Caractéristiques de contrainte-déformation des aciers

The term steel is commonly used to denote a material that is principally iron (Fe), often in the 95% to 98% range. Pure iron is allotropic, with a body-centered cubic (BCC) structure at room temperature that changes into a face-centered cubic (FCC) structure above 912°C. The empty spaces in the FCC structure and imperfections in the crystal structure allow for other atoms, such as carbon (C) atoms, to be added or removed through diffusion from the interstitial (or empty) spaces. These additions, and the subsequent development of different crystal structures, are the result of heating and cooling at different rates and temperature ranges, a process known as heat treatment. This technology has been known for over 2000 years, but kept secret for many years in applications such as Damascus steel, which utilized Wootz steel from India (≈300AD).

If we expand the open circles in the FCC structure until the spheres begin to touch, and then cut a basic cube for this atomic structure, the result is the unit cell. Spheres with 41.4% of the iron atom diameter can be added before these new spheres begin to touch the iron ones. Carbon atoms are 56% of the diameter of iron ones, so the new structure becomes distorted as carbon atoms are introduced. The properties of steel can be manipulated by changing the size, frequency, and distribution of these distortions.

Wrought iron, one of the most useful predecessors of steel, has a carbon content of more than 2%. It turns out that the optimum carbon content for steels from civil applications is the range of 0.2% to 0.5%. Many of the early metallurgical treatment process were aimed at bringing carbon contents to these levels in volumes that were economical to produce. The Bessemer process in the USA and the Siemens process in the UK are two of the more successful examples of those early techniques. The processes most commonly used today are the electric arc furnace and the basic oxygen furnace. In addition to carbon, most modern steels contain manganese (Mn), chromium (Cr), molybdenum (Mo), copper (Cu), nickel (Ni), and other metals in small amounts to improve strength, deformability, and toughness. A simple example of the effect of these alloys on engineering properties is the so-called carbon equivalent (CE):

The CE is a useful index in determining the weldability of a particular steel; typically, a CE < 0.4% is representative of a steel that is weldable. As many connections in metal structures are made by welding, this is a useful index to remember when specifying materials for construction.

As noted in the JoVE video regarding "Material Constants" , for modeling purposes we need to establish some relationship between stress and strains. The best simple description of the behavior of many materials is given by a stressstrain curve (Fig.1). As a result of problems with buckling when loading in compression and difficulties in loading a material uniformly in more than one direction, a uniaxial tensile test is usually run to determine a stress-strain curve. This test provides basic information on the main engineering characteristics primarily of homogeneous metallic materials.

The typical tension test is described by ASTM E8. ASTM E8 defines the type and size of test specimen to be used, typical equipment to be used, and data to be reported for a metal tension test.

Figure 1: Stress-strain curve for low carbon steel.

Since we need to measure through very large plastic strains, the strain measurement cannot always be made with strain gages over the entire deformation range (up to 40%); the glue will almost always fail before the specimen fractures. An extensometer, which consists of a small C-frame with cantilevered arms instrumented with strain gages and appropriately calibrated, is typically used up to about 20%. Since the extensometer is an expensive and delicate instrument, it needs to be removed before the specimen fractures; the test will be stopped, and the extensometer removed shortly after the specimen reaches its maximum stress and the maximum deformation estimated from marks on the specimen.

The main properties of interest are (Fig. 2):

Proportional limit: The proportional limit is the maximum stress for which stress remains linearly proportional to strain, i.e. for which Hooke's law is strictly applicable (JoVE video – "Material Constants"). This value is generally determined by looking at changes in stress rate when the test is run under constant crosshead speed conditions. In the linear elastic range, the stress rate is proportional to the strain rate and is, ideally, constant. As the material begins to plastify, as evidenced by an increase in the strain rate, the stress rate begins to decrease. The proportional limit is taken as the stress when the initial stress rate begins to decrease.

Yield point: Many metals exhibit a sharp yield point or stress at which the strains continue to increase rapidly without any increase in stress. This is evidenced by a horizontal line, or yield plateau, in the stress-strain curve. The yield point corresponds roughly to the load at which slip begins to occur in the atomic lattices. This slip is triggered by reaching some critical shear force and is much lower than can be calculated from first principles because of the numerous imperfections in the crystal structure. In some materials, such as the mild steel tested in this experiment, there is a small but noticeable decrease in stress before the material reaches the yield plateau, giving rise to upper and lower yield points. For materials that do not exhibit a clear yield point, an equivalent yield strength is used. We will look at this definition in detail in the JoVE video regarding "Stress Strain Characteristics of Aluminum", which deals with these properties in aluminum.

