Physical criteria of materials' and structural elements' stress-strain state assessment

Dr. V.T. Vlasov, Dr., Professor A.A. Dubov

Many times we paid attention to the paradoxicality of the situation formed in solution of the most crucial problem of assessment of structural materials’ stress-strain state and of the residual life of complex technical objects. On the one hand, the modern diagnostics offers "the entire arsenal" of means for measuring of materials’ mechanical characteristics, internal stresses and even the residual, applying various non-destructive physical methods, which quite complies with the urgency of the problem. And on the other hand, along with the extremely high demand for the means of assessment of the actually formed state of the material, practically the complete refusal of fracture mechanics experts to use the offered means during the residual life determination is observed. And the motivation of the refusal is quite fair and based on the well-known and little known objective reasons.

The many-years experimental and practical experience, gained in the course of the development and practical application of the magnetic memory method during the diagnostics of various objects, revealed and proved the objectiveness of "discrepancies" between the actual values of physical internal stress parameters and the "usual" limiting values of mechanical characteristics, for example, to the time strength limit.

The results of theoretical investigations of the rules of physical strains distribution allowed explaining the observed "discrepancies" and proved the delusiveness of the well-known criterion of the actual material’s state assessment in local zones of the developing damaging by the degree of closeness to the reference limiting mechanical characteristics of the material.

In fact, this was known long ago, because by variation of the strained specimen’s shape we can easily judge about the non-uniformity of the strains distribution on the specimen (see fig.1). However it did not allow speaking about the quantitative ratios of strains in different areas of the specimen. We managed to solve this problem.

Fig.1. Areas of the uniform and non-uniform straining and the "neck"of the destroyed specimen: ε_lim - the limiting strain in the neck area; ε_ave - the average strain along the entire specimen length; ε_uni - uniform straining along the length L_r ; ε_non-uni – non-uniform straining along the length L_n .

Investigations of the strain-force characteristics of ninety-seven specimens of various steels and alloys (see table 1) showed that the values of the limiting external specific forces, reduced to the non-uniform and uniform strain areas, will notably differ from those for the entire specimen.

Table 1.

Fig.2 and 3 show the values of the ratios of the limiting load in the non-uniform and uniform strain areas to the time strength limit, depending on the average value of the maximum strain in the non-uniform strain area.

Fig.2. The ratios of the limiting values of the specific load in the non-uniform strain area to the time strength limit for various materials.	Fig.3. The ratios of the limiting values of the specific load in the uniform strain area to the time strength limit for various materials.

If we speak about strain during the specimens’ tension, then at time strength the average strain values in the non-uniform and uniform strain areas differ even more from the reference values of the material’s relative strains. Fig.4 shows the values of the ratios of average longitudinal strains in the non-uniform strain area of the specimen to the average values in the uniform strain area when the external load value corresponds to the time strength limit. And fig.5 shows the ratios of transverse strains.

Fig.4. The ratio of average longitudinal strains in the non-uniform strain area to the values in the uniform strain area.	Fig.5. The ratio of average transverse strains in the non-uniform strain area to the values in the uniform strain area.

At the same time in local areas the strain values will already differ by orders! Not only the experimental data but also the obtained by us functions of local strains distribution across the specimen thickness indicate this (see fig.6). And it means that the limiting state criteria, obtained in the course of simple mechanical tests of the specimens, cannot reflect the limiting state of the material and, particularly, the limiting state of the structural element.

Fig.6. Dependence of the ratio of maximum strain values in the local failure initiation area to the average strain on the size of the local area.

But, in order to realize this, it is necessary to overcome the well-established idea about the internal stresses and to remember that the stresses - "sigmas", which we are all so much used to – are not stresses. This is an external specific force applied to the specifically shaped specimen, which changes the internal stresses. Thus, this is a conditional equivalent of internal stresses!

"Conditional" - because the indispensable conditions are: the specific shape of the specimen and the specific procedure of testing.

Only upon understanding the physics of the process of the material’s resistance to straining it is possible to realize what the internal stresses are and how and when they occur. Here the well-known from fracture mechanics concept of a "structural element" - an elementary volume, in which the characteristic changes of the material take place during its straining – is of great help.

Fig.7 shows sequentially, how the external specific force, while "splitting" to components, influences the material, straining it in different directions (glide, normal, width) and rotating it in the space, which causes occurrence of the appropriate internal forces of the material’s resistance to straining and which finally determines the input of the material’s proper energy for resistance to the external load. It is obvious that the strains, which we can measure (longitudinal and transverse), represent the algebraic sums of projections of the internal physical strain vectors onto the usual for us directions – along and across the applied force axis.

