Effect of the stress state on ultimate strain energy density in the failure of reinforced epoxy resin

Epoxy materials are used in construction, mechanical engineering, and household appliances as protective coatings, sealers, glues, impregnating and repair materials. In the processes, the structural components may experience a complex stress-strain state under conditions of both low and high temperatures. In view of the trend for an everincreasing use of these materials in state-of-the-art structures and mechanisms, it is relevant to predict their failure under service loads. The stress state in compressive, tensile, bending, and other tests regulated by standards is generally different from that under service conditions. Therefore, the results of these tests can be used only for qualitative or comparative estimation of material serviceability. The effect of the stress state on the failure of adhesives is being actively studied in terms of fracture mechanics, which studies conditions for the evolution of existing cracks appearing on the interfaces of adhesive joints or in laminated metal-polymer materials (see recent reviews [1-2]).
Significant experience in the quantitative description of limiting states under conditions of different stress states has been gained in studying the failure of metal materials. In order to predict the risk of failure under planned loading conditions, the fracture criteria depending on the invariant characteristics of the stress state are generally used. Experimental procedures in which, by varying the shape of specimens made of materials under study (mainly metal ones), one can change the stress state and use the experiment results to identify the fracture criteria or damage models are described in the literature [3-5]. As a rule, the stress triaxiality factor $k_0$$k_0$k_(0)k_{0}$k_0$ and the Lode-Nadai coefficient ${\mu }_{\sigma }$${\mu }_{\sigma }$mu_(sigma)\mu_{\sigma}${\mu }_{\sigma }$ (or the Lode normed angle $\overline{\theta }$$\overline{\theta }$bar(theta)\bar{\theta}$\overline{\theta }$ ) are currently being used in fracture testing to characterize the stress state [4-9], which are calculated by the formulas:

