Introduction Objectives and Applications of Solid Mechanics
Solid Mechanics is a collection of physical laws,
mathematical techniques, and computer algorithms that can be used to predict
the behavior of a solid material that is subjected to mechanical or thermal
loading. The field has a wide range of applications, including
FEA model of a knee joint from the
website. They have several nice
Geomechanics modeling the shape of planets; tectonics; and
designing foundations or structures;
engineering designing load bearing components for
vehicles; power generation and transmission;
engineering designing metal and polymer forming processes;
Biomechanics designing implants; bone mechanics; as well as
modeling stress driven phenomena controlling celular and molecular processes;
designing composites; alloy microstructures,
thin films, and developing materials processing;
Microelectronics designing failure resistant packaging and
Nanotechnology stress driven self-assembly on surfaces;
manufacturing processes such as nano-imprinting; modeling atomic-force
This chapter describes how solid mechanics can be used
to solve practical problems. The
remainder of the book contains a more detailed description of the physical laws
that govern deformation and failure in solids, as well as the mathematical and
computational methods that are used to solve problems involving deformable
Chapter 2 covers the
mathematical description of shape changes and internal forces in solids;
Chapter 3 discusses
constitutive laws that are used to relate shape changes to internal forces;
4 contains analytical solutions to a series of simple problems involving
5 provides a short summary of analytical techniques and solutions for linear
Chapter 6 describes
analytical techniques and solutions for plastically deforming solids;
Chapter 7 gives an
introduction to finite element analysis, focusing on using commercial software;
Chapter 8 has a more
complete discussion of the implementation of the finite element method;
Chapter 9 describes how
to use solid mechanics to model material failure;
10 discusses solids with special geometries (rods, beams, membranes, plates and
Solid mechanics is incomprehensible without some
background in vectors, tensors and index notation. These topics are reviewed briefly in the
1.1 Defining a Problem in Solid
of the application, the general steps in setting up a problem in solid
mechanics are always the same:
Decide upon the
goal of the problem and desired information;
geometry of the solid to be modeled;
loading applied to the solid;
physics must be included in the model;
calibrate) a constitutive law that
describes the behavior of the material;
Choose a method of
step in the process is discussed in more detail below.
1.1.1 Deciding what
An animation showing one of the vibration modes of
the Mars Orbiter Laser Altimiter, from a NASA website.
This seems a rather silly question but at some point of their careers, most
engineers have been told by their manager `Why don’t you just set up a finite
element model of our (crank-case; airframe; material., etc, etc) so we can stop
it from (corroding; fatiguing; fracturing, etc).’ If you find yourself in this
situation, you are doomed. Models can certainly be helpful in preventing
failure, but unless you have a very clear idea of why the failure is occurring,
you won’t know what to model.
Here is a list of of some of the things that can
typically be calculated very accurately using solid mechanics:
shape of a structure or component subjected to mechanical, thermal or
required to cause a particular shape change;
The stiffness of
a structure or component;
forces (stresses) in a structure or component;
forces that lead to failure by structural instability (buckling);
model of rupture during tube hydroforming.
frequencies of vibration for a structure or component.
In addition, solid mechanics can be used to model a
variety of failure mechanisms. Failure
predictions are more difficult, however, because the physics of failure can
only be modeled using approximate constitutive equations. These must be calibrated experimentally, and
do not always perfectly characterize the failure mechanism. Nevertheless, there are well established
procedures for each of the following:
critical loads to cause fracture in a brittle or ductile solid containing a
fatigue life of a component under cyclic loading;
Predict the rate
of growth of a stress-corrosion crack in a component;
Predict the creep
life of a component;
Find the length
of a crack that a component can contain and still withstand fatigue or
Predict the wear
rate of a surface under contact loading;
fretting or contact fatigue life of a surface.
FEA model of the evolution of grain structure of a
polycrystal during deformation. From ORNL
Solid mechanics is increasingly being used for
applications other than structural and mechanical engineering design. These are active research areas, and some are
better developed than others.
properties (e.g. elastic modulus, yield stress, stress-strain curve; fracture
toughness, etc) of a composite material in terms of those of its constituents.
influence of the microstructure (e.g. texture; grain structure; dispersoids;
etc) on the mechanical properties of metals such as modulus, yield stress,
strain hardening, etc.
physics of failure in materials, including fracture, fatigue, plasticity, and
wear, and using the models to design failure resistant materials
materials processing examples include casting and solidification;
alloy heat treatments; thin film and surface coating deposition (e.g. by
sputtering, vapor deposition or electroplating);
biological phenomena and processes, such as bone growth; cell mobility; cell
wall/particle interactions; and bacterial mobility.
