Intrinsic Length Parameter Research Articles

On configurational forces for gradient-enhanced inelasticity

In this paper we discuss how configurational forces can be computed in an efficient and robust manner when a constitutive continuum model of gradient-enhanced viscoplasticity is adopted, whereby a suitably tailored mixed variational formulation in terms of displacements and micro-stresses is used. It is demonstrated that such a formulation produces sufficient regularity to overcome numerical difficulties that are notorious for a local constitutive model. In particular, no nodal smoothing of the internal variable fields is required. Moreover, the pathological mesh sensitivity that has been reported in the literature for a standard local model is no longer present. Numerical results in terms of configurational forces are shown for (1) a smooth interface and (2) a discrete edge crack. The corresponding configurational forces are computed for different values of the intrinsic length parameter. It is concluded that the convergence of the computed configurational forces with mesh refinement depends strongly on this parameter value. Moreover, the convergence behavior for the limit situation of rate-independent plasticity is unaffected by the relaxation time parameter.

Acoustic emission assisted fracture zone analysis of cellulose fibre materials

An acoustic emission technique is used to quantify and position microfracture events ahead of a growing opening mode crack in paper materials containing different amounts of added starch. A mechanical model based on gradient-enhanced elasticity, containing an intrinsic length parameter reflecting the fibre-based materials microstructure, is applied to analyse the results. It is found in experiments that the addition of starch increases the tensile strength of paper significantly while the level of onset of microfracture nucleation at the crack-tip is only slightly increased. It is also found that the height of the process zone (zone in which microfractures ahead of the crack predominantly take place), measured from the crack plane, decreases with increasing amount of starch. The experimental and analytical results suggest that adding cationic starch to paper reduces the material’s sensitivity to gradients in the stress and strain fields and making the fibre network material more ‘continuum-like’. The experimental observations are shown to be qualitatively in agreement with the numerical results and lend confidence to the applied model.

Strain gradient plasticity analysis of transformation induced plasticity in multiphase steels

“To what extent do plastic strain gradients affect the strengthening resulting from the transformation of small metastable inclusions into hard inclusions within a plastically deforming matrix?” is the central question addressed here. Though general in the approach, the focus is on the behavior of TRIP-assisted multiphase steels. A two-dimensional embedded cell model of a simplified microstructure composed of a single metastable austenitic inclusion surrounded by a soft ferritic matrix is considered. The cell is inserted in a large homogenized medium. The transformation of a fraction of the austenite into a hard martensite plate is simulated, accounting for a transformation strain, and leading to complex elastic and plastic accommodation. The size of a transforming plate in real multiphase steels is typically between 0.1 and 2 μm, a range of size in which plastic strain gradient effects are expected to play a major role. The single parameter version of the Fleck–Hutchinson strain gradient plasticity theory is used to describe the plasticity in the austenite, ferrite and martensite phases. The higher order boundary conditions imposed on the plastic flow have a large impact on the predicted strengthening. Using realistic values of the intrinsic length parameter setting the scale at which the gradients effects have an influence leads to a noticeable increase of the strengthening on top of the increase due to the transformation of a volume fraction of the retained austenite. The geometrical parameters such as the volume fraction of retained austenite and of the transforming zone also bring significant strengthening. Strain gradient effects also significantly affect the stress state inside the martensite plate during and after transformation with a potential impact on the damage resistance of these steels.

Size effects on void growth in single crystals with distributed voids

The effect of void size on void growth in single crystals with uniformly distributed cylindrical voids is studied numerically using a finite deformation strain gradient crystal plasticity theory with an intrinsic length parameter. A plane strain cell model is analyzed for a single crystal with three in-plane slip systems. It is observed that small voids allow much larger overall stress levels than larger voids for all the stress triaxialities considered. The amount of void growth is found to be suppressed for smaller voids at low stress triaxialities. Significant differences are observed in the distribution of slips and on the shape of the deformed voids for different void sizes. Furthermore, the orientation of the crystalline lattice is found to have a pronounced effect on the results, especially for the smaller void sizes.

Determination of the Material Intrinsic Length Scale of Gradient Plasticity Theory

The enhanced gradient plasticity theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical plasticity and the classical continuum plasticity. The later cannot predict the size effects since it does not posses an intrinsic length scale. To assess the size effects, it is indispensable to incorporate an intrinsic material length parameter l into the constitutive equations. However, the full utility of gradient-type theories hinges on one's ability to determine the constitutive length-scale parameter l that scales the gradient effects. Thus, the definition and magnitude of the intrinsic length scale are keys to the development of the new theory of plasticity that incorporates size effects. The classical continuum plasticity is also unable to predict properly the evolution of the material flow stress since the local deformation gradients at a given material point are not accounted for. The gradient-based flow stress is commonly assumed to rely on a mixed type of dislocations: those that are initially randomly or statistically distributed, which are referred to as statistically stored dislocations (SSDs), and those formed to account for the additional strengthening mechanism associated with the deformation gradients, which are referred to as geometrically necessary dislocations (GNDs). In this work two micromechanical models to assess the coupling between SSDs and GNDs are discussed. One in which the SSDs and GNDs are simply summed (model-I) and one in which, implicitly, their accompanying strength are added (model-II). These two dislocation interaction models, which are based on Taylor's hardening law, are then used to identify the deformation-gradient-related intrinsic length-scale parameter l in terms of measurable microstructural physical parameters. The paper also presents a method for identifying the material intrinsic length parameter l from micro hardness results obtained by conical or pyramidal indenters.

NON-COMMUTATIVE SPACE–TIME OF DOUBLY SPECIAL RELATIVITY THEORIES

Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity. We observe that there are infinitely many DSR constructions of the energy–momentum sector, each of whose can be promoted to the κ-Poincaré quantum (Hopf) algebra. Then we use the co-product of this algebra and the Heisenberg double construction of κ-deformed phase space in order to derive the non-commutative space–time structure and the description of the whole of DSR phase space. Next we show that contrary to the ambiguous structure of the energy momentum sector, the space–time of the DSR theory is unique and related to the theory with non-commutative space–time proposed long ago by Snyder. This theory provides non-commutative version of Minkowski space–time enjoying ordinary Lorentz symmetry. It turns out that when one builds a natural phase space on this space–time, its intrinsic length parameter ℓ becomes observer-independent.

How fast can a crack go?

The assumption of a material specific relation between the energy dissipation at the edge of a crack propagating under small scale yielding conditions leads to upper limits of the crack velocity: the Rayleigh wave velocity for modes I and II, and theS wave velocity for mode III. If a mode II crack can pass the “forbidden” subsonic super-Rayleigh region, then the upper limit would be theP wave velocity. Experiments invariably show lower maximum speeds, typically less than 60% of theS wave velocity, but they also frequently show accelerations to a constant velocity, depending on the acceleration history, and increasing surface roughness, even during the constant velocity phase. Other experiments show that the energy dissipation at the crack edge usually is several times larger at very high than at very low velocities. These results indicate a considerable growth of the process region, so much that the intrinsic length parameter which determines its height at very low crack speeds becomes insignificant, and then the rationale for a material specific relation between energy dissipation and velocity disappears. This disappearance can contribute to the explanation of the experimental results.