Analytical solution of the problem of repeated imposition of large tension-compression and torsion deformations in a multilayered viscoelastic cylinder

Abstract

The problem of repeated imposition of large tension-compression and torsion deformations is solved for a sequentially formed multilayered cylinder, each layer of which is made of a viscoelastic incompressible material. After attaching each successive layer, tension-compression and torsion of the composite cylinder is performed. Based on the description of the formation and deformation of the cylinder, a mathematical formulation of the problem is given for the quasi-static case and the main steps of its exact analytical solution are described. The results of numerical calculations are presented.

Keywords

Viscoelastic material large deformations multi-stage deformation exact solution analytical solution

1. Introduction

In nonlinear mechanics of a deformable solid, great importance is attached to the study of effects that arise during torsion of cylindrical bodies. One of these effects is the Poynting effect [1,2], which consists in changing the length of the cylinder during torsion in the absence of axial forces applied to the bases of the cylinder. Another manifestation of this effect is the occurrence of stresses directed along its axis during torsion of the cylinder, when the bases of the cylinder cannot move along this axis.

When solving the problem of torsion of a cylinder made of a nonlinear elastic material, exact analytical solutions are of great importance. For incompressible isotropic nonlinear elastic materials, an exact solution to the problem of torsion and tension-compression of a circular cylinder is known, which is one of the universal solutions [2 –9]. Torsion of cylinders of non-circular cross-section under large deformations is studied by Zubov and Bogachkova [10], in which, in particular, approximate analytical solutions for cylinders of elliptical and rectangular cross-sections were obtained. The influence of surface effects on the stress-strain state of rods during torsion and tension-compression was studied by Sigaeva and Czekanski [11].

For compressible nonlinear elastic materials, solving the problem of torsion of rods is more complicated. Exact analytical solutions are known for the case of pure torsion for some special classes of materials [12]. Approximate analytical solutions using series were proposed by De Pascalis [13]. Dell’Isola et al. [14,15] applied the Signorini method to the approximate calculation of the stress-strain state of pre-deformed rods. Hajhashemkhani and Hematiyan [16] formulated the problem of inflation, stretching and torsion of a functionally gradient compressible hyperelastic tube as a one-dimensional problem and solved by the finite difference method.

Effects similar to the Poynting effect have been revealed at large deformations for inelastic materials. In plasticity theory, this effect is known as the “Swift effect” [17]. An experimental and numerical study of this effect for polymers in a glassy state was carried out by Wu and van der Giessen [18]. Bruhns et al. [19] obtained an approximate analytical solution to the problem of torsion of a circular cylinder made of an elastoplastic material within the framework of a plasticity model with isotropic hardening. Cazacu et al. [20] studied within the framework of the elastic-plasticity model, the Swift effect during cyclic deformation of a sample. Colak and Krempl [21] simulated the Swift effect in a hollow cylinder made of viscoplastic material, including the case of cyclic deformation. Khan et al. [22] studied the torsion of a polymer material within the framework of nonlinear viscoelasticity models. The combined action of high pressure and torsion is used to create metals with very small grain sizes [23]. To model this phenomenon, both numerical methods and approximate analytical solutions are used [24,25]. An analytical solution to the problem of large elastoplastic deformations of a cylinder under the combined action of pressure and torsion was obtained by Sevastyanov [26].

When studying joint torsion and tension-compression, problems of loading of bodies in several stages are of interest, when at each stage a new part of the body is attached to the loaded body. These problems belong to the class of problems on the surface growth of deformable solids. A detailed literature review of the problems on surface growth is given by Sozio and Yavari [27]. The models of surface growth (accretion) include the models of continuous growth and the models of discrete growth [28 –31]. If the deformations are sufficiently large, then the theory of multiple superposition of large deformations is used when these problems are formulated and solved [32 –34]. Zingerman and colleagues [35,36] considered the problems of tension-compression and torsion of composite cylinders made of nonlinear elastic materials. These problems were solved for the case of discrete growth when one or more cylindrical layers were attached to the cylinder. Similar problems have been solved for hypoelastic materials [37].

The continuous surface growth of nonlinear-elastic circular cylindrical bars subjected to finite torsion is analyzed by Yavari and Pradhan [38]. The residual stresses are computed, and nonlinear effects are estimated. The surface growth of cylindrical bars, growing continuously in radial direction and subjected to axial stretch or the action of axial force, is investigated by Yavari et al. [39]. In these papers, it is assumed that strains are finite. For viscoelastic bodies the models of continuous accretion under finite and small strains are developed by Drozdov [40]. In particular, the continuous growth of viscoelastic bars in this book is analyzed under small torsion strains.

The models of surface accretion can be used for the description of growth of biological tissues [41,42] and for the strength analysis of additive manufacturing products [43 –45].

