FATIGUE OF METALLIC MATERIALS AND STRUCTURES VS. COMPOSITES IN GENERAL STRENGTH THEORY

by

© Ph. D. & Dr. Sc. Lev Gelimson

RUAG Aerospace Services GmbH, Germany

Mechanical Monograph

The “Collegium” All World Academy of Sciences Publishers,

Munich (Germany), 2006

Abstract: Critical (limiting, ultimate) state criteria in solid mechanics and, consequently, so-called strength theories, by their physical sense, should be universal laws of nature. But well-known strength criteria are separate for diverse materials types, have nothing in common with simple and universal fundamental laws of nature, and possess evident defects. For known criteria applicable to some very special cases only, generalization and correction methods to consider any (ductile or brittle, isotropic or anisotropic) materials under stationary or variable loading are founded. For this, scalar or vectorial reduced relative stresses are introduced that are equal to the reciprocals to sign-preserving individual safety factors. Each of these relative stresses is defined as a usual stress divided by the modulus of its limit (ultimate value) of the same sign by vanishing all the remaining stresses under the same other loading conditions. For the first time, general strength theory is developed and entire hierarchies of universal strength laws of nature are discovered.

1. CRITICAL STATE CRITERIA

FOR AN ISOTROPIC DUCTILE MATERIAL

WITH EQUAL STRENGTH IN TENSION AND COMPRESSION

A critical state criterion for an isotropic ductile material is

σ_e = F(σ₁ , σ₂ , σ₃) = σ_L (1)

where σ_e is the equidangerous uniaxial tensile stress that is equivalent to the triaxial stress state σ₁ , σ₂ , σ₃ under consideration by the degree of the danger of the closest critical state (by yield, failure, etc.); F is a certain function of the principal stresses σ₁ , σ₂ , σ₃ (σ₁ ≥ σ₂ ≥ σ₃ by vanishing the shear stresses τ₁₂ , τ₂₃ , τ₃₁) and possibly of some material constants; σ_L is the uniaxial limiting stress in a corresponding critical state such as yield or failure of a solid's material, e.g., the yield stress, σ_y , or the ultimate strength, σ_u . The safety factor

n_L = σ_L/σ_e = σ_L/F(σ₁ , σ₂ , σ₃) (2)

in every triaxial stress state σ₁ , σ₂ , σ₃ is equal to 1 if and only if the state is critical. Function F in formulas (1) and (2) is specified by diverse criteria. For the theory of maximum shearing stresses [1-3], of potential energy of distortion [1-3-8], and the Hosford’s criterion [5],

σ_e = σ₁ - σ₃ = σ_L , (3)

σ_e = σ_i ={[(σ₁ - σ₂)² + (σ₂ - σ₃)² + (σ₃ - σ₁)²]/2}^1/2 = σ_L (σ_i the intensity of stresses), (4)

σ_e = σ_i ={[(σ₁ - σ₂)^γ + (σ₂ - σ₃)^γ + (σ₃ - σ₁)^γ]/2}^1/γ = σ_L (γ constant). (5)

2. UNIVERSALIZATION OF CRITICAL STATE CRITERIA

BY TRANSFORMING THE PRINCIPAL STRESSES

For criterion (3), its formula includes the specific value of the only constant σ_L for a given isotropic ductile material. Suppose criterion (3) be a specific manifestation of some unknown universal law of nature as applied to a given material. It is not the formula (3) itself that allows no generalization and so needs a transformation. Divide each principal stress by the modulus σl of its ultimate value ±σ_L in uniaxlal state:

σ₁⁰ = σ₁/σ_L , σ₂⁰ = σ₂/σ_L , σ₃⁰ = σ₃/σ_L , σ_e⁰ = σ_e/σ_L = n_L^-1. (6)

Transformed criteria (3), (4), (5), and, generally, (1), respectively, become

σ_e/σ_L = σ₁/σ_L - σ₃/σ_L = σ_L/σ_L = 1; σ_e⁰ = σ₁⁰ - σ₃⁰ = 1, (7)

σ_e⁰ = σ_i⁰ = {[(σ₁⁰ - σ₂⁰)² + (σ₂⁰ - σ₃⁰)² + (σ₃⁰ - σ₁⁰)²]/2}^1/2 = 1, (8)

σ_e⁰ = {[(σ₁⁰ - σ₂⁰)^γ + (σ₂⁰ - σ₃⁰)^γ + (σ₃⁰ - σ₁⁰)^γ]/2}^1/γ = 1, (9)

σ_e⁰ = F(σ₁⁰, σ₂⁰, σ₃⁰) = 1. (10)

It is important for their verifying that criteria (7) - (9) coincide with each other when σ₂⁰ is equal to σ₁⁰ or σ₃⁰. Transformation (6) is independent of any criterion. Transformed criteria (7) - (10) have no evident material constant and so allow to impart a generalized sense (in comparison with formulas (6)) to the reduced (relative) principal stresses σ₁⁰, σ₂⁰, σ₃⁰ according to the specific character of the strength of any given material.

