It is known that the type of system of partial differential equations governing finite elastostatics can change type from elliptic to non-elliptic at sufficiently large deformations for some verifiable materials. We show that this one-dimensional case does not have a classically smooth solution for certain ranges.
CHAPTER I
CHAPTER 2
The nominal stresses are now given by .. where p~) is a scalar field that arises due to the misfit constraint. To prove the first part of the result, let X be a principal frame for the positive definite symmetric tensor FF T. l is the matrix of components of the tensor ~l in the frame ~l . where we have exploited the fact that det F = 1. It is clear that we can assume A.;;:: 1 without loss of generality.
CHAPTER 3
A physical interpretation of the ellipticity condition (3.21) can be obtained in terms of the concept of the local rate of shear, introduced in Section 2. Now suppose that W'(I) is f 0 at the point of interest and that the ellipticity goes lost due to the fact that the second of (3. 21) is violated.
CHAPTER 4
To investigate questions related to the existence of piecewise homogeneous elastostatic shocks we pose the following problem. We can now pose the following problem, which is equivalent to the one posed previously.
CHAPTER 5
Equation (5. 19) must necessarily be valid if there will exist a family of one-parameter elastostatic shocks of the type under consideration. In case, corresponding to a given I and p, there is a family of one-parameter shocks of the type under consideration, jumps of various physical quantities along the shock can easily be.
CHAPTER 6
First, we introduce the following terminology. 2.40) and (3. 21) that there may be a loss of ellipticity of the equilibrium displacement equations at a certain strain and at some point if and. A strict ellipticity error of the displacement balance equations at a given homogeneous strain and pressure at TI is sufficient. We will show that it corresponds to a given homogeneous strain on TI + so that the associated value of p is in accordance with (6. 28) and.
This implies a loss of ellipticity of the displacement equations for equilibrium at a homogeneous deformation, where the deformation gradient F is such that F AF A= I,. We draw attention to the fact that Theorem 2 does not imply that if ellipticity is strictly lost at the given deformation, then the corresponding configuration of the body must involve a shock. Likewise, a loss of ellipticity at the given deformation is not necessary for a corresponding elastostatic shock to exist.
CHAPTER 7
However, if there is an elastostatic shock in the domain :i9, then (7.1) does not hold automatically and therefore imposes a local constraint on the jumps of the field quantities at each point of the shock. It can be shown that, if at some point t (7. 1) holds with strict inequality for all subdomains ;ig that intersect £, then. i) the motion of the shock line £ at that instant is translational in a direction not parallel to itself. Conversely, if at some time t the quasi-static family of solutions conforms to (i), then (7.1) holds at that time with strict inequality.
On the other hand, we can show that if at the instant t (7. with equality holds, then either. ii) the shock line £ is instantaneously stationary at that instant, or. iii) the shock line £ is instantaneously in a state of translation parallel to itself at that instant, or. in which case the shock line movement is not limited to translation). Conversely, if the quasi-static family of solutions at some time corresponds to one of (ii), (iii), and (iv), then (7. 1) holds with equality at that instant. As one would expect, and as verified by Knowles [4], these results remain locally true in the general case of a curved shock in a non-homogeneous elastic field, with the exception that the shock motion is no longer limited to translation may not be. .
CHAPTER 8
Recall that a corresponding piecewise homogeneous elastostatic shock exists if and only if the function If, then, at the moment when the family of shocks coincides with this given shock, the shock line -£ translates into IT, it is con-+. Then, if at the moment when the family of shocks coincides with this given shock, the shock line -£ translates into IT, it is.
In this case, the sign of the shock strength and the permissible direction of the quasi-. Note in particular that the allowed direction of quasi-static shock motion, for dissipativeness, is governed solely by whether the spatial.
CHAPTER 9
For sufficiently small and large values of the prescribed rotation we have a unique smooth solution where the displacement equations of equilibrium are elliptic. In all cases there are ranges of values of the prescribed twist for which we do not find a solution. We then prove that there are in fact no smooth solutions in these regions of the prescribed twist.
