On the Behavior of a Body (Structure) under Conditions of “Ideal Plasticity”

G. A. Doshchinskii

(Presented by the scientific seminar of the Department of Strength of Materials)

The behavior of a body under the action of loads is predetermined on the basis of the equations of the theory of elasticity and plasticity, which make it possible, by calculation, to predict the character and magnitude of reversible and irreversible deformations at each moment of the loading process. The mechanical properties of the materials used are introduced into the calculation on the basis of preliminary testing of material specimens under laboratory conditions. In these tests the tensile diagram is most often recorded, expressing the relation between the stress and the elongation of the specimen as elastic and plastic deformation develops.

Among the most important and widely used structural materials are low-carbon steels. Initial tensile tests often reveal for these steels (especially in the annealed state) a characteristic property—an intensive increase in irreversible deformations at a certain unchanged value of the tensile stress, called the yield point—$\sigma_t$. On the tensile diagram this feature is expressed by a horizontal segment (yield plateau) on the tensile curve. Accordingly, the so-called Prandtl diagram, shown in Fig. 1, a, is often taken as the basis for calculations of structures made of low-carbon steels.

For a body in a complex state of stress it is assumed that yielding occurs when a certain combination of all stresses is reached (some function of the three principal stresses), established by the plasticity condition of a definite magnitude:

\[ F(\sigma_1,\sigma_2,\sigma_3)<0 \quad \text{within the elastic range,} \]

\[ F(\sigma_1,\sigma_2,\sigma_3)=0 \quad \text{under plastic deformation.} \]

The behavior of a body (structure) within the elastic range (up to the yield point with the Prandtl diagram) is determined sufficiently reliably by the relations of the theory of elasticity.

When the state corresponding to the yield point is reached, the development of irreversible plastic deformations begins, and the behavior of the body must be determined by the relations of the theory of plasticity. However, describing the behavior of a body (or structure) under so-called “ideal plasticity,” corresponding to the horizontal segment of the Prandtl diagram, is associated with certain mathematical inconsistencies. Let us note the principal such difficulties in calculations for

based on the deformation theory of plasticity (i.e., the theory that considers finite, rather than differential, relations between stresses and strains).

As the principal condition establishing the relation of stresses under plastic deformation, the Mises–Hencky plasticity condition is regarded here as the most reliable:

\[ \sigma_i=\sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2-\sigma_1\sigma_2-\sigma_2\sigma_3-\sigma_3\sigma_1}=\sigma_T . \tag{1} \]

The deformation theory also assumes that for each material there exists a definite dependence between the generalized stress (the stress intensity, \(\sigma_i\)) and the generalized strain (the strain intensity, \(e_i\)), whose form is established by the initial tensile test. However, in fact the dependence between the stress and strain intensities

\[ \sigma_i=f\!\left(\frac{3}{2(1+\mu)}\,e_i\right) \]

is reduced to the form \(\sigma_i=f(e_i)\), coinciding with the tensile diagram, only if the transverse-strain coefficient is assumed to be equal to \(0.5\) [1], [2]. This value is certainly inconsistent with the elastic deformability of steels, for which Poisson’s ratio \(\mu \approx 0.3\).

Fig. 1.

Such a discontinuity in the transition from the elastic stage to the plastic stage introduces a certain violation of the monotonicity of the functions determining the values of the strain components, and does not correspond to the actual change in the transverse-strain coefficient, which increases monotonically beyond the elastic limit from \(0.3\) and asymptotically approaches the value \(0.5\).

The fundamental relations of deformation theory have received theoretical and experimental justification only for conditions of simple loading, i.e., when all stress components increase in the course of loading proportionally to a single parameter—\(\lambda\). However, simple loading itself becomes incompatible with the condition of plasticity on the horizontal segment of the diagram. Here one must assume either a discontinuity (a stop) in the monotonicity of the loading process, or else consider that the plasticity condition is satisfied through a change in the ratio between stresses; but this will contradict proportional loading.

The greatest difficulty, perhaps, will be associated with the circumstance that on the horizontal segment of the relation there is no one-to-one connection between the generalized stress and the strain. One and the same value of \(\sigma_i\) corresponds to different values of \(e_i\), as a result of which the deformation within the range of ideal yielding becomes indeterminate. This permits one to assume a certain arbitrariness in the behavior of the body (structure) and an impossibility of predicting this behavior under loading. In fact, however, when bodies are loaded under such conditions, a quite definite repeatability of the deformation pattern is observed.

