Mechanism of memory effect of paste which dominates desiccation crack patterns

Abstract

When a densely packed colloidal suspension, called a paste, behaves as plastic fluid, it can remember the direction of its motion it has experienced, such as vibrational motion and flow. These memories kept in paste can be visualized as the morphology of crack patterns that appear when the paste is dried. For example, when a paste remembers the direction of vibrational motion, all primary desiccation cracks propagate in the direction perpendicular to the direction of the vibrational motion that the paste has experienced. On the other hand, when a paste remembers the direction of flow motion, all primary cracks propagate along the flow direction. To find out the mechanism of memory effect of vibration, we perform experiments to rewrite memory in paste by applying additional vibration to the paste along a different direction before the paste is dried. By investigating the process of rewriting memory in paste, we find the competitive phenomena between quasi-linear effect and nonlinear effect, which were studied in each theoretical model based on residual tension theories. That is, at the initial stage of the memory-imprinting process of the vibrational motion, the mechanism predicted by the quasi-linear analysis based on residual tension theory holds, but, as the paste is vibrated repeatedly, the mechanism shown by the nonlinear analysis gradually come to play a dominant role in the memory effect.

This article is part of the theme issue ‘Statistical physics of fracture and earthquakes’.

1. Introduction

We usually face with fracture phenomena in our daily life, but it is very difficult to control fracturing processes. In many cases, such as earthquakes and metal fatigue, it is difficult to predict when a fracture will happen. And, since fracturing process is an irreversible phenomenon, once cracks start to appear, crack propagation at later stage strongly depends on the history of earlier-stage crack formation. Therefore, fracture is still one of the difficult topic of nonlinear non-equilibrium physics in science and its control is very important in the field of technology.

When we drop a glass onto the floor, it breaks at the floor and each fragment takes different shapes with different sizes. However, even though such crack formations due to the rapid impacts are usually random and stochastic phenomena, it is reported that size distributions of fragments obey statistical laws, depending on the shape of the original object [1]. There is also a famous experiment proposed by Yuse and Sano in which they control the morphology of cracks in quasi-static fracture [2]. In the experiment, a narrow glass plate is heated and then quenched, and thus cracks are formed. By increasing the cooling gradient carefully, the propagation of straight cracks becomes unstable, and the transition from straight to oscillatory cracks is obtained [2,3]. That is, the morphology of crack patterns can be controlled by cracking conditions.

Now let us consider about morphology of desiccation crack patterns [4]. When a densely packed colloidal suspension, called a paste, is poured into a container and is dried at room temperature, the paste shrinks as the water evaporates, and if the paste is stick to the bottom of the container, desiccation patterns emerge so as to release the stress induced by the shrinkage, as are often seen on dried lake. Usually the morphology of desiccation cracks become isotropic and cellular and the final spacing between cracks is proportional to the thickness of the paste as long as the paste is not so thick [5]. However, anisotropic desiccation cracks can also be observed in nature and in the laboratory when the paste is dried under a drying gradient.

For example, when the thickness of the paste is large, a crack front propagates from the surface towards the inside along the drying direction, and, as for the case of starch paste, columnar joints of hexagonal structure emerge, which look similar to the columnar joints of cooled lava [6–10]. In each case, materials shrink under drying or cooling, and the directions of the drying or cooling gradient produce an anisotropic crack structure along the gradient direction, such as columnar joints. The diameters of these columns are determined by the value of drying or cooling gradient. Therefore, the directions of these columns indicate in which direction the paste was dried and the lava was cooled. Even if the thickness of the paste is smaller, there appears a drying gradient at the thinner perimeter of the droplet of the paste and, in such a case, stripe cracks are formed from the perimeter towards the central part along the drying gradient [4].

In this article, we first review the case that, even without any drying gradient, we can control and get regular anisotropic desiccation crack patterns by using memory effect of paste [11–15]. Since a paste behaves as a plastic fluid, it remembers the direction of its motion, such as vibration or flow, and the memory of such motion determines the preferential direction for cracks to propagate. When a paste remembers its vibrational motion, all primary desiccation cracks propagate in the direction perpendicular to the direction of the vibration that the paste has experienced, while all primary cracks propagate along the flow direction when a paste remembers its flow motion. Two types of residual tension theories are presented to explain the memory effect of paste, one is based on quasi-linear effects and the other on nonlinear effects [16–18]. Thus, after the review sections, we report our recent experiments, called memory-rewriting experiments, to validate which mechanism is dominant in memory formation. Through memory-rewriting experiments, we will see the competition between the quasi-linear effect and the nonlinear effect in memory formation.

