questions:
1.
What is the critical combination of normal
and tangential forces cause the surfaces to start to slide?
2. If
the two surfaces do start to slip, what tangential force is required to keep them sliding?
3. If the surfaces are slipping, how does the
tangential force vary with sliding velocity and normal pressure, and how does
the surface respond to changes in
sliding velocity and pressure?
4. How does friction depend on the contact area, the
properties of the two contacting surfaces; surface roughness; environment;
lubricant films, etc?
The
results of these experiments show that, for most engineering surfaces which
make contact over a nominal area exceeding or so:
 The critical tangential
traction required to initiate sliding between two surfaces is proportional to
the normal pressure. If the normal
force is zero, the contact can’t support any tangential force. Doubling the normal force will double the
critical tangential force that initiates slip.
Surface roughness has a
very modest effect on friction.
Doubling the surface roughness might change the friction force by a
few percent.
Contaminants or
lubricant on the two surfaces has a big effect on friction. Even a little moisture on the surfaces can
reduce friction by 20-30%. If there is
a thin layer of grease on the surfaces it can cut friction by a factor of
10. If the contaminants are removed, friction forces can be huge, and the two
surfaces can seize together completely.
Friction forces depend
quite strongly on what the two surfaces are made from. Some materials like to bond with each other
(metals generally bond well to other metals, for example) and so have high
friction forces. Some materials (e.g.
Teflon) don’t bond well to other materials.
In this case friction forces will be smaller.
If the surfaces start
to slide, the tangential force often (but not always) drops slightly. Thus, kinetic friction forces are often a
little lower than static friction forces.
Otherwise, kinetic friction forces behave just like static friction they are proportional to the normal force,
etc.
The steady-state kinetic
friction force usually (but not always) decreases slightly as the sliding
speed increases. Increasing sliding
speed by a factor of 10 might drop the friction force by a few percent.
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The transient
response to changes in sliding speed has been extensively studied in geological
materials, motivated by the need to understand earthquakes. In these
materials, increasing the sliding speed causes an instantaneous increase in
shear traction, which then gradually decays to a lower steady-state value,
as illustrated in the figure. If
the sliding speed is reduced, there is an instantaneous drop in the
friction force, which subsequently increases towards a steady state. The transients occur over a sliding
distance of order 10-50 microns.
This behavior has been observed in other materials (including
metals) as well, but is not universal for example Gearing et al In J.
Plasticity 17 p. 237, (2001) observe an increase in steady
state friction forces with sliding speed in sliding of Al against steel.
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The transient
response to a change in contact pressure has not been studied as
extensively as the response to changes in sliding speed. The data of Prakash (Journal of
Tribology, 120, 1998 p. 27) indicates that when the contact
pressure is suddenly increased, the shear traction is initially unchanged,
and subsequently asymptotes towards a value proportional to the new contact
pressure as the relative distance of sliding between the two surfaces
increases, as illustrated in the figure.
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These
trends can be attributed to the effects of surface roughness. All surfaces are rough, and when brought
into contact meet only at highest points on the two surfaces. The
true area of contact between the two surfaces is much less than the nominal
contact area, and increases roughly in proportion to the nominal contact
pressure acting between the surfaces.
The nominal tangential traction is proportional to the product of the
true contact area and the shear strength of the contacting surfaces, and is
therefore approximately proportional to the nominal contact pressure.
There are some
situations where the true area of contact approaches the nominal contact
area. Examples include (i) the tip of an atomic force microscope, which has
roughness comparable to atomic scale dimensions; (ii) friction between the
tool and workpiece in metal forming applications. In these situations the traction acting
tangent to the surface is relatively insensitive to the contact
pressure.
Kinematics: Constitutive laws for friction must account
for large relative motion between the contacting surfaces. Consequently, the
contact is best characterized by the relative position and motion of the two
surfaces in the deformed configuration.
1. One of the two surfaces is arbitrarily designated
the `master’ surface, and labeled . The other surface is designated the
‘slave’, and is labeled . Note that in some friction models (e.g. the
plasticity model described below) exchanging the master and slave surface
will have a small influence on the behavior of the interface.
2. At a representative point on ,
we let n denote a unit vector
normal to and introduce two mutually perpendicular
unit vectors that are tangent to . We take to point along a characteristic material
direction in ,
i.e. if is a unit vector tangent to in the undeformed slave surface, in the deformed surface.
3. The gap between the two surfaces is characterized by
the points on the two surfaces that lie along n, i.e.
4.
The relative
velocity of the two surfaces is defined as
It is convenient to separate the relative velocity
into components normal and tangent to the surface , .
5.