where $\sigma =\frac{{\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}}{3}$$\sigma =\frac{{\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}}{3}$sigma=(sigma_(11)+sigma_(22)+sigma_(33))/(3)\sigma=\frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3}$\sigma =\frac{{\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}}{3}$ is the mean normal (hydrostatic) stress; $T=\frac{1}{\sqrt{6}}\sqrt{{({\sigma }_{11}-{\sigma }_{22})}^2+{({\sigma }_{22}-{\sigma }_{33})}^2+{({\sigma }_{33}-{\sigma }_{11})}^2}$$T=\frac{1}{\sqrt{6}}\sqrt{{\left({\sigma }_{11}-{\sigma }_{22}\right)}^2+{\left({\sigma }_{22}-{\sigma }_{33}\right)}^2+{\left({\sigma }_{33}-{\sigma }_{11}\right)}^2}$T=(1)/(sqrt6)sqrt((sigma_(11)-sigma_(22))^(2)+(sigma_(22)-sigma_(33))^(2)+(sigma_(33)-sigma_(11))^(2))T=\frac{1}{\sqrt{6}} \sqrt{\left(\sigma_{11}-\sigma_{22}\right)^{2}+\left(\sigma_{22}-\sigma_{33}\right)^{2}+\left(\sigma_{33}-\sigma_{11}\right)^{2}}$T=\frac{1}{\sqrt{6}}\sqrt{{({\sigma }_{11}-{\sigma }_{22})}^2+{({\sigma }_{22}-{\sigma }_{33})}^2+{({\sigma }_{33}-{\sigma }_{11})}^2}$ is tangential stress intensity; ${\sigma }_{11},{\sigma }_{22}$${\sigma }_{11},{\sigma }_{22}$sigma_(11),sigma_(22)\sigma_{11}, \sigma_{22}${\sigma }_{11},{\sigma }_{22}$ and ${\sigma }_{33}$${\sigma }_{33}$sigma_(33)\sigma_{33}${\sigma }_{33}$ are principal stresses; $J_2$$J_2$J_(2)J_{2}$J_2$ and $J_3$$J_3$J_(3)J_{3}$J_3$ are the second and third invariants of the
stress deviator. Normal tensile stresses prevail when $k>0$$k>0$k > 0k>0$k>0$, compressive ones being dominant when $k<0$$k<0$k < 0k<0$k<0$. The LodeNadai coefficient characterizes the form of the stress state. The values ${\mu }_{\sigma }=\pm 1$${\mu }_{\sigma }=\pm 1$mu_(sigma)=+-1\mu_{\sigma}= \pm 1${\mu }_{\sigma }=\pm 1$ correspond to the axisymmetric compression/tension state, and there is a plane strain state when ${\mu }_{\sigma }=0$${\mu }_{\sigma }=0$mu_(sigma)=0\mu_{\sigma}=0${\mu }_{\sigma }=0$. The values $\overline{\theta }=\pm 1$$\overline{\theta }=\pm 1$bar(theta)=+-1\bar{\theta}= \pm 1$\overline{\theta }=\pm 1$ correspond to the axisymmetric tension/compression state, whereas $\overline{\theta }=0$$\overline{\theta }=0$bar(theta)=0\bar{\theta}=0$\overline{\theta }=0$ corresponds to the plane strain state.
The effect of the parameters defined in Eqs. (1) and (2) on ultimate fracture characteristics are studied in detail for metal materials ([5-9] etc.), but these dependences for polymers and adhesives have yet to be studied. There are separate data on the effect of the stress triaxiality factor $k$$k$kk$k$ in the region of tensile stresses, which were obtained in tensile and bending testing of specimens with stress concentrators [10-11], in tension and compression [12]. Complex testing of epoxy resin by various methods were discussed in [13]. In [14] thin-walled tubular specimens made of hardened epoxy resin were subjected to torsion combined with axial loading, external and internal pressure. The features of the experimental procedures made it impossible to reach high hydrostatic pressure ( $p<17.24\mathrm{MPa}$$p<17.24\mathrm{MPa}$p < 17.24MPap<17.24 \mathrm{MPa}$p<17.24\mathrm{MPa}$ at a yield stress of $\approx 60\mathrm{MPa}$$\approx 60\mathrm{MPa}$~~60MPa\approx 60 \mathrm{MPa}$\approx 60\mathrm{MPa}$ ). This may be why the authors concluded that hydrostatic pressure had no effect on ultimate strain properties, although this effect was significantly manifested when the tensile and compressive test results were compared.
Testing in high-pressure chambers was formerly widely used to construct fracture loci of metal materials. The test results enabled the influence of the parameters defined by Eqs. (1) and (2) on ultimate strain properties [15-16, etc.] to be correctly separated; however, due to the complexity of conducting such experiments, the majority of studies now use testing of differently shaped specimens. Besides, this was favored by the development of numerical methods for stress-strain analysis, which make it possible to calculate the values of the stress state parameters in testing.
Besides widespread testing for uniaxial tension, compression, and torsion, where $k=\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}$$k=\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}$k=(1)/(sqrt3),-(1)/(sqrt3)k=\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}$k=\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}$ and $0,{\mu }_{\sigma }=-1,+1$$0,{\mu }_{\sigma }=-1,+1$0,mu_(sigma)=-1,+10, \mu_{\sigma}=-1,+1$0,{\mu }_{\sigma }=-1,+1$ and 0 , respectively, irregularly shaped specimens are used to extend the range of variation of the stress state. Thus, e.g., butterflyshaped specimens are used for testing at ${\mu }_{\sigma }=0$${\mu }_{\sigma }=0$mu_(sigma)=0\mu_{\sigma}=0${\mu }_{\sigma }=0$ [17-18].
These specimens allow $k$$k$kk$k$ to be varied in a fairly wide range, between -0.33 and 1.73 . However, to test these specimens, it is necessary to use a testing machine with two independent perpendicularly positioned drives. For testing under conditions of the conservative value ${\mu }_{\sigma }=1$${\mu }_{\sigma }=1$mu_(sigma)=1\mu_{\sigma}=1${\mu }_{\sigma }=1$ Nakajima specimens are used [19], the value $k=1.09$$k=1.09$k=1.09k=1.09$k=1.09$ being also constant. The fracture locus is plotted from the test results, showing the amount of strain accumulated before fracture ${\epsilon }_f$${\epsilon }_f$epsi_(f)\varepsilon_{f}${\epsilon }_f$ as a function of the stress coefficients.
In our earlier papers we proposed a set of tests enabling us to study the deformability of metal materials in the range of variation of the stress parameters $-1\le {\mu }_{\sigma }\le 1$$-1\le {\mu }_{\sigma }\le 1$-1 <= mu_(sigma) <= 1-1 \leq \mu_{\sigma} \leq 1$-1\le {\mu }_{\sigma }\le 1$ and $-0.6\le k\le 1.2$$-0.6\le k\le 1.2$-0.6 <= k <= 1.2-0.6 \leq k \leq 1.2$-0.6\le k\le 1.2$ [7]. Besides standard tensile and compressive testing of cylindrical specimens, the set includes tensile and compressive testing of bell-shaped specimens and dishing of thick-walled cup-shaped specimens. The set of tests was tried out on studying a number of aluminum alloys and metal matrix composites [7, 20-22], and even at high temperatures. It is of interest to extend the proposed approach to studying the effect of the stress state on the characteristics of the cohesive failure of organic polymer materials. Epoxy materials are known for their low plastic properties; therefore, it is difficult to use strain ${\epsilon }_f$${\epsilon }_f$epsi_(f)\varepsilon_{f}${\epsilon }_f$ as a limiting characteristic to plot a fracture locus for them since its values are small. Therefore, in this paper, the notion of fracture locus includes strain energy density instead of strain or amount of strain.
The intention of this study is to modify and adapt the set of tests [7] for studying the effect of the stress parameters and temperature on the cohesive failure of epoxy resin in order to represent the research results in the form of a fracture locus.

