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hould be considered furthermore in the high temperature region 3 t 0 7 t m έ can be low contributing to low τ crss however plastic flow will still occur due to thermally activated high temperature time dependent plastic deformation mechanisms such as nabarro herring nh and coble diffusional flow through the lattice and along the single crystal surfaces respectively as well as dislocation climb glide creep stages of time independent plastic flow post yielding edit the three stages of time independent plastic deformation of single crystals during the easy glide stage 1 the work hardening rate defined by the change in shear stress with respect to shear strain dτ dγ is low representative of a small amount of applied shear stress necessary to induce a large amount of shear strain facile dislocation glide and corresponding flow is attributed to dislocation migration along parallel slip planes only i e one slip system moderate impedance to dislocation migration along parallel slip planes is exhibited according to the weak stress field interactions between these dislocations which heightens with smaller interplanar spacing overall these migrating dislocations within a single slip system act as weak obstacles to flow and a modest rise in stress is observed in comparison to the yield stress during the linear hardening stage 2 of flow the work hardening rate becomes high as considerable stress is required to overcome the stress field interactions of dislocations migrating on non parallel slip planes i e multiple slip systems acting as strong obstacles to flow much stress is required to drive continual dislocation migration for small strains the shear flow stress is directly proportional to the square root of the dislocation density τ flow ρ ½ irrespective of the evolution of dislocation configurations displaying the reliance of hardening on the number of dislocations present regarding this evolution of dislocation configurations at small strains the dislocation arrangement is a random 3d array of intersecting lines moderate strains correspond to cellular dislocation structures of heterogeneous dislocation distribution with large dislocation density at the cell boundaries and small dislocation density within the cell interior at even larger strains the cellular dislocation structure reduces in size until a minimum size is achieved finally the work hardening rate becomes low again in the exhaustion saturation of hardening stage 3 of plastic flow as small shear stresses produce large shear strains notably instances when multiple slip systems are oriented favorably with respect to the applied stress the τ crss for these systems may be similar and yielding may occur according to dislocation migration along multiple slip systems with non parallel slip planes displaying a stage 1 work hardening rate typically characteristic of stage 2 lastly distinction between time independent plastic deformation in body centered cubic transition metals and face centered cubic metals is summarized below comparison between the time independent plastic deformation of body centered cubic transition metals and face centered cubic metals highlighting the critical resolved shear stress work hardening rate and necking strain during tensile testing body centered cubic transition metals face centered cubic metals critical resolved shear stress high relatively strongly temperature dependent critical resolved shear stress low relatively weakly temperature dependent work hardening rate temperature independent work hardening rate temperature dependent necking strain increases with temperature necking strain decreases with temperature time independent yielding and plastic flow in polycrystals edit plasticity in polycrystals differs substantially from that in single crystals due to the presence of grain boundary gb planar defects which act as very strong obstacles to plastic flow by impeding dislocation migration along the entire length of the activated slip plane s hence dislocations cannot pass from one grain to another across the grain boundary the following sections explore specific gb requirements for extensive plastic deformation of polycrystals prior to fracture as well as the influence of microscopic yielding within individual crystallites on macroscopic yielding of the polycrystal the critical resolved shear stress for polycrystals is defined by schmid s law as well τ crss σ y ṁ where σ y is the yield strength of the polycrystal and ṁ is the weighted schmid factor the weighted schmid factor reflects the least favorably oriented slip system among the most favorably oriented slip systems of the grains constituting the gb grain boundary constraint in polycrystals edit the gb constraint for polycrystals can be explained by considering a grain boundary in the xz plane between two single crystals a and b of identical composition structure and slip systems but misoriented with respect to each other to ensure that voids