3. CRITICAL STATE CRITERIA

FOR AN ISOTROPIC BRITTLE MATERIAL

WITH UNEQUAL STRENGTH IN TENSION AND COMPRESSION

Again, divide each principal stress by the modulus of its ultimate value in the corresponding uniaxial state, namely σ_t in tension and σ_c > 0 in compression:

σ_j⁰ = σ_j/σ_t (σ_j ≥ 0), σ_j⁰ = σ_j/σ_c (σ_j ≤ 0); j ∈ {1, 2, 3, e}. (11)

Transformation (11) 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. Test (11) 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) collected in [2]. Dividing the principal stresses by σ_t [2] unifies all data (the points) without their separation for diverse materials only in the first quadrant σ₁ ≥ 0, σ₂ ≥ 0 (Fig. 1) but not in the fourth one σ₁ ≥ 0, σ₂ ≤ 0 (Fig. 2 with straight line (3) and curve (4)) that are both borrowed from [2]. However, reduction (11) unifies all the data (and they can be approximated by criteria (7, the broken line) and (8, the curve)) in both the first and the fourth quadrants (see Fig. 1).

Figs. 1, 2

Therefore, transformation (11) is adequate and allows to use universal criteria (7) - (10) for any isotropic brittle material. (Further we shall call a solid material natural one.) Their expressions are usual and invariant in the space of the reduced (relative) principal stresses, σ₁⁰, σ₂⁰, σ₃⁰, but depend on the combination of the signs of the usual principal stresses, σ₁⁰, σ₂⁰, σ₃⁰, in their space. The corresponding expressions of criterion (7)

σ_e = σ₁ - σ₃ = σ_c (0 ≥ σ₁ ≥ σ₂ ≥ σ₃);

σ_e = σ₁ - χσ₃ = σ_t (σ₁ ≥ 0 ≥ σ₃ , χ = σ_t/σ_c);

σ_e = σ₁ - σ₃ = σ_t (σ₁ ≥ σ₂ ≥ σ₃ ≥ 0)

have obvious physical sense: If the signs of all the nonzero principal stresses are identical, a natural material having unequal strength in tension and compression is similar to the two model materials. Each of them has equal moduli (either σ_t or σ_c , both instead of σ_L in criterion (3)) of the limiting stresses in tension and compression and is used when all the principal stresses are either nonnegative or nonpositive, respectively. If there are principal stresses with distinct signs, the critical states of a natural material are described by criterion (7) that coincides with Coulomb’s criterion [2,6] as the linear approximation of Mohr’s theory [2,6]

σ_e = σ₁ - χσ₃ = σ_t (12)

in this case only. This can be regarded as the suggestion to determine the applicability range of that theory, which is not quite obvious [6]. In fact, the well-known method [6] to obtain formula (12) is latently based on the distinction between the signs of the principal stresses. Otherwise, the proportional increase of the corresponding Mohr circle based on segment [σ₃ , σ₁] results in its contacts with the approximating straight line (12) not between its contacts with the Mohr circles based on segments [-σ_c , 0] and [0, σ_t]. Furthermore, if σ₁ < 0, then the equivalent stress, σ_e , by (12) can be negative and should be compared not with σ_t > 0 but with -σ_c < 0 like it is in the theory of maximum normal stresses [2,3,6]

-σ_c ≤ σ_j ≤ σ_t (j = 1, 2, 3). (13)

Besides that, by criterion (12), each stress state with σ₁ = χσ₃ seems to be quite safe, and uniform triaxial compression does not. Such contradictions do not concern criterion (7). The last is natural if the signs of the nonzero principal stresses are identical and coincides with (12) if not all these signs coincide, in particular in biaxial stress state, and is corroborated by verifying (12). If σ_t = σ_c , then criterion (13) is transformed by (11) into criterion

