Elena Pasternak

Generalised homogenisation procedures for granular materials

Engineering materials are generally non-homogeneous, yet standard continuum descriptions of such materials are admissible, provided that the size of the non-homogeneities is much smaller than the characteristic length of the deformation pattern. If this is not the case, either the individual non-homogeneities have to be described explicitly or the range of applicability of the continuum concept is extended by including additional variables or degrees of freedom. In the paper the discrete nature of granular materials is modelled in the simplest possible way by means of finite-difference equations. The difference equations may be homogenised in two ways: the simplest approach is to replace the finite differences by the corresponding Taylor expansions. This leads to a Cosserat continuum theory. A more sophisticated strategy is to homogenise the equations by means of a discrete Fourier transformation. The result is a Kunin-type non-local theory. In the following these theories are analysed by considering a model consisting of independent periodic 1D chains of solid spheres connected by shear translational and rotational springs. It is found that the Cosserat theory offers a healthy balance between accuracy and simplicity. Kunin’s integral homogenisation theory leads to a non-local Cosserat continuum description that yields an exact solution, but does not offer any real simplification in the solution of the model equations as compared to the original discrete system. The rotational degree of freedom affects the phenomenology of wave propagation considerably. When the rotation is suppressed, only one type of wave, viz. a shear wave, exists. When the restriction on particle rotation is relaxed, the velocity of this wave decreases and another, high velocity wave arises.

Topological interlocking of platonic solids: A way to new materials and structures

The structural integrity of natural and engineered materials relies on chemical or mechanical bonding between the building blocks of which they consist. Materials whose building blocks are not joined, but rather interlocked topologically, possess remarkable mechanical and functional properties. We show that identical elements, in the shape of the five platonic solids, can be arranged into layer-like structures in which they are interlocked topologically. It is shown that truncated icosahedra (buckyballs) can also be arranged in a layer with topological interlocking. The geometrical possibility of such assemblies opens up interesting avenues in the design of structures and materials.

Toughening by fragmentation – How topology helps

In this brief overview, we present recent work on the novel concept of topological interlocking as a means of designing new materials and structures. Starting from a special self-supporting arrangement of tetrahedron-shaped elements that was discovered first, we proceed with the introduction of other shapes exhibiting topological interlocking. Unusual mechanical properties of assemblies of tetrahedrons that were studied both experimentally and theoretically are discussed. Possible applications can range from large scale mortar free construction in civil engineering to novel advanced materials based on microscale interlocking elements.

A new concept in design of materials and structures: assemblies of interlocked tetrahedron-shaped elements

Youth Centre of Scienceand Technology, Moscow, Russia(Received January 3, 2001)(Accepted in revised form February 15, 2001)Keywords: Assembly of interlocked tetrahedrons; Structural behaviour; Mechanical properties;Toughness; CompositesIntroductionIn this communication we propose a new material architecture based on regular assemblies of identicalinterlocked elements. The packing arrangement is such that each individual element is prevented frombreaking out by its immediate neighbours. It is claimed that the existence of such interlocked assembliesopens up a new direction for creating special strong and flexible composite materials with high impactresistance.In conventional composites, the reinforcement particles are held together by a matrix or a binderphase. Consequently, the overall strength is restricted by the strength of the matrix no matter how strongthe particles are.In principle, the strength of a composite could be increased if the particles were directly connectedto each other. In this case, the matrix or the binder phase would only play the role of a ‘buffer’,inhibiting the propagation of dislocations or cracks from one particle to another. The matrix could alsoprovide additional properties, such as thermal or acoustic insulation.Interlocking can, of course, be achieved by equipping reinforcement particles or building blocks withspecial locking ‘keys’ (this is particularly the case in the building industry, cf., e.g., self-locking bricks).Obviously, the ‘keys’ are stress concentrators that impose severe limitations on the overall strength ofthe structure, not to mention the additional technological difficulties associated with the manufacturingof blocks with ‘keys’. We propose a different approach to the production of interlocked structures. Itis based on the use of the topological possibility of establishing self-locking in assemblies of simpleconvex shaped elements free of stress concentrators. In what follows, a particular example of aninterlocked layer structure consisting of identical tetrahedron-shaped elements packed in a special wayis discussed. The first results of mechanical tests done on an experimental specimen assembledaccording to this recipe are then presented. Finally, a brief outlook on new perspectives resulting fromthis novel approach to materials design is given.

