##
**Structure of the set of equilibrated loads in nonlinear elasticity and applications to existence and nonexistence.**
*(English)*
Zbl 0654.73029

The paper is divided into three sections. In Section I, we describe the geometrical structure of the set of equilibrated loads found in the traction problem of nonlinear elasticity. Using this structure, we present in Section II existence theorems for nonlinear traction problems, and in Section III nonexistence theorems for the linear traction problem.

In the first Section, we consider Signorini’s condition of compatibility. This condition asserts that a deformation \(\phi\) and a corresponding load (b,t), where b and t are force densities (which may or may not depend on \(\phi)\) on the reference configuration \(\Omega\) and its boundary \(\partial \Omega\), may be in equilibrium only if \[ (0)\quad \int_{\Omega}\phi \wedge b+\int_{\partial \Omega}\phi \wedge t=0, \] so that the total torque vanishes. In the dead load traction problem of nonlinear elasticity, relation (0) cannot be imposed in advance as a condition to be satisfied by the load (by contrast with the linear case) since it involves the unknown deformation \(\phi\). Signorini suggested that condition (0) be regarded as a ‘condition of compatibility’ to be checked a posteriori once the problem is solved. On the contrary, we consider here that (0) is an a priori condition to be satisfied by the admissible triples (\(\phi\),b,t). More precisely, we show that condition (0) defines a subset M of the product space \(W^{m+2,p}(\Omega)^ 3\times W^{m,p}(\Omega)^ 3\times W^{m+1-1/p,p}(\partial \Omega)^ 3\), \(p>3\), a dense subset of which, \(M\setminus N\), has a \(C^{\infty}\)- manifold structure. A further subset, \(M\setminus N'\), is actually endowed with a natural vector bundle structure, and condition (0) simply says that (b,t) must belong to the fiber attached to \(\phi\). The space \[ {\mathcal L}_ e=\{(b,t),\quad \int_{\Omega}x\wedge b+\int_{\partial \Omega}x\wedge t=0\}, \] which appears in the classical linear theory and in Signorini’s and Stoppelli’s theorems [C. C. Wang and C. Truesdell, Introduction to rational elasticity (1973; Zbl 0308.73001)], is the fiber of M at \(\phi =Id\). Moreover, the operator of nonlinear elasticity \[ E:\quad \phi \to (\phi,-divT_ R(x,\nabla \phi),\quad T_ R(x,\nabla \phi)v), \] where \(T_ R(x,\nabla \phi)\) is the first Piola- Kirchhoff stress tensor, defines (under suitable smoothness assumptions) a \(C^ 1\)-section of the vector bundle M. If we assume that the reference configuration is natural (i.e. stress free), then usual ellipticity assumptions on the associated linear elasticity operator imply that this section is transversal to the trivial section \(\phi\) \(\to (\phi,0,0).\)

In Section II, we use this property of transversality to give a simple version of Stoppelli’s existence theorem [F. Stoppelli, Ric. Mat. 3, 247-267 (1954; Zbl 0058.397)] for the dead load traction problem, when the loads depend on a parameter \(\lambda\) lying in some Banach space \(\Lambda\), under assumptions that reduce to the classical assumption of no axis of equilibrium when the load takes the form (\(\lambda\) b,\(\lambda\) t) with \(\lambda\in {\mathbb{R}}.\)

We proceed as follows: we construct another \(C^ 1\)-section of the vector bundle M by using the natural action of the group of rotations SO(3) on the loads, i.e. by rotating the load in such a way that the rotated load satisfies condition (0); the assumption that there is no axis of equilibrium is used here. Then, any element of the intersection of the images of these two sections gives rise to a solution of the equilibrium problem, and vice versa. If for \(\lambda =0\), the section of loads reduces to the trivial section, then the transversality of the nonlinear elasticity operator with the trivial section yields the existence when \(\| \lambda \|\) is small. Basically, this approach is only slightly different from that of Stoppelli, who uses the action of SO(3) to map the loads in \({\mathcal L}_ e\) rather than in M (as we do here).

In Section III, we give nonexistence and nonuniqueness results for linear elasticity with nonzero residual stress. The main result is that relation (0) (which in the classical case, i.e. with zero residual stress, induces the usual nonexistence and dual nonuniqueness in linear elasticity) induces in general nonexistence if and only if the residual load is a parallel load. This is a consequence of the previously noted fact that the nonlinear elasticity operator defines a section of M. Therefore, the range of the differential of this section, which is closely related to the linear elasticity operator, is contained in the tangent space to M at the point under consideration, for example \((Id,b_ 0,t_ 0)\) where \((b_ 0,t_ 0)\) denotes the residual load. It necessarily follows that the image of the linear elasticity operator is contained in the projection of the tangent space \(T_{(Id,b_ 0,t_ 0)}M\) on the space of loads \(W^{m,p}(\Omega)^ 3\times W^{m+1-1/p,p}(\partial \Omega)^ 3\). Obviously, the properties of this projection are solely due to the geometrical structure of the manifold \(M\setminus N\) so that we get ‘universal’ results, in the sense that they are valid for any constitutive law. More precisely, we show that this projection is a proper subspace of the space of loads and thus we get a priori nonexistence, if and only if the residual load \((b_ 0,t_ 0)\) is a parallel load. We also obtain the dual properties of the kernel of the linear elasticity operator (i.e., nonuniqueness) without using Betti’s reciprocity theorem, and thus in particular without the assumption of hyperelasticity.

