On the residual stress possible in an elastic body with material symmetry

Continuum Mech. Thermodyn.

2015

In this paper we provide a new example of the solution of a finite deformation boundaryvalue problem for a residually-stressed elastic body. Specifically, we analyze the problem of the combined extension, inflation and torsion of a circular cylindrical tube subject to radial and circumferential residual stresses and governed by a residual-stress dependent nonlinear elastic constitutive law. The problem is first of all formulated for a general elastic strain-energy function, and compact expressions in the form of integrals are obtained for the pressure, axial load and torsional moment required to maintain the given deformation. For two specific simple prototype strain-energy functions that include residual stress the integrals are evaluated to give explicit closed-form expressions for the pressure, axial load and torsional moment. The dependence of these quantities on a measure of the radial strain is illustrated graphically for different values of the parameters (in dimensionless form) involved, in particular the tube thickness, the amount of torsion and the strength of the residual stress. The results for the two strain-energy functions are compared and also compared with results when there is no residual stress.

Section: Introductionmentioning

confidence: 99%

Extension, inflation and torsion of a residually stressed circular cylindrical tube

Merodio

Continuum Mech. Thermodyn.

2015

“…The initial stress satisfies the equilibrium equation Divτ = 0 in the absence of body forces, and is symmetric in the absence of intrinsic couple stresses, Div being the divergence operator on B r . If the initial stress is a residual stress, in the sense of Hoger (1985), then it also satisfies the zero traction boundary condition τ N = 0 on ∂B r , where N is the unit outward normal to ∂B r . According to this definition residual stresses are necessarily inhomogeneous, and they have a strong influence on the material response relative to B r .…”

Section: Kinematics and Stressmentioning

confidence: 99%

The effect of initial stress on the propagation of surface waves in a layered half-space

Nam

Merodio

International Journal of Solids and Structures

et al. 2016

In this paper the propagation of small amplitude surface waves guided by a layer with a finite thickness on an incompressible half-space is studied. The layer and half-space are both assumed to be initially stressed. The combined effect of initial stress and finite deformation on the speed of Rayleigh waves is analyzed and illustrated graphically. With a suitable simple choice of constitutive law that includes initial stress, it is shown that in many cases, as is to be expected, the effect of a finite deformation (with an associated pre-stress) is very similar to that of an initial stress (without an accompanying finite deformation). However, by contrast, when the finite deformation and initial stress are considered together independently with a judicious choice of material parameters different features are found that don't appear in the separate finite deformation or initial stress situations on their own.

“…If the traction on the boundary @B r of B r vanishes pointwise then T is referred to as a residual stress, and it is necessarily non-uniform [Hoger 1985;. If the traction is not zero then we refer to T as an initial stress or pre-stress, and in general this may be accompanied by some prior deformation required to reach the configuration B r from an unstressed state.…”

Section: Equations Of Motionmentioning

confidence: 99%

“…This requires, in particular, that the initial stress T is uniform. We recall, however, that a residual stress cannot be uniform [Hoger 1985;, so the following analysis does not apply if the initial stress is a residual stress. Suppose further that 0 .X / is a homogeneous deformation.…”

Section: Equations Of Motionmentioning

confidence: 99%

“…The term initial stresses embraces situations in which the stress is accompanied by finite deformation from an unstressed configuration, in which case the term pre-stresses is commonly used, as, for example, in [Ieşan 1989], and situations in which the initial stress arises from some other process, such as manufacturing or growth, and is present in the absence of applied loads. In this latter case the initial stress is referred to as residual stress according to the definition of [Hoger 1985].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Propagation of waves in an incompressible transversely isotropic elastic solid with initial stress: Biot revisited

J. Mech. Mater. Struct.

Singh²

2011

In this paper, the general constitutive equation for a transversely isotropic hyperelastic solid in the presence of initial stress is derived, based on the theory of invariants. In the general finite deformation case for a compressible material this requires 18 invariants (17 for an incompressible material). The equations governing infinitesimal motions superimposed on a finite deformation are then used in conjunction with the constitutive law to examine the propagation of both homogeneous plane waves and, with the restriction to two dimensions, Rayleigh surface waves. For this purpose we consider incompressible materials and a restricted set of invariants that is sufficient to capture both the effects of initial stress and transverse isotropy. Moreover, the equations are specialized to the undeformed configuration in order to compare with the classical formulation of Biot. One feature of the general theory is that the speeds of homogeneous plane waves and surface waves depend nonlinearly on the initial stress, in contrast to the situation of the more specialized isotropic and orthotropic theories of Biot. The speeds of (homogeneous plane) shear waves and Rayleigh waves in an incompressible material are obtained and the significant differences from Biot's results for both isotropic and transversely isotropic materials are highlighted with calculations based on a specific form of strain-energy function.