Memory Diagrams Research Articles

Modeling for nonlinear acoustic imaging of an isolated crack via standing waves in a 2D solid

This paper is concerned with numerical modeling for nondestructive imaging of defects in solids via standing waves excited by a periodic sinewave signal. The stationary solution, purely sinusoidal in an intact sample, contains higher harmonics when damage is present. These harmonics are generated by contact acoustic nonlinearity and form their own standing waves whose intensity maximum usually indicates the position of damage, in a way similar to resonant vibrometry experiments. The key point of the developed numerical tool that describes those wave phenomena is a model of planar damage (crack, delamination) considered here as an inner contact with rough surfaces and friction. The corresponding boundary conditions are given by the previously developed contact model based on the Method of Memory Diagrams (MMD) capable of automating the account for hysteretic frictional effects. Combination of the MMD for boundary conditions and a finite element formulation for waves in a volume (MMD-FEM model) provides a complete description which represents a numerical code applied here for nonlinear standing waves simulations. We present a number of examples obtained for idealized 2D geometry and reveal conditions in which both position and extent of damage are clearly seen as well as cases where only partial detection is possible.

On Applications of Semi-Analytical Methods of Contact Mechanics

This paper is concerned with possible applications of semi-analytical methods of frictional contact mechanics. The semi-analytical solutions, such as the Method of Memory Diagrams, enables to write the load-displacement relationship for contact of two axisymmetric bodies with friction as an analytical expression with parameters calculated via a numerical procedure. As a result, a complex history-dependent solution is obtained for an arbitrary loading history in a computationally efficient way. This fact allows one to calculate hysteretic responses to extremely complex loading histories such as random vibrations. Another case when complex loading histories appear is a harmonic excitation of a dynamic contact system in which inertia is taken into account. The both examples are considered here. The random excitation case can be used as a basis for modeling for wear in frictional contacts while the second one may be extended to describe coupled dynamic contact systems, stick-slip phenomena, or friction-induced instabilities.

Friction-induced energy losses in mechanical contacts subject to random vibrations

In this paper, we apply the previously developed Method of Memory Diagrams (MMD) to the description of an axisymmetric mechanical contact with friction subject to random vibrations. The MMD belongs to a family of semi-analytical methods of contact mechanics originating from the classical Cattaneo–Mindlin solution; it allows one to efficiently compute mechanical and energetic responses to complex excitation signals such as random or acoustic ones. For an axisymmetric contact driven by random normal and tangential displacements having fractal statistical properties, we calculate the friction-induced mechanical energy loss averaged over a large number of realizations. In the considered problem, this energy depends on a very restrained number of parameters: on the rms of random displacements, on the fractal dimension, and on the upper cut-off frequency of the fractal spectrum. In addition, a radial distribution of the dissipated energy has been obtained that has a direct relation to wear in the contact system. For small displacement amplitudes, wear should be expected in an annulus inside of a mean contact circle whereas for large displacements it will start at the contact center.

Theoretical calculation of the instantaneous friction-induced energy losses in arbitrarily excited axisymmetric mechanical contact systems

In this paper, we present the theoretical formula for calculating a friction-induced energy loss in a mechanical contact system consisting of two convex axisymmetric bodies subject to arbitrarily varying oblique loading. Non-flat contact geometry engenders a particular regime, partial slip, in which weakly compressed areas of the contact zone slip, while strongly compressed ones stick. Mechanical energy is transformed into heat only in the slip area in which two field characteristics should be calculated: the local infinitesimal slip distance distribution and shear stress. Their product equals the spatial distribution of the energy loss or heat source density, while the integration over the slip zone produces the global energy loss. The energetic characteristics are obtained in a general situation. In the proposed method, energy losses can be calculated at any time during a loading protocol, in the general case of an arbitrary load expressed in terms of normal and tangential displacements. This is a step forward in comparison to more traditional approaches in contact mechanics allowing to calculate the energy loss per period in the case where the contact is excited by a single periodic input. In this work, the loading is limited to two dimensions: one normal and one tangential. Therefore, the tangential loading is colinear with the tangential displacement. All required contact characteristics, including displacement and stress distributions, are provided by the semi-analytical Method of Memory Diagrams (MMD) that has initially been developed to obtain memory-dependent solutions to the problem of frictional contact between axisymmetric profiles and has later been generalized for rough surfaces. Correspondingly, the proposed solution is exact for axisymmetric bodies and is approximately valid in the case of rough surfaces for which the energy loss per unit nominal area is calculated. The obtained theoretical results for energy dissipation are compared to analytical and numerical calculations for periodic and non-periodic loading situations, illustrating the potential of this method in realistic contact settings.

Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction – Part I: Theoretical background

Our study aims at the creation of a numerical toolbox that describes wave propagation in samples containing internal contacts (e.g. cracks, delaminations, debondings, imperfect intergranular joints) of known geometry with postulated contact interaction laws including friction. The code consists of two entities: the contact model and the solid mechanics module. Part I of the paper concerns the modeling of internal contacts (called cracks for brevity), while part II is related to the integration of the developed contact model into a solid mechanics module that allows the description of wave propagation processes. The contact model is used to produce normal and tangential load-displacement relationships, which in turn are used by the solid mechanics module as boundary conditions at the internal contacts. Due to friction, the tangential reaction curve is hysteretic and memory-dependent. In addition, it depends on the normal reaction curve. An essential feature of the proposed contact model is that it takes into account the roughness of the contact faces. On one hand, accounting for roughness makes the contact model more complicated since it gives rise to a partial slip regime when some parts on the contact area experience slip and some do not. On the other hand, as we will show, the concept of contact surfaces covered by asperities receding under load makes it possible to formulate a consistent contact model that provides nonlinear load-displacement relationships for any value of the drive displacements and their histories. This is a strong advantage, since this way, the displacement-driven model allows for a simple explicit procedure of data exchange with the solid mechanics module, while more traditional flat-surface contacts driven by loads generate a complex iterative procedure. More specifically, the proposed contact model is based on the previously developed method of memory diagrams that allows one to automatically obtain memory-dependent solutions to frictional contact problems in the particular case of partial slip. Here we extend the solution onto cases of total sliding and contact loss which is possible while using the displacement-driven formulation. The method requires the knowledge of the normal contact response obtained in our case as a result of statistical consideration of roughness of contact faces.

Two dimensional modeling of elastic wave propagation in solids containing cracks with rough surfaces and friction – Part II: Numerical implementation

Our study aims at the creation of a numerical toolbox that describes wave propagation in samples containing internal contacts (e.g. cracks, delaminations, debondings, imperfect intergranular joints) of known geometry with postulated contact interaction laws including friction. The code consists of two entities: the contact model and the solid mechanics module. Part I of the paper concerns an in-depth description of a constitutive model for realistic contacts or cracks that takes into account the roughness of the contact faces and the associated effects of friction and hysteresis. In the crack model, three different contact states can be recognized: contact loss, total sliding and partial slip. Normal (clapping) interactions between the crack faces are implemented using a quadratic stress-displacement relation, whereas tangential (friction) interactions were introduced using the Coulomb friction law for the total sliding case, and the Method of Memory Diagrams (MMD) in case of partial slip. In the present part of the paper, we integrate the developed crack model into finite element software in order to simulate elastic wave propagation in a solid material containing internal contacts or cracks. We therefore implemented the comprehensive crack model in MATLAB® and introduced it in the Structural Mechanics Module of COMSOL Multiphysics®. The potential of the approach for ultrasound based inspection of solids with cracks showing acoustic nonlinearity is demonstrated by means of an example of shear wave propagation in an aluminum sample containing a single crack with rough surfaces and friction.

Solution to the frictional contact problem via the method of memory diagrams for general 3D loading histories

This paper is concerned with the method of memory diagrams developed for solving a problem of frictional elastic contact. Our goal is to establish a link between contact force and displacement for a general plane contact between two arbitrarily-shaped bodies; rolling and torsion are not considered. Description of mechanical interactions between two solids in contact in the presence of friction is a nontrivial task since the desired force-displacement relationships have hysteretic (memory-dependent) character. Arbitrarily changing applied force (or displacement) creates a cumbersome shear stress distribution in the contact zone that has to be adequately parameterized and accounted for. In that regard, it is suggested to consider, instead of complex shear stress distributions, a simpler functional form called memory diagram that contains the same memory information. We have established two integral relationships that link the force and displacement vectors with that internal functional dependency. The integral relationships are supplemented with two other evolution rules for memory diagrams that eventually follow from the Coulomb friction law. The memory diagram is updated with the help of these rules following a given force history. Then the calculated memory diagram is used to update the history of displacement i.e. to produce the desired force-displacement relationship.

Method of memory diagrams for mechanical frictional contacts subject to arbitrary 2D loading

In this paper, we present a solution to the contact problem for two elastic bodies with friction loaded by an arbitrarily varying force in 2D. The general statement of the problem concerns the following two important challenges. On one hand, the consideration of an arbitrary contact geometry implies that a multitude of individual arbitrarily-shaped contact zones appears, which can split and merge as a result of changes in the normal loading. On the other hand, the existence of friction invokes the presence of a tangential force and, consequently of a complicated traction distribution inside each of the contact zones. The complexity of these traction distributions results from the fact that, generally, each contact spot contains stick and slip zones with various boundary conditions defined by the previous tangential contact interaction forces. The presence of two classes of boundary conditions makes the problem memory-dependent and hysteretic. In spite of this, by using both conventional and original methods of contact mechanics, it is possible to drastically simplify the problem. First of all, the geometric difficulty residing in the description of arbitrary multiple contacts can be successfully dealt with by applying the reduced friction principle that expresses the tangential solution through the normal loading curve, for both normal and tangential forces kept constant. The principle is valid for a broad range of geometries and makes it possible to replace an arbitrary contact geometry by an equivalent system of two axisymmetric bodies. Secondly, the hysteretic character of the equivalent axisymmetric problem can be addressed through the method of memory diagrams which replaces the complex traction distribution in the contact zone by a simple internal functional dependency comprising the same memory information. Following any change of loading, the internal “memory function” evolves in accordance to a definite procedure that is based on the force balance equation written for small increments of the normal and tangential forces. In this paper, we formulate the method, prove its validity, and illustrate its numerical implementation with some simulated examples of hysteretic load–displacement curves. These obtained results might be of use for applications related to solids with cracks and contacts showing acoustical nonlinearity, and for any modeling requiring physics-based boundary conditions to describe and predict the effect of mechanical contacts with friction.