In solving practical problems, the difficulties indicated are partly bypassed by introducing a number of assumptions. The approximate picture of the behavior of the body (structure) that follows from such a calculation turns out to a considerable extent to have been prescribed in advance by the consequences of the assumptions made, and at times may deviate substantially from reality.

It may be expected that the use, for calculation, of theories free from the inconveniences indicated above can reveal new qualitative features of the behavior of structures under ideal yielding, features that are concealed by the assumptions used previously. In this connection we shall show some possibilities that follow from the theory of elastic-plastic deformation developed by the author, certain propositions of which are presented in works [3], [4], [5].

The fundamental relations of this theory, establishing the connection between stresses and strains, may be represented by the following relations given below.

The components of the principal strains are connected, for an isotropic material under simple loading, with the stresses by means of two characteristics \(E(\lambda)\) and \(\mu(\lambda)\), corresponding to the secant modulus of the tensile diagram and to the coefficient of total (elastic-plastic) transverse strain,

\[ e_n=\frac{1}{E(\lambda)} \left[\sigma_n-\mu(\lambda)(\sigma_m+\sigma_k)\right] \qquad (n,m,k=1,2,3), \]

\[ E(\lambda)=\frac{\sigma}{e}, \qquad \mu(\lambda)=\frac{e_{\text{transv.}}}{e_{\text{longit.}}}. \]

At the same time, the stress state as a whole is connected with the deformed state by the following condition, expressing the basic mechanical characteristic of the material:

\[ \sqrt{ \sigma_1^2+\sigma_2^2+\sigma_3^2 - \frac{4\mu(\lambda)-2\mu(\lambda)^2}{1+2\mu(\lambda)^2} (\sigma_1\cdot\sigma_2+\sigma_3\cdot\sigma_2+\sigma_3\cdot\sigma_1) } = \]

\[ =E(\lambda)\cdot \sqrt{ \frac{e_1^2+e_2^2+e_3^2}{1+2\mu(\lambda)^2} }. \]

This condition coincides completely with the tensile diagram, without any assumptions.

The corresponding condition of elastic-plastic deformation, equivalent to the plasticity condition of the deformation theory of plasticity for the yield segment, will have the form

\[ \sqrt{\sigma_1^2+\sigma_2^2+\sigma_3^2 -\frac{4\mu(\lambda)-2\mu(\lambda)^2}{1+2\mu(\lambda)^2} \left(\sigma_1\sigma_2+\sigma_2\sigma_3+\sigma_3\sigma_1\right)} =\sigma_{\mathrm{T}} . \tag{2} \]

By taking into account the actual change of \(\mu(\lambda)\), this condition establishes a one-to-one relation between the stressed and deformed states also on the yield segment, where the state of the material is characterized by the value of \(\mu(\lambda)\) corresponding to each instant of the deformation process (Fig. 1, в).

Condition (2) makes it possible, by calculation, to predict the change in the resistance of the body, expressed by the change in the stresses \(\sigma_1,\sigma_2,\sigma_3\) during the yield process. This change will in fact be the driving factor determining the development of the deformation process under the given conditions. At the same time, the constancy of the stresses during yielding, established by plasticity conditions of the Saint-Venant or Mises type, is associated with certain difficulties in explaining the cause of motion, since deformation during yielding often does not have the character of steady motion.

It is also necessary here to give certain explanations of the concept of simple loading that we use.

The concept of simple loading introduced by A. A. Ilyushin [1] corresponds to theories of plasticity based on plasticity conditions of the Saint-Venant or Mises–Hencky type. The use of other theories may also require a corresponding clarification of the concept of types of loading.

Defining simple loading as loading with an increase of all components of the stressed state (the complete components, and not only the deviatoric ones) in proportion to a single parameter \(\lambda\), we impose no conditions whatsoever on the deformations, the magnitude and ratios of which are determined solely by the properties of the material.

It is not difficult to imagine that if, while defining simple loading, one were simultaneously to impose some conditions on the deformations, this would constitute a certain prescription to real materials of the interrelation between stress and deformation. In other words, the objective relation would acquire a certain conditional character.

By analogy with simple loading, one may introduce the concept of simple deformation, with an increase of all components of deformation in proportion to a single parameter. In this case the magnitude of the stresses and their ratios will be determined by the properties of the material itself.

In the sense indicated, various testing devices can realize either conditions of simple loading or of simple deformation.