Promising ${\mathrm{TiO}}_2$${\mathrm{TiO}}_2$TiO_(2)\mathrm{TiO}_{2}${\mathrm{TiO}}_2$-reinforced epoxy resin is used as the research material. The ED-20-4,4'-isopropylidenediphenol commercial epoxy-diane resin with the epoxy number $21.1\mathrm{\%}$$21.1\mathrm{\%}$21.1%21.1 \%$21.1\mathrm{\%}$ was made at Sverdlov Plant, Dzerzhinsk, Russia. Titanium oxide ( ${\mathrm{TiO}}_2$${\mathrm{TiO}}_2$TiO_(2)\mathrm{TiO}_{2}${\mathrm{TiO}}_2$ ) nanoparticles ( $99.5\mathrm{\%}$$99.5\mathrm{\%}$99.5%99.5 \%$99.5\mathrm{\%}$ purity, an average size 21 nm ) were made by Sigma-Aldrich (Germany). The epoxy resin was preliminarily dissolved in tetrahydrofuran, $10\mathrm{wt}\mathrm{\%}$$10\mathrm{wt}\mathrm{\%}$10wt%10 \mathrm{wt} \%$10\mathrm{wt}\mathrm{\%}$ of ${\mathrm{TiO}}_2$${\mathrm{TiO}}_2$TiO_(2)\mathrm{TiO}_{2}${\mathrm{TiO}}_2$ was added to the resulting solution, and this was processed in a ball mill for 4 hours. Polyethylenepolyamine was added in a $10:1$$10:1$10:110: 1$10:1$ ratio for curing, and, after mixing, this was poured into split Teflon molds, and this excludes the appearance of mechanical stress concentrators on the specimens. The reinforcement technology applied enabled us to obtain a uniform distribution of ${\mathrm{TiO}}_2$${\mathrm{TiO}}_2$TiO_(2)\mathrm{TiO}_{2}${\mathrm{TiO}}_2$, predominantly forming agglomerates with an average size of $1.2\mu \mathrm{m}$$1.2\mu \mathrm{m}$1.2 mum1.2 \mu \mathrm{~m}$1.2\mu \mathrm{m}$, as is shown in fig. 1 . Complete curing occurred in 24 hours at ${25}^{\circ }\mathrm{C}$${25}^{\circ }\mathrm{C}$25^(@)C25^{\circ} \mathrm{C}${25}^{\circ }\mathrm{C}$.

The curing conditions were set according to the recommendations of the resin producer, and no other conditions were studied. The effect of ${\mathrm{TiO}}_2$${\mathrm{TiO}}_2$TiO_(2)\mathrm{TiO}_{2}${\mathrm{TiO}}_2$ on the degrees of cure and the mechanical properties were studied earlier and discussed in [23]. Besides reinforced epoxy resin specimens, similar pure epoxy resin ones were made in order to detect the effect of adding the reinforcing agent on the deformability of the epoxy resin at the same time.

Bell-shaped tensile and compressive specimens and thick-walled cup-shaped specimens to be tested under axisymmetric deformation were made by analogy with those described in [7], though their geometric dimensions were appropriately adjusted (fig. 2 a, b, c). The specimens (fig. 2 d) reported in [23] were used for shear testing under plane strain. The cylindrical compressive specimens had a diameter of 4 mm and a height of 6 mm .

The specimens were deformed to fracture in an Instron 8801 servohydraulic testing machine at an actuator velocity of $1\mathrm{mm}/\min$$1\mathrm{mm}/\min$1mm//min1 \mathrm{~mm} / \mathrm{min}$1\mathrm{mm}/\min$. The machine has the following characteristics: a maximum loading force of 100 kN ; a load measurement accuracy of $0.25\mathrm{\%}$$0.25\mathrm{\%}$0.25%0.25 \%$0.25\mathrm{\%}$ from the measured value; a displacement gauging accuracy of $0.1\mathrm{\%}$$0.1\mathrm{\%}$0.1%0.1 \%$0.1\mathrm{\%}$ from the measured value. The tests were performed in a temperature chamber enabling us to test the specimens at temperatures ranging from -70 to $+{300}^{\circ }\mathrm{C}$$+{300}^{\circ }\mathrm{C}$+300^(@)C+300^{\circ} \mathrm{C}$+{300}^{\circ }\mathrm{C}$. The Solidol-Zh grease was used to decrease friction between the deforming tool and the specimen.
The normal elastic modulus, the stress-strain curve, and the glass transition temperature were determined by means of a DMA Eplexor 100 N test system for dynamic mechanical analysis, which determines the mechanical and thermomechanical material properties in the range of static loads from 0 to 1500 N and oscillating loads between 0 and 100 N at frequencies ranging between 0.1 and 100 Hz . The device is equipped with a climate chamber for testing at temperatures from -160 to $+{500}^{\circ }\mathrm{C}$$+{500}^{\circ }\mathrm{C}$+500^(@)C+500^{\circ} \mathrm{C}$+{500}^{\circ }\mathrm{C}$. The experiment results were recorded and processed by means of the EPLEXOR ${}^{\circledR }9$${}^{\circledR }9$^(®)9{ }^{\circledR} 9${}^{\circledR }9$ specialized software.
To make certain that the specimen geometry allows us to achieve the required stress state during loading, we first constructed finite-element models of the specimens and simulated loading.
In order to achieve a satisfactory reproducibility of the tests, a lot of three specimens was used in each type of testing and the obtained results were averaged. Since the plastic strains before failure are insignificant, the reproducibility of the experiments was assessed from the value of the displacement $U$$U$UU$U$ of the loading tool at the moment of failure, not from the geometric parameters of the failed specimens. A statistical analysis of the sample of the obtained values was made. The following was determined within the statistical analysis:

The coefficient of variation for each specimen lot did not exceed 4.33%.
The relative deviation from the mean in terms of the value of ultimate strain before failure calculated by finite element simulation did not exceed $\pm 8.5\mathrm{\%}$$\pm 8.5\mathrm{\%}$+-8.5%\pm 8.5 \%$\pm 8.5\mathrm{\%}$.