Defining the geometry of the solid
Again, this seems rather obvious surely the shape of the solid is always
known? True but it is usually not obvious how much of the component to model, and
at what level of detail. For example,
in a crash simulation, must the entire vehicle be modelled, or just the front
part? Should the engine block be
included? The passengers?
At the other extreme, it is often not obvious how much
geometrical detail needs to be included in a computation. If you model a component, do you need to
include every geometrical feature (such as bolt holes, cutouts, chamfers,
etc)? The following guidelines might be
brittle fracture, fatigue failure, or for calculating critical loads required
to initiate plastic flow in a component, it is very important to model the
geometry in great detail, because geometrical features can lead to stress
concentrations that initiate damage.
creep damage, large scale plastic deformation (eg metal forming), or vibration
analysis, geometrical details are less important. Geometrical features with dimensions under
10% of the macroscopic cross section can generally be neglected.
features often only influence local
stresses they do not have much influence far away. Saint Venant’s principle, which will be
discussed in more detail in Chapter 5, suggests that a geometrical feature with
characteristic dimension L (e.g. the
diameter of a hole in the solid) will influence stresses over a region with
dimension around 3L surrounding the
feature. This means that if you are
interested in the stress state at a particular point in an elastic solid, you
need not worry about geometrical features that are far from the region of
interest. Saint-Venants principle
strictly only applies to elastic solids, although it can usually also be
applied to plastic solids that strain harden.
As a general rule, it is best to start with the
simplest possible model, and see what it predicts. If the simplest model answers your question,
you’re done. If not, the results can
serve as a guide in refining the calculation
are five ways that mechanical loads can be induced in a solid:
can be subjected to a prescribed displacement or motion;
can be subjected to a distribution of pressure normal to the surface, or
frictional traction tangent to the surface;
A boundary may be
subjected to a combination of displacement and traction (“mixed”) boundary
conditions for example, you could prescribe horizontal
displacements, together with the vertical traction, at some point on the
The interior of
the solid can be subjected to gravitational or electromagnetic body forces;
The solid can
contact another solid, or in some cases can contact itself.
Stresses can be
induced by nonuniform thermal expansion in the solid, or some other materials
process such as phase transformation that causes the solid to change its shape.
When specifying boundary conditions, you must follow
In a 3D analysis,
you must specify three components of
either displacement or traction (but not both) at each point on the boundary.
You can mix these so for example you could prescribe or ,
but exactly three components must always be prescribed. This includes free
surfaces, where the tractions are prescribed to be zero.
Similarly, in a
2D analysis you must prescribe two components of displacement or traction at
each point on the boundary.
If you are
solving a static problem with only tractions prescribed on the boundary, you
must ensure that the total external force acting on the solid sums to zero
(otherwise a static equilibrium solution cannot exist).
In practice, it can be surprisingly difficult to find
out exactly what the loading on your system looks like. For example, earthquake loading on a building
can be modeled as a prescribed acceleration of the building’s base but what acceleration should you apply? Pressure loading usually arises from wind or
fluid forces, but you might need to do some sophisticated calculations just to
identify these forces. In the case of
contact loading, you’ll need to be able to estimate friction coefficients. For nano-scale or biological applications,
you may also need to model attractive forces between the two contacting
This is where standards
are helpful. For example, building
codes regulate civil engineernig structures; NHTSA specify design requirements for
vehicles, and so on.
You can also avoid the need to find exactly what
loading a structure will experience in service by simply calculating the
critical loads that will lead to failure, or the fatigue life as a function of
loading. In this case some other
unfortunate engineer will have to decide whether or not the failure loads are
Deciding what physics to include in the model
are three decisions to be made here:
1. Do you need to calculate additional field quantities,
such as temperature, electric or magnetic fields, or mass/fluid diffusion
through the solid? Temperature is the
most common additional field quantity. Here are some rough guidelines that will
help you to decide whether to account for heating effects.
As a rough guide, the stress induced by temperature variation in a component is ,
where E is the Young’s modulus of the
material; is its thermal expansion coefficient, and T is temperature. The symbols denote the maximum and minimum values of the
product in the component. You need to account for temperature
variations if is a significant fraction of the stress
induced by mechanical loading.