In finite
element computations it is sometimes convenient to introduce a small elastic
compliance for the interface. In this
case, the relative velocity of the surfaces is divided into a reversible
elastic part and an irreversible (plastic) part by defining
Kinetics: The forces acting between
the two surfaces are characterized as follows:
1. The points on the two surfaces at position and are assumed to exert equal and opposite
tractions on one another. We let ,
denote the tractions on and ,
respectively.
2. Since ,
the tractions can be characterized by the three scalar components of in the basis .
Constitutive
equations for sliding friction must specify relationships between and . Various alternatives are summarized briefly
below.
Coulomb
Friction: This is the most
familiar friction law. For this model
(i) The
interface separates, with an indeterminate if ;
(ii) The surfaces are prevented from
inter-penetrating if ;
(iii) No slip occurs between the surfaces if ,
where is the coefficient of friction;
(iv) The tangential traction is proportional to the
normal pressure and opposes the direction of slip if the two surfaces slide .
The table below (taken
from `Engineering Materials’ by Ashby and Jones, Pergammon, 1980) lists rough
values for friction coefficients for various material pairs. These are rough guides only friction coefficients for a given material
can by highly variable (for example, friction for a steel/steel contact can
vary anywhere between 0.001 to 3), and can even vary significantly with time or
sliding distance during an experiment.
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Material
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Approx friction
coefficient
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Clean metals in air
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0.8-2
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Clean metals in wet air
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0.5-1.5
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Steel on soft metal (lead, bronze, etc)
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0.1-0.5
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Steel on ceramics (sapphire, diamond, ice)
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0.1-0.5
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Ceramics on ceramics (eg carbides on
carbides)
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0.05-0.5
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Polymers on polymers
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0.05-1.0
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Metals and ceramics on polymers (PE, PTFE,
PVC)
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0.04-0.5
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Boundary lubricated metals (thin layer of
grease)
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0.05-0.2
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High temperature lubricants (eg graphite)
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0.05-0.2
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Hydrodynamically lubricated surfaces (full
oil film)
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0.0001-0.0005
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HEALTH WARNING: Descriptions of Coulomb friction in elementary
mechanics and physics texts often distinguish between kinetic and static
friction coefficients. It is not
advisable to do adopt this approach when posing a boundary value problem in
continuum mechanics, as it is likely to make the problem ill-posed (with
either no solution, or an infinite number of solutions). In fact, even with a single friction
coefficient the Coulomb friction model can be ill-posed and should be used
with caution.
Coulomb
Friction with a shear cutoff: In metal forming applications contacting
surfaces can be subjected to extremely high pressure, with the result that
the true area of contact approaches the nominal area. Under these conditions, the shear traction
is no longer proportional to the contact pressure. Behavior at high pressure is often
approximated by truncating the shear traction at a critical value (usually
taken to be somewhat lower than the shear yield strength of the softer of the
two contacting surfaces). The modified
friction has the following constitutive equations:
(i) The interface separates, with an indeterminate if ;
(ii) The surfaces are prevented from
inter-penetrating if ;
(iii) We introduce the shear resistance of the
interface defined as
(iii) No slip occurs between the surfaces if ,
where is the coefficient of friction;
(iv) The tangential traction is proportional to the
normal pressure and opposes the direction of slip if the two surfaces slide .
Rate
and state variable models of friction: The variation of friction with
sliding velocity and transient behavior following changes in contact pressure
play an important role in controlling the stability of sliding on an
interface. Several friction laws have
been developed to describe this behavior, and are widely used in geophysics
applications. As a representative
example, we outline a constitutive law based loosely on work of Dieterich,
J.H.,. J. Geophys. Res. 84, 2161 (1979),
Ruina, A.L.,. J. Geophys. Res. 88,
10359 (1983) and Prakash, V., J. Tribol. 120,
97 (1998).
The
transient behavior of sliding friction is modeled by introducing two `state
variables’ for each material point on . The state variables evolve according to
where
,
are material properties with units of
length, and are material properties with units of
time. The two surfaces and may have different properties. To interpret
these equations, note that
1. Both and evolve with time and the sliding distance.
2. If the surfaces slide at constant speed, then in the steady state, while if the surfaces
are subjected to a time independent normal traction .
3. The two constants
control the time-scale associated with this
evolution for a static contact; while control the distance required for and to reach their steady state values under a
rapidly sliding contact.
HEALTH WARNING: Note that (i) state variables must be introduced
to characterize both contacting
surfaces, because coincident points on the two surfaces experience different
histories of contact pressure and slip velocity. To see this, note that as you slide your
finger over the surface of a table, a point on your finger sees a constant
contact pressure, while a point on the table experiences a cycle of loading. (ii) the time derivatives of the state
variables should be interpreted as the rate of change experienced by an
observer traveling with a particular material particle in each surface.
