Finite element formulation with embedded weak discontinuities for strain localization under dynamic conditions

Abstract

We present an explicit finite element formulation designed for the treatment of strain localization under highly dynamic conditions. A material stability analysis is used to detect the onset of localization behavior. Finite elements with embedded weak discontinuities are employed with the aim of representing subsequent localized deformation accurately. The formulation and its algorithmic implementation are described in detail. Numerical results are presented to illustrate the usefulness of this computational framework in the treatment of strain localization under highly dynamic conditions, and to examine its performance characteristics in the context of two-dimensional plane-strain problems.

Appendices

Appendix A: The $\bar{\varvec{\mathsf {B}}}$ operator

Denoting by $\varvec{\mathsf {B}}$ the array whose elements are the spatial gradients of the shape functions,

$$\begin{aligned} \mathsf {B}_{ijA}=\frac{\partial \mathsf {N}_{iA}}{\partial x_{j}}, \end{aligned}$$

(74)

we split this operator into symmetric and skew-symmetric components, which can be used to compute the rate of deformation tensor and the spin tensor, respectively, from the nodal velocities:

$$\begin{aligned} \varvec{\mathsf {B}}=\varvec{\mathsf {B}}^{\text {sym}}+\varvec{\mathsf {B}}^{\text {skw}}. \end{aligned}$$

(75)

Then, the symmetric part is split into volumetric and deviatoric parts,

$$\begin{aligned} \varvec{\mathsf {B}}^{\text {sym}}=\varvec{\mathsf {B}}^{\text {vol}}+\varvec{\mathsf {B}}^{\text {dev}}, \end{aligned}$$

(76)

and the deviatoric part is split further into shear and normal (diagonal) parts:

$$\begin{aligned} \varvec{\mathsf {B}}^{\text {dev}}=\varvec{\mathsf {B}}^{\text {shr}}+\varvec{\mathsf {B}}^{\text {nrm}}. \end{aligned}$$

(77)

Combining Eqs. (75)–(77), we obtain

$$\begin{aligned} \varvec{\mathsf {B}}=\varvec{\mathsf {B}}^{\text {vol}}+\varvec{\mathsf {B}}^{\text {shr}}+\varvec{\mathsf {B}}^{\text {nrm}}+\varvec{\mathsf {B}}^{\text {skw}}. \end{aligned}$$

(78)

The selective reduced integration technique of Ref. [64] is then used to alleviate volumetric locking due to (nearly) incompressible material response. Selective reduced integration is also used to mitigate shear locking in bending dominated problems (e.g. see Ref. [65]). This is achieved by defining the $\bar{\varvec{\mathsf {B}}}$ operator,

$$\begin{aligned} \bar{\varvec{\mathsf {B}}}=\varvec{\mathsf {B}}_{0}^{\text {vol}}+\varvec{\mathsf {B}}_{0}^{\text {shr}}+\varvec{\mathsf {B}}^{\text {nrm}}+\varvec{\mathsf {B}}^{\text {skw}}, \end{aligned}$$

(79)

where, $\varvec{\mathsf {B}}_{0}^{\text {vol}}$ and $\varvec{\mathsf {B}}_{0}^{\text {shr}}$ are evaluated at the centroid of the four-noded quadrilateral elements used here, and treated as constant over the element. This is equivalent to the use of single-point quadrature for integration of volumetric and shear effects. Finally, to suppress hourglass modes that may pollute the solution as a result of using reduced quadrature rules, following Ref. [1], we redefine the $\bar{\varvec{\mathsf {B}}}$ operator as follows:

$$\begin{aligned} \bar{\varvec{\mathsf {B}}}= & {} \left( 1-\epsilon _{\text {stb}}\right) \left[ \varvec{\mathsf {B}}_{0}^{\text {vol}}+\varvec{\mathsf {B}}_{0}^{\text {shr}}\right] +\epsilon _{\text {stb}}\left[ \varvec{\mathsf {B}}^{\text {vol}}+\varvec{\mathsf {B}}^{\text {shr}}\right] \nonumber \\&+\,\varvec{\mathsf {B}}^{\text {nrm}}+\varvec{\mathsf {B}}^{\text {skw}}, \end{aligned}$$