The memory effect of paste is attracting attention as a new phenomenon in fields of rheology and crack control. Therefore, much research is performed to control the morphology of desiccation crack patterns by using the memory effect of electrical fields [19,20] and by magnetic fields [21–23]. It is expected that the memory effect of paste has great technological applications, since it now becomes possible to control crack formation by controlling memories in paste.

2. Memory of paste and its effect on crack patterns

In this section, we review the memory effect of paste and show experimental results on how the morphology of desiccation crack patterns is controlled by the memory of its motion before drying.

(a). Memory of vibration and resultant striped crack pattern

Figure 1a shows a typical isotropic and cellular crack pattern which we obtain when we dry a water-rich mixture of powder and water. Here the median diameter of powder particles in a mixture is a few micrometres, the thickness of the mixture is 10 mm, and the final spacing between cracks is proportional to the thickness of the mixture [4,5]. The morphology of desiccation crack patterns is usually a random, isotropic and cellular structure.

Figure 1.

Desiccation crack patterns, where sample (a) is a water-rich mixture of magnesium carbonate hydroxide powder and samples (b) and (c) are water-poor pastes of calcium carbonate. The diameter of each circular acrylic container is 500 mm and the thickness of each paste is about 10 mm. The arrows in (b) and (c) indicate the direction of the vibration applied to the paste before drying. (a) Random, isotropic and cellular crack pattern. (b) Radial crack pattern. (c) Striped crack pattern. In (b) and (c), all primary cracks propagate in the direction perpendicular to the vibrational motion the paste had experienced before drying (memory of vibration) [11,12].

On the other hand, regular desiccation crack patterns, such as radial and striped structures, can be produced by using the memory effect of a water-poor mixture of powder particles and water, called a paste [11,12]. For example, a water-poor paste of calcium carbonate with solid volume fraction of 44% is poured into a circular acrylic container. Since the paste has a plasticity, we have to vibrate the paste horizontally to make it flat; then we dry the paste to get a desiccation crack pattern. In figure 1b,c, we show results of desiccation crack patterns, where we had vibrated the paste in an angular direction in figure 1b and in one direction in figure 1c, before we dried each paste. We see that the paste remembers the direction of the vibration it had experienced before drying, and all primary desiccation cracks propagate in the direction perpendicular to the direction of its vibration. Therefore, we would like to call this effect the ‘memory effect of paste’.

(b). Plasticity of paste and its role on memory effect

Why can the paste remember the direction of its vibrational motion and why does the direction of crack propagation depend on such memory? We consider that plasticity of water-poor paste plays an important role in remembering its motion. A water-rich mixture cannot remember any of its motion, while a water-poor paste can remember its vibrational motion and the morphology of desiccation crack patterns depends on what kind of memory the paste has. Therefore, the ratio of mixing powder to water is an important parameter to characterize the rheology of the mixture.

The fluidity of a Newtonian fluid like a water is expressed as , where and σ[Pa] represent shear rate and shear stress, respectively, and the coefficient η[Pa · s] is called viscosity. When the value of the solid volume fraction is low, such a water-rich mixture behaves as a Newtonian fluid. As we increase the value of the solid volume fraction ϕ in the mixture, the value of viscosity increases, and at a certain threshold called the liquid-limit, such a water-poor mixture (paste) starts to show plasticity with non-zero yield stress. To fluidize the paste, we have to apply a shear stress larger than the value of the yield stress of the paste. The value of the yield stress σ_Y[Pa] increases as the value of the solid volume fraction ϕ increases, and at a certain threshold called the plastic-limit, the value of the yield stress diverges and the paste enters semi-solid state.