In the first Section, we consider Signorini’s condition of compatibility. This condition asserts that a deformation \(\phi\) and a corresponding load (b,t), where b and t are force densities (which may or may not depend on \(\phi)\) on the reference configuration \(\Omega\) and its boundary \(\partial \Omega\), may be in equilibrium only if \[ (0)\quad \int_{\Omega}\phi \wedge b+\int_{\partial \Omega}\phi \wedge t=0, \] so that the total torque vanishes. In the dead load traction problem of nonlinear elasticity, relation (0) cannot be imposed in advance as a condition to be satisfied by the load (by contrast with the linear case) since it involves the unknown deformation \(\phi\). Signorini suggested that condition (0) be regarded as a ‘condition of compatibility’ to be checked a posteriori once the problem is solved. On the contrary, we consider here that (0) is an a priori condition to be satisfied by the admissible triples (\(\phi\),b,t). More precisely, we show that condition (0) defines a subset M of the product space \(W^{m+2,p}(\Omega)^ 3\times W^{m,p}(\Omega)^ 3\times W^{m+1-1/p,p}(\partial \Omega)^ 3\), \(p>3\), a dense subset of which, \(M\setminus N\), has a \(C^{\infty}\)- manifold structure. A further subset, \(M\setminus N'\), is actually endowed with a natural vector bundle structure, and condition (0) simply says that (b,t) must belong to the fiber attached to \(\phi\). The space \[ {\mathcal L}_ e=\{(b,t),\quad \int_{\Omega}x\wedge b+\int_{\partial \Omega}x\wedge t=0\}, \] which appears in the classical linear theory and in Signorini’s and Stoppelli’s theorems [C. C. Wang and C. Truesdell, Introduction to rational elasticity (1973; Zbl 0308.73001)], is the fiber of M at \(\phi =Id\). Moreover, the operator of nonlinear elasticity \[ E:\quad \phi \to (\phi,-divT_ R(x,\nabla \phi),\quad T_ R(x,\nabla \phi)v), \] where \(T_ R(x,\nabla \phi)\) is the first Piola- Kirchhoff stress tensor, defines (under suitable smoothness assumptions) a \(C^ 1\)-section of the vector bundle M. If we assume that the reference configuration is natural (i.e. stress free), then usual ellipticity assumptions on the associated linear elasticity operator imply that this section is transversal to the trivial section \(\phi\) \(\to (\phi,0,0).\)

In Section II, we use this property of transversality to give a simple version of Stoppelli’s existence theorem [F. Stoppelli, Ric. Mat. 3, 247-267 (1954; Zbl 0058.397)] for the dead load traction problem, when the loads depend on a parameter \(\lambda\) lying in some Banach space \(\Lambda\), under assumptions that reduce to the classical assumption of no axis of equilibrium when the load takes the form (\(\lambda\) b,\(\lambda\) t) with \(\lambda\in {\mathbb{R}}.\)

We proceed as follows: we construct another \(C^ 1\)-section of the vector bundle M by using the natural action of the group of rotations SO(3) on the loads, i.e. by rotating the load in such a way that the rotated load satisfies condition (0); the assumption that there is no axis of equilibrium is used here. Then, any element of the intersection of the images of these two sections gives rise to a solution of the equilibrium problem, and vice versa. If for \(\lambda =0\), the section of loads reduces to the trivial section, then the transversality of the nonlinear elasticity operator with the trivial section yields the existence when \(\| \lambda \|\) is small. Basically, this approach is only slightly different from that of Stoppelli, who uses the action of SO(3) to map the loads in \({\mathcal L}_ e\) rather than in M (as we do here).

In Section III, we give nonexistence and nonuniqueness results for linear elasticity with nonzero residual stress. The main result is that relation (0) (which in the classical case, i.e. with zero residual stress, induces the usual nonexistence and dual nonuniqueness in linear elasticity) induces in general nonexistence if and only if the residual load is a parallel load. This is a consequence of the previously noted fact that the nonlinear elasticity operator defines a section of M. Therefore, the range of the differential of this section, which is closely related to the linear elasticity operator, is contained in the tangent space to M at the point under consideration, for example \((Id,b_ 0,t_ 0)\) where \((b_ 0,t_ 0)\) denotes the residual load. It necessarily follows that the image of the linear elasticity operator is contained in the projection of the tangent space \(T_{(Id,b_ 0,t_ 0)}M\) on the space of loads \(W^{m,p}(\Omega)^ 3\times W^{m+1-1/p,p}(\partial \Omega)^ 3\). Obviously, the properties of this projection are solely due to the geometrical structure of the manifold \(M\setminus N\) so that we get ‘universal’ results, in the sense that they are valid for any constitutive law. More precisely, we show that this projection is a proper subspace of the space of loads and thus we get a priori nonexistence, if and only if the residual load \((b_ 0,t_ 0)\) is a parallel load. We also obtain the dual properties of the kernel of the linear elasticity operator (i.e., nonuniqueness) without using Betti’s reciprocity theorem, and thus in particular without the assumption of hyperelasticity.

### MSC:

74B20 | Nonlinear elasticity |

58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |

57R22 | Topology of vector bundles and fiber bundles |

74G30 | Uniqueness of solutions of equilibrium problems in solid mechanics |

74H25 | Uniqueness of solutions of dynamical problems in solid mechanics |

### Keywords:

geometrical structure; set of equilibrated loads; Signorini’s condition of compatibility; vector bundle structure; property of transversality; a priori nonexistence
Full Text:
DOI

### References:

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