Wave-propagation in an incompressible hollow elastic cylinder with residual stress

A study is presented to observe the effect of residual stress on waves in an incompressible, hyper-elastic, thick and hollow cylinder of infinite length. The problem is based on the non-linear theory of infinitesimal deformations occurring after a finite deformation. A prototype model of strain energy function is used which adequately includes the effects of residual stress and deformation. The expressions for internal pressure and the axial load are calculated and graphical illustrations are presented. Analysis of infinitesimal wave propagation is carried for the axisymmetric case in the considered cylinder. Numerical solution is obtained in the undeformed configuration and analyzed for the two-point boundary-value problem. Dispersion curves are plotted for varying choice of parameters.

1. Introduction. The study of waves in solids is motivated by numerous fields such as (but not limited to) ultrasonics, borehole testing, medicine and technology, nondestructive testing of materials, composite materials etc., to name a few. A solid material may not be completely free of stresses in its reference state. The manufacture and construction of a material may introduce stresses that remain locked inside the material after when the source of producing stress is removed. This stress is now termed in literature as an initial stress. This initial stress is used along with the linear theory of elasticity by [5,6] who explored the propagation of elastic waves and presented specialized results in geographical context. Here, we use the term initial stress to represent the cases of both the prestress or the residual stress. A pre-stress is the stress produced in the elastic material due to a finite deformation which results from an applied load. An initial stress is a residual stress when it is the result of a process other than a finite deformation. For example, as the case of growth in tissues or tectonic events in rocks, etc. This concept of residual stress is credited to [10]. For further study on residual stress and development of basic constitutive equations for residually-stressed materials, we refer to [10,11,12,13,14,15,16]. It was found that the mechanical properties of elastic materials are inhomogeneous and anisotropic in the presence of inhomogeneous residual stress. The reader is also referred to [32] and the references therein. [17] followed the work in [10] and presented generalized results which relate to much earlier work of [6].
In this paper, a study on the propagation of small amplitude waves in a residuallystressed incompressible (hollow) cylinder is presented. The theoretical formulation is followed from [28]. To illustrate various results numerically, a prototype strain energy function is used which is a function of initial stress and the finite deformation through their respective second rank tensors. The theory of small (static or time dependent) deformations, which are usually referred to as incremental deformations, superimposed on large or finite deformations is used to analyze the effect of the initial stress on the speed of wave in the material. The components of (the fourth order) elasticity tensor appearing in this theoretical formulation depends on the residual stress tensor components and the components of deformation gradient tensor (See [27]). The approach acquired by [5,6,32] is different from this more generalized approach since they used the linearized theory and a very specialized assumed form of the constitutive equation. Also, [17] assumed the initial stress to be small in magnitude so that the terms are linear in the initial stress and made use of a different form of elasticity tensor in calculations. More recently, the reader is referred to the work done by [3,24,33,4].
Basic equations for a residually-stressed elastic material are presented and also the discussion on the elasticity tensor for residually-stressed materials is carried out. The focus is on development of he constitutive law for an initially stressed material that has no intrinsic material symmetry, i.e. its response relative to the undeformed configuration is considered isotropic in the absence of initial stress. The constitutive law of the material is based on a strain-energy function that depends on the combined invariants of the initial stress tensor and the right Cauchy-Green deformation tensor. For an incompressible material in the three-dimensional case, 9 such independent invariants exist. Expressions for the Cauchy stress and nominal stress tensors and the elasticity tensor are given in general forms for incompressible materials are given. These are then specialized for a strain energy model. In the later part of the paper, problem of wave propagation is analyzed for a hollow cylindrical tube with inhomogeneous residual stress. Numerical solution to the boundary value problem is presented and behavior of wave speed in the presence of residual stress is presented through graphical illustrations. Various results are specialized to match the results in the linear theory for isotropic materials.
2. Problem formulation: Constitutive laws and elasticity tensor in terms of invariants. In this section, the reader is generally referred to ( [28], [29]) for details on the basics of the theory and various expressions involved in the problem formulation.
In this problem, an incompressible material with an initial stress is considered. Let the elastic response of this material is measured by the strain energy function P(C, ζ). Here, C = Λ T Λ is the usual right Cauchy-Green tensor and ζ is the initial stress tensor and Λ is the the deformation gradient tensor. Let ζ ii , (i = 1, 2, 3) and ζ ij , i ̸ = j, (i, j ∈ {1, 2, 3}) are the normal and shear components of the initial stress, respectively. Let λ 1 , λ 2 , λ 3 are the principal stretches corresponding the principal axes x 1 , x 2 and x 3 , respectively, when the material is subjected to a pure homogeneous pre-strain. The strain energy is considered invariant under an orthogonal rotation if it depends on the following invariants , det(ζ)}, I 5 = tr(Cζ), I 6 = tr(C 2 ζ), I 7 = tr(Cζ 2 ), which reduce to in the undeformed configuration. For further details on invariants of tensors, we refer to [28,30].