Fig.7. Structural element strain at uniaxial tension.

Schematically the process of the material’s resistance to the external load can be represented as follows (see fig.8). Any external exposure (from simple uniaxial to the most complex one) of a specific object made from the investigated material, including the specimen or a complex part, always "splits" in the material to three force and three moment (rotational) components, except for the uniaxial loading of a cylindrical specimen, when it "splits" to two force and one moment component. The presented scheme shows that the internal stresses – the difference of the internal energy density in the local and the adjacent to it areas - represent the difference of potentials.

Moreover, now we can say that internal stresses - are a special unified energy characteristic of the material’s equilibrium state, which is determined (see fig.8) by the set of physical strain-force parameters reflecting various types of the internal energy variation as a result of various types of exposure of the specifically shaped material.

Fig.8. The material’s reaction to external force exposure: 1 - external specific force; 2 - force components of external exposure inside the material; 3 - an aggregate of local strains and material’s resistance – parameters of internal stresses; 4 - external characteristics of the material’s reaction; 5 - internal stresses - variations of internal energy density.

Expressly for "gourmets" the following definitions of the material’s state can be offered. Any material possessed a proper internal energy characterized by the average energy density, which could be represented by the two zero-rank tensors - scalar and vector potentials. The energy distribution in the volume of even "isotropic" material is non-uniform. But it is characterized by the strict order along each of the possible, quite definite directions of variation of the initial energy value - by the three linear (ordinary) vectors (the first-rank tensors) and one axial (rotational) vector. All these - are individual qualities or properties of the material, determined by one characteristic - the average energy density, which depends neither on the object’s shape nor on the nature of the external exposure. But at the same time the "quite definite" linear directions of variation of the initial energy value - normal, shear and width, determined by the glide plane position in the force or another external field space - already depend on the shape of the object made of the investigated material.

While resisting to the external exposure, the material uses its proper energy. The input of this energy can be evaluated by the work of the external field - the strain-force parameters expressed by two complete second-rank tensors (force and strain), or by two pairs of linear (symmetrical) and rotational (antisymmetric) tensors. It should be noted that the loss of an antisymmetric rotational tensor in the theory of materials’ resistance led to the deeply wrong idea about the existence of "principal stresses" and "principal strains". It is impossible either theoretically (in case no mistake is made) or in real conditions to find such a "plane", where there will be no shear forces and angular moments! It is simpler to understand it physically: the material’s energy consists of the two, practically equal by quantity components - potential (electrostatic), determining the "repulsion" of atoms, and quantum, determining the "attraction" of atoms. And it follows that during any impact on the material in any of its areas both fields - quantum (attraction) and potential (repulsion) - always "work". So, the pair of antisymmetric tensors, lost by the theory of materials’ resistance, exactly describes the input of the quantum component of the material’s internal energy for resistance to the external exposure. And, as a matter of fact, what would remain from the material if the forces, attracting atoms to each other, were really cast away?!

The obtained by us investigation results allow "seeing" how the material resists the external exposure.

Variations of the internal forces of the material’s resistance in the non-uniform specimen strain area to the external tensile force are shown in fig.9, where for the sake of comparison the graph of the external specific force variation is presented.

Fig.9. Dependence of internal forces of resistance to elastic and plastic strains St.6 on the specific force in the non-uniform strain area: σ^ext - stresses due to the specific force of the external load; σ_long^Pl_int - the graph of stresses due to the normal longitudinal plastic (mechanical) strain; σ_n^Pl_int - the graph of stresses due to the normal longitudinal plastic (physical) strain; σ_n^avePl_int - the graph of variation of the average across the specimen value of the internal force of resistance to the longitudinal strain; Мn_α^Pl_int - the moment of resistance to plastic variation of the glide angle.

As it is seen, the force physical characteristics of the material’s resistance to straining depend on the external specific force, which is taken for internal stresses. Their appearance is not simple and they are far from being proportional to it! Moreover, the special or critical points, which can be seen on the graphs, can hardly be linked with the yield strength or the time strength limit.

If we return to the latest ideas of the classical fracture mechanics about the process of failure development, formed by the 80-s of the XX^th century and based on various experimental and theoretical investigations, it can be seen that, "to put it mildly", they do not rely on the well-known mechanical characteristics of materials either (though in absence of anything else, they are used in calculations).

