Chapter 6

modulus of elasticity

= E = young's modulus is the constant of proportionality for Hooke's law

Poisson's ratio

A parameter termed ____, ν, is defined as the ratio of the lateral and axial strains, or ν=−εx/εz=−εy/εz

permanent

A structure or component that has plastically deformed—or experienced a ____ change in shape—may not be capable of functioning as intended. It is therefore desirable to know the stress level at which plastic deformation begins, or where the phenomenon of yielding occurs.

plastic deformation

After yielding, the stress necessary to continue ____ in metals increases to a maximum, point M, and then decreases to the eventual fracture, point F

tensile

All deformation up to this point of maximum stress is uniform throughout the narrow region of the ____ specimen. However, at this maximum stress, a small constriction or neck begins to form at some point, and all subsequent deformation is confined at this neck

safe stress

Alternatively, a safe stress or working stress, σw, is used instead of design stress.

modulus of elasticity

As a consequence, the magnitude of the ____ is a measure of the resistance to separation of adjacent atoms, that is, the interatomic bonding forces.

stress

As a result of elastic elongation, there will be constrictions in the lateral (x and y) directions perpendicular to the applied ____; from these contractions, the compressive strains εx and εy may be determined

elastic behavior

As would be expected, the imposition of compressive, shear, or torsional stresses also evokes ____

plastic deformation

At the upper yield point, ____ is initiated with an apparent decrease in engineering stress. Continued deformation fluctuates slightly about some constant stress value, termed the lower yield point; stress subsequently rises with increasing strain.

grain

Because the ____ orientation is random in most polycrystalline materials, these may be considered to be isotropic; inorganic ceramic glasses are also isotropic.

metals

For ____, the anelastic component is normally small and is often neglected

slip

For crystalline solids, plastic deformation is accomplished by means of a process called ____, which involves the motion of dislocations

anisotropic

For elastically ____ materials, the elastic properties are completely characterized only by the specification of several elastic constants, their number depending on characteristics of the crystal structure Even for isotropic materials, for complete characterization of the elastic properties, at least two constants must be given

Poisson's ratio

For isotropic materials, shear and elastic moduli are related to each other and to ____ according to E=2G(1+ν)

Poisson's ratio

For many metals and other alloys, values of ____ range between 0.25 and 0.35

yield strength

For materials having a nonlinear elastic region, use of the strain offset method is not possible, and the usual practice is to define the ____ as the stress required to produce some amount of strain (e.g., ε = 0.005)

yield point phenomenon

For metals that display the ____, the yield strength is taken as the average stress that is associated with the lower yield point because it is well defined and relatively insensitive to the testing procedure. Thus, it is not necessary to employ the strain offset method for these materials.

stress strain

For metals that experience this gradual elastic-plastic transition, the point of yielding may be determined as the initial departure from linearity of the ____ curve; this is sometimes called the proportional limit and represents the onset of plastic deformation on a microscopic level

elastic deformation

For most metallic materials, ____ persists only to strains of about 0.005. As the material is deformed beyond this point, the stress is no longer proportional to strain (Hooke's law ceases to be valid), and permanent, nonrecoverable, or plastic deformation occurs.

stress strain

For nonlinear elastic behavior, either the tangent or secant modulus is normally used. The tangent modulus is taken as the slope of the ____ curve at some specified level of stress, whereas the secant modulus represents the slope of a secant drawn from the origin to some given point of the σ-ε curve.

structural

For virtually all ____ materials, εx and εz will be of opposite sign; therefore, the negative sign is included in the preceding expression to ensure that ν is positive

plastic deformation

From an atomic perspective, ____ corresponds to the breaking of bonds with original atom neighbors and then the re-forming of bonds with new neighbors as large numbers of atoms or molecules move relative to one another; upon removal of the stress, they do not return to their original positions

stress

If the applied ____ is uniaxial (only in the z direction) and the material is isotropic, then εx = εy

metals

In most ____, G is about 0.4E; thus, if the value of one modulus is known, the other may be approximated BUT PROF SAYS v = 0.33 FOR METALS

elastic strain

In most engineering materials, there will exist a time-dependent ____ component—that is, elastic deformation will continue after the stress application, and upon load release, some finite time is required for complete recovery.

anisotropic

Many materials are elastically ____; that is, the elastic behavior (the magnitude of E) varies with crystallographic direction

elastic deformation

Most structures are designed to ensure that only ____ will result when a stress is applied.

elastic strain

On an atomic scale, macroscopic ____ is manifested as small changes in the interatomic spacing and the stretching of interatomic bonds.

