Why Many Catastrophes Occur

© Lev Gelimson

Modern engineering (astronautics, aircraft building, ship building, deep-sea industry, power engineering, electronics, chemical industry, building, etc.) requires optimal design of structural elements. It is based on a rational control of the necessary and sufficient strength of such elements and corresponding materials. Their types are diverse: ductile materials like metals; brittle alloys and nonmetals (glass, crystalline glass, concrete, and stone); anisotropic materials (fiber-reinforced ones and other composites). They are intended for extreme exploiting conditions (variable loading, high pressure, high or low temperature, radiation, etc.).

It is universally recognized that the most effective approach to solving strength problems in modern engineering is phenomenological. It provides considering more or less adequate mathematical, mechanical, physical, etc. models of materials and structural elements instead of real ones and uses mathematical methods to determine the usually triaxial stress state (in the stationary case) or process (in nonstationary loading) at each point of a structural element. It remains to compare the diverse triaxial stress states at all points of a solid (structural element) with one another by the degree of danger to reach the closest critical state (initiation of yielding, fracture, etc.). Therefore, it is necessary to use so-called critical (limiting, ultimate) state (process) criteria (elasticity criteria, yield criteria, failure criteria, etc.) that reduce the problems to the simplest ones dealing with uniaxial stress states only investigated enough.

The well-known critical state and process criteria are separate for diverse materials types, have nothing in common with simple and universal fundamental laws of nature, and possess evident defects. If a material has unequal strength in tension and compression, the criteria possess obviously restricted ranges of applicability and do not allow comparing arbitrary stress states with one another. No critical process criterion for anisotropic materials under variable loading, when the directions of the principal stresses at a solid's point under consideration can arbitrarily turn, is known at all.

Consequently, there was no strength theory satisfying the complex of modern requirements for inherent unity, consistency, sufficient completeness, universality, naturalness, simplicity, and convenience for practical application.

That is why developing such fundamental mechanical and strength sciences is of great interest and importance. Its kernel is critical (limiting, ultimate) state theory considering critical state (process) criteria as some manifestations of certain fundamental laws of nature, which are mathematically described by universal critical state (process) criteria.

The present generalizations are founded on the principle of tolerable simplicity analytically specified as a choice criterion. The cardinal idea of the generalizations is the universality of critical state (process) criteria as some manifestations of fundamental laws of nature.

In the simplest stationary case of an isotropic ductile material with equal strength in tension and compression, a critical state criterion can be usually expressed in the following form:

The equidangerous uniaxial tensile stress that is equivalent to the triaxial stress state under consideration by the degree of the danger of the closest critical state (by yield, failure, etc.) and represented by a certain function of the principal stresses by vanishing the shear stresses and possibly of some material constants is equal to the uniaxial limiting stress in a corresponding critical state such as yield or failure of a solid's material, e.g., the yield stress or the ultimate strength. 

That function is specified by diverse criteria. According to the theory of maximum shearing stresses, it is the difference of the ultimate principal stresses. By the theory of potential energy of distortion, it is the intensity of stresses.

The essence of the proposed method to generalize critical state criteria for arbitrary materials and loading conditions might be shown in the first of these two criteria. Its formula includes the specific value of the only constant, the uniaxial limiting stress, for a given isotropic ductile material and cannot be immediately applied even to such a material with another value of this constant. Therefore, it is natural to suppose the criterion to be a specific manifestation of some unknown universal law of nature as applied to a given material. The problem aroused is to determine a formula expressing that law. It is not the known criterion itself that allows no generalization. Its simplest transformation is to divide each principal stress by the modulus, the uniaxial limiting stress, of its ultimate values in uniaxlal tension and compression when a principal stress under consideration is sole and the other two principal stresses vanish. Such dividing is coordinated with similarity and dimensionality theories.

This transformation is independent of any criterion. The transformed criteria in the relative stresses have no evident material constant and so allow imparting a generalized sense (in comparison with the known criteria) to the reduced (relative) principal stresses according to the specific character of the strength of any given material.

