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Combustion, Explosion, and Shock Waves, Vol. 48, No. 2, pp. 226–235, 2012. c L.A. Merzhievskii, M.S. Voronin. Original Russian Text

Modeling of Shock-Wave Deformation of Polymethyl Metacrylate L. A. Merzhievskiia and M. S. Voronina

UDC 539.3

Translated from Fizika Goreniya i Vzryva, Vol. 48, No. 2, pp. 113–123, March–April, 2012. Original article submitted July 27, 2011.

Abstract: A model of a Maxwellian elastoplastic body is constructed to describe the behavior of polymethyl metacrylate (C5 O2 H8 )n under loading. A principal feature of this model is supplementing the governing equations with the relaxation time of shear stresses in the form of a continuous dependence on parameters characterizing the state of the medium. The analytical form of the dependence is chosen with allowance for microstructural and mesostructural mechanisms of irreversible deformation. Another specific feature of the model is the equation of state of the medium, which includes the dependence of the internal energy on the first and second invariants of the strain tensor. Such an approach allows obtaining a unified mathematical description of all physical states of polymers. Particular attention is paid at the stage of model verification to comparisons of model predictions with experimental data for the temperature of the shock-compressed material and decay of the shock wave due to its interaction with overtaking and side rarefaction waves. This comparison shows that the model provides an adequate description of shock-wave processes in polymethyl metacrylate. Keywords: polymethyl metacrylate, shock compression, modeling, Maxwellian approach. DOI: 10.1134/S0010508212020128

INTRODUCTION In the course of deformation under external loading, polymer media demonstrate a complicated behavior, which is due to specific features of their structure. These features are manifested in the variety of structural deformation mechanisms realized at different structural levels. For amorphous polymers, three different physical states are distinguished: glassy, highly elastic, and viscoplastic states [1]. Each state has its own set of microstructural, mesostructural, and macrostructural mechanisms of irreversible deformation. Deformation mechanisms related not only to flexibility of macromolecules and conformal transitions, but also to displacements and reconstructions of supermolecular structures are consecutively activated. In quasi-static processes, the transition from one physical state to another involves a change in temperature and is accoma

panied by changes in microscopic and macroscopic processes. Various approaches are used for constructing models of dynamic deformation of polymers. Rheological models taking into account the clearly expressed relaxation character of the behavior of polymers during their deformation are most widely used [2]. In this case, the canonical models of Maxwell [3] and Voigt [4] are taken as a basis; for the results obtained to be quantitatively consistent with experimental data, models consisting of several consecutively and parallelly connected elements of elasticity and viscosity as structural elements are constructed [5]. Thus, several relaxation times are formally introduced into the model, and each relaxation time corresponds to a certain relaxation mechanism. Actually, this means inclusion of additional empirical constants (fitting constants) into the model.

Lavrent’ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia; [email protected].

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c 2012 by Pleiades Publishing, Ltd. 0010-5082/12/4802-0226

Modeling of Shock-Wave Deformation of Polymethyl Metacrylate Under shock-wave loading accompanied by an increase in entropy and temperature, the relaxation transitions and possible destruction of polymers are manifested in the form of structural phase transitions reflected in the shock adiabat shape [6] (see Fig. 3). In calculating the shock-wave motion and its interaction with rarefaction waves, the process cannot be accurately described if such structural phase transitions are ignored, in particular, the wave decay rate can be underestimated. This fact is clearly seen in comparisons of propagation and decay of shock waves in polymethyl metacrylate (PMMA) calculated by traditional models, where the calculated pressures on the decaying wave front can be thrice higher than the experimentally determined values [7]. The interest in such problems is caused by numerous applications of PMMA in structures that experience intense force and thermal loads. In studying shock-wave and detonation processes, Plexiglas is often used for fabricating elements of experimental assemblies and as a reference material with a known shock adiabat. In particular, it is often used as an inert target in studying the sensitivity of explosives by the so-called Gap test [8]. Therefore, the task of constructing and testing models that ensure an adequate description of the PMMA behavior under shock loading becomes rather urgent. The results of the activities aimed at constructing such models mainly based on nonlinear elasticity principles are summarized in [9]. The data obtained in [10, 11] were used as an experimental basis for these works. A somewhat different approach was used in [12]: the calculated results were compared there with experimental data on the shock adiabat and propagation of shock pulses [10]. In this paper, to describe the behavior of polymethyl metacrylate (C5 O2 H8 )n , we construct a Maxwellian elastoviscous body model, which differs principally from other models by inclusion of the relaxation time of shear stresses in the form of a continuous dependence on parameters characterizing the state of the medium into the governing relations. The analytical form of this dependence is chosen with allowance for microstructural and mesostructural mechanisms of irreversible deformation. Another specific feature of the model is the equation of state of the medium, which includes the dependence of the internal energy on the first and second invariants of the strain tensor. Such an approach allows us to obtain a unified mathematical description of all physical states of polymers. This approach was tested on models of deformation of polycrystalline media where it ensured good results in terms of the description of shock-wave processes [13].

