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Прикладная математика и механика, 2022, T. 86, № 4, стр. 477-487

О НЕКОТОРЫХ КРИТЕРИЯХ ПРЕДЕЛЬНОГО СОСТОЯНИЯ В МЕХАНИКЕ ТВЕРДОГО ТЕЛА

Х. Альтенбах a, *

a Институт механики, Отто фон Герике университет Магдебурга
Магдебург, Германия

* E-mail: holm.altenbach@ovhgu.de

Поступила в редакцию 01.05.2022
После доработки 15.05.2022
Принята к публикации 25.05.2022

Полный текст (PDF)

Аннотация

Предельные состояния поведения материала могут быть различными. В литературе предельные состояния определяются, в частности, как переход от упругого к неупругому диапазону (например, предел текучести), а также как потеря прочности. Для описания предельных состояний используются понятия напряжения, деформации, а также энергии и мощности. Доминирующими до сих пор являются модели, основанные на напряжениях. Одним из наиболее важных вопросов здесь является формулировка гипотез эквивалентности, которые позволяют сравнивать материальные параметры, которые сами являются скалярными величинами, с соответствующими эквивалентными величинами на основе тензора напряжений. В статье представлены некоторые гипотезы эквивалентности, возможная классификация и открытые вопросы.

Ключевые слова: критерии предельного состояния, эквивалентные напряжения, основные испытания материалов

1. Motivation. Since the time of Galilei, Hooke and others, the question of limit behavior has been in the context of modeling material behavior. External loadings cause reactions in the material which, within the framework of phenomenological approaches, can be expressed by stresses, strains, but also by energy or power. From a certain stress level, depending on the material and external conditions, one can observe phenomena of material failure such as yielding or brittle fracture [1, 2]. The description of failure modes is complicated, especially since there is no rational approach to it. The reasons for this include the different failure modes (plastic flow, creep, fatigue, fracture, etc.) as well as the multitude of influencing factors (stress level, temporal change of the stresses, temperature, environmental influences, etc.). Numerous experimental studies have shown that the stress level and the type of loading have a particular influence with respect to brittle or ductile failure [3–10], among others. Failure states in the narrower sense are all critical states immediately before fracture (brittle materials) or the transition to plastic flow (ductile materials) [39]. The models of the limit behavior are connected with the formulation of limit criteria (e.g. in the stress space). In the first case (brittle failure) one speaks of strength criteria, in the second case (plastic flow) of yield or plasticity criteria. A discussion of failure criteria in the broader sense (consideration of material damage, failure due to cyclic loading) was made in the following references [11–16], among others.

The formulation of phenomenological theories of failure is controversial. In addition to the view that failure cannot, in principle, be described within the framework of phenomenological models, one also finds special articles and monographs which demonstrate the possibilities of phenomenological concepts and in which failure criteria are formulated primarily for isotropic materials [17–28]. Depending on the number of basic experiments (or the parameters to be determined), criteria can be classified [28]. Thus, 1-parameter criteria, 2-parameter criteria, etc. are obtained.

In the present paper we discuss phenomenological limit criteria. The following limiting assumptions are made:

• The considered materials are macroscopically isotropic and homogeneous.

• The state of stress is characterized exclusively by the resulting state of stress in the material, non-mechanical influencing factors are not taken into account.

• The material is monotonically loaded.

• Only plastic flow or loss of strength (fracture) are accepted as possible failure states.

The following discussions cover some classic criteria, although the goal is not to give a score on the criteria. The main idea is to introduce the criteria and make a possible classification. Before formulation the criteria, some elementary mathematical basics must be introduced.

2. Mathematical Basics. Restricting ourself to stress-based limit criteria for isotropic materials and static monotonous loading, the stress tensor invariants and principal stresses, among others, are of particular importance. Besides the stress tensor invariants, the stress deviator invariants, but also other quantities sometimes should be included in the formulation of criteria. The mathematical fundamentals are briefly given below.

2.1. Principal Stresses and Directions. Let us introduce the stress tensor $\sigma $11. The eigenvalue problem for this tensor is defined by the following equation

(2.1.1)
$\sigma \cdot {\mathbf{a}} = \lambda {\mathbf{a}}\quad with\quad {\mathbf{a}} \ne {\mathbf{0}}$
${\mathbf{a}}$ and $\lambda $ are the eigenvector and the eigenvalue of the problem, · denotes the scalar product. The solution can estimated from
(2.1.2)
$(\sigma - \lambda {\mathbf{I}}) \cdot {\mathbf{n}} = {\mathbf{0}}\quad with\quad {\mathbf{n}} \cdot {\mathbf{n}} = 1$
${\mathbf{n}}$ is a orthonormal eigenvector and I is the sekond rank unit tensor. The characteristic equation yields the eigenvalues

(2.1.3)
$\det (\sigma - \lambda {\mathbf{I}}) = 0$

As shown, for example [29], in the case of symmetric stress tensor one distinguish three real solutions following from

(2.1.4)
${{\lambda }^{3}} - {{\lambda }^{2}}{{J}_{1}}(\sigma ) + \lambda {{J}_{2}}(\sigma ) - {{J}_{3}}(\sigma ) = 0$
with the principal invariants

(2.1.5)
$\begin{gathered} {{J}_{1}}(\sigma ) = \sigma \cdot \cdot \,{\mathbf{I}} = {\text{tr}}\sigma \\ {{J}_{2}}(\sigma ) = \frac{1}{2}[J_{1}^{2}(\sigma ) - {{J}_{1}}({{\sigma }^{2}})] \\ {{J}_{3}}(\sigma ) = \det \sigma = \frac{1}{3}\left[ {{{J}_{1}}({{\sigma }^{3}}) + 3{{J}_{1}}(\sigma ){{J}_{2}}(\sigma ) - J_{1}^{3}(\sigma )} \right] = \\ = \frac{1}{3}\left[ {{{J}_{1}}({{\sigma }^{3}}) - \frac{1}{2}{{J}_{1}}({{\sigma }^{2}}){{J}_{1}}(\sigma ) + \frac{1}{6}J_{1}^{3}(\sigma )} \right] \\ \end{gathered} $

Equation (2.1.) has three solutions (principal stresses) ${{\sigma }_{{\text{I}}}} \geqslant {{\sigma }_{{{\text{II}}}}} \geqslant {{\sigma }_{{{\text{III}}}}}$:

• three distinguished solutions

• one distinguished solution and one double solution and

• one triple solution.

