Ch. 7: Mechanical Properties
What is the main difference between macroscopic deformation of semicrystalline polymers compared to ductile metals?
In semicrystalline polymers, necking occurs when chains become oriented parallel to the direction of elongation. this leads to localized strengthening and elongation occurs as this neck of aligned regions extends further across the gauge length. this is different for ductile metals because in metals, once a neck forms, all deformation occurs within the neck region.
What is strain hardening (or cold-working) in the context of elastic/plastic deformation?
Initially, a load is applied that results in plastic deformation. Then the load is released, and reapplied; after the sceond time, the yield strength is greater than the initial strength.
Tensile Strength
Maximum possible engineering stress in tension (before this point, all deformation occurs uniformly: after this point, "necking" occurs, decreasing the cross-sectional area, preventing any greater engineering stress, and eventually leading to fracture in this location). This is also the *maximum load-bearing capacity.*
Is the modulus of polymers constant?
Not necessarily; varies with time and strain rate. as a result, need to report strain rate and fracture strain before fracture.
Poisson's ratio is given as v = -εx/εz = -εy/εz. What is the meaning of this, and why is the sign negative?
Poisson's ratio gives the ratio of the lateral and axial strains. In general the axial strain is opposite in sign to the lateral strain, which is why there is the negative sign. also, remember: this is (-width strain)/(axial strain), that is, -(∆w/w)/(∆l/l). strain defined as usual
How are tensile strength and yield strength defined in (plastic) polymers?
Tensile strength: *stress at which fracture occurs.* Yield strength: point at which engineering stress is at a maximum, immediately after the termination of the elastic region of the curve. Although this "seems" slightly counterintuitive, it really isn't: in a metal stress-strain curve, the yield strength occurs "before" the tensile strength (ie, right after the linear region), just as it does for polymers.
what conditions must apply to the system for the given equations converting engineering stress/strain --> true stress/strain to apply? and how long are these equations valid?
The volume must remain the same before and after deformation. These equations can only apply up to the beginning of necking.
In general, what is the appropriate approach for complex states of stress/strain in 3D?
apply Hooke's Law in direction parallel to applied stress; apply poisson effect to direction perpendicular to stress. Use linear superposition to combine the effects.
Define viscoelastic creep.
deformation as a function of time when stress is kept constant. Creep modulus: E(t) = (stress)/(strain(t))
Define ductility: what does it mean if a material is "brittle"? how is ductility expressed? How does temperature influence this property?
ductility == the degree of plastic deformation that a material has sustained at fracture. if a material is brittle, this means it experiences very little plastic deformation before fracturing. In general, increasing temperature increases ductility, and decreasing temperature makes the material more brittle (for metals). Ductility is expressed as either %elongation or %reduction in area; in both cases this corresponds to final state at fracture (length at fracture or area at fracture) in relation to the original state (original length/original area).
Define tangent modulus for polymers.
for polymers, the stress-strain curve is not linear. Tangent modulus is the slope of the tangent line to the curve at some specified amount of stress. This is appropriate if you know the amount of stress a material might be subjected to during real use.
tension
force "pulling apart" the ends of the material
compression; sign convention?
force "pushing in" on the ends of the material. by convention, stress/strain for compression are *negative.*
What is the "strain offset method" for determining the yield strength?
(1) Choose a starting "offset" (often 0.002 strain) (2) draw a line parallel to the linear region (3) yield strength is where the dotted line crosses the stress-strain curve
why is the "dog bone" the conventional shape for stress-strain testing apparatus?
remember that stress == force/area. So by reducing the cross-sectional area, you increase the stress, thus increasing the likelihood that the item will fracture at the desired area.
strain
response of the material to stress (ie, physical deformation). *engineering strain* == ∆l/l0 (change in length/original length)
what is the proportional limit?
the point of beginning of deviation from linear behavior.
Mathematically, how is toughness in metals measured?
toughness is the area under the stress-strain curve up to the point of fracture.
Define the relaxation modulus.
viscoelastic behavior depends on both time and temperature. the relaxation modulus is one of two analogues to the *elastic modulus* in metals, for viscoelastic polymers. It is defined as the *stress as a function of time* necessary to maintain a *constant amount of strain*. Since viscoelasticity depends on temperature, a different temperature will result in a different relaxation modulus. It is called the "relaxation modulus" because over time it decreases. As *molecular relaxation* occurs, it requires *less applied stress* to maintain the *same amount of strain* in the material.
shear stress
when force is applied in opposite directions on opposite sides of the material
shear stress/strain equations:
τ = F/A; γ = tan(ø) where ø is the angle
Define resilience: computationally, what does the given integral represent for this property? What is Ur? (and the second formula for Ur?)
Resilience is the ability of a material to absorb energy during elastic deformation, then recover the energy when the load is removed. Graphically this is represented by the area under the elastic region: so the integral, which gives the *modulus of resistance,* is simply the area under the stress-strain curve *up to the yielding point*. The second formula for Ur gives the modulus of resistance for linear elastic behavior. the physical meaning of Ur is "energy per unit volume required to stretch the material from the unloaded state to the point of yielding."
Define secant modulus for polymers.
Secant modulus is the average modulus (total ∆σ/total ∆ε) between the origin and some specified strain ε1.
what is the yield point phenomenon? How do you measure yield strength in this situation?
Some metals exhibit the yield point phenomenon. What this means is that initially, with some stress, there is a sudden break from linear behavior, and plastic deformation suddenly occurs (so that the *engineering stress* decreases. Then, continuing strain occurs around an approximately constant stress value, until eventually stress/strain go back to rising nonlinearly. In this case, use the *lower, approximately constant stress value* as the yield strength: no need to use strain-offset method.