In the current work, the discrete accretion of a composite viscoelastic cylinder subjected to tension/compression and torsion is analyzed under finite strains. The cylinder is considered as a body subjected to the imposition of large deformations. The mechanical properties of the cylinder material are described by constitutive relations of integral type, which, in the absence of viscosity, are reduced to the constitutive relations for a neo-Hookean material [46,47].

2. Notation

In this work, the notation of the theory of repeated superposition of large deformations is used [32 –37]. The states (configurations) of the body are numbered from 0 (initial state) to n (final state).

$σ_{l, m}$ —the Cauchy stress tensor during transition from state l to m.

$Σ_{l, m}$ —generalized stress tensor during transition from state l to m.

$C_{l, m} = F_{l, m}^{T} \cdot F_{l, m}$ —the Green deformation tensor during transition from state l to m.

$F_{l, m}$ —deformation gradient during transition from state l to m.

$E$ —unit tensor.

$tr X = X \cdot \cdot E$ —trace of a rank two tensor X.

$p_{m}$ —pressure that is not expressed in terms of strain, in state m.

$x_{s}^{(m)}$ —Cartesian coordinates of particles in a compound cylinder in state m.

$i_{s}^{(m)}$ —coordinate vectors of Cartesian coordinates in state m.

$r_{m}, φ_{m}, z_{m}$ —cylindrical coordinates of particles in a compound cylinder in the state m.

$R_{m}^{(l)}$ —the outer radius of the cylindrical layer l in state m.

Relationships between cylindrical and Cartesian coordinates:

\begin{matrix} x_{1}^{(m)} = r_{m} \cos φ_{m} & x_{2}^{(m)} = r_{m} \sin φ_{m} & x_{3}^{(m)} = z_{m} \end{matrix}

$e_{r}^{(m)}, e_{φ}^{(m)}, e_{z}^{(m)}$ —unit vectors tangent to the coordinate lines of cylindrical coordinates in the state m:

\begin{matrix} e_{r}^{(m)} = i_{1}^{(m)} \cos φ_{m} + i_{2}^{(m)} \sin φ_{m} & e_{φ}^{(m)} = - i_{1}^{(m)} \sin φ_{m} + i_{2}^{(m)} \cos φ_{m} & e_{z}^{(m)} = i_{3}^{(m)} \end{matrix}

3. The mechanical problem statement

The problem is formulated within the framework of the theory of repeated superposition of large strains [32 –35]. There is a cylindrical bar made of incompressible isotropic viscoelastic material. Initially, there are no stresses and deformations in this bar. Then, at a given time interval, the bar undergoes an initial torsional deformation together with tension or compression in the axial direction under the action of a torque and axial force. This deformation can change over time according to a given rule. The lateral surface of the cylinder is unloaded. Due to the incompressibility of the material, tension or compression of the rod along its axis is accompanied, respectively, by compression or tension of the rod in the radial direction. It is assumed that at the end of the given time interval the bar is in the first intermediate state. In this state, a hollow undeformed viscoelastic cylinder (layer) joins it. The inner radius of this hollow cylinder coincides with the radius of the bar after the initial deformation, and the length coincides with the length of the bar after the initial deformation. The cylinders are glued along their common boundary surface (for the initial cylinder this is the outer and only boundary, for the attached cylinder—the inner one), and in what follows it is assumed that the conditions of ideal contact are satisfied on this surface.

Furthermore (during the next time interval), there is a joint additional deformation of the cylinders. It is also a deformation of torsion together with tension or compression along the axis, which occurs as a result of the action of a torque together with an axial force. At the end of the second time interval, the compound cylinder passes to the second intermediate state. Then the next hollow viscoelastic cylinder (layer) is attached to it. This layer is not initially deformed.

This procedure of sequential connection of the cylinders with their subsequent deformation of torsion and tension-compression can be repeated several times.

The solution is obtained for the quasi-static processes. It is assumed that each layer is added instantly, in sense of a study by Christensen [48], and the dynamical effects are not taken into account.

4. Calculation of strains and stresses in a composite cylinder

We consider a multilayered circular cylinder deformed over several stages. The mth stage ( $m = 1, \dots, n$ ) occurs over a given period of time $t_{m - 1} \leq t \leq t_{m}$ , $t_{0} = 0$ .

At the beginning of the $m th$ stage the next mth undeformed layer with an outer radius $R_{m}^{(m - 1)}$ is added to the multilayered cylinder. Furthermore, during the stage, the multilayered cylinder is subjected to tension-compression and torsion, deforming as a single whole.