Considering that the strain energy function P is a function of the above mentioned ten invariants, we have where P r = ∂P/∂I r (See [28]). For the considered material (with I 3 = 1), nominal stress tensor, denoted S, is and Cauchy stress, denoted T is where p is the Lagrange multiplier associated with the constraint and Ξ = ΛζΛ T and B * = I 1 B − B 2 . in the undeformed configuration, Eq. (5) gives where p 0 is the value of p evaluated with respect to reference configuration. This suggests where the derivatives of P in Eq. (7) are evaluated in reference configuration. Considering the principal axes of B, Ξ ij = λ i λ j ζ ij and therefore Eq. (5) in its component form is with no summation on the repeated indices on the right hand side. The (updated) elasticity tensor is given by where P rs = ∂ 2 W/∂I r ∂I s . For an incompressible material, Eq. (9) gives For brevity, the dependence of P on I 7 and I 8 is omitted here. It can be assumed that Ξ ij = 0, i ̸ = j and this results in the following non-vanishing components For the initial configuration. Eq. (10) reduces to while Eq. (7) are being satisfied. Here α 1 = 2(P 1 + P 2 ), α 2 = 2P 6 , α 3 = 4(P 55 + 4P 56 + 4P 66 ), If the initial stress is assumed to vanish, Eq. (13) gives The classical isotropic strain energy function, in terms of invariants, is where λ and µ are the Lamé moduli. From Eq. (16) we have where µ is the shear modulus in B r . The difference between the expressions in Eq. (17) and any alternative expression is absorbed byṗ, which is the incremental Lagrange multiplier.

Radial inflation and axial extension.
In this section, a thick walled circular hollow cylinder is considered. The geometry in the undeformed configuration is where A and B are the internal and outer radii, respectively, and L is the length measured in the coordinate system R, Θ, Z. Let the cylinder is finitely deformed so that the circular cylindrical shape is maintained and the new coordinates are r, θ, z Under the finite deformation, the basis vectors {e R , e Θ , e Z } transform to {e r , e θ , e z }. Therefore, the geometry in the deformed configuration is where a and b are the internal and outer radii, respectively, l is the length after deformation. In this case, we consider the deformation where λ z denotes the uniform axial stretch. Let the unit basis vectors associated with coordinates ζ, z, r, are e 1 , e 2 , e 3 , respectively whereas λ 1 , λ 2 , λ 3 denote the corresponding principal stretches. The incompressibility condition (λ 1 λ 2 λ 3 = 1) and Eq. (29) give where λ is the azimuthal stretch which, from Eq. (29), is a function of r (or R) . It follows from Eqs. (29) and (30) where Therefore, for fixed λ z , we have which is satisfied during inflation of the tube. Equality holds in Eq. (33), for which is the case of simple tension.
It is convenient to assume that residual stress has only non-zero diagonal components with the chosen cylindrical polar axes. Let the principal residual Cauchy stresses are denoted ζ 1 , ζ 2 , ζ 3 . Then, from Eq. (1), the invariants can be rewritten as Considering P to be independent of I 5 , I 6 , I 9 and I 10 , the principal components of Cauchy stress are Considering ζ ij = 0, i ̸ = j and using the relation it can be easily deduced that Using Eq. (34) and rewriting P as a function of λ 1 = λ and λ 2 = λ z and ζ 1 , ζ 2 , ζ 3 asP It may be noted that generally P is not symmetric in λ, λ z . Using Eq. (39), we have For brevity, we can assume ζ 2 = 0 whereas the non-zero residual stresses satisfy (since Div ζ = 0) with which are the boundary conditions in the undeformed state of the cylinder. After the deformation occurs, Eq. (44) becomes with as the boundary conditions. Here P is the internal pressure for inflating the tube. After some rearrangements and using Eqs. (29), (30) and (31), we have Integrating Eq. (46) and incorporating the boundary conditions (47), we get where Eq. (48) is being used to change the independent variable r with λ.