In the first case one can identify three principal directions, in the second case one principal direction and in the third case no principal direction.

2.2. Other Invariants. Instead of Eqs. (2.1.5) other sets of invariants are possible [12, 28]. Each set of three linear-independent invariants is a unique set of invariants, so one can use the invariant sets equally. An important alternative set of invariants are the so-called irreducible invariants [12, 30, 31]:

(2.2.1)
$\begin{gathered} {{I}_{1}}(\sigma ) = \operatorname{tr} \sigma = \sigma \cdot \cdot \,{\mathbf{I}}\quad linear\;invariant \\ {{I}_{2}}(\sigma ) = \sigma \cdot \cdot \,\sigma \quad quadratic\;invariant \\ {{I}_{3}}(\sigma ) = (\sigma \cdot \sigma ) \cdot \cdot \,\sigma \quad cubic\;invariant \\ \end{gathered} $
The conversion of the principal invariants to the irreducible invariants and vice versa are given in [12, 30], among others. Further invariant were introduced by Novozhilov [32, 33] or Lode [34, 35], among others.

Finally, let us present the principal invariants for the stress deviator

(2.2.2)
${\mathbf{s}} = \sigma - \frac{1}{3}\operatorname{tr} \sigma {\mathbf{I}}$

Now we have

(2.2.3)
$\begin{gathered} {{J}_{1}}({\mathbf{s}}) = {\mathbf{s}} \cdot \cdot \,{\mathbf{I}} = 0 \\ {{J}_{2}}({\mathbf{s}}) = - \frac{1}{2}{{J}_{1}}({{{\mathbf{s}}}^{2}}) \\ {{J}_{3}}({\mathbf{s}}) = \det {\mathbf{s}} = \frac{1}{3}{{J}_{1}}({{{\mathbf{s}}}^{3}}) \\ \end{gathered} $

3. Classical Criteria

3.1. The Very First Criteria. Possibly the most widespread equivalence hypothesis is the hypothesis of Huber [36, 37], von Mises [38] and Hencky [39]

(3.1.1)
${{\sigma }_{{{\text{eq}}}}} = {{\sigma }_{{{\text{vM}}}}} = \sqrt {\frac{3}{2}{\mathbf{s}} \cdot \cdot \,{\mathbf{s}}} $

This condition was earlier formulated by J.C. Maxwell in a private letter to prospect Lord Kelvin [40]. One possible interpretation of the criterion (3.1.1) is that maximum distortion energy yields a limit state (yielding of a ductile material) which begins when the second invariant of the deviatoric stress ${{J}_{2}}({\mathbf{s}})$ reaches a critical value [38]. This critical value can be identified experimentally, for example, in the tension test (${{\sigma }_{{\text{Y}}}}$) or torsion test (${{\tau }_{{\text{Y}}}}$). The critical shear stress ${{\tau }_{Y}}$ is related to the critical tension stress ${{\sigma }_{{\text{Y}}}}$ by

(3.1.2)
${{\tau }_{{\text{Y}}}} = \frac{{{{\sigma }_{{\text{Y}}}}}}{{\sqrt 3 }}$

It is obvious that this criterion does not consider the whole stress state expressed by the stress tensor. One can show that the first principal invariant is ignored which means compressibility of the material is excluded.

Possibly the oldest criterion is the maximum principal stress criterion. If ${{\sigma }_{{\text{I}}}}$ is the maximum principal stress a critical state occurs if [41]

(3.1.3)
${{\sigma }_{{\text{I}}}} \leqslant {{\sigma }_{{\text{T}}}}$

Here ${{\sigma }_{{\text{T}}}}$ is the positive tensile limit stress when failure occurs. The hypothesis is applied to materials which fail with separation fracture, without yielding:

• brittle materials (e.g. gray cast iron or welds) under predominantly static tensile loading and

• brittle and ductile materials under impact loading.

Note that the criterion is valid only in the case of maximum principal tension stresses. Otherwise the criteria contradicts Kachanov’s statement concerning failure under compression [42]. It is obvious that also this criterium does not consider the whole stress state.

Henry Tresca [43] introduced a criterium assuming that the maximum shear stress is responsible for failure

(3.1.4)
${{\sigma }_{{{\text{eq}}}}} = 2{{\tau }_{{{\text{max}}}}}$

With the maximum principal stresses the equivalent stress can be expressed as

(3.1.5)
${{\sigma }_{{{\text{eq}}}}} = {\text{max}}\left( {\left| {{{\sigma }_{{\text{I}}}} - {{\sigma }_{{{\text{II}}}}}} \right|,\left| {{{\sigma }_{{{\text{II}}}}} - {{\sigma }_{{{\text{III}}}}}} \right|,\left| {{{\sigma }_{{{\text{III}}}}} - {{\sigma }_{{\text{I}}}}} \right|} \right)$

The hypothesis is used for ductile materials with sliding fracture.

3.2. Criteria with One and More Parameters. A simple form of systematization of limit state criteria is according to the number of parameters to be determined experimentally for the limit state. However, it should be noted that there may be restrictions on the experiments that can be performed. If one takes the basic tests of materials testing [44], there is only a limited number of tests that produce homogeneous stress or distortion states in the material sample. In the following, we will again restrict ourselves to isotropic material behavior.

3.2.1. One-Parameter Criteria. The so-called one-parameter criteria are particularly simple, since only one test needs to be used.