The glass transition temperature $T_g=78{}^{\circ }\mathrm{C}$$T_g=78{}^{\circ }\mathrm{C}$T_(g)=78^(@)CT_{g}=78{ }^{\circ} \mathrm{C}$T_g=78{}^{\circ }\mathrm{C}$ ( ${\mathrm{TiO}}_2$${\mathrm{TiO}}_2$TiO_(2)\mathrm{TiO}_{2}${\mathrm{TiO}}_2$-reinforced epoxy resin) and ${74}^{\circ }\mathrm{C}$${74}^{\circ }\mathrm{C}$74^(@)C74^{\circ} \mathrm{C}${74}^{\circ }\mathrm{C}$ (pure epoxy resin) was determined by dynamic mechanical analysis according to ASTM D7028 by the test temperature dependence of the storage module at a static load of 20 N and an oscillating measuring load of 1.2 N with a frequency of 1 Hz . Therefore, at testing temperatures of -50 and ${25}^{\circ }\mathrm{C}$${25}^{\circ }\mathrm{C}$25^(@)C25^{\circ} \mathrm{C}${25}^{\circ }\mathrm{C}$, the epoxy materials were vitrified.
To determine the stress-strain curves, cylindrical specimens were compressed by means of a DMA Eplexor 100 N device using the Universal Test option, which provided testing under conditions of simple quasi-static loading. The use of this equipment was caused by the impossibility of using Instron 8801 to deform such small specimens.
Strain $\epsilon$$\epsilon$epsi\varepsilon$\epsilon$ and flow stress ${\sigma }_S$${\sigma }_S$sigma_(S)\sigma_{S}${\sigma }_S$ were calculated by the formulas:

where $h_0$$h_0$h_(0)h_{0}$h_0$ and $h$$h$hh$h$ are the initial and current specimen heights; $P$$P$PP$P$ is the current compressive force; $F$$F$FF$F$ is the current specimen cross-sectional area determined from the volume constancy condition by the formula:

The stress-strain curves obtained from the tests of specimens made of each material performed at -50 and ${25}^{\circ }\mathrm{C}$${25}^{\circ }\mathrm{C}$25^(@)C25^{\circ} \mathrm{C}${25}^{\circ }\mathrm{C}$, which are limited by maximum stress corresponding to ultimate strength, are shown in fig. 3. The specimens were loaded at rates ranging between 1 and $10{\mathrm{s}}^{-1}$$10{\mathrm{s}}^{-1}$10s^(-1)10 \mathrm{~s}^{-1}$10{\mathrm{s}}^{-1}$. The analysis of the results has shown that the change of strain rate by an order of magnitude causes a $3\mathrm{\%}$$3\mathrm{\%}$3%3 \%$3\mathrm{\%}$ change in strain resistance at a test temperature of $-50{}^{\circ }\mathrm{C}$$-50{}^{\circ }\mathrm{C}$-50^(@)C-50{ }^{\circ} \mathrm{C}$-50{}^{\circ }\mathrm{C}$ and an $8\mathrm{\%}$$8\mathrm{\%}$8%8 \%$8\mathrm{\%}$ change at ${25}^{\circ }\mathrm{C}$${25}^{\circ }\mathrm{C}$25^(@)C25^{\circ} \mathrm{C}${25}^{\circ }\mathrm{C}$, and this is below the $10\mathrm{\%}$$10\mathrm{\%}$10%10 \%$10\mathrm{\%}$ allowable natural variation of the strength properties for structural materials [24]. Therefore, averaged stress-strain curves were used in the simulation, without regard for the effect of strain rate. The average values of the normal elastic moduli $E$$E$EE$E$ (Tab. 1) were determined on the linear lengths of the stress-strain curves of the specimens in the selected stress range of 20 to 60 MPa . The stress-strain curves were approximated by Eqn. (5) on the portion of linear elasticity and by the polynomial dependence represented by Eqn. (6) on the nonlinear one in the Statistica 8.0 software:

where ${\epsilon }_{eq}$${\epsilon }_{eq}$epsi_(eq)\varepsilon_{e q}${\epsilon }_{eq}$ and ${\sigma }_{eq}$${\sigma }_{eq}$sigma_(eq)\sigma_{e q}${\sigma }_{eq}$ are equivalent stress and equivalent strain; $a_3,a_2,a_1$$a_3,a_2,a_1$a_(3),a_(2),a_(1)a_{3}, a_{2}, a_{1}$a_3,a_2,a_1$ and $a_0$$a_0$a_(0)a_{0}$a_0$ are approximation coefficients, whose values are shown in Tab. 1.

where ${\epsilon }_{11},{\epsilon }_{22}$${\epsilon }_{11},{\epsilon }_{22}$epsi_(11),epsi_(22)\varepsilon_{11}, \varepsilon_{22}${\epsilon }_{11},{\epsilon }_{22}$ and ${\epsilon }_{33}$${\epsilon }_{33}$epsi_(33)\varepsilon_{33}${\epsilon }_{33}$ are total principal strains, ${\sigma }_{11},{\sigma }_{22}$${\sigma }_{11},{\sigma }_{22}$sigma_(11),sigma_(22)\sigma_{11}, \sigma_{22}${\sigma }_{11},{\sigma }_{22}$ and ${\sigma }_{33}$${\sigma }_{33}$sigma_(33)\sigma_{33}${\sigma }_{33}$ are total principal stresses.