(80)

where $\epsilon _{\text {stb}}$ is a small stabilization parameter. The value $\epsilon _{\text {stb}}=0.05$ has been used throughout the present work. Note that setting $\epsilon _{\text {stb}}=1$ recovers the conventional displacement-based formulation (78), whereas setting $\epsilon _{\text {stb}}=0$ recovers the selective reduced integration technique (79).

Appendix B: Iterative solution of the traction continuity equation

The traction continuity condition (32) can be written

(81)

Our goal is to obtain the values of $\alpha ^{\text {M}}$ and $\varvec{m}$ at $t= {{}t}_{(n+1)}$ by solving this nonlinear equation using a Newton scheme. We assume that the $k\text {-th}$ iterates denoted by ${{}(\cdot )}_{(n+1)}^{(k)}$ are known. Using the truncated Taylor expansion ${{}\varvec{\sigma }}_{(n+1)}^{(k+1)} \approx {{}\varvec{\sigma }}_{(n+1)}^{(k)} + \delta \varvec{\sigma }$ in Eq. (81) yields

(82)

Then, the stress is linearized as follows:

$$\begin{aligned} \delta \sigma _{ij}= & {} \left[ \frac{\partial {{}\sigma _{ij}}_{(n+1)}^{(k)} }{\partial {{}\mathrm {\Delta }\mathrm {d}_{kl}}_{(n+{\frac{1}{2}})}^{(k)}}\, \frac{\partial {{}\mathrm {\Delta }\mathrm {d}_{kl}}_{(n+{\frac{1}{2}})}^{(k)}}{\partial {{}\mathrm {\Delta }\mathrm {l}_{pq}}_{(n+{\frac{1}{2}})}^{(k)}}\right. \nonumber \\&\left. +\, \frac{\partial {{}\sigma _{ij}}_{(n+1)}^{(k)} }{\partial {{}\mathrm {\Delta }\mathrm {w}_{kl}}_{(n+{\frac{1}{2}})}^{(k)}}\, \frac{\partial {{}\mathrm {\Delta }\mathrm {w}_{kl}}_{(n+{\frac{1}{2}})}^{(k)}}{\partial {{}\mathrm {\Delta }\mathrm {l}_{pq}}_{(n+{\frac{1}{2}})}^{(k)}}\right] \delta \mathrm {\Delta }\mathrm {l}_{pq}. \end{aligned}$$

(83)

This can be written in the more compact form,

$$\begin{aligned} \delta \sigma _{ij}= {{}\left[ \mathbb {C}_{ijkl}\,\mathbb {P}^{\text {Sym.}}_{klpq} +\mathbb {G}_{ijkl}\,\mathbb {P}^{\text {Skw.}}_{klpq} \right] }_{(n+1)}^{(k)} \delta \mathrm {\Delta }\mathrm {l}_{pq}, \end{aligned}$$

(84)

where the tangent stiffness moduli,

$$\begin{aligned} {{}\mathbb {C}_{ijkl}}_{(n+1)}^{(k)} \,{:=}\,\frac{ \partial {{}\sigma _{ij}}_{(n+1)}^{(k)} }{ \partial {{}\mathrm {\Delta }\mathrm {d}_{kl}}_{(n+{\frac{1}{2}})}^{(k)} }, \end{aligned}$$

(85)

are obtained through the stress update procedure (Algorithm 1), and

$$\begin{aligned} {{}\mathbb {G}_{ijkl}}_{(n+1)}^{(k)} \,{:=}\,\frac{ \partial {{}\sigma _{ij}}_{(n+1)}^{(k)} }{ \partial {{}\mathrm {\Delta }\mathrm {w}_{kl}}_{(n+{\frac{1}{2}})}^{(k)} } \end{aligned}$$

