Work hardening

The primary species responsible for work hardening are dislocations. Dislocations interact with each other by generating stress fields in the material. The interaction between the stress fields of dislocations can impede dislocation motion by repulsive or attractive interactions. Additionally, if two dislocations cross, dislocation line entanglement occurs, causing the formation of a jog which opposes dislocation motion. These entanglements and jogs act as pinning points, which oppose dislocation motion. As both of these processes are more likely to occur when more dislocations are present, there is a correlation between dislocation density and shear strength.

The shear strengthening provided by dislocation interactions can be described by:^[6]

${\displaystyle \Delta \tau _{d}=\alpha Gb{\sqrt {\rho _{\perp }}}}$

where ${\displaystyle \alpha }$ is a proportionality constant, ${\displaystyle G}$ is the shear modulus, ${\displaystyle b}$ is the Burgers vector, and ${\displaystyle \rho _{\perp }}$ is the dislocation density.

Dislocation density is defined as the dislocation line length per unit volume:

${\displaystyle \rho _{\perp }={\frac {\ell }{\ell ^{3}}}}$

Similarly, the axial strengthening will be proportional to the dislocation density.

${\displaystyle \Delta \sigma _{y}\propto {Gb{\sqrt {\rho _{\perp }}}}}$

This relationship does not apply when dislocations form cell structures. When cell structures are formed, the average cell size controls the strengthening effect.^[6]

Increasing the dislocation density increases the yield strength which results in a higher shear stress required to move the dislocations. This process is easily observed while working a material (by a process of cold working in metals). Theoretically, the strength of a material with no dislocations will be extremely high (${\displaystyle \sigma \approx {\frac {G}{10}}}$) because plastic deformation would require the breaking of many bonds simultaneously. However, at moderate dislocation density values of around 10⁷-10⁹ dislocations/m², the material will exhibit a significantly lower mechanical strength. Analogously, it is easier to move a rubber rug across a surface by propagating a small ripple through it than by dragging the whole rug. At dislocation densities of 10¹⁴ dislocations/m² or higher, the strength of the material becomes high once again. Also, the dislocation density cannot be infinitely high, because then the material would lose its crystalline structure.

This is a schematic illustrating how the lattice is strained by the addition of interstitial solute. Notice the strain in the lattice that the solute atoms cause. The interstitial solute could be carbon in iron for example. The carbon atoms in the interstitial sites of the lattice creates a stress field that impedes dislocation movement.

This is a schematic illustrating how the lattice is strained by the addition of substitutional solute. Notice the strain in the lattice that the solute atom causes.

Solid solution strengthening and alloying

For this strengthening mechanism, solute atoms of one element are added to another, resulting in either substitutional or interstitial point defects in the crystal (see Figure on the right). The solute atoms cause lattice distortions that impede dislocation motion, increasing the yield stress of the material. Solute atoms have stress fields around them which can interact with those of dislocations. The presence of solute atoms impart compressive or tensile stresses to the lattice, depending on solute size, which interfere with nearby dislocations, causing the solute atoms to act as potential barriers.

The shear stress required to move dislocations in a material is:

${\displaystyle \Delta \tau =Gb{\sqrt {c}}\epsilon ^{3/2}}$

where ${\displaystyle c}$ is the solute concentration and ${\displaystyle \epsilon }$ is the strain on the material caused by the solute.

Increasing the concentration of the solute atoms will increase the yield strength of a material, but there is a limit to the amount of solute that can be added, and one should look at the phase diagram for the material and the alloy to make sure that a second phase is not created.

In general, the solid solution strengthening depends on the concentration of the solute atoms, shear modulus of the solute atoms, size of solute atoms, valency of solute atoms (for ionic materials), and the symmetry of the solute stress field. The magnitude of strengthening is higher for non-symmetric stress fields because these solutes can interact with both edge and screw dislocations, whereas symmetric stress fields, which cause only volume change and not shape change, can only interact with edge dislocations.