Thermal stress analysis of a turbine casing, from
the consulting firm IDAC
To decide whether you need to do a transient heat conduction analysis, note
that the temperature rise at a distance r
from a point source of heat of intensity in an infinite solid is ,
where erfc() denotes the complementary error function, is the material’s thermal conductivity, and is its thermal diffusivity, with the mass density and the specific heat capacity. This suggests that
a solid with dimension L will reach
its steady state temperature in time . If the time-scale of interest in your problem
is significantly larger than this, and heat flux is contstant, you can use the
steady-state temperature distribution. If
not, you must account for transients.
Finally to decide whether you need to account for heat generated by plastic flow, note
that the rate of heat generation per unit volume is of order where is the material yield stress, and is the plastic strain rate. The temperature rise due to rapid (adiabatic)
plastic heating is thus of order ,
where is the strain increment applied to the
A dynamic analysis of a leg kicking a soccer ball.
From the Fluent
2. Do you need to do a dynamic analysis, or a static
analysis? Here are some rough
guidelines that will help you to decide:
The speed of a shear wave propagating through an
elastic solid is ,
where is the mass density of the solid, is its shear modulus and its Poisson’s ratio. The time taken for a wave to propagate across
a component with size L is of order . In many cases, stresses decay to their static
values after about 10L/c. If the loading applied to the component does
not change significantly during this time period a quasi-static computation
(possibly including accelerations as body forces) should suffice.
The stress induced by acceleration (e.g. in a rotating
component) is of order ,
where L is the approximate size of
the component, is its mass density, and a is the magnitude of the acceleration. If this stress is negligible compared with
other forces applied to the solid, it can be neglected. If not, it should be included (as a body
force if wave propagation can be neglected).
A coupled fluid-solid analysis from CEI
3. Are you solving a coupled
fluid/solid interaction problem?
These arise in aeroelasticity (design of flexible aircraft wings or
helicopter rotor blades; or very long bridges); offshore structures; pipelines;
or fluid containers. In these
applications the fluid flow has a high Reynold’s number (so fluid forces are
dominated by inertial effects). Coupled
problems are also very common in biomedical applications such as blood flow or
cellular mechanics. In these applications the Reynolds number for the fluid
flow is much lower, and fluid forces are dominated by viscous effects. Different analysis techniques are available
for these two applications. Such problems are beyond the scope of this book.
Defining material behavior
Choosing the right equations to describe material
behavior is the most critical part of setting up a solid mechanics
calculation. Using the wrong model, or
inaccurate material properties, will always completely invalidate your
predictions. Here are a few of your
choices, with suggested applications:
FEA model of a rubber tire, using a hyperelastic constitutive
equation. From the LOZIK website
linear elasticity (familiar in one
dimension as ) good for polycrystalline metals, ceramics,
glasses, and polymers udergoing small deformations and subjected to low loads
(less than the material yield stress).
Only two material constants are required to characterize the material,
and material data is highly accurate and readily available.
linear elasticity (similar to
isotropic linear elasticity, but models materials which are stiffer in some
directions than others). Good for reinforced composites; wood; single crystals
of metals and ceramics. Between 3 and 21
material properties must be determined. Material data is highly accurate and
readily available.Isotropic or anisotropic linear elasticity is good for the
vast majority of engineering design calculations, where components cannot
safely exceed yield. It can be used for
deflection calculations, fatigue analysis, and vibration analysis.
3. `Hyperelasticity’ used to model rubber and foams, which can
sustain huge, reversible, shape changes.
There are several models to choose from.
The simplest model (the incompressible Neo-Hookean solid, which, in
uniaxial stress, has a true stress true strain relation given by ) has only a single material constant. More complex models have several parameters,
and it may be difficult to find values for your material in the published
literature. Experimental calibration
will almost certainly be required.
FEA model of a die-forming process, from the DIPRO website. A sophisticated plasticity model would be
required to accurately predict the wall-thickness of the part and its elastic
4. Viscoelasticity. Used to model materials which exhibit a gradual
increase in strain when loaded at constant stress (with stress rate-v-strain
rate ) or which show hysteresis during cyclic
loading (with stress-v-strain rate of form
Usually used to model polymeric materials and polymer based composites,
and biological tissue. Can also model
slow creep in amorphous solids such as glass.
Constitutive equations contain at least 3 parameters, and usually many
more. Material behavior varies widely between materials and is highly
temperature dependent. Experimental
calibration will almost certainly be required to obtain accurate predictions.
