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Cauchy elastic material
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<h2 id="mathematical-definition">Mathematical definition</h2> <p>Formally, a material is said to be Cauchy-elastic if the <a href="/facts/Cauchy_stress_tensor/5MsXPQVH">Cauchy stress tensor</a> σ {\displaystyle {\boldsymbol {\sigma }}} is a function of the <a href="/facts/Strain_tensor/j4CJFdsU">strain tensor</a> (<a href="/facts/Deformation_gradient/rEjpC9oD">deformation gradient</a>) F {\displaystyle {\boldsymbol {F}}} alone: </p> σ = G ( F ) {\displaystyle \ {\boldsymbol {\sigma }}={\mathcal {G}}({\boldsymbol {F}})} <p>This definition assumes that the effect of temperature can be ignored, and the body is homogeneous. This is the <a href="/facts/Constitutive_equation/IXTjEAbL">constitutive equation</a> for a Cauchy-elastic material. </p><p>Note that the function G {\displaystyle {\mathcal {G}}} depends on the choice of reference configuration. Typically, the reference configuration is taken as the relaxed (zero-stress) configuration, but need not be. </p><p><a href="/facts/Material_frame-indifference/aMA8KsKD">Material frame-indifference</a> requires that the constitutive relation G {\displaystyle {\mathcal {G}}} should not change when the location of the observer changes. Therefore the <a href="/facts/Constitutive_equation/IXTjEAbL">constitutive equation</a> for another arbitrary observer can be written σ ∗ = G ( F ∗ ) {\displaystyle {\boldsymbol {\sigma }}^{*}={\mathcal {G}}({\boldsymbol {F}}^{*})} . Knowing that the <a href="/facts/Stress_(physics)/ErAXDbbd">Cauchy stress tensor</a> σ {\displaystyle \sigma } and the <a href="/facts/Deformation_gradient/rEjpC9oD">deformation gradient</a> F {\displaystyle F} are objective quantities, one can write: </p> σ ∗ = G ( F ∗ ) ⇒ R ⋅ σ ⋅ R T = G ( R ⋅ F ) ⇒ R ⋅ G ( F ) ⋅ R T = G ( R ⋅ F ) {\displaystyle {\begin{aligned}&{\boldsymbol {\sigma }}^{*}&=&{\mathcal {G}}({\boldsymbol {F}}^{*})\\\Rightarrow &{\boldsymbol {R}}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {R}}^{T}&=&{\mathcal {G}}({\boldsymbol {R}}\cdot {\boldsymbol {F}})\\\Rightarrow &{\boldsymbol {R}}\cdot {\mathcal {G}}({\boldsymbol {F}})\cdot {\boldsymbol {R}}^{T}&=&{\mathcal {G}}({\boldsymbol {R}}\cdot {\boldsymbol {F}})\end{aligned}}} <p>where R {\displaystyle {\boldsymbol {R}}} is a proper orthogonal tensor. </p><p>The above is a condition that the <a href="/facts/Constitutive_equation/IXTjEAbL">constitutive law</a> G {\displaystyle {\mathcal {G}}} has to respect to make sure that the response of the material will be independent of the observer. Similar conditions can be derived for <a href="/facts/Constitutive_equation/IXTjEAbL">constitutive laws</a> relating the <a href="/facts/Deformation_gradient/rEjpC9oD">deformation gradient</a> to the first or second <a href="/facts/Piola-Kirchhoff_stress_tensor/ErAXDbbd">Piola-Kirchhoff stress tensor</a>. </p> <h2 id="isotropic-cauchy-elastic-materials">Isotropic Cauchy-elastic materials</h2> <p>For an isotropic material the <a href="/facts/Stress_(physics)/ErAXDbbd">Cauchy stress tensor</a> σ {\displaystyle {\boldsymbol {\sigma }}} can be expressed as a function of the <a href="/facts/Finite_strain_theory/rEjpC9oD">left Cauchy-Green tensor</a> B = F ⋅ F T {\displaystyle {\boldsymbol {B}}={\boldsymbol {F}}\cdot {\boldsymbol {F}}^{T}} . The <a href="/facts/Constitutive_equation/IXTjEAbL">constitutive equation</a> may then be written: </p> σ = H ( B ) . {\displaystyle \ {\boldsymbol {\sigma }}={\mathcal {H}}({\boldsymbol {B}}).} <p>In order to find the restriction on h {\displaystyle h} which will ensure the principle of material frame-indifference, one can write: </p> σ ∗ = H ( B ∗ ) ⇒ R ⋅ σ ⋅ R T = H ( F ∗ ⋅ ( F ∗ ) T ) ⇒ R ⋅ H ( B ) ⋅ R T = H ( R ⋅ F ⋅ F T ⋅ R T ) ⇒ R ⋅ H ( B ) ⋅ R T = H ( R ⋅ B ⋅ R T ) . {\displaystyle \ {\begin{array}{rrcl}&{\boldsymbol {\sigma }}^{*}&=&{\mathcal {H}}({\boldsymbol {B}}^{*})\\\Rightarrow &{\boldsymbol {R}}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {R}}^{T}&=&{\mathcal {H}}({\boldsymbol {F}}^{*}\cdot ({\boldsymbol {F}}^{*})^{T})\\\Rightarrow &{\boldsymbol {R}}\cdot {\mathcal {H}}({\boldsymbol {B}})\cdot {\boldsymbol {R}}^{T}&=&{\mathcal {H}}({\boldsymbol {R}}\cdot {\boldsymbol {F}}\cdot {\boldsymbol {F}}^{T}\cdot {\boldsymbol {R}}^{T})\\\Rightarrow &{\boldsymbol {R}}\cdot {\mathcal {H}}({\boldsymbol {B}})\cdot {\boldsymbol {R}}^{T}&=&{\mathcal {H}}({\boldsymbol {R}}\cdot {\boldsymbol {B}}\cdot {\boldsymbol {R}}^{T}).\end{array}}} <p>A <a href="/facts/Constitutive_equation/IXTjEAbL">constitutive equation</a> that respects the above condition is said to be <a href="/facts/Isotropic/mc3QEY2x">isotropic</a>. </p> <h2 id="non-conservative-materials">Non-conservative materials</h2> <p>Even though the stress in a Cauchy-elastic material depends only on the state of deformation, the work done by stresses may depend on the path of deformation. Therefore a Cauchy elastic material in general has a non-conservative structure, and the stress cannot necessarily be derived from a scalar "elastic potential" function. Materials that are conservative in this sense are called <a href="/facts/Hyperelastic_material/uyLxmXpN">hyperelastic</a> or "Green-elastic". </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>R. W. Ogden, 1984, Non-linear Elastic Deformations, Dover, pp. 175–204. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>