Strength and Fracture Laws of Nature

© Lev Gelimson

Modern engineering requires optimal design and rational control of resistant structural elements of ductile and brittle, isotropic and anisotropic materials under extreme stationary and variable loading. This needs adequately determining the danger of the real spatial stress process at any point of a solid with respect to the closest critical (limiting, ultimate) stress process. Critical (limiting, ultimate) spatial stress processes should be obtained via critical process criteria by using strength data available in simple experiments. The dangers of real spatial stress processes should be given by measures of the proximity of a real stress process to the closest critical one.

A usual stress is not a pure number, depends on the choice of physical dimensions (units) for a force and a length, is not numerically invariant by unit transformations, and alone represents no degree of the danger of itself even in stationary loading. If a solid’s material is not isotropic with equal strength in tension and compression, it is not reasonable even in stationary loading to compose functions of different stresses without their adequate weighing because of mixing their values having distinct limits and hence diverse degrees of danger. If loading is variable, the same holds even for different values of the process of a stress alone.

The known critical state criteria separate for diverse materials types, unlike simple and universal fundamental laws of nature, have contradictions, restricted and vague ranges of adequacy, sometimes lose physical sense, not always bring a suitable equivalent stress, and are applicable in the stationary case only. For an isotropic ductile material with equal strength in tension and compression even under stationary loading, the criteria ignore considerable strength increase in uniform triaxial compression. The only known attempts to propose a critical process criterion are reduced to very special cases of uniaxial stress cycles and of combined cyclic bending and twisting a bar. For anisotropic materials under variable loading, when the directions of the principal stresses at a solid's point under consideration can arbitrarily turn, there has been no attempt to propose a critical process criterion at all.

The only known measure of the proximity of a real stress process to the closest critical one is a safety factor as the ratio of a limiting stress to an equivalent stress. This could suffice only if all the principal stresses are directly proportional to a common variable parameter. Otherwise, a usual safety factor does not determine the permissible combinations of the initial data in a strength problem, can overestimate actual reserves by an order of magnitude, and is manifestly insufficient.

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

The boundaries of the quality (e. g., linearity, elasticity, plasticity, and failure) domains for solids are determined by corresponding critical (limiting, ultimate) criteria (for example, yield criteria and failure ones) that should be universal laws of nature applicable to any anisotropic solids under arbitrary nonstationary loading conditions.

The critical stress states at a point of a solid under loading form a critical surface, e.g., the yield or failure boundary, between the elastic, plastic, and fracture domains in the conventional space of the three principal stresses in the principal directions with vanishing the three shear stresses. The critical criteria determine the conventional uniaxial stress equivalent (equidangerous) to the usually triaxial stress state at a point of a solid as a function of the three principal stresses there. Its value is then equal to the uniaxial critical stress, e.g., a yield stress or an ultimate strength, as a constant of a certain material. In the simplest case of an isotropic solid (especially with equal strength in tension and compression) by static loads, some classical criteria by Galilei, Tresca, von Mises, Coulomb, etc. are suitable. There are also specific criteria for some particular cases of an anisotropic (usually orthotropic) material (with using Einstein's tensor notation) and cyclic loads. They are usually too complicated and have nothing in common with universal laws of nature always simple enough in their initial forms. For the general case of an arbitrarily anisotropic solid under any variable loading with turning the principal directions of the stress state, there was earlier no known proposition and hence no known law of nature.

That is why the known critical criteria being indisputable achievements of many prominent scientists have great theoretical and practical significance but are only manifestations, or special cases, of some laws of nature as applied to certain solids and loads.

Thus the purpose of fundamental mechanical and strength sciences is clarifying universal forms of such laws applicable even to any anisotropic solids arbitrarily extremely loaded in modern engineering (astronautics, aircraft building, deep-sea industry, power engineering, building, etc.).

