Fully Coupled Peridynamic Thermomechanics

Abstract

This chapter concerns the derivation of the coupled peridynamic (PD) thermomechanics equations based on thermodynamic considerations. The generalized peridynamic model for fully coupled thermomechanics is derived using the conservation of energy and the free-energy function. Subsequently, the bond-based coupled PD thermomechanics equations are obtained by reducing the generalized formulation. These equations are also cast into their nondimensional forms. After describing the numerical solution scheme, solutions to certain coupled thermomechanical problems with known previous solutions are presented.

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This chapter concerns the derivation of the coupled peridynamic (PD) thermomechanics equations based on thermodynamic considerations. The generalized peridynamic model for fully coupled thermomechanics is derived using the conservation of energy and the free-energy function. Subsequently, the bond-based coupled PD thermomechanics equations are obtained by reducing the generalized formulation. These equations are also cast into their nondimensional forms. After describing the numerical solution scheme, solutions to certain coupled thermomechanical problems with known previous solutions are presented.

Thermomechanics concerns the influence of the thermal state of a solid body on the deformation and the influence of the deformation on the thermal state. In many cases, the effect of the deformation field on the thermal state may be ignored. This leads to a decoupled or uncoupled thermomechanical analysis, for which only the effect of the temperature field on the deformation is present. However, the uncoupled thermomechanics may not be satisfactory for certain transient problems. Experimental verification of the influence of the deformation on the thermal state exists. It was shown that an adiabatic solid experiences a temperature drop when it is strained in tension (Chadwick 1960; Fung 1965). Also, elastic bodies under tensile loading experience cooling below the yield stress; however, beyond the yield stress the bodies heat up due to the irreversible nature of plasticity (Nowinski 1978).

Also, the temperature field induced by structural loading may not be uniform. For example, when a beam with an initially uniform temperature is under bending, part of the beam is in tension while the other part is in compression. Due to the thermomechanical coupling, the part of the beam that is in tension cools and the region that is in compression heats up, establishing a thermal gradient. This leads to the onset of heat diffusion. The heat flow is irreversible; thus, some of the mechanical energy supplied to bend the beam is dissipated through its conversion to heat energy. This phenomenon is called thermoelastic damping and it plays a critical role in vibrations and wave propagation.

It is well known that during fracture in metals a plastic region, in which the material has locally yielded, occurs ahead of the crack tip. As a result, the mechanical energy is dissipated as heat and the temperature rises in the local region ahead of the crack tip. A slightly different phenomenon is observed for fracture in polymers. During fracture in polymers, it was experimentally observed that thermoelastic cooling is followed by a temperature rise due to the plastic zone and/or fracture process itself, which exposes new surfaces (Rittel 1998). Consequently, in order to accurately model fracture, especially the crack tip, thermal consideration needs to be taken into account and a coupled thermomechanical analysis becomes necessary. The thermal and structural interaction becomes especially important for high-speed impact and penetration fracture problems (Brünig et al. 2011).

The derivation of the classical thermomechanics equation from a thermodynamic perspective did not occur till the mid 1950s (Biot 1956). Biot used generalized irreversible thermodynamics to formulate the classical thermomechanical laws in variational form, with the corresponding Euler equations representing the coupled momentum and energy equations.

The fully coupled thermomechanical equations based on the classical theory are well established. The classical equations of thermoelasticity are comprised of the deformation equation of motion with a thermoelastic constitutive law and the heat transfer equation with a structural (or deformational) heating and cooling term contributing to the thermal energy. For isotropic materials, the thermoelastic constitutive law includes the thermal stresses, which are related to the temperature gradient, while the structural heating and cooling are dependent on the thermal modulus and rate of dilatation. Depending on the structural idealization, the thermal modulus is defined as

$$ {\beta_{cl }}=(3\lambda +2\mu )\alpha =\frac{{E\alpha }}{{1-2\nu }}\ \mathrm{ for}\ \mathrm{ three}\ \mathrm{ dimensions}, $$

(13.1a)

$$ {\beta_{cl }}=(2\lambda +2\mu )\alpha =\frac{{E\alpha }}{{\left( {1-\nu } \right)}}\ \mathrm{ for}\ \mathrm{ two}\ \mathrm{ dimensions}, $$

(13.1b)

$$ {\beta_{cl }}=(2\mu )\alpha =E\alpha\ \mathrm{ for}\ \mathrm{ one}\ \mathrm{ dimension}, $$

(13.1c)

in which $E$ is the elastic modulus, $\alpha$ is the coefficient of thermal expansion, and $\nu$ is the Poisson’s ratio. The parameters $\lambda$ and $\mu$ are Lamé’s constants.

Typically, the strength of coupling is measured via the nondimensional quantity known as the coupling coefficient and defined as

$$ \mathtt{\epsilon}=\frac{{{\beta_{cl}}^2{\Theta_0}}}{{\rho\;{c_v}(\lambda +2\mu )}}, $$

(13.2)

for which $\rho$ is the mass density, ${c_v}$ is the specific heat capacity, and ${\Theta_0}$ is the reference temperature at which the stress in the body is zero (Nowinski 1978). The coupling coefficients of metals are significantly lower than those of plastics. Steel, for example, has a coupling coefficient of about 0.011 while certain plastics have a value of $\mathtt{\epsilon} = 0.43$.

13.1 Local Theory

Various researchers analytically examined plane waves in thermoelastic solids (Chadwick and Sneddon 1958; Deresiewicz 1957). In a one-dimensional formulation, they showed that the presence of thermal and elastic waves are dispersed and attenuated. They also studied the effect of frequency on the phase velocity, attenuation, and damping. Later, Chadwick (1962) extended the analysis to two dimensions and investigated the propagation of thermoelastic waves in thin plates. Paria (1958) determined the temperature and stress distribution of a two-dimensional half-space problem using Laplace and Hankel transforms. Laplace transforms have also been used by Boley and Hetnarski (1968) to characterize propagating discontinuities in various one-dimensional coupled thermoelastic problems. Fourier transforms were employed by Boley and Tolins (1962) to determine the mechanical and thermal response of a one-dimensional semi-infinite bar with transient boundary conditions. The major challenge with transform methods is in finding the analytical inverse transforms—in many cases this is not possible and numerical inversion is necessary. Other analytical solution methods have been adopted to solve coupled thermoelastic problems. Soler and Brull (1965) used perturbation techniques and more recently Lychev et al. (2010) determined a closed-form solution by an expansion of the eigenfunctions generated by the coupled equations of motion and heat conduction.

Numerical approximations to the classical thermoelastic equations have been very commonly found using the finite element (FE) method. A transient thermoelastic FE model was developed by Nickell and Sackman (1968) and Ting and Chen (1982). The approximations from their FE model were compared against analytical solutions for various one-dimensional semi-infinite problems. Oden (1969) and Givoli and Rand (1995) developed dynamic thermoelastic FE models. Additionally, Chen and Weng (1988, 1989a, b) modeled various thermoelastic problems such as the transient response of an axisymmetric infinite cylinder and an infinitely long plate using a finite element formulation in the Laplace transform domain. Hosseini-Tehrani and Eslami (2000) presented solutions for thermal and mechanical shocks in a finite domain based on the boundary element method (BEM) in conjunction with the Laplace-transform method in a time domain. They provided results for small time durations (early stages of the shock loads) using the numerical inversion of the Laplace-transform method.