• Maximum distortion criterion (Maxwell, Huber, von Mises, Hencky)

(3.2.1)
${{\sigma }_{{{\text{eq}}}}} = {{\sigma }_{{{\text{vM}}}}} = \sqrt {\frac{3}{2}{\mathbf{s}} \cdot \cdot \,{\mathbf{s}}} \leqslant {{\sigma }_{{\text{u}}}}$

Here is ${{\sigma }_{{\text{u}}}}$ is the ultimate tensile stress when the limit case occurs. Equation (3.2.1) was presented earlier in this paper as (3.1.1).

• Maximum shear stress criterion (Coulomb, Tresca, St. Venant)

(3.2.2)
${{\tau }_{{{\text{max}}}}} = \frac{1}{2}({{\sigma }_{{\text{I}}}} - {{\sigma }_{{{\text{III}}}}}) \leqslant \frac{1}{2}{{\sigma }_{{\text{u}}}}$
or

(3.2.3)
${{\sigma }_{{{\text{eq}}}}} = ({{\sigma }_{{\text{I}}}} - {{\sigma }_{{{\text{III}}}}}) \leqslant {{\sigma }_{{\text{u}}}}$

This criterion was introduced earlier in this paper by Eq. (3.1.4).

• Maximum principal strain criterion (Lamé, Clebsch, Rankine [41])

(3.2.4)
${{\sigma }_{{{\text{eq}}}}} = {{\sigma }_{{\text{I}}}} - \frac{1}{2}({{\sigma }_{{\text{I}}}} - {{\sigma }_{{{\text{III}}}}}) \leqslant {{\sigma }_{{\text{u}}}}$

• Maximum principal stress criterion (Galilei, Leibniz)

(3.2.5)
${{\sigma }_{{{\text{eq}}}}} = {{\sigma }_{{\text{I}}}} \leqslant {{\sigma }_{{\text{u}}}}$

Note that the criterion is applicable only if ${{\sigma }_{{\text{I}}}} > 0$. In the previous section this criterion was given by Eq. (3.1.3).

• Maximum linear deformation criterion (St. Venant, Bach)

Let us assume the maximal linear strain ${{\varepsilon }_{{\text{I}}}}$ as

(3.2.6)
${{\varepsilon }_{{\text{I}}}} = {{\sigma }_{{\text{I}}}} - \nu ({{\sigma }_{{{\text{II}}}}} + {{\sigma }_{{{\text{III}}}}})$

The criterion is expressed by using the Hooke’s law. Finally, we have

(3.2.7)
${{\sigma }_{{{\text{eq}}}}} = {{\sigma }_{{\text{I}}}} - \nu ({{\sigma }_{{{\text{II}}}}} + {{\sigma }_{{{\text{III}}}}}) \leqslant {{\sigma }_{{\text{u}}}}$

Note that the last expression contains two parameters ($\nu $ and ${{\sigma }_{{\text{u}}}}$), but only one is related to the limit state (${{\sigma }_{{\text{u}}}}$).

• Sdobyrev criterion [45, 46]

(3.2.8)
${{\sigma }_{{{\text{eq}}}}} = \frac{1}{2}({{\sigma }_{{{\text{vM}}}}} + {{\sigma }_{{\text{I}}}}) \leqslant {{\sigma }_{{\text{u}}}}$

The Sdobyrev criterion is an averaging of the maximum distortion criterion and the maximum principal stress criterion. The limitations concerning the maximum principal stress are not presented in the original paper of Sdobyrev.

3.2.2. Two-Parameter Criteria. Now we are looking on two-parameter criteria based on two independent limit state tests.

• Mohr-Coulomb criterion [47–50]

(3.2.9)
${{\sigma }_{{{\text{eq}}}}} = {{\sigma }_{{\text{I}}}} - \xi {{\sigma }_{{{\text{III}}}}} \leqslant {{\sigma }_{{\text{u}}}}$

This criterion is a modification of maximum shear stress criterion (3.2.2). The additional parameter $\xi $ is related to the ratio ${{\sigma }_{{\text{u}}}}{\text{/}}{{\sigma }_{{\text{c}}}}$. That means the compression test should be performed and the ${{\sigma }_{{\text{c}}}}$ as the limit state stress at compression should be estimated. The Mohr–Coulomb criterion describes the response of brittle materials such as concrete, or rubble piles, to shear stress as well as normal stress. The theory applies to materials for which the compressive limit stress far exceeds the tensile limit stress.

• Botkin–Mirolyubov or Drucker–Prager criterion [51–54]

(3.2.10)
${{\sigma }_{{{\text{eq}}}}} = {{\sigma }_{{\text{I}}}} - \chi {{\sigma }_{{{\text{III}}}}} \leqslant {{\sigma }_{{\text{u}}}}$

• Pisarenko–Lebedev criterion [26, 55]

(3.2.11)
${{\sigma }_{{{\text{eq}}}}} = \frac{1}{2}[(1 + \chi ){{\sigma }_{{{\text{vM}}}}} + (1 - \chi ){{I}_{1}}] \leqslant {{\sigma }_{{\text{u}}}}$

This criterion can describe the von Mises equivalent stress and the influence of the hydrostatic stress state. Similar expression were given by Schleicher [56], Klebowski [57] and Nadai [58].

• Sandel criterion [59–61]

(3.2.12)
${{\sigma }_{{{\text{eq}}}}} = {{\sigma }_{{\text{I}}}} - \frac{1}{2}(1 - \chi ){{\sigma }_{{{\text{II}}}}} - \chi {{\sigma }_{{{\text{III}}}}} \leqslant {{\sigma }_{{\text{u}}}}$

The Sandel criterion takes into account all principal stresses instead of the maximum principal stress (3.2.5) or the first and the third principal stress.