Figure 2: Definitions of variables at low strains.

Elastic modulus: The modulus of elasticity of a material is defined as the slope of the straight-line portion of the stress-strain diagram as shown in Fig.2. This property was discussed in the JoVE video regarding "Material Constants". E is a relatively large number: 30 x 10⁶ psi (210Gpa) for steel; 10 x 10⁶  psi (70 GPa) for aluminum; 1.5 X 10⁶  psi (10.5 GPa) for oak; and 0.5 x 10⁶  psi (3.5 GPa) for plexiglass.

Modulus of resilience: The modulus of resilience is the area under the elastic portion of the stress-strain diagram and has units of energy per unit of volume. The modulus of resilience measures the capacity of a material to absorb energy without undergoing permanent deformations.

Strain hardening modulus: As the slip, or dislocation movements, that triggered the yield plateau begin to reach the grain boundaries (or areas where the lattices are oriented at different angles), the dislocations begin to "pile up", and additional energy is required to propagate their movement into other grains. This leads to a stiffening in the stress-strain behavior, although the strain hardening modulus is usually at least one order of magnitude below Young's modulus.

Ultimate strength: This is the maximum value of the engineering stress reached during the test and occurs shortly before the specimen begins to neck (or change area) appreciably (Fig. 3).

Maximum strain: This value is taken as the strain value when the specimen fractures. Since the extensometer generally has been removed by the time we get to this point in the test and the deformation has localized (necking) into a very short distance along the length of the specimen, this value is very difficult to measure experimentally. For this reason, both a uniform elongation and a percent elongation are often used when specifying materials instead of a maximum strain value.

Figure 3: Definitions at large strains.

Uniform Elongation: The percent elongation is defined as the percent elongation (change in length/original length) of the specimen just before necking occurs.

Percent Elongation: Generally two marks, nominally 2 in. apart, are made on the specimen before testing. After the test, the two pieces of the fractured specimen are put together as best as possible, and the final deformation between the marks is remeasured. This is a crude, but useful way of specifying minimum elongation for materials in an engineering context.

Percent area: Similarly to percent elongation, it is possible to try to make a measurement of the final area of the fractured specimen. By dividing the force just before fracture by this area, it is possible to obtain an idea of the true strength of the material.

Toughness: Toughness is defined to be the total area under the stress-strain diagram. It is a measure of the capability of a material to undergo large, permanent deformations prior to fracture. Its units are the same as those for the modulus of resilience.

The properties described above can be used to assess how well a given material will conform to the performance criteria discussed in the JoVE video regarding "Material Constants". Insofar as safety is concerned, the strength and deformation capacity characteristics are key; these characteristics are usually grouped under the term of ductile behavior. Ductile behavior implies that a material will yield and be able to maintain its strength over a large plastic deformation regime. A large toughness is desirable, which in practice means that a structure will give signs of impending failure, e.g. very large visible deformation before a catastrophic collapse occurs, allowing its occupants time to evacuate the structure.

In contrast, materials that exhibit brittle behavior, will generally fail in a sudden, catastrophic way. This is the case of cementatious and ceramic materials, which exhibit poor tensile capacity. A concrete beam will fail in this fashion because it is very weak in tension. To remedy this pitfall, one places reinforcing steel bars in the tensile region of concrete beams, turning them into reinforced concrete beams.

It is important to realize that brittle and ductile behavior is not an inherent material behavior. As we will see in the JoVE video regarding "Rockwell Hardness Test", subjecting a carbon steel that is ductile at room temperature and under a low strain loading rate conditions to very fast strain loading condition (impact) at low temperatures can result in brittle behavior. In addition, it is important to recognize that some materials, e.g., cast iron, can be very brittle in tension, but ductile in compression.

Two other important material characteristic that need to be defined at this point, as they influence our choice of material modeling, are isotropy and homogeneity. A material is said to be isotropic if its elastic properties are the same in all directions. Most engineering materials are made from crystals that are small compared to the dimensions of the entire body. These crystals are randomly oriented, so statistically the behavior of the material can be considered isotropic. Other materials, such as wood and other fibrous materials, can have similar elastic properties in two directions only (orthotropic) or in all three directions (anisotropic).

On the other hand, a material is said to be homogeneous if its elastic properties are the same throughout the body. For design purposes, most construction materials are assumed as homogeneous. This is valid even for materials like concrete that have different phases (mortar and stones), as we generally are talking about characterizing much larger volumes, which can be considered statistically homogeneous.