It follows from Tab. 1 and fig. 3 that the introduction of $10\mathrm{\%}$$10\mathrm{\%}$10%10 \%$10\mathrm{\%}$ of ${\mathrm{TiO}}_2$${\mathrm{TiO}}_2$TiO_(2)\mathrm{TiO}_{2}${\mathrm{TiO}}_2$ nanoparticles enhances the strength properties of epoxy resin; namely, the value of the normal elastic modulus increases by 14 and $11\mathrm{\%}$$11\mathrm{\%}$11%11 \%$11\mathrm{\%}$ at $T=-50$$T=-50$T=-50T=-50$T=-50$ and ${25}^{\circ }\mathrm{C}$${25}^{\circ }\mathrm{C}$25^(@)C25^{\circ} \mathrm{C}${25}^{\circ }\mathrm{C}$, respectively, and the value of ultimate compressive strength increases by 25 and $35\mathrm{\%}$$35\mathrm{\%}$35%35 \%$35\mathrm{\%}$ at $T=-50$$T=-50$T=-50T=-50$T=-50$ and ${25}^{\circ }\mathrm{C}$${25}^{\circ }\mathrm{C}$25^(@)C25^{\circ} \mathrm{C}${25}^{\circ }\mathrm{C}$, respectively.
In order to evaluate the characteristics of the stress-strain state, mathematical simulation was made by the finite element method. The Ansys finite element software was used for the simulation. The material of the deforming tool was treated as a perfectly rigid body, and the epoxy materials were considered to be isotropic elastoplastic materials with the deformation behavior described by Eqs. (5) and (6). The associated flow rule and the von Mises yield criterion were applied in the plastic region, the stress-strain curves were specified in the tabular form of the MISO model by using the results of calculations by Eqn. (6). The boundary conditions were set in displacements ( $U$$U$UU$U$ ). Friction between the deforming tool and the specimen was described by the Amontons-Coulomb friction law with the friction coefficient $\mu =0.1$$\mu =0.1$mu=0.1\mu=0.1$\mu =0.1$. In view of the obvious axial symmetry of deformation, a half of the specimen cross-section was simulated. PLANE183 finite elements were used to construct a grid model. Besides, a grid convergence study was conducted in order to avoid the effect of finite element sizes on the simulation results. The element sizes were varied between 1 and 0.1 mm . The solution results were compared by equivalent strain in the most probable specimen fracture zone (in the middle of the lateral surface). Fig. 4 exemplifies the dependence of equivalent strain on the sizes of the finite elements of the model. Staring from element sizes of 0.25 and smaller, the solution results differ by less than $1\mathrm{\%}$$1\mathrm{\%}$1%1 \%$1\mathrm{\%}$. Finally, the number of finite elements in the model was 192, with an element size of 0.25 mm .

The adequacy of the constructed finite element models was also verified by comparing the experimentally obtained deformation forces with those calculated in the simulation. Fig. 5 exemplifies this comparison for compressive testing of cylindrical specimens of reinforced epoxy resin at ${25}^{\circ }\mathrm{C}$${25}^{\circ }\mathrm{C}$25^(@)C25^{\circ} \mathrm{C}${25}^{\circ }\mathrm{C}$.

The simulation resulted in the evaluation of the changes in the stress and strain tensor components, which are necessary to calculate the parameters $k_0$$k_0$k_(0)k_{0}$k_0$ and ${\mu }_{\sigma },{\epsilon }_{eq}$${\mu }_{\sigma },{\epsilon }_{eq}$mu_(sigma),epsi_(eq)\mu_{\sigma}, \varepsilon_{e q}${\mu }_{\sigma },{\epsilon }_{eq}$ and ${\sigma }_{eq}$${\sigma }_{eq}$sigma_(eq)\sigma_{e q}${\sigma }_{eq}$ at each loading step.
As an example, fig. 6a shows a finite element model of cylindrical specimen compression and the cross-sectional distribution of ${\epsilon }_{eq}$${\epsilon }_{eq}$epsi_(eq)\varepsilon_{e q}${\epsilon }_{eq}$ at fracture. Although the highest strain is localized in the central region of the specimen, the disruptive crack occurs in the middle of the lateral surface of the specimen, this being caused by the appearance of tensile tangential stresses with surface warping due to the inhibiting effect of the friction forces at the tool-specimen interface. Herewith, the parameters $k_0$$k_0$k_(0)k_{0}$k_0$ and ${\mu }_{\sigma }$${\mu }_{\sigma }$mu_(sigma)\mu_{\sigma}${\mu }_{\sigma }$ change from their initial values ${\mu }_{\sigma }=1$${\mu }_{\sigma }=1$mu_(sigma)=1\mu_{\sigma}=1${\mu }_{\sigma }=1$ and $k=-\frac{1}{\sqrt{3}}$$k=-\frac{1}{\sqrt{3}}$k=-(1)/(sqrt3)k=-\frac{1}{\sqrt{3}}$k=-\frac{1}{\sqrt{3}}$ (fig. 6 b).
The above-discussed approach to constructing a finite-element model was used to study the stress-strain state in other types of testing, which are discussed in what follows.