• Koval’chuk criterion [62]

(3.2.13)
${{\sigma }_{{{\text{eq}}}}} = \zeta {{\sigma }_{{{\text{vM}}}}} - (1 - \zeta )({{\sigma }_{{\text{I}}}} - {{\sigma }_{{{\text{III}}}}}) \leqslant {{\sigma }_{{\text{u}}}}$

The additional parameter $\zeta $ is defined as $\zeta = 2 - \frac{\varphi }{2} - \sqrt 3 $ with $\varphi = \frac{{{{\sigma }_{{\text{u}}}}}}{{{{\tau }_{{\text{t}}}}}}$ ${{\tau }_{{\text{t}}}}$ is the limit stress under torsion. This criterion is a combination of the von Mises criterion and the maximum shear stress criterion.

3.2.3. Three-Parameter and Four-Parameter Criteria. There are few three and four parameters criteria. The reason for this is the significantly increasing experimental effort. Let us introduce the following two criteria:

• Paul criterion [25, 63]

(3.2.14)
${{\sigma }_{{{\text{eq}}}}} = {{a}_{1}}{{\sigma }_{{\text{I}}}} + {{a}_{2}}{{\sigma }_{{{\text{II}}}}} + {{a}_{3}}{{\sigma }_{{{\text{III}}}}} \leqslant {{\sigma }_{{\text{u}}}}$

Here the three principal stresses are included independently.

• Birger criterion [64]

(3.2.15)
${{\sigma }_{{{\text{eq}}}}} = {{a}_{1}}{{\sigma }_{{\text{I}}}} + {{a}_{2}}{{\sigma }_{{{\text{II}}}}} + {{a}_{3}}{{\sigma }_{{{\text{III}}}}} + {{a}_{4}}{{\sigma }_{{{\text{vM}}}}} \leqslant {{\sigma }_{{\text{u}}}}$

Here the three principal stresses and the von Mises stress are included independently.

3.3. Final Comments. There are suggested much more simple criteria in the literature [7, 65–74], among others. However, it is difficult overall to determine the parameters experimentally. This has led to the use of the simplest possible criterion, where experiments can be easily planned and carried out. The possibilities of extending the classical criteria have been extensively discussed in recent years, with comprehensive studies on the development of new criteria cited in [13–15, 75–80], among others. However, with the increasing number of parameters and necessary tests the uniqueness of parameter identification is under question.

4. One Possible Classification

4.1. Generalized Formulation. Possible classifications of generalized limit criteria are widely discussed in the literature [12, 81] among others. Let us introduce here one simple classification including limit state criteria up to 6 parameters presented in [82]. The starting point is

(4.1.1)
${{\sigma }_{{{\text{eq}}}}} \leqslant {{\sigma }_{{\text{u}}}},$
with ${{\sigma }_{{{\text{eq}}}}}$ as a function of the stress tensor $\sigma $. Assuming isotropic material behavior, the function ${{\sigma }_{{{\text{eq}}}}} = {{\sigma }_{{{\text{eq}}}}}(\sigma )$ can be simplified since in this particular case ${{\sigma }_{{{\text{eq}}}}}$ is a function of three linear independent invariants. Any set of invariants can be used – all are equal. Here we introduce the following invariants

(4.1.2)
${{I}_{1}} = \sigma \cdot \cdot \,{\mathbf{I}},\quad {{\sigma }_{{{\text{vM}}}}} = \sqrt {\frac{3}{2}{\mathbf{s}} \cdot \cdot \,{\mathbf{s}}} ,\quad sin\xi = - \frac{9}{2}\frac{{({\mathbf{s}} \cdot {\mathbf{s}}) \cdot \cdot \,{\mathbf{s}}}}{{\sigma _{{{\text{vM}}}}^{3}}}\quad {\text{with}}\quad \xi \leqslant \frac{\pi }{6}$

The last invariant is similar to the Lode angle [34, 35] and discussed in [32, 33]. The equivalent stress now is defined as

(4.1.3)
${{\sigma }_{{{\text{eq}}}}} = {{\lambda }_{1}}{{\sigma }_{{{\text{vM}}}}}sin\xi + {{\lambda }_{2}}{{\sigma }_{{{\text{vM}}}}}cos\xi + {{\lambda }_{3}}{{\sigma }_{{{\text{vM}}}}} + {{\lambda }_{4}}{{I}_{1}} + {{\lambda }_{5}}{{I}_{1}}sin\xi + {{\lambda }_{6}}{{I}_{1}}cos\xi $
where ${{\lambda }_{i}}$ $(i = 1, \ldots ,6)$ are scalar parameters which should be determined by experiments. Possible experiments are uniaxial tension with the limit stress ${{\sigma }_{{\text{u}}}}$, uniaxial compression with the limit stress ${{\sigma }_{{\text{c}}}}$, pure torsion with the limit stress ${{\tau }_{{\text{u}}}}$, thin-walled tubular specimen under inner pressure, biaxial tension and uniaxial tension under superposed hydrostatic pressure. Only the first three test belongs to the basic tests in mechanical testing of materials [44] realizing homogeneous states. The remaining three tests can be exchanged at will.