The coordinates of particles of the cylinder during the mth stage of deformation are related to the coordinates of these particles in the previous state as follows [36,37]:

\begin{matrix} r_{m} = λ_{m} (t) r_{m - 1}, & φ_{m} = φ_{m - 1} + α_{m} (t), & z_{m} = z_{m - 1} λ_{m}^{- 2} (t) \end{matrix}

(1)

where $λ_{m} (t)$ is the specified elongation factor at stage m; $α_{m} (t) = k_{m} (t) z_{m - 1}$ ; $k_{m} (t)$ –specified characteristic of torsional deformation at stage m. Relations (1) ensure the fulfillment of the incompressibility condition. Note that when $t \leq t_{m - 1}$ , relations $λ_{m} (t) = 1$ , $k_{m} (t) = 0$ must be satisfied. When $t \geq t_{m}$ , assumptions $λ_{m} (t) = λ_{m} (t_{m})$ , $k_{m} (t) = k_{m} (t_{m})$ must be fulfilled.

Considering relations (1) for several successive stages, we obtain:

\begin{matrix} r_{m} = Λ_{l, m} r_{l - 1}, & φ_{m} = φ_{l - 1} + α_{l, m}, & z_{m} = z_{l - 1} \cdot Λ_{l, m}^{- 2} \end{matrix}

(2)

where

\begin{matrix} Λ_{l, m} = λ_{l} λ_{l + 1} \dots λ_{m - 1} λ_{m}, \\ α_{l, m} = k_{l, m} z_{l - 1}, \\ k_{l, m} = k_{l} + k_{l + 1} Λ_{l, l}^{- 2} + \dots + k_{m} Λ_{l, m - 1}^{- 2} \end{matrix}

To characterize the deformation of the lth layer during the mth stage, the deformation gradient is used:

\begin{matrix} F_{l, m} = Λ_{l, m} (e_{r}^{(m)} \otimes e_{r}^{(l)} + e_{φ}^{(m)} \otimes e_{φ}^{(l)}) + \\ + k_{l, m} r_{m} e_{z}^{(m)} \otimes e_{φ}^{(l)} + Λ_{l, m}^{- 2} e_{z}^{(m)} \otimes e_{z}^{(l)} \end{matrix}

(3)

The Green deformation tensor during transition from state l to m is computed as follows:

C_{l, m} = F_{l, m}^{T} \cdot F_{l, m}

It is assumed that the material of the multilayered cylinder is viscoelastic. The generalized stress tensor satisfies the constitutive relations:

\begin{matrix} {Σ_{l, m} |}_{t = t_{m}} = - p_{m} C_{l, m}^{- 1} + \\ + \sum_{q = l}^{m} \int_{t^{(q - 1)}}^{t^{(q)}} (μ + η e^{- β (t_{q} - τ)}) (E - \frac{1}{3} \frac{\partial (C_{l, q}^{- 1} tr C_{l, q})}{\partial τ}) d τ \end{matrix}

(4)

where μ, η, β are the mechanical parameters of the material, $p_{m}$ is the Lagrange multiplier (pressure). If it is necessary to calculate $Σ_{l, m}$ at an arbitrary moment of the mth stage of deformation, $t_{m}$ should be replaced by t in the last integral (at $q = m$ ). The relations of this type are proposed by Adamov and colleagues [46,47] and used for solution of some stress concentration problems under finite strains [33,49].

The true stress tensor is related to the generalized stress tensor as follows:

σ_{l, m} = F_{l, m} \cdot Σ_{l, m} \cdot F_{l, m}^{T}

(5)

When writing formula (5), the incompressibility of the material is taken into account.

Using formulas (3)–(5), we can write down the stresses $σ_{l, m}$ at an arbitrary moment t from the interval $t_{0} \leq t \leq t_{m}$ up to the factor $p_{m}$ , which is determined from the equilibrium equations and boundary conditions of contact between the layers.

We write the equilibrium equations for the mth state in the form:

\frac{\partial {(σ_{l, m})}_{rr}}{\partial r_{m}} + \frac{{(σ_{l, m})}_{rr} - {(σ_{l, m})}_{φ φ}}{r_{m}} = 0

(6)

Differential equation (6) is supplemented with boundary conditions. For the outer layer $l = m$ this condition has the form:

{{(σ_{m, m})}_{rr} |}_{r_{m} = R_{m}^{(m)}} = 0

(7)

For the internal layers the boundary conditions are written as follows:

{{(σ_{l, m})}_{rr} |}_{r_{m} = R_{m}^{(l)}} = {{(σ_{l + 1, m})}_{rr} |}_{r_{m} = R_{m}^{(l)}}

(8)

By solving equation (6) with boundary condition (7) or (8) for each $l -$ th layer, we determine $p_{m}$ , and, as a consequence, $σ_{l, m}$ . Thus, we obtain a complete description of the deformation process at each stage.

The axial force and torque in the mth state are computed using the relations [29 –31]

N_{0} = 2 π \int_{0}^{R_{m}^{(m)}} σ_{zz} r_{m} d r_{m}

M_{0} = 2 π \int_{0}^{R_{m}^{(m)}} σ_{φ z} r_{m}^{2} d r_{m}

For the verification of the obtained results, the computations were performed for the case of pure elastic material [ $η = 0$ in the constitutive relations (4)]. The results are found to be the same as the results for neo-Hookean material [35,50].

