Split-Hopkinson Pressure Bar test (SHPB) for dynamic compression

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Strength differential effect – SD effect

In order to get a comprehensive characterization of a material there should be performed not only the tension test (which is commonly used to determine normative properties of ma-terial) but also compression and shear. Extended experimental procedure allows to exhibit different behaviours for different loading types of examined material. Difference in charac-teristics obtained due to compression and tension have been reported by [Hirth and Cohen 1970] for martensitic steels which are stronger in uniaxial compression than in tension. This observation was confirmed in works of [Chait 1972; Spitzig et al. 1979; Spitzig and Rich-mond 1979] for other high strength steels. The examples of martensitic, quenched, tempered or maraging steels were studied. In [Drucker 1973] the phenomenon of asymmetry of elas-tic ranges was more generally formulated for metals and plastics. Experimental studies on the effect of superimposed hydrostatic pressure on the deformation behaviour of iron based materials [Spitzig et al. 1979], as well as of aluminum [Spitzig and Richmond 1984], have proved that strength differential phenomenon exists for many kinds of materials. The as-sumption that tensile and compressive yield strengths are equal diminishes when polymers are under consideration, what is proved in [Raghava et al. 1973]. The SD effect has been studied extensively also in recent years. Further examples which show the different response in tension and compression for different materials are presented in, e.g. [Casey and Sullivan 1985; Altenbach and Zolochevsky 1995; Wilson 2002; Altenbach and Zolochevsky 1996; Bai and Wierzbicki 2008]. Recently, the strength differential effect has attracted extensive interest due to the increasing adoption of hexagon-closed packed (HCP) metals and alloys to satisfy the requirement for the high ratio of strength to density. For example, the tensile yield stress is much higher than the one in compression for magnesium alloys, [Lou et al. 2007; Cheng-wen et al. 2007; Yoshikawa et al. 2008; Rusinek 2011].
The phenomenon of inequality of the tensile and compressive yield strengths is known as the strength-differential (SD) effect. It can be also denoted as an asymmetric effect. The SD parameter is defined by the following relation: C YT= Y (2.1).
In the studies [Spitzig and Richmond 1984] the SD effect was interpreted as a direct con-sequence of the dependence of the flow stress on the hydrostatic stress (mean stress) in the specimen. The SD effect and a sensitivity of the flow stress to the hydrostatic pressure are related to volume expansion of the material as a consequence of plastic deformation, [Spitzig et al. 1979; Mahnken 2001].
According to [Fletcher and M. Cohen 1974; Casey and Jahedmotlagh 1984] for the SD ef-fect is not related to frictional effect in compression testing, residual stresses from quenching and phase transformations, microscopic void formation or transfrormation-induced Bauschinger effect. Therefore, only a difference in the inherent response to tensile and compressive defor-mation can be responsible.
In [Hirth and Cohen 1970; Casey and Jahedmotlagh 1984; Mahnken 2001] there are pre-sented following observation considreding SD effect: it remains constant over a considerable range of the plastic strain and it is not influenced by the strain rate if the tension and compres-sion tests are performed with the same constant « , [Duckett et al. 1970].
In [Ghorbel 2008] based on the investigation for polymers for which difference in yield strengths is characteristic, it is reported that is independent on the temperature, what is es-tablished for different polymeric materials (for PET this observation was made by [Duckett et al. 1970], for PC information are collected in [Bauwens-Crowet et al. 1972]). The further information about SD effect are presented in, e.g. [Chait 1972; Drucker 1973; Casey and Jahedmotlagh 1984; Mahnken 2001; Lou et al. 2013].

