UDC 539.3

THEORY OF ELASTICITY

V. M. PANFEROV

THEORY OF ELASTICITY AND THE DEFORMATION THEORY OF PLASTICITY FOR SOLIDS WITH DIFFERENT PROPERTIES IN COMPRESSION, TENSION, AND TORSION

(Presented by Academician L. I. Sedov, July 28, 1967)

Let us consider a homogeneous solid which, in the case of elastoplastic deformation, has different properties in uniaxial tension, uniaxial compression, and torsion, under the condition that the material remains isotropic with respect to each type of stress state.

We describe such a medium in a rectangular Cartesian coordinate system \(x_i,\ i=1,2,3\). The elastoplastic strains \(e_{ij}\) are assumed small, and the time interval \(t\) of deformation is such that the phenomena of creep and stress relaxation \(\sigma_{ij}\) may be neglected.

In the case where elastoplastic deformation of the body takes place, the laws for proportional or nearly proportional loading are taken in the form

\[ \sigma_{ij}=2G_1(\beta)\psi(T/T_0)(1-\omega(\bar e_u)) \left(e_{ij}-{}^1/_3 e_{kk}\delta_{ij}\right) +3K_0(T/T_0,e_1/|e_1|)e_1\delta_{ij} \]
\[ (i,j=1,2,3), \tag{1} \]

\[ 3e_1=e_{kk}-3\alpha(T-T_0),\qquad e_u=\bar e_u e_{s1}(\gamma)\chi(T'/T_0),\qquad \beta(1+\gamma)=\gamma, \]

\[ \gamma=e_1/e_u, \]

if

\[ \bar e_u>1,\qquad \partial_t(\bar e_u)>0,\qquad e_u^2={}^2/_3\left(e_{ij}-{}^1/_3 e_{kk}\delta_{ij}\right)^2,\qquad \partial_t(\bar e_u)=\partial \bar e_u/\partial t . \]

If \(\bar e_u\leq 1\), then the function \(\omega(\bar e_u)\) is identically equal to zero and elastic deformation takes place under any loading. The functions \(\omega\), \(e_{s1}\), \(G_1\) must satisfy the conditions

\[ 0\leq \omega\leq \omega+\bar e_u\partial_{\bar e_u}^{2}\omega<1-\delta^2,\qquad \partial_{\bar e_u}\omega>0,\qquad 2\partial_{\bar e_u}\omega>-\bar e_u\partial_{\bar e_u}^{2}\omega,\qquad \delta^2=\text{const} \tag{2} \]

and conditions (13), (14), (15), (16), (17). In formulas (1), the coefficient of linear expansion is denoted by \(\alpha\), and the initial temperature of the body, constant at all points of the body, is denoted by \(T_0\). For this temperature, if no loads act on the body, the stress components at every point of the body are equal to zero.

The modulus \(G_1(\beta)\psi(T'/T_0)>0\) and the yield limit (limit of proportionality) \(e_{s1}(\gamma)\chi(T'/T_0)>0\) are determined from four experiments: uniaxial compression, tension, torsion, and uniaxial compression with all-round pressure. The function \(\omega(\bar e_u)\) is determined from a uniaxial-compression experiment.

The laws of proportional unloading, or of unloading close to it, for \(\partial_t\bar e_u\leq 0\) are taken in the form

\[ \tilde\sigma_{ij}=\sigma_{ij}^{*}-\sigma_{ij}^{0} =3G_1(\beta)\psi(T'/T_0)\left(\tilde e_{ij}-{}^1/_3\tilde e_{kk}\delta_{ij}\right) +K_0(\tilde e_{kk}-3\alpha T), \]
\[ \tilde e_{ij}=e_{ij}^{*}-e_{ij}^{0}. \tag{3} \]

The components \(\sigma_{ij}^{0}, e_{ij}^{0}\) are reckoned from the moment at which unloading begins, marked by the superscript “asterisk.” For a justification of the proposed

relations (1) in the case of proportional loading, we shall make use of the laws of nonequilibrium thermodynamics of a solid body that does not exchange mass with other systems and is chemically inert for the process of small deformation. Suppose that the value of the temperature \(T\) at any point of the body differs little from \(T_0\).

In the case of elastic (reversible) deformation, the free energy \(\varphi\) is constructed, from which relations (1) follow for \(\omega \equiv 0\).