The simplest model to represent a plastic fluid is called a Bingham fluid, expressed by

2.1

where η_p is a positive constant and the rheological behaviour of a Bingham fluid is described in figure 2a. To measure the plasticity of a mixture of colloidal particles of calcium carbonate and water, we perform the rheological measurement of the mixture using a rheometer (Physica MCR301, Anton Paar, Graz, Austria). The mixture is sandwitched between two parallel plates and the value of the shear rate of the mixture is measured as a function of the applied shear stress to the mixture. Figure 2b is a result of a water-poor paste of calcium carbonate with its solid volume fraction of 40%, which is obtained by increasing the value of a shear stress. We see that the value of the yield stress of the paste is given by σ_Y = 1.0 Pa [4].

Figure 2.

Plasticity of paste is expressed as non-zero yield stress. In both figures, a shear rate is shown as a function of a shear stress applied to paste. (a) Bingham model of plastic fluid. (b) Experimental result of a water-poor paste of calcium carbonate with its solid volume fraction of 40%. The value of the yield stress of the paste is given by σ_Y = 1.0 Pa [4].

By performing rheological measurements, we obtain the value of the yield stress σ_Y as a function of the solid volume fraction ϕ of the mixture, as is shown in figure 3a. When the value of the solid volume fraction is smaller than the value of liquid-limit of 25% the water-rich mixture can be regarded as a Newtonian viscous fluid. As the value of the solid volume fraction increases, the value of the yield stress of the water-poor paste, called a paste, increases until it diverges at plastic-limit of 54%. When the value of solid volume fraction exceeds the value of plastic-limit, the paste is regarded as semi-solid with many cracks and we cannot mix powder with water homogeneously any more. Therefore, a mixture of powder and water can be regarded as a plastic fluid when the value of the solid volume fraction is between the liquid-limit and the plastic-limit.

Figure 3.

Rheology and memory effect of calcium carbonate paste. In both figures, the dotted and dash-dotted guidelines correspond to the liquid-limit and plastic-limit of the mixture, respectively. (a) Yield stress of a mixture is shown as a function of the solid volume fraction. (b) Morphological phase diagram of desiccation crack patterns as a function of solid volume fraction and the strength of the vibration applied to the container. The solid curve represents the situation when the value of the shear stress applied to the paste equals to that of the yield stress of the paste. Open circles denote isotropic and cellular crack patterns with no memory of its motion. Filled squares denote striped crack patterns which were produced by the memory effect of vibration. Triangles represent combination of isotropic cellular pattern and striped pattern. In region B just above the solid yield stress line, the paste can remember the direction of its vibrational motion, and all primary desiccation cracks propagate in the direction perpendicular to the direction of the vibration [11,12].

(c). Experimental results on the condition of memory effect

To find a condition for a paste to show the memory effect, we performed experiments in which we systematically changed the value of the solid volume fraction of the mixture ϕ(%) and the strength of the vibration applied to the container in which the paste was set, expressed as a maximum acceleration 4π²rf² (m s⁻²). Here r is the amplitude of the vibration and set to r = 15 mm, and we vary the value of the frequency f(Hz) of the vibration. The mass of calcium carbonate contained in each mixture is fixed at 360 g, and we vary the amount of water to add to control the solid volume fraction of the mixture. We pour each mixture into square acrylic containers with 200 mm sides, which was set on a horizontal shaker. Then we shake the container horizontally for 60 s, and dry each mixture at room temperature of 25°C and low humidity of 30%.

Figure 3b shows a morphological phase diagram of desiccation crack patterns, which represents the condition for the memory effect of vibration. A paste can be regarded as a plastic fluid when the value of the solid volume fraction is between the liquid-limit and the plastic-limit. In region A below the solid yield stress line, the paste does not move at all, because the shear stress applied to the paste is smaller than the yield stress of the paste. On the other hand, in region B just above the solid yield stress line, the shear stress applied to the paste exceeds the yield stress of the paste, the paste can be deformed under vibration, it can remember the direction of its vibrational motion, and all primary desiccation cracks propagate in the direction perpendicular to the direction of the vibration (memory effect of vibration) [11,12,15]. In region C, we observe a flow motion of paste during the shaking protocol, and only obtain random and isotropic cellular crack patterns. We find that, once a paste of calcium carbonate flows, it cannot remember any flow motion.