Define yield strength (σy): how is this *different* from the proportional limit?
Strength required to produce a specified amount of plastic deformation. *proportional limit* is the 'actual' point of deviation from linear behavior; but that is not logistically easy to measure, so instead, the yield strength is found by strain offset method.
Hooke's law (given) is σ= Eε. What does it mean?
Stress = (young's modulus)*(strain). So young's modulus is a measure of the material's resistance to elastic deformation (stiffness). On a stress-strain curve: E is the slope in the linear region.
Define engineering stress
Stress = σ = (Applied Force)/(Initial Area) ** "initial" area is significant!
Define *stiffness, strength, hardness, ductility, resilience, and toughness*.
(1) STIFFNESS:*resistance to elastic deformation*: this depends *only* on the types of bonds; so microstructure doesn't effect it. (2) STRENGTH: resistance to *plastic deformation.* Characterized by the yield strength (3) HARDNESS: a material's resistance to scratching/indentation: roughly proportional to strength. (this is measured as a parameter primarily because the measurement is easy and non-destructive). (4) DUCTILITY: degree of plastic deformation before fracture -- characterized by %EL and %RA. (5) RESILIENCE: ability of the material to absorb energy during *elastic deformation*. Characterized by the *modulus of resilience,* which is the area under the elastic region. (6) TOUGHNESS: total ability of the material to absorb energy before fracture-- characterized by integral over the entire stress-strain curve.
What are the three basic types of stress-strain behavior for polymers?
(1) brittle material; fractures with very little plastic deformation. (2) plastic material. Similar to metals in that there is initial elastic deformation, then yielding and plastic deformation. (3) totally elastic; large recoverable strains at very low levels of stress. these are *elastomers*
What is the difference between viscous behavior, elastic behavior, and viscoelasticity?
(1) elastic behavior: instantaneous strain in response to applied stress, complete recovery when stress is no longer applied. (2) viscous behavior: delayed (wrt time) response to applied stress, not reversible (3) viscoelastic behavior: combination of viscous and elastic, similar to superposition of an elastic response with viscous response. Some immediate elastic deformation, then subsequent time-dependent viscous transformation.
In general what are the 4 regions of polymer behavior (although not all 4 are always present) and the relaxation modulus in this region?
(1) glassy region: rigid and brittle. relaxation modulus is initially independent of temp. (2) glass transition region: time-dependent deformation, incomplete recovery when load is released. relaxation modulus decreases rapidly with temp in this region. (3) rubbery region: both viscous and elastic, easy to deform. relaxation modulus decreases slowly with T. (4) viscous flow region: highly viscous liquid, deformation is completely viscous. relaxation modulus decreases rapidly with T once again until reaching melting T.
3 important characteristics of plastic deformation:
(1) permanent (2) nonlinear stress-strain relation (3)caused by bonds stretching and planes shearing
3 important characteristics of Elastic Deformation:
(1) reversible (2) linear stress-strain relation (3) on the microscale, causes by slightly stretching bonds in the lattice
What is the difference between *engineering stress* and *true stress*? And what are the equations for these 2 since they are not given?
*engineering stress* is always measured relative to the initial cross-sectional area. *true stress* is measured with the instantaneous cross-sectional area (so these two diverge when the cross sectional area changes significantly). True stress: σt = F/Ai, where Ai = instantaneous area. True strain == ln(li/l0), where li == instantaneous length and l0 == original length.
What is the relationship between modulus of elasticity and force-separation curves for materials?
*strongly bonded* materials means there must be a large energy barrier between equilibrium bond length and increasing bond length beyond that. so the slope from r0 to increasing r is large, that is, lots of energy required to increase r; so the modulus of elasticity will be large since strain does not increase easily. In general, *the slope of the force-interatomic separation curve, at the equilibrium bond length, is proportional to the modulus of elasticity.*
In general, how are mechanical properties of ceramics determined?
3-point bend test: sample supported on either side, force applied between the two supports. Measure modulus of elasticity from the change in the midpoint.
%EL is given, and %RA. what do these mean?
Both of these are measures of ductility, that is, the amount of plastic deformation a material undergoes before fracture. %elongation == (change in length)/(original length. %RA: (change in cross-sectional area)/(original cross-sectional area). These values are independent of the original values for length or area.
In general, why do the mechanical properties of ceramics differ from those of metals?
Ceramics are more brittle than metals. This is because for plastic deformation to occur, there must be dislocations moving through the material. In ceramics it is more difficult for dislocations to occur, because there are fewer slip systems and ions with like charge don't want to move past each other.
Define hardness. How does this relate to tensile strength.
Hardness is defined as resistance to permanently indenting the surface. large hardness means a *large resistance to plastic deformation or cracking during compression*. Tensile strength and hardness both relate to a material's resistance to plastic deformation; so these two are roughly proportional.
What two properties make a material resilient?
High yield strength (lots of elastic deformation before plastic deformation) and low modulus of elasticity (small stress == large strain). Can re-derive this by plugging in (sigma/E) for strain into given equations.
What is the factor of safety N?
In design, you don't want to push the limit of the material. so: (working stress) == (yield stress)/(factor of safety) to ensure that there is enough space between the stress the material will actually experience and its limit.
In general, what is the effect of temperature on the modulus of elasticity?
In most cases E decreases with increasing temperature (more thermal energy to contribute to stretching bonds)