Fracture in polymers

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Polymer fracture is the study of the fracture surface of an already failed material to determine the method of crack formation and extension in polymers both fiber reinforced and otherwise. [1] Failure in polymer components can occur at relatively low stress levels, far below the tensile strength because of four major reasons: long term stress or creep rupture, cyclic stresses or fatigue, the presence of structural flaws and stress-cracking agents. Formations of submicroscopic cracks in polymers under load have been studied by x ray scattering techniques and the main regularities of crack formation under different loading conditions have been analyzed. The low strength of polymers compared to theoretically predicted values are mainly due to the many microscopic imperfections found in the material. These defects namely dislocations, crystalline boundaries, amorphous interlayers and block structure can all lead to the non-uniform distribution of mechanical stress.

Contents

Long term stress or Creep Failure

Taking into account the viscoelastic path at small strain based on thermally activated rate processes. When strain attains higher values, high enough to lead to failure, its slope versus time exhibits an abrupt change. At this specific time the creep function appears a minimum. [2] In most cases DMTA (Dynamic mechanical thermal analysis) can be used to determine the viscoelastic behavior of samples as a function of time. A classic case is when the rubber hose ruptures due to creep after many years of service. DMTA can be used for o-rings and gaskets to measure the creep rates.

Fatigue Failure

The term fatigue refers to the effect of cyclic or intermittent loads. Cyclic loading due to either oscillating mechanical stress or to alternate heating and cooling is more detrimental than static loading. Under cyclic load, cracks initiate at localized sites within the part and these extend in size during cycling. Ultimately they expand and join to such an extent that the material can no longer hold and support the stress. Fractures can be characterized by a series of concentric crack growth bands that grow from the surface initiation site. Cyclic loading can bring about failure in polymer due to: chain scission, built up heat due to hysteresis, recrystallization of material and cumulative crack generation.

Chain scission

Chain scission occurs in a polymer as a result of intense localized heat. The chemical bond in a polymer backbone may be broken with the generation of free radicles by heat, ionizing irradiation, mechanical stress and chemical reactions. These scissions multiply in number and cause a fracture tip initialization to occur followed by its growth. [3]

Built-up heat from hysteresis

Polymers are viscoelastic by nature, and exhibit mechanical hysteresis even at moderate strains due to continuous elongation and contraction. Some of this inelastic deformation energy is dissipated as heat within the polymer, and consequently the materials temperature will rise as a function of frequency, testing temperature, the stress cycle and the type of polymer. As the temperature within the polymer rises, the stiffness and yield strength will fall, and thermal failure becomes a possibility as deformation levels become excessive.

Fracture Mechanics in Polymers

Fracture mechanics in polymers has become an increasingly concerning field as many industries transition to implementing polymers in many critical structural applications. As industries make the shift to implementing polymeric materials, a greater understanding of failure mechanisms for these polymers is needed . Polymers may exhibit some inherently different behaviors than metals when cracks are subject to loading. This is largely attributed to their tough and ductile mechanical properties. Microstructurally, metals contain grain boundaries, crystallographic planes and dislocations while polymers are made up of long molecular chains. In the same instance that fracture in metals involves breaking bonds, the covalent and van der Waals bonds need to be broken for fracture to occur. These secondary bonds (van der Waals) play an important role in the fracture deformation at crack tip. Many materials, such as metals, use linear elastic fracture mechanics to predict behavior at the crack tip. For some materials this is not always the appropriate way to characterize fracture behavior and an alternate model is used. Elastic-plastic fracture mechanics relates to materials that show a time independent and nonlinear behavior or in other words plastically deform. The initiation site for fracture in these materials can often occur at inorganic dust particles where the stress exceeds critical value.

Under standard linear elastic fracture mechanics, Griffiths law can be used to predict the amount of energy needed to create a new surface by balancing the amount of work needed to create new surfaces with the sample's stored elastic energy. His popular equation below provides the necessary amount of fracture stress required as a function of crack length. E is the young's modulus of the material, γ is the surface free energy per area and a is the crack length.