Initial anisotropy

Manufacturing processes of metals and alloys produce often, due to the evolution of tex-ture, change of mechanical properties that they lose their isotropic character. Technological processes of production (ex. forging, stamping, rolling, extruding) results in the massive reori-entation of structure during the large strain forming operation. The presence of anisotropy in structure may lead to premature failure or unexpected shear localization, [Arruda et al. 1993]. Two main sources of initial anisotropy result from the thermo-mechanical processes involved in metal production: crystallographic texture and intragranular dislocation structure. The ma-terial in the ’as-received’ state becomes usually anisotropic because manufacturing processes associated with the initial forming of metal sheets, bars or metal profiles stock. Proceeding with such materials must not only account for the inhomogeneous and direction-dependent properties but should also anticipate undesirable features of anisotropic behaviour, such as premature fracture and undesirable shear banding. The development of a strong texture will lead to orientation-dependent mechanical characteristics because the crystallographic struc-ture of grains is intrinsically anisotropic. Concerning the intragranular structure, dislocation interactions depend on the Burgers vectors as well as on the slip direction. This leads to such effects like latent hardening or Bauschinger softening, [Rauch 1998]. It is difficult, however, to measure and describe in the direct way the initial anisotropy of elastic properties and yield behaviour. For initially anisotropic materials exact descriptions of its yielding behaviour be-comes rather difficult.
Loading direction or stress state may result in a different response on the orientation of dominant slip-plane and the critical value of shearing stress. The yielding state of materials is often physically interpreted as observation that yielding of element in a material occurs when the shearing stress on a dominant slip-plane reaches its critical value, [Hu 2007]. If material is treated as isotropic, such physical interpretation can be described by Huber-Mises-Hencky criterion (Huber 1904: [Huber 2004], Mises 1913: [von Mises 1913]) for which the shear limit fulfils the relation given by Eq. (2.2): H Y Y = p where YH is a yield strength in shear according to Huber-Mises approach and Y strength in tension.

Oxygen Free High Conductivity Copper – OHFC Cu

Copper has the highest conductivity of electricity and heat from all the ’base metals’. In addition, its other properties, such as relatively high strength, toughness, ease of processing and design, corrosion resistance, durability, non-toxicity make it an extremely valuable mate-rial in the electrical, electronics, energetic industries, as well as in construction and technical installations. Performance of copper recycling industry can reduce use of natural resources and reduce waste generation during disposal of used products – copper material are environ-mentally friendly, highly re-producting material, e.g.: [Publications 1998; Rosenberg et al. 2009]. The development of the electrical industry requires the use of increasingly refined copper characterized by increasingly better chemical, physical, technological and utility prop-erties.
High Conductivity Oxygen Free Copper – OFHC Cu – is a material with specially shaped structure, small particle size, high purity – Class 5N. The amount of pollution does not exceed 25 ppm by weight. The reason of importance of purity of copper is that the most harmful impurities can significantly decrease electrical conductivity, increase the mechanical strength of the annealed wire, retard recrystallization, and will sometimes induce hot shortness during the hot rolling process in the production of rod, [www.copper.org].
Oxygen is intentionally alloyed with copper to act as a scavenger for dissolved hydrogen and sulfur to form the gases and melt. The presence of oxygen in the copper leads to reduce the service time of cables and conductors, it is the so-called ’hydrogen embrittlement’. Annealing of wires at high temperature leads to the reaction of hydrogen with oxygen and due to the resulting reaction, water vapor contributes to the cracking of the copper grain boundaries. In the case of oxygen free copper hydrogen embrittlement does not occur, neither other effects caused by the presence of oxygen. Obtaining OFHC requires highly advanced production techniques, the process of melting high purity feedstock is performed under pressure, in the conditions that eliminate the access of oxygen and other pollution. A continuous casting and rolling process produces virtually all copper rod. Benefits of continuous casting include less microsegregation of impurities, reduction of copper oxide particles on the surface, fewer steel inclusions resulting from con-tact with mill rolls, almost total elimination of welds, and lower overall processing costs, [www.copper.org].
The thermal stability of the production determines the reproducibility of physical and chemical properties. Obtained chemical purity determines the high electrical conductivity. Copper exhibits high conductivity because its electrons show relatively little resistance to movement under an electric field. Copper in particular is an excellent conductor because out-ermost electrons have a large mean free path (about 100 atomic spacings) between collisions.
The electrical resistivity is inversely related to this mean free path. Signal running through the guide should face the least amount of obstacles, so that the sound is not distorted – this conditions is fulfilled in the production processes of OFHC Cu. Negligible presence of oxides and fewer grain boundaries in oxygen-free copper provide lossless transmission of sound and image signals. Small amounts of oxygen leads to improved performance properties, partic-ularly ductility and corrosion resistance, the recrystallization temperature is also increased, [www.kghm.pl, www.copper.org, www.Philips.com].
Due to the specific properties OFHC Cu is used in modern, more demanding, computer and electronic technologies. For example, in hi-fi technology, cryogenics, semiconductor heat sinks, in diodes and hygristors for heavy current operations, to production of wires and cables, busbars, commutators, welding electrodes, contacts, contact springs, printed circuit boards, semiconductors, high vacuum and other electronic devices, tuyeres, heat exchangers, cables and busbars, www.kghm.pl, www.copper.org.