A new principle in design of composite materials: reinforcement by interlocked elements

We propose a new materials design concept based on the use of regular assemblies of topologically interlocked elements. A particular implementation of this concept, viz. a layer of tetrahedron-shaped elements, was studied in some detail. The packing arrangement in the layer is such that each individual element is held in place by its immediate neighbours. This structure can provide a load-bearing skeleton of a composite material. A second phase, serving as a matrix or binder, can be selected to provide special structural or functional properties such as thermal or sound insulation, fluid transport, controlled electrical conductivity, etc. It is envisaged that strong and flexible composite materials with high impact resistance can be created on this basis. A model specimen assembled according to this topological principle was tested with respect to its stiffness and load bearing capacity. First experimental and theoretical results show that a layer consisting of many interlocked elements has a much larger mechanical compliance than its monolithic counterpart, and can withstand considerable deformations. Other possible shapes of three-dimensional elements interlocked into a monolayer and the principles of their generation are discussed. The design principle proposed opens up new avenues for creating multifunctional composite materials.

Topological interlocking of protective tiles for the space shuttle

We propose a new concept of design of heat- and impact-resistant tile covering for space shuttles based upon topological interlocking of tiles. The key features of this type of tiling are as follows: firstly, the tiles are kept in place by virtue of the geometry of their contacting surfaces alone, so that no binding agent needs to be used; secondly, failure of an individual element does not compromise the structural integrity of the tile assembly and, moreover, a failed tile will be kept in place by its neighbours, thus maintaining its thermal protection function; thirdly, topological interlocking removes the need for connectors or other stress concentrators deleterious in extreme conditions of space flight and re-entry. Topological interlocking is scale and material independent, which permits combining tiles made from different materials in any proportion.

Mechanical effect of rotating non-spherical particles on failure in compression

When non-spherical particles (grains, blocks) in a rock or geomaterial under compression are sufficiently detached to enable rolling under shear stress the moment equilibrium dictates that further increase in displacement requires reduced shear stress. Thus the rolling non-spherical particle produces an effect of apparent negative stiffness whose value is proportional to the magnitude of the compressive stress. We model geomaterials with rolling particles at a macroscale as a matrix containing inclusions with negative shear modulus. We consider two extreme types of inclusions: spherical inclusions and shear cracks with negative stiffness shear springs inside. We show that there exists a critical concentration of inclusions at which the effective shear modulus abruptly becomes negative and the geomaterial loses stability. Furthermore, there exists a special value of negative shear modulus/stiffness of inclusions (and hence the magnitude of the compressive stress) at which the critical concentration is zero, such that the first rolling particle induces the global instability. This stress is of the order of the shear modulus the geomaterial exhibits at the advanced stage of fracturing when the particles are sufficiently detached to enable the rolling.

Sustained acoustic emissions following tensile crack propagation in a crystalline rock

We show that simple breakage of a crystalline rock (gabbro) in tension begets further breakage of rock in the area around the first crack that is self-sustaining and spontaneous and that is detected via sustained acoustic emissions (AE). The result is a sequence of AE events that is statistically similar to aftershocks from earthquakes, that scales with the size of the main crack, and that we were able to observe for days following the initial breakage in laboratory-scale experiments. A new model for aftershock generation that is based on residual strain relaxation is shown to be consistent with the observed hyperbolic decay of the event rate with time and with the manner in which the decay law scales with the size of the main rupture.

Physical Modelling of Stress-dependent Permeability in Fractured Rocks

This paper presents the results of laboratory experiments conducted to study the impact of stress on fracture deformation and permeability of fractured rocks. The physical models (laboratory specimens) consisted of steel cubes simulating a rock mass containing three sets of orthogonal fractures. The laboratory specimens were subjected to two or three cycles of hydrostatic loading/unloading followed by the measurement of displacement and permeability. The results show a considerable difference in both deformation and permeability trends between the first loading and the subsequent loading/unloading cycles. However, the micrographs of the contact surfaces taken before and after the tests show that the standard deviation of asperity heights of measured surfaces are affected very little by the loadings. This implies that both deformation and permeability are rather controlled by the highest surface asperities which cannot be picked up by the conventional roughness characterization technique. We found that the dependence of flow rate on mechanical aperture follows a power law with the exponent n smaller or larger than three depending upon the loading stage. Initially, when the maximum height of the asperities is high, the exponent is slightly smaller than 3. The first loading, however, flattens these asperities. After that, the third loading and unloading yielded the exponent of around 4. Due to the roughness of contact surfaces, the flow route is no longer straight but tortuous resulting in flow length increase.

Path independent integrals for cosserat continua and application to crack problems

We generalise Noethers theorem to include Cosserat Continua and derive corresponding conservation laws and path independent boundary integrals. From translational invariance of the deformation energy we optain the Cosserat generalisation of the J-integral. In a Cosserat Continuum, additional integrals follow from rotational invariance.