Here P is regarded as a function of λ z and λ a due to Eq. (31). Differentiating Eq. (49) with respect to λ a , we have Using Eq. (41) in Eq. (46) and integrating, we get where Eq. (48) is being used. For a specific strain energy functiton, the above expression gives the radial Cauchy stress component. Integrating Eq. (44), we have Using (45), we must have In [22] and [23], a uniform circumferential stress is assumed. The results show that the radial residual stress (here ζ 3 ) is small in magnitude. It is negative except at the boundaries where its value is zero due to Eq. (45). The circumferential stress (here ζ 1 ) is of tensile nature at the outer radius and compressive at the inner radius. The reader can find a similar discussion in [7], [31] and [25]. However, in [21], uniform circumferential strain distribution and uniform circumferential stress are considered which result in opposite signs of the residual radial and residual circumferential stresses as compared to those in [22] and [23]. Further, the results in [22] and [23] match those presented in [26] for a particular choice of the circumferential growth stretch ratio. This is when growth is considered in addition to the uniform circumferential stress. [8] have considered a different strain energy function and the growth in the material results in the development of residual stresses. The results thus obtained very similar results to those in [23] .
In view of the above cited papers, for the problem considered here, a particular behavior of residual circumferential stress is expected inside a residually-stressed thick-walled tube. For simplicity only, let ζ 1 (R) is linear in R given by where k 1 and k 2 are constants. Since B > A, it can be easy to infer that which is in accordance with the expected behavior of the residual circumferential stress that is negative(positive) on the inner(outer) boundary. From Eq. (53), we have which implies k 1 = 2k 2 . Integration of Eq. (52) gives which vanishes at R = A and R = B.
Equations (54) and (57) can be rewritten as For a fixed tube thickness, the behavior of these stress components (in dimensionless form) is shown in Figure 1. The behaviors are very similar to those observed in [23], [22], [31], [25], [26] and [8]. A simple non-linear model. In this section, a prototype model is chosen to understand the effect of a non-homogeneous residual stress on wave propagation in an incompressible elastic cylinder. A finite deformation is considered from a configuration that is subject to a non-homogeneous initial stress. Following [28], a simple strain-energy function of the form where µ andμ are material constants.and Eq. (34) gives the invariants I 1 , I 41 and I 7 . Rewriting Eq. (60) in terms of λ 1 , λ 2 , λ 3 and ζ 1 , ζ 2 , ζ 3 as Using (60), for the deformed configuration of an incompressible material, various required derivatives are which reduce to in the undeformed configuration.
3.2. Pressure inside in a thick-walled tube with residual stress. In this section, an explicit expression for calculating the pressure P inside a thick-walled tube is presented in terms of the principal stretches and principal residual stress components. Equations (30) and (60) give and Eq. (64) therefore gives which reduces to in the undeformed configuration.
Also, from Eq. (61), we have which, in the undeformed configuration, reduces to Making use Eq. (31) in Eqs. (58) and (59), ζ 1 and ζ 3 can be rewritten aŝ where the dimensionless quantity is being introduced. It may be noted here λ serves as a variable of integration and the residual stress is not explicitly dependent on the stretches. It can be written from Eq. (50) where and β 0 = µμ. The value of β 2 is given by Eq. (71). The dimensionless pressure is calculated using Eqs. (64) and (70) in (49) and is given by where β 0 is defined above. Figure 2 is the plot of dP * dλa versus λ a . After using Eqs. (31) and (49), it may be noted in Fig. 3 that for λ a = λ −1/2 z , the pressure is zero at λ a = λ −1/2 z in the absence of residual stress. With increase in the value of λ a , the pressure tends to remain constant. This is a similar behavior as observed in the case of neo-Hookean or Mooney-Rivlin material models. This behavior is obvious in plots (a) and (b) in Fig. 2. The plots (c) and (d) in Fig. 2 are for the derivative of pressure with nonzero residual stress. It is observed that the vanishing of the pressure takes place at a shifted value of λ a (for fixed axial stretch) depending on the values of β 1 , β 2 and the ratio B/A. Also, with variation in the values of the parameters, the pressure is expected to increases or decreases. The plot in Fig. 3 is the behavior of pressure for zero residual stress whereas Figs. 4-6 show the graphs for various values of λ z , B/A, β 1 and β 2 with non-zero residual stress. It is observed that for β 1 > 0, β 2 > 0, the pressure increases whereas for β 1 < 0, β 2 < 0, the pressure decreases with increasing values of λ a . For fixed axial stretch and wall thickness, Fig.  6 illustrates the increasing or decreasing trend of pressure for different combinations of β 1 and β 2 characterizing the presence of residual stress.

3.3.