Figure 2: A schematic illustrating how the dislocations can interact with a particle. It can either cut through the particle or bow around the particle and create a dislocation loop as it moves over the particle.

Precipitation hardening

In most binary systems, alloying above a concentration given by the phase diagram will cause the formation of a second phase. A second phase can also be created by mechanical or thermal treatments. The particles that compose the second phase precipitates act as pinning points in a similar manner to solutes, though the particles are not necessarily single atoms.

The dislocations in a material can interact with the precipitate atoms in one of two ways (see Figure 2). If the precipitate atoms are small, the dislocations would cut through them. As a result, new surfaces (b in Figure 2) of the particle would get exposed to the matrix and the particle-matrix interfacial energy would increase. For larger precipitate particles, looping or bowing of the dislocations would occur and result in dislocations getting longer. Hence, at a critical radius of about 5 nm, dislocations will preferably cut across the obstacle, while for a radius of 30 nm, the dislocations will readily bow or loop to overcome the obstacle.

The mathematical descriptions are as follows:

For particle bowing- ${\displaystyle \Delta \tau ={Gb \over L-2r}}$

For particle cutting- ${\displaystyle \Delta \tau ={\gamma \pi r \over bL}}$

Figure 3: A schematic roughly illustrating the concept of dislocation pile up and how it effects the strength of the material. A material with larger grain size is able to have more dislocation to pile up leading to a bigger driving force for dislocations to move from one grain to another. Thus, less force need be applied to move a dislocation from a larger, than from a smaller grain, leading materials with smaller grains to exhibit higher yield stress.

Dispersion strengthening

Dispersion strengthening is a type of particulate strengthening in which incoherent precipitates attract and pin dislocations. These particles are typically larger than those in the Orowon precipitation hardening discussed above. The effect of dispersion strengthening is effective at high temperatures whereas precipitation strengthening from heat treatments are typically limited to temperatures much lower than the melting temperature of the material.^[7] One common type of dispersion strengthening is oxide dispersion strengthening.

Grain boundary strengthening

In a polycrystalline metal, grain size has a tremendous influence on the mechanical properties. Because grains usually have varying crystallographic orientations, grain boundaries arise. While undergoing deformation, slip motion will take place. Grain boundaries act as an impediment to dislocation motion for the following two reasons:

1. Dislocation must change its direction of motion due to the differing orientation of grains.^[4]
2. Discontinuity of slip planes from grain one to grain two.^[4]

The stress required to move a dislocation from one grain to another in order to plastically deform a material depends on the grain size. The average number of dislocations per grain decreases with average grain size (see Figure 3). A lower number of dislocations per grain results in a lower dislocation 'pressure' building up at grain boundaries. This makes it more difficult for dislocations to move into adjacent grains. This relationship is the Hall-Petch relationship and can be mathematically described as follows:

${\displaystyle \sigma _{y}=\sigma _{y,0}+{k \over {d^{x}}}}$,

where ${\displaystyle k}$ is a constant, ${\displaystyle d}$ is the average grain diameter and ${\displaystyle \sigma _{y,0}}$ is the original yield stress.

The fact that the yield strength increases with decreasing grain size is accompanied by the caveat that the grain size cannot be decreased infinitely. As the grain size decreases, more free volume is generated resulting in lattice mismatch. Below approximately 10 nm, the grain boundaries will tend to slide instead; a phenomenon known as grain-boundary sliding. If the grain size gets too small, it becomes more difficult to fit the dislocations in the grain and the stress required to move them is less. It was not possible to produce materials with grain sizes below 10 nm until recently, so the discovery that strength decreases below a critical grain size is still finding new applications.

Transformation hardening

This method of hardening is used for steels.

High-strength steels generally fall into three basic categories, classified by the strengthening mechanism employed. 1- solid-solution-strengthened steels (rephos steels) 2- grain-refined steels or high strength low alloy steels (HSLA) 3- transformation-hardened steels

Transformation-hardened steels are the third type of high-strength steels. These steels use predominantly higher levels of C and Mn along with heat treatment to increase strength. The finished product will have a duplex micro-structure of ferrite with varying levels of degenerate martensite. This allows for varying levels of strength. There are three basic types of transformation-hardened steels. These are dual-phase (DP), transformation-induced plasticity (TRIP), and martensitic steels.