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GOVERNING EQUATIONS OF THE MODEL The form of the full system of equations of the Maxwellian elastoviscous medium model used in this paper was constructed in [14]. In the general case, the governing equations (of motion and energy) in the differential form are 1 ∂σij σij ∂ui dE dui − − = 0, = 0. dt ρ ∂rj dt ρ ∂rj They are supplemented with equations that describe the evolution of the components of the effective elastic strain tensor. If the metric tensor G = gij is chosen as a measure of effective strains, the evolution equations are written as dG 1 3ρ + GW + W0 G + I = 0. G+ dt τ 2R Here gij are the components of the metric tensor of effective elastic strains, ∂uj ∂ui , R = ∂ρ , , W0 = W = ∂rj ∂ri ∂gkk σij are the components of the stress tensor, ui are the components of the velocity vector, ρ0 and ρ are the initial and current density, E is the specific elastic energy, τ is the relaxation time of shear stresses, t is the time, rj are the spatial coordinates (i, j = 1, 2, 3), I is the unit tensor, and ∂ ∂ d = + uk , dt ∂t ∂rk √ ∂E ρ = ρ0 det G, σij = −2ρgik . ∂gki The choice of the metric tensor as a measure of final strains is not of principal importance; it is also possible to formulate evolutionary equations for other tensors of final strains. In this case, the continuity equation (mass conservation law) follows from the presented equations. The system is closed by the equation relating the change in the specific internal energy E to the components of the non-spherical strain tensor and the entropy S E = E(gij , S), and the dependence for the relaxation time of shear stresses τ = τ (gij , S), which is used to take into account microstructural mechanisms of irreversible deformation. For the model considered here, these equations can be taken as the equations of state of the medium, which take into account energy dissipation due to the work on shear strains and

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reflecting the kinetics of establishment of a thermodynamically equilibrium state in the course of deformation. In the one-dimensional case considered further, the system has the form ∂(ρrν ) ∂(ρurν ) + = 0; ∂t ∂r ∂(ρurν ) ∂[(ρu2 − σ1 )rν ] + + νrν−1 σ2 = 0; ∂t ∂r ∂[ρ(E + u2 2)rν ] ∂[(ρu(E + u2 /2) − σ1 u)rν ] + = 0; ∂t ∂r ∂h2 ν(3 − ν) u d2 ∂h2 +u − = − ; di = hi − q; ∂t ∂r 2 r τ ∂h3 ν(ν − 1) u d3 ∂h3 +u − =− ; (1) ∂t ∂r 2 r τ 1 q = (h1 + h2 + h3 ); 3 ∂E ∂E ; E = E(δ, Δ, S); σi = ρ ; T = ∂hi ∂S ρ τ = τ (δ, Δ, S); δ = = exp(−h1 − h2 − h3 ); ρ0 1 2 (d + d22 + d23 ). 2 1 Here ν is the index of symmetry (ν = 0, 1, and 2 correspond to plane, cylindrical, and spherical symmetry, respectively), u is the velocity, hi is the logarithm of relative extension of the medium element along the principal axes (component of the Hencky strain tensor), hi = ln(ki ), where ki is the coefficient of compression/extension of the medium element along the coordinate axes, T is the temperature, and δ and Δ are the first and second invariants of the strain tensor. The equations are written in the principal axes of the stress and strain tensors (they coincide in the case considered). Δ=