A punch was placed into the cylindrical hollow of the specimen (fig. 7 a), and this assembly was then put on a ring with an inner diameter slightly exceeding the diameter of the lower part of the specimen (fig. 7 a). The specimen was loaded by applying a compressive force to the punch. As a result, the material volume in the transition part between the upper and lower parts of the specimen experienced localized shear strain followed by fracture (fig. 7 b).
Fig. 8 shows the finite-element model of the testing process and the cross-sectional distribution of ${\epsilon }_{\text{eq}}$${\epsilon }_{\text{eq }}$epsi_("eq ")\varepsilon_{\text {eq }}${\epsilon }_{\text{eq }}$ at fracture. The number of finite elements in the model is 821 . The element size is 0.2 mm .
The results of the visual examination of the specimens after testing (fig. 7 b) were compared with the simulation results (fig. 8). The comparison testifies that the specimens fracture at the site of maximum equivalent strain on the external surface of the transition zone. The moment of fracture was identified in the testing by a sharp decrease in the deformation force.
Fig. 9 shows the behavior of the stress triaxiality parameter $k_0$$k_0$k_(0)k_{0}$k_0$ and the Lode-Nadai coefficient ${\mu }_{\sigma }$${\mu }_{\sigma }$mu_(sigma)\mu_{\sigma}${\mu }_{\sigma }$ at the site of failure in the testing of a bell-shaped pure epoxy resin specimen at ${25}^{\circ }\mathrm{C}$${25}^{\circ }\mathrm{C}$25^(@)C25^{\circ} \mathrm{C}${25}^{\circ }\mathrm{C}$.

The specimens were mounted onto an immobile flat die in the testing machine and subjected to a compressive force applied to the upper end face of the specimen (fig. 10 a). Fracture occurred near the internal surface of the specimen transition zone (fig. 10 b).
Fig. 11 shows the finite-element model of the testing process and the cross-sectional distribution of ${\epsilon }_{\text{eq}}$${\epsilon }_{\text{eq }}$epsi_("eq ")\varepsilon_{\text {eq }}${\epsilon }_{\text{eq }}$ at fracture. The number of finite elements in the model is 670 . The element size is 0.2 mm . The simulation (fig. 11) shows that the highest strain is accumulated on the external surface of the transition zone, although fracture occurred in another zone of the specimen. This must be due to more favorable stress state implemented there ( $k=-0.86$$k=-0.86$k=-0.86k=-0.86$k=-0.86$ in the initial stage of deformation, and this parameter subsequently decreases to $k<-1$$k<-1$k < -1k<-1$k<-1$ ), which prevents the appearance of a disruptive crack (fig. 12 a). On the internal surface of the transition zone the stress triaxiality parameter is twice as high at the beginning of deformation, $k=-0.42$$k=-0.42$k=-0.42k=-0.42$k=-0.42$ (fig. 12 b). The values of the Lode-Nadai parameter ${\mu }_{\sigma }$${\mu }_{\sigma }$mu_(sigma)\mu_{\sigma}${\mu }_{\sigma }$ become stabilized near 0 with developed plastic strain.

The test specimen 1 (fig. 13 a) was placed on the base 2 in the workspace of the testing machine and fixed by a cap 3 threaded to the base. A steel ball 4 was put inside the specimen and then a force from the testing machine was applied through a punch 5. The design of the specimen provided the appearance of the initial fracture zone in the center of the external surface of the bottom, as shown in the photograph (fig. 13 b).
Fig. 14 shows the finite element model of the testing process and the cross-sectional distribution of ${\epsilon }_{\text{eq}}$${\epsilon }_{\text{eq }}$epsi_("eq ")\varepsilon_{\text {eq }}${\epsilon }_{\text{eq }}$ at fracture. The number of finite elements in the model is 2734 . The element size is 0.15 mm .
The analysis of the results of simulating the testing process shows that the most unfavorable stress-strain state is found near the middle of the external surface of the dished bottom (fig. 14). The highest value of accumulated strain ${\epsilon }_{\text{eq}}$${\epsilon }_{\text{eq }}$epsi_("eq ")\varepsilon_{\text {eq }}${\epsilon }_{\text{eq }}$ and intensive tensile stresses $(k=1.15)$$(k=1.15)$(k=1.15)(k=1.15)$(k=1.15)$ are responsible for the appearance of incipient failure here. The Lode-Nadai coefficient is the same as for upsetting of cylindrical specimens, ${\mu }_{\sigma }=1$${\mu }_{\sigma }=1$mu_(sigma)=1\mu_{\sigma}=1${\mu }_{\sigma }=1$ (fig. 15).