(86)

are obtained via application of the chain rule of differential calculus to Eq. (55). This process is straightforward but somewhat cumbersome, since it requires linearization of the exponential map (59). In addition,

$$\begin{aligned} \mathbb {P}^{\text {Sym.}}_{klpq}&\,{:=}\,\frac{1}{2}(\delta _{kp}\delta _{lq} + \delta _{kq}\delta _{lp} ), \end{aligned}$$

(87)

$$\begin{aligned} \mathbb {P}^{\text {Skw.}}_{klpq}&\,{:=}\,\frac{1}{2}(\delta _{kp}\delta _{lq} - \delta _{kq}\delta _{lp} ), \end{aligned}$$

(88)

are symmetric and skew symmetric projectors, respectively. Note that due to the symmetry of $\mathbb {C}_{ijkl}$ and the skew symmetry of $\mathbb {G}_{ijkl}$ with respect to the indices k and l, we have

$$\begin{aligned} \mathbb {C}_{ijkl}\,\mathbb {P}^{\text {Sym.}}_{klpq}&= \mathbb {C}_{ijpq}, \end{aligned}$$

(89)

$$\begin{aligned} \mathbb {G}_{ijkl}\,\mathbb {P}^{\text {Skw.}}_{klpq}&= \mathbb {G}_{ijpq}. \end{aligned}$$

(90)

Hence, Eq. (84) is written

$$\begin{aligned} \delta \sigma _{ij}= {{}\left[ \mathbb {C}_{ijkl} + \mathbb {G}_{ijkl}\right] }_{(n+1)}^{(k)} \delta \mathrm {\Delta }\mathrm {l}_{kl}, \end{aligned}$$

(91)

and the traction continuity condition (82) becomes

$$\begin{aligned}&{{}n_{i}}_{(n+1)} \left( {{}\left[ \mathbb {C}_{ijkl}^{\text {B}} + \mathbb {G}_{ijkl}^{\text {B}}\right] }_{(n+1)}^{(k)} \delta \mathrm {\Delta }\mathrm {l}_{kl}^{\text {B}}\right. \nonumber \\&\quad \left. -{{}\left[ \mathbb {C}_{ijkl}^{\text {M}} + \mathbb {G}_{ijkl}^{\text {M}}\right] }_{(n+1)}^{(k)} \delta \mathrm {\Delta }\mathrm {l}_{kl}^{\text {M}} \right) \nonumber \\&\quad = -{{}n_{i}}_{(n+1)} {{} \left[ \sigma _{ij}^{\text {B}}-\sigma _{ij}^{\text {M}}\right] }_{(n+1)}^{(k)} . \end{aligned}$$

(92)

Using Eq. (27) in the matrix region, we obtain

$$\begin{aligned} {{}\mathrm {\Delta }\mathrm {l}_{kl}^{\text {M}}}_{(n+{\frac{1}{2}})}^{(k)} = \left( \delta _{kr}\delta _{ls} - {{}\left[ \alpha ^{\text {M}}T_{kl} T_{rs}\right] }_{(n+1)}^{(k)} \right) \frac{\partial {{}\mathrm {\Delta }u_r}_{(n+1)}}{\partial {{}x_{s}}_{(n+{\frac{1}{2}})}}, \end{aligned}$$

(93)

where ${{}\mathrm {\Delta }u_r}_{(n+1)}={{}u_r}_{(n+1)}-{{}u_r}_{(n)}$ is the incremental displacement, and

$$\begin{aligned} {{}T_{kl}}_{(n+1)}^{(k)} = {{}m_{k}}_{(n+1)}^{(k)} {{}n_{l}}_{(n+1)}. \end{aligned}$$

(94)