Griffith Law

While many ideas from the linear elastic fracture mechanics (LEFM) models are applicable to polymers there are certain characteristics that need to be considered when modeling behavior. Additional plastic deformation should be considered at crack tips as yielding is more likely to occur in plastics.

Yielding Mechanisms

Figure 1 - Crack-tip Craze Zone Crack-tip Craze Zone.png
Figure 1 - Crack-tip Craze Zone

As metals yield through dislocation motions in the slip planes, polymer yield through either shear yielding or crazing. [4] In shear yielding, molecules move with respect to one another as a critical shear stress is being applied to the system resembling a plastic flow in metals. Yielding through crazing is found in glassy polymers where a tensile load is applied to a highly localized region. High concentration of stress will lead to the formation of fibrils in which molecular chains form aligned sections. This also creates voids which are known as cavitation and can be seen at a macroscopic level as a stress-whitened region as shown in Figure 1. These voids surround the aligned polymer regions. [5] The stress in the aligned fibrils will carry majority of the stress as the covalent bonds are significantly stronger than the van der Waals bonds. The plastic like behavior of polymers leads to a greater assumed plastic deformation zone in front of the crack tip altering the failure process. [6]

Crack Tip Behavior

Just as in metals, when the stress at the crack tip approaches infinity, a yield zone will form at this crack tip front. Craze yielding is the most common yielding method at the crack front under tension due to the high triaxial stresses being applied in this local region. The Dugdale- Barenblatt strip-yield model is used to predict the length of the craze zone. [7] KI represent the stress intensity factor, s is the crazing stress being applied to the system (perpendicular to the crack in this situation), and r is the crazing zone length.

Dugdale-Barenblatt strip-yield model

The equation for stress intensity factor for a specimen with a single crack is given in the following equation where Y is a geometric parameter, s is the stress being applied and a is the crack length. For an edge crack ‘a’ is the total length of the crack where as a crack not on the edge has a crack length of ‘2a’.

Stress Intensity Equation

As the fibrils in the crack begin to rupture the crack will advance in either a stable, unstable or critical growth depending on the toughness of the material. To accurately determine the stability of a crack growth and R curve plot should be constructed. A unique tip of fracture mode is called stick/slip crack growth. This occurs when an entire crack zone ruptures at some critical crack tip opening displacement (CTOD) followed by a crack arrest and then the formation of a new crack tip.

Critical Stress Intensity Factor

The critical stress intensity factor (KIC) can be defined as the threshold value of stress intensity base on the material properties. Therefore, the crack will not propagate so long as KI is less than KIC. Since KIC is a material property it can be determined through experimental testing. [8] ASTM D20 provides a standard testing method for determining critical stress of plastics. Although KIC is material dependent it can also be a function of thickness. Where plane stress is dominant in low thickness samples increasing the critical stress intensity. As your thickness increases the critical stress intensity will decrease and eventually plateau. This behavior is caused by the transitioning from the plane stress to plain strain conditions as the thickness increases. Fracture morphology is also dependent on conditions at the located at the crack tip.

Fatigue

Figure 2 - Hysteresis Stress Strain Curve Hystersis.png
Figure 2 - Hysteresis Stress Strain Curve

As the needs for polymers for engineering purposes are increasing, the fatigue behavior of polymers is receiving more attentions. Polymer fatigue life is affected by multiple factors, including temperature, oxidation, crystallization and so on. [9] Therefore, the need becomes vital for people to study and predict the mechanical performances of polymers under different environments.