For an isotropic brittle material with unequal strength in tension and compression, it is natural to reduce each principal stress by dividing it by the modulus of its ultimate value in the corresponding uniaxial state again. But that value now depends on the sign of the stress and is equal to the uniaxial limiting stress in tension for nonnegative principal stresses and to the uniaxial limiting stress in compression for negative principal stresses. Furthermore, a generalization of that transformation has to keep the sign of each stress. So it is divided namely by the modulus of its ultimate value whose sign coincides with the stress sign in the corresponding uniaxial state (either tension or compression).

This transformation again independent of any critical state criterion is an immediate expression of the proposed method generalizing critical state criteria for any isotropic material with unequal strength in tension and compression. It is possible to test this transformation by the known experimental data on biaxial critical stress states in diverse ductile and brittle isotropic materials (commercial quality steel, hard steel, copper, nickel, gray iron, gypsum, porous iron, and concrete). The well-known division of the principal stresses independently of their signs by the uniaxial limiting stress in tension allows unifying all the data (the points) without their separation for diverse materials only if all the three principal stresses are nonnegative, which is quietly natural. At the same time, our transformation allows unifying all the data (the points) without their separation for diverse materials always, i.e. independently of the signs of the three principal stresses. Therefore, this transformation is adequate and allows using universal criteria for any isotropic brittle material. Their expressions are usual and invariant in the space of the reduced (relative) principal stresses, but depend on the combination of the signs of the usual principal stresses in their space.

The transformed theory of maximum shearing stresses has obvious physical sense as follows. If the signs of all the nonzero principal stresses are identical, a natural material having unequal strength in tension and compression is similar to two model materials with equal strength in tension and compression. Namely, if all the principal stresses are nonnegative, a natural material is modeled by a material whose equal strength in tension and compression coincides with the strength of a natural material in tension. And if all the principal stresses are nonpositive, a natural material is modeled by a material whose equal strength in tension and compression coincides with the strength of a natural material in compression. But if there are principal stresses with distinct signs, the critical states of a natural material are described by a criterion that coincides with the Coulomb linear approximation of Mohr’s theory in this case only. This can be regarded as the suggestion to determine the applicability range of that theory, which is not quite obvious. In fact, the well-known method to obtain that criterion is latently based on the distinction between the signs of the principal stresses.

For any anisotropic material (with generally unequal strength in tension and compression in each direction) and arbitrary static loading, it is natural to reduce each principal stress by dividing it by the modulus of its ultimate value in the corresponding direction by uniaxial state. Again that value now depends on the sign of the stress and is equal to the uniaxial limiting stress in tension for nonnegative principal stresses and to the uniaxial limiting stress in compression for negative principal stresses. Furthermore, a generalization of that transformation has to keep the sign of each stress. So it is divided namely by the modulus of its ultimate value whose sign coincides with the stress sign in the corresponding direction by uniaxial state (either tension or compression).

In contrast to well-known criteria, universal critical state criteria in the reduced (relative) principal stresses always conserve their simple forms like all fundamental laws of nature.

This transformation and its particular cases are natural but not the only. If a material has unequal strengths in tensions and compressions in the principal directions of the stress state at a solid’s point under consideration, there is also another possibility. It is not less natural even if loading is static, but cannot be discovered within the bounds of statics. In the case of cyclic loading, the limiting amplitude of stresses reaches its peak in the diagram when the cycle is asymmetric and possibly the cycle mean stress is nonzero. So the stress state when all the principal stresses equal to that cycle mean stress (as opposed to the zero stress state when all the principal stresses vanish) can be considered as the initial one instead of the zero state. Such a situation can be caused by the effect of microstresses and submicrostresses as a phenomenological macroresult. So a material having unequal strengths in tension and compression in some direction can be considered as one having not only equal strength in each tension and compression but also the corresponding initial stresses. Then that cycle mean stress should be taken off from each principal strength and its limiting value of each sign in any direction before defining sign, taking a modulus, and dividing.