3. Residual tension theories for memory effect of vibration

In this section, we review theoretical approaches based on residual tension theories which were presented to explain the memory effect of vibration. Here we have a question: how can the paste remember its motion of vibration and how is the morphology of the desiccation crack patterns determined by the memory? To answer this question, we have to explain why the criteria for the memory effect can be described by the yield stress line as in figure 3b and why all primary desiccation cracks propagate in the direction perpendicular to the direction of the vibration that the paste had experienced. For this reason, Prof. Michio Otsuki and Prof. Ooshida Takeshi presented theoretical models based on residual tension theories and insisted that residual tension induced by plastic shear deformation under vibration plays an important role in the memory effect [16–18].

The schematic illustration of the deformation of a paste under vibration is drawn in figure 4a. It is assumed that a horizontal stretching and compression emerge at the upper layer of the paste, while the lower layer suffers a shear deformation. In the following subsections, first a quasi-linear analysis of the elastoplastic model is performed to explain the mechanism that residual tension is produced by the stretching deformation due to non-uniform shear at the upper layer of the paste. Then, a nonlinear analysis is performed to explain that residual tension can be produced even under uniform shear deformation in the lower layer of the paste [4,16–18,24]. We will validate the efficiency of these residual tension theories by rewriting-memory experiments in the next section.

Figure 4.

Shear deformation of paste under horizontal vibration. (a) Schematic illustration of the deformation of a paste under vibration. At the upper layer of the paste horizontal stretching and compression emerge, while the lower layer suffers a shear deformation. (b) Visualization by dyeing experiment. The centre of the paste of calcium carbonate is dyed vertically straight by black carbon powder. Then the paste experiences sudden ‘Go and Stop’ motion horizontally in one direction from left to right. We see that the lower layer of the paste suffers a shear deformation [4].

(a). Quasi-linear analysis

As is shown in figure 4b the coordinate system is fixed on the container, with the x-axis taken horizontally in the direction of the vibration and the z-axis vertically upwards. The paste is assumed to be a thin layer with the thickness of the paste h much more smaller than the horizontal size of the paste 2L, and the shallow water approximation h≪L becomes valid, because typically 2L = 200 mm and h = 9 mm. Therefore, we can regard that the paste moves only in the horizontal direction. Denoting the inertial force due to the vibration by F_x = F_x(t), the equation of motion for the displacement X = X(x, z, t) is given by

3.1

where σ = (σ_xx(x, z, t), σ_xz(x, z, t)) denotes the stress and the constant D is related to viscosity and h. The normal stress and shear stress are expressed as

3.2

and

3.3

in terms of the Láme elastic coefficients of the paste, λ and μ. Here, c = c(t) is the reference strain, which represents the desiccation shrinkage and is treated as a given function of time. Equation (3.3) gives an elastoplastic linear decomposition, and the plastic shear strain β = β(x, z, t) is modelled by Bingham plasticity as [16]

3.4

To perform numerical simulation to obtain desiccation crack patterns, equations (3.1)–(3.4) are extended to include y-directional displacement and the stress components, such as σ_yz, with appropriate boundary conditions, and a model is obtained to simulate the whole process of the memory effect, i.e. a shaking protocol and a drying process. During a shaking protocol, F_x(t) is assumed to be a sinusoidal function, σ_Y takes positive and finite value, and c = 0. And, in a drying process, σ_Y diverges, c(t) increases monotonically as time goes on, and cracks are formed when normal stresses σ_xx or σ_yy exceed the Griffith criteria. Numerical results are shown in figure 5. When the paste is vibrated strongly enough before drying, the first crack propagates in the direction perpendicular to the applied vibration. This numerical result is accordance with the experimental result on the memory effect of vibration [4].

Figure 5.

Desiccation crack patterns obtained by performing numerical simulations based on quasi-linear analysis. The symbol g denotes the direction of gravity. If a shape of a sample is rectangular, a first crack tends to propagate along the shortest path to divide the sample into two fragments. When the paste is vibrated weakly, a first crack propagates along the shortest path, as is expected. On the other hand, when the paste is vibrated strongly, the first crack propagates in the direction perpendicular to the applied vibration, which is not the same as the direction of a shortest path. (The figure presented here is through the courtesy of Michio Otsuki.) [4].