The experimental methods to study polymer fatigue vary, including pure shear test, simple extension test, single edge crack test and tearing test, [10] among which the most widely used geometry people adopt is mode I cyclic tension test under pure geometry. [11] [12] This is due to the fact that polymers have viscoelastic behavior and poor conductivity of heat, and they are more sensitive to their cyclic loading conditions than metal. [13] Unlike metals and other materials, polymer do not cyclic harden; rather, polymers perform cyclic softening for most of the time, and the extent of which usually depends on the loading conditions of the experimental setup. In a rare case, polymers can also remain cyclically stable under small strain deformations, during which the polymer remain linearly elastic. [13] However, mode III geometry has also been applied to test on twisted rubber disks for further understanding of its fracture behaviors. [14] [15] [16]

In reinforced polymers, crack initiation usually occurs at the interface of polymer fiber and the matrix. Fatigue performances in polymers caused by cyclical loading usually go through two stages: crack initiation/nucleation and crack growth. Hence, a lot of researcher design experiments to study the fatigue behaviors of polymers according to these two phases, especially for rubber fatigue.

Crack Nucleation Approach [17]

The crack nucleation approach considers that polymers will eventually crack under a history of stress and strains. Study under this proposal is first adapted by August Wöhler in the 1860s, [18] who aimed to study railroad axles. Two important parameters are involved in this conversation: maximum principal strain and strain energy density. [12] As strain is easier to be determined by controlling displacements in actual experimental tests, it is natural to set controlled strain. The alternating and maximum values of the maximum principle strain decides the nucleation life in polymer, but there is not yet any theory to quantitatively describe the strain conditions to fatigue life in lab tests. The strain energy density is defined as the energy invested into the volume unit of polymeric material for deformation, also known as the elastic energy per unit volume in the solid. In crack nucleation approach, no crack is introduced into the sample initially, and crack initiation is observed under loadings.

Crack Growth Approach

Unlike crack initiation approach, the crack growth approach introduces a pre-existing crack to the specimen, and calculate for the crack propagation rate under cyclic loadings. Because of the pre-existing crack, the elastic energy released by fracture will associate with the surface energy at new crack surfaces, [12] and it was found that the crack propagation rate can be described by a function of the energy release rate, depending on which regime of crack propagation this relation is in. [11] Four regimes were proposed: the steady-state regime, the linear regime, the power-law regime and the fourth regime where crack grows catastrophically. In the power-law regime, the relation is found to be similar to the Paris law, which was found by Paris et al. [19] in metals.

Hysteresis heating and Chain scission

Fatigue in polymers, controlled by cyclical loading, is caused by two general mechanisms: hysteresis heating and chain scission. If the polymer is relatively brittle, it will exhibit fatigue crack growth through chain scission. In this mechanism, the crack-tip yielding is limited by the brittle material properties and each loading cycle breaks a specific amount of bonds allowing the crack front to advance. Polymers with viscoelastic behavior fatigue by the hysteresis heating mechanism. In this mechanism, during loading and unloading, the polymer the stress-strain curve will act as a hysteresis loop as shown in Figure 2, creating energy on the material as discussed before. Because energy is dissipated to the crack tip, this process is different from the cyclic loading of an elastic material where the loading and unloading paths are the same and strain energy can be recovered. The work inputted into the material (area of the hysteresis loop) is converted to heat raising the temperature of the material, possibly above the glass transition temperature. This creates a localized melting at the crack tip allowing the crack to advance. The magnitude at which the crack front will advance is largely dependent on the amount/magnitude of cycles, glass transition temperature of the material and the thermal conductivity of the polymer. A polymer that has a high thermal conductivity will dissipate the heat much fast than a material with a low coefficient.

Figure 3 - Cumulative Loading S-N Curve Cumulative loading.png
Figure 3 - Cumulative Loading S-N Curve

A S-N curve represents the amount of cycles being applied along with the stress amplitude and can derived from the Goodman relationship.

(Goodman Relationship)

Where σf is the fatigue stress, σm is mean stress, σa is the amplitude stress and σt is the tensile stress of the sample being tested. In certain applications of polymers, materials will experience cyclic loading at different stress levels. Figure 3 gives an S-N diagram of cumulative cycles being applied at different stress amplitudes. The variable n represents the number of cycles being applied at the designated stress level and N is the fatigue life at the same stress level.