σ_e⁰ = max{|σ₁⁰|, |σ₂⁰|, |σ₃⁰|} = 1 (14)

that is equivalent to (13) for any σ_t and σ_c . For the criterion by Pisarenko and Lebedev [2]

σ_e⁰ = (1 - χ)σ₁ + χσ_i = σ_t , (15)

σ_e⁰ = (1 - χ)max{|σ₁⁰|, |σ₂⁰|, |σ₃⁰|} + χσ_i⁰ = 1. (16)

In the space of σ₁⁰, σ₂⁰, σ₃⁰, the physical sense, expression, and graphical interpretation of each general criterion (10) are usual and invariant. In the space of σ₁ , σ₂ , σ₃ , the expression and graphical interpretation of (10) are not invariant and can be obtained from their invariant images by inverting the transformation (11). In the both cases (in particular in Figs. 1 and 2), the continuity of the corresponding geometrical images is achieved by introducing the unregulated principal stresses, σ_1u , σ_2u , σ_3u , instead of σ₁ ≥ σ₂ ≥ σ₃ (similarly to [12]) and the unregulated reduced (relative) principal stresses, σ_1u⁰, σ_2u⁰, σ_3u⁰, instead of σ₁⁰ ≥ σ₂⁰ ≥ σ₃⁰. In particular, criterion (7) is represented in the space of σ_1u , σ_2u , σ_3u by two similar semi-infinite hexahedral Tresca’s prisms placed out of the layer -σ_c ≤ σ_1u + σ_2u + σ_3u ≤ σ_t and connected by Mohr’s truncated pyramid within this layer; criterion (8) is represented by circumscribing two semi-infinite von Mises’ cylinders and the connecting truncated cone (Fig. 3). In biaxial state σ_1u , σ_2u (σ_3u = 0), criteria (3, interrupted broken line), (4, interrupted curve), (7, broken line), and (8, curve) are interpreted in the plane (Fig. 4); (7) and (12) coincide. There are local failures of convexity in the limiting surfaces in the space of σ_1u , σ_2u , σ_3u but those failures of Drucker’s postulate [7] in plasticity theory are directly proportional to the degree of the brittleness of the material and are well-known even in biaxial state for gypsum, graphite, and crystalline glass [8]. Perhaps a certain convexity postulate in the reduced (relative) principal stresses holds. Criteria (7) and (8) give values 1/(1 + χ) and (1 + χ + χ²)^-1/2, respectively, for the ratio of pure shear ultimate strength τ_u to σ_t standing many tests.

Fig. 3

Fig. 4

4. CRITICAL STATE CRITERIA FOR AN ORTHOTROPIC MATERIAL WHEN ITS BASIC DIRECTIONS COINCIDE

WITH THE PRINCIPAL DIRECTIONS OF A STRESS STATE

Generalize (11) for such a material with limiting stresses σ_t1 , σ_t2 , σ_t3 in uniaxial tensions, σ_c1 , σ_c2 , σ_c3 in uniaxial compressions in the basic and simultaneously principal directions 1, 2, 3:

σ_j⁰ = σ_j/σ_tj (σ_j ≥ 0), σ_j⁰ = σ_j/σ_cj (σ_j ≤ 0); j ∈ {1, 2, 3, e}. (17)

Universal criteria (7) - (10), (14), and (16) in the reduced principal stresses are invariant. But (17) does not keep the regulation of the principal stresses, and their renumbering that gives

σ₁⁰ ≥ σ₂⁰ ≥ σ₃⁰ (18)

is necessary for criterion (7) but not essential for (8). A particular case of (17) was perhaps implied by the Hu and Marin generalization [9] of criterion (4) as applied to yielding an orthotropic material having equal yield stress σ_yj in tensions and compressions [12]:

σ₁²/(σ_y1)² + σ₂²/(σ_y2)² + σ₃²/(σ_y3)² - σ₁σ₂/(σ_y1σ_y2) - σ₂σ₃/(σ_y2σ_y3) - σ₃σ₁/(σ_y3σ_y1) = 1.

If such a material has unequal strength in tension and compression in some direction, then generalization (8) of criterion (4) in σ₁ , σ₂ , σ₃, depends on their signs combination. For instance, if σ₁ ≥ 0, σ₂ ≤ 0, and σ₃ ≤ 0, then

(σ_e⁰)² = σ₁²/(σ_t1)² + σ₂²/(σ_c2)² + σ₃²/(σ_c3)² - σ₁σ₂/(σ_t1σ_c2) - σ₂σ₃/(σ_c2σ_c3) - σ₃σ₁/(σ_c3σ_t1) = 1.