5. Rate independent
metal plasticity. Used to model permanent deformation in metals
loaded above their yield point. A wide
range of models are available. The
simplest is a rigid perfectly plastic solid, which changes its shape only if
loaded above its yield stress ,
and then deforms at constant stress. An elastic-perfectly plastic solid deforms
according to linear elastic equations when loaded below the yield stress, but
deforms at constant stress if yield is exceeded. These models would be appropriate to predict
energy dissipation in a crash analysis, or to calculate tool forces in a metal
cutting operation, for example. Data for material yield stress are readily
available, but are sensitive to material processing and microstructure and so
should be used with caution. More sophisticated models describe strain hardening in some way (the change
in the yield stress of the solid with plastic deformation). These are used in modeling ductile fracture,
low cycle fatigue (where the material is repeatedly plastically deformed), and
when predicting residual stresses and springback in metal forming operations.
Finally, the most sophisticated plasticity models attempt to track the
development of microstructure or damage in the metal. For example, the Gurson plasticity model (developed at Brown University
by then-graduate student A.L Gurson) models the nucleation and growth of voids
in a metal, and is widely used to simulate ductile fracture. Such models typically have a large number of
parameters, and can differ widely in
their predictions. They must be very carefully chosen and calibrated to get
FEA simulation of fracture in a weld using a Gurson
platicity model, which accounts for ductile rupture by void nucleation and
growth. From the NW numerics website
6. Viscoplasticity: similar in structure to metal plasticity, but
accounts for the tendency of the flow stress of a metal to increase when
deformed at high strain rates. These
would be used in modeling high-speed machining, for example, or in applications
involving explosive shock loading.
Viscoplastic constitutive equations are also used to model creep the steady accumulation of plastic strain in a
metal when loaded below its yield stress, and subjected to high
temperatures. The simplest viscoplastic
constitutive law has a uniaxial strain
rate versus stress response of the form and so has only two parameters. More complex models account for elastic
deformation and strain hardening. Data
for the simple models is quite easy to find, but more sophisticated and
accurate models would have to be calibrated experimentally.
7. Crystal plasticity: Used to model
anisotropic plastic flow in a single crystal of a metal. Mostly used in materials science calculations
and in modeling some metal forming processes.
These models are still under development, material data is not easy to
find, and very laborious and expensive to measure.
Gradient Plasticity: A formulation
developed in the last 5-10 years to model the behavior very small volumes of a
metal (i.e. less than 100 ).
Typically, small volumes of metal are stronger than bulk samples. These models are still under development, are
difficult to calibrate, and don’t always work very well.
9. Discrete Dislocation Plasticity: A
technique to model plastic flow in very small volumes of material by tracking
the nucleation, motion and annihiliation of individual dislocations in the
solid. DDP models contain a large number
of material parameters that are very hard to calibrate. Currently a research
Discrete dislocation plasticity simulation of
dislocations near the tip of a propagating crack. From the GM/Brown CRL
state plasticity (cam-clay). Used to model soils, whose behavior depends
on moisture content. Somewhat similar in
structure to metal plasticity, except that the yield strength of a soil is
highly pressure dependent (it increases with compressive pressure). Simple models contain only 3 or 4 material
parameters that can be calibrated quite accurately.
viscoplasticity. Similar to critical state plasticity, in that these
models account for changes in flow stress of a material with confining
pressure. Used to model granular
materials, and some polymers and composite materials (typically in modeling
processes such as extrusion or drawing).
models. Intended to model the
crushing (in compression) or fracture (in tension) of concrete (obviously!).
The mathematical structure resembles that of pressure dependent plasticity.
Quasicontinuum simulation of a crack approaching a
bi-material interface. From the Quasicontinuum website.
models. Replace traditional
stress-strain laws with a direct calculation of stress-strain behavior using
embedded atomic scale simulations. The
atomic scale computations use empirical potentials to model atom interactions,
or may approximate the Schrodinger wave equation directly. Techiques include
the `Quasi-Continuum’ method, and the Coupled-Atomistic-Discrete Dislocation
Method. Their advantage is that they capture the physics of material behavior
extremely accurately; their disadvantage is that they currently can only model
extremely small material volumes (20-100nm or so). Atomistic models based on
empirical potentials contain a large number of adjustable parameters these are usually calibrated against known
quantities such as elastic moduli and stacking fault energies, and can also be
computed using ab-initio techniques. The
accuracy of the the predictions depends strongly on the accuracy of the
potentials. Mostly a research tool for nanotechnology and materials design
This is by no means an exhaustive list additional models are available for materials
such as shape memory alloys, and piezoelectric materials, for example.
