Description of Residual Stress and Strain Fields in FGM Hollow Disc Subject to External Pressure

Elastic/plastic stress and strain fields are obtained in a functionally graded annular disc of constant thickness subject to external pressure, followed by unloading. The elastic modulus and tensile yield stress of the disc are assumed to vary along the radius whereas the Poisson’s ratio is kept constant. The flow theory of plasticity is employed. However, it is shown that the equations of the associated flow rule, which are originally written in terms of plastic strain rate, can be integrated with respect to the time giving the corresponding equations in terms of plastic strain. This feature of the solution significantly facilitates the solution. The general solution is given for arbitrary variations of the elastic modulus and tensile yield stress along the radial coordinate. However, it is assumed that plastic yielding is initiated at the inner radius of the disc and that no other plastic region appears in the course of deformation. The solution in the plastic region at loading reduces to two ordinary differential equations. These equations are solved one by one. Unloading is assumed to be purely elastic. This assumption should be verified a posteriori. An illustrative example demonstrates the effect of the variation of the elastic modulus and tensile yield stress along the radius on the distribution of stresses and strains at the end of loading and after unloading. In this case, it is assumed that the material properties vary according to power-law functions.


Introduction
Stress and strain analyses of solid and hollow circular discs have long been an important topic in the mechanics of solids. The motivation of doing such analyses is that circular discs subject to mechanical, thermal, and inertial loading are used in many sectors of industry. The performance of such discs under service conditions can be improved by using functionally graded materials. The material may be continuously graded or be piecewise homogeneous. It is assumed in the present paper that the distribution of all material properties is axisymmetric. Discs made of homogeneous materials are not discussed. Discs made of functionally graded materials have been the subject of intense research. A linearly elastic solution under plane stress and plane strain conditions has been given in [1], assuming that the disc is loaded by external or internal pressure. It has been concluded that the stress response of the functionally graded disc is significantly different from that of the homogeneous disc. Another plane stress solution of this boundary value problem has been obtained in [2] and another plane strain solution in [3]. A thermoelastic stress solution for a disc of variable thickness has been found in [2]. A thermoelastic analysis of a disc subject to a steady-state temperature distribution together with external and internal pressures has been provided in [4]. It has been assumed in this

Statement of the Problem
Consider a thin hollow disc of functionally graded material subject to uniform pressure p 0 over its outer radius b 0 , followed by unloading. The inner radius of the disc is denoted as a 0 . The thickness of the disc is constant. The mechanical properties of the disc are classified in terms of the yield stress tension σ Y , Poisson's ratio ν, and Young's modulus E. It is assumed that the Poisson's ratio is constant, whereas the value of both σ Y and E vary with radius. It is convenient to use a cylindrical coordinate system (r, θ, z) whose z-axis coincides with the axis of symmetry of the disc. The normal stresses in this coordinate system are the principal stresses. The state of stress is plane (i.e., the axial stress in the cylindrical coordinate system vanishes). Therefore, Hooke's law can be written as here σ r is the radial stress, σ θ is the circumferential stress, ε e r , ε e θ , and ε e z are the elastic strains referred to the cylindrical coordinate system. Plastic yielding is controlled by the von Mises yield criterion. Under a plane stress condition, this criterion reads The flow theory of plasticity is adopted. The flow rule associated with the yield criterion (2) is here ξ p r , ξ p θ , and ξ p z are the plastic strain rates referred to the cylindrical coordinate system and λ is a non-negative multiplier. The total strain components in the cylindrical coordinate system are given by and the equation of strain compatibility of the form The boundary conditions at the stage of loading are for r = b 0 and σ r = 0 for r = a 0 . The boundary conditions at the stage of unloading will be formulated in Section 6. It is convenient to introduce the following dimensionless quantities: here σ 0 is the value of σ Y at r = b 0 and E 0 is the value of E at r = b 0 . Then, the variation of σ Y and E with ρ can be represented as where Φ(ρ) and η(ρ) are arbitrary functions of ρ satisfying the conditions Φ(ρ) = 1 and η(ρ) = 1 at ρ = 1. In what follows, it is assumed that these functions are such that plastic yielding initiates at the inner radius of the disc and no other plastic region appears in continued deformation.