4.2. Example. It can be shown that all previous discussed criteria can be expressed by (4.1.3). In the case of the Birger criterion this is shown in [83]. To determine the conditions of macroscopic fracture in a complex stress state, the critical (limit) stress ${{\sigma }_{{{\text{limit}}}}}(T)$ must be compared with the equivalent stress ${{\sigma }_{{{\text{eq}}}}}$. $T$ is the temperature. Let us consider that the temperature is constant. The further presentation of the problem of fracture criteria for a complex stress state will be carried out based on the ideas developed [64], where the value of ${{\sigma }_{{{\text{eq}}}}}$ is defined as a function of principal stresses ${{\sigma }_{i}}$, $i = {\text{I}},{\text{II}},{\text{III}}$, material parameters ${{\lambda }_{j}}$, $j = 0,1,2,3$ and internal state variables ${{\beta }_{k}}$, $k = 1,2$, …, reflecting the loading history and other effects [13, 84]. Rejecting the consideration of internal state variables (no influence of ${{\beta }_{k}}$) and accepting the linear dependence ${{\sigma }_{{{\text{eq}}}}} = f({{\sigma }_{1}},{{\sigma }_{2}},{{\sigma }_{3}})$ we have a four-parameter fracture criterion (in a similar way a six-parameter criterion was introduced in [85])

(4.2.1)
${{\sigma }_{{{\text{eq}}}}} = {{\lambda }_{0}}{{\sigma }_{{{\text{vM}}}}} + {{\lambda }_{1}}{{\sigma }_{1}} + {{\lambda }_{2}}{{\sigma }_{2}} + {{\lambda }_{3}}{{\sigma }_{3}} \leqslant {{\sigma }_{{{\text{limit}}}}} \equiv {{\sigma }_{{\text{u}}}}$

To define ${{\lambda }_{0}}$, ${{\lambda }_{1}}$, ${{\lambda }_{2}}$, ${{\lambda }_{3}}$ four basic experiments22 are required. Using experiments on tension, compression, torsion and testing of thin-walled tube under inner pressure as such experiments, we obtain

(4.2.2)
$\begin{gathered} {{\lambda }_{0}} = \frac{1}{{2 - \sqrt 3 }}\left( {1 + \frac{{{{\sigma }_{{{\text{tension}}}}}}}{{{{\sigma }_{{{\text{compression}}}}}}} - \frac{{{{\sigma }_{{{\text{tension}}}}}}}{{{{\tau }_{{{\text{torsion}}}}}}}} \right) \\ {{\lambda }_{1}} = 1 - {{\lambda }_{0}},\quad {{\lambda }_{2}} = 2\frac{{{{\sigma }_{{{\text{tension}}}}}}}{{{{\sigma }_{{{\text{inner}}\,{\text{pressure}}}}}}} - 1 + \frac{{{{\sigma }_{{{\text{tension}}}}}}}{{{{\sigma }_{{{\text{compression}}}}}}} - \frac{{{{\sigma }_{{{\text{tension}}}}}}}{{{{\tau }_{{{\text{torsion}}}}}}} \\ {{\lambda }_{3}} = \frac{1}{{2 - \sqrt 3 }}\left[ {1 + (\sqrt 3 + 1)\frac{{{{\sigma }_{{{\text{tension}}}}}}}{{{{\sigma }_{{{\text{compression}}}}}}} - \frac{{{{\sigma }_{{{\text{tension}}}}}}}{{{{\tau }_{{{\text{torsion}}}}}}}} \right], \\ \end{gathered} $
where ${{\sigma }_{{{\text{tension}}}}}$, ${{\sigma }_{{{\text{compression}}}}}$, ${{\tau }_{{{\text{torsion}}}}}$ and ${{\sigma }_{{{\text{inner}}{\kern 1pt} {\text{pressure}}}}}$ are ultimate strengths (maximum of the stress-strain diagrams) for the listed types of loading. If we accept the values of the relations ${{\sigma }_{{{\text{tension}}}}}{\text{/}}{{\sigma }_{{{\text{compression}}}}}$, ${{\sigma }_{{{\text{tension}}}}}{\text{/}}{{\tau }_{{{\text{torsion}}}}}$ and ${{\sigma }_{{{\text{tension}}}}}{\text{/}}{{\sigma }_{{{\text{inner}}{\kern 1pt} {\text{pressure}}}}}$ constant, then instead of (4.2.2) we have one-parameter criteria33.

Using as basic two experiments (for example, tension and compression or tension and torsion), from which we determine the ratios $\lambda = {{\sigma }_{{{\text{tension}}}}}{\text{/}}{{\sigma }_{{{\text{compression}}}}}$ or $\lambda = {{\sigma }_{{{\text{tension}}}}}{\text{/}}{{\tau }_{{{\text{torsion}}}}}$, we obtain two-parameter criteria. The most popular criteria are one- and two-parameter criteria. They are obtained from the ratio (4.2.1) at various values of the ratio of strength characteristics, Table 1. Another example is given for the assumed 6-parameter criterion in [81].

Table 1
Criterion Expression $\frac{{{{\sigma }_{{\operatorname{tension} }}}}}{{{{\sigma }_{{{\text{compression}}}}}}}$ $\frac{{{{\sigma }_{{\operatorname{tension} }}}}}{{{{\sigma }_{{{\text{torsion}}}}}}}$ $\frac{{{{\sigma }_{{\operatorname{tension} }}}}}{{{{\sigma }_{{{\text{inner}}\;{\text{pressure}}}}}}}$
1 ${{\sigma }_{{\operatorname{tension} }}} = {{\sigma }_{{\operatorname{vM} }}}$ 1 $\sqrt 3 $ $\frac{{\sqrt 3 }}{2}$
2 ${{\sigma }_{{\operatorname{tension} }}} = {{\sigma }_{1}} - {{\sigma }_{3}} = 2{{\tau }_{{\operatorname{torsion} }}}$ 1 2 1
3 ${{\sigma }_{{\operatorname{tension} }}} = {{\sigma }_{1}}$ 0 1 1
4 ${{\sigma }_{{\operatorname{tension} }}} = {{\sigma }_{1}} - \frac{1}{2}\left( {{{\sigma }_{2}} + {{\sigma }_{3}}} \right)$ $\frac{1}{2}$ $\frac{3}{2}$ $\frac{3}{4}$
5 ${{\sigma }_{{\operatorname{tension} }}} = \frac{1}{2}\left( {{{\sigma }_{{\operatorname{vM} }}} + {{\sigma }_{1}}} \right)$ $\frac{1}{2}$ $\frac{1}{2}\left( {\sqrt 3 + 1} \right)$ $\frac{1}{2}\left( {\sqrt 3 + 2} \right)$
6 ${{\sigma }_{{\operatorname{tension} }}} = {{\sigma }_{1}} - \chi {{\sigma }_{3}}$ $\chi $ $1 + \chi $ 1
7 $\begin{gathered} {{\sigma }_{{\operatorname{tension} }}} = \frac{1}{2}\left( {1 + \chi } \right){{\sigma }_{{\operatorname{vM} }}} + \hfill \\ + \;\frac{1}{2}\left( {1 - \chi } \right)\left( {{{\sigma }_{1}} + {{\sigma }_{2}} + {{\sigma }_{3}}} \right) \hfill \\ \end{gathered} $ $\chi $ $\frac{{\sqrt 3 }}{2}\left( {1 + \chi } \right)$ $\frac{1}{4}\left( {\sqrt 3 \left( {1 + \chi } \right) + 3\left( {1 - \chi } \right)} \right)$
8 ${{\sigma }_{{\operatorname{tension} }}} = \chi {{\sigma }_{{\operatorname{vM} }}} + \left( {1 - \chi } \right){{\sigma }_{1}}$ $\chi $ $1 + \left( {\sqrt 3 - 1} \right) - \chi $ $1 - \left( {1 - \frac{{\sqrt 3 }}{2}} \right)\chi $