(b). Theoretical consideration on the condition for memory effect

The threshold for realization of the memory effect is determined by the yield stress line. To understand, equations (3.3) and (3.4) are combined together to derive the equation of relaxation as

3.5

(see [4] for the derivation). Here ν(ϵ) is an inverse of the relaxation time and is given as a function of the elastic strain energy ϵ∝σ²_xz as is shown in figure 6. Figure 6 means that ν takes a positive finite value when |σ_xz| > σ_Y and vanishes for |σ_xz| < σ_Y. When the strength of the vibration is weak and a shear stress induced by the vibration cannot exceed the yield stress of paste as in region A of figure 3b, ν is always zero even under vibration, and no memory is imprinted into the paste. If the strength of the vibration is strong enough so that a shear stress exceeds the yield stress as in region B of figure 3b, plastic deformation of the paste emerges under vibration according to equation (3.5). Then, what will happen when we stop the vibration? If the paste behaves as a Newtonian fluid with no plasticity as in the region left of the liquid-limit line of figure 3b, the value of ν is still positive even after we stop vibrating the paste, and equations (3.1)–(3.3) and (3.5) show that the paste relaxes into stress-free state with no memory. On the other hand, if the paste behaves as a plastic fluid, ν vanishes before the stress induced by vibration relaxes to zero, and the plastic shear strain β remains as a memory of vibration. Considering that the plastic shear strain β remains even after the vibration stops and a stick boundary condition holds at the bottom of the paste, the displacement of paste X is frozen and there remains tension of σ_xx > 0 somewhere in the system.

Figure 6.

Plasticity is expressed as a form of the inverse of the relaxation time, ν, and is given as a function of the elastic strain energy ϵ. Here σ_Y is an yield stress of the paste and μ is the Láme elastic coefficient [4,24].

From the above explanation, we understand that the non-uniformity of the displacement X with respect to x is essential for the production of residual tension. If some part of the paste remains stretched as ∂X/∂x > 0 even after the vibration stops, the residual tension remain along the direction of the vibration, and primary cracks tend to propagate in the direction perpendicular to the direction of the vibration. Therefore, quasi-linear analysis can explain why the criteria for memory effect can be described by the yield stress line as in figure 3b and why all primary desiccation cracks propagate in the direction perpendicular to the direction of the vibration that the paste had experienced [4,24].

However, experimental results, such as figure 3b,c, show that, when a paste was vibrated horizontally with a shear stress larger than the yield stress of the paste, all primary cracks in all regions propagate in the direction perpendicular to the direction of the vibration that the paste had experienced. But geometrically it is impossible that all regions of the paste remain stretched after the vibration stops; some part of the paste should be compressed instead. To solve this paradox, nonlinear analysis is performed, as will be explained in the following subsection.

(c). Nonlinear analysis

In this subsection, nonlinear analysis is explained to solve the paradox that, when a paste has a memory of vibration, all primary cracks in all regions propagate in the direction perpendicular to the direction of the vibration. Under the horizontal vibration, the deformation of paste is assumed to be only in the x-direction. It is also assumed that the deformation of paste is uniform in the x-direction, and the constitutive equations go back to those of uniform elastic media in the limit of . Here a simple model is adopted among such sets of constitutive equations that can take into account the effect of finite deformation. Then, the plastic strain can be characterized by tensors, one of which is the plastic shear strain β, and the other is the plastic normal strain, which should appear in the right-hand side of equation (3.2). The effect of plastic normal strain can be characterized by α, where represents a component of natural metric of paste in the direction of horizontal vibration, i.e. in the x-direction, and α = 0 corresponds to the initial state before the horizontal vibration is applied. Then, the relaxation of α can be roughly given by

3.6

Since equation (3.6) contains a term proportional to σ²_xz, this term becomes always positive during a horizontal vibration. Then, once the shear stress exceed the yield stress and the relation ν > 0 holds, the value of α increases monotonically as is schematically shown in figure 7. When we stop the horizontal vibration, α tends to relax to zero, but, since suddenly vanishes as soon as we stop vibration, the value of α remains positive even after the horizontal vibration stops. This is the mechanism why the x-directional tension remains anywhere in paste and all primary cracks in all regions propagate in the direction perpendicular to the direction of the vibration [4,17,18,24].

Figure 7.