We now find a solution that corresponds to each value of the prescribed spin, but unfortunately many solutions correspond to certain values. Finally, in section 14 we investigate the stability of each of the available solutions against perturbational perturbations of arbitrary size. For sufficiently small and large values of applied twist, this unique stable solution is smooth and elliptical.
CHAPTER 10
It is convenient to express field quantities at any point. r, 8, z) with respect to the components in the rectangular Cartesian coordinate frame X. The matrix of components of the strain gradient tensor F =Vy in the frame X / is easily calculated from. Since, in addition to p, the pressure p may also depend on the coordinates $ and S, the Cauchy stress tensor follows.
It is sufficient for our purposes to specify the response of the material in shnple shear alone. Observe from ( 10. 24) that the plane strain elastic potential W(I) is completely determined by the function f, so that the response in simple shear is, in fact, determined. completely the in-plane response in all plane deformations. i) f is continuously differentiable on {- oo, oo),. It is clear that for the particular class of materials just described, f has no single-valued inverse.
CHAPTER 11
We can demonstrate this in a completely analogous way, if the prescribed twist is given. In this article we will consider in detail the case when the dimensions of the pipe and the constitutive law of the material are such. For a given material, one might see that (11.23) requires that the thickness of the tube be sufficiently small.
In the next section, we show that there are in fact no smooth solutions when the twist is prescribed in these regions. We now establish a sequence of lemmas that lead to a result that is actually stronger than the one asserted at the end of the last one. Thus, we have shown that for certain ranges of the prescribed twist there is no solution in the classical sense of the problem under consideration.
CHAPTER 12
We will, without loss of generality, restrict attention to the first quadrant of We will show that for every fixed choice of the written down i,j = 1,2,3, if j , there is a single connected closed region A. The other cases - corresponding to the remaining choices of the written down i, j - may possibly also be investigated.
However, we are now faced with the unsatisfactory situation in which there are an infinite number of acceptable solutions at certain values of the prescribed curve. IO(b) how the local amount of shear varies continuously on either side of the shock but undergoes a jump discontinuity along it. In general, the weak solution (i, j) is associated with the ith and /h branches of f(k), with the part of the pipe inside the strike line corresponding to the ith branch.
CHAPTER 13
We will refer to the piecewise smooth oriented1. curve r in the torque-twist plane defined by ¢. According to the hypothesis, for all values of t that are sufficiently close to t1, the loading path r is in A. 1)) be the point on r corresponding to t=t. The arrows in Figure 6(i) indicate the allowable orientations of a loading path at some typical points in A. 8 and 9, where each allowable loading path starting from 0 1 is also limited to · O'R for the subsequent time.
0 exceeds the ¢x value, however, the loading path can lie anywhere in PQYX, and we have no criteria to decide which path to follow. Likewise, during a steady decrease in applied bias, the charge path would be constrained to 0 1QY, then allowed to follow an arbitrary path (consistent with dissipativization) in XYRS, and finally constrained to SO. It is interesting to note that if in either case the loading path lies on the curve OXYO'.
CHAPTER 14
Assume first that we have a dead load on the inner surface of the pipe while the outer surface is fixed so that the torque T remains constant during the virtual displacement. Now consider the case where the inner and outer surfaces of the pipe are fixed during apparent displacement. We therefore have that there exists a unique stable solution ¢(r) of the boundary value problem in its weak formulation corresponding to each value of the twist applied.
Knowles and Eli Sternberg, On the Ellipticity of the Equations of Nonlinear Elastostatics for a Special Material, Journal of. Knowles and Eli Sternberg, Asymptotic finite-:-strain analysis of the elastostatic field near a crack tip, Journal of Elasticity, l p.67. Knowles and Eli Sternberg, Finite strain analysis of an elastostatic field near a crack tip: Remodeling and higher order results, Journal of Elasticity, i p.
SOLUTION ADMISSIBLE (I,}) REGION An
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