Test 2 Study Guide
Chapter 5: Atom and Ion Movements in Materials (Continued)
Composition Profile (Fick’s Second Law):
∙ Describes the dynamic diffusion of atoms. Used for non-steady state systems.
∂ t=∂∂ x(D∂ c
∙ The solution to this past equation depends on the boundary conditions for a particular situation. One of the solutions is:
cs = Constant concentration of the diffusing atoms at the surface of the material. co = Initial uniform concentration of the diffusing atoms in the material. cx = Concentration of the diffusing atom at location x below the surface after time t.
∙ This previous equation assumes a one dimensional model (atoms move in the x-direction).
∙ The erf function is called the error function and it can be evaluated from Table 5-3 or Figure 5-19. The mathematical function for this function is: erf ( x )=2√ᴨ∫0xexp(−y2) dy
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∙ These solutions to Fick’s second law allow us to determine the concentration of diffusing species as a function of time (t) and distance (x).
Diffusion and Materials Processing:
∙ Melting and Casting:
- One of the most used processes to process metal, alloys, plastics, and glasses that mainly consists in the melting and casting of
materials into alloys.
- Diffusion is an important factor in the solidification of metals. For example, difference in diffusion of dopants in both molten and solid We also discuss several other topics like What is the definition of a classification pathway?
forms must be taken into account during the growth of
semiconductors’ single crystals.
- High temperature treatment that causes particles to join, which gradually reduces the volume of pore space between them.
- Frequent step in the manufacturing of ceramic components. Also, very important in the production of metallic parts by powder metallurgy (process in which metal powders are pressed and sintered into monolithic components).
- Liquid phase sintering is the process where a small amount of liquid forms and assists densification.
∙ Grain Growth:
- Movement of grain boundaries that allows larger grains to grow at the expense of smaller grains.
- Used to lower overall energy in the material by reducing the area of the grain boundary (driving force).
- In normal grain growth, the average grain size increases steadily and the width of the grain size distribution is not severely affected. On the other hand, disproportionate grain growth tends to occur when we are dealing with an abnormal situation.
∙ Diffusion Bonding:
- Method used to join materials that occurs in three main steps. - The first step consists in the increase of temperature and pressure in order to force the two surfaces together, which causes the flattening of surface, fragmentation of impurities, and production of high atom to atom contact area. If you want to learn more check out What is the meaning of active transport?
- The second step is the atom diffusion along grain boundaries to the remaining voids and the posterior condensation and reduction of any voids’ size at the interface. This step occurs very quickly because of the fast grain boundary diffusion; however, grain growth eventually isolates the remaining voids from the grain boundaries.
- The third step volume diffusion must occur in order to eliminate the voids. This diffusion process is often used for joining reactive metals, dissimilar metals and materials, and ceramics. Don't forget about the age old question of Who uses accounting information?
Chapter 6 : Mechanical Properties: Part One
Terminology for Mechanical Properties:
∙ Stress: Force acting per unit area over which the force is applied. The main types of stress are: tension, compression, and shear. It is typically expressed in psi or Pa.
∙ Strain: Change in dimension per unit length. It has dimension and it is usually expressed in in/in or cm/cm.
The Tensile Test: Use of Stress-Strain Diagram
∙ Tensile test measures the resistance of a material to a static or slowly applied force. A strain gage is used to measure the amount that the specimen stretches between the gage marks when the force is applied. ∙ Engineering Stress:
∙ Engineering Strain:
F = Force/load applied
Ao = Original cross-sectional area of the specimen before the test begins Lo = Original distance between the gage marks
∆L = Change in length after force
Properties Obtained from Tensile Test:
∙ Yield Strength: Don't forget about the age old question of What is meiosis summary?
- As stress is applied to a material, the material initially shows elastic deformation; however, as this applied stress increases, the material is eventually going to yield both elastic and plastic deformations.
- Elastic deformations are reversible whereas plastic deformations are permanent. The critical stress value required to start plastic
deformations is known as the elastic limit.