Schematic illustration of the nonlinear effect of shear motion under horizontal vibration. The shape of the ellipse represents plastic strain of the natural metrics under horizontal vibration. Owing to the effect of second order of shear stress σ_xz, the natural metric of the paste is elongated along vertical z-direction, and at the same time it shrinks in horizontal x-direction as α > 0 due to the incompressibility of paste. This effect produces a x-directional tension that induces striped cracks along horizontal y-direction [4].

4. Memory-rewriting experiment and its role to validate residual tension theories

In this section, we report our recent experiments in which we rewrite memories in pastes by applying additional vibration to the paste along a different direction before the paste is dried. By investigating the process of rewriting memory in paste, we find that these experiments can be evidence to validate the efficiency of theoretical models based on residual tension theories.

First, figure 8a represents a typical striped crack pattern obtained by memory of vibration. We tried to visualize the vibrational motion of the paste at the shaking protocol in figure 8b,c by checking the deformation of a letter M under vibration. In contrast to the case of flow motion, we could observe only slight deformation and the form of the letter M can still be read clearly. Therefore, we tried to visualize a shear deformation inside the paste under vibration and the result is shown in figure 4b. We can confirm that the lower layer of the paste suffers a shear deformation, as is assumed in figure 4a [4,24].

Figure 8.

Memory effect of vibration. The paste is a water-poor dense mixture of magnesium carbonate hydroxide and water with its solid volume fraction of 12.5%. The sides of each container are 200 mm and the arrow indicates the direction of the applied vibration. (a) Striped cracks in the direction perpendicular to its vibrational motion. (b,c) Visualization of the deformation of paste under vibration by checking the deformation of a letter M written by black carbon powder on the surface of the paste. Here (b) is taken before vibration and (c) under vibration (d) schematic image of the formation of congested structure under vibration [13,14].

In the previous section, it was shown that nonlinear analysis based on the residual tension theory can explain why striped cracks appear everywhere in the system as is shown in figure 1b,c, even though some regions of the paste were not stretched at the final motion of the horizontal vibration. Then, can the nonlinear analysis explain every crack pattern which results from the memory effect of vibration?

In the quasi-linear analysis based on equations (3.1)–(3.4), most of the nonlinear effects, except the yield stress, are neglected, but, by regarding the ratio L/h as finite, the theory takes into account the non-uniformity of the deformation in the x-direction. In contrast, in the nonlinear analysis, equation (3.5) neglects the non-uniformity of the deformation in the x-direction, but the theory takes into account the nonlinear effects. Which effect becomes important in the memory effect? The answer will depend on the situation of the experiment. It will be shown from the following memory-rewriting experiments that both effects play important roles in the memory-imprinting process.

The mass of calcium carbonate in each mixture is fixed at 360 g and 180 g, respectively, so the solid volume fraction of the mixture is 44%. We pour each water-poor mixture (paste) into square acrylic containers with 200 mm each side. First, a paste is primarily vibrated horizontally in one direction at amplitude of r = 20 mm and frequency of f = 52 r.p.m., i.e. at the maximum acceleration of 0.59 m s⁻², for 1 min, a short but sufficient duration to remember the direction of the primary vibration. Then the paste is additionally vibrated horizontally in a different direction, the angle of which is π/4 different from the first one, at amplitude of r = 15 mm and frequency of f = 60 r.p.m., i.e. at the same maximum acceleration of 0.59 m s⁻². We check how the memory of the primary vibration is replaced by the memory of the additional vibration. Here, the upper panels in figure 9 indicate the protocol of rewriting-memory experiments, and the lower panels indicate how we understand the results obtained by rewriting-memory experiments, because arrows in the lower panels denote which direction of vibration each region of each paste remembers, first primary one or next additional one. By changing the duration of the additional vibration, it is shown experimentally in figure 9 that two additional shakes are enough to rewrite new memory on old memory everywhere in the system. This result also indicates verification of the nonlinear analysis.

Figure 9.