The diagram of testing is shown in fig. 16. Before testing, the specimen was placed on a die 2 to be deformed by an upper die 3, through which a force is translated to the specimen. The horizontal displacement of the dies and the lateral surfaces of the specimen must be restricted. For this purpose, the specimen and the dies were placed in a container 4 (shown as a sectional view in fig. 16). More intricately shaped dies 2 and 3 can be used to apply additional compressive stresses (fig. 16 b), which restrict metal flow in the direction perpendicular to the motion of the upper die (the dies with a lateral support). During testing, the strain is localized in the central part of the specimen. The stress state in the deformation zone essentially depends on the ratio of the dimensions of the central part, namely central part thickness $S$$S$SS$S$, the spherical radius $R$$R$RR$R$, the
pitch $\alpha$$\alpha$alpha\alpha$\alpha$ of the line connecting the spherical radii. Specimens of three dimension types were made for testing: 1) $R=2\mathrm{mm}$$R=2\mathrm{mm}$R=2mmR=2 \mathrm{~mm}$R=2\mathrm{mm}$, $S=2\mathrm{mm},\alpha ={30}^{\circ }$$S=2\mathrm{mm},\alpha ={30}^{\circ }$S=2mm,alpha=30^(@)S=2 \mathrm{~mm}, \alpha=30^{\circ}$S=2\mathrm{mm},\alpha ={30}^{\circ }$; 2) $\mathrm{R}=3\mathrm{mm},S=2.5\mathrm{mm},\alpha ={50}^{\circ }$$\mathrm{R}=3\mathrm{mm},S=2.5\mathrm{mm},\alpha ={50}^{\circ }$R=3mm,S=2.5mm,alpha=50^(@)\mathrm{R}=3 \mathrm{~mm}, S=2.5 \mathrm{~mm}, \alpha=50^{\circ}$\mathrm{R}=3\mathrm{mm},S=2.5\mathrm{mm},\alpha ={50}^{\circ }$; 3) $\mathrm{R}=2\mathrm{mm},S=5\mathrm{mm},\alpha ={80}^{\circ }$$\mathrm{R}=2\mathrm{mm},S=5\mathrm{mm},\alpha ={80}^{\circ }$R=2mm,S=5mm,alpha=80^(@)\mathrm{R}=2 \mathrm{~mm}, S=5 \mathrm{~mm}, \alpha=80^{\circ}$\mathrm{R}=2\mathrm{mm},S=5\mathrm{mm},\alpha ={80}^{\circ }$.
Figs. 17 a and 17 b show photographs of dimension type 1 specimens before and after testing as an example. Fig. 17 c shows the finite-element model used to analyze the stress-strain state in the course of testing. The front end face of the model corresponds to the symmetry plane at the half-width of the specimen. The finite element models of the specimens have the following characteristics.
Type I specimens: the number of finite elements is 5210 and the element size is 0.3 mm .
Type II specimens: the number of finite elements is 5160 and the element size is 0.3 mm .
Type III specimens: the number of finite elements is 6840 and the element size is 0.3 mm .

The maximum strain is localized on the specimen-die interface at point A (fig. 17 d ). However, most probably, fracture is initiated in the central part of the specimen (point B in fig. 17 d) since a more unfavorable stress state occurs here (fig. 18). The Lode-Nadai coefficient is close to zero at both points.

train energy density $W_f$$W_f$W_(f)W_{f}$W_f$ is used as the quantitative parameter characterizing the limiting state of epoxy resin in cohesive failure:

where ${\epsilon }_{eq}^{\ast }$${\epsilon }_{eq}^{\ast }$epsi_(eq)^(**)\varepsilon_{e q}^{*}${\epsilon }_{eq}^{\ast }$ is ultimate strain at the site of failure.
The value of $W_f$$W_f$W_(f)W_{f}$W_f$ for each material depends on the stress state characterized by the totality of the parameters $k$$k$kk$k$ and ${\mu }_{\sigma }$${\mu }_{\sigma }$mu_(sigma)\mu_{\sigma}${\mu }_{\sigma }$, as well as temperature and strain rate. Rigorously, to construct the fracture locus, $W_f$$W_f$W_(f)W_{f}$W_f$ must be determined in experiments conducted at constant values of $k$$k$kk$k$ and ${\mu }_{\sigma }$${\mu }_{\sigma }$mu_(sigma)\mu_{\sigma}${\mu }_{\sigma }$. However, it follows from the above dependences that the stress state is hardly invariable during testing; therefore, the quantitative dependence $W_f(k,{\mu }_{\sigma })$$W_f\left(k,{\mu }_{\sigma }\right)$W_(f)(k,mu_(sigma))W_{f}\left(k, \mu_{\sigma}\right)$W_f(k,{\mu }_{\sigma })$ can be established only by using the averaged values of the stress parameters or by the identification procedure. The latter way is used in this study. Strain energy density $W^{\ast }$$W^{\ast }$W^(**)W^{*}$W^{\ast }$ at the crack initiation site is evaluated from the simulation of the testing process with account taken of the stress-strain history.
The values of ${\epsilon }_{eq}^{\ast }$${\epsilon }_{eq}^{\ast }$epsi_(eq)^(**)\varepsilon_{e q}^{*}${\epsilon }_{eq}^{\ast }$ and $W^{\ast }$$W^{\ast }$W^(**)W^{*}$W^{\ast }$ reached at the site of failure and averaged from the simulation results are shown in Tab. 2.