From condition (2) there follows a certain possibility of conducting a process of simple loading also on the yield segment. A proportional change of stresses at constant \(\sigma_{\mathrm{T}}\) here can be achieved by a change of \(\mu(\lambda)\), i.e., ultimately in connection with a redistribution of deformations.

Conversely, by specifying conditions of simple deformation, one can obtain on the yield segment a change of stresses which, depending on the type of deformation, may have either an active character (increase of stresses) or a passive one (decrease of stresses).

Noting the possibilities for calculating the behavior of a body (or structure) under “ideal plasticity” on the basis of condition (2) (which has been verified by us experimentally and is confirmed by the data of other investigators), we shall give, in conclusion, two simplest examples.

In calculating thin-walled vessels, the stresses along the edges of an element cut out from the wall are determined from the equilibrium conditions of the element (Laplace equations) and the equilibrium of a part of the vessel.

For a cylindrical vessel subjected to a uniform internal pressure, the tangential and meridional stresses are determined, respectively, as

\[ \sigma_t=\frac{qD}{2t}, \qquad \sigma_m=\frac{qD}{4t}. \]

For the radial stress one may take, on the average,

\[ \sigma_r=-\frac{q}{2}. \]

Denoting the ratio of the vessel diameter to the thickness by \(c\), we have

\[ \sigma_t=\frac{q}{2}c;\qquad \sigma_m=\frac{q}{4}c;\qquad \sigma_r=-\frac{q}{2}. \]

For “ideal plasticity,” from the Mises–Hencky plasticity condition it follows that plastic deformation will develop at the constant pressure

\[ q=\frac{4\sigma_T}{\sqrt{3c^2+6c+4}}. \]

In calculation by condition (2) we obtain

\[ q=\frac{4\sigma_T}{\sqrt{(5c^2+4)-\frac{4\varphi(\lambda)-2\varphi(\lambda)^2}{1+2\varphi(\lambda)^2}(2c^2+6c)}}, \]

whence it follows that, in order to conduct the process of plastic expansion of the cylinder, a certain (though small, but systematic) increase in pressure is necessary.

Below, in the table, are given test data for a cylinder of annealed steel (St. 3), which in tension gave a horizontal portion of the diagram of the order of \(e_{\text{plast.}}=1.7\%\) at \(\sigma_T=2.1\cdot 10^8\ \frac{\text{N}}{\text{m}^2}\), \(D=41\ \text{mm}\), \(t=1\ \text{mm}\). The calculated increase of pressure in the process of flow \(q_{\text{theor.}}\) is confirmed by the experimental data. The table gives the pressure values as a function of the change in the cylinder diameter \(\Delta D\), as well as the values of the equivalent relative longitudinal deformation in simple tension \(e_{\text{plast.}}\), and the corresponding values of the coefficient of transverse deformation. The internal pressure in the cylinder, plugged at the ends, creates reliable conditions of simple loading under the given conditions. The pressure was measured with the aid of a reference pressure gauge (Cl. 0.35).

For the case of shear (torsion) the principal stresses are respectively equal to

\[ \sigma_1=\tau,\quad \sigma_2=0,\quad \sigma_3=-\tau. \]

According to condition (2), for a material in which, in a tensile test, ...

Fig. 2.

Fig. 3.

under tension a yield plateau is found, corresponding to the shear stress

\[ \tau=\frac{\sigma_{\mathrm{T}}}{\sqrt{\dfrac{2[1+\mu(\lambda)]^{2}}{1+2\mu(\lambda)^{2}}}}, \]

i.e., the possibility is assumed of a certain drop in stresses within the range \((0.590 \div 0.577)\sigma_{\mathrm{T}}\).

Figures 2 and 3 show oscillograms of the processes of tension and torsion of steel specimens. The torque on the testing machine was measured by the deflection of a pendulum (conditions were realized approximately corresponding to simple deformation). Some

drop in stress was accompanied by a corresponding decrease in the deflection of the balancing pendulum. The decrease in torque (in a solid specimen) is visible on the oscillogram (Fig. 3).

REFERENCES

1. A. A. Ilyushin. Plasticity. Gostekhizdat, 1948.
2. M. M. Filonenko-Borodich et al. Course of Strength of Materials. Vol. 2, 1950.
3. G. A. Doshchinskii. On the theory of elastic-plastic deformation. Izvestiya TPI, vol. 85, 1957.
4. G. A. Doshchinskii. A general law of the relation between stresses and deformations. Izvestiya TPI, vol. 114, 1963.
5. G. A. Doshchinskii. On the law of volumetric deformation, the present collection.