ORMAT Chapter 11. Mechanical Behavior of Materials
11.1 Introduction 2
11.2. Mechanical Testing 2
11.2.1. Uniaxial Tensile Loading Test 2
11.2.2. Biaxial Testing: Tube Burst 6
11.2.3. The Von Mises Equivalent Stress 7
11.2.4. Hardness Testing 10
11.2.5. Impact Testing 11
11.3 Microstructural Aspects of Deformation 12
11.3.1. Critical Resolved Shear Stress and the Yield Stress 12
11.3.2. Dislocations as cause of work hardening 15
11.3.3. Yield Strength Increase Mechanisms 15
11.4. Creep Deformation 17
Larson-Miller Plots 18
11.4.1 Creep Mechanisms 20
Thermally enhanced glide 20
Thermally induced dislocation glide (Climb and glide mechanism) 20
Stress Induced Diffusional Flow Difference (Nabarro-Herring Creep): 20
Coble Creep: 20
11.5. Material Fracture 22
11.5.1. Ductile Failure by Diffuse Necking: The Considère Criterion 22
11.5.2. Ductile Fracture 23
11.5.3. Fracture due to Crack Growth 26
Griffith Fracture Theory 26
Problems 30
References 33
11.1 Introduction
When in service, materials may be subjected to loads of various intensities, types and duration. The response of the material to these applied loads is termed the mechanical behavior of the material, and it is one of the most important factors to be considered for materials design. The most important questions to be answered are:
How and when does the material undergo permanent deformation?
When does the material fracture or otherwise fail?
These apparently simple questions encompass the range of questions addressed in the field of mechanical behavior of materials. The questions have different answers depending on whether the load is applied quickly or over a period of time, whether the material is pulled in one or in more than one direction at once (uniaxial or multi-axial loading), whether the load is cyclic, and whether there are pre-existing flaws on the material. The ability of the material to resist deformation and failure under such conditions is measured by various mechanical properties, such as strength, ductility and toughness. These macroscopic material properties are discussed in this chapter as well as their link to the material microstructure.
11.2. Mechanical Testing
Our quantitative knowledge of materials behavior comes from subjecting such materials to various mechanical tests and deriving measurable material properties. To instruct the discussion, various mechanical tests and their mechanical properties are described in the following
11.2.1. Uniaxial Tensile Loading Test
In a tensile test a specimen of uniform cross section (such as a cylinder) is uniaxially loaded and its deformation measured as a function of the applied load (Figure 11.1).
Figure 11.1 A schematic depiction of specimen deformation during a uniaxial tensile loading test.
In figure 11.1, a specimen of initial gauge length lo and cross-sectional area Ao is subjected to an increasing tensile load P, applied along its axis and the resulting specimen deformation is measured. Upon application of the load the specimen length increases to l and the cross-sectional area is reduced to A. The engineering stress is defined as:
ORMAT T 1)
but since during deformation the specimen cross sectional area is reduced from Ao to A, the true stress is actually
T 2)
The engineering strain is
T 3)
However, the true strain is the integral of the increments of strain along the specimen length:
T 4)
In a tension test the true strain is always somewhat larger than the engineering strain as long as deformation is uniform along gauge length. From volume conservation
T 5)
Note that eeng is the strain in the loading direction; by Poisson’s ratio this implies a strain of et=neeng in the two transverse directions; if the material is isotropic, volume conservation yields et=-0.5eeng. Although volume is conserved during plastic deformation, it is not so during elastic deformation.
From and
T 6)
We are now ready to examine the results of a tensile test. If the applied stress is plotted against the specimen strain we obtain the stress-strain curve, a schematic example of which is shown in Figure 11.2.
Figure 11.2: Schematic stress-strain behavior for an austenitic steel.
As load is applied the material initially starts to deform such that the strain is linearly proportional to the load, according to Hooke’s Law, i.e.
T 7)
where E is Young’s modulus. The region of validity of Hooke’s law is the elastic region. This region is characterized by small reversible deformation: that is, once the load is removed, the strain disappears. In that region, the engineering stress/engineering strain curve is coincident with the true stress/true strain curve. Some typical values of the elastic moduli are shown in Table 11.1.