1 – Huber, von Mises, Hencky; 2 – Coulomb, Tresca, St. Venant; 3 – maxiumum principal stress (the stress must be positive), Galilei, Leibniz; 4 – maximum principal strain (special case if ν = 0.5), Rankine, Lamé, Clebsch; 5 – Sdobyrev; 6 – Mohr; 7 – Nadai, Botkin–Mirolyubov, Schleicher, Prager–Drucker; 8 – Pisarenko–Lebedev

Conclusions. The formulation of criteria will not lose its importance in the future, since mechanical engineering designs, but also in other applications, require simple engineering concepts for the evaluation of limit states. In contrast to the balance equations of continuum mechanics, which can be rationally justified, the formulation of limit criteria depends on many factors. Material or material, load, but also other factors play a role. Knowledge of the microstructure and its evolution explains some limit cases better, but for a macroscopic analysis one would quickly reach calculation times that are not realistic for industrial practice. Thus, there is also a great need for future research work on the limiting criteria. It would be important to expand the data basis so that more complex models can also find their way into practice.

Further research is also needed to overcome the limits of the isotropic model. The focus should be on two directions:

• Extension in the direction of anisotropic boundary criteria

Here, in particular, the question of mastering the rapidly increasing number of parameters needs to be addressed.

• Cyclic processes

Structures are often subject to cyclic stresses. Consequently, it has to be clarified here whether the equivalent stress expressions can be transferred to this case.

Список литературы

  1. Altenbach H., Krause J., Zolochevsky A. Uber ein verallgemeinertes Versagenskriterium der Theorie des Grenzzustandes isotroper Werkstofe // Techn. Mech., 1991, vol. 12, no. 2, pp. 113–117.

  2. Burth K., Brocks W. Plastizität, Vieweg, Braunschweig, Wiesbaden, 1992.

  3. Bridgman P.W. Studies in Large Plastic Flow and Fracture. N.Y.: McGraw-Hill, 1949.

  4. Brostow W. Science of Materials. Krieger Pub. Co., 1985.

  5. Cristescu N. Rock Rheology. Dordrect: Kluwer, 1989.

  6. French I.E., Weinrich P.F., Weaver C.W. Tensile fracture of free machining brass as a function of hydrostatic pressure // Acta Metall., 1973, vol. 21, no. 8, pp. 1045–1049.

  7. Hayhurst D.R. Creep rupture under multi-axial states of stress // J. Mech.&Phys. Solids, 1972, vol. 20, no. 6, pp. 381–390.

  8. Hunsche U., Albrecht H. Results of true triaxial strength tests on rock salt // Engng. Fracture Mech., 1990, vol. 35, no. 4, pp. 867–877.

  9. Launay P., Gachon H. Strain and ultimate strength of concrete under triaxial stress // Amer. Concrete Inst., 1970, vol. 34, no. 13, pp. 269–282.

  10. Mills L.L., Zimmerman R.M. Compressive strength of plain concrete under multiaxial loading conditions // Amer. Concrete Inst., 1970, vol. 67, no. 10, pp. 802–807.

  11. Altenbach H. Strength hypotheses – a never ending story // Czasopismo Techn., 2010, vol. 107, no. 20, pp. 5–15.

  12. Altenbach H., Altenbach J., Zolochevsky A. Erweiterte Deformationsmodelle und Versagenskriterien der Werkstofmechanik. Deutscher Verlag für Grundstofndustrie, 1995.

  13. Altenbach H., Bolchoun A., Kolupaev V.A. Phenomenological yield and failure criteria, Plasticity of Pressure-Sensitive Materials. Berlin: Springer, 2014. pp. 49–152.

  14. Altenbach H., Kolupaev V.A. Classical and non-classical failure criteria, Failure and Damage Analysis of Advanced Materials // CISM Int. Centre for Mech. Sci., vol. 560, Wien: Springer, 2015, pp. 1–66.

  15. Kolupaev V.A. Equivalent Stress Concept for Limit State Analysis, Advanced Structured Materials. Vol. 86. Cham: Springer, 2018.

  16. Kolupaev V.A., Bolchoun A., Altenbach H. Aktuelle Trends beim Einsatz von Festigkeit-shypothesen // Konstr., 2009, vol. 5, pp. 59–66.

  17. Chen W.F. Plasticity for Reinforced Concrete. N.Y.: McGraw-Hill, 1982.

  18. de Boer R. Failure conditions for brittle and granular materials // Quart. Appl. Math., 1986, vol. 44, no. 1, pp. 71–79.

  19. de Boer R., Dresenkamp H.T. Constitutive equations for concrete in failure state // J. Engng. Mech., 1989, vol. 115, no. 8, pp. 1591–1608.