The annealing process for dual -phase steels consists of first holding the steel in the alpha + gamma temperature region for a set period of time. During that time C and Mn diffuse into the austenite leaving a ferrite of greater purity. The steel is then quenched so that the austenite is transformed into martensite, and the ferrite remains on cooling. The steel is then subjected to a temper cycle to allow some level of marten-site decomposition. By controlling the amount of martensite in the steel, as well as the degree of temper, the strength level can be controlled. Depending on processing and chemistry, the strength level can range from 350 to 960 MPa.

TRIP steels also use C and Mn, along with heat treatment, in order to retain small amounts of austenite and bainite in a ferrite matrix. Thermal processing for TRIP steels again involves annealing the steel in the a + g region for a period of time sufficient to allow C and Mn to diffuse into austenite. The steel is then quenched to a point above the martensite start temperature and held there. This allows the formation of bainite, an austenite decomposition product. While at this temperature, more C is allowed to enrich the retained austenite. This, in turn, lowers the martensite start temperature to below room temperature. Upon final quenching a metastable austenite is retained in the predominantly ferrite matrix along with small amounts of bainite (and other forms of decomposed austenite). This combination of micro-structures has the added benefits of higher strengths and resistance to necking during forming. This offers great improvements in formability over other high-strength steels. Essentially, as the TRIP steel is being formed, it becomes much stronger. Tensile strengths of TRIP steels are in the range of 600-960 MPa.

Martensitic steels are also high in C and Mn. These are fully quenched to martensite during processing. The martensite structure is then tempered back to the appropriate strength level, adding toughness to the steel. Tensile strengths for these steels range as high as 1500 MPa.

Polymer

Polymers fracture via breaking of inter- and intra molecular bonds; hence, the chemical structure of these materials plays a huge role in increasing strength. For polymers consisting of chains which easily slide past each other, chemical and physical cross linking can be used to increase rigidity and yield strength. In thermoset polymers (thermosetting plastic), disulfide bridges and other covalent cross links give rise to a hard structure which can withstand very high temperatures. These cross-links are particularly helpful in improving tensile strength of materials which contain much free volume prone to crazing, typically glassy brittle polymers.^[8] In thermoplastic elastomer, phase separation of dissimilar monomer components leads to association of hard domains within a sea of soft phase, yielding a physical structure with increased strength and rigidity. If yielding occurs by chains sliding past each other (shear bands), the strength can also be increased by introducing kinks into the polymer chains via unsaturated carbon-carbon bonds.^[8]

Adding filler materials such as fibers, platelets, and particles is a commonly employed technique for strengthening polymer materials. Fillers such as clay, silica, and carbon network materials have been extensively researched and used in polymer composites in part due to their effect on mechanical properties. Stiffness-confinement effects near rigid interfaces, such as those between a polymer matrix and stiffer filler materials, enhance the stiffness of composites by restricting polymer chain motion.^[9] This is especially present where fillers are chemically treated to strongly interact with polymer chains, increasing the anchoring of polymer chains to the filler interfaces and thus further restricting the motion of chains away from the interface.^[10] Stiffness-confinement effects have been characterized in model nanocomposites, and shows that composites with length scales on the order of nanometers increase the effect of the fillers on polymer stiffness dramatically.^[11]

Increasing the bulkiness of the monomer unit via incorporation of aryl rings is another strengthening mechanism. The anisotropy of the molecular structure means that these mechanisms are heavily dependent on the direction of applied stress. While aryl rings drastically increase rigidity along the direction of the chain, these materials may still be brittle in perpendicular directions. Macroscopic structure can be adjusted to compensate for this anisotropy. For example, the high strength of Kevlar arises from a stacked multilayer macrostructure where aromatic polymer layers are rotated with respect to their neighbors. When loaded oblique to the chain direction, ductile polymers with flexible linkages, such as oriented polyethylene, are highly prone to shear band formation, so macroscopic structures which place the load parallel to the draw direction would increase strength.^[8]