EQUATION OF STATE Equations of state based on the Mie–Gr¨ uneisen approach are most often used to describe shock-wave and high-temperature processes in wide ranges of thermodynamic parameters in crystalline and polycrystalline media [15, 16]. According to this semi-empirical approach, a certain thermodynamic potential (e.g., internal or free energy) is presented as a sum of the cold (elastic) component corresponding to substance compression at 0 K and thermal terms determined by thermal excitation. The main hypothesis here is the assumption that the medium atoms perform small oscillations around the equilibrium state and they can be considered as a

system of harmonic oscillators. The thermal components, in turn, are presented as a sum of terms reflecting the contribution of the thermal motion of atoms or molecules of the crystalline lattice and of thermally excited electrons. A particular form of individual terms depends on the level of generality of the equation being constructed. At comparatively low temperatures, the equations of state contain two unknown functions Pc (V ) and γ(V ) (elastic component of pressure and Gr¨ uneisen coefficient) related by a differential equation [15, 16]. In practice, these functions are constructed independently, on the basis of the general theoretical concepts, and are specified with the use of experimental data (usually, shock adiabats). As the temperature increases, thermal oscillations of atoms become anharmonic, which has to be taken into account in the equations of state [17]. Owing to good results in the description of experimental data and real processes in polycrystalline media, which were provided by this approach, the latter was extended to amorphous media as well. This extension involved a number of additional specific features owing to the structure of amorphous polymer media. It is necessary to take into account that there are two principal types of bonds and, correspondingly, two types of oscillations contributing to the thermal component of the thermodynamic potential. Therefore, two Gr¨ uneisen coefficients are introduced [18]: thermodynamic coefficient corresponding to intrachain oscillations and lattice coefficient reflecting the contribution of interchain oscillations. In deriving semi-empirical equations of state, unified dependences including both contributions are constructed, which is ensured by an appropriate choice of the coefficients [17, 19, 20]. Using the principles formulated in [15] and constricting our considerations to the pressure range within ≈100 GPa, we present the free energy as F = Fc + Fd + FT + Fe , where Fc , Fd , FT , and Fe are the elastic (cold), deviatoric, thermal, and electron components, respectively. The elastic component is usually chosen in the form of an interaction potential or an empirical relation verified on the basis of experimental data. In the case considered, we use a dependence of the form m−1 aδ bδ n−1 ρ + − Fc0 , δ = , Fc (δ) = V0 m−1 n−1 ρ0 which implies that Pc (δ) = aδ m + bδ n − Pc0 . Here a, b, m, n, Fc0 , and Pc0 are constants whose values are determined by a method similar to that described in [19] with the use of data accumulated in [21]. The

Modeling of Shock-Wave Deformation of Polymethyl Metacrylate deviatoric term is chosen in the form of a linear dependence on Δ, which was justified in constructing similar equations of state of polycrystalline media [22]: Fd (δ, Δ) = K(δ)Δ, K(δ) = 2c2l δ ξ0 (cl is the transverse velocity of sound and ξ0 is a constant). In constructing the dependence for the thermal component, it is necessary to identify those parts of the spectral distribution in vibrational spectra of polymers that correspond to vibrational modes of the chains (acoustic branch) and to modes of chain links and their elements (optical branch). The greatest problem is to analyze and determine all vibrational modes of the optical branch, because the number of such modes in polymers with complicated chain links can be rather large. It is possible to get a simplified phenomenological description whose principles described in [23] were used to construct the dependence for FT . The final formulation of the thermal term of the proposed equation of state and some of its corollaries have the form 3 FT (δ, T ) = Rm T Ni ln(1 − exp(−xi )), i=1

θi (δ) , θi = θ0i δ γ0i , T whence it follows that 3 3 N i θi = EiT , ET (δ, T ) = Rm exp x − 1 i i=1 i=1 xi =

PT (δ, T ) = ρ0 δ

3

γ0i EiT ,

i=1

ST (δ, T ) = Rm

3 i=1

Ni

xi − ln(1 − exp(−xi )) , exp xi − 1

where Rm is the universal gas constant divided by the molar weight, θ0i are the characteristic temperatures determined experimentally, γ0i have the sense of analogs of the Gr¨ uneisen coefficients for the corresponding vibrational modes, and Ni are interpolation constants satisfying the condition Ni = N (N is the total number of vibrational modes). The electron component is approximated by the conventional relation 1 Ee (δ, T ) = β(δ)T 2 , β(δ) = β0 δ −Γe . 2 The parameters in the dependences are chosen from the condition of the best fitting of the entire set of available experimental data mainly found in [6, 19–21]. The values of the parameters in the equation of state used