5. Numerical studies

Calculations were performed for the case of $n = 2$ layers of viscoelastic material. The following notation is used: R is the radius of the inner layer in the first state, T is the duration of the first stage of deformation $(R = R_{1}^{(1)}, T = t_{1}); r = r_{1} / R$ . Material constants are taken as follows: $η = μ$ , $β = 1 / T$ . The outer radius of the second layer at the moment of attachment $R_{2}^{(1)} = 2 R$ .

The formulas for the deformation parameters have the form

\begin{matrix} λ_{1} (t) = 0, 2 H (t) + 1, & λ_{2} (t) = 1 + (L - 1) H (t - t_{1}), \\ k_{1} (t) = KH (t), & k_{2} (t) = KH (t - t_{1}), \end{matrix}

where $H (t)$ is the Heaviside unit function; L and K are numerical parameters, the values of which are different for the different graphs below.

The distribution of true stresses in the cylinder at the final moment of time $t = t_{2} = 2 T$ is shown in Figures 1 –8 for different values of K, L. The stresses are referred to the modulus μ. The discontinuities in Figures 3 –8 are associated with the fact that the first (internal) layer is preliminarily strained, and the second (external) layer has no preliminary strains. So, there is a jump of total strains (and, respectively, a jump of total stresses, with the except of radial stresses) at the boundary between the layers.

Figure 1.

Distribution of $σ_{rr}$ in the cylinder at time $t = 2 T$ at $K = 1$ for various values of L (see graph legend).

Figure 2.

Distribution of $σ_{rr}$ in the cylinder at time $t = 2 T$ at $L = 1$ for various values of K (see graph legend).

Figure 3.

Distribution of $σ_{φ φ}$ in the cylinder at time $t = 2 T$ at $K = 1$ for various values of L (see graph legend).

Figure 4.

Distribution of $σ_{φ φ}$ in the cylinder at time $t = 2 T$ at $L = 1$ for various values of K (see graph legend).

Figure 5.

Distribution of $σ_{zz}$ in the cylinder at time $t = 2 T$ at $K = 1$ for various values of L (see graph legend).

Figure 6.

Distribution of $σ_{zz}$ in the cylinder at time $t = 2 T$ at $L = 1$ for various values of K (see graph legend).

Figure 7.

Distribution of $σ_{φ z}$ in the cylinder at time $t = 2 T$ at $K = 1$ for various values of L (see graph legend).

Figure 8.

Distribution of $σ_{φ z}$ in the cylinder at time $t = 2 T$ at $L = 1$ for various values of K (see graph legend).

The dependences of hoop stresses at the outer boundary of the cylinder on the deformation parameters K and L at various times are shown in Figures 9 and 10. The values of time, referred to T, are given in the graph legends.

Figure 9.

Dependence of $σ_{φ φ}$ at the outer boundary of the cylinder on L at different times (see graph legend).

Figure 10.

Dependence of $σ_{φ φ}$ at the outer boundary of the cylinder on K at different times (see graph legend).

The dependences of axial force and torque on the deformation parameters K and L at various times are shown in Figures 11 –14. The values of time, referred to T, are given in the graph legends.

Figure 11.

Dependence of $N_{0}$ on L at different times (see graph legend).

Figure 12.

Dependence of $N_{0}$ on K at different times (see graph legend).

Figure 13.

Dependence of $M_{0}$ on L at different times (see graph legend).

Figure 14.

Dependence of $M_{0}$ on K at different times (see graph legend).

One can see from Figures 9 –13 that significant nonlinear effects take place for the given values of parameters. The dependence of torque on the deformation parameter K is linear (Figure 14). Figures 9 –14 show that the hoop stress, axial force, and torque are decreased in time by absolute value. This effect is related with the relaxation processes.

6. Conclusion

An exact analytical solution is constructed for the problem of tension-compression and torsion of composite viscoelastic cylinder undergoing large strains. It is assumed that the cylinder is composed of some layers and these layers are added successively. The problem is formulated using the theory of multiple imposition of large strains. The results of numerical studies are obtained. Significant nonlinear effects are found. The effects of viscosity are analyzed. The results of solution can be used for the verification and validation of software packages for engineering analysis. For example, this solution can be used to verify calculations of the stress–strain state of structural elements manufactured using additive technologies [45,51]. It is of interest to take into account surface effects [52]. It is possible to apply the proposed approach to modeling the surface growth of deformable bodies [28,29,31] and to modeling phase transitions [53].

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was completed at Moscow State University and supported by the Russian Science Foundation (project no. 22-11-00110).

ORCID iD

VA Levin

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