Numerical solution schemes for thermomechanical problems are divided into two categories—monolithic schemes and staggered schemes. In monolithic schemes, the differential equations for different variables are solved simultaneously. On the other hand, for staggered or partitioned schemes, the solutions of the different variables are determined separately. In general, the staggered schemes have been favored over monolithic schemes, as the monolithic systems can be very large, making it unfeasible to solve practical problems. In addition, the mechanical and thermal parts of a thermomechanical problem may have very different time scales, hence requiring different time stepping schemes. However, the very nature of monolithic schemes renders this impossible.

One of the major issues associated with staggered numerical analysis of coupled thermomechanics is the concern of stability. When conditionally stable techniques are used to solve the coupled momentum and energy equations, a small time step size is required, which may be computationally impractical for certain problems. Even when various unconditionally stable methods are used to solve the equation of motion and heat transfer equation, the overall solution to the coupled problem may still be conditionally stable. A substantial amount of work has been done to combat this issue and to develop unconditionally stable staggered algorithms. Examples of such algorithms based on the finite element method include an adiabatic split scheme by Armero and Simo (1992) and various implicit-implicit and implicit-explicit schemes (Farhat et al. 1991; Liu and Zhang 1983; Liu and Chang 1985).

13.2 Nonlocal Theory

Research into nonlocal coupled thermomechanics is undoubtedly emerging. Classical nonlinear constitutive equations for nonlocal fully coupled thermoelasticity have been presented by Huang (1999). Ardito and Comi (2009) developed a fully nonlocal thermoelastic model that has an internal length scale. They analytically solved the nonlocal equations in order to determine the dissipation in microelectromechanical resonators. Comparison of their results with experimental observations revealed that the nonlocal model is able to capture the size effect that the standard local thermoelastic analysis is unable to capture. The work by Ardito and Comi (2009) illustrates the importance of nonlocality in small-scale problems. With the peridynamic thermomechanical model, not only are the problems that require nonlocality solvable, such as the microelectromechanical problems, but also the problems with discontinuities can be readily modeled. A crack that forms and propagates in a body with a varying temperature or temperature gradient is an example of such a problem. Therefore, the peridynamic approach to thermomechanics is advantageous as it not only accounts for nonlocality but also allows for coupled deformation and temperature fields to be determined in spite of cracks and other discontinuities. Uncoupled thermomechanics using the bond-based theory was developed within the realm of peridynamics by Kilic and Madenci (2010). However, no work has been published on fully coupled thermomechanics within the peridynamic framework.

13.3 Peridynamic Thermomechanics

Similar to the derivation of classical thermomechanical equations (Nowinski 1978), the generalized peridynamics for fully coupled thermomechanics is based on irreversible thermodynamics, i.e., the conservation of energy and the free-energy density function.

13.3.1 Peridynamic Thermal Diffusion with a Structural Coupling Term

The first law of thermodynamics based on peridynamic quantities, accounting for the conservation of mechanical and thermal energy, has been given by Silling and Lehoucq (2010) as

$$ {{\dot{\varepsilon}}_s}=\underset{\scriptscriptstyle-}{\mathbf{T}}\cdot \mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}+{{\bar{Q}}_b}+{s_b}, $$

(13.3)

where ${{\dot{\varepsilon}}_s}$ is the time rate of change of the internal energy storage density, and ${s_b}$ is the prescribed volumetric heat generation per unit mass. The term $\underset{\scriptscriptstyle-}{\mathbf{T}}\cdot \mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}$ represents the absorbed power density; it is the dot product of the force state and the time rate of deformation state. The absorbed power density in peridynamics is analogous to the stress power, ${\mathbf{\upsigma}} \cdot \dot{\mathbf{F}}$ in classical continuum mechanics, where ${\mathbf{\upsigma}}$ is the Piola stress tensor and $\mathbf{F}$ is the deformation gradient tensor. The variable $\bar{Q}$ is the rate of heat energy exchange with other material points, and it is given by

$$ \bar{Q}=\int\limits_H {\left( {{\underline{h}}\left( {\mathbf{x},t} \right)\left\langle {\mathbf{x}^{\prime}- \mathbf{x}} \right\rangle -{\underline{h}}\left( {{\mathbf{x}}^{\prime},t} \right)\left\langle {\mathbf{x}-\mathbf{x} \mathbf{^{\prime}}} \right\rangle } \right)d{V}^{\prime}}, $$

(13.4)

in which ${\underline{h}}$ is the heat flow scalar state. The quantity $\bar{Q}$ is related to ${{\bar{Q}}_b}$ as $\bar{Q}=\rho {{\bar{Q}}_b}$.

The free-energy density function is defined as (Silling and Lehoucq 2010)

$$ \psi ={\varepsilon_s}-\Theta \eta, $$

(13.5)

where $\Theta$ is the absolute temperature and $\eta$ is the entropy density. The time derivative of Eq. 13.5 becomes

$$ \dot{\psi}={{\dot{\varepsilon}}_s}-\dot{\Theta} \eta -\Theta \dot{\eta}. $$

(13.6)

Substituting for ${{\dot{\varepsilon}}_s}$ in Eq. 13.6 from the conservation of energy given in Eq. 13.3 leads to the following expression:

$$ \dot{\psi}=\underset{\scriptscriptstyle-}{\mathbf{T}}\cdot \mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}+{{\bar{Q}}_b}+{s_b}-\dot{\Theta} \eta -\Theta \dot{\eta}. $$

(13.7)

The functional dependency of the free-energy density and the entropy density can be defined in terms of the deformation state, time rate of change of the deformation state, and the temperature in the form

$$ \psi =\psi \left( {\underset{\scriptscriptstyle-}{\mathbf{Y}},\mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}},\Theta} \right), $$

(13.8a)

$$ \eta =\eta \left( {\underset{\scriptscriptstyle-}{\mathbf{Y}},\mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}},\Theta} \right). $$

(13.8b)

In conjunction with the chain rule, the time rate of change of the free-energy density can be expressed as

$$ \dot{\psi}={\psi_{,}}_{{\underset{\scriptscriptstyle-}{\mathbf{Y}}}}\cdot \mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}+{\psi_{{,\mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}}}}\cdot \mathbf{\ddot{{\underset{\scriptscriptstyle-}{Y}}}}+{\psi_{{,\Theta}}}\dot{\Theta}, $$

(13.9)

where the variable after the subscript comma indicates differentiation. If it is a state variable, its differentiation is known as the Fréchet derivative, as explained in the Appendix.

Substituting from Eq. 13.7 into Eq. 13.9 results in

$$ \left( {\Theta \dot{\eta}-{{\bar{Q}}_b}-{s_b}} \right)+\left( {{\psi_{{,\Theta}}}+\eta } \right)\dot{\Theta} +\left( {{\psi_{{,\underset{\scriptscriptstyle-}{\mathbf{Y}}}}}-\underset{\scriptscriptstyle-}{\mathbf{T}}} \right)\cdot \mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}+{\psi_{{,\mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}}}}\cdot \mathbf{\ddot{{\underset{\scriptscriptstyle-}{Y}}}}=0. $$

(13.10)

Adopting the assumption of Nowinski (1978) that $\dot{\underline{\mathbf{Y}}}$, $\ddot{\underline{\mathbf{Y}}}$, and $\dot{\Theta}$ vary independently, Eq. 13.10 leads to

$$ \Theta \dot{\eta}-{{\bar{Q}}_b}-{s_b}=0, $$

(13.11a)

$$ \eta =-{\psi_{{,\Theta}}}, $$

(13.11b)

$$ \underset{\scriptscriptstyle-}{\mathbf{T}}={\psi_{{,\underset{\scriptscriptstyle-}{\mathbf{Y}}}}}, $$