proportional limit

P The position of this point P is difficult to measure precisely As a consequence, a convention has been established by which a straight line is constructed parallel to the elastic portion of the stress-strain curve at some specified strain offset, usually 0.002. The stress corresponding to the intersection of this line and the stress-strain curve as it bends over in the plastic region is defined as the yield strength, σy (MPa or psi)

yield point phenomenon

Some steels and other materials exhibit the tensile stress-strain behavior in which the elastic-plastic transition is very well defined and occurs abruptly in what is termed a ____

atomic bonding

The differences in the moduli of elasticity for the different types of materials are a direct consequence of the different types of ____ in the three materials types.

plastic deformation

The magnitude of the yield strength for a metal is a measure of its resistance to ____

plastic deformation

The mechanism of ____ is different for crystalline and amorphous materials.

modulus of elasticity

The stress-strain characteristics at low stress levels are virtually the same for both tensile and compressive situations, to include the magnitude of the ____

stress strain

The tensile strength TS (MPa or psi) is the stress at the maximum on the engineering ____ curve. This corresponds to the maximum stress that can be sustained by a structure in tension; if this stress is applied and maintained, fracture will result.

plastic deformation

The transition from elastic to plastic is a gradual one for most metals; some curvature results at the onset of ____, which increases more rapidly with rising stress.

Poisson's ratio

Theoretically, ____ for isotropic materials should be ¼; furthermore, the maximum value for ν (or the value for which there is no net volume change) is 0.50

stress strain

There are some materials (i.e., gray cast iron, concrete, and many polymers) for which this elastic portion of the ____ curve is not linear; hence, it is not possible to determine a modulus of elasticity with Hooke's law

modulus of elasticity

Values of the ____ for ceramic materials are about the same as for metals; for polymers they are lower

tensile

When a ____ stress is imposed on a metal specimen, an elastic elongation and accompanying strain εz result in the direction of the applied stress (arbitrarily taken to be the z direction)

materials and metallurgical engineers

____ are concerned with producing and fabricating materials to meet service requirements as predicted by these stress analyses; this necessarily involves an understanding of the relationships between the microstructure (internal features) of materials and their mechanical properties

mechanical properties

____ are of concern to a variety of parties (producers and consumers of materials, research organizations, government agencies) that have differing interests

torsional

____ forces produce a rotational motion about the longitudinal axis of one end of the member relative to the other end

plastic deformation

____ in noncrystalline solids (as well as liquids) occurs by a viscous flow mechanism

torsion

____ is a variation of pure shear in which a structural member is twisted

safe stress

____ is based on the yield strength of the material and is defined as the yield strength divided by a factor of safety, N, or σw=σyN

elastic deformation

____ is nonpermanent, meaning that when the applied load is released, the piece returns to its original shape

modulus of elasticity

____ may be thought of as stiffness, or a material's resistance to elastic deformation, and is an important design parameter for computing elastic deflections

mechanical behavior

____ of a material reflects its response or deformation in relation to an applied load or force

mechanical properties

____ of materials are ascertained by performing carefully designed lab experiments that replicate as nearly as possible the service conditions; factors to be considered include the nature of the applied load and its duration, and environmental conditions

Shear

____ stress and strain are proportional to each other through the expression τ=Gγ G = the shear modulus

tensile

____ tests are more common because they are easier to perform; for most materials used in structural applications, very little additional information is obtained from compressive tests

torsional

____ tests are normally performed on cylindrical solid shafts or tubes

compressive

____ tests are used when a material's behavior under large and permanent (plastic) strains is desired, as in manufacturing applications or when the material is brittle in tension

Yield strength

____ values may range from 35 MPa (5000 psi) for a low-strength aluminum to greater than 1400 MPa (200,000 psi) for high-strength steels

compression

a ____ test is conducted in a manner similar to the tensile test, except that the force is compressive and the specimen contracts along the direction of the stress

stress strain

a ____ test typically takes several minutes to perform and is destructive, meaning that the test specimen is permanently deformed and usually fractured

shear stress

a ____ τ is a function of the applied torque T, whereas shear strain γ is related to the angle of twist φ (phi)

tensile

a specimen is deformed, usually to fracture, with a gradually increasing ____ load that is applied uniaxially along the long axis of a specimen; normally the cross section is circular, but rectangular specimens are also used; this "dogbone" specimen configuration was chosen so that during testing the deformation is confined to the narrow center region (which has a uniform cross section along its length) and also to reduce the likelihood of fracture at the ends of the specimen

compressive

by convention, ____ force is taken to be negative, which yields a negative stress; because l₀ is greater than li, ____ strains are necessarily also negative