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Recovery and recrystallization in OFHC

OFHC Cu demonstrates softening due to static and dynamic restoration processes, static thermal recovery and recrystallization, [Tanner 1998; Tanner and McDowell 1999; Doherty et al. 1997]. Recrystallization is an important issue in the thermomechanical processing of copper because it restores a worked metal to an unworked and formable state. It is possible to find in literature a OFHC copper called ’annealed’. In this case a heat treatment is done allowing process of recrystallization. The yield stress reduced and changing hardening.
The grains after manufacturing processes get elongated in the direction normal to the di-rection of applied force. The material gets strain hardened, i.e. its yield strength and hardness increase while ductility decreases. The strain hardening occurs because the dislocation density increases due to cold deformation. With increase in temperature the movement of dislocations gets easier and they readjust due to stresses locked in the lattice. Some dislocations having op-posite sign may annihilate each other. This is a recovery process in which the residual stresses are reduced, however, the enhanced properties due to cold working are only little affected, [Juneja 2010]. Due to the recovery in OFHC Cu, the density of local dislocations continu-ously increases, which creates instabilities in the substructure and recrystallization processes become possible, [McQueen and Jonas 1975].
Recrystallization is the process of new grains forming and growing through the material resulting in new regions of low dislocation density. Large numbers of dislocations are elim-inated during recrystallization. Dynamic recrystallization occurs when the nucleation and growth occurs concurrent with straining under stress, [Kostryzhev 2009].
The average grain size decreased during the deformation conditions when the flow stress exhibited a single peak, which is consistent with grain refinement. During the deformation conditions when oscillations in the flow stress occurred, the average grain size increased, consistent with grain coarsening. These observations indicate that the softening observed at temperatures above 270 C, [Tanner 1998], are due to the changes in grain structure resulting from recrystallization. The effect of recrystallization is to change the grain structure which results in the reduction of the steady state flow stress. The initial peak strain, the strain corre-sponding to the peak in flow stress, decreases with increasing temperature at the same strain rate. The observed softening is consistent with the softening data shown in Fig. 2.3. A greater amount of softening occurs with increasing time.

Mechanical properties of Polycarbonate (PC)

Polycarbonate (PC) is an amorphous polymer with a glass transition temperature above room temperature. The PC glass transition temperature is of the order 148 C, and a beta relaxation temperature of the order -60 C. Amorphous glassy polymers are main compo-nents in many industrial and commercial products, such as automotive lighting, bus windows, http://www.poliweglan.com. PC is a light-transmissive material with a high resistance to heat and acids, widely used for its optical and mechanical properties. It has a good thermal resis-tance what makes it useful in the building industry, it is used for noise barriers, roofs, roof-windows, http://www.poliweglan.com. The material is also used in extreme temperature and loading condition such as impact-resistant aircraft windows. The increase in using polymers for several industrial applications leads to a strong need for the development of constitutive models dedicated to the simulation of the behavior during the manufacturing process or dur-ing the in-service phase. Since the beginning of polymer science, numerous experimental studies have been carried out on polymers to characterize the mechanical behaviour as a function of temperature and strain rate. Among these studies, a great deal of attention has been given to the yield stress, e.g.: [Bauwens-Crowet et al. 1969; 1972; Shaban et al. 2007; Altenbach and Zolochevsky 1996; Altenbach and Tushtev 2001; Mulliken and Boyce 2006; Richeton et al. 2006; Ghorbel 2008].
Polymers are known to exhibit a strongly pronounced rate and temperature-dependent be-havior and display non-linear responses during loading and unloading. Significant inelastic deformation, denoted as viscoplastic, may be observed even at very small deformation lev-els, [Shaban et al. 2007; Ghorbel 2008; Matadi Boumbimba et al. 2012]. The yield stress increases for the low temperatures as well as for the high strain rates. Strain rate and temper-ature are known to significantly influence the mechanical behavior of polymers. Concerning the stress–strain behavior the mechanical response at the low strain rates first is that an ini-tial elastic response followed by yielding, strain softening and then a strain hardening. The yield stress increases with an increasing strain rate especially at high strain rates. The initial Young’s modulus appears also to be strain rate dependent. Whereas, the yield stress and the initial Young’s modulus are found to decrease with an increasing temperature; a similar effect is also observed for the strain hardening rate. This corroborates that the temperature rises at high strain rates controls the strain hardening rate, which depends itself on temperature. The closer the temperature is to the glass transition temperature, the lower the strain hardening rate.
For polycarbonate the yield stress in compression is greater than that in tension as observed e.g. in [Spitzig and Richmond 1979]. Many authors, e.g.: [Bauwens-Crowet et al. 1969; 1972; Shaban et al. 2007; Altenbach and Zolochevsky 1995; Mulliken and Boyce 2006; Richeton et al. 2006; Ghorbel 2008] have similar conclusions concerning also other polymers, like polymethil methacrylate (PMMA).