Axial load for a thick-walled tube with residual stress. An axial load, say N, must be applied to the ends of the tube to keep the axial stretch λ z fixed. The value of N is given by Using Eqs. (33) and (41)-(48), the above expression gives where A ′ = πµA 2 . Figure 7 is a plot of axial load versus λ a for various wallthicknesses and values of parameters. The load tends to remain constant for increasing values of λ a for zero residual stress. This behavior is similar to that of neo-Hookean materials and Mooney-Rivlin materials. Figure 8 shows the effect of residual stress on the axial load for a given wall-thickness. With increasing values of λ a , the axial load may decrease or increase which also depends on the values of the parameters that specify the magnitude of residual stress.
where x and X are the positions of the particle in the deformed and the undeformed configurations, respectively. Let the gradient of displacement is such that Λ = I + G.
(80) Consider now infinitesimal deformations (which are also time dependent) in the cylindrical tube after it undergoes a finite deformation. It can therefore be inferred from Eq. (80) thatΛ for small G.
If u = ve ζ + we z + ue r , we have where [G] is the matrix of components of G and (r, ζ, z) in the subscripts denote the partial derivatives. Since the axisymmetric case is under consideration, Eq. (85) gives where the subscripts represnt the derivative axxording to the respective variable. Also, the incompressibility condition, G pp = 0, gives or Equation (88) thus allows a potential function ϕ = ϕ(r, z), such that For i = 2 and i = 3 From Eq. (83), we have respectivelẏ In the expanded form, Eqs. (90) and (91) givė p z + ρP tt = A 03232 P rr + A 02222 P zz + (rA ′ 03232 + A 03232 )P r /r +(A 02233 + A 03223 )u rz respectively. The symbol ' ′ ' denotes the derivative with respect to 'r'. Taking r-derivative of Eq. (92), z-derivative of Eq. (93) and subtracting, we get From Eq. (89), we have Using the above expressions and their various derivatives in Eq. (94), we get Considering a solution for ϕ of the form where ω is the frequency and k is the wave number. Using Eqs. (97) Therefore, Eq. (98) becomes The pressure loading boundary condition in the undeformed configuration is where P is the pressure on the boundary per unit area of the deformed configuration and N is the unit normal to the area. In its incremental form, Eq. (103) (after updating to the deformed configuration), we havė Equation (104) can now be specialized for an infinite cylindrical tube with the outer boundary as traction free and the inner boundary subject to pressure P . For i = 2, 3, in Eq. (104) withṖ = 0, the boundary conditions arė which for the considered case, with Eq. (87), gives on r = a, b Using Eqs. (10), (97), (39) and (47), Eq. (106) becomes and respectively.

4.1.
Analysis for a special model. The theory developed in the previous section may be used now for the special model in Eq. (60). The principal residual stress components from Eqs. (58) and (59) using Eq. (29) in the deformed configuration, areζ and β 2 = k 2 A/µ. Using these notations, Eq. (100), in the deformed configuration, becomesγ whereω which, from Eq. (115) gives along with the four boundary conditionŝ

Numerical solution of the boundary value problem for an infinite residually-stressed thick-walled cylindrical tube.
A numerical approach is adopted to find the solution in the undeformed configuration of an infinite thickwalled cylindrical tube. Using Eqs. (117)-(122) in Eq. (100) and (108), we get the differential equations to be solved with the special model (60). For brevity, it is assumed that ζ 2 ≡ 0 in the undeformed configuration. For so, the stretches are equal to unity and also the derivative with respect to r vanishes. Therefore, from Eqs. (117)-(122),γ where the principal (residual) stresses arê in the dimensionless form. Using Eq. (102), the expression for p ′′ 0 (R), in the undeformed configuration, is which is evaluated using Eq. (70) and we find that p ′′ 0 = 0 for this special case. In order to have dimensionless expressions, the following notations are introduced,R Using these notations and Eqs. (100), (133)-(136), the equation of motion for the special model in the undeformed configuration iŝ and from Eqs. (133), we get and the derivatives arê respectively. Also, p ′ 0 /µ, in reference configuration, is which vanishes in this special case. As ζ 3 is at the boundary, Eqs. (108) and (109), appropriately made dimensionless, specialize tô respectively, and both the above equations hold atR = 1, andβ.
In order to find a numerical solution of Eq. (138), the problem is transformed to a system of first order linear ordinary differential equations. Suppose which, from Eq. (149) give Equations (149) 4.3. Isotropy. As a special case it is considered that when the material is not residually stressed, it is isotropic. The boundary value problem equations above is reduced to the to the classical problem in linear elasticity and can be solved analytically. For so, β 1 = 0 = β 2 . Dropping the notation defined in Eq. (137), we have γ 1 = 1, γ 2 = −2, γ 4 = 1, γ 6 = 1, The equation of motion (138) therefore becomes with boundary conditions from Eqs. (153)-(156) Equation (157) can be rewritten as where The solution of Eq. (162) is given by where A 1 , A 2 , A 3 , A 4 are arbitrary constants which can be determined using Eqs.