Now let us consider the elastoplastic deformation of a body. Here, in addition to the two basic laws of thermodynamics, we shall use the hypotheses (4), which are a certain generalization of Onsager’s principle.

Following (4), we represent the generalized forces (components of the stress tensor) as the sum of two terms

\[ \sigma_{ij}=\sigma_{ij}^{(o)}+\sigma_{ij}^{(\mathrm{н})}, \]

where the increment of the reversible part of the work \(W^{(o)}\) is determined by the formula

\[ dW^{(o)}=\sigma_{ij}^{(o)}\,de_{ij}. \]

We shall assume that the irreversible part of the work \(W^{(\mathrm{н})}\) is determined by the relation

\[ W^{(\mathrm{н})}=D=\sigma_{ij}^{(\mathrm{н})}\dot{\vartheta}_{ij}\geqslant 0, \]

where the components \(\vartheta_{ij}^{(\mathrm{н})}\) are equal to

\[ \vartheta_{ij}^{(\mathrm{н})} = \left( \frac{\partial D}{\partial \dot{\vartheta}_{nm}}\dot{\vartheta}_{nm} \right)^{-1} D\,\frac{\partial D}{\partial \dot{\vartheta}_{ij}}. \]

In order to be entitled to apply this formula, it is necessary that the system be stable, i.e., that \(\vartheta_{u}^{(\mathrm{н})}\) increase with increasing \(\vartheta_u\). We shall use this condition below.

We shall consider proportional loading. In this case we take the dissipative function \(D\) in the form

\[ D=2G_1(\beta)\psi_1(\bar e_u)e_u \sqrt{\,{}^3\!/_{2}\dot{\vartheta}_{ij}\dot{\vartheta}_{ij}\,} = \sigma_{ij}^{(\mathrm{н})}\dot{\vartheta}_{ij} = 3G_1(\beta)\psi_1(\bar e_u)e_u\dot e_u. \tag{4} \]

Then the values of the reversible part of the stresses \(\vartheta_{ij}^{(o)}\) as functions of the components \(\vartheta_{ij}\) will be equal to

\[ \vartheta_{ij}^{(o)} = \frac{\partial}{\partial \vartheta_{ij}} \int_{0}^{\sigma_u/3G} \sigma_u\,d\frac{\sigma_u(\vartheta_{ij})}{3G(\beta)}. \tag{5} \]

Thus, the components of the stress tensor, according to formulas (4), (5), are equal to

\[ \sigma_{ij} = \sigma_{ij}^{(\mathrm{н})}+\sigma_{ij}^{(o)} = 2G(\beta)(\psi_1+\chi_1)(e_{ij}-e\delta_{ij}) + 3K_0 e_1\delta_{ij}, \tag{6} \]

or, introducing the function \(\omega(e_u)\), they have the form according to formulas (1). In formulas (6) the function \(\chi_1\) is introduced on the basis of (5).

Now let us compute the stress components \(\vartheta_{ij}^{(o)}\) by formula (5), using the expression for the function \(\sigma_u\) according to formulas (1). Then

\[ \vartheta_{ij}^{(o)} = \sigma_u \frac{\partial \sigma_u/3G}{\partial e_u} \frac{\partial e_u}{\partial \vartheta_{ij}} = 2G(1-\omega) \left[ 1-\omega-e_u\frac{\partial\omega}{\partial e_u} \right] (e_{ij}-e\delta_{ij}). \tag{7} \]

Hence, the irreversible stress components are equal to

\[ \sigma_{ij}^{(\mathrm{н})} = \sigma_{ij}-\sigma_{ij}^{(o)} = \frac{2\sigma_u}{3e_u} \left( 1-\frac{\partial \sigma_u/3G}{\partial e_u} \right) (e_{ij}-e\delta_{ij}), \]

\[ \sigma_u^{(\mathrm{н})} = \vartheta_u^{(\mathrm{н})} = \sigma_u\left[1-\partial_{e_u}(\sigma_u/3G)\right]. \tag{8} \]

Now let us obtain the basic inequalities that follow from the thermodynamic analysis. These inequalities are obtained from the conditions:

a) the dissipative function is positive

\[ D=\sigma_u[1-\partial_{e_u}(\sigma_u/3G)]\dot e_u \geq 0; \tag{9} \]

b) the intensity of the irreversible stress components \(\sigma_u^{(\mathrm{н})}\) increases with increasing intensity \(\bar e_u\), i.e.