Mixing polymers is another method of increasing strength, particularly with materials that show crazing preceding brittle fracture such as atactic polystyrene (APS). For example, by forming a 50/50 mixture of APS with polyphenylene oxide (PPO), this embrittling tendency can be almost completely suppressed, substantially increasing the fracture strength.^[8]

Interpenetrating polymer networks (IPNs), consisting of interlacing crosslinked polymer networks that are not covalently bonded to one another, can lead to enhanced strength in polymer materials. The use of an IPN approach imposes compatibility (and thus macroscale homogeneity) on otherwise immiscible blends, allowing for a blending of mechanical properties. For example, silicone-polyurethane IPNs show increased tear and flexural strength over base silicone networks, while preserving the high elastic recovery of the silicone network at high strains.^[12] Increased stiffness can also be achieved by pre-straining polymer networks and then sequentially forming a secondary network within the strained material. This takes advantage of the anisotropic strain hardening of the original network (chain alignment from stretching of the polymer chains) and provides a mechanism whereby the two networks transfer stress to one another due to the imposed strain on the pre-strained network.^[13]

Glass

Many silicate glasses are strong in compression but weak in tension. By introducing compression stress into the structure, the tensile strength of the material can be increased. This is typically done via two mechanisms: thermal treatment (tempering) or chemical bath (via ion exchange).

In tempered glasses, air jets are used to rapidly cool the top and bottom surfaces of a softened (hot) slab of glass. Since the surface cools quicker, there is more free volume at the surface than in the bulk melt. The core of the slab then pulls the surface inward, resulting in an internal compressive stress at the surface. This substantially increases the tensile strength of the material as tensile stresses exerted on the glass must now resolve the compressive stresses before yielding.

${\displaystyle \sigma _{y=modified}=\sigma _{y,0}+\sigma _{compressive}}$

Alternately, in chemical treatment, a glass slab treated containing network formers and modifiers is submerged into a molten salt bath containing ions larger than those present in the modifier. Due to a concentration gradient of the ions, mass transport must take place. As the larger cation diffuses from the molten salt into the surface, it replaces the smaller ion from the modifier. The larger ion squeezing into surface introduces compressive stress in the glass's surface. A common example is treatment of sodium oxide modified silicate glass in molten potassium chloride. Examples of chemically strengthened glass are Gorilla Glass developed and manufactured by Corning, AGC Inc.'s Dragontrail and Schott AG's Xensation.

Fiber reinforcement

Fiber-reinforced composites (FRCs) consist of a matrix of one material containing parallel embedded fibers. There are two variants of fiber-reinforced composites, one with stiff fibers and a ductile matrix and one with ductile fibers and a stiff matrix. The former variant is exemplified by fiberglass which contains very strong but delicate glass fibers embedded in a softer plastic matrix resilient to fracture. The latter variant is found in almost all buildings as reinforced concrete with ductile, high tensile-strength steel rods embedded in brittle, high compressive-strength concrete. In both cases, the matrix and fibers have complimentary mechanical properties and the resulting composite material is therefore more practical for applications in the real world.

For a composite containing aligned, stiff fibers which span the length of the material and a soft, ductile matrix, the following descriptions provide a rough model.

Laminar reinforcement

Molecular dynamics simulation assisted studies

The molecular dynamics (MD) method has been widely applied in materials science as it can yield information about the structure, properties, and dynamics on the atomic scale that cannot be easily resolved with experiments. The fundamental mechanism behind MD simulation is based on classical mechanics, from which we know the force exerted on a particle is caused by the negative gradient of the potential energy with respect to the particle position. Therefore, a standard procedure to conduct MD simulation is to divide the time into discrete time steps and solve the equations of motion over these intervals repeatedly to update the positions and energies of the particles.^[15] Direct observation of atomic arrangements and energetics of particles on the atomic scale makes it a powerful tool to study microstructural evolution and strengthening mechanisms.

