<p>For theories working with fully implicit constitutive relations we find conditions for strict hyperbolicity for the system of elastodynamic equations in one space dimension. To bypass the fact that stress is only implicitly related to strain and not explicitly, we utilize three partial differential equations to form our system: the momentum equation, the compatibility equation as well as the time differentiated fully implicit constitutive law. This way we built up a system of three quasilinear partial differential equations of first order for the velocity (<InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>t</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$u_{t}$</EquationSource> </InlineEquation>), the displacement gradient (<InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>x</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$u_{x}$</EquationSource> </InlineEquation>) (or the strain (<InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>e</mi> </math></EquationSource> <EquationSource Format="TEX">$e$</EquationSource> </InlineEquation>)) as well as the stress (<InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> <EquationSource Format="TEX">$\sigma $</EquationSource> </InlineEquation>). For this system we find conditions for strict hyperbolicity using the characteristic polynomial. We do that for cases where the implicit constitutive laws are either of the form <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mi>f</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>,</mo> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$f(\sigma , u_{x})=0$</EquationSource> </InlineEquation>, or <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <mi>f</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$f(\sigma , e)=0$</EquationSource> </InlineEquation>. The same method applies also to constitutive laws of the form <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <mi>e</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$e=g(\sigma )$</EquationSource> </InlineEquation>.</p>

A Note on Hyperbolicity for Fully Implicit Theories in One Space Dimension

期刊论文
发表日期：

2026年2月12日

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D. Sfyris,
R. Bustamante

摘要

For theories working with fully implicit constitutive relations we find conditions for strict hyperbolicity for the system of elastodynamic equations in one space dimension. To bypass the fact that stress is only implicitly related to strain and not explicitly, we utilize three partial differential equations to form our system: the momentum equation, the compatibility equation as well as the time differentiated fully implicit constitutive law. This way we built up a system of three quasilinear partial differential equations of first order for the velocity ( $u_{t}$ $u_{t}$ ), the displacement gradient ( $u_{x}$ $u_{x}$ ) (or the strain ( $e$ $e$ )) as well as the stress ( $σ$ $\sigma $ ). For this system we find conditions for strict hyperbolicity using the characteristic polynomial. We do that for cases where the implicit constitutive laws are either of the form $f (σ, u_{x}) = 0$ $f(\sigma , u_{x})=0$ , or $f (σ, e) = 0$ $f(\sigma , e)=0$ . The same method applies also to constitutive laws of the form $e = g (σ)$ $e=g(\sigma )$ .