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to perform the calculations described below are ρ0 = 1.2 g/cm3 , a = −38.32, b = 38.16, m = 2.67, n = 2.88, 8.3144 · 10−3 R = kJ/(g · K), cl = 1.4, ξ0 = 3, Rm = μ 100.1164 γ1 = 0.1, γ2 = 1.70, γ3 = 0.6, N1 = 12.10, N2 = 12.65, N3 = 20.23, θ1 = 3.72 · 103 K, θ2 = 2.83 · 102 K, θ3 = 1.42 · 103 K, β0 = 0.9 · 106 kJ/(g · K), and Γe =2/3.

DEPENDENCE FOR THE RELAXATION TIME OF SHEAR STRESSES Taking into account relaxation processes induced by external actions is of principal importance for the adequate description of real properties of polymer materials [1, 2]. In the present model, this is done by means of the dependence of the relaxation time of shear stresses on parameters characterizing the state of the medium. Being induced by mechanical actions, relaxation transitions are associated with various types of thermal motion of structural elements of the polymer, which are characterized by their own spectra of relaxation times. As any thermal-fluctuating process, relaxation is characterized by the activation energy Eia ; for the corresponding time τi , we can use the Boltzmann– Arrhenius formula Ea τi = τ0i exp i , kT where τ0i is the characteristic time of the relaxation transition. The total relaxation time can be presented as a sum of terms corresponding to relaxation mechanisms at different structural levels: Ea τi = τ0i exp i . τ= kT i i When the polymer is loaded, the arising stresses change the activation energy and reduce the potential barrier of the relaxation transition. Taking into account this fact, we obtain E a − α0i σint , τ0i exp i τ= kT i where σint is the intensity of shear stresses and α0i is the effective activation volume treated as the volume of the activated structural element [18]. In the general case, α0i can be considered as a function of temperature and strain rate, whereas Eia is a more complicated function of the process characteristics. An analysis of the basic relaxation mechanisms [24–26] and preliminary calculations show that it is sufficient to leave two terms in the formula for τ . The values of the activation energy of the selected mechanisms were given in [26]. To specify other parameters of this dependence, we use a method developed

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and tested previously for polycrystalline media [27]. It is based on solving a problem of deformation of a thin bar, with the result being the strain diagram for this material. The solution (diagram) depends on the parameters in the formula for the relaxation time. Comparing the calculated diagrams or their elements with experimental data (e.g., by minimizing the functional of the root-mean-square deviations of the characteristic variables), we can find the values of the parameters involved into this formula. The main problem is the small amount of available experimental diagrams for polymers, including PMMA, obtained at high strain rates corresponding to conditions of shock-wave loading. The situation is aggravated by disintegration of the sample in the region of elastic deformation (split Hopkinson pressure bar) under loading with strain rates above 103 s−1 [28], and only a small segment of shockwave loading can be detected in experiments. For this reason, the parameters of the dependence for the relaxation time have to be additionally corrected in calculating the PMMA shock adiabat.

Fig. 1. Strain diagrams for PMMA: the solid curves and points show the calculated and experimental [5] results, respectively.

RESULTS OF SOLVING THE PROBLEMS Calculation of Strain Diagrams The strain diagrams calculated with the chosen values of the parameters of the dependence for the relaxation time are compared in Figs. 1 and 2 with experimental data [5, 29]. Unfortunately, the range of the strain rates in the experiments was limited to ≈103 s−1 . The results of solving the problems of shock-wave deformation show that the constructed dependence is valid in this case as well. Shock Adiabat and Unloading Adiabat The main characteristic of the material behavior under shock-wave loading is the shock adiabat. The experimental points of the shock adiabat (data of [6, 19, 21, 30]) and the calculated data are compared in Fig. 3 in the coordinates “mass velocity U –shock-wave velocity D.” The calculations include both finding the shock compression parameters from the constructed equation of state and direct numerical calculations of propagation of a plane steady shock wave in a half-space in accordance with the formulated model. One objective of direct calculations is to verify the adequacy of the numerical method used. The calculated results almost coincide. As it follows from the figure, the shock adiabat in the coordinates (D–U ) can be presented in the form of at least three linear dependences corresponding