For a fixed β, a simple code in MAPLE is used to solve Eq. (165) and the builtin command 'implicitplot' is used to obtain the dispersion curves. Figure 9 illustrates the comparison between the results for linear elasticity for an isotropic material and the results obtained through Eq. (152)-(156) with no residual stress. It is found that the numerical results obtained here are in good accordance with the classical results. In the case of more than a single solution, BVP solvers in softwares require an initial value for a parameter for the solver to converge to the nearest solution. There can be a single set, a finite number of (possible) sets, or an infinite number of possible sets, of parameters associated with the solution of a BVP.
To find a numerical solution of Eq. (152) with Eqs. (153)-(156), a built-in MAT-LAB function 'bvp4c' is used. This input of this function is a system of first order ordinary differential equations and an initial guess for the unknown parameters. For fixedβ, β 1 , β 2 andk, the solver gives a solution and therefore dispersion curves are obtained which are graphed for illustration purposes. The code is mentioned in the Appendix.
For the case when there is no residual stress, β 1 = 0 = β 2 . For this instance, dispersion curves are plotted in Fig. 9 which are in accordance with the analytic results in Section 4.3. These results are also similar to those presented in [18] for variable wall thickness and different frequencies for waves in a (hollow) elastic rod. In [18], the displacements are expanded in a series of orthogonal polynomials which are a function of the radial coordinate and only the earliest terms in the series are retained. The error because of omission of terms is reduced by various adjustment factors for the frequency spectrum to match the exact theory.
In [3], the authors have considered a pre-stretched and an initially stressed (hollow) cylinder. The results are obtained for study of wave propagation with and without the presence of initial stress. Figure 10 shows dispersion curves that are similar to those found in [3] for varying wall thicknesses when initial stress is absent. Further, in the same paper, plots of dispersion curves are presented to study the influence of pre-stretch and initial stress on the wave speed. In our discussion, the graphs show modes without any more branches in contrast to the graphs in [3] since the value of pre-stretch is assumed as unity. Apart from this, the behavior of curves is similar in the first few modes. Figures 11 -13 illustrate the effect of residual stress for different modes. For zero residual stress, the strain energy function follows the behavior of the neo-Hookean type materials. In this case, first few modes are shown in Fig. 10 and it may be noted that for small values of dimensionless wave number, wave speed is same with increasing wall thickness. The plots in Fig. 11 show a different trend. With increasing wall thickness and for smallk, the first modes in Fig. 11 have different phase speeds. From Fig. 12, for varying parameters and fixed wall thickness, first few modes are shown. The graph ′ a ′ shows the behavior for zero residual stress which thus the case of neo-Hookean type material. It is observed that as β 1 and β 2 increase above zero, the phase speed is reduced from that observed for a neo-Hookean type materials. For decreasing values, say β 1 < 0 or β 2 < 0, as shown in plots (d) and (e), the phase speed increases. First four modes are shown in Fig. 13 in the case when the residual stress is present, shown in the continuous plots, and when the residual stress is zero (dashed plots).

5.
Conclusions. In this paper, nonlinear theory of elasticity is used to study the effect of initial stress on the wave propagation in an incompressible hollow cylinder. The initial stress is considered non-homogeneous and the material bears properties of isotropic materials in the absence of this initial stress. The hollow cylindrical tube is considered to first undergo a finite deformation and later an infinitesimal deformation. The pressure boundary conditions are used and explicit expression for pressure inside the tube is calculated. This pressure is a function of the finite deformation through the stretches and also of the residual stress. Variation in pressure is observed for a given wall thickness is observed for varying parameters.
A prototype strain energy function is used to study the effect of residual stress on wave propagation in the cylindrical tube. The two point boundary value problem is solved numerically and analyzed. It is noted that for small values of dimensionless wave number, wave speed is same with increasing wall thickness. With increasing wall thickness and for small wave number, the first modes for varying values of parameters have different phase speeds. A comparison is also drawn when the residual stress vanishes (a neo-Hookean type material). It is found that for positive values of parameters, the phase speed is reduced from that observed for a neo-Hookean type materials. For decreasing values the phase speed increases for a fixed wall thickness. pleted at the University of Glasgow and funded through the Faculty development program, Pakistan. Also, the help provided by Dr. Steven Roper, lecturer, University of Glasgow, UK, in software code for the results presented here is highly appreciated.
Appendix. The following file mentions the system of differential equations and the boundary conditions.