To obtain critical process criteria for any anisotropic material (with generally unequal strengths in tension and compression in each direction) and arbitrary variable loading, begin with any initial critical state criterion for a model isotropic material with equal strength in tension and compression under stationary loading. Then the corresponding general critical process criterion at any point of a solid of any natural material possibly anisotropic under arbitrary variable loading can be created due to the following algorithm:

1) for each unregulated principal stress by a stationary numeration on the whole time interval of loading, the reserve (safety) factor for the uniaxial stress process is obtained from the condition that the uniaxial stress process similar to the realized one is limiting. The processes are called similar if their variable directions at a point under consideration in a solid of a natural material possibly anisotropic are synchronized, i.e., coincide at every moment just as the values of these processes are proportional to each other with a factor constant on the whole time interval. At every moment of time, it is possible to take into account the damage accumulation in limiting uniaxial stress process on the previous time subinterval;

2) each principal stress is synchronously reduced via the above transformation always using time dependence under the same other loading conditions such as speed, temperature, radiation level, etc.;

3) using the least upper bound and the greatest lower bound of the stationary reduced (relative) stress in uniaxial stress process in a model material, the stationary reduced (relative) mean stress is determined;

4) in a model material, a stationary reduced amplitude stress in the uniaxial stress cycle, whose mean stress is just the above mean stress and which is as safe as the uniaxial stress process in a natural material on the time interval is determined as the least upper bound of such values that every uniaxial stress process in a direction coinciding with the direction of the real process at every moment in the natural material and is reduced by the above transformation to some cycle having the above mean stress and the above amplitude stress in the model material, has a safety factor not less than the above safety factor. If the set of such values is empty, then the mean cycle stress in the model material is so changed that the modulus (absolute value) of the change is as small as possible, and we take a stationary reduced amplitude stress to be zero.

Let us assume that the both limiting stresses (in tension and in compression) at the natural solid's point under consideration are stationary. This holds, e.g., in the case of damage nonaccumulation and stationary temperature, radiation level, etc. if the directions of the principal stresses in the natural material are stationary or if it is isotropic. In that stationary case for the natural material, the diagram of the limiting stresses (the maximum and the minimum) in the uniaxial stress cycle, whose direction coincides with the direction of the principal stress at every moment and which is as safe as its process, can be directly determined. Then both the abscissa and the ordinate of each point in the diagram are separately reduced by the above transformation and give the interrupted curve consisting of the upper and lower branches. Further each couple of the diagram's points having equal abscissas are replaced in the vertical direction to straighten the diagram's middle line so that their common abscissa and the difference of their ordinates are invariant and the middle of the segment connecting those replaced points lies on the principal diagonal of the first and the third quadrants. The obtained diagram of the limiting stresses allows determining Haigh's diagram of the limiting amplitude stresses by halving that difference for each abscissa and to directly obtain the amplitude of the uniaxial stress cycle;

5) the stationary vector for the model material in the limiting amplitude diagram in the cycle, which is as safe as the variable uniaxial stress process in the natural material, is determined;

6) the range of triaxial stress processes stationarily safe is determined using the synchronous values of the uniaxial principal stress processes;

7) the range of triaxial stress processes variably safe at the natural solid’s point under consideration is determined according to the rules of vector algebra by selecting the most dangerous stationary permutation of the stationary stress indexes. In the modulus of the corresponding possibly vectorial function, the three independent (irrespectively of their synchronization) uniaxial stress processes are replaced with the stress cycles as safe as the corresponding processes. These cycles hold in the model material and are described by the three stationary vectorial reduced stresses. Interpreting them by complex numbers might be acceptable geometrically but not algebraically because a sum of squared complex numbers can vanish not only by vanishing each of those numbers. This might lead to the illusion that limiting stress processes can be absolutely safe;

8) the range of safe triaxial stress processes at the natural solid's point under consideration is determined by combining the last two results. First the three (relative) uniaxial principal stress processes synchronously reduced in the model material at every moment are taken into account in assumption that each triaxial stress process is considered stationary. Uniting this with the combination of the three independent (irrespective of their synchronization) uniaxial stress processes (the both conditions in the aggregate generalize the well-known verification of static strength in cyclic loading), we obtain the critical process criterion for any anisotropic material (with generally unequal strength in tension and compression) under arbitrary variable loading (as a basis for hierarchies of strength laws of nature) corresponding to the initial critical state criterion for an isotropic material with equal strength in tension and compression under static loading.

The proposed stress transformation methods are independent of critical criteria themselves, unify them, and hence are an immediate expression of limiting criteria generalization methods.

Analyzing experimental data for different ductile and brittle isotropic and anisotropic solids convincingly shows that, for the first time, fundamental mechanical and strength sciences bring universal strength and fracture laws of nature.

For formulae and details, see the scientific website of the author and especially his fundamental mechanical and strength sciences.