Rewriting experiments of memory of vibration. We pour a water-poor paste of calcium carbonate into each square acrylic container with 200 mm each side. Upper panels indicate the protocol of rewriting-memory experiments. First, the paste is primarily vibrated horizontally in one direction for 60 shakes. Next, it is additionally vibrated horizontally but in a different direction, the angle of which is π/4 different from the first one. In a series of experiments, the duration of the additional vibration is regarded as a controlling parameter in each experiment. The durations of the additional vibration are (a) 0 (no more shake), (b) shake, (c) 1 shake, (d) shake and (e) 2 shakes, respectively. Lower panels indicate how we understand results obtained by rewriting-memory experiments. Arrows in lower panels denote which vibration each region of each paste remembers, first, primary one or the next additional one. Regions represented as open rectangles surrounded by bold lines are places where the memory of primary vibration is replaced by the memory of additional vibration. These figures show that two additional shakes is enough to rewrite new memory on old memory everywhere in the system. On the other hand, when the duration of the additional vibration is shorter than two shakes, the memory of the primary vibration is replaced by the memory of the additional vibration only at the rear region where the upper layer of the paste is stretched horizontally at the final motion of the additional vibration, just like a train at emergency braking. See electronic supplementary material for more informations.

On the other hand, when the duration of the additional vibration is shorter than two shakes, the memory of the primary vibration is replaced by the memory of the additional vibration only at the region where the upper layer of the paste is stretched horizontally at the final motion of the additional vibration (which corresponds to a rear part of a train at emergency braking), and the memory of primary vibration still remains at a region where the upper layer of the paste is compressed at the final motion of the additional vibration (which corresponds to a front part of a train at emergency braking). That is, at the initial stage of the memory-imprinting process, the mechanism predicted by the quasi-linear analysis holds. On the other hand, as the paste is vibrated repeatedly and the number of shakes increases, the change of the plastic shear strain β cancels due to the symmetric cycle of increase (stretching) and decrease (compression) under repeated vibrations, but the increase of α is accumulated since the nonlinear term in the r.h.s. of equation (3.6) is proportional to σ²_xz and does not depend on the sign of σ_xz. Thus, the mechanism shown by the nonlinear analysis gradually comes to play a dominant role in the memory effect.

5. Discussion

By investigating the process of rewriting memory in paste, experimental results are found to be consistent with theoretical models which are based on residual tension theories. Here we would like to discuss our recent experimental approaches to investigate the mechanism of memory effect. We carried out μCT measurements for Lycopodium paste with a memory of vibration, and found a statistical anisotropy of particle arrangements in paste. This anisotropy depends on the direction of the horizontal vibration in such a manner that the number of neighbouring particles increases in the direction perpendicular to the shaking [25], as is schematically drawn in figure 8. It is also shown by measuring the time evolution of stress in paste during the drying process that the stress difference induced by the memory of vibration, such as σ_xx − σ_yy, is not so large just after the vibration stops, but the difference gradually increases as the paste dries and finally it plays an important role in inducing which direction cracks have to propagate [26]. More challenges are required to incorporate these effects into theoretical models based on residual tension theories.

6. Conclusion

Owing to its plasticity, a paste can remember the direction of the motion it has experienced, such as vibration and flow. These memories stored in paste can be visualized as the morphology of desiccation crack patterns. For example, when a paste remembers the direction of its vibration, all primary desiccation cracks propagate in the direction perpendicular to the direction of the vibration that the paste has experienced. To find out the mechanism of the memory effect of vibration, we performed rewriting-memory experiments by applying additional vibration to the paste along a direction different from the primary one. By investigating the process of rewriting memory in paste, we found competitive phenomena between the quasi-linear effect and the nonlinear effect, which were studied in each theoretical model based on residual tension theories. That is, at the initial stage of the memory imprinting process of its vibrational motion, the mechanism predicted by the quasi-linear analysis holds, but, as the paste is vibrated repeatedly, the mechanism shown by the nonlinear analysis gradually comes to play a dominant role in the memory effect.

Data accessibility

Supporting data are made available as electronic supplementary materials.

Authors' contributions

A.N., T.H., R.H. and Y.M. carried out the experiments. A.N. performed the data analysis. A.N. and S.K. compared experimental results with theoretical predictions. A.N. drafted the manuscript and all authors read and approved the manuscript.

Competing interests

We declare that we have no competing interests.

Funding

A.N. and S.K. acknowledge JSPS KAKENHI grant nos. (B)22340112, (C)16K05485 and (C)18K03560. A.N. and S.K. acknowledge supports by HAS-JSPS International Joint Research Project and by International Exchange Schemes between The Royal Society and JSPS.