Nodally integrated thermomechanical RKPM: Part II—generalized thermoelasticity and hyperbolic finite-strain thermoplasticity

Abstract

In this two-part paper, a stable and efficient nodally-integrated reproducing kernel particle method (RKPM) approach for solving the governing equations of generalized thermomechanical theories is developed. Part I investigated quadrature in the weak form using classical thermoelasticity as a model problem, and a stabilized and corrected nodal integration was proposed. In this sequel, these methods are developed for generalized thermoelasticity and generalized finite-strain plasticity theories of the hyperbolic type, which are more amenable to explicit time integration than the classical theories. Generalized thermomechanical models yield finite propagation of temperature, with a so-called second sound speed. Since this speed is not well characterized for common engineering materials and environments, equating the elastic wave speed with the second sound speed is investigated to obtain results close to classical thermoelasticity, which also yields a uniform critical time step. Implementation of the proposed nodally integrated RKPM for explicit analysis of finite-strain thermoplasticity is also described in detail. Several benchmark problems are solved to demonstrate the effectiveness of the proposed approach for thermomechanical analysis.

Appendices

Appendix A: Small-strain thermoelasticity

With $\eta ^p=0$, the Helmholtz free energy function (6) for thermoelasticity can be written as

$$\begin{aligned} \begin{aligned} \phi =e-\theta \eta \end{aligned} \end{aligned}$$

(97)

For small-strain thermoelasticity, the free energy function is a function of strain and the temperature $\phi (\varvec{\varepsilon },\theta )$ with

$$\begin{aligned} \begin{aligned} \dot{\phi }(\varvec{\varepsilon },\theta )=\frac{\partial \phi }{\partial \varvec{\varepsilon }}\dot{\varvec{\varepsilon }}+\frac{\partial \phi }{\partial \theta }\dot{\theta }. \end{aligned} \end{aligned}$$

(98)

With the assumption (97) and (98), the dissipation inequality (2a) becomes

$$\begin{aligned} \begin{aligned} \Omega _{\text {thermech}}=(\frac{1}{\rho }\varvec{\sigma }-\frac{\partial \phi }{\partial \varvec{\varepsilon }}):\dot{\varvec{\varepsilon }}+(\eta +\frac{\partial \phi }{\partial \theta })\dot{\theta }\ge 0. \end{aligned} \end{aligned}$$

(99)

Which yields the constitutive relations

$$\begin{aligned} \begin{aligned} \eta =-\frac{\partial \phi }{\partial \theta } \end{aligned} \end{aligned}$$

(100)

and

$$\begin{aligned} \begin{aligned} \varvec{\sigma }=\rho \frac{\partial \phi }{\partial \varvec{\varepsilon }}. \end{aligned} \end{aligned}$$

(101)

Assuming $\varvec{\varepsilon }=\varvec{0}$ and $\theta =\theta _0$ in the reference state, (101) becomes

$$\begin{aligned} \begin{aligned}&\varvec{\sigma }=\varvec{D}:\varvec{\varepsilon }-\varvec{\beta }\theta ,\quad \text {or} \quad \varvec{\sigma }=\varvec{D}:(\varvec{\varepsilon }-\varvec{\alpha }\theta ),\\&\quad \,\text {or}\,\varvec{\sigma }=2\mu \text {dev}(\varvec{\varepsilon })+K \text {tr}(\varvec{\varepsilon })-3\alpha (\theta -\theta _0)\varvec{I}\end{aligned} \end{aligned}$$

(102)

where $\varvec{\alpha }$ contains the thermal expansion coefficients, K is bulk modulus and

$$\begin{aligned} \varvec{D}&=\rho \frac{\partial ^2\phi }{\partial \varvec{\varepsilon }^2} , \end{aligned}$$

(103a)

$$\begin{aligned} \varvec{\beta }&=\varvec{D}:\varvec{\alpha }=-\rho \frac{\partial ^2\phi }{\partial \varvec{\varepsilon }\partial \theta } . \end{aligned}$$

(103b)

Also, the entropy for small strain can be expressed as

$$\begin{aligned} \begin{aligned} \eta (\varvec{\varepsilon },\theta )=-\frac{\partial \phi }{\partial \theta }=\left. \begin{array}{l}-\frac{\partial \phi }{\partial \theta }\end{array}\right| _0-\left. \begin{array}{l}\frac{\partial ^2\phi }{\partial {\varvec{\varepsilon }}\partial \theta }\end{array}\right| _0:{\varvec{\varepsilon }}-\left. \begin{array}{l}\frac{\partial ^2\phi }{\partial \theta _0^2}\end{array}\right| _0(\theta -\theta _0) \end{aligned} \end{aligned}$$

(104)

where $|_0$ means that the quantity is evaluated at the reference state. With (100), (102), (103), and $({\theta -\theta _0})/{\theta }\approx ({\theta -\theta _0})/{\theta _0}$, and The specific heat capacity is $c_p=\theta \frac{\partial \eta }{\partial \theta }\approx \theta _0\frac{\partial \eta }{\partial \theta }$ the entropy (104) can be written as

$$\begin{aligned} \eta (\varvec{\varepsilon },\theta )&=\eta _0+\frac{3K\alpha }{\rho }\text {tr}(\varvec{\varepsilon })+\frac{\theta -\theta _0}{\theta _0}c_p, \end{aligned}$$

(105a)

$$\begin{aligned} \dot{\eta }(\varvec{\varepsilon },\theta )&=\frac{3K\alpha }{\rho }\text {tr}(\dot{\varvec{\varepsilon }})+\frac{\dot{\theta }}{\theta _0}c_p. \end{aligned}$$