The main idea of created fundamental mechanical and strength sciences is that, in principle, critical criteria (distinct in their usual forms for different materials and loading conditions) must be sufficiently universal fundamental laws of nature. Assume a criterion be an expression of a certain temporarily unknown sufficiently general criterion now applied to a certain material and loading. Then try to determine, for such a desired criterion, its form that may not include this specific constant of a certain material. By dimensionality and similarity theories, divide principal stresses by that critical stress. The criterion becomes pure (dimensionless) without evident constants of a material and holds for any ductile material with equal strength in tension and compression independently of the specific value of that stress. Such a circumstance makes it possible, to further generalize the criterion by giving, as compared with that transformation, a more general meaning to the reduced relative stresses introduced in this way. For a brittle solid with unequal strength in tension and compression divide each nonnegative principal stress by the critical stress in tension, and each negative principal stress by the modulus of the critical stress in compression. For orthotropic materials, the principal directions of a stress state coinciding with the basic orthotropy directions at the same solid's point, use the critical stress of the same sign and direction as a divisor. The same holds for any stationarily loading an arbitrary anisotropic solid.

By nonstationarily loading, first define the three processes of the reduced principal stresses by using the simultaneous critical stresses. Then define the reserve and the equidangerous cycle of each of the processes by its similar critical process of the reduced principal stresses. Further define a stationary vectorial reduced stressdue to a limiting amplitude diagram. Ultimately, the postulate on the universality of a criterion function gives (as a basis for hierarchies of strength laws of nature) a result criterion with choosing the most dangerous, possibly depending on time, permutations of the stationary stress indexes.

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

The known yield and failure criteria cannot be universal laws of nature and, for an anisotropic material, determine no equivalent stress necessary for solving strength problems.

A relative stress is introduced as a stress divided by the modulus of its limiting value of the same sign in the same direction by vanishing all the other stresses at the same solid's point by the same other conditions. A relative stress is the reciprocal to a sign-preserving individual safety factor independently of choosing a limiting criterion and stress unit and expresses the degree of the danger of a stress better than this factor, the usual individual safety factor, and this stress itself.

A limiting criterion in the relative stresses is obtained from a yield or failure criterion by passage to the relative stresses, naturally extends the initial physical sense of that criterion, expresses a relative equivalent stress, being the reciprocal to a safety factor, as a function of the sign-preserving individual safety factors, has a universal form like laws of nature, and applies to any solid arbitrarily loaded.

The passage to the relative stresses unites the experimental data on the strengths of different materials, raises the reliability of the results by their clustering, and shows the equal adequacy of limiting criteria in the relative stresses as applied to different types of materials.

Limiting criteria in the relative stresses determine the applicability ranges of yield and failure criteria and extend them to universal laws of nature, which are verified by many known theoretical and experimental data and systematize them.

Limiting criteria in the relative stresses are naturally extended to translimiting criteria in the relative stresses discriminating the subcritical, critical, and supercritical stress states of solids.

Fundamental mechanical and strength sciences being developed are based on the principle of tolerable simplicity specified analytically and the postulate stating the essential universality of critical process criterion functions. Fundamental mechanical and strength sciences include a transformation method for usual stresses, a generalization method for critical state (process) criteria, a correction method for critical state (process) criteria, and a generalization method for safety factors.

The transformation method for usual stresses is (essentially) reducing them to relative ones. A relative stress stationarily reduced is a number-value function of time, which is introduced as a usual stress divided by the modulus of its limiting value of the same sign in the same direction by vanishing all the other stresses at the same solid's point at the same time instant by the same other conditions. A relative stress nonstationarily reduced is a stationary vector of real-number length, whose abscissa and ordinate are the mean and amplitude relative stress values, respectively, of a cycle equidangerous to the corresponding usual stress process stationarily indexed. A relative stress is numerically invariant by any unit transformations. It is the reciprocal to a sign-preserving individual safety factor independently of choosing a limiting criterion and stress unit and expresses the degree of the danger of a stress process better than this factor, the usual individual safety factor, and this stress process itself. A passage to the relative stresses unites the experimental data on the strengths of different materials and raises the reliability of the results by their clustering. The relative stresses open many new ways in strength measurement and investigation to discover mechanical and physical laws of nature.