(13.11c)

$$ \psi_{\dot{\underline{\mathbf{Y}}}}=0. $$

(13.11d)

By using the free-energy density, the first law of thermodynamics, and the Clausius-Duhem inequality, Silling and Lehoucq (2010) also determined Eqs. 13.11b and 13.11d. In addition, they obtained expressions for the equilibrium, ${{\underset{\scriptscriptstyle-}{\mathbf{T}}}^e}$, and dissipative, ${{\underset{\scriptscriptstyle-}{\mathbf{T}}}^d}$, parts of force vector state as

$$ {{\underset{\scriptscriptstyle-}{\mathbf{T}}}^e}\left( {\underset{\scriptscriptstyle-}{\mathbf{Y}},\Theta} \right)={\psi_{{,\underset{\scriptscriptstyle-}{\mathbf{Y}}}}}\left( {\underset{\scriptscriptstyle-}{\mathbf{Y}},\Theta} \right), $$

(13.12a)

$$ {{\underset{\scriptscriptstyle-}{\mathbf{T}}}^d}\left( {\underset{\scriptscriptstyle-}{\mathbf{Y}},\mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}},\Theta} \right)\cdot \mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}\geq\;0. $$

(13.12b)

Using Eqs. 13.11b, 13.11d, and 13.8b in conjunction with the chain rule, the time derivative of the entropy density may be rewritten in the form

$$ \dot{\eta}=-{\psi_{{,\Theta}}}_{{\underset{\scriptscriptstyle-}{\mathbf{Y}}}}\cdot \mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}-{\psi_{{,\Theta \Theta}}}\dot{\Theta} . $$

(13.13)

Substituting from Eq. 13.13 into Eq. 13.11a and multiplying by $\rho$ leads to

$$ \rho \Theta {\psi_{{,\Theta}}}_{{\underset{\scriptscriptstyle-}{\mathbf{Y}}}}\cdot \mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}+\rho \Theta {\psi_{{,\Theta \Theta}}}\dot{\Theta} +\bar{Q}+\rho {s_b}=0. $$

(13.14)

Based on the classical theory (Nowinski 1978), the specific heat capacity, ${c_v}$, can be related to the classical free-energy density, $\bar{\psi}$, as

$$ \Theta {{\bar{\psi}}_{{,\Theta \Theta}}}=-{c_v}. $$

(13.15)

The assumption of the classical free-energy density at a point being equal to the peridynamic free-energy density, $\psi$ leads to

$$ \Theta {\psi_{{,\Theta \Theta}}}=-{c_v}. $$

(13.16)

Based on this observation, it is evident that the specific heat capacity has a similar meaning in the peridynamic theory as in the classical theory. Therefore, the term $\Theta {\psi_{{,\Theta \Theta}}}$ in Eq. 13.14 can be replaced by $-{c_v}$.

Based on the classical theory (Fung 1965), the thermal modulus ${\beta_{ij }}$ can be related to the classical free-energy density $\bar{\psi}$ through

$$ {\beta_{cl}}_{ij }=\rho \frac{{{\partial^2}\bar{\psi}}}{{\partial {e_{ij }}\partial \Theta}}, $$

(13.17)

where ${e_{ij }}$ is the strain tensor. Note that ${\beta_{cl}}_{ij }={\beta_{cl }}{\delta_{ij }}$ for isotropic materials.

Analogus to the thermal modulus of the classical theory thermal modulus state, a vector state, $\underset{\scriptscriptstyle-}{\mathbf{B}}$, can be defined as

$$ \underset{\scriptscriptstyle-}{\mathbf{B}}=\rho {\psi_{\Theta}}_{{\underset{\scriptscriptstyle-}{\mathbf{Y}}}}. $$

(13.18)

Substituting from Eqs. 13.4, 13.16, and 13.18 into Eq. 13.14 and after rearranging some of terms results in

$$ \begin{array}{ll} \rho {c_v}\dot{\Theta} \left( {\mathbf{x},t} \right)=\int\limits_H {\left( {{\underset{\scriptscriptstyle-}{h}}\left( {\mathbf{x},t} \right)\left\langle {\mathbf{x} \mathbf{^{\prime}-x}} \right\rangle -{\underset{\scriptscriptstyle-}{h}}\left( {{\mathbf{x}}^{\prime},t} \right)\left\langle {\mathbf{x}-\mathbf{x} \mathbf{^{\prime}}} \right\rangle } \right)d{V}^{\prime}} \hfill \\ \kern4em +\Theta \left( {\mathbf{x},t} \right)\underset{\scriptscriptstyle-}{\mathbf{B}}\left( {\mathbf{x},t} \right)\cdot \mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}\left( {\mathbf{x},t} \right)+\rho {s_b}\left( {\mathbf{x},t} \right)\ . \hfill \\ \end{array} $$

(13.19)

Applying the definition of the vector state dot product (see Appendix) renders the equation

$$ \begin{array}{ll} \rho {c_v}\dot{\Theta} \left( {\mathbf{x},t} \right)=\int\nolimits_H {\left( {\left( {{\underset{\scriptscriptstyle-}{h}}\left( {\mathbf{x},t} \right)\left\langle {\mathbf{x} \mathbf{^{\prime}-x}} \right\rangle -{\underset{\scriptscriptstyle-}{h}}\left( {{\mathbf{x}}^{\prime},t} \right)\left\langle {\mathbf{x}-\mathbf{x} \mathbf{^{\prime}}} \right\rangle } \right)} \right.} \hfill \\ \left. {\kern6em +\Theta \left( {\mathbf{x},t} \right)\underset{\scriptscriptstyle-}{\mathbf{B}}\left\langle {\mathbf{x} \mathbf{^{\prime}-x}} \right\rangle \cdot \mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}\left\langle {\mathbf{x} \mathbf{^{\prime}-x}} \right\rangle } \right)d{V}^{\prime}+\rho {s_b}\left( {\mathbf{x},t} \right), \hfill \\ \end{array} $$

(13.20)

in which the term $\underset{\scriptscriptstyle-}{\mathbf{B}}\cdot \mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}$ represents the effect of deformation on temperature. The final form of this equation can be obtained by defining $\underline{\dot{\mathbf{Y}}}$ and $\underset{\scriptscriptstyle-}{\mathbf{B}}$ in terms of the time rate of change of the extension scalar state, $\dot{{\underline{e}}}$, and the thermal modulus scalar state, ${\underline{\beta }}$, as

$$ \underset{\scriptscriptstyle-}{\mathbf{B}}\left\langle {\mathbf{x} \mathbf{^{\prime}-x}} \right\rangle ={\underset{\scriptscriptstyle-}{\beta }}\left\langle {\mathbf{x} \mathbf{^{\prime}-x}} \right\rangle \frac{{{\mathbf{y}}^{\prime}-\mathbf{y}}}{{\left| {{\mathbf{y}}^{\prime}-\mathbf{y}} \right|}}, $$

(13.21a)

$$ \mathbf{\dot{{\underset{\scriptscriptstyle-}{Y}}}}\left\langle {\mathbf{x} \mathbf{^{\prime}-x}} \right\rangle =\dot{{\underset{\scriptscriptstyle-}{e}}}\left\langle {\mathbf{x} \mathbf{^{\prime}-x}} \right\rangle \frac{{{\mathbf{y}}^{\prime}-\mathbf{y}}}{{\left| {{\mathbf{y}}^{\prime}-\mathbf{y}} \right|}}, $$

