Residual Stress in Biological Structures

In solid mechanics, many of us are familiar with the physical quantities - normal and shear stress in designing structural members. But perhaps not much with the quantities – primary, secondary and residual stresses. The stresses derived from the bending moment and shear force either by hand calculations or structural analysis software are mostly primary stresses. Upon failing to distinguish the stresses in detailed 3D analysis, a FE analyst often endup considering the combination of all the stresses as primary stress leading to over-conservative design. Therefore, in addition to these normal and shear stresses, it is essential to comprehend and distinguish the primary and secondary stresses designing a structure efficiently.

DEFINITIONS

Citing to 1971 edition of ASME Boiler & Pressure Vessel Code; the stresses are mainly categorised as primary and secondary stress. According to the B&PV code, the primary stress is defined as [1],

“Normal stress or a shear stress developed by an imposed loading that is necessary to satisfy the laws of equilibrium in terms of the external and internal forces and moments.”?

Meaning: To satisfy the law of equilibrium, the stress must increase as the loading increases to counter the effect otherwise it will be total collapse. The secondary stress is described as,

"Normal or shear stress developed by the constraint of adjacent parts or by self-constraint of a structure. The basic characteristic of a secondary stress is that it is self-limiting.”

Meaning: Stress is a result of strain incompatibility between adjacent parts, which facilitates redistribution rather than complete collapse following the yielding or cracking.

One common characteristic between the two stress kinds is self-limiting. Secondary stresses are self-limiting and primary stresses are not because, the latter donot redistribute and reduce upon yielding or cracking [2,3]. Primary stresses are the consequence of persistent loading, as they increase, deformation doesn’t reduce them. They are present as long as the load is applied and will not diminish with time or as deformation takes place. Secondary stresses are those developed due to geometric discontinuities or stress concentration. In addition to these two stresses, there is another stress kind which reside in the structural materials after their manufacture and even in the absence of external loading known as residual stresses. In isotropic metals, they are usually the consequence of inelastic deformation, thermal gradient and phase transformation (material science) during its manufacturing process [4]. In anisotropic composites, they are generally due to the strain incompatibility between two different materials. Since residual stresses self-equilibrate and attenuate around the local yielding or cracked area, they are typically regarded as secondary stresses [5]. A simple way of deciphering them is through identifying and studying their existence in nature. One such instance is in fruits and vegetables.

?ANALOGY

Let us consider a simple bio system - tomato (arguably a fruit and a vegetable too!). However, before investigating the stress state in a tomato, let's take a quick look at the mechanism by which stress develops in a balloon.

?Figure 1. Stress State in Balloon and Laplace Law?????????????

A balloon filled with air exerts pressure on the internal surface of the skin resulting tensile stress in it (Figure. 2). The behaviour is similar to a pressure vessel where the stress in the skin keep raises as the pressure intensifies and finally dissipate upon rupture of the skin (burst). Stresses do not redistribute or equilibrate following the balloon rupture, they can therefore be categorically classified as primary stresses and not as a secondary or residual stresses. However, now returning to the case in tomato, following few interesting questions arise to address,

1. whether the pressure developed inside tomato is a primary loading?
2. can the stresses developed in tomato skin be classified as residual stresses?

To answer the above questions, we need to look at the different growth stages and their corresponding physiological changes the tomato undergoes during its evolution from raw stage to ripened stage. The three common stages in tomato are illustrated in the cover Figure 2.?

1. Raw tomato as unstressed cuticle/skin
2. Ripened tomato with stressed skin?
3. Fissured tomato with stressed skin relieved by fracture?

During its ripening stage, the biomechanical properties of cuticle changes and starts experiencing the tensile stress due to inevitable growth of the core. Similar to but not quite like a balloon as the tomato is a composite material with its core as semi solid. The hydrostatic pressure developed inside fruit is termed as turgor pressure and is always balanced in cuticle in its uncracked state. Initiation of crack happens when there is an incompatible strain rate between the cuticle and the inner core. The incompatibility in strain rate is either due to a) sudden growth of the core or/and b) fluctuations in ambient weather [9]. Some authors also studied the effect of genetic differences on cracking but the physiological basis of these differences is still not clear [6, 8].

a) The incompatible growth of core is a consequence of rapid influx of solute (water) due to over watering at the time of ripening. Therefore, the stress development is attributed to the inability of skin to expand at the growth rate of core and its gradual evolution of elastic properties to endure the turgor pressure and expand appropriately.?

b) In case of unexpected precipitation, the condition would be similar to case (a): the sudden influx of water from roots in saturated soil. In contrast, during summer, due to sudden raise in ambient temperature the skin loses its moisture and tries to contract and develops compressive thermal stress. It is worth noting here, the thermal stresses are always considered as residual stresses as they self-limit upon deformation or cracking.?

These requirements of strain compatibility in natural composite between the skin and core is what categorizes the stress developed in the tomato skin as secondary stress or residual stress. Once the crack is nucleated there will be no further increase in stress even with the increase in turgor pressure inside the tomato. However, both the stresses due to rapid influx of water and thermal stresses perhaps exceptional if they are found long-range and extreme form of splitting that penetrates deep into the pulp [10] is observed. Similar to structural integrity, fruit cracking is considered as a physical failure (physiological disorder) of the fruit that manifests as a fracture in the skin [10]. In horticultural science, it is considered as loss of marketable yield. Once the turgor pressure crosses beyond the critical stress intensity factor/fracture toughness of the cuticle, the residual stress in it relieved and is not considered as balanced residual stress anymore.

CONCLUSIONS

In a nutshell, under the condition of incompatible growth of vegetables, higher the water content in core, greater will be the tensile residual stresses [9]. Higher the ambient temperature, the higher will be the compressive thermal residual stress in the skin. The topic is however lies in a grey area and can be debatable whether the stress generated can be categorized as a residual stress as no external load acting upon it or as an internal stress due to turgor pressure.?

For future works, it would be interesting to measure these residual stresses to predict incipient failure in biological structures. Some of the methods could be indentation deformation, cutting or puncturing resistance of tomato skin using a needle or blade to correlate tensile residual stress with it.

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