In biaxial stress state σ₁ ≥ σ₂ = 0 ≥ σ₃ , this criterion (σ_e⁰)² = σ₁²/(σ_t1)² - σ₁σ₃/(σ_t1σ_c3) + σ₃²/(σ_c3)² = 1 approximates the experimental data on tubes of textolite and bakelite with cotton [3].

5. CRITICAL STATE CRITERIA FOR ANY ANISOTROPIC MATERIAL AND ARBITRARY STATIC LOADING

In this case, the reduction (17) can be generalized by transformation

σ_j⁰ = σ_j/|σ_Lj|. (19)

where σ_Lj is the limiting value of a sole (uniaxial) principal stress σ_j . That value has the direction and sign of stress σ_j and acts at the same solid’s point under the same other loading conditions. This is a new generalized re-comprehension of reduction (17) if σ_tj and σ_cj mean the limiting stresses in tension and compression both in the direction of the principal stress σ_j but not indispensably in the basic directions of the anisotropic material, which are not obliged to exist. If a material is orthotropic, these limiting stresses in every direction might be determined in the way [6]. In contrast to the well-known criteria [1-3], universal critical state criteria in the reduced (relative) principal stresses always conserve their simple forms like all fundamental laws of nature.

Transformation (19) 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, σ_aj (Fig. 5), reaches its peak, σ_ajmax , in the diagram when the cycle is asymmetric and possibly the cycle mean stress, σ_m0j , is nonzero.

Fig. 5

So the stress state when the principal stresses, σ_j = σ_m0j , j = 1, 2, 3, as opposed to the zero stress state (with σ_j = 0), can be considered as initial one instead of the zero state. 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 instead of (19),

σ_j⁰ = (σ_j - σ_m0j)/|σ_Lj - σ_m0j| (20)

generalizes (18) that is a particular case of (20) by σ_m0j = 0. By σ_m0j = (σ_tj - σ_cj)/2, (20) gives

σ_j⁰ =(2σ_j + σ_cj - σ_tj)/(σ_tj + σ_cj) (21)

that coincides with Lode-Nadai’s parameter [10]

μ_σ = (2σ₂ - σ₁ - σ₃)/(σ₁ - σ₃) (22)

when σ₁ = σ_tj , σ₂ = σ_j , σ₃ = -σ_cj , -σ_cj ≤ σ_j ≤ σ_tj (these inequalities not necessarily hold in triaxial stress). If a material is isotropic, indexation of σ_tj and σ_cj by j becomes unnecessary.

6. CRITICAL PROCESS CRITERIA FOR ANY

ANISOTROPIC MATERIAL AND ARBITRARY VARIABLE LOADING

As always, the unique postulate is: If a stress process at any solid’s point is critical then, for every critical state criterion, the corresponding functional dependence between the reduced (relative) principal stresses is universal.

We can use any initial critical state criterion (1) 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, σ_j(t), by a stationary numeration on the whole time interval T = [t₀ , t₁] (t₀ ≤ t ≤ t₁) of loading, the reserve (safety) factor n_j for the uniaxial stress process σ_j(t) is obtained from the condition that the uniaxial stress process n_jσ_j(t) is limiting. This process is similar to the realized one, σ_j(t), and 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 t ∈ T = [t₀ , t₁] just as the values of these processes are proportional to each other. At every moment t , it is possible to take into account the damage accumulation in limiting uniaxial stress process n_jσ_j(t’) on the previous time subinterval [t₀ , t];

2) each principal stress σj(t) is synchronously reduced to

σ_j⁰(t) = [σ_j(t) - σ_m0j(t)]/|σ_Lj(t) - σ_m0j(t)| (23)

where σ_m0j(t) is the mean cycle stress that secures the maximum limiting amplitude of stresses, σ_ajmax(t), at the same moment t ∈ T = [t₀ , t₁] in a cycle of the value of the sole (uniaxial) principal stress σ_j(t) only in the stationary direction coinciding with the variable direction of the stress σ_j(t) at the selected moment t only, if the triaxial stress state σ_m01(t), σ_m02(t), σ_m03(t) is considered initial;

σ_m0j(t) = 0 for every moment t ∈ T = [t₀ , t₁] and for each j = 1, 2, 3 if the zero stress state is regarded as the initial one;