The transformation method for critical process criteria is essentially a passage in an initial critical state criterion to the relative stresses. First they are reduced stationarily, then nonstationarily with further using the modulus of the vector value of a criterion function by rules of vector algebra. Equalizing to unit the maximum of the two values of the criterion function by choosing the most dangerous permutation of the stress indexes in the both cases and  the most dangerous instant of time for the criterion function of the relative stresses reduced stationarily gives the corresponding critical process criterion. It naturally generalizes the physical sense of the initial critical state criterion and expresses a relative equivalent stress, being the reciprocal to the result reserve (safety) factor, as a function of the individual reserve (safety) factors. They are taken with the signs (in stationary reducing) or the vector directions (in nonstationary reducing) of the corresponding relative stresses. Each general critical process criterion has a universal form like laws of nature and applies to any solid arbitrarily loaded. Critical process criteria in the relative stresses determine the applicability ranges of the known yield and failure criteria and extend them to universal strength laws of nature, which are verified by many known theoretical and experimental data and systematize them. Critical process criteria in the relative stresses may be naturally extended to transcritical criteria in the relative stresses naturally discriminating subcritical, critical, and supercritical stress processes.

The correction method for critical process criteria is essentially based on the natural hypotheses that in critical state, the value of a critical state criterion function of the stresses either usual or relative is equal to the value of another function of the same nature. In particular, the last function might be a certain linear function of the principal stresses either usual or relative. This method first allows taking into account the well-known considerable influence (established by Bridgman) of uniform triaxial compressions on reaching critical states. The method first establishes a considerable influence of uniform triaxial tensions on reaching critical states and allows to indirectly determine such limiting states and processes.

The generalization method for safety factors as a deep extension of the generalization method for limiting criteria is essentially worst-case measuring the proximity of a real process to the closest critical one and is not restricted to stress problems with stress processes. In any problem, for all input and output parameters, using such natural methods as additive and multiplicative ones, the ranges of the parameters are expressed through their design values and individual reserves being introduced. Applying worst-case approach to functional dependences expressing the output parameters through the input ones, the corresponding dependences correlating their individual reserves by the worst realizable combinations of the parameters values in the parameters ranges determined by these reserves are obtained. These reserves and their dependences realistically estimate tolerable deviations of the actual values of the parameters from their design values. It is possible to introduce some additional correlations and attach a priori values to some reserves. In particular, a result reserve might be expressed through the other ones as some functions of one real number, which makes it possible, to realistically determine all the other reserves, given a value of the result reserve. This allows to rationally control the authentic reserves of structures.

Fundamental mechanical and strength sciences first created as a source to discover many strength laws of nature are based on the principle, ideas, postulate, and hypotheses that generalize many well-known theoretical propositions and experimental data. These sciences give simple (as far as it is possible in complicated problems to be solved) and practically convenient final results. Fundamental mechanical and strength sciences are very suitable, allow generally representing and processing measurement data, reduce time and cost expense, and bring many further advantages in mutual completion of difficultly obtained experimental data on the strength of different materials in spatial stress states under arbitrarily changeable loading. Fundamental mechanical and strength sciences build scientific foundations for rational control of the strength of modern materials and structural elements intended for extreme loading conditions. These sciences make clear physical sense, cover all stages in solving strength problems, develop many well-known results as manifestations of the discovered laws of nature, and found a new promising scientific trend of great importance in solid mechanics.

It is possible to propose the following rational algorithm for analytically approximating the experimental data on critical states of materials under consideration by adequate critical state criteria:

1) if a material is not isotropic with equal strength in tension and compression, the principal stresses are divided by their appropriate limiting values and interpreted in the space of the reduced principal stresses possibly in some planes rationally chosen;

2) the reduced data are examined to abstract from their spread, expel obvious outliers, and determine the general character of a limiting surface and its curves in the planes;

3) the representative sample of the possibly least size is chosen by the principle of tolerable simplicity;

4) critical state criteria are tested to select a criterion that is adequate enough and has the least number of constants (parameters of a material). The methods for estimating and optimizing unierrors and reserves can be used;

5) desired critical state criteria in the usual principal stresses are determined reversing the formulae reducing them by static or nonstationary loading.

For formulae and details, see the scientific website of the author and especially his works under "Strength".