Plastic collapse mechanisms in thin disks subject to thermo-mechanical loading
© Alexandrov and Pham; licensee Springer. 2014
Received: 22 November 2013
Accepted: 23 January 2014
Published: 29 April 2014
A new solution for plastic collapse of a thin annular disk subject to thermo-mechanical loading is presented.
It is assumed that plastic yielding is controlled by Hill's quadratic orthotropic yield criterion. A distinguished feature of the boundary value problem considered is that there are two loading parameters. One of these parameters is temperature, and the other is pressure over the inner radius of the disk.
The general qualitative structure of the solution at plastic collapse is discussed in detail.
It is shown that two different plastic collapse mechanisms are possible. One of these mechanisms is characterized by strain localization at the inner radius of the disk. The entire disk becomes plastic according to the other collapse mechanism. In addition, two special regimes of plastic collapse are identified. According to one of these regimes, plastic collapse occurs when the entire disk is elastic except its inner radius. According to the other regime, the entire disk becomes plastic at the same values of the loading parameters at which plastic yielding starts to develop.
Thin plates and disks with holes and embedded inclusions have many structural applications. A significant amount of analytical and numerical research for various material models has been carried out in the area of stress and strain analysis of such structures (see [1–20] among many others). An excellent review of previous works devoted to the problem of enlargement of a circular hole in thin plates has been given in . The assumptions made regarding yield criterion, strain hardening and unloading have a significant effect on the predicted response and residual stress and strain fields . Even though closed-form solutions involve more assumptions than numerical solutions, the former are necessary for studying qualitative effects and verifying numerical codes. Typical qualitative effects under plane stress conditions are the singularity of the velocity field and non-existence of the solution under certain conditions [9–12, 17, 20]. These features of boundary value problems can cause difficulties with their treatment by means of standard commercial numerical codes. In particular, some specific difficulties with numerical solution for plane stress problems have been mentioned in .
In the present paper, the effect of temperature and pressure over the inner radius of a thin hollow disk on plastic collapse is investigated. The outer radius of the disk is fixed. The state of stress is plane. The classical Duhamel-Neumann law is adopted to connect the thermal and elastic portions of the strain tensor and stress components. Plastic yielding is controlled by Hill's quadratic orthotropic yield criterion . It is assumed that the principal axes of anisotropy coincide with the base vectors of a cylindrical coordinate system (r, θ, z) whose z-axis coincides with the axis of symmetry of the disk. Therefore, the boundary value problem is axisymmetric and its solution is independent of the polar angle. It is shown that the general qualitative structure of the plastic collapse solution is rather complicated. In particular, two different plastic collapse mechanisms have been found. According to one of these mechanisms, the plastic collapse occurs because the entire disk becomes plastic. The other plastic collapse mechanism is characterized by localization of plastic deformation at the inner radius of the disk.
Statement of the problem
where p = P/σ0 and a = a0/b0. As a result of an increase in τ, or p or both, a plastic zone can appear at the inner radius of the disk.
General stress solution in the plastic zone
where β is a dummy variable of integration and φ a is the value of φ at ρ = a.
Taking into account this equation, the solution to the system of Equations (15) and (16) gives a relation between p and τ when the entire disk becomes plastic. However, a difficulty is that this system may have no solution.
General structure of the solution at plastic collapse
to leading order. It follows from this equation that the plastic zone cannot develop if φ0 = 0. For Equation (22) is valid at any elastic/plastic boundary ρ = ρ c at which φ = φ c = 0. In this case, the plastic zone occupies the domain a ≤ ρ ≤ ρ c . Therefore, ρ - ρ c < 0 in the plastic zone, which contradicts the left-hand side of Equation (22). Putting ρ c → a, it is possible to conclude that this statement is true at the initiation of plastic yielding as well. The physical interpretation of this mathematical feature of the solution is that plastic deformation is localized within a layer of infinitesimal thickness at ρ = a. This corresponds to another mechanism of plastic collapse as compared to the state in which the entire disk is plastic. A remarkable property of the set of parameters at point k (Figure 2) is that the disk losses its load-bearing capacity without any plastic deformation in the domain a < ρ ≤ 1.
It is evident that Equation (13) has a special solution φ = φ s which is not obtainable from (14). Since φ s is constant, it follows from (12) that the stresses σ r and σ θ are independent of ρ. The physical meaning of this mathematical feature of Equation (13) is that the plastic zone occupies the entire disk once the plastic zone has initiated at ρ = a.
Since the value of φ m has been found, the corresponding value of τ can be determined from this equation with no difficulty. Thus, the dependence of p on τ is obtained in parametric form. This dependence is illustrated by curve 3 in Figure 3 for a = 1/2, ν = 0.3 and G = H = F. Curves 1 and 3 have the same tangent line at point s.
Results and discussion
Figure 3 is a geometric illustration of the state of plastic collapse. In what follows, it is assumed that the increase in the loading parameters does not lead to unloading. It is convenient to divide curve 1 into three segments, namely qk, ks and sw. If the initiation of plastic yielding occurs at some point of the segment qk, then the only possible mechanism of plastic collapse is localization of plastic deformation at ρ = a. In Figure 3, this mechanism of plastic collapse corresponds to a point of the line tk. If the initiation of plastic yielding occurs at some point of the segment sw, then the only possible mechanism of plastic collapse is the fully plastic disk. In Figure 3, this mechanism of plastic collapse corresponds to a point of the curve sd. Finally, if the initiation of plastic yielding occurs at some point of the segment ks, then either mechanism of plastic collapse is possible. Moreover, both mechanisms occur simultaneously if the values of the loading parameters correspond to point f. A special feature of the solution corresponding to point k is that the plastic collapse occurs by strain localization when the entire disk is elastic (except the line ρ = a). A special feature of the solution corresponding to point s is that the entire disk becomes plastic at the values of the loading parameters corresponding to the initiation of plastic yielding.
A new semi-analytical solution for the state of plastic collapse of a thin annular plastically orthotropic disk subject to thermo-mechanical loading has been found. The numerical part of the solution reduces to solving Equation (16) for φ m . Plastic yielding is controlled by Hill's quadratic orthotropic criterion. The study has emphasized qualitative features of the plastic collapse solution whose general structure is illustrated in Figure 3. It has been shown that there are two plastic collapse mechanisms. According to one of these mechanisms, the load-bearing capacity of the disk is lost because of strain localization at its inner radius. According to the other plastic collapse mechanism, the entire disk becomes plastic. In addition to these two general cases, there are three special cases of great interest for both numerical solutions of similar problems and the interpretation of elastic/plastic stress solutions for thin plastically anisotropic structures. These special cases are denoted by symbols k, s and f in Figure 3. If the state of stress corresponds to point k, then the disk losses its load-bearing capacity by plastic localization at its inner radius whereas the entire disk (except the inner radius) is elastic. If the state of stress corresponds to point s, then the disk becomes plastic at the same values of the loading parameters at which plastic yielding initiates. A distinguished feature of point f is that both of the aforementioned plastic collapse mechanisms occur simultaneously. It is expected that these qualitative features of the solution are rather common for a class of plastically anisotropic thin-walled structures and they can cause some difficulties with finding numerical solutions for such structures.
The research described in this paper has been supported by RFBR (Russia) and VAST (Vietnam), Project RFBR-14-01-93000.
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