As the objective function for the identification of the dependence $W_f=f(k,{\mu }_{\sigma })$$W_f=f\left(k,{\mu }_{\sigma }\right)$W_(f)=f(k,mu_(sigma))W_{f}=f\left(k, \mu_{\sigma}\right)$W_f=f(k,{\mu }_{\sigma })$, the criterion for the failure of the linear model of material damage $\omega$$\omega$omega\omega$\omega$ under deformation is used in the following interpretation:

During deformation, damage $\omega$$\omega$omega\omega$\omega$ varies from 0 before the deformation starts to 1 at the moment of crack initiation, and the intermediate values of $\omega$$\omega$omega\omega$\omega$ characterize the amount of microscopic material damage by strain-induced defects. Assume that, at a fixed test temperature, the dependence $W_f=f(k,{\mu }_{\sigma })$$W_f=f\left(k,{\mu }_{\sigma }\right)$W_(f)=f(k,mu_(sigma))W_{f}=f\left(k, \mu_{\sigma}\right)$W_f=f(k,{\mu }_{\sigma })$ can be described by the approximating linear function:

where $C_0(k)$$C_0(k)$C_(0)(k)C_{0}(k)$C_0(k)$ and $C_1(k)$$C_1(k)$C_(1)(k)C_{1}(k)$C_1(k)$ are the unknown functions of the stress triaxiality factor $k$$k$kk$k$. To describe the $k$$k$kk$k$ dependence of $W_f$$W_f$W_(f)W_{f}$W_f$ at fixed values of ${\mu }_{\sigma }$${\mu }_{\sigma }$mu_(sigma)\mu_{\sigma}${\mu }_{\sigma }$, similarly to [7], the study uses the exponential law:

where $\alpha$$\alpha$alpha\alpha$\alpha$ and $\beta$$\beta$beta\beta$\beta$ are empirical coefficients.
Let ${\alpha }_0,{\alpha }_{+1},{\beta }_0$${\alpha }_0,{\alpha }_{+1},{\beta }_0$alpha_(0),alpha_(+1),beta_(0)\alpha_{0}, \alpha_{+1}, \beta_{0}${\alpha }_0,{\alpha }_{+1},{\beta }_0$ and ${\beta }_{+1}$${\beta }_{+1}$beta_(+1)\beta_{+1}${\beta }_{+1}$ have the values of the coefficients $\alpha$$\alpha$alpha\alpha$\alpha$ and $\beta$$\beta$beta\beta$\beta$ when ${\mu }_{\sigma }=0$${\mu }_{\sigma }=0$mu_(sigma)=0\mu_{\sigma}=0${\mu }_{\sigma }=0$ and ${\mu }_{\sigma }=1$${\mu }_{\sigma }=1$mu_(sigma)=1\mu_{\sigma}=1${\mu }_{\sigma }=1$, respectively, then it follows from Eqn. (12) that:

Substituting the right-hand sides of Eqs. (13) and (14) and the corresponding values of ${\mu }_{\sigma }$${\mu }_{\sigma }$mu_(sigma)\mu_{\sigma}${\mu }_{\sigma }$ into Eqn. (11), after some transformations we find expressions for $C_0(k_{)}$$C_0\left(k_{)}\right.$C_(0)(k_()):}C_{0}\left(k_{)}\right.$C_0(k_{)}$and $C_1(k_{)}$$C_1\left(k_{)}\right.$C_(1)(k_()):}C_{1}\left(k_{)}\right.$C_1(k_{)}$and obtain the interpolation formula:

For the parametric identification of Eqn. (15), we use the data from Tab. 2 and the dependences $k\left({\epsilon }_{eq}\right)$$k\left({\epsilon }_{eq}\right)$k(epsi_(eq))k\left(\varepsilon_{e q}\right)$k\left({\epsilon }_{eq}\right)$ and ${\mu }_{\sigma }\left({\epsilon }_{eq}\right)$${\mu }_{\sigma }\left({\epsilon }_{eq}\right)$mu_(sigma)(epsi_(eq))\mu_{\sigma}\left(\varepsilon_{e q}\right)${\mu }_{\sigma }\left({\epsilon }_{eq}\right)$ during the testing, which are partially shown in figs. $6,9,12,15$$6,9,12,15$6,9,12,156,9,12,15$6,9,12,15$, and 18 . The functional of squared deviations (16) of the objective function (10) from 1 is minimized by varying the coefficients ${\alpha }_0,{\alpha }_{+1},{\beta }_0$${\alpha }_0,{\alpha }_{+1},{\beta }_0$alpha_(0),alpha_(+1),beta_(0)\alpha_{0}, \alpha_{+1}, \beta_{0}${\alpha }_0,{\alpha }_{+1},{\beta }_0$ and ${\beta }_{+1}$${\beta }_{+1}$beta_(+1)\beta_{+1}${\beta }_{+1}$.

where $n$$n$nn$n$ is the number of types of testing at the given temperature.
The coefficients are varied by the Hooke-Jeeves method [25]. As a result of searching, it is possible to hit the local minimum of the functional (16); therefore, we select not one initial search point, but a certain set of points. The Hooke-Jeeves procedure is performed for each initial search point. The found values of the coefficients are presented in Tab. 3. When calculating by Eqn. (10), as the upper limit of integration, we substitute the values of ${\epsilon }_{\text{eq}}$${\epsilon }_{\text{eq }}$epsi_("eq ")\varepsilon_{\text {eq }}${\epsilon }_{\text{eq }}$ from Tab. 2, which were reached at the site of failure during the tests. The identification error is less than $12\mathrm{\%}$$12\mathrm{\%}$12%12 \%$12\mathrm{\%}$.