(13.21b)

in which the extension scalar state, ${\underline{e}}$, and thermal modulus scalar state, ${\underline{\beta }}$, are defined as

$$ {\underline{e}}={\underline{y}}-{\underline{x}}, $$

(13.21c)

$$ {\underline{\beta }}=\rho {\psi_{,}}_{{\Theta{\underline{e}}}}, $$

(13.21d)

with ${\underset{\scriptscriptstyle-}{y}}=\left| {\underset{\scriptscriptstyle-}{\mathbf{Y}}} \right|$ and ${\underset{\scriptscriptstyle-}{x}}=\left| {\underset{\scriptscriptstyle-}{\mathbf{X}}} \right|$. Thus, Eq. 13.20 can be recast as

$$ \begin{array}{lll} \rho {c_v}\dot{\Theta} \left( {\mathbf{x},t} \right) =\int\nolimits_H {\left( {\left( {{\underline{h}}\left( {\mathbf{x},t} \right)\left\langle {\mathbf{x} ^{\prime}-\mathbf{x}} \right\rangle -{\underline{h}}\left( {{\mathbf{x}}^{\prime},t} \right)\left\langle {\mathbf{x}-\mathbf{x} {^{\prime}}} \right\rangle } \right)} \right.} \hfill \\ \quad +\left. {\Theta \left( {\mathbf{x},t} \right){\underline{\beta }}\left\langle {\mathbf{x} ^{\prime}-\mathbf{x}} \right\rangle \dot{{\underline{e}}}\left\langle {\mathbf{x} ^{\prime}-\mathbf{x}} \right\rangle } \right)d{V}^{\prime}+\rho {s_b}\left( {\mathbf{x},t} \right). \end{array} $$

(13.22)

13.3.2 Peridynamic Deformation with a Thermal Coupling Term

Based on the classical linear theory of thermoelasticity (Nowinski 1978), the free-energy density is a potential function given by

$$ \bar{\psi}=\bar{\psi}\left( {{e_{ij }},T} \right)=\frac{1}{2}{c_{ijkl }}{e_{ij }}{e_{kl }}-{\beta_{cl}}_{ij }{e_{ij }}T-\frac{{{c_v}}}{{2{\Theta_0}}}{T^2}, $$

(13.23)

where ${c_{ijkl }}$ is the elastic moduli of the material, $T=\Theta -{\Theta_0}$, and ${\Theta_0}$ is the reference temperature. A similar approach is adopted herein for the derivation of the peridynamic deformation equation with a thermal coupling term.

Silling (2009) developed a linearized form of the state-based peridynamics for small elastic deformation by introducing the force vector state, $\underset{\scriptscriptstyle-}{\mathbf{T}}$, as

$$ \underset{\scriptscriptstyle-}{\mathbf{T}}=\underset{\scriptscriptstyle-}{\mathbf{T}}\left( {\underset{\scriptscriptstyle-}{\mathbf{U}}} \right), $$

(13.24)

where $\underset{\scriptscriptstyle-}{\mathbf{U}}$ is the displacement vector state. The free-energy density function is expressed in terms of $\underset{\scriptscriptstyle-}{\mathbf{U}}$ as

$$ \psi \left( {\underset{\scriptscriptstyle-}{\mathbf{U}},T} \right)=\psi \left( {{{{\underset{\scriptscriptstyle-}{\mathbf{Y}}}}^0}} \right)+{{\underset{\scriptscriptstyle-}{\mathbf{T}}}^0}\cdot \underset{\scriptscriptstyle-}{\mathbf{U}}+\frac{1}{2}\underset{\scriptscriptstyle-}{\mathbf{U}}\cdot \boldsymbol{{\underset{\scriptscriptstyle-}{\mathbb{K}}}}\cdot \underset{\scriptscriptstyle-}{\mathbf{U}}, $$

(13.25)

where ${{\underset{\scriptscriptstyle-}{\mathbf{Y}}}^0}$ and ${{\underset{\scriptscriptstyle-}{\mathbf{T}}}^0}$ are defined as the equilibrated deformation and force states, respectively. The double state $\boldsymbol{{\underset{\scriptscriptstyle-}{\mathbb{K}}}}$ is called the modulus state, and it is given by Silling (2009) as

$$ \boldsymbol{{\underset{\scriptscriptstyle-}{\mathbb{K}}}}={{\underset{\scriptscriptstyle-}{\mathbf{T}}}^0}_{{,\underset{\scriptscriptstyle-}{\mathbf{Y}}}}. $$

(13.26)

For linear thermoelastic material response, in accordance with Eq. 13.23, this form of the free energy is modified by including $T$ and $\underset{\scriptscriptstyle-}{\mathbf{U}}$ as

$$ \psi \left( {\underset{\scriptscriptstyle-}{\mathbf{U}},T} \right)=\psi \left( {{{{\underset{\scriptscriptstyle-}{\mathbf{Y}}}}^0}} \right)+{{\underset{\scriptscriptstyle-}{\mathbf{T}}}^0}\cdot \underset{\scriptscriptstyle-}{\mathbf{U}}+\frac{1}{2}\underset{\scriptscriptstyle-}{\mathbf{U}}\cdot \boldsymbol{{\underset{\scriptscriptstyle-}{\mathbb{K}}}}\cdot \underset{\scriptscriptstyle-}{\mathbf{U}}-\underset{\scriptscriptstyle-}{\mathbf{B}}\cdot \underset{\scriptscriptstyle-}{\mathbf{U}}T-\frac{{{c_v}}}{{2{\Theta_0}}}{T^2}. $$

(13.27)

Invoking this equation into Eq. 13.11c results in the explicit form of the force state as

$$ \underset{\scriptscriptstyle-}{\mathbf{T}}=\boldsymbol{{\underset{\scriptscriptstyle-}{\mathbb{K}}}}\cdot \underset{\scriptscriptstyle-}{\mathbf{U}}-\underset{\scriptscriptstyle-}{\mathbf{B}}T. $$

(13.28)

It represents the state-based constitutive relation for a linearized peridynamic thermoelastic material. Substituting from Eq. 13.28 into the peridynamic equation of motion, Eq. 2.22a results in the following:

$$ \begin{array}{lll} \rho \ddot{\mathbf{u}} =\int\limits_H {\left[ {\left( {\boldsymbol{{\underset{\scriptscriptstyle-}{\mathbb{K}}}}\cdot \underset{\scriptscriptstyle-}{\mathbf{U}}-\underset{\scriptscriptstyle-}{\mathbf{B}}T} \right)\left( {\mathbf{x},t} \right)\left\langle {{\mathbf{x}}^{\prime}-\mathbf{x}} \right\rangle -\left( {\boldsymbol{{\underset{\scriptscriptstyle-}{\mathbb{K}}}}\cdot \underset{\scriptscriptstyle-}{\mathbf{U}}-\underset{\scriptscriptstyle-}{\mathbf{B}}T} \right)\left( {{\mathbf{x}}^{\prime},t} \right)\left\langle {\mathbf{x}-{\mathbf{x}}^{\prime}} \right\rangle } \right]d{V}^{\prime}} \hfill \\ \kern3em +\mathbf{b}(\mathbf{x},t), \hfill \\ \end{array} $$

(13.29)

in which the term $\underset{\scriptscriptstyle-}{\mathbf{B}}\left\langle {{\mathbf{x}}^{\prime}-\mathbf{x}} \right\rangle T$ represents the effect of the thermal state on deformation. For a nonlinear elastic material model, the free energy is composed of a thermal and a mechanical component. Therefore, one possible form of the force state can be

$$ \underset{\scriptscriptstyle-}{\mathbf{T}}={\underset{\scriptscriptstyle-}{\nabla }}W-\underset{\scriptscriptstyle-}{\mathbf{B}}T $$