Nanostructure fabrication for material strengthening

Based on the mechanism of strengthening discussed in the previous contents, nowadays people are also working on enhancing the strength by purposely fabricating nanostructures in materials. Here we introduce several representative methods, including hierarchical nanotwined structures, pushing the limit of grain size for strengthening and dislocation engineering.

Hierarchical nanotwinned structures

As mentioned in the previous content, hindering dislocation motion renders great strengthening to materials. Nanoscale twins – crystalline regions related by symmetry have the ability to effectively block the dislocation motion due to the microstructure change at the interface.^[23] The formation of hierarchical nanotwinned structures pushes the hindrance effect to the extreme, due to the construction of a complex 3D nanotwinned network. Thus, the delicate design of hierarchical nanotwinned structures is of great importance for inventing materials with super strength. For instance, Yue et al. constructed a diamond composite with hierarchically nanotwinned structure by manipulating the synthesis pressure. The obtained composite showed the higher strength than typical engineering metals and ceramics.

Pushing the limit of grain size for strengthening

The Hall-Petch effect illustrates that the yield strength of materials increases with decreasing grain size. However, many researchers have found that the nanocrystalline materials will soften when the grain size decreases to the critical point, which is called the inverse Hall-Petch effect. The interpretations of this phenomenon is that the extremely small grains are not able to support dislocation pileup which provides extra stress concentration in the large grains.^[24] At this point, the strengthening mechanism changes from dislocation-dominated strain hardening to growth softening and grain rotation. Typically, the inverse Hall-Petch effect will happens at grain size ranging from 10 nm to 30 nm and makes it hard for nanocrystalline materials to achieve a high strength. To push the limit of grain size for strengthening, the hindrance of grain rotation and growth could be achieved by grain boundary stabilization. The construction of nanolaminated structure with low-angle grain boundaries is one method to obtain ultrafine grained materials with ultra-strength. Lu et al.^[25] applied a very high rate shear deformation with high strain gradients on the top surface layer of bulk Ni sample and introduced nanolaminated structures. This material exhibits an ultra-high hardness, higher than any reported ultrafine-grained nickel. The exceptional strength is resulted from the appearance of low-angle grain boundaries, which have low-energy states efficient for enhancing structure stability. Another method to stabilize grain boundaries is the addition of nonmetallic impurities. Nonmetallic impurities often aggregate at grain boundaries and have the ability to impact the strength of materials by changing the grain boundary energy. Rupert et al.^[26] conducted first-principles simulations to study the impact of the addition of common nonmetallic impurities on Σ5 (310) grain boundary energy in Cu. They claimed that the decrease of covalent radius of the impurity and the increase of electronegativity of the impurity would lead to the increase of the grain boundary energy and further strengthen the materials. For instance, boron stabilized the grain boundaries by enhancing the charge density among the adjacent Cu atoms to improve the connection between two grain boundaries.

Dislocation engineering

Previous studies on the impact of dislocation motion on materials strengthening mainly focused on high density dislocation, which is effective for enhancing strength with the cost of reducing ductility. Engineering dislocation structures and distribution is promising to comprehensively improve the performance of material. Solutes tend to aggregate at dislocations and are promising for dislocation engineering. Kimura et al.^[27] conducted atom probe tomograph and observed the aggregation of niobium atoms to the dislocations. The segregation energy was calculated to be almost the same as the grain boundary segregation energy. That's to say, the interaction between niobium atoms and dislocations hindered the recovery of dislocations and thus strengthened the materials. Introducing dislocations with heterogeneous characteristics could also be utilized for material strengthening. Lu et al.^[28] introduced ordered oxygen complexes into TiZrHfNb alloy. Unlike the traditional interstitial strengthening, the introduction of the ordered oxygen complexes enhanced the strength of the alloy without the sacrifice of ductility. The mechanism was that the ordered oxygen complexes changed the dislocation motion mode from planar slip to wavy slip and promoted double cross-slip.