Introducing the vector $\hat{\varvec{m}}$, defined by

$$\begin{aligned} {{}\hat{m}_{k}}_{(n+1)}^{(k)} \,{:=}\,{{}\hat{\alpha }}_{(n+1)}^{(k)} {{}m_{k}}_{(n+1)}^{(k)}, \end{aligned}$$

(95)

in terms of its magnitude,

$$\begin{aligned} {{}\hat{\alpha }}_{(n+1)}^{(k)} \,{:=}\,{{}\alpha ^{\text {M}}}_{(n+1)}^{(k)} {{}T_{rs}}_{(n+1)}^{(k)} \frac{\partial {{}\mathrm {\Delta }u_r}_{(n+1)}}{\partial {{}x_{s}}_{(n+{\frac{1}{2}})}}, \end{aligned}$$

(96)

we write Eq. (93) in the form,

$$\begin{aligned} {{}\mathrm {\Delta }\mathrm {l}_{kl}^{\text {M}}}_{(n+{\frac{1}{2}})}^{(k)} = \frac{\partial {{}\mathrm {\Delta }u_k}_{(n+1)}}{\partial {{}x_{l}}_{(n+{\frac{1}{2}})}} - {{}\hat{m}_{k}}_{(n+1)}^{(k)}\,{{}n_{l}}_{(n+1)}, \end{aligned}$$

(97)

which is linearized as follows:

$$\begin{aligned} \delta \mathrm {\Delta }\mathrm {l}_{kl}^{\text {M}} = - \delta \hat{m}_{k}\,{{}n_{l}}_{(n+1)}. \end{aligned}$$

(98)

Similarly, using the relationship (29) between $\alpha ^{\text {M}}$ and $\alpha ^{\text {B}}$, we obtain for the localized region:

$$\begin{aligned} \delta \mathrm {\Delta }\mathrm {l}_{kl}^{\text {B}} = \left( \frac{h}{\omega }-1\right) \delta \hat{m}_{k}\,{{}n_{l}}_{(n+1)}. \end{aligned}$$

(99)

Substituting Eqs. (98)–(99) into the traction continuity condition (92), we arrive at the following linearized equation,

$$\begin{aligned} {{}A_{jk}}_{(n+1)}^{(k)} \delta \hat{m}_{k}= {{}g_{j}}_{(n+1)}^{(k)}, \end{aligned}$$

(100)

where

$$\begin{aligned} {{}A_{jk}}_{(n+1)}^{(k)}= & {} {{}n_{i}}_{(n+1)} \bigg [ \left( \frac{h}{\omega }-1\right) \,{{}\left[ \mathbb {C}_{ijkl}^{\text {B}} + \mathbb {G}_{ijkl}^{\text {B}}\right] }_{(n+1)}^{(k)}\nonumber \\&+\,{{}\left[ \mathbb {C}_{ijkl}^{\text {M}} + \mathbb {G}_{ijkl}^{\text {M}}\right] }_{(n+1)}^{(k)} \bigg ]{{}n_{l}}_{(n+1)}, \end{aligned}$$

(101)

and

$$\begin{aligned} {{}g_{j}}_{(n+1)}^{(k)} = -{{}n_{i}}_{(n+1)}{{}\left[ \sigma _{ij}^{\text {B}}-\sigma _{ij}^{\text {M}}\right] }^{(k)}. \end{aligned}$$

(102)

In each iteration of this scheme, $\delta \hat{\varvec{m}}$ is computed using Eq. (100), and the update formula ${{}\hat{\varvec{m}}}_{(n+1)}^{(k+1)}={{}\hat{\varvec{m}}}_{(n+1)}^{(k)}+\delta \hat{\varvec{m}}$ is applied. Then, the variables ${{}\varvec{m}}_{(n+1)}^{(k+1)}$ and ${{}\alpha ^{\text {M}}}_{(n+1)}^{(k+1)}$ are obtained from ${{}\hat{\varvec{m}}}_{(n+1)}^{(k+1)}$ via Eqs. (95)–(96).