(13.30)

in which W is the deformational strain energy density and ${\underline{\nabla }}W$ is its Fréchet derivative. The part of the force state, ${{\underset{\scriptscriptstyle-}{\mathbf{T}}}_s}$, that includes only the structural deformation can be defined as

$$ {{\underset{\scriptscriptstyle-}{\mathbf{T}}}_s}={\underset{\scriptscriptstyle-}{\nabla }}W. $$

(13.31)

Substituting from these equations into the peridynamic equation of motion, Eq. 2.22a can be recast as

$$ \begin{array}{lll} \rho \ddot{\mathbf{u}} \left( {\mathbf{x},t} \right)=\int\nolimits_H {\left[ {\left( {{{{\underline{\mathbf{T}}}}_s}\left\langle {{\mathbf{x}}^{\prime}-\mathbf{x}} \right\rangle -\underline{\mathbf{B}}\;\left\langle {{\mathbf{x}}^{\prime}-\mathbf{x}} \right\rangle T} \right)} \right]}\,\hfill \\ \kern3.5em \left. {-\left( {{{{\mathbf{{{\underset{\scriptscriptstyle-}{T}}}^{\prime}}}}_s}\left\langle {\mathbf{x}-{\mathbf{x}}^{\prime}} \right\rangle -\mathbf{{{\underset{\scriptscriptstyle-}{B}}}^{\prime}}\left\langle {\mathbf{x}-{\mathbf{x}}^{\prime}} \right\rangle\;{T}^{\prime}} \right)} \right]d{V}^{\prime}+\mathbf{b}(\mathbf{x},t), \hfill \\ \end{array} $$

(13.32)

where ${{\underline{\mathbf{T}}}_s}={{\underline{\mathbf{T}}}_s}\left( {\mathbf{x},t} \right)$ and ${{\underline{\mathbf{T}}}^{\prime}}_s={{\underline{\mathbf{T}}}_s}\left( {{\mathbf{x}}^{\prime},t} \right)$; similar notation is used for $\underset{\scriptscriptstyle-}{\mathbf{B}}$ and $T$.

Substituting from Eq. 4.8 into Eq. 2.22b in conjunction with Eqs. 4.11 and 4.12 results in the bond-based PD equation for an isotropic material including the effect of temperature as

$$ \begin{array}{lll} \rho \ddot{\mathbf{u}} \left( {\mathbf{x},t} \right)=\int\nolimits_H {\left\{ {\left( {\frac{c}{2}\frac{{\left| {{\mathbf{y}}^{\prime}-\mathbf{y}} \right|-\left| {\mathbf{x}'-\mathbf{x}} \right|}}{{\left| {\mathbf{x}'-\mathbf{x}} \right|}}-\frac{c}{2}\alpha\,T\,} \right)\frac{{{\mathbf{y}}^{\prime}-\mathbf{y}}}{{\left| {{\mathbf{y}}^{\prime}-\mathbf{y}} \right|}}} \right.} \hfill \\ \left. {\kern4.25em -\left( {\frac{c}{2}\frac{{\left| {\mathbf{y}-{\mathbf{y}}^{\prime}} \right|-\left| {\mathbf{x}-\mathbf{x}'} \right|}}{{\left| {\mathbf{x}-\mathbf{x}'} \right|}}-\frac{c}{2}\alpha\,{T}^{\prime}\,} \right)\,\frac{{\mathbf{y}-{\mathbf{y}}^{\prime}}}{{\left| {\mathbf{y}-{\mathbf{y}}^{\prime}} \right|}}} \right\}d{V}^{\prime}+\mathbf{b}\left( {\mathbf{x},t} \right)\ . \hfill \\ \end{array} $$

(13.33)

Comparison of this equation with Eq. 13.32 leads to the explicit forms of

$$ {{\underline{\mathbf{T}}}_s}\left\langle {{\mathbf{x}}^{\prime}-\mathbf{x}} \right\rangle =\frac{c}{2}\frac{{\left| {{\mathbf{y}}^{\prime}-\mathbf{y}} \right|-\left| {\mathbf{x}'-\mathbf{x}} \right|}}{{\left| {\mathbf{x}'-\mathbf{x}} \right|}}\frac{{{\mathbf{y}}^{\prime}-\mathbf{y}}}{{\left| {{\mathbf{y}}^{\prime}-\mathbf{y}} \right|}} $$

(13.34a)

and

$$ \underline{\mathbf{B}}\;\left\langle {{\mathbf{x}}^{\prime}-\mathbf{x}} \right\rangle =\frac{c}{2}\alpha \frac{{{\mathbf{y}}^{\prime}-\mathbf{y}}}{{\left| {{\mathbf{y}}^{\prime}-\mathbf{y}} \right|}}. $$

(13.34b)

Comparison of Eq. 13.34b with Eq. 13.21a results in the expression for the thermal modulus scalar state ${\underline{\beta }}$ as

$$ {\underline{\beta }}\;\left\langle {{\mathbf{x}}^{\prime}-\mathbf{x}} \right\rangle =\frac{c}{2}\alpha. $$

(13.35)

13.3.3 Bond-Based Peridynamic Thermomechanics

The difference between the generalized heat transfer equation, Eq. 12.51, and thermomechanical heat transfer equation for an isotropic material, Eq. 13.22, is due to the deformational heating and cooling term, $({\underline{\beta }}\cdot \dot{{\underline{e}}})$. In light of this difference, the bond-based heat transfer equation, Eq. 12.51, can be modified to include the deformational heating and cooling term. Therefore, the bond-based coupled thermomechanical heat transfer equation can be cast as

$$ \rho {c_v}\dot{\Theta} \left( {\mathbf{x},t} \right)=\int\nolimits_H {\left( {{f_h}-\Theta \frac{c}{2}\alpha \dot{e}\;} \right)d{V}^{\prime}} +\rho {s_b}\left( {\mathbf{x},t} \right), $$

(13.36)

where $\dot{e}$ is the time rate of change of the extension between the material points, and it is defined as

$$ e=\left| {{\mathbf{\upeta}} +\boldsymbol{ \mathbf{\upxi}}} \right|-\left| \boldsymbol{ \mathbf{\upxi}} \right|, $$

(13.37a)

with its time rate of change

$$ \dot{e}=\frac{{{\mathbf{\upeta}} +\boldsymbol{ \mathbf{\upxi}}}}{{\left| {{\mathbf{\upeta}} +\boldsymbol{ \mathbf{\upxi}}} \right|}}\cdot \dot{{\mathbf{\upeta}}}, $$

(13.37b)

where $\dot{{\mathbf{\upeta}}}$ is the time rate of change of the relative displacement vector. Equation 13.36 can be rewritten in terms of the change in temperature, $T=\Theta -{\Theta_0}$, by replacing $\Theta$ with $T+{\Theta_0}$ and $\dot{\Theta}$ with $\dot{T}$ as

$$ \rho {c_v}\dot{T}\left( {\mathbf{x},t} \right)=\int\nolimits_H {\left( {{f_h}-\left( {T+{\Theta_o}} \right)\frac{c}{2}\alpha \dot{e}\;} \right)d{V}^{\prime}} +\rho {s_b}\left( {\mathbf{x},t} \right), $$