It should be noted that the most terrible consequence of the established mistake in realizing of the internal stress physics, one might say, the craftiness of the mistake shows itself only now, when the problem of the material’s state and efficiency determination - an important component of the problem of complex technical objects’ residual life assessment - has become so urgent.

The danger consists in the fact that the mechanical characteristics obtained in the course of the specimens testing – the reference limiting values of the specimens strains and specific forces – the conditional equivalents of internal stresses, are considered to be proper characteristics of the material, determining its resistance to any external loads independently from the shape of the product made of this material. This was the source of the principal mistake being peculiar to all methods of internal stress "measurement" without exception.

Thus, the obtained by us results of experimental and theoretical investigations allows "materializing" the conclusions, which the fracture mechanics approached practically in real earnest long ago, as well as presenting the limiting states and the inseparably linked with them background concepts, as follows:

• the limiting state of the material - is the minimum possible density of the internal energy – a limiting potential determined only by the value of the average density of the internal energy, being an individual quality of the material. It depends neither on the dimensions of the structural element nor on its loading conditions;

• the limiting state of the structural element is determined by the ratio of the dimensions of the local are, in which the material reached the limiting state, as well as by the dimensions of the structural element’s area, in which this local area is located;

• the local area dimensions are determined by the material’s individual qualities, the structural element’s dimensions and its loading conditions and, in turn, they determine the nature of the actual distribution of local strain in the structural element’s volume;

• the actual state of the material in the local area - the value of the actual density of the internal energy - is an actual potential determined by the individual qualities of the material, the location site of the investigated area in the structural element’s volume and by its loading conditions;

• the internal stresses - the difference of potentials - is the difference of the internal energy density in the local and the adjacent to it areas.

As it can be seen, the commonly known values of the limiting states - yield and time strength, obtained in the course of simple mechanical testing of the specimens, cannot reflect the limiting state of the material and, particularly, the limiting state of a structural element.

Thus, the investigation results analysis brings us to the known from the fracture mechanics conclusion that we should already speak about several different criteria of failure: the limiting value of normal strain at uniaxial tension, the limiting value of strain by width at uniaxial compression and the limiting value of shear strain at torsion or bending, as well as various combinations of the limiting values at complex loading.

All this requires more attentive approach to the diagnostics of the material’s stress-strain state and to the procedure of assessment of the degree of the material’s actual state closeness in the structural element’s local area to the limiting state, both for the material and for the entire structural element, because now it is clear that these are far from being the same!

It is quite obvious that prediction of possible periods of safe operation of real "ageing" structural elements (the main type of damages development) based on the results of the material’s SSS diagnostics using calibration dependences, obtained in the course of simple mechanical testing of specimens without evaluation of a fatigue failure development time and rate in a specific object and under specific conditions, is not simply useless but extremely dangerous!

The point is that all the known methods react to elastic strains. And in a real structure elastic strains never exceed the values, which correspond to the yield strength.

Moreover, taking into account the acute (from units to several tens of micrometers) locality of the fatigue damaging development process, the peculiarities of the local physical strains distribution and their ratios with average values of strains, one might state that, using conventional active methods of diagnostics, possessing the large averaging base (10 mm at best), most probably, we shall not simply detect the damaging development area, not to mention the possibility of the developing damaging parameters determination.

The obtained results of investigation of the rules of the physical strains distribution directly indicate the necessity of development of the new normative documentation, which would regulate certification of means for structural materials’ stress-strain state diagnostics, as well as the techniques for "adjustment" of means for SSS diagnostics.

We used the term "adjustment" instead of the more usual term "calibration", because we wanted to stress that the obtained by us results of experimental and theoretical investigations make it possible to carry out the diagnostics without the preliminary testing of the specimens, the shape of which is most often selected based on the conditions of convenience of the diagnostic means’ sensors installation. And, as a rule, it does not comply with the standard for performing of mechanical tests.

Thus, the new State Standard of the Russian Federation GOST R 52330-2005 "Non-destructive testing. Stress-strain state test on industrial objects and transport. General requirements", which was put in effect in 2005, is the first little but, perhaps, the most difficult and important step on the way of turning of structural materials’ SSS diagnostic methods and means from the spectacular but self-sufficient (and therefore useless) field of diagnostics into an effective - really necessary and useful - tool for actual state assessment of structural materials and structures.