Figure 4 - Power Law Regime Paris law number 2 ex.png
Figure 4 - Power Law Regime

Many times, polymeric materials containing a crack are subject to cyclic loading in service. This decreases the life expectancy of the sample drastically and should be taken into consideration. In cases where polymers such as PVC follow the rules of linear elastic fracture mechanics, Paris law may be used to relate the fatigue crack propagation rate to the magnitude of stress intensity being applied. Below a certain stress intensity, crack propagation slowly increases until stable crack propagation is reached from higher levels of stress intensity. Higher levels of stress intensity leads to an unstable crack rate as shown in Figure 4. This figure is a log plot of the crack propagation rate versus the max stress intensity example. The stable crack growth regime represents the linear region of the red curve which is described using the Power Law model where ‘A’ is a pre-exponential factor.

(Power Law Regime Equation)

Recrystallization

This process can be caused as a consequence of extensive movement of chain segments like in case or work hardening of materials.

Fatigue in Nylon

When a Nylon component is subjected to conditions of tensile fatigue, failure occurs when a minimum strain is reached. this means that the lifetime of nylon material is dictated by the time under load and not on the number of cycles

Fatigue of Short-Fibre-Reinforced Plastics

Fatigue failure in these reinforced polymers is due to the formation of micro cracks that are easily initiated, and which coalesce into one crack, causing the final failure [20]

Impact Fracture

A good polymer is generally defined as one capable of absorbing a large amount of energy before failure. Polycarbonates have one of the highest impact resistance values. However, amorphous polymers exhibit brittle behaviour under impact, especially if the component is notched or is too thick relative to a corner radius. The occurrence of brittle failure can be decreased by: increasing the molecular weight, inclusion of rubber phase, inducing orientation in the polymer and reducing internal defects and contaminants.

Measuring impact strength

Conventional Izod tests are used to measure the energy required to break a notched specimen. however, this is not considered as a satisfactory test. Major limitation being that most polymers as notch sensitive and fail readily under izod test.

Blends

Blended materials can have an increased fracture toughness with balanced stiffness and strength. Usually these are formed from copolymerization or modification with a suitable elastomer. However, the mechanical properties of blends, especially the modulus, follow the ‘rule of mixture’ Voigt model and the morphologies show coarsed dispersion. [21]

Related Research Articles

In engineering, deformation refers to the change in size or shape of an object. Displacements are the absolute change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain is the relative internal change in shape of an infinitesimal cube of material and can be expressed as a non-dimensional change in length or angle of distortion of the cube. Strains are related to the forces acting on the cube, which are known as stress, by a stress-strain curve. The relationship between stress and strain is generally linear and reversible up until the yield point and the deformation is elastic. The linear relationship for a material is known as Young's modulus. Above the yield point, some degree of permanent distortion remains after unloading and is termed plastic deformation. The determination of the stress and strain throughout a solid object is given by the field of strength of materials and for a structure by structural analysis.

<span class="mw-page-title-main">Fracture</span> Split of materials or structures under stress

Fracture is the appearance of a crack or complete separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displacement develops perpendicular to the surface, it is called a normal tensile crack or simply a crack; if a displacement develops tangentially, it is called a shear crack, slip band, or dislocation.

In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to plasticity, in which the object fails to do so and instead remains in its deformed state.

The field of strength of materials typically refers to various methods of calculating the stresses and strains in structural members, such as beams, columns, and shafts. The methods employed to predict the response of a structure under loading and its susceptibility to various failure modes takes into account the properties of the materials such as its yield strength, ultimate strength, Young's modulus, and Poisson's ratio. In addition, the mechanical element's macroscopic properties such as its length, width, thickness, boundary constraints and abrupt changes in geometry such as holes are considered.