(13.38a)

or

$$ \rho {c_v}\dot{T}\left( {\mathbf{x},t} \right)=\int\nolimits_H {\left( {{f_h}-{\Theta_o}\left( {\frac{T}{{{\Theta_o}}}+1} \right)\frac{c}{2}\alpha \dot{e}\;} \right)d{V}^{\prime}} +\rho {s_b}\left( {\mathbf{x},t} \right). $$

(13.38b)

As suggested by Nowinski (1978), if the temperature change, $T$, is very small when compared with the reference temperature, ${\Theta_o}$, Eq. 13.38b can be approximated as

$$ \rho {c_v}\dot{T}\left( {\mathbf{x},t} \right)=\int\nolimits_H {\left( {{f_h}-{\Theta_o}\frac{c}{2}\alpha \dot{e}\;} \right)d{V}^{\prime}} +\rho {s_b}\left( {\mathbf{x},t} \right). $$

(13.39)

Substituting for the thermal response (heat flow density) function from Eq. 12.55 leads to its final form as

$$ \rho {c_v}\dot{T}\left( {\mathbf{x},t} \right)=\int\nolimits_H {\left( {\kappa \frac{\tau }{{\left| \boldsymbol{ \mathbf{\upxi}} \right|}}-{\Theta_o}\frac{c}{2}\alpha \dot{e}\;} \right)d{V}^{\prime}} +\rho {s_b}\left( {\mathbf{x},t} \right). $$

(13.40)

From Eq. 13.33, the bond-based PD equation of motion including the effect of temperature can be rewritten as

$$ \rho \ddot{\mathbf{u}} \left( {\mathbf{x},t} \right)=\int\nolimits_H {\frac{{\boldsymbol{ \mathbf{\upxi}} +{\mathbf{\upeta}}}}{{\left| {\boldsymbol{ \mathbf{\upxi}} +{\mathbf{\upeta}}} \right|}}\left( {cs-c\alpha {T_{avg }}} \right)d{V}^{\prime}} + \mathbf{b}\left( {\mathbf{x},t} \right), $$

(13.41)

in which c is the peridynamic material parameter. The initial relative position and relative displacement vectors are defined as $\boldsymbol{ \mathbf{\upxi}} ={\mathbf{x}}^{\prime}-\mathbf{x}$ and ${\mathbf{\upeta}} =\mathbf{u} \mathbf{^{\prime}-\mathbf{u}}$, respectively. The parameter $s$ represents the stretch between material points ${\mathbf{x}}^{\prime}$ and x, and ${T_{avg }}$ is the mean value of the change in temperatures at material points ${\mathbf{x}}^{\prime}$ and x defined as

$$ {T_{avg }}=\frac{{T + {T}^{\prime}}}{2}. $$

(13.42)

Introducing $\beta$ as the bond-based peridynamic thermal modulus, the final form of the fully coupled bond-based thermomechanical equations becomes

$$ \rho {c_v}\dot{T}\left( {\mathbf{x},t} \right)=\int\nolimits_H {\left( {\kappa \frac{\tau }{{\left| \boldsymbol{ \mathbf{\upxi}} \right|}}-{\Theta_o}\frac{\beta }{2}\dot{e}\;} \right)d{V}^{\prime}} +{h_s}\left( {\mathbf{x},t} \right), $$

(13.43a)

with ${h_s}=\rho {s_b}$ representing the heat source due to volumetric heat generation, and

$$ \rho \ddot{\mathbf{u}} \left( {\mathbf{x},t} \right)=\int\nolimits_H {\frac{{\boldsymbol{ \mathbf{\upxi}} +{\mathbf{\upeta}}}}{{\left| {\boldsymbol{ \mathbf{\upxi}} +{\mathbf{\upeta}}} \right|}}\left( {cs-\beta {T_{avg }}} \right)d{V}^{\prime}} + \mathbf{b}\left( {\mathbf{x},t} \right), $$

(13.43b)

with

$$ \beta =c\alpha. $$

(13.43c)

The first equation is the conservation of thermal energy (i.e., the heat transfer equation) with a contribution from deformational heating and cooling, and the second equation is the conservation of linear momentum (i.e., the equation of motion) with a thermoelastic constitutive relation.

13.4 Nondimensional Form of Thermomechanical Equations

The nondimensional form of an equation or system of equations involves eliminating the units associated with the variables and parameters. For coupled systems, various parameters may differ in size and the effects of certain parameters may not be apparent. The nondimensional form of equations can permit the effects of the different parameters to become more evident. The appropriate scaling, relative measure of quantities, and characteristic properties of the system, such as time constants, length scales, and resonance frequencies, can be revealed through nondimensionalization.

13.4.1 Characteristic Length and Time Scales

The characteristic length/time quantity for heat conduction is the diffusivity defined as

$$ \gamma =\frac{k}{{\rho {c_v}}}=\frac{{{\ell^{*2 }}}}{{{t^{*}}}}, $$

(13.44)

where ${\ell^{*}}$ and ${t^{*}}$ represent the characteristic length and time, respectively. For the equation of motion, the characteristic length/time is the elastic wave speed. The square of the elastic wave speed, $\tilde{a}$, is

$$ {{\tilde{a}}^2}=\frac{{\left( {\lambda +2\mu } \right)}}{\rho }=\frac{{{\ell^{*2 }}}}{{{t^{*2 }}}}, $$

(13.45)

where $\lambda$ and $\mu$ are Lamé’s constants. Combining the characteristic length/time scale from Eqs. 13.44 and 13.45 leads to the characteristic length and time for thermomechanics as

$$ {\ell^{*}}=\frac{\gamma }{\tilde{a}}\;\;\;\mathrm{ and}\;\;\;{t^{*}}=\frac{\gamma }{{{{\tilde{a}}^2}}}. $$

(13.46)

The characteristic length and time are typically employed in the non dimensionalization of the thermomechanical equations.

13.4.2 Nondimensional Parameters

The nondimensional form of Eq. 13.43a can be achieved by adopting the approach by Nickell and Sackman (1968) using Eqs. 13.44 and 13.45 for thermal diffusivity and the square of the elastic wave speed. The nondimensional variables are denoted with an overscore. The nondimensionalization procedure for length-related variables, i.e., x, $\delta$, A, and V (the volume), employs the characteristic length, and they are defined as

$$ x=\left( {\frac{\gamma }{\tilde{a}}} \right)\bar{x},\;\;\;\delta =\left( {\frac{\gamma }{\tilde{a}}} \right)\bar{\delta},\;\;\;A={{\left( {\frac{\gamma }{\tilde{a}}} \right)}^2}\bar{A}\;\;\mathrm{ and}\;\;\;V={{\left( {\frac{\gamma }{\tilde{a}}} \right)}^3}\bar{V}. $$

(13.47)

The displacement is nondimensionalized as

$$ u=\left( {\frac{\gamma }{\tilde{a}}} \right)\frac{{{\beta_{cl }}{\Theta_o}}}{{\left( {\lambda +2\mu } \right)}}\bar{u}. $$

(13.48)

The stretch is nondimensionalized as

$$ s=\frac{{{\beta_{cl }}{\Theta_o}}}{{\left( {\lambda +2\mu } \right)}}\bar{s}. $$

(13.49)

The time is scaled using the characteristic length as

$$ t=\left( {\frac{\gamma }{{{{\tilde{a}}^2}}}} \right)\bar{t}. $$

(13.50)

