Viscoelastic deformation of a thick-walled tube acted on by internal pressure and axial tension, with allowance for strain hardening of the material

Abstract

In this article we consider the viscoelastic deformation of a thick-walled tube, with headers rigidly attached to the ends, under the simultaneous action of internal pressure and longitudinal tension. This type of problem arises in considering the stress-deformation state of the rock near a circular working (e.g., a tunnel) where the external forces can be reduced to axial tension along the working and internal and external pressure. In [1] I attacked this problem by starting with the theory of ideal plasticity ['2]; but in this paper I shall make allowance for strain hardening of the material during plastic deformation.I shall use the model of a viscoelastic body [2] in which the condition of ideal plasticity is replaced by a condition allowing for strain hardening of the material If we know the directions of the principal axes and ol, o8, o 3 (o 1 > o 8 > o3) are the principal stresses and 8 t, e 8, s 3 the principal deformations, then as the condition of plasticity we can take the Trask-Saint-Venant condition, Ol-O 3 = 2rs, where r s is the yield point under tension or compression, and distinguish between states of complete and incomplete plasticity. Plastic deformation is a pure shear in the planes of principal axes 1 and 3.The magnitude of this shear is ?'max = sl'e3; the greatest tangential stress is rma x = (o,-o3)/2. Plastic deformation arises as soon as rma x reaches some limiting value rs; under elastic deformation, rmax = /~ ?'max; rma x = r s = const for plastic deformation and any values of ?'max.Real materials do not obey the law of ideal plasticity; as the plastic deformation increases, the plastic resistance of the material rises. As the law of strain hardening we take a relation between the tangential stress and the shear in the form [3] rma x = f(Ymax). It is considered that this law satisfactorily represents the behavior of viscoelastic bodies under loads such that the principal axes of the stress and deformation tensors coincide and do not vary in spatial orientation.The form of the strain-hardening law follows naturally from the Trask-Saint-Venant theory of flow for an ideal plastic body, provided that we take account of strain hardening. When we do so, the condition of plasticity becomes where )'e is the limiting elastic shear (the shear at the onset of plastic flow), /~p is the plastic shear modulus, and we shall assume that ~p = const during plastic deformation.In the direction of the middle principal stress o 8 no plastic deformations will occur; o 8 is related by the elastic law to the deformation [2].Let us take the internal radius of the tube as unity, and its external radius as R, and let us take the generators of the tube as the direction of the z-axis. The tube will expand under the action of the internal pressure P; at its ends we apply equal and opposite forces of magnitude F. We shall assume that the tube is long so that the distributions of stresses and deformations in normal cross sections sufficiently far from the ends can be assumed identical. From the condition of constancy...

Two problems in rock mechanics arising out of the working of deep ore or coal deposits

“…In analysis of the state of the rocks in the neighborhood of a development working with allowance for changes of depth and the possible influence on extraction operations on the state of the working, it is necessary to take account of the form of the state of stress in conformity with the parameter (23) and to estimate the role of areas with tangential stresses T, T~, and T2 on which shears develop to breaking point. For this purpose it is important to take account of the difference between the initial state of stress and the hydrostatic state and to know the strength standards of the rocks under triaxlal loads.…”

“…This remarkable closing of the model, unfortunately, has been almost forgotten by contemporary investigators of strength, but has now been dls-, cussed by a number of authors [20,23,24,28].…”

“…Experiments show that anisotropy of the deformability and strength properties is very important both in metals [20,23] and rocks [24]. This enables us to study the strength standards in the form (1.7).…”