<span class="mw-page-title-main">Fatigue (material)</span> Initiation and propagation of cracks in a material due to cyclic loading

In materials science, fatigue is the initiation and propagation of cracks in a material due to cyclic loading. Once a fatigue crack has initiated, it grows a small amount with each loading cycle, typically producing striations on some parts of the fracture surface. The crack will continue to grow until it reaches a critical size, which occurs when the stress intensity factor of the crack exceeds the fracture toughness of the material, producing rapid propagation and typically complete fracture of the structure.

<span class="mw-page-title-main">Fracture mechanics</span> Study of propagation of cracks in materials

Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.

<span class="mw-page-title-main">Work hardening</span> Strengthening a material through plastic deformation

In materials science, work hardening, also known as strain hardening, is the strengthening of a metal or polymer by plastic deformation. Work hardening may be desirable, undesirable, or inconsequential, depending on the context.

<span class="mw-page-title-main">Fracture toughness</span> Stress intensity factor at which a cracks propagation increases drastically

In materials science, fracture toughness is the critical stress intensity factor of a sharp crack where propagation of the crack suddenly becomes rapid and unlimited. A component's thickness affects the constraint conditions at the tip of a crack with thin components having plane stress conditions and thick components having plane strain conditions. Plane strain conditions give the lowest fracture toughness value which is a material property. The critical value of stress intensity factor in mode I loading measured under plane strain conditions is known as the plane strain fracture toughness, denoted . When a test fails to meet the thickness and other test requirements that are in place to ensure plane strain conditions, the fracture toughness value produced is given the designation . Fracture toughness is a quantitative way of expressing a material's resistance to crack propagation and standard values for a given material are generally available.

Corrosion fatigue is fatigue in a corrosive environment. It is the mechanical degradation of a material under the joint action of corrosion and cyclic loading. Nearly all engineering structures experience some form of alternating stress, and are exposed to harmful environments during their service life. The environment plays a significant role in the fatigue of high-strength structural materials like steel, aluminum alloys and titanium alloys. Materials with high specific strength are being developed to meet the requirements of advancing technology. However, their usefulness depends to a large extent on the degree to which they resist corrosion fatigue.

The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material. The theoretical concept of J-integral was developed in 1967 by G. P. Cherepanov and independently in 1968 by James R. Rice, who showed that an energetic contour path integral was independent of the path around a crack.

In materials science the flow stress, typically denoted as Yf, is defined as the instantaneous value of stress required to continue plastically deforming a material - to keep it flowing. It is most commonly, though not exclusively, used in reference to metals. On a stress-strain curve, the flow stress can be found anywhere within the plastic regime; more explicitly, a flow stress can be found for any value of strain between and including yield point and excluding fracture : .

<span class="mw-page-title-main">Embrittlement</span> Loss of ductility of a material, making it brittle

Embrittlement is a significant decrease of ductility of a material, which makes the material brittle. Embrittlement is used to describe any phenomena where the environment compromises a stressed material's mechanical performance, such as temperature or environmental composition. This is oftentimes undesirable as brittle fracture occurs quicker and can much more easily propagate than ductile fracture, leading to complete failure of the equipment. Various materials have different mechanisms of embrittlement, therefore it can manifest in a variety of ways, from slow crack growth to a reduction of tensile ductility and toughness.

Material failure theory is an interdisciplinary field of materials science and solid mechanics which attempts to predict the conditions under which solid materials fail under the action of external loads. The failure of a material is usually classified into brittle failure (fracture) or ductile failure (yield). Depending on the conditions most materials can fail in a brittle or ductile manner or both. However, for most practical situations, a material may be classified as either brittle or ductile.

Crack closure is a phenomenon in fatigue loading, where the opposing faces of a crack remain in contact even with an external load acting on the material. As the load is increased, a critical value will be reached at which time the crack becomes open. Crack closure occurs from the presence of material propping open the crack faces and can arise from many sources including plastic deformation or phase transformation during crack propagation, corrosion of crack surfaces, presence of fluids in the crack, or roughness at cracked surfaces.