The nondimensionalization for the velocity-related variables is achieved by

$$ v=\frac{{{\beta_{cl }}{\Theta_o}}}{{\left( {\lambda +2\mu } \right)}}\tilde{a}\bar{v}\kern1.25em \mathrm{ and}\kern1em \dot{e}=\frac{{{\beta_{cl }}{\Theta_o}}}{{\left( {\lambda +2\mu } \right)}}\tilde{a}\bar{\dot{e}}. $$

(13.51)

Finally, the temperature and temperature difference are nondimensionalized as

$$ T={\Theta_o}\bar{T}\kern1.25em \mathrm{ and}\kern0.5em \tau ={\Theta_o}\bar{\tau}. $$

(13.52)

It is worth noting that the definitions of thermal modulus, bulk modulus, Lamé constants, shear modulus, peridynamic parameters, and microconductivity depend on the structural idealization. Their definitions for one-, two-, and three-dimensional analysis are summarized as:

One-dimensional analysis

$$ \lambda =0,\mu =\frac{E}{2},\alpha =\frac{{{\beta_{cl }}}}{{2\mu }},c=\frac{2E }{{A{\delta^2}}},\kappa =\frac{2k }{{A{\delta^2}}}. $$

(13.53a)

Two-dimensional analysis

$$ \lambda =\frac{{E\nu }}{{\left( {1-\nu } \right)\left( {1+\nu } \right)}},\ \mu =\frac{E}{{2\left( {1+\nu } \right)}},\ \alpha =\frac{{{\beta_{cl }}}}{{2\lambda +2\mu }},\ c=\frac{9E }{{\pi h{\delta^3}}},\ \kappa =\frac{6k }{{\pi h{\delta^3}}}. $$

(13.53b)

Three-dimensional analysis

$$ \lambda =\frac{{E\nu }}{{\left( {1+\nu } \right)\left( {1-2\nu } \right)}},\ \mu =\frac{E}{{2\left( {1+\nu } \right)}},\ \alpha =\frac{{{\beta_{cl }}}}{{3\lambda +2\mu }},\ c=\frac{12E }{{\,\pi\,{\delta^4}}},\ \kappa =\frac{6k }{{\pi {\delta^4}}}. $$

(13.53c)

Equating the coefficient of thermal expansion from Eqs. 13.1 and 13.43c leads to the thermal modulus as

$$ \beta =\frac{{{\beta_{cl }}}}{{3\lambda +2\mu }}c\ \mathrm{ for}\ \mathrm{ three}\ \mathrm{ dimensions}, $$

(13.54a)

$$ \beta =\frac{{{\beta_{cl }}}}{{2\lambda +2\mu }}c\ \mathrm{ for}\ \mathrm{ two}\ \mathrm{ dimensions}, $$

(13.54b)

$$ \beta =\frac{{{\beta_{cl }}}}{{2\mu }}c\ \mathrm{ for}\ \mathrm{ one}\ \mathrm{ dimension}. $$

(13.54c)

Substituting from Eqs. 13.47, 13.48, 13.49, 13.50, 13.51, and 13.52 with the dimensional considerations from Eq. 13.53, the fully coupled bond-based thermomechanical equations in the absence of body force and heat source can be cast into their nondimensional forms:

One-dimensional analysis

$$ \frac{{{\partial^2}\bar{\mathbf{u}}}}{{\partial {{\bar{t}}^2}}}=\frac{2}{{{{\bar{\delta}}^2}\bar{A}}}\int\nolimits_H {\frac{{\boldsymbol{ \mathbf{\upxi}} +\boldsymbol{ \mathbf{\upeta}}}}{{\left| {\boldsymbol{ \mathbf{\upxi}} +\boldsymbol{ \mathbf{\upeta}}} \right|}}\left( {\bar{s}-{{\bar{T}}_{avg }}} \right)d{{\bar{V}}_{{{\mathbf{x}}^{\prime}}}}} +\bar{\mathbf{b}}, $$

(13.55a)

$$ \frac{{\partial \bar{T}}}{{\partial \bar{t}}}=\frac{2}{{{{\bar{\delta}}^2}\bar{A}}}\int\nolimits_H {\left( {\frac{\bar{\tau}}{{\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}} \right|}}-\mathtt{\epsilon}\frac{{\bar{\dot{e}}}}{2}} \right)d{{\bar{V}}_{{{\mathbf{x}}^{\prime}}}}} +{{\bar{h}}_s}, $$

(13.55b)

Two-dimensional analysis

$$ \frac{{{\partial^2}\bar{\mathbf{u}}}}{{\partial {{\bar{t}}^2}}}=\frac{{9\left( {1-\nu } \right)}}{{\pi {{\bar{\delta}}^3}\bar{h}}}\int\nolimits_H {\frac{{\boldsymbol{ \mathbf{\upxi}} +\boldsymbol{ \mathbf{\upeta}}}}{{\left| {\boldsymbol{ \mathbf{\upxi}} +\boldsymbol{ \mathbf{\upeta}}} \right|}}\left( {\left( {1+\nu } \right)\bar{s}-{{\bar{T}}_{avg }}} \right)d{{\bar{V}}_{{{\mathbf{x}}^{\prime}}}}} +\bar{\mathbf{b}}, $$

(13.56a)

$$ \frac{{\partial \bar{T}}}{{\partial \bar{t}}}=\frac{6}{{\pi {{\bar{\delta}}^3}\bar{h}}}\int\nolimits_H {\left( {\frac{\bar{\tau}}{{\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}} \right|}}-\frac{3}{4}(1-\nu )\mathtt{\epsilon}\bar{\dot{e}}} \right)d{{\bar{V}}_{{{\mathbf{x}}^{\prime}}}}} +{{\bar{h}}_s}, $$

(13.56b)

Three-dimensional analysis

$$ \frac{{{\partial^2}\bar{\mathbf{u}}}}{{\partial {{\bar{t}}^2}}}=\frac{6}{{\pi {{\bar{\delta}}^4}}}\int\nolimits_H {\frac{{\boldsymbol{ \mathbf{\upxi}} +\boldsymbol{ \mathbf{\upeta}}}}{{\left| {\boldsymbol{ \mathbf{\upxi}} +\boldsymbol{ \mathbf{\upeta}}} \right|}}\left( {\frac{{1+\nu }}{{1-\nu }}\bar{s}-{{\bar{T}}_{avg }}} \right)d{{\bar{V}}_{{{\mathbf{x}}^{\prime}}}}} +\bar{\mathbf{b}}, $$

(13.57a)

$$ \frac{{\partial \bar{T}}}{{\partial \bar{t}}}=\frac{6}{{\pi {{\bar{\delta}}^4}}}\int\nolimits_H {\left( {\frac{\bar{\tau}}{{\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}} \right|}}-\frac{1}{2}\mathtt{\epsilon}\bar{\dot{e}}} \right)d{{\bar{V}}_{{{\mathbf{x}}^{\prime}}}}} +{{\bar{h}}_s}, $$

(13.57b)

in which the nondimensional coupling coefficient, $\mathtt{\epsilon}$, body force density, $\bar{\mathbf{b}}$, and heat source due to volumetric heat generation, ${{\bar{h}}_s}$, are defined as

$$ \mathtt{\epsilon}=\frac{{\beta_{cl}^2{\Theta_0}}}{{\rho {c_v}\left( {\lambda +2\mu } \right)}}, $$

(13.58a)

$$ \bar{\mathbf{b}} =\frac{{\gamma \left( {\lambda +2\mu } \right)}}{{\rho {{\tilde{a}}^3}\beta_{cl }{\Theta_0}}}\mathbf{b}, $$

(13.58b)

$$ {{\bar{h}}_s}=\frac{\gamma }{{\rho {c_v}{{\tilde{a}}^2}{\Theta_0}}}{h_s}. $$

(13.58c)

The coupling coefficient $\mathtt{\epsilon}$ measures the strength of thermal and deformation coupling and it appears in the nondimensional thermomechanical equations associated with the heating and cooling term due to deformation. The coupling coefficient transpires out of the nondimensional form of these peridynamic equations in a similar manner as it does out of the classical thermomechanical equations, as illustrated by Nickell and Sackman (1968). The nondimensional equation represents decoupled thermomechanics for $\mathtt{\epsilon}=0$. It is worth noting that the equation of motion still contains the effect of temperature even if $\mathtt{\epsilon}=0$.