  20. Gol’denblat I.I., Kopnov V.A. Kriterii prochnosti i plastichnosti konstrukcionnykh materialov // in: Criteria for Strength and Plasticity of Structural Materials. Moscow: Mashinostroenie, 1968.

  21. Goldscheider M. Grenzbedingung und Fließregel von Sand // Mech. Res. Commun., 1976, vol. 3, no. 6, pp. 463–468.

  22. Gross D., Seelig T. Bruchmechanik. Berlin: Springer / Vieweg, 2016.

  23. Kreissig R. Einführung in die Plastizitätstheorie. Leipzig: Fachbuchverlag, 1992.

  24. Ottosen N.S. A failure criterion for concrete // J. Engng. Mech. Division, 1977, vol. 103, no. 4, pp. 527–535.

  25. Paul B. Macroscopic plastic flow and brittle fracture, Fracture: An Advanced Treatise. Vol. II / Ed. by Liebowitz H. N.Y.: Acad. Press, 1968. pp. 313–496.

  26. Pisarenko G.S., Lebedev A.A. Deformation and Strength of Materials under Complex Stress State, Kiev: Naukowa Dumka, 1976.

  27. Sauter J., Wingerter N. Neue und alte statische Festigkeitshypothesen, Fortschritt-Berichte VDI Reihe 1: Konstrukt. / Masch., Vol. 191. Düsseldorf: VDI-Verlag, 1990.

  28. Zyczkowski M. Combined Loadings in the Theory of Plasticity. Warsaw: PWN, 1981.

  29. Lurie A.I. Theory of Elasticity, Foundations of Engineering Mechanics, Berlin: Springer, 2005.

  30. Betten J. Tensorrechnung für Ingenieure. Stuttgart: B.G. Teubner, 1987.

  31. Itskov M. Tensor Algebra and Tensor Analysis for Engineers. Cham: Springer, 2018.

  32. Novozhilov V.V. On the principles of processing the results of static tests of isotropic materials // PMM, 1951, vol. 15, no. 6, pp. 709–722.

  33. Novozhilov V.V. On the relationship between stresses and strains in a nonlinear-elastic medium // PMM, 1951, vol. 15, no. 2, pp. 183–194.

  34. Lode W. Versuche über den Einfuß der mittleren Hauptspannung auf die Fließgrenze // ZAMM J. Appl. Math.&Mech., 1925, vol. 5, no. 2, pp. 142–144.

  35. Lode W. Versuche u¨ber den Einfuß der mittleren Hauptspannung auf das Fließen der Metalle Eisen, Kupfer und Nickel // Zeitschrift fur Physik, 1926, vol. 36, no. 11–12, pp. 913–939.

  36. Huber M.T. Specific strain work as a measure of material effort // Czasopismo Techniczne, Lwow, Organ Towar. Politechn. we Lwowie, 1904, vol. 22, pp. 34–40, 49–50, 61–62, 80–81.

  37. Huber M.T. Specific work of strain as a measure of material effort // Arch. Mech., 2004, vol. 56, no. 3, pp. 173–190.

  38. Hencky H. Zur Theorie plastischer Deformationen und der hierdurch im Material her-vorgerufenen Nachspannungen // ZAMM J. Appl. Math.&Mech., 1924, vol. 4, no. 4, pp. 323–334.

  39. von Mises R. Mechanik des festen Körpers im plastischen deformablen // Zust. Nachrichten der König. Gesellschaft der Wissenschaften Gottingen, Math.-phys. Klasse, 1913, pp. 582–592.

  40. Rychlewski J. Elastic energy decomposition and limit criteria // Engng. Trans., 2014, no. 1, pp. 31–63.

  41. Rankine W.J.M. Manual of Applied Mechanics. L.: Griffin, 1876.

  42. Kachanov L.M. Fundamentals of Fracture Mechanics. Moscow: Nauka, 1974.

  43. Tresca H. Memoire sur l’ecoulement des corps solides // Memoires Pres. par Div. Savants, 1868, vol. 18, pp. 733–799.

  44. Blumenauer H. Werkstofprüfung. Leipzig: Deutscher Verlag für Grundst., 1996.

  45. Sdobyrev V.P. Long-term strength of El-437b alloy in a complex stress state // Izv. AN SSSR. Otd. Tekhn. Nauk, Mekh. i Mashin., 1958, no. 4, pp. 92–97.

  46. Sdobyrev V.P. Criterion for the long term strength of some heat-resistant alloys at a multiaxial loading // Izv. AN SSSR. Otd. Tekhn. Nauk, Mekh. i Mashin., 1959, no. 6, pp. 93–99.

  47. Coulomb C.A. Essai sur une application des regles des maximis et minimis a quelques problemes de statique relatifs, a la architecture // Mem. Acad. Roy. Div. Sav., 1776, vol. 7, pp. 343–387.

  48. Mohr O. Welche Umstände bedingen die Elastizitätsgrenze und den Bruch eines Materials // Zeitschrift des VDI, 1900, vol. 45, pp. 1524–1530.

  49. Mohr O. Welche Umstände bedingen die Elastizitätsgrenze und den Bruch eines Materials // Zeitschrift des VDI, 1900, vol. 46, pp. 1572–1577.

  50. Mohr O. Abhandlungen aus dem Gebiete der Technischen Mechanik. Berlin: Wilhelm&Sohn, 1914.

  51. Botkin A.I. Equilibrium of granular and brittle materials // Trans. Sci. Res. Inst. Hydrotechn., 1940, vol. 28, pp. 189–211.

  52. Botkin A.I. Theories of elastic failure of granular and of brittle materials // Trans. Sci. Res. Inst. Hydrotechn., 1940, vol. 26, pp. 205–236.

  53. Drucker D.C., Prager W. Soil mechanics and plastic analysis or limit design // Quart. Appl. Math., 1952, vol. 10, pp. 157–165.

