Symmetric unisolvent equations for linear elasticity purely in stresses

Abstract

In this work we introduce novel stress-only formulations of linear elasticity with special attention to their approximate solution using weighted residual methods. We present four sets of boundary value problems for a pure stress formulation of three-dimensional solids, and in two dimensions for plane stress and plane strain. The associated governing equations are derived by modifications and combinations of the Beltrami-Michell equations and the Navier-Cauchy equations. The corresponding variational forms of dimension $d \in \{2,3\}$ allow to approximate the stress tensor directly, without any presupposed potential stress functions, and are shown to be well-posed in $H^1 \otimes \mathrm{Sym}(d)$ in the framework of functional analysis via the Lax-Milgram theorem, making their finite element implementation using $C^0$-continuous elements straightforward. Further, in the finite element setting we provide a treatment for constant and piece-wise constant body forces via distributions. The operators and differential identities in this work are provided in modern tensor notation and rely on exact sequences, making the resulting equations and differential relations directly comprehensible. Finally, numerical benchmarks for convergence as well as spectral analysis are used to test the limits and identify viable use-cases of the equations.

Figures (12)

• Figure 2.1: The domain $V \subset \mathbb{R}^3$ with Dirichlet $A_D\subset \mathbb{R}^2$ and Neumann $A_N\subset \mathbb{R}^2$ boundaries under internal body forces $\mathbf{f}$ and tractions $\bm{\mu}$ with the outer normal vector $\mathbf{n}$.
• Figure 2.2: The three-dimensional elasticity sequence. The space of rigid body motions is the kernel of the symmetrised gradient operator $\ker(\mathop{\mathrm{\mathrm{sym}}}\nolimits\mathrm{D}) = \mathit{RM}(V)$, and the range of the latter is the kernel of the incompatibility operator $\mathop{\mathrm{\mathrm{sym}}}\nolimits \mathrm{D} [\mathit{C}^\infty(V)]^3 = \ker(\mathop{\mathrm{\mathrm{Inc}}}\nolimits)$. The range of the incompatibility operator is exactly the kernel of the divergence operator $\mathop{\mathrm{\mathrm{Inc}}}\nolimits[\mathit{C}^\infty(V)\otimes \mathrm{Sym}(3)] = \ker(\mathop{\mathrm{\mathrm{Div}}}\nolimits)$ for symmetric tensors, and the range of the divergence operator is a surjection onto the last space in the sequence $\mathop{\mathrm{\mathrm{Div}}}\nolimits[\mathit{C}^\infty(V)\otimes \mathrm{Sym}(3)] = [\mathit{C}^\infty(V)]^3$.
• Figure 2.3: The three-dimensional de Rham sequence. The space of constants is the kernel of the gradient operator $\mathbb{R} = \ker(\nabla)$, and the range of the latter is the kernel of the curl operator $\nabla \mathit{C}^\infty(V) = \ker(\mathop{\mathrm{\mathrm{curl}}}\nolimits)$. The range of the curl operator is exactly the kernel of the divergence operator $\mathop{\mathrm{\mathrm{curl}}}\nolimits[\mathit{C}^\infty(V)]^3 = \ker(\mathop{\mathrm{\mathrm{div}}}\nolimits)$, of which the range is a surjection onto the last space $\mathop{\mathrm{\mathrm{div}}}\nolimits[\mathit{C}^\infty(V)]^3 = \mathit{C}^\infty(V)$.
• Figure 2.4: Two planar elasticity sequences derived from the reduction of the incompatibility operator. For two-dimensional tensors, the incompatibility operator turns into the $\mathop{\mathrm{\mathrm{rot}}}\nolimits \mathop{\mathrm{\mathrm{Rot}}}\nolimits$-operator and for scalars into the airy operator. In the first sequence the two-dimensional space of rigid body motions is the kernel of the $\mathop{\mathrm{\mathrm{sym}}}\nolimits\mathrm{D}$-operator $\mathit{RM}(A) = \ker(\mathop{\mathrm{\mathrm{sym}}}\nolimits \mathrm{D})$, of which the range is the kernel of the $\mathop{\mathrm{\mathrm{rot}}}\nolimits \mathop{\mathrm{\mathrm{Rot}}}\nolimits$-operator $\mathop{\mathrm{\mathrm{sym}}}\nolimits \mathrm{D} [\mathit{C}^\infty(A)]^2 = \ker(\mathop{\mathrm{\mathrm{rot}}}\nolimits \mathop{\mathrm{\mathrm{Rot}}}\nolimits)$. Finally, the $\mathop{\mathrm{\mathrm{rot}}}\nolimits\mathop{\mathrm{\mathrm{Rot}}}\nolimits$-operator yields a surjection onto the last space in the sequence $\mathop{\mathrm{\mathrm{rot}}}\nolimits \mathop{\mathrm{\mathrm{Rot}}}\nolimits[\mathit{C}^\infty(A) \otimes \mathrm{Sym}(2)] = \mathit{C}^\infty(A)$. In the second sequence the kernel of the $\mathop{\mathrm{\mathrm{airy}}}\nolimits$-operator is given by the linear polynomial space $\mathit{P}^1(A) = \ker(\mathop{\mathrm{\mathrm{airy}}}\nolimits)$, and the $\mathop{\mathrm{\mathrm{airy}}}\nolimits$-operator maps the kernel the divergence operator for symmetric tensors $\mathop{\mathrm{\mathrm{airy}}}\nolimits \mathit{C}^\infty(A) = \ker(\mathop{\mathrm{\mathrm{Div}}}\nolimits)$. Lastly, the divergence yields a surjection onto the last space in the sequence $\mathop{\mathrm{\mathrm{Div}}}\nolimits [\mathit{C}^\infty(A) \otimes \mathrm{Sym}(2)] = [\mathit{C}^\infty(A)]^2$.
• Figure 2.5: One of two possible two-dimensional de Rham sequences derived by the reduction of the curl operator to two dimensions. The space of constants is the kernel of the rotated gradient operator $\mathbb{R} = \ker(\nabla^\perp)$, and the range of the latter is the kernel of the divergence operator $\nabla^\perp\mathit{C}^\infty(A) = \ker(\mathop{\mathrm{\mathrm{div}}}\nolimits)$. The range of the divergence operator is a surjection onto the last space $\mathop{\mathrm{\mathrm{div}}}\nolimits[\mathit{C}^\infty(A)]^2 = \mathit{C}^\infty(A)$.

Theorems & Definitions (44)

• Remark 2.1: Vanishing right-hand side
• Remark 2.2: Specialised stress functions
• Remark 2.3: Vanishing planar right-hand side
• definition 3.1: The pure stress boundary value problem for solids I
• definition 3.2: Variational form of the pure stress problem I
• definition 3.3: Variational functional in pure stresses I
• Lemma 3.1: Upper bound of $\| {\mathrm{D} \bm{\sigma}} \|_{\mathit{L}^2}$
• proof
• Lemma 3.2: Upper bound of $\| {\mathop{\mathrm{\mathrm{skw}}}\nolimits \mathop{\mathrm{\mathrm{Curl}}}\nolimits \bm{\sigma}} \|$
• proof