The
variation of steady-state friction coefficient with sliding velocity is
modeled by introducing a friction coefficient that is a function of the state
variables
where
,
and are all material properties. The constant represents the limiting value of the
friction coefficient as sliding velocity approaches infinity (it can be
interpreted as the kinetic friction coefficient), while is the steady-state value of friction
coefficient for a static contact (and can be interpreted as the static
friction coefficient). The two
constants control the rate at which the friction
transitions from one value to the other.
The
friction law can be conveniently expressed as a relationship between the
tractions and the relative velocity of the contact, as follows. For the surfaces are traction free . For ,
1. The elastic part of the relative velocity is related
to the traction components by
where and are two elastic stiffnesses. Note that these equations assume that the
elastic distorsion of the interface occurs on the slave surface (the time
derivatives correspond to the traction rate experienced by an observer fixed
to the slave surface).
2. The irreversible part of the normal component of
velocity
3. The irreversible part of the tangential component of
velocity is calculated from
where ,
,
and are two constants that control the variation
of shear stress to a step change in sliding velocity.
To
interpret this equation, suppose that the interface is subjected to a
constant (i.e. time independent) pressure, and is constrained to slip at a
rate . The magnitude of the shear traction follows
as
The
steady state value is . Following an instantaneous increase in
sliding speed, the shear traction first jumps to a new, higher value, then
progressively decreases to a lower steady-state value as approaches the new value of sliding
speed. Similarly, if the sliding speed
is suddenly reduced, the shear stress first drops to a lower value, and
subsequently increases gradually to a higher steady-state.
Representative values of material
properties for state variable model of friction. The subtle features of friction
captured by this constitutive equation are very sensitive to the materials
involved, the surface finish, and the environment. Extensive tests are required to
characterize a particular contacting pair.
As a rough guide to the orders of magnitudes of the various material
parameters, the table below lists rough estimates for parameters based on a
discussion by Coker, Needleman and Rosakis (J. Mech. Phys. solids, 53 p.884 (2005)) of transient
friction in Homalite.
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Representative values of parameters for a simple
state variable friction law
(the two contacting surfaces are assumed to have
identical properties)
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n
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300
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100
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0.5
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0.6
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26
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1.2
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m
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20
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20
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100
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5
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Friction laws based on plasticity theory:
The general framework of viscoplasticity can easily be adapted to
construct friction laws that approximate the variation of friction with
sliding speed and the evolution of friction with slip. Laws of this kind are often used in metal
forming simulations. Several such
models exist and will not be described in detail here. Instead, we will illustrate the general
idea by adapting the critical state theory of plasticity outlined in Section
3.10, together with the viscoplasticity law described in Section 3.7 to
construct a friction law that captures the transient behavior of a sliding
interface, as follows:
1. The normal traction must satisfy :
if the surfaces separate and
2. The relative velocity of the two surfaces is divided
into elastic and plastic parts
3. The elastic part of the relative velocity is related
to the traction components by
where and are two elastic stiffnesses. Note that these equations assume that the
elastic distorsion occurs on the slave surface (the time derivatives
correspond to the traction rate experienced by an observer fixed to the slave
surface).
4. Three state variables are introduced to track the history of
contact pressure, sliding speed and sliding distance on each of the two
contacting surfaces. The state
variables evolve according to
where ,
and are material properties. Naturally, the two surfaces may have
different values of . The governing equations for the evolution
of the state variables have been designed so that and under conditions of steady sliding.
5. The variation of relative velocity between the
surfaces with traction is approximated using a slip potential (similar to the
viscoplastic potential described in 3.7), defined as
where are material properties which control the
response of the interface to an instantaneous change in traction, ,
and is a representative shear strength which may
be a function of one or more of the state variables, as discussed further
below. The state variable plays the role of a in the critical state soil model outlined in Section 3.10.
6. The plastic part of the relative velocity between
the surfaces is related to g through
an associated flow law
Evaluating the derivatives gives
7. Finally, the variation of the tangential force with
contact pressure, sliding speed, and slip distance must be specified by an
appropriate equation for . Any sensible function can be chosen,
depending on the behavior that you would like to approximate. For example,
Setting
will produce Coulomb friction like behavior,
with a delayed response to changes in contact pressure. To see this, note that the model behaves
like the critical state soil model discussed in Section 3.10, with
`volumetric strain’ replaced by the normal separation between the surfaces,
and .
Setting
,
where is the root of the equation will give a Coulomb-like friction law with a
velocity dependent friction coefficient similar to the rate-and
state-variable model outlined earlier.
There
is very little to distinguish the rate- and state-variable model from the
plasticity based model. The plasticity
model has some advantages for numerical simulations, because (a) the
transition from stick to slip is gradual; (b) The plasticity model has a
`soft’ relationship between the normal displacement of the surfaces and the
normal pressure; (c) the plasticity model has an associated flow rule. All these tend to stabilize numerical
computations.
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