σ_Lj(t) is the limiting value of the uniaxial stress in the stationary direction coinciding with the variable direction of the principal stress σ_j(t) just at the selected moment t ∈ T = [t₀ , t₁], the sign of the value coinciding with the sign of the difference σ_j(t) - σ_m0j(t) just at the selected moment t , under the same other loading conditions such as speed, temperature, radiation level, etc. At every moment t ∈ T = [t₀ , t₁], it is possible to take into account the damage accumulation in uniaxial stress process σ_Lj(t’) on the previous time subinterval [t₀ , t];

3) the stationary reduced (relative) mean stress

σ_mj⁰ = [sup_t∈Tσ_j⁰(t) + inf_t∈Tσ_j⁰(t)]/2

using its least upper bound and greatest lower one in uniaxial stress process σ_j⁰(t) in a model material is determined;

4) in a model material, a stationary reduced amplitude stress σ_aj⁰ in the uniaxial stress cycle, whose mean stress is just σ_mj⁰ and which is as safe as the uniaxial stress process σ_j(t) in a natural material on the time interval T = [t₀ , t₁], is determined as sup σ_aj⁰⁰. These values σ_aj⁰⁰ are such that every uniaxial stress process σ_j’(t), which has a direction coinciding with the direction of the real process σ_j(t) at every moment t ∈ T = [t₀ , t₁] in the natural material and is reduced by transformation (23) to some cycle having the mean stress σ_mj⁰ and the amplitude stress σ_aj⁰ in the model material, has a safety factor not less than n_j . If the set of such amplitude stresses σ_aj⁰⁰ is empty (there is no σ_aj⁰⁰ ≥ 0), then the mean cycle stress σ_mj⁰ in the model material is so changed that the modulus (absolute value) of the change is as small as possible, and we take σ_aj⁰ = 0.

Let us assume that the both limiting stresses σ_Lj(t) (σ_tj(t) in tension and σ_cj(t) 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 (Figs. 6 and 7) of the limiting stresses (the maximum σjmax and the minimum σjmin) in the uniaxlal stress cycle, whose direction coincides with the direction of the principal stress σj(t) at every moment t ∈ T = [t₀ , t₁] and which is as safe as the stress σ_j(t), can be directly determined. Then both the abscissa and the ordinate of each point in the diagram are separately reduced by the transformation (23) and give the interrupted curve consisting of the upper branch for σ_jmax(t) and the lower branch for σ_jmin(t) (Fig. 6 by σ_m0j = 0 and Fig. 7 by σ_m0j ≠ 0). 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 or their ordinates are invariant and the middle of the segment connecting those replaced points lies on the principal diagonal σ_j = σ_m0j of the first and the third quadrants. The obtained diagram of the limiting stresses σ_jmax⁰ and σ_jmin⁰ (continuous lines in Figs. 6 and 7) allows to determine Haigh's diagram [11] of the limiting amplitude stresses σ_aj⁰ (see Figs. 6 and 7) by halving that difference for each abscissa and to directly obtain the amplitude σ_aj⁰ of the uniaxial stress cycle;

Fig. 6

Fig. 7

5) the stationary vector

σ_j⁰ = (σ_mj⁰, σ_aj⁰) (24)

for the model material in the limiting amplitude diagram (see Figs. 6 and 7) in the cycle, which is as safe as the variable uniaxial stress process σ_j(t) in the natural material, is determined;

6) the range of triaxial stress processes stationarily safe

σ_es⁰(t) = max_ju(t)F[σ_1u⁰(t), σ_2u⁰(t), σ_3u⁰(t)] = max_ju(t)F[_{j = 1}³σ_ju⁰(t)] ≤ 1

is determined using the synchronous values of the uniaxial principal stress processes σ₁(t), σ₂(t), σ₃(t). Such a permutation (depending on t) 1u , 2u , 3u of indices 1, 2, 3 at every moment t that function F takes its maximum value should be chosen to satisfy inequalities (18). The stress state process at any moment t is considered stationary. Introduce designating a set of indexed elements, in particular, of arguments of a function. On T = [t₀ , t₁] (Fig. 8), the range is σ_es⁰ = sup_t∈Tσ_es⁰(t) ≤ 1;

Fig. 8

7) the range of triaxial stress processes variably safe at the natural solid’s point under consideration (Fig. 9)

Fig. 9

σ_ev⁰ = max_ju|F[_{j = 1}³σ_ju⁰]| ≤ 1

is determined according to the rules of vector algebra by selecting such a stationary permutation 1u , 2u , 3u of the indexes 1, 2, 3 that the modulus is maximum. In the modulus of possibly vectorial function F , the three independent (irrespective 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 σ₁⁰, σ₂⁰, σ₃⁰. Interpreting σ_ju⁰, j = 1, 2, 3, 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 t ∈ T = [t₀ , t₁] 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 [12]), we obtain

σ_e⁰ = max{σ_es⁰, σ_ev⁰} ≤ 1.