stress strain

compression ____ tests may be conducted if in-service forces are of this type

tensile

consider the the cylindrical tensile specimen that is subjected to a ____ stress σ applied parallel to its axis, and the plane p-p' oriented at some arbitrary angle θ to the ends of the specimen upon the plane p-p', the applied stress is no longer a pure ____ one; a more complex stress state is present that consists of a ____ (or normal) stress σ' that acts normal to the plane and a shear stress τ' that acts parallel to the plane σ' = σ*cos²θ = σ[(1+cos(2θ)) / 2] τ' = σ*sinθ*cosθ = σ(sin(2θ) / 2) these same mechanics principles allow the transformation of stress components from one coordinate system to another coordinate system with a different orientation

elastic deformation

deformation in which stress and strain are proportional is called ____ a plot of stress vs strain results in a linear relationship with a slope of E

tension

for most metals that are stressed in ____ and at relatively low levels, stress and strain are proportional to each other through the relationship σ = Eε

anelasticity

for some polymeric materials, the magnitude of the ____ is significant; in this case it is termed viscoelastic behavior

shear stress

for tests performed using a pure shear force, the ____ is computed according to: τ = F/A₀ F = load or force imposed parallel to the upper and lower faces A₀ = area of each face

static

if a load is ____ or changes relatively slowly with time and is applied uniformly over a cross section or surface of a member, the mechanical behavior may be ascertained by a simple stress-strain test; these are most commonly conducted for materials at room temperature

shear

in engineering practice many loads are torsional rather than pure ____

strain

independent of unit system

tests

it is imperative that there be some consistency in the manner in which ____ are conducted and in the interpretation of their results, which is accomplished by using standardized testing techniques

load

it is possible for the ____ to be tensile, compressive, or shear, and its magnitude may be constant with time or it may fluctuate continuously; application time may be only a fraction of a second, or it may extend over a period of many years; service temperature may be an important factor

load

it requires twice the ____ to produce the same elongation if the cross sectional area of the specimen is doubled

mechanical properties

key ____ are stiffness, strength, hardness, ductility, and toughness

loads

many materials are subjected to forces or ____ when in service; in such situations it is necessary to know the characteristics of the material and to design the member from which it is made such that any resulting deformation will not be excessive and fracture will not occur

structural

materials are frequently chosen for ____ applications because they have desirable combinations of mechanical characteristics

tension

one of the most common mechanical stress-strain tests is performed in ____

plastic deformation

permanent, nonrecoverable deformation

sperately

polymers and ceramics are treated ____ because they are, to a large degree, mechanically different from metals

modulus of elasticity

range of ____ values for metals is between 45GPa and 407 GPa dependent on temperature

strain

sometimes expressed as a percentage, with the value multiplied by 100

dogbone specimen

the ____ is mounted by its ends into the holding grips of the testing apparatus

modulus of elasticity

the ____ is proportional to the slope of the interatomic force-separation curve at the equilibrium spacing E ∝ (dF/dr) at r₀

stress

the ____ state is a function of the orientations of the planes upon which the stresses are taken to act

tension

the ____ test can be used to ascertain several mechanical properties that are important in design

tensile

the ____ testing machine is designed to elongate the specimen at a constant rate, and to continuously and simultaneously measure the instantaneous applied load with a load cell and the resulting elongations with an extensometer

stress

the degree to which a structure deforms or strains depends on the magnitude of an imposed ____

modulus of elasticity

the greater the ____, the stiffer the material, and the smaller the elastic strain that results from an applied stress

tensile

the output of a ____ test is recorded as load or force vs elongation; these load-deformation characteristics depend on the specimen size

structural engineers

the role of ____ is to determine stresses and stress distributions within members that are subjected to well-defined loads; this may be accomplished by experimental testing techniques and/or by theoretical and mathematical stress analyses

shear modulus

the slope of the linear elastic region of the shear stress-strain curve

dogbone specimen

the standard diameter is about 12.8mm/0.5in. whereas the reduce section length should be at least 4 times this diameter: 60mm/2.25in. is common; guage length is used in ductility computations with a standard value of 50mm/2.0in.

load

there are 3 principle ways in which a ____ may be applied: tension, compression, shear

anelasticity

time-dependent elastic behavior is known as ____, and it is due to time-dependent microscopic and atomistic processes that are attendant to the deformation

engineering

to minimize these geometrical factors, load and elongation are normalized to the respective parameters of ____ stress and ____ strain

modulus of elasticity

with increasing temperature, the ____ decreases

shear strain

γ = tanθ = ∆x/L defined as the tangent of the strain angle θ

engineering strain

ε = (li - l₀)/l₀ = ∆l/l₀ (unitless or m/m or in./in.) l₀ = original length before any load is applied li = instantaneous length

Hooke's law

σ = Eε

engineering stress

σ = F/A₀ (in MPa, 1 MPa = 10⁶ N/m², or psi) F = the instantaneous load applied perpendicular to the specimen cross section (N or lb force) A₀ = original cross sectional area before any load is applied (m² or in.²)