do not form between individually deforming grains the gb constraint for the bicrystal is as follows ε xx a ε xx b the x axial strain at the gb must be equivalent for a and b ε zz a ε zz b the z axial strain at the gb must be equivalent for a and b and ε xz a ε xz b the xz shear strain along the xz gb plane must be equivalent for a and b in addition this gb constraint requires that five independent slip systems be activated per crystallite constituting the gb notably because independent slip systems are defined as slip planes on which dislocation migrations cannot be reproduced by any combination of dislocation migrations along other slip system s planes the number of geometrical slip systems for a given crystal system which by definition can be constructed by slip system combinations is typically greater than that of independent slip systems significantly there is a maximum of five independent slip systems for each of the seven crystal systems however not all seven crystal systems acquire this upper limit in fact even within a given crystal system the composition and bravais lattice diversifies the number of independent slip systems see the table below in cases for which crystallites of a polycrystal do not obtain five independent slip systems the gb condition cannot be met and thus the time independent deformation of individual crystallites results in cracks and voids at the gbs of the polycrystal and soon fracture is realized hence for a given composition and structure a single crystal with less than five independent slip systems is stronger exhibiting a greater extent of plasticity than its polycrystalline form the number of independent slip systems for a given composition primary material class and structure bravais lattice 10 11 bravais lattice primary material class independent slip systems face centered cubic metal 5 ceramic covalent 5 ceramic ionic 2 body centered cubic metal 5 simple cubic ceramic ionic 3 hexagonal metal 2 ceramic mixed 2 implications of the grain boundary constraint in polycrystals edit although the two crystallites a and b discussed in the above section have identical slip systems they are misoriented with respect to each other and therefore misoriented with respect to the applied force thus microscopic yielding within a crystallite interior may occur according to the rules governing single crystal time independent yielding eventually the activated slip planes within the grain interiors will permit dislocation migration to the gb where many dislocations then pile up as geometrically necessary dislocations this pile up corresponds to strain gradients across individual grains as the dislocation density near the gb is greater than that in the grain interior imposing a stress on the adjacent grain in contact when considering the ab bicrystal as a whole the most favorably oriented slip system in a will not be the that in b and hence τ a crss τ b crss paramount is the fact that macroscopic yielding of the bicrystal is prolonged until the higher value of τ crss between grains a and b is achieved according to the gb constraint thus for a given composition and structure a polycrystal with five independent slip systems is stronger greater extent of plasticity than its single crystalline form correspondingly the work hardening rate will be higher for the polycrystal than the single crystal as more stress is required in the polycrystal to produce strains importantly just as with single crystal flow stress τ flow ρ ½ but is also inversely proportional to the square root of average grain diameter τ flow d ½ therefore the flow stress of a polycrystal and hence the polycrystal s strength increases with small grain size the reason for this is that smaller grains have a relatively smaller number of slip planes to be activated corresponding to a fewer number of dislocations migrating to the gbs and therefore less stress induced on adjacent grains due to dislocation pile up in addition for a given volume of polycrystal smaller grains present more strong obstacle grain boundaries these two factors provide an understanding as to why the onset of macroscopic flow in fine grained polycrystals occurs at larger applied stresses than in coarse grained polycrystals mathematical descriptions edit deformation theory edit an idealized uniaxial stress strain curve showing elastic and plastic deformation regimes for the deformation theory of plasticity there are several mathematical descriptions of plasticity 12 one is deformation theory see e g hooke s law where the cauchy stress tensor of order d 1 in d dimensions is a function of the strain tensor although this description is accurate when a small part of matter is subjected to increasing loading such as strain loading this theory cannot account for irreversibility ductile materials can sustain large plastic deformations without fracture however even ductile metals will fracture when the strain becomes large enough this is as a result of work hardening of the material which