These material models are intended primarily to approximate
stress-strain behavior. Special
constitutive equations have also been developed to model the behavior of
contacting surfaces or interfaces between two solids (Coulomb friction is a
simple example). In addition, if you need to model damage (fracture or
fatigue), you may need to select and calibrate additional material models. For example, to model brittle fracture, you
would need to know the fracture toughness of the material. To model the growth of a fatigue crack, you would
probably use Paris’
grack growth law and would need data for the Paris constant C and exponent n. There are several other stress- or strain-based
fatigue laws in common use. These models
are often curve fits to experimental data, and are not based on any detailed
physical understanding of the failure mechanism. They must therefore be used with caution, and
material properties must be measured carefully.
representative initial value problem in solid mechanics
The result of the decisions made in Sections
1.1.1-1.1.5 is a Boundary value problem (for
static problems) or initial value problem
(for dynamic problems). This consists of
a set of partial differential equations, together with initial and boundary
conditions, that must be solved for the displacement and stress fields, as well
as any auxiliary fields (such as temperature) in the solid. To illustrate the structure of these
equations, in this section we list the governing equations for a representative
initial value problem.
a representative example, we state the initial value problem that governs
elastic wave propagation in a linear elastic solid.
The shape of the
solid in its unloaded condition
modulus E and Poisson’s ratio for the solid
expansion coefficient for the solid, and temperature distribution in the solid (for simplicity we assume that
the temperature does not vary with time)
displacement field in the solid ,
and the initial velocity field
A body force
distribution (force per unit volume) acting on the solid
conditions, specifying displacements on a portion and tractions on a portion of the boundary of R
Calculate displacements ,
strains and stresses satisfying the governing equations of linear
strain-displacement (compatibility) equation
elastic stress-strain law
- The equation of motion for a continuum (F=ma)
The fields must
satisfy initial conditions
As well as
Choosing a method of analysis
Once you have set up the problem, you will need to
solve the equations of motion (or equilibrium) for a continuum, together with
the equations governing material behavior, to determine the stress and strain
distributions in the solid. Several
methods are available for this purpose.
Exact solutions: There is a good chance that you can find an exact
Analytical solution to the stress field around a
hypotrochoidal hole in an elastic solid
2D (plane stress
or plane strain) linear elastic solids, particularly under static loading. Solution techniques include transforms,
stress function methods, and complex variable methods. Dynamic solutions are
also possible, but somewhat more difficult.
elasticitity problems can be solved, usually using integral transforms, if they
are simple enough.
2D (plane strain)
deformation of rigid plastic solids (using slip line fields)
Naturally, analytical solutions are most easily found
for solids with a simple geometry (e.g. an infinite solid containing a crack;
loading applied to a flat surface, etc).
In addition, special analytical techniques can be used for problems
where the solid’s geometry can be approximated in some way. Examples include membrane theory; shell and
plate theory; beam theory; and truss analysis.
Even when you can’t find an exact solution to the
stress and strain fields in your solid, you can sometimes get the information
you need using powerful mathematical theorems.
For example, bounding theorems allow you to estimate the plastic
collapse loads for a structure quickly and easily.
Solutions: These are used for most
engineering design calculations in practice. These include
element method which will be discussed in detail in this book. This is the most widely used technique, and
can be used to solve almost any problem in solid mechanics, provided you understand
how to model your material, and have access to a fast enough computer.
methods somewhat similar to FEM but much less widely
integral equation method (or boundary element method) is a more efficient
computer technique for linear elastic problems, but is less well suited to nonlinear
materials or geometry.
methods: Used more in computational fluid dynamics than in solids, but good for
problems involving very large deformations, where the solid flows much like a
methods: used in nanotechnology applications to model material behavior at the
atomic scale. Molecular Dynamic
techniques integrate the equations of motion (Newton’s laws) for individual atoms;
Molecular statics solve equilibrium equations to calculate atom positions. The forces between atoms are computed using
empirical constitutive equations, or sometimes using approximations to quantum
mechanics. These computations can only
consider exceedingly small material volumes (up to a few million atoms) and
short time-scales (up to a nanosecond).