XIII G: Role of Mathematics behind patterns of nagging cracks.

XIII G: Role of Mathematics behind patterns of nagging cracks.
           [Contd. A Journey to the Wonderland of Math.by Ajay Kumar Chaudhuri.]
             "Who has cracked?" ----"Mr. Nobody"
      
          Have you ever seen crack of any kind on any thing around you? It seems to be a silly question,as all of us have definitely observed it,aware of it and even scare of its harmful effect.
        But what does cause these cracks and how?Who is behind this mischievous act is oblivious to us and appears to be mysterious. In this context let us go back to our early school days as kids to remember a funny poem" Mr Nobody" by an anonymous poet. The first few lines are as:
                              I know a funny little man ,
                                 As quiet as a mouse,
                              Who does the mischief that is done
                                In everybody`s house!
                             There`s no one ever sees his face,
                              And yet we all agree
                             That every plate we break was cracked
                            By Mr. Nobody.

      Yes, crack is as mischievous as Mr. Nobody.Yet we are to confront him, since crack is one of the patterns of nature.

All of us are very familiar with such cracks,  which are seen around us on the surfaces of many things. In nature ordered crack patterns are found in places ranging from cracks in soil beds of fields during drought or in dry season. We get worried when cracks are detected on the walls of our buildings, on bridges, on dams or railway tracks and at many such vulnerable surfaces.

A crack, as it appears, is a line on the surface of something along which it splits without breaking to relieve stress. The physics behind these patterns is , when an elastic* material stretches or shrinks uniformly, it reaches its breaking strength and then fails suddenly in all directions, creating cracks with 120 degrees joint and so three cracks meet at a point, called ‘node’. Conversely, when an inelastic material fails its breaking strength, straight cracks form to relieve stress. Further stress in the same direction would then simply open the existing cracks; stress at right angles can create new cracks, at 90 degrees to the old ones. Thus pattern of cracks indicates whether the material is elastic or not.

* [‘Elasticity’ means ability of a deformed material body to return to its original shape and size when the forces causing deformation, called ‘Stress’ are removed. It is a fundamental property of a material and is called ‘elasticity’. The body which possesses this property is called an ‘elastic body’.

To a greater or lesser extent most solid material exhibit elastic behavior, but there is a limit to the magnitude of the force and the accompanying deformation within which elastic recovery is possible for any given material. This limit is called ‘elastic limit’. So, contrary to the concept of elasticity in daily life, in physics, elasticity stands for opposition to change. Qualitatively, more rigid a body, move elastic, it is said to be. for this reason, a steel rod is more elastic than a rubber band. Conversely, the material which does not possess the properties of elastic bodies is called

an ‘inelastic body’. That means when the stress is removed, it will not come to its original shape and size, and will undergo permanent deformation. The putty or paraffin waxes are examples of such bodies.]

Surface crack patterns occur on a variety of materials, including glass, mud and ceramic glaze. These cracks are often results of shrinkage of the object’s surface area. For example mud in a riverbed dries faster on the surface than in the underlying soil, causing stress to build. The mud develops cracks to relieve this stress.

In ceramics, glaze with a different co-efficient of thermal expansion (The co-efficient of thermal expansion of a material is the change in length, area or volume of it for a change of 1 degree in temperature) than that of the pottery will accumulate stress during cooling process. When this stress is too high the glaze cracks, we have developed a method to model the crack patterns created by these types of processes, as demonstrated by the example of ‘Crackle glass’ shown in Pic. No. 21a.

               

               Pic. No. 21a

                                                    Some items of crackle glass.

To a degree these patterns can also be controlled as the artistic ‘Craquelure’ of paintings and pottery. Craquelure is the fine pattern of dense cracking formed on the surface of the materials as a part of the process of ageing of their original formation or production.  This term is more often used to refer to oil paintings, a sign of age, and is also sometimes induced in forgeries, and ceramics where it is often deliberate and usually called ‘Crackle’. It can also develop in old ivory carvings and painted miniature on an ivory backing are prone to craquelure.

Since energy can be related by widening of a crack, anywhere along its path, the entire history of the crack affects how it will grow at any instant. This means that whether a crack will grow or not is the question, whether a bridge will fall down or an air plane wing snap off, is solved.

Cracks and splits in tree trunks are fairly common and may occur for various reasons. One of the most common reasons for cracks and splits on tree trunk is cold temperature. These are called frost cracks and are caused when the inner and outer wood in the tree’s trunk expands and contracts at different rates when temperatures change. This happens when winter temperatures plummet below zero especially after a Sunny day when a tree’s trunk has been warmed by the Sun. The different expansion rates between the inner and outer wood can cause such a strain on the trunk that a crack develops. Frost cracks occur suddenly, can be several feet long and are often accompanied by a loud rifle shot sound. They often originate at a point where the trunk has been physically injured in the past. Maples, the ornamental trees, and several types of tree but with somewhat similar leaf form, called sycamores are the most prone to frost cracks. Apples, ornamental crab apple, walnut, oak, maples, beach, tulip tree etc. are also susceptible.

Another form of injury that can result in cracks and splits is caused by exposure to excessive sunlight, which is called ‘Sun scald’. It occurs usually in the winter. The damage takes place when the cells in the living tissue beneath the bark break dormancy on warm, sunny days and then rupture and die when the temperature drop below freezing at night. The tree is injured when enough cells in a given area are killed. The following spring these dead areas will appear discoloured and sunken. In time the bark killed by sun-scald will split and peel. These areas also provide entry points for insects and diseases.

The crack patterns are also perceivable in leaves of plants. The growth of leaf veins resembles how crack patterns form in drying films. Now it is a baffling question whether this similarity is apparent or has some deeper reasons. Botanists are trying to unravel the mystery.

Cracks on the peel or skin of many fruits are found very often. Fruit cracking appears and that spread out in radial or concentric patterns. Depending on the cultivar (cultivable variety of wild plants), the cracks can appear from mature green to the ripe stage of maturity. Fruit cracking may be caused by alterations in growth rate or fluctuations in moisture and temperature.

One of the main disorders that widely limit fruit quality and quantity is fruit cracking or splitting that is observed on fruit skin and flesh in pre-harvest stage. Besides, cracking can occur during post harvest in some fruits, mostly attributed to the environmental conditions of storage. Many fruits such as apple, sweet cherry, grape, plum, litchis, citrus fruits like orange, lemon, lime, banana and many other fruits as well as tomato can crack or split. The crack patterns on the skin of melons are vividly displayed in Pic. No. 21b.

            Pic.No21b. 

                                                              Cracks on Melon skin.
 

                                                                                                                                [To continue]

        Reference Internet:
                                        All the images are downloaded from Public Domain.