Table of contents :

1 Framework of the Thesis 
1.1 Introduction
1.2 Motivation
1.3 Objectives
1.4 Contents of the Thesis
1.5 Original contributions
1.6 Applications
2 Materials 
2.1 Introduction
2.1.1 Strength differential effect – SD effect
2.1.2 Initial anisotropy
2.2 Oxygen Free High Conductivity Copper – OHFC Cu
2.3 E335 – high strength steel
2.4 Mechanical properties of chosen polymers
2.5 Concluding remarks
3 Experimental techniques and their methodology 
3.1 Introduction
3.1.1 Schematic representation of stress states
3.1.2 Determining the yield strength
3.2 Uniaxial tension and compression test
3.2.1 FEM analysis of the tension and compression test
3.3 Shear test
3.3.1 The double shear test
3.3.2 FEM analysis of the double shear test
3.4 Biaxial compression test
3.4.1 Preliminary results of biaxial compression test
3.4.2 FEM analysis of biaxial compression test
3.5 Complex stress state test
3.6 Split-Hopkinson Pressure Bar test (SHPB) for dynamic compression
3.7 Concluding remarks
4 Burzy´ nski material effort hypothesis 
4.1 Introduction
4.2 General formulation of the yield condition
4.3 Visualization of yield criteria
4.4 Yield criteria in the literature
4.4.1 Traditional yield criteria
4.4.2 Criteria accounting for stress invariants J2; I1 and J2; I1; J3
4.4.3 Phenomenological criteria
4.5 Burzy´nski material effort hypothesis and the resulted criteria
4.5.1 Elastic energy
4.5.2 Hypothesis of variable limit energy of volume change and distortion .
4.5.3 Paraboloidal Burzy´nski criterion for isotropic materials
4.5.4 Burzy´nski criterion for materials with initial anisotropy
4.5.5 Smoothness of criterion for initially anisotropic materials
4.5.6 Phenomenological formulation of the Burzy´nski criterion
4.5.7 Concluding remarks
4.6 User subroutine for the Burzy´nski isotropic criterion
4.6.1 Analysis of the tensile test of notched specimens for E335 steel
4.6.2 Experimental and numerical results
4.6.3 Concluding remarks
5 Experimental results and applications of the Burzy´ nski criteria 
5.1 Introduction
5.2 Application for various materials
5.2.1 Yield surface basing on the complex state test results
5.2.2 Grey cast-iron
5.2.3 Titanium alloy – Ti6Al4V
5.2.4 Mg alloy
5.2.5 Metallic glasses
5.3 Application for FCC material – OFHC Copper
5.3.1 Results of strength tests
5.3.2 Yield surface for OFHC Cu
5.4 Applications for BCC materials – E335 steel
5.4.1 Results of strength tests for E335
5.4.2 Yield surface for E335
5.5 Applications for amorphous materials
5.5.1 SD effect in the polymers and its influence on the yield state
5.5.2 Results for polycarbonate
5.5.3 Results for PLA/PBAT
6 Conclusions 
Appendix 1 – Constitutive models of strain rate sensitivity
Appendix 2 – Algorithm of Burzy´nski paraboloidal criterion
Appendix 3 – Project of biaxial compression set-up


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