(105b)

Substitution of (105b) and $\beta =3K\alpha$ into (7) with Fourier’s law (14) yields

$$\begin{aligned} \begin{aligned} \rho c_p\frac{\theta }{\theta _0} \dot{\theta }+\theta \beta \text {tr}(\dot{\varvec{\varepsilon }})=k\nabla \cdot \nabla \theta +Q. \end{aligned} \end{aligned}$$

(106)

With the assumption $\theta \approx \theta _0$, the energy equation becomes

$$\begin{aligned} \begin{aligned} \rho c_p \dot{\theta }+\theta _0\beta \text {tr}(\dot{\varvec{\varepsilon }})=k\nabla \cdot \nabla \theta +Q \end{aligned} \end{aligned}$$

(107)

which is the same as (12).

Appendix B: Finite-strain thermoplasticity

First, taking time derivative of the Helmholtz free energy function one obtains

$$\begin{aligned} \begin{aligned} \dot{\phi }=\dot{e}-\dot{\theta }\eta -\theta \dot{\eta }+\dot{\theta }\eta ^p+\theta \dot{\eta }^p. \end{aligned} \end{aligned}$$

(108)

With the assumption (108) and (40), the dissipation inequality (2a) becomes

$$\begin{aligned} \begin{aligned}&\Omega _{\text {thermech}}=\rho (\varvec{P}-\frac{\partial \phi ^e}{\partial \varvec{F}^e}\frac{\varvec{F}^e}{\varvec{F}}):\dot{\varvec{F}}+(-(\eta -\eta ^p)+\rho \frac{\partial \phi }{\partial \theta })\dot{\theta }\\&\quad +\rho \frac{\partial \phi ^e}{\partial \varvec{F}^e}\frac{\varvec{F}^e}{\varvec{F}^p}:\dot{\varvec{F}^p}+\rho \frac{\partial \phi ^p}{\partial \upsilon }\dot{\upsilon }+\theta \dot{\eta }^p\ge 0. \end{aligned} \end{aligned}$$

(109)

where $\varvec{F}=\varvec{F}^e\varvec{F}^p$; $\varvec{F}^e$ is the deformation gradient; $\varvec{F}^e$ and $\varvec{F}^p$ are the elastic and plastic part of the deformation gradient, respectively. Since $\varvec{F}$, $\dot{\varvec{F}}$, $\theta$, and $\dot{\theta }$ are arbitrary values, we obtain the constitutive relations of

$$\begin{aligned} \begin{aligned} \varvec{P}=\frac{\partial \phi }{\partial \varvec{F}^e}\frac{\partial \varvec{F}^e}{\partial \varvec{F}}, \quad \eta -\eta ^p=\rho \frac{\partial \phi }{\partial \theta } \end{aligned} \end{aligned}$$

(110)

where $\varvec{P}$ is the first Piola-Kirchhoff stress.

Here we define (109) into two parts

$$\begin{aligned} \text {D}_{\text {mech}}=&\rho \frac{\partial \phi ^e}{\partial \varvec{F}^e}\frac{\varvec{F}^e}{\varvec{F}^p}:\dot{\varvec{F}^p}+\rho \frac{\partial \phi ^p}{\partial \upsilon }\dot{\upsilon }=\varvec{\Sigma }: \varvec{D}^p+\rho \frac{\partial \phi ^p}{\partial \upsilon }\dot{\upsilon }, \end{aligned}$$

(111a)

$$\begin{aligned} \text {D}_{\text {ther}}=&\theta \dot{\eta }^p \end{aligned}$$

(111b)

where $\varvec{\Sigma }=2\varvec{C}^e\frac{\partial \phi ^e}{\partial \varvec{C}^e}$ is the Mandel stress tensor; $\varvec{C}=\varvec{F}^T \varvec{F}$ denotes the right Cauchy-Green tensor and $\varvec{C}^e$ is the elastic part, and $\varvec{D}^p=\text {sym}(\dot{\varvec{F}}^p \varvec{F}^{p-1})$ is the symetric part of the plastic velocity gradient. Substitution (108), (42) , and (111) into (7) with the Fourier’s law (4) yields

$$\begin{aligned} \begin{aligned} \rho c_p \dot{\theta }=k\nabla \cdot \nabla \theta +Q+\text {D}_{\text {mech}}+\theta \frac{\partial (\varvec{P}:{\dot{\varvec{F}}}-\text {D}_{\text {mech}})}{\partial \theta } \end{aligned} \end{aligned}$$

(112)

where the last term is the elasto-plastic heating.

The simplification can be written with a dissipation factor $\chi$ and $\text {D}_{\text {mech}}-\theta \frac{\partial \text {D}_{\text {mech}}}{\partial \theta }$ replaced with the total plastic power $\dot{w}^p$

$$\begin{aligned} \begin{aligned} \rho c_p \dot{\theta }=k\nabla \cdot \nabla \theta +Q+\theta \frac{\partial (\varvec{P}:{\dot{\varvec{F}}})}{\partial \theta }+\chi \dot{w}^p. \end{aligned} \end{aligned}$$

(113)

In applications of metal thermoplasticity, the plastic dissipation is much greater than thermoelastic heating. Therefore, we rewrite the energy equation neglecting the thermoelastic heating

$$\begin{aligned} \begin{aligned} \rho c_p \dot{\theta }=k\nabla \cdot \nabla \theta +Q+\chi \dot{w}^p \end{aligned} \end{aligned}$$

which is (43).