Materials that are used for biomedical or clinical applications are known as biomaterials. The following article deals with fifth generation biomaterials that are used for bone structure replacement. For any material to be classified for biomedical applications, three requirements must be met. The first requirement is that the material must be biocompatible; it means that the organism should not treat it as a foreign object. Secondly, the material should be biodegradable ; the material should harmlessly degrade or dissolve in the body of the organism to allow it to resume natural functioning. Thirdly, the material should be mechanically sound; for the replacement of load-bearing structures, the material should possess equivalent or greater mechanical stability to ensure high reliability of the graft.

<span class="mw-page-title-main">Crack tip opening displacement</span>

Crack tip opening displacement (CTOD) or is the distance between the opposite faces of a crack tip at the 90° intercept position. The position behind the crack tip at which the distance is measured is arbitrary but commonly used is the point where two 45° lines, starting at the crack tip, intersect the crack faces. The parameter is used in fracture mechanics to characterize the loading on a crack and can be related to other crack tip loading parameters such as the stress intensity factor and the elastic-plastic J-integral.

In materials science, toughening refers to the process of making a material more resistant to the propagation of cracks. When a crack propagates, the associated irreversible work in different materials classes is different. Thus, the most effective toughening mechanisms differ among different materials classes. The crack tip plasticity is important in toughening of metals and long-chain polymers. Ceramics have limited crack tip plasticity and primarily rely on different toughening mechanisms.

Solder fatigue is the mechanical degradation of solder due to deformation under cyclic loading. This can often occur at stress levels below the yield stress of solder as a result of repeated temperature fluctuations, mechanical vibrations, or mechanical loads. Techniques to evaluate solder fatigue behavior include finite element analysis and semi-analytical closed-form equations.

Fracture of biological materials may occur in biological tissues making up the musculoskeletal system, commonly called orthopedic tissues: bone, cartilage, ligaments, and tendons. Bone and cartilage, as load-bearing biological materials, are of interest to both a medical and academic setting for their propensity to fracture. For example, a large health concern is in preventing bone fractures in an aging population, especially since fracture risk increases ten fold with aging. Cartilage damage and fracture can contribute to osteoarthritis, a joint disease that results in joint stiffness and reduced range of motion.

<span class="mw-page-title-main">Crack growth equation</span>

A crack growth equation is used for calculating the size of a fatigue crack growing from cyclic loads. The growth of a fatigue crack can result in catastrophic failure, particularly in the case of aircraft. When many growing fatigue cracks interact with one another it is known as widespread fatigue damage. A crack growth equation can be used to ensure safety, both in the design phase and during operation, by predicting the size of cracks. In critical structure, loads can be recorded and used to predict the size of cracks to ensure maintenance or retirement occurs prior to any of the cracks failing. Safety factors are used to reduce the predicted fatigue life to a service fatigue life because of the sensitivity of the fatigue life to the size and shape of crack initiating defects and the variability between assumed loading and actual loading experienced by a component.