- The transition from elastic deformation to plastic flow is abrupt (yield point phenomenon). As plastic deformation begin, the stress value drops from the upper yield point and then it oscillates around an average value called lower yield point (See Figure 6-8).
∙ Tensile Strength: Stress obtained at the highest applied force, which can be located in the engineering stress-strain curve as the maximum stress. ∙ Elastic Properties: Modulus of elasticity (Young’s modulus) ,E, is the slope of the stress-strain curve in the elastic region. The relationship between stress and strain is known as Hooke’s Law:
S = Stress
E = Strain
∙ Tensile Toughness: Energy absorbed by a material prior to fracture, which is sometimes measured as the area under the true stress-strain curve
(work fracture). Engineers usually equate tensile toughness to the area under the stress-strain curve since it is easier to measure stress-strain. ∙ Ductility: Ability of a material to be permanently deformed without breaking when a forced is applied. There are two common measures of ductility: Don't forget about the age old question of Does photosynthesis occur in autotrophs or heterotrophs?
- Percent Elongation: Quantifies the permanent plastic deformation at failure by measuring the distance between gage marks on the
specimen before and after the test.
lf = Distance between gage marks after the specimen breaks
lo = Distance between gage marks before breaking
- Percent Reduction in Area: Measure the percent change in the cross sectional area at the point of fracture before and after the test. It describes the amount of thinning undergone by the specimen during the test.
Af = Final cross-sectional area at the fracture surface
Ao = Initial cross-sectional area at the fracture surface
- Measure of the ability of a material to absorb energy without fracture. It is used to describe the combination properties of strength and ductility, and it has the units of J/m3 or MPa.
- High toughness means that there is high yield strength and ductility. True Stress and True Strain:
∙ True Stress:
∙ True Strain:
l=ln ?¿ l
A = Instantaneous area
F = Force applied
l = Instantaneous sample length
lo = Initial length
The Bend Test for Brittle Materials:
∙ Many brittle materials may crack when placed in the tensile testing machine, so the bend test is used. This test consists in the application of a load at three points causing bending, so a tensile force acts on the material opposite the midpoint.
∙ Flexural Strength: Also called modulus of rupture, it describes the material’s strength
Flexural strength for three point bend test: σbend=3FL
F = Fracture load
L = Distance between the two outer points
w = Width of the specimen
h = Height of the specimen
∙ Flexural Modulus: Modulus of elasticity in bending
δ = Deflection of the beam
∙ The test can also be conducted using the four-point bend test, which maximum stress is given by:
L1 = L/4
NOTE: Check Figure 6-7 for the complete engineering stress-strain curve. Chapter 7: Mechanical Properties: Part Two
∙ Discipline concerned with the behavior of materials containing cracks or other small flaws> this latter term refers to features such as small pores, inclusions, or microcracks.
∙ Fracture Toughness: Measures the ability of a material containing a flaw to resist an applied load. The typical test is performed by applying a tensile stress to a specimen prepared with a flaw of known size and geometry.
K = Stress intensity factor
f = Geometry factor for the specimen and flaw
σ = Applied stress
a = Flaw size
NOTE: When infinite width is assumed, f ~ 1
∙ The value of K that causes the flaw to grow and cause failure can be determined by performing a test on a specimen with a known flaw size. This critical stress intensity factor is known as the fracture toughness (Figure 7-2):
Kc=K required for a crack ¿ propagate
∙ Fracture toughness depends on the thickness of the sample ( As thickness increases, Kc decreases to a constant value). This constant is known as the plane strain fracture toughness K1c, which is the one that is usually reported as the property of a material. This latter term doesn’t depend on the thickness of the sample.
Importance of Fracture Mechanics:
∙ Brittle Fracture: Any small crack or imperfection limits the capability of a ceramic to resist a tensile stress. This occurs when a crack, usually
called Griffith flaw, concentrates and magnifies the applied stress (Figure 7-4):
σactual ? 2σ√a/r
σ = Tensile stress
r = Very thin cracks
a = Long cracks
NOTE: If σactual exceeds the yield strength, the crack is going to grow and eventually cause failure, even though the nominal applied stress σ is small.