Material / E (Gpa) / G(GPa) / n (Poisson's ratio)Aluminum / 70.3 / 26.1 / 0.345
Gold / 78 / 27 / 0.44
Iron / 211.4 / 81.6 / 0.293
Nickel / 199.5 / 76 / 0.312
Tungsten / 411 / 160.6 / 0.28
Zirconium 1120 dir / 99 / 32.43 (ave.)
Zirconium 0001 dir / 125
Table 11.1 Typical elastic and shear moduli and Poisson’s ratios of different materials
Eventually the material deformation starts to deviate from Hooke’s law as a result of plastic, or irreversible, deformation. In order to establish a definite onset of plastic deformation, , a minimum value of deviation from Hooke’s law (normally 0.2%) is taken to be the point at which plastic deformation starts to occur. The stress at which this happens is called the yield stress, sy, and is an important property of engineered materials.
Beyond the yield point, the material starts to deform plastically. The stress required for further deformation continues to increase because of work hardening, (also called strain hardening) often described by an equation of the type:
T 8)
where n is the work hardening coefficient and K is a constant, called the strength coefficient (note that the equation describes only the work hardening region, starting at yield.) At any given strain the stress required to continue plastic deformation is called the flow stress. Thus the flow stress is equal to the yield stress at the yield point but is higher than the yield stress in the work hardening region.
The material deformation occurring in the elastic region and in the beginning of the work hardening region is uniform, that is, all the material within the gauge section participates equally in the deformation process. At some point diffuse necking sets in, causing deformation to become non-uniform. A small variation in cross sectional area in a section of the material can cause a slightly larger stress, which causes larger deformation in turn further diminishing the area and so on.
Once specimen necking starts, the material fails, because the cross-sectional area in the necked region becomes progressively smaller, and thus no additional load is needed to cause further deformation. The stress at which necking occurs is called the ultimate tensile strength of the material or sUTS. This is the maximum load-bearing capacity of the material. The strain at which necking occurs is the uniform strain, euniform, and is one measure of the ductility of the material (i.e. how much can the material deform before failing). As deformation proceeds, the neck becomes progressively thinner until the material fractures. The strain at which this occurs is the fracture strain, ef, and represents another measure of material ductility.
Another quantity that can be obtained from Figure 11.2 is the toughness of the material, represented by the area under the curve, associated with plastic deformation. This area is the full integrated area under the curve minus the shaded region on the right, which represents the elastic energy stored in the material and which is recovered upon fracture. The more the material can deform before fracturing, the tougher the material is.
Notice that in a tensile test the true-stress/true-strain curve is always higher than the engineering stress- engineering strain curve. Although at small strains the two curves are similar, as the strains become larger they start to deviate from each other, with the true-stress/true-strain curve not showing the apparent stress decrease shown in the engineering curve.
11.2.2. Biaxial Testing: Tube Burst
Although the uniaxial tensile test described above is experimentally convenient and its theoretical interpretation is made straightforward by the fact that there is only one nonzero component of the stress tensor (the normal stress in the direction of the stress, sxx), the relevant states of stress for the loading of fuel cladding loading and other core materials are often different and exhibit greater degree of stress triaxiality. For example the stress-state of fuel-element cladding loaded internally by fission-gas pressure more closely resembles that in a long thin-walled cylindrical tube closed at both ends and pressurized by a gas. This state of stress is a biaxial state of stress, in which the cladding wall is simultaneously stressed in the axial and hoop directions.
Such a state of stress is best studied by a burst test. Creep rupture testing of unirradiated and irradiated steel tubing has also been performed by pressurizing closed tubing with an inert gas.
According to elasticity theory (see equations 6.32-6.34, Chapter 6) the stresses in such a condition (thin wall tubing) are
T 9)
T 10)
T 11)
where p is the internal pressure, R is the tube radius and t is the tube thickness.
To analyze the plastic deformation of the material and to derive a yield condition in such multiaxial stress configurations, it is necessary to define an equivalent stress.
11.2.3. The Von Mises Equivalent Stress
By definition, a specimen subjected to uniaxial loading yields when the stress reaches sy. When a specimen is subjected to multiaxial loading, a yield criterion based on an equivalent stress must be developed. Such a criterion has been developed by von Mises, and is derived here for the case of the tube burst test. The criterion is based on the principal stresses in a material. [1]
Von Mises’ general yield criterion is based on the difference between the total energy density under a multiaxial state of stress and the energy density resulting in the material when subjected to the mean of the three principal stresses, . When this difference reaches a critical value , i.e., when
T 12)
the material will undergo plastic deformation. The mean of the three principal stresses is given by
T 13)
.