Fig. 2. Strain diagrams for PMMA: the solid curves and points show the calculated and experimental [29] results, respectively; curves 1–7 shows the results for the strain rates of 2 · 10−4 , 2 · 10−3 , 2 · 10−2 , 2 · 10−1 , 3, 45, and 760 s−1 , respectively.

to the increase in the shock-wave amplitude. This is a consequence of the above-discussed structural transitions. Thus, the constructed model reflects the specific features of the PMMA behavior associated with the evolution of microstructural mechanisms.

Modeling of Shock-Wave Deformation of Polymethyl Metacrylate

Fig. 3. Shock adiabat.

Fig. 4. Shock adiabat and unloading adiabat: the solid curves and points show the calculated and experimental [6, 19, 21, 30] results, respectively; the dashed curves are the unloading adiabats constructed on the basis of the equation of state [19, 21].

The shock adiabat and the unloading adiabats from certain states on the shock adiabat, as well as the experimental data [6, 19, 21, 30] and unloading adiabats constructed on the basis of the equation of state [19, 21] are shown in Fig. 4 in the coordinates (σ1 –U ). One parameter of shock compression, which is difficult to describe, is the temperature of the shock-

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Fig. 5. Temperature versus the shock-wave amplitude: the curves show the results calculated by the equation of state constructed in this work (curve 1) and in [20] (curve 2) and [30] (curve 3); the points are the experimental data.

Fig. 6. Profiles of steady shock waves.

compressed substance. The first measurements of the PMMA temperature behind the shock-wave front were performed in [31]. The data obtained there differ from the results of later measurements [32]. A comparison of experimental data with the values calculated by the equation of state is one of the most important tests for equations being constructed. Such a comparison is illustrated in Fig. 5. A significant scatter of experimental data [31–33] evidences first of all that the problem

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of correct measurements of the temperature of shockcompressed materials has not been solved yet. This fact is responsible for the difference in the measured temperatures and, as a consequence, for the difference in the temperature values calculated by different equations of state, as is shown in Fig. 5. The results calculated by our equation of state occupy an intermediate position and perfectly agree with experimental data for the range of moderate shock-wave amplitudes. Propagation of Steady Shock Waves Essential information about the medium behavior can be extracted from the profiles of the fronts of steady shock waves. Figure 6 shows the calculated profiles of the propagating steady shock waves with three amplitudes. The values of these amplitudes are chosen for the state of the medium behind the front to be glassy, highly elastic, and viscoplastic (curves 1–3 in Fig. 6, respectively). We solve an initial-boundary problem with a given constant velocity on one of the boundaries in a plane one-dimensional formulation. In the first case, the shock-wave front structure consists of a sudden (elastic) shock and a subsequent relaxation region converting the material to the final state behind the front. In the second case, a two-wave configuration is formed, where the front decomposes into an elastic precursor and a plastic wave. In the third case, a jumplike transition to the final state occurs. In the latter case, the relaxation processes have enough time to proceed in the shockwave front. These observations are confirmed by Fig. 7, which shows the evolution of the shear stresses in each shock wave. Impact of Plates In experimental investigations, the shock wave in the examined sample is most often generated by an impact of a plate accelerated to a high velocity. To obtain independent verification of the applicability of the constructed model for calculating shock-wave processes, we solved problems that reproduced arrangement of experiments [10, 11] for studying shock-wave processes in PMMA. In the first case, we modeled an impact of a Plexiglas projectile (plate) on a Plexiglas target. The dependence of the mass velocity on time at the contact boundary between the target and the Plexiglas window marked by depositing a thin aluminum layer was recorded in experiments by a laser interferometer. The profiles of the shock pulses for three different amplitudes at a fixed distance from the impact surface are compared in Fig. 8 with the calculated values. The predictions adequately reproduce all specific features of the experimental profiles. In the second case, we studied the evolution of a compression wave with a given

Fig. 7. Behavior of the shear stress in the shock-wave front.

