3.5 Response of Materials to Stress
55
Of course, in nature, some bodies are ‘softer’ than others. We can sometimes
approximate the hard ones as if they were acting rigidly. We use the term rigid
only as an approximation, in cases where the internal relative displacements are not
important to the problem being investigated. In this discussion, we are interested
particularly in those displacements which distort a material.
Even in equilibrium, internal distortions of the material may generate forces to
oppose the external surface and body forces acting on a given material element
and cause internal forces of one material element on the next. For example, the
response of a bone to a set of distorting forces is an opposing set of forces while
the bone changes in shape. The response of an elastic band to forces at its ends
is to stretch. Stretching produces internal forces on each piece of the rubber. The
action of increased pressure on gas bubbles results in a decreased gas volume,
thereby increasing the outward force on the agents causing the pressure. Confined
sidewalks can explode from internal stress as the temperature of the concrete rises.
The concrete attempts to expand, but may experience large opposing forces, and so
the concrete might become compressed beyond its fracture limit.
To characterize the effect of forces in distorting a body, we can apply Newton’s
2nd law 30 to each small element within that body, and Newton’s 3rd law to follow
the effects from one element to the next. The choice of the elements is arbitrary, and
they need not be physical. However, for simplicity, the elements are usually taken
sufficiently small that relevant quantities within do not vary appreciably (to some
approximation acceptable in a calculation).
In the mathematical descriptions of the mechanical behavior of a system of
entities, an enumeration of each entity is often given by specifying its spatial
position in a selected coordinate system. Strategic choice of a coordinate system
according to its symmetries can simplify calculations. For example, the elements in
a bone with approximate cylindrical symmetry are best followed by their location
in a set of appropriate cylindrical coordinates. As the system responds to forces, the
elements may distort in shape as well as move. To be definite, we will circumscribe
each element with an imaginary surface which follows all the atoms in the initial
volume of the element. Keeping the total number and kind of atoms fixed also keeps
the total mass of the element constant to within the precision of ordinary laboratory
instruments.
In any set of ‘curvilinear coordinates’ (i.e. ones whose axes are locally perpendicular), small elements in a body can be selected to be initially approximate
rectangular solids with surfaces perpendicular to coordinate lines. As we described,
forces acting on such elements can be divided into two types: body forces and
surface forces. Body forces act on all parts within the volume of the elements.
The body forces acting on each element will be taken together as a single vector.
30 A useful alternative to studying forces is to consider the energy changes in a body as it is
distorted. One advantage of starting with energy is the fact that energy is a number, while force
has both size and direction. However, our intuition is built first with the effect of forces, so we will
continue with them.
55
Of course, in nature, some bodies are ‘softer’ than others. We can sometimes
approximate the hard ones as if they were acting rigidly. We use the term rigid
only as an approximation, in cases where the internal relative displacements are not
important to the problem being investigated. In this discussion, we are interested
particularly in those displacements which distort a material.
Even in equilibrium, internal distortions of the material may generate forces to
oppose the external surface and body forces acting on a given material element
and cause internal forces of one material element on the next. For example, the
response of a bone to a set of distorting forces is an opposing set of forces while
the bone changes in shape. The response of an elastic band to forces at its ends
is to stretch. Stretching produces internal forces on each piece of the rubber. The
action of increased pressure on gas bubbles results in a decreased gas volume,
thereby increasing the outward force on the agents causing the pressure. Confined
sidewalks can explode from internal stress as the temperature of the concrete rises.
The concrete attempts to expand, but may experience large opposing forces, and so
the concrete might become compressed beyond its fracture limit.
To characterize the effect of forces in distorting a body, we can apply Newton’s
2nd law 30 to each small element within that body, and Newton’s 3rd law to follow
the effects from one element to the next. The choice of the elements is arbitrary, and
they need not be physical. However, for simplicity, the elements are usually taken
sufficiently small that relevant quantities within do not vary appreciably (to some
approximation acceptable in a calculation).
In the mathematical descriptions of the mechanical behavior of a system of
entities, an enumeration of each entity is often given by specifying its spatial
position in a selected coordinate system. Strategic choice of a coordinate system
according to its symmetries can simplify calculations. For example, the elements in
a bone with approximate cylindrical symmetry are best followed by their location
in a set of appropriate cylindrical coordinates. As the system responds to forces, the
elements may distort in shape as well as move. To be definite, we will circumscribe
each element with an imaginary surface which follows all the atoms in the initial
volume of the element. Keeping the total number and kind of atoms fixed also keeps
the total mass of the element constant to within the precision of ordinary laboratory
instruments.
In any set of ‘curvilinear coordinates’ (i.e. ones whose axes are locally perpendicular), small elements in a body can be selected to be initially approximate
rectangular solids with surfaces perpendicular to coordinate lines. As we described,
forces acting on such elements can be divided into two types: body forces and
surface forces. Body forces act on all parts within the volume of the elements.
The body forces acting on each element will be taken together as a single vector.
30 A useful alternative to studying forces is to consider the energy changes in a body as it is
distorted. One advantage of starting with energy is the fact that energy is a number, while force
has both size and direction. However, our intuition is built first with the effect of forces, so we will
continue with them.