Purely Elastic Solution
When p is small enough, the entire disc is elastic. In this case, the total strains are equal to the elastic strains. The system of equations comprises Hooke' law, the equilibrium equation, and the equation of strain compatibility. Using (9) it is possible to rewrite Equations (5) and (6) as ∂σ r ∂ρ  (12) Eliminating the strains in this equation by means of (1) and using (9) results in ∂σ θ ∂ρ Equations (11) and (13) comprise the system for determining the distribution of stresses in the purely elastic disc. Then, the distribution of strains can be readily found from (1) and (9). However, the purely elastic solution is not of interest in the case under consideration. Therefore, the solution to Equations (11) and (13) is only necessary to determine the value of p at which plastic yielding is initiated. This value of p is denoted as p e . By assumption, plastic yielding is initiated at ρ = a. It follows from the boundary condition (8) and the yield criterion (2) that σ θ = −σ Y at ρ = a at the initiation of plastic yielding. Using (10), this condition can be rewritten as σ θ = −σ 0 Φ(a) at ρ = a. This is one of the boundary conditions of the boundary value problem to be solved. The other boundary condition is given by (8). Equations (11) and (13) should be solved together with these boundary conditions. The value of p e is readily found from this solution as p e = −σ r (1)/σ 0 . In what follows, it is assumed that p > p e .

Elastic/Plastic Stress Solution
If p > p e , then the disc consists of two regions, elastic and plastic. The elastic region occupies the domain ρ c ≤ ρ ≤ 1 and the plastic region the domain a ≤ ρ ≤ ρ c . Here, ρ c is the elastic/plastic boundary. Consider the plastic region. The yield criterion (2) is satisfied by the following substitution: here ψ is a new function of ρ. This function should be found from the solution. Substituting (14) into (11) yields The boundary condition to this equation follows from (8) and (14). In particular, σ r = 0 if ψ = 0 or ψ = π. It is evident that σ θ < 0 at ρ = a. Then, the boundary condition to Equation (15) is for ρ = a. Solving Equation (15) together with this boundary condition supplies the variation of ψ with ρ. This solution and (14) determine the distribution of the stresses in the plastic region. Let p p be the value of p at which the entire disc becomes plastic. Putting in the solution for the radial stress ρ = 1 gives the value of p p as p p = −σ r (1)/σ 0 . In what follows, it is assumed that p < p p . In this case, a < ρ c < 1. The value of σ r on the plastic side of the elastic/plastic boundary is denoted as σ c r and the value of σ θ on the plastic side of the elastic/plastic boundary as σ c θ . These values are readily found from the solution of (15) and (14). Equations (11) and (13) are valid in the elastic region. The radial and circumferential stresses must be continuous across the elastic/plastic boundary. Therefore, the boundary conditions to Equations (11) and (13) are for ρ = ρ c . Solving Equations (11) and (13) together with these boundary conditions supplies the distribution of the radial and circumferential stresses in the elastic region. In particular, the value of p involved in (7) is determined from the equation p = −σ r (1)/σ 0 . Therefore, the solution found connects p and ρ c . One of these parameters should be prescribed. Then, the other parameter is determined from the solution.

Elastic/Plastic Strain Solution
Consider the plastic region, a ≤ ρ ≤ ρ c . Eliminating λ between the equations in (3) gives Using (14), the stresses in these equations can be expressed in terms of ψ. Then, taking into account that ξ here t is the time. It is seen from the structure of Equation (15) and the boundary condition (16) that ψ is independent of t. Therefore, the coefficients of ∂ε p θ /∂t in (19) are also independent of t, and the equations in (19) can be immediately integrated with respect to the time to give It has been taken into account here that ε  (1) and (14) with the use of (9) and (10). As a result, here Λ = Φ/η. The total strains are found from (4), (20), and (21) as It follows from these equations that These equations and Equation (6), in which r should be replaced with ρ by means of (9), combine to give The derivative dψ/dρ in this equation can be eliminated by means of (15). Then, Since ψ has already been determined as a function of ρ in Section 4, (23) is a linear differential equation for ε p θ . The boundary condition to this equation is for ρ = ρ c . Once Equation (23) has been solved, the distribution of ε p r and ε p z in the plastic region is found from (20) and the distribution of the total strains from (22).
The distribution of strains in the elastic region is determined from the solution for stress found in Section 4 and Hooke's law.