causes it to become brittle heat treatment such as annealing can restore the ductility of a worked piece so that shaping can continue flow plasticity theory edit main article flow plasticity theory in 1934 egon orowan michael polanyi and geoffrey ingram taylor roughly simultaneously realized that the plastic deformation of ductile materials could be explained in terms of the theory of dislocations the mathematical theory of plasticity flow plasticity theory uses a set of non linear non integrable equations to describe the set of changes on strain and stress with respect to a previous state and a small increase of deformation yield criteria edit comparison of tresca criterion to von mises criterion main article yield engineering if the stress exceeds a critical value as was mentioned above the material will undergo plastic or irreversible deformation this critical stress can be tensile or compressive the tresca and the von mises criteria are commonly used to determine whether a material has yielded however these criteria have proved inadequate for a large range of materials and several other yield criteria are also in widespread use tresca criterion edit the tresca criterion is based on the notion that when a material fails it does so in shear which is a relatively good assumption when considering metals given the principal stress state we can use mohr s circle to solve for the maximum shear stresses our material will experience and conclude that the material will fail if σ 1 σ 3 σ 0 displaystyle sigma _ 1 sigma _ 3 geq sigma _ 0 where σ 1 is the maximum normal stress σ 3 is the minimum normal stress and σ 0 is the stress under which the material fails in uniaxial loading a yield surface may be constructed which provides a visual representation of this concept inside of the yield surface deformation is elastic on the surface deformation is plastic it is impossible for a material to have stress states outside its yield surface huber von mises criterion edit the von mises yield surfaces in principal stress coordinates circumscribes a cylinder around the hydrostatic axis also shown is tresca s hexagonal yield surface main article von mises yield criterion the huber von mises criterion 13 is based on the tresca criterion but takes into account the assumption that hydrostatic stresses do not contribute to material failure m t huber was the first who proposed the criterion of shear energy 14 15 von mises solves for an effective stress under uniaxial loading subtracting out hydrostatic stresses and states that all effective stresses greater than that which causes material failure in uniaxial loading will result in plastic deformation σ v 2 1 2 σ 11 σ 22 2 σ 22 σ 33 2 σ 11 σ 33 2 6 σ 23 2 σ 31 2 σ 12 2 displaystyle sigma _ v 2 tfrac 1 2 sigma _ 11 sigma _ 22 2 sigma _ 22 sigma _ 33 2 sigma _ 11 sigma _ 33 2 6 sigma _ 23 2 sigma _ 31 2 sigma _ 12 2 again a visual representation of the yield surface may be constructed using the above equation which takes the shape of an ellipse inside the surface materials undergo elastic deformation reaching the surface means the material undergoes plastic deformations see also edit atterberg limits creep deformation deformation mechanics deformation engineering plastometer poisson s ratio yield engineering references edit lubliner jacob 2008 plasticity theory dover isbn 978 0 486 46290 5 bigoni davide 2012 nonlinear solid mechanics bifurcation theory and material instability cambridge university press isbn 978 1 107 02541 7 jirásek milan bažant zdeněk p 2002 inelastic analysis of structures john wiley and sons isbn 0 471 98716 6 chen wai fah 2008 limit analysis and soil plasticity j ross publishing isbn 978 1 932159 73 8 yu mao hong ma guo wei qiang hong fu zhang yong qiang 2006 generalized plasticity springer isbn 3 540 25127 8 chen wai fah 2007 plasticity in reinforced concrete j ross publishing isbn 978 1 932159 74 5 ziegenhain gerolf and urbassek herbert m reversible plasticity in fcc metals in philosophical magazine letters 89 11 717 723 2009 doi 10 1080 09500830903272900 maaß robert derlet peter m january 2018 micro plasticity and recent insights from intermittent and small scale plasticity acta materialia 143 338 363 arxiv 1704 07297 bibcode 2018acmat 143 338m doi 10 1016 j actamat 2017 06 023 s2cid 119387816 courtney thomas 2005 mechanical behavior of materials second ed long grove illinois waveland press inc isbn 978 1 57766 425 3 partridge peter 1969 deformation and fatigue of hexagonal close packed metals university of surrey cite book cs1 maint location missing publisher link groves geoffrey w kelly anthony 1963 independent slip systems in crystals philosophical magazine 8 89 877 887 bibcode 1963pmag 8 877g doi 10 1080 14786436308213843 hill rodney 1998 the mathematical theory of plasticity oxford university press isbn 0 19 850367 9 von mises richard 1913 mechanik der festen körper im plastisch deformablen zustand nachrichten von der gesellschaft der wissenschaften zu göttingen mathematis...