The elastic strain energy Eel as a result of applied stresses is given by equation 6.31:
T 14)
which for a solid under hydrostatic stress (no shear stresses) simplifies to
T 15)
Grouping the elements together yields the value of the homogeneous (average) elastic energy density
T 16)
Subtracting equation from we obtain
T 17)
By setting equation equal to the case where there is only one applied stress allows to obtain the equivalent stress seq
T 18)
from where we obtain the von Mises equivalent stress
T 19)
The equivalent stress for the uniaxial tension case can be computed by noting that , which yields . For the tube burst case, substituting equations to into yields
T 20)
Since the equivalent stress derived above defines yielding in a multiaxial stress state then equation defines a yield locus, which is a surface in three-dimensional space. The yield locus for the von Mises condition for the case in which s3=0 (a plane stress condition) is illustrated in Figure 11.3
Figure 11.3: Yield locus generated by the von Mises equivalent stress criterion [2]. The dotted lines indicate the Tresca yield criterion .
For an isotropic material, a uniaxial tensile test results in a strain along the principal stress axis equal to twice that of the transverse directions and thus in the elastic regime
T 21)
We note here that the yield locus shown in Fig.11.3 is symmetric with respect to the deformation direction, i.e. it is the same in tension or compression as well as along the two orthogonal in-plane directions. This does not always have to be the case, as materials such as zirconium are anisotropic and show higher yield stresses along particular directions of deformation. Clearly, the yield locus depends on the mechanisms of deformation at work and it is possible that the operation of other plastic deformation mechanisms could intervene to cut the yield locus (i.e. achieve plastic deformation at a lower stress than the von Mises criterion). Figure 11.4 shows the yield locus for textured hexagonal close packed sheet material, which is seen not to obey equation . Shown in Figure 11.4 is also a line showing that under compressive states of stress twinning is favored over slip.
Figure 11.4 Yield Loci for slip and twinning in textured hexagonal close packed sheet, after [3]
11.2.4. Hardness Testing
Hardness testing is a comparatively easy means to obtain information about the deformation behavior of a material. In hardness testing, an indenter –which can be of various shapes, but typically a ball or a pyramid- is applied with a certain force to a material. The indenter is much harder than the material to be tested, so that only the material tested plastically deforms. For a given force, the indenter will penetrate a certain depth into the material. The depth of indenter penetration can be measured microscopically by the area of indentation. Naturally, the depth of penetration and indentation area will be significantly larger in softer materials.
As the indenter penetrates the material, plastic flow takes place under the indenter. The stress required for such flow to occur is the flow stress (defined above as the stress required to sustain deformation at a given strain). The presence of the elastic region nearby the indent causes a constraint on the deformation with the result that a much larger pressure than the yield stress is needed to cause plastic flow. For a spherical indenter the relationship between the two is [2].
T 22)
where p is the pressure on the indenter, P is the applied load, d is the (fixed) diameter of eth indentation. Thus sy can be estimated from the test above.
Several types of hardness testing exist, depending on indenter shape and whether indent area or depth is measured. Typical examples includeBrinell, Meyer, Vickers and Rockwell. It is also possible to perform this test in a microscale, for example by applying an indenter that is smaller than the dimensions of one grain.
11.2.5. Impact Testing
Another type of test that is of great relevance to the nuclear industry is the Charpy Impact test. Impact tests can be used to measure the fracture resistance of the material and can be at least qualitatively related to the toughness of the material. The Charpy test has been widely used in the nuclear industry for assessing the degree of embrittlement suffered by the pressure vessel after exposure to neutron irradiation. In the Charpy test a large hammer swung from a pendulum is released from an initial height h1 towards a specimen in which a v-notch groove has been fabricated (Figure 11.5). The high speed of the hammer and the presence of the notch cause the material to be deformed at a high strain rate and in a triaxial state of stress, both of which favor fracture. The kinetic energy of the hammer is sufficient to break the sample. Subsequent to specimen fracture the hammer continues along its arc, rising to a height h2, lower than h1. The energy absorbed by the specimen during fracture is given by