Unloading
Let p f be the value of p at the end of loading. Then, the boundary conditions for the stage of unloading are ∆σ r = 0 for ρ = a and ∆σ r = σ 0 p f for ρ = 1. Here, ∆σ r is the increment of the radial stress after unloading (∆σ θ will stand for the increment of the circumferential stress). It is assumed that unloading is purely elastic. This assumption should be verified a posteriori. Equations (11) and (13), in which σ r should be replaced with ∆σ r and σ θ with ∆σ θ , are valid. An iterative procedure should be used for solving this system of equations together with the boundary conditions (25) and (26). Once this boundary value problem has been solved, the increment of strains is determined from Hooke's law as The distribution of residual stresses, σ res r and σ res θ , is given by Here, σ is the distribution of the radial stress and σ ( f ) θ is the distribution of the circumferential stress at the end of loading. These distributions have been found in Section 4. Substituting (28) into (2) provides the condition to verify that the process of unloading is purely elastic in the form Here, Equation (10) has been taken into account. The distribution of residual strains, ε res r , ε res θ and ε res z , is given by Here, ε ( f ) r is the distribution of the radial strain, ε ( f ) θ is the distribution of the circumferential strain and ε ( f ) z is the distribution of the axial strain at the end of loading. These distributions have been found in Section 5.

Illustrative Example
It is often assumed that material properties vary according to a power law along the radius of the disc [1,[9][10][11][12][13]. In the case under consideration one possible variant of this law reads Φ(ρ) = ρ m and η(ρ) = ρ n In all calculations, ν = 0.3, a = 0.3, and n = 0.3. The value of m varies in the range 0 ≤ m ≤ 0.3. It is worthy of note that there is no need to prescribe the values of σ 0 and E 0 for numerical analysis since the stress components are proportional to σ 0 , and the strain components are proportional to k. It is assumed that ρ c = 0.8. Then, the value of p f has been found from the stress solution given in Section 4.
The dependence of p f on m is presented in Table 1. The variation of the radial and circumferential stresses with ρ at p = p f is depicted in Figures 1 and 2, respectively. It has been verified that the yield criterion (2) is not violated in the elastic region. The variation of the radial, circumferential, and axial strains with ρ at p = p f is depicted in Figures 3-5, respectively.
Using the values of p f found ( Table 1) the system of Equations (11) and (13) together with the boundary conditions (25) and (26) has been solved for ∆σ r /σ 0 and ∆σ θ /σ 0 . Having this solution and the radial distribution of the radial and circumferential stresses at the end of loading (Figures 1 and 2) the radial distribution of the residual stresses is determined from (28). The variation of the residual radial and circumferential stresses with ρ is shown in Figures 6 and 7, respectively. Then, it has been verified that the inequality (29) is satisfied in the range a ≤ ρ ≤ 1. Having the solution for ∆σ r /σ 0 and ∆σ θ /σ 0 , the increment of the strains is determined from (27). It is evident from (10) and (27) that these increments are proportional to k introduced in (9). The radial distribution of the residual strains is determined from (30). The variation of the residual radial, circumferential, and axial strains with ρ is shown in Figures 8-10, respectively.                       ε , with the dimensionless radius, ρ .

Discussion
The paper presents a general solution for the distribution of stress and strain in an functionally graded hollow disc subject to external pressure, followed by unloading. The solution is valid for any variation of the yield stress and Young's modulus with radius if the plastic region initiates at, and then propagates from, the inner radius of the disc. The purely elastic solution is valid if p p e ≤ . This solution is known, and therefore is not considered in the present paper. However, the general elastic solution is used in the elastic region, 1 c ρ ρ ≤ ≤ . The stress solution follows from the solution of equation (13). Then, the distribution of strains is determined from Hooke's law shown in (1).
The constitutive equations of the classical flow theory of plasticity are adopted. In particular, Hooke's law shown in (1) is used to connect the stress components and the elastic strain components. This law is valid in the elastic region. The von Mises yield criterion (2) is adopted in the plastic region. In this region, the stress components are connected to the strain rate components rather than to the strain components. The corresponding constitutive equation is the associated flow rule (3). The total strain components in the plastic region are given by (4). A detailed description of this material model can be found in any textbook on plasticity theory (for example, [29,30]).  (15) and (23), should be solved numerically. These equations can be solved one by one. In particular, the dependence of ψ on ρ is found from (15). Then, this numerical function is substituted into (23). As a result, a linear differential equation for the circumferential plastic strain, p θ ε , is obtained. The solution of equation (15) supplies the distribution of stresses in the plastic region according to (14). A remarkable feature of the strain solution is that the equations in (18), which are derived from the associated flow rule written in terms of plastic strain rates, can be immediately integrated with respect to the time to result in the equations in terms of plastic strains (equation 20). This feature of the solution significantly facilitates the solution. Another remarkable feature of the strain solution is that all strain components are proportional to k introduced in (9). It is seen from equations (22) and (23). Therefore, simple scaling of any strain solution provides the strain solution for similar discs of material with the same other properties and geometry, but any value of k. To calculate the distribution of strains, it is first necessary to solve equation (23) for the circumferential plastic strain. Then, the radial and axial plastic strains are readily found from (20) and the solution of