  54. Mirolyubov I.N. On the generalization of the strengt theory based on the octaedral stresses in the case of brittle materials // Tr. Leningr. Technol. Inst., 1953, pp. 42–52.

  55. Pisarenko G.S., Lebedev A.A. Deformation and Fracture of Materials under Combined Stress. Kiev: Naukowa Dumka, 1969.

  56. Schleicher F. Der Spannungszustand an der Fließgrenze (Plastizitatsbedingung) // ZAMM J. Appl. Math.&Mech., 1926, vol. 6, no. 3, pp. 199–216.

  57. Klebowski Z. Obecny stan wytrzymalosciowego obliczenia materjalow o wlasnosciach uogölnionych // Przeglad Techn., 1934, no. 11, pp. 373–376.

  58. Nadai A. Theory of Flow and Fracture in Solids. N.Y.: McGraw-Hill, 1950.

  59. Sandel G.D. Ueber die Festigkeitsbedingungen: Ein Beitrag zur Lösung der Frage der zulässigen Anstrengung der Konstruktionsmaterialen // Diss. Techn. Hochsch. zu Stuttgart, 1919.

  60. Sandel G.D. Uber die Festigkeitsbedingungen: Ein Beitrag zur Losung der Frage der zulassigen Anstrengung der Konstruktionsmaterialen. Leipzig: Janecke, 1925.

  61. Sandel G.D. Die Anstrengungsfrage // Schweizerische Bauzeitung, 1930, vol. 95, no. 26, pp. 335–338.

  62. Koval’chuk B.I. Criterion for the limiting state of some hull steels under combined strain at room and elevated temperatures // Strength Mater., 1981, vol. 13, no. 5, 548–554.

  63. Paul B. Generalized pyramidal fracture and yield criteria // Int. J. Solids&Struct., 1968, vol. 4, no. 2, pp. 175–196.

  64. Birger I.A. On a criterion for fracture and plasticity // Izv. AN SSSR. MTT, 1977, vol. 9, no. 4, pp. 143–150.

  65. Burzynski W. Study on Material Efort Hypotheses. Lwow: Akad. Nauk Techn, 1928.

  66. Burzynski W. Über die Anstrengungshypothesen // Schweizerische Bauzeitung, 1929, vol. 94, no. 21, pp. 259–262.

  67. Burzynski W. Theoretical foundations of the hypotheses of material effort // Engng. Trans., 2008, vol. 56, Spec. Iss., pp. 9–45.

  68. Ishlinsky A.Yu. Hypothesis of strength of shape change // Uch. Zap. Mosk. Univ. Mekh., 1940, vol. 46, pp. 104–114.

  69. Schmidt R. Über den Zusammenhang von Spannungen und Formanderungen im Verfesti-gungsgebiet // Ingenieur-Archiv, 1932, vol. 3, no. 3, pp. 215–235.

  70. Tarasenko I.I. On criterion of brittle strength of metals // Sb. Nauchn. Tr. Leningr. Inzh.-Stroit. Inst., 1957, vol. 26, pp. 161–166.

  71. Tsvelodub I.Yu. Stability Postulate and Its Application to the Theory of Creep of Metals. Novosibirsk: Inst. Hydrodyn. SB RAS, 1991.

  72. Yagn Yu.I. New methods of strength prediction // Vestn. Inzh. i Tekhn., 1931, vol. 6, pp. 237–244.

  73. Yu M.-H. General behaviour of isotropic yield function // Sci.&Technol. Res. Paper of Xi’an Jiaotong Univ., 1961, pp. 1–11.

  74. Haythornthwaite R.M. Range of yield condition in ideal plasticity // Proc. ASCE. J. Eng. Mech. Division EM6 87, 1961, pp. 117–133.

  75. Altenbach H., Kolupaev, V.A. General forms of limit surface: Application for isotropic materials, Material Modeling and Structural Mechanics / Ed. by Altenbach H., Beitelschmidt M. et al. Cham: Springer Int. Publ., 2022. pp. 19–94.

  76. Advances in strength theories for materials under complex stress state in the 20th century // Appl. Mech. Rev., 2002, vol. 55, no. 5, pp. 169–218.

  77. Yu M.-H. Unifed Strength Theory and Its Applications. Berlin: Springer, 2004.

  78. Yu M.-H. Unifed strength theory (UST), Rock Mechanics and Engineering. Vol. 1: Principles (Leiden) / Ed. by Feng X.-T. Balkema: CRC Press, 2017. pp. 425–452.

  79. Unifed Strength Theory and Its Applications. Singapore: Springer, 2018.

  80. Yu M.-H., Yu S.-Q. Introduction to Unifed Strength Theory. Balkema;L.: CRC Press, 2019.

  81. Kolupaev V.A. 3D-Creep Behaviour of Parts Made of Non-Reinforced Thermoplastics // Diss., Martin-Luther-Universität Halle-Wittenberg, Halle-Saale, 2006.

  82. Altenbach H., Zolochevsky A. A generalized failure criterion for three-dimensional behaviour of isotropic materials // Engng. Fracture Mech., 1996, vpl. 54, no. 1, pp. 75–90.

  83. Getsov L.B. Materials and Strength of Gas Turbine Parts. Vol. 1. Materials, Properties, Damage, Deformation and Fracture Models / Ed. by Altenbach H., Naumenko K. // in: Adv. Struct. Mater., 2021, vol. 150.

  84. Volkov S.D. Basics of the statistical theory of strength // in: Nekotorye problemy prochnosti tverdogo tela / Ed. by Iofe A.F., Kurdjymov G.B., 1959. pp. 325–333.

  85. Altenbach H., Lauschke U., Zolochevsky A.A. Ein verallgemeinertes Versagenskriterium und seine Gegenüberstellung mit Versuchsergebnissen // ZAMM Zeitschrift für angewandte Math.&Mech., 1993, vol. 73, no. 4, T372–T375.

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