11. Behaviors specific to 1D elements

11.1. Keyword factor ECRO_ASYM_LINE (cf. [R5.03.09])

It makes it possible to model linear isotropic work hardening behavior, but with different elasticity limits and work hardening modules in tension and compression. This is used by the 1D behavior model VMIS_ASYM_LINE, which can be used for bar elements.

The elastic behavior under traction and compression is the same: same Young’s modulus.

There are two isotropic work hardening domains defined by \({R}_{T}\) and \({R}_{C}\). The two domains are independent of each other. We adopt a \(T\) index for traction and \(C\) for compression.

\({\sigma }_{\mathrm{YT}}\)	Limited traction effort. In absolute value.
\({\sigma }_{\mathrm{YC}}\)	Limited compression force. In absolute value.
\({p}_{T}\)	Cumulative plastic deformation under traction. Algebraic value.
\({p}_{C}\)	Cumulative plastic deformation under compression. Algebraic value.
\({E}_{\mathrm{TT}}\)	Work hardening slope under traction.
\({E}_{\mathrm{TC}}\)	Compression work hardening slope.

The equations of the behavior model are:

math:

{begin {array} {c} {c} {dot {epsilon}} ^ {p} =dot {epsilon} -stackrel {mathrm {.}} -stackrel {mathrm {.}} {stackrel {} {{E} ^ {-1}sigma}} - {dot {epsilon}} ^ {mathit {th}}\ {dot {epsilon}}} ^ {p}} ^ {p} = {dot {epsilon}} = {dot {epsilon}} = {dot {epsilon}}} ^ {p} = {dot {epsilon}}} ^ {p} = {dot {epsilon}} = {dot {epsilon}} = {dot {epsilon}} ^ {p} = {dot {epsilon}} = {dot {epsilon}}} ^ {p} = {dot {epsilon}} hfill\ {dot {epsilon}}} _ {C} ^ {p}} =dot {{p} _ {C}}frac {sigma} {left|sigmaright|}hfill\sigma -}}hfill\ sigma -}}hfill\right|}}}hfill\ right|}}hfill\ right|}}hfill\ right|}}hfill\right|}}hfill\sigma -}}hfill\sigma -}}hfill\ sigma -}}hfill\ sigma -}}hfill\ sigma - {R} _ {C} ({p} _ {C})le 0hfillend {array} `:math:begin {array} {c}text {with}hfill\ (begin {array} {c} {c}dot {{p}}dot {p}}dot {p} _ {C}dot {p}}dot {p} _ {C}dot {p} _ {C}) <0hfill\dot {{p} _ {C}}ge 0text {si}} -sigma = {R} _ {C} ({p} _ {C})hfill\ dot {{p} _ {P} _ {T}}} =0text {si}}} =0text {si}}} =0text {si}} -sigma - {R} _ {C})hfill\ dot {{p} _ {T}) <0hfill\dot {{p} _ {T}}ge 0ge 0text {si}sigma = {R} _ {T} ({p} _ {T})hfillend {array}}end {array} `

where:

\({\dot{\varepsilon }}_{C}^{p}\): speed of plastic deformation in compressions,

\({\dot{\varepsilon }}_{T}^{p}\): speed of plastic deformation under traction.

\({\varepsilon }_{\mathrm{th}}\): thermal deformation: \({\varepsilon }_{\mathrm{th}}=\alpha (T-{T}_{\mathrm{ref}})\). \(\alpha\) is defined under ELAS.

Note that it is not possible to simultaneously have plasticization in tension and in compression: either \(\dot{{p}_{C}}=0\), or \(\dot{{p}_{T}}=0\), or both are zero.