Discussion
The paper presents a general solution for the distribution of stress and strain in an functionally graded hollow disc subject to external pressure, followed by unloading. The solution is valid for any variation of the yield stress and Young's modulus with radius if the plastic region initiates at, and then propagates from, the inner radius of the disc. The purely elastic solution is valid if p ≤ p e . This solution is known, and therefore is not considered in the present paper. However, the general elastic solution is used in the elastic region, ρ c ≤ ρ ≤ 1. The stress solution follows from the solution of Equation (13). Then, the distribution of strains is determined from Hooke's law shown in (1).
The constitutive equations of the classical flow theory of plasticity are adopted. In particular, Hooke's law shown in (1) is used to connect the stress components and the elastic strain components. This law is valid in the elastic region. The von Mises yield criterion (2) is adopted in the plastic region. In this region, the stress components are connected to the strain rate components rather than to the strain components. The corresponding constitutive equation is the associated flow rule (3). The total strain components in the plastic region are given by (4). A detailed description of this material model can be found in any textbook on plasticity theory (for example, [29,30]).
For any given functions Φ(ρ) and η(ρ) involved in (10), the distribution of stress and strain in the plastic region, a ≤ ρ ≤ ρ c , can be calculated as follows. Two ordinary differential equations, Equations (15) and (23), should be solved numerically. These equations can be solved one by one. In particular, the dependence of ψ on ρ is found from (15). Then, this numerical function is substituted into (23). As a result, a linear differential equation for the circumferential plastic strain, ε p θ , is obtained. The solution of Equation (15) supplies the distribution of stresses in the plastic region according to (14). A remarkable feature of the strain solution is that the equations in (18), which are derived from the associated flow rule written in terms of plastic strain rates, can be immediately integrated with respect to the time to result in the equations in terms of plastic strains (Equation (20)). This feature of the solution significantly facilitates the solution. Another remarkable feature of the strain solution is that all strain components are proportional to k introduced in (9). It is seen from Equations (22) and (23). Therefore, simple scaling of any strain solution provides the strain solution for similar discs of material with the same other properties and geometry, but any value of k. To calculate the distribution of strains, it is first necessary to solve Equation (23) for the circumferential plastic strain. Then, the radial and axial plastic strains are readily found from (20) and the solution of (15). The elastic portions of strain in the plastic region are determined from (14) and Hooke's law shown in (1). Finally, the total strains follow from (4).
The general solution found is used to find a numerical solution, assuming that the power laws shown in (31) are valid. The effect on m-value on the radial and circumferential stresses in an a = 0.3 disc at ν = 0.3, n = 0.3, and ρ c = 0.8 is illustrated in Figures 1 and 2, respectively. The corresponding values of the external pressure are depicted in Table 1. It is seen from these figures that this effect is quite significant, especially on the circumferential stress. The associated distribution of strains is illustrated in Figures 3-5 (the radial strain is depicted in Figure 3, the circumferential strain in Figure 4 and the axial strain in Figure 5). The effect of m-value on these distributions is not so pronounced as compared to the stress distributions. This is associated with the imposed condition that ρ c = 0.8 in all cases. If the value of p f were fixed, then the effect of m-value on the strain distributions would be more significant as compared to its effect on the stress distribution. In particular, it is seen from Table 1 that the value of p f is quite sensitive to the value of m.
Residual stress and strain fields after purely elastic unloading are also obtained. These distributions are given by Equations (28) and (30). The validity of the pure elastic solution at unloading should be verified by means of Equation (29). The distribution of residual radial and circumferential stresses is illustrated in Figures 6 and 7, respectively. As in the case of the stress distributions at the end of loading, the effect of m-value is most significant on the circumferential stress. The effect of this value on the distribution of the residual radial strain is small (Figure 8). It is moderate in the case of the residual circumferential strain ( Figure 9) and large in the case of the residual axial strain ( Figure 10).