So any critical process criterion for any anisotropic material under arbitrary variable loading is

σ_e⁰ = max{σ_es⁰, σ_ev⁰}= 1,

σ_e⁰ = max{sup_t∈Tmax_ju(t)F[_{j = 1}³σ_ju⁰(t)], max_ju|F[_{j = 1}³σ_ju⁰]|} = 1. (25)

Formula (25) for criteria (3) and (4) give known results 1/(1 + χ) and (1 + χ + χ²)^-1/2 [2,3], respectively, for the ratio τ_-1/σ_-1 of the limiting amplitudes of stresses in symmetric cycles under torsion and bending and the well-known formula by Gough and Pollard n_στ^-2 = n_σ^-2 + n_τ^-2 for a cyclic safety factor nστ by bending (with a partial safety factor n_σ) and torsion (with a partial safety factor n_τ) combined.

BASIC RESULTS AND CONCLUSIONS

1. The known strength criteria have nothing in common with universal laws of nature. They have evident contradictions, restricted and vague ranges of adequacy and applicability.

2. The proposed general strength theory including generalization and correction methods for critical state criteria and safety factors gives entire hierarchies of universal strength laws of nature. They are applicable to the general case of any anisotropic solid with different strengths in tensions and compressions in any directions under arbitrary nonstationary loading.

BIBLIOGRAPHY

[1] Handbuch Struktur-Berechnung (1998), Industrie-Ausschuss-Struktur-Berechnungsunterlagen, Prof. Dr.-Ing. Schwarmann, L. (Ed.), Bremen.

[2] Pisarenko, G. S. and Lebedev, A. A. (1976), Deformation and Strength of Materials with a Complex Stress State [In Russian], Naukova Dumka Publishers, Kiev.

[3] Lebedev, A. A., Kovalchuk, B. I., Giginyak, F. F., and Lamashevskii, V. P. (1983), Mechanical Properties of Structural Materials with a Complex Stress State [In Russian], Naukova Dumka Publishers, Kiev.

[4] Bridgman, P. W. (1964), Collected Experimental Papers, Vols. 1 to 7, Harvard University Press, Cambridge, Massachusetts.

[5] Hosford, W. F. (1972), A generalized isotropic yield criterion, Trans. A. S. M. E., Ser. E 2, p. 290-292.

[6] Goldenblatt, I. I. and Kopnov, V. A. (1968), Strength and Plasticity Criteria for Structural Materials [In Russian], Mashinostroenie Publishers, Moscow.

[7] Drucker, D. C. (1950), Some implication of work hardening and ideal plasticity, Quart. Appl. Math., 7, No. 4, p. 414-418.

[8] Okhrimenko, G. M. (1984), Strength or porcelain in biaxial compression, Problemy Prochnosti, 5, p. 70-75.

[9] Marin, J. (1957), Theories of strength for combined stresses and nonisotropic materials, J. Aeronaut. Sci., 4.

[10] Nadai, A. L. (1950), Theory of Flow and Fracture of Solids, McGraw-Hill, N. Y.

[11] Forrest, P. G. (1962), Fatigue or Metals, Pergamon Press, Oxford.

[12] Haigh (1920), The strain energy function and the elastic limit, Engineering, 109, p. 158.

[13] Timoshenko, S. P. (1956), Strength of Materials, Van Nostrand Co., Inc., Princeton, N. J.

[14] Lev Gelimson (2004), Elastic Mathematics. General Strength Theory, The ”Collegium” International Academy of Sciences Publishers, Munich (Germany).

[15] Lev Gelimson (2005), Providing helicopter fatigue strength: flight conditions [elastic mathematics] . In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Vol. II, p. 405-416, Dalle Donne, C. (Ed.), Hamburg.

[16] Lev Gelimson (2003), General Strength Theory. In: Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin, 3, p. 56-62, Mairanowski, V. (Ed.), Berlin.

[17] Lev Gelimson (2005), Providing helicopter fatigue strength: unit loads [general strength theory]. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Vol. II, p. 589-600, Dalle Donne, C. (Ed.), Hamburg.