References

  1. ↑ John Scheirs, “john wiley and sons”, 30-oct-2000 “[Compositional and Failure Analysis of Polymers: A Practical Approach]”
  2. ↑ G. Spathis, E. Kontou, “Creep failure time prediction of polymers and polymer composites”
  3. ↑ Robert Oboigbaotor Ebewele,"CRC Press,2000" "polymer science and technology"
  4. ↑ Hertzberg, Richard (2013). Deformation and Fracture Mechanics of Engineering Materials. Danvers, MA: John Wiley & Sons Inc.
  5. ↑ Arencon, David (November 2009). "Fracture Toughness of Polypropylene-Based Particulate Composites". Materials. 2 (4): 2046. Bibcode:2009Mate....2.2046A. doi: 10.3390/ma2042046 . hdl: 2117/6841 – via MDPI.
  6. ↑ Zeidi, Mahdi; Kim, Chun IL; Park, Chul B. (2021). "The role of interface on the toughening and failure mechanisms of thermoplastic nanocomposites reinforced with nanofibrillated rubbers". Nanoscale. 13 (47): 20248–20280. doi:10.1039/D1NR07363J. ISSN   2040-3372. PMID   34851346. S2CID   244288401.
  7. ↑ Anderson, T.L. (2005). Fracture Mechanics Fundamentals and Applications Third Addition. Boca Raton, FL: Taylor and Francis Group.
  8. ↑ Hawinkels, R.J.H. (August 30, 2011). "Fatigue Crack Propagation in Polycarbonate". Einfhoven.
  9. ↑ Mars, W. V.; Fatemi, A. (July 2004). "Factors that Affect the Fatigue Life of Rubber: A Literature Survey". Rubber Chemistry and Technology. 77 (3): 391–412. doi:10.5254/1.3547831. ISSN   0035-9475.
  10. ↑ Aliha, M.R.M.; Bahmani, A.; Akhondi, Sh. (July 2016). "A novel test specimen for investigating the mixed mode I+III fracture toughness of hot mix asphalt composites – Experimental and theoretical study". International Journal of Solids and Structures. 90: 167–177. doi: 10.1016/j.ijsolstr.2016.03.018 . ISSN   0020-7683.
  11. 1 2 Mzabi, Samy; Berghezan, Daniel; Roux, Stéphane; Hild, Francois; Creton, Costantino (2011-09-07). "A critical local energy release rate criterion for fatigue fracture of elastomers" (PDF). Journal of Polymer Science Part B: Polymer Physics. 49 (21): 1518–1524. Bibcode:2011JPoSB..49.1518M. doi:10.1002/polb.22338. ISSN   0887-6266.
  12. 1 2 3 MARS, W; FATEMI, A (September 2002). "A literature survey on fatigue analysis approaches for rubber". International Journal of Fatigue. 24 (9): 949–961. doi:10.1016/s0142-1123(02)00008-7. ISSN   0142-1123.
  13. 1 2 Courtney, Thomas H. (2013). Mechanical Behavior of Materials. McGraw Hill Education (India). ISBN   978-1259027512. OCLC   929663641.
  14. ↑ Aboutorabi, H.; Ebbott, T.; Gent, A. N.; Yeoh, O. H. (March 1998). "Crack Growth in Twisted Rubber Disks. Part I: Fracture Energy Calculations". Rubber Chemistry and Technology. 71 (1): 76–83. doi:10.5254/1.3538474. ISSN   0035-9475.
  15. ↑ De, D. K.; Gent, A. N. (March 1998). "Crack Growth in Twisted Rubber Disks. Part II: Experimental Results". Rubber Chemistry and Technology. 71 (1): 84–94. doi:10.5254/1.3538475. ISSN   0035-9475.
  16. ↑ Gent, A. N.; Yeoh, O. H. (November 2003). "Crack Growth in Twisted Rubber Disks. Part 3. Effects of Crack Depth and Location". Rubber Chemistry and Technology. 76 (5): 1276–1289. doi:10.5254/1.3547802. ISSN   0035-9475.
  17. ↑ Verron, E; Huneau, B; Beurrot, S (September 2009), "In-situ SEM study of fatigue crack growth mechanism in carbon black-filled natural rubber" (PDF), Constitutive Models for Rubber VI, CRC Press, pp. 319–324, doi:10.1201/noe0415563277.ch52, ISBN   9780415563277
  18. ↑ Wöhler, A (1867). "Wöhler's experiments on the strength of metals". Engineering. 4: 160–161.
  19. ↑ Paris, P.; Erdogan, F. (1963). "A Critical Analysis of Crack Propagation Laws". Journal of Basic Engineering. 85 (4): 528. doi:10.1115/1.3656900. ISSN   0021-9223.
  20. ↑ Mandell and Lang
  21. ↑ Wolfgang Grellmann, Sabine Seidler, “Springer 2001” “Deformation and Fracture Behaviour of Polymers”
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