Force chain splitting in granular materials:a mechanism for large scale pseudo-elastic behaviour

Definition .

A Lie algebra $\mathfrak{g}$ is called decomposable if it can be written

\mathfrak{g}=\mathfrak{s}\oplus\mathfrak{t}\oplus\mathfrak{n}

where $\mathfrak{s}$ is a Levi subalgebra, $\mathfrak{n}$ the nilradical and $\mathfrak{t}$ an abelian subalgebra whose elements are $ad$ -semisimple and which satisfies $\left[\mathfrak{s},\mathfrak{t}\right]=0.$

Definition 2.1

Let ${\mathcal{K}}$ be a real Hilbert space. The algebra $\mathcal{A}({\mathcal{K}})$ is the free unital ${}^{*}$ -algebra with generators $\omega(h)$ for all $h\in{\mathcal{K}}$ , divided by the relations:

\omega(af+bg)=a\omega(f)+b\omega(g),\qquad\omega(f)=\omega(f)^{*}

(2.1)

for all $f,g\in{\mathcal{K}}$ and $a,b\in\mathbb{R}$ .

Definition 2.2

Let $\mathcal{H}$ be a complex Hilbert space. The algebra $\mathcal{C}(\mathcal{H})$ is the free unital ${}^{*}$ -algebra with generators $a(h)$ and $a^{*}(h)$ for all $h\in\mathcal{H}$ , divided by the relations:

a^{*}(\lambda f+\mu g)=\lambda a^{*}(f)+\mu a^{*}(g),\qquad a^{*}(f)=a(f)^{*}

(2.2)

for all $f,g\in\mathcal{H}$ and $\lambda,\mu\in\mathbb{C}$ .

Definition 5.1 (T-Norm)

t(a,1)=a\quad\text{(boundary condition)}

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b\leq d\;\text{implies}\;t(a,b)\leq t(a,d)\quad\text{(monotonicity)}

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t(a,b)=t(b,d)\quad\text{(commutativity)}

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t(a,t(b,d))=t(t(a,b),d)\quad\text{(associativity)}

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