\[ \partial_t\sigma_u^{(\mathrm{н})}=\dot\sigma_u^{(\mathrm{н})}>0,\qquad \partial_t\bar e_u=\dot{\bar e}_u>0,\qquad \bar e_u>1. \tag{10} \]

In what follows we consider a medium for which the conditions are satisfied

\[ -p=\frac{1}{e_s}\partial_\gamma e_s<0,\qquad -m=\frac{1}{3G}\partial_\beta 3G<0. \tag{11} \]

To ensure inequalities a) and b), it is sufficient that the following inequalities hold:

\[ 0\leq \omega<\omega+\bar e_u\partial_{e_u}^{-}\omega<1-\delta^2,\qquad 2\partial_{e_u}^{-}\omega>e_u\partial_{e_u}^{2}\omega+\delta_2^2>0, \]

\[ 3G=3G_1\psi(T/T_0); \tag{12} \]

\[ \dot e_u>0,\qquad \dot{\bar e}_u>0,\qquad \bar e_u\geq 1; \tag{13} \]

\[ 1>K_1> \frac{\bar e_u p\gamma \partial_{e_u}^{-}\omega}{\omega+(1+p\gamma)\bar e_u\partial_{e_u}^{-}\omega} + \left(1+\frac{m(1-\gamma^2)}{p(1+\gamma^2)}\right) \frac{p\bar e_u\partial_t\gamma}{\partial_t\bar e_u}c_1+c_2c_3>0; \tag{14} \]

\[ K_2(1-p\gamma)\geq c_3c_4\partial_{e_u}^{-}\omega>0; \tag{15} \]

\[ K_1=\frac{1-\omega-\bar e_u\partial_{e_u}^{-}\omega}{1-\omega},\qquad K_2=2\partial_{e_u}^{-}\omega+\bar e_u\partial_{e_u}^{2}\omega,\qquad c_4=1+\gamma\partial_\gamma e_u p; \]

\[ c_2-1=\delta_3^2\ll 1; \]

\[ K_1>c_2c_3^0\ \text{for }\gamma<0;\qquad c_3^0=p\gamma/(1+p\gamma); \tag{16} \]

\[ c_4-1=\delta_4^2\ll 1\quad \text{for }\gamma>0. \tag{17} \]

Note that conditions (16) ensure that the strain components \(e_{ij}\) are uniquely expressed through the stress components \(\sigma_{ij}\), both in the case of elastic-plastic and elastic deformations.

Conditions (12), (13), (14), (15), (16), (17) guarantee that, with increasing shear deformation referred to the yield-limit deformation, the stress intensity referred to the yield limit increases. We shall call such deformation active. It is accompanied by an increase in elastic-plastic deformations.

Fig. 1. Deformation curves for ZrO\(_2\) P-47 in coordinates \(\sigma/\sigma_s=f(e/e_s)\). \(a\)—compression, \(T=1600^\circ\); \(b\)—tension, \(T=1600^\circ\); \(v\)—compression, \(T=1400^\circ\).

We determine the temperature field independently of the deformation pattern for the case when the strain rate is small and the rate of input of the external heat flux is small.

We shall assume that laws (1), (3) are also valid for more significant gradients of the temperature field in the body. Indeed, in these materials the compression modulus is 30% greater than the tensile modulus, while the compressive yield limit is an order of magnitude greater than the tensile yield limit (Fig. 1).

At the same time, the hypotheses of similarity are also experimentally confirmed in the case of uniaxial compression and tension, i.e.,

\[ \sigma_u / \sigma_s = \bar{\sigma}_u = \bar{e}_u [1 - \omega(\bar{e}_u)], \qquad G = G_1(\beta)\psi(T/T_0), \]

\[ e_s = e_{s_1}(\gamma)\chi(T/T_0). \]

Thus, the first relation means that the curves of uniaxial compression and tension, represented in the coordinates \(\bar{\sigma}_u, \bar{e}_u\), coincide (Fig. 1).

Institute of Mechanics
of Moscow State University
named after M. V. Lomonosov

Received
5 VII 1967

REFERENCES

1. S. I. Groot, Nonequilibrium Thermodynamics, Moscow, 1964.
2. A. A. Ilyushin, Plasticity, Moscow, 1948.
3. L. I. Sedov, Introduction to Continuum Mechanics, Moscow, 1962.
4. G. Ziegler, Some Extremal Principles of Irreversible Thermodynamics and Their Application to Continuum Mechanics, 1963.