Shift and torsion contact problems for arbitrary axisymmetric normal stress distributions

In this paper, we analyze the contact interaction of axisymmetric particles subject to a subsequent application of a constant normal load and a tangential or rotational force. A rigorous solution to the frictional contact problem is given by the known Jäger theorem that presents a relationship between shear and normal stress distributions provided the latter is an exact solution to a normal contact problem. However, in the case of strong loading, when the normal displacement reaches a value of 5–10% of the spheres’ diameter, exact solutions for the normal problem are absent; some model concepts exist instead. For instance, the rod model describes strong normal loading of spheres as a sort of combination of the Hertz problem (weak loading of spheres) with a compression of a pair of confined cylinder of the same radius as the Hertz contact spot. Here we propose a method that is based on considerations similar to the Jäger theorem but is appropriate for any model (or empirical) normal stress distribution. The resulting integral representations describe both shearing and torsion of prestressed particles with axisymmetric profiles. Rolling of particles, as well as plasticity and adhesion of the particles’ material, are not considered. We also analyze the asymptotic behavior of the integral representations for weak and strong strains. The obtained general solutions allow us to use the method of memory diagrams in order to calculate the reaction of the system on arbitrarily varying tangential or rotational actions.

Hertz–Mindlin problem for arbitrary oblique 2D loading: General solution by memory diagrams

In this paper we present a new general solution to the fundamental problem of frictional contact of two elastic spheres, also known as the Hertz–Mindlin (HM) problem. The description of spheres in contact is a central topic in contact mechanics. It became a foundation of many applications, such as the friction of rough surfaces and the mechanics of granular materials and rocks, etc. Moreover, it serves as a theoretical background in modern nonlinear acoustics and elasticity, e.g. seismology and nondestructive testing. However, despite many efforts, a rigorous analytical solution for the general case when arbitrary normal and tangential forces are present is still missing, mainly because the traction distribution within the contact zone is convoluted and hardly tractable, even under relatively simple external action. Here, accepting a number of traditional limitations such as 2D loading and the existence of a functional dependence between normal and tangential forces, we propose an original way of replacing the complex traction distributions by simple graphical counterparts called memory diagrams, and we formulate a procedure that enables initiating and maintaining these memory diagrams following an arbitrary loading history. For each memory diagram, the solution can be expressed by closed-form analytical formulas that we have derived using known techniques suggested by Mindlin, Deresiewicz, and others. So far, to the best of our knowledge, arbitrary loading histories have been treated only numerically. Implementation of the proposed memory diagram method provides an easy-to-use computer-assisted analytical solution with a high level of generality. Examples and results illustrate the variety and richness of effects that can be encountered in a geometrically simple system of two contacting spheres.

CS1 assessment using memory diagrams

Understanding the execution of an object-oriented program can be a challenge for a student starting a CS1 course. We believe that a type of diagram that we call a memory diagram can aid the student in understanding object-oriented programming and can assist the instructor in assessing the student's understanding.Memory diagrams focus on how, in an abstract sense, the memory of the machine changes as the program executes. Though memory diagrams are a simple idea, by careful use of shape and placement, a number of key points about the meaning of a program fragment can be conveyed visually. We have found a correlation between a student's ability to construct these diagrams and that student's comprehension of object-oriented concepts. We feel that this correlation indicates that memory diagrams can be used as an assessment technique that, in turn, can be used to improve student learning.

Simple method for estimating the memory diagram in single mode semiconductor lasers

The authors propose a simple method, based on the probability distribution functions of the switch-on time, to predict the appearance of pattern effects at the output of single-mode semiconductor lasers. The method allows one to evaluate the overlap between the switch-on time probability distribution functions under repetitive gain-switching and periodic '...1111...' sequences of input bits in the return to zero scheme for any operating point of the laser. If the overlap is large enough, no-pattern effects will appear at the output when the laser is modulated with a pseudorandom sequences of input bits. If the contrary happens, pattern effects will be observed at the output. Nonmemory bands in the memory region are also obtained.

Memory diagram of single-mode semiconductor lasers

A memory diagram, i.e., regions where pattern effects appear at the output of the laser, in the bias-current-versus-modulation-period plane is obtained, both by numerical simulation of the rate equations and by using an analytical approach. A simple method, based on the superposition of the turn-on time probability distribution for the periodic sequence . . .1111. . . of input bits and the turn-on time probability distribution obtained for repetitive gain switching, is used to describe the response of the laser to a pseudorandom word modulation of the injection current. >