front shape shown in the inset in Fig. 9. The wave was generated by a fused silica projectile. In the calculations, the pulsed imparted to the sample was set as a boundary condition. The shape of the pulse front was unaffected because its evolution was traced under conditions where the signal from the sample–projectile interface did not reach the wave front. A comparison of the experimentally recorded and predicted shapes of the wave front at a distance of 6.35 mm from the impact surface (see Fig. 9) reveals some minor differences between the calculated and experimental data, which can be attributed to different PMMA brands used in these experiments and in experiments aimed at obtaining strain diagrams used as a basis for constructing the dependence for the relaxation time. Decay of the Shock Wave The problem of decay of the shock wave due to its interaction with the overtaking rarefaction wave is of particular interest from the viewpoint of verification of the applicability of the models to describe shock-wave processes. The studies [7, 34] showed that many models used to describe shock-wave processes actually do not yield a correct description of the decay of the amplitude of the shock wave interacting with the overtaking rarefaction wave. This means that the use of such models for solving problems of shock-wave deformation can lead to significant quantitative and qualitative differences of the final results from those observed in practice (in experiments).

Modeling of Shock-Wave Deformation of Polymethyl Metacrylate

Fig. 8. Calculated profiles (curves) and experimental profiles [10] (points) of propagating shock pulses.

Fig. 9. Evolution of the compression-wave front: the curve and points show the calculated and experimental [10] data, respectively.

One factor affecting the rate of shock-wave decay is the process geometry (plane, cylindrical, or spherical symmetry). This fact is illustrated in Fig. 10, which shows the calculated wave decay rates for all these types of symmetry, which were obtained for equivalent initial data. Naturally, verification of the model adequacy is based on appropriate experimental data. There are no available publications on experiments aimed at studying

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Fig. 10. Decay of the shock wave in problems with plane (1), cylindrical (2), and spherical (3) symmetry.

Fig. 11. Decay of the shock wave.

decay of waves generated by an impact of plates. Decay of the shock wave in PMMA initiated by blasting an explosive charge (the charge diameter was smaller than the sample diameter) on the surface of a Plexiglas cylinder was studied in [35, 36]. The data of these experiments are compared with the calculated results in Fig. 11. The boundary conditions here were set as the velocity of motion of the contact surface with time, where the initial value of this velocity was determined

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Fig. 12. Profiles of the decaying shock wave at the time instants of 0.333, 1, 2.06, 3.12, 4.12, 5.93, 7.04, 9.11, 10.4, 13.1, 15.5, and 21.3 µs.

mer are taken into account in constructing the equation of state and the dependence for the relaxation time of shear stresses, which are necessary to close the model. To specify the closing relations, experimental data on the diagrams of dynamic deformation and on the shock adiabat for PMMA are used. The results predicted by the formulated model are carefully compared with appropriate experimental data. The calculations correctly reproduce the strain diagrams, the shock adiabats, and the unloading adiabats. Particular attention is paid to comparisons of the calculated and experimental data on the temperature of the shock-compressed material and decay of the shock wave interacting with the overtaking and side rarefaction waves. The results of comparisons of the solutions of particular problems with the corresponding experimental data show that the constructed model ensures an adequate description of shock-wave processes in PMMA. This work was supported by the Integration Project No. 115 of the Siberian Branch of the Russian Academy of Sciences. REFERENCES

by solving the Riemann problem on the interface between the detonation products and the sample. The velocity was assumed to decrease in accordance with the isentrope of the detonation products. The results calculated in the plane case appreciably deviate from the experimental points [35] beginning from the distance of ≈3 cm (dashed curve in Fig. 11), but agree well with the data obtained in [36]. A simple estimate with the use of the velocity of sound in the compressed substance for the explosive charge size used in [35] shows that the side rarefaction waves merge on the axis of symmetry approximately at this distance. We can naturally assume that the type of symmetry of the problem is changed at this instant, and the wave decays in accordance with the pattern for spherical symmetry. A corresponding correction in the calculation yields a solid curve located sufficiently close to the experimental points. As charges of a larger diameter were used in [36], this process is described within the framework of the plane approximation. The transformation of the pulse propagating over the sample in the first case is illustrated in Fig. 12. CONCLUSIONS A model of dynamic deformation of polymethyl metacrylate is formulated in the paper. The model is based on the Maxwellian concepts of the mechanisms of irreversible deformation of continuous media. Specific features and variety of interactions inside the molecular chains and between the chains forming the poly-

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