13.5 Numerical Procedure

For numerically approximating the solution to the classical fully coupled equations for thermoelasticity, one of two different time stepping strategies is generally employed by researchers. The monolithic or simultaneous scheme is one time stepping strategy. For a monolithic algorithm, the time stepping scheme is applied simultaneously to the full system of equations and the unknown variables are solved for at the same time. If the time stepping scheme for the monolithic algorithm is implicit, unconditional stability is usually achieved. However, monolithic algorithms can result in practical large systems, in spite of their unconditional stability. For the staggered or partitioned scheme, the coupled system of equations are split, typically according to two different fields, the displacement and temperature fields. Each field is then individually treated with a different time stepping algorithm. Staggered algorithms generally circumvent the shortcomings of their monolithic counterparts; however, this is often accomplished at the expense of the unconditional stability. In many scenarios, even when unconditionally stable time stepping schemes are used to solve each partitioned equation, the overall stability of the thermomechanical system of equations is only conditional (Wood 1990). As a result, a good deal of work has been performed to successfully develop unconditionally stable staggered algorithms for thermoelasticity (Armero and Simo 1992; Farhat et al. 1991; Liu and Chang 1985).

For the numerical treatment of the fully coupled thermoelastic peridynamic system of equations, a staggered strategy is adopted. The system is partitioned naturally according to the structural and thermal fields; thus, the equation of motion is solved for the displacement field and the heat transfer equation is solved for the temperature field. Explicit time stepping schemes are utilized to approximate the solutions to both equations.

In order to illustrate the numerical implementation, one-dimensional peridynamic thermoelastic equations, Eq. 13.55a,b, are considered, and they can be discretized in the forms

$$ \bar{\ddot{u}}_{(i)}^n=\frac{2}{{{{\bar{\delta}}^2}\bar{A}}}\sum\limits_{j=1}^N {\frac{{\bar{\boldsymbol{ \mathbf{\upxi}}}_{(i)(j)}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{(i)(j)}^n}}{{\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}_{(i)(j)}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{(i)(j)}^n} \right|}}\left( {{{\bar{s}}^n}_{(i)(j) }-{{\bar{T}}^n}_{(i)(j) }} \right){{\bar{V}}_{(j) }}} $$

(13.59a)

and

$$ \bar{\dot{T}}_{(i)}^n=\frac{2}{{{{\bar{\delta}}^2}\bar{A}}}\sum\limits_{j=1}^N {\left( {\frac{{{{\bar{\tau}}^n}_{(i)(j) }}}{{\left| {{{\bar{\mathbf{\upxi}}}^n}_{(i)(j) }} \right|}}-\epsilon \frac{{{{{\bar{\dot{e}}}}^n}_{(i)(j) }}}{2}\;} \right){{\bar{V}}_{(j) }}}, $$

(13.59b)

in which the term $2/({{\bar{\delta}}^2}\bar{A})$ is assumed to be constant throughout the domain, $n$ represents the time step number, $i$ is the collocation point that is being solved for, and $j$ represents the collocation points within the horizon of $i$. The nondimensional volume of the subdomain represented by the collocation point $j$ is denoted by ${{\bar{V}}_{(j) }}$.

The discretization of a one-dimensional domain is illustrated in Fig. 13.1. The one-dimensional domain is discretized into subdomains, with the collocation points at the center of each subdomain.

Fig. 13.1

Discretization of one-dimensional domain with collocation points

The horizon is $\bar{\delta}=3\bar{\Delta}$, where $\bar{\Delta}$ is the nondimensional spacing between material points. The material point of interest is denoted by i and it interacts with the three points to its left and right. Thus, points j within the horizon of i are i−3, i−2, i−1, i + 1, i + 2, and i + 3, as shown in Fig. 13.1

The nondimensional displacement, velocity, and temperature of all the collocation points are known at the n ^th time step, i.e., the current time step. Based on Fig. 13.1, Eq. 13.55a can be discretized as

$$ \begin{array}{lll} \bar{\ddot{u}}_{(i)}^n =\frac{2}{{{{\bar{\delta}}^2}\bar{A}}}\left[ {\frac{{\bar{\boldsymbol{ \mathbf{\upxi}}}_{{(i)\left( {i+3} \right)}}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{{(i)\left( {i+3} \right)}}^n}}{{\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}_{{(i)\left( {i+3} \right)}}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{{(i)\left( {i+3} \right)}}^n} \right|}}\left( {{{\bar{s}}^n}_{{(i)\left( {i+3} \right)}}-{{\bar{T}}^n}_{{(i)\left( {i+3} \right)}}} \right){{\bar{V}}_{{\left( {i+3} \right)}}}} \right. \hfill \\ \quad+\frac{{\bar{\boldsymbol{ \mathbf{\upxi}}}_{{(i)\left( {i+2} \right)}}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{{(i)\left( {i+2} \right)}}^n}}{{\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}_{{(i)\left( {i+2} \right)}}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{{(i)\left( {i+2} \right)}}^n} \right|}}\left( {{{\bar{s}}^n}_{{(i)\left( {i+2} \right)}}-{{\bar{T}}^n}_{{(i)\left( {i+2} \right)}}} \right){{\bar{V}}_{{\left( {i+2} \right)}}} \hfill \\ \quad +\frac{{\bar{\boldsymbol{ \mathbf{\upxi}}}_{{(i)\left( {i+1} \right)}}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{{(i)\left( {i+1} \right)}}^n}}{{\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}_{{(i)\left( {i+1} \right)}}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{{(i)\left( {i+1} \right)}}^n} \right|}}\left( {{{\bar{s}}^n}_{{(i)\left( {i+1} \right)}}-{{\bar{T}}^n}_{{(i)\left( {i+1} \right)}}} \right){{\bar{V}}_{{\left( {i+1} \right)}}} \hfill \\ \quad +\frac{{\bar{\boldsymbol{ \mathbf{\upxi}}}_{{(i)\left( {i-3} \right)}}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{{(i)\left( {i-3} \right)}}^n}}{{\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}_{{(i)\left( {i-3} \right)}}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{{(i)\left( {i-3} \right)}}^n} \right|}}\left( {{{\bar{s}}^n}_{{(i)\left( {i-3} \right)}}-{{\bar{T}}^n}_{{(i)\left( {i-3} \right)}}} \right){{\bar{V}}_{{\left( {i-3} \right)}}} \hfill \\ \quad +\frac{{\bar{\boldsymbol{ \mathbf{\upxi}}}_{{(i)\left( {i-2} \right)}}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{{(i)\left( {i-2} \right)}}^n}}{{\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}_{{(i)\left( {i-2} \right)}}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{{(i)\left( {i-2} \right)}}^n} \right|}}\left( {{{\bar{s}}^n}_{{(i)\left( {i-2} \right)}}-{{\bar{T}}^n}_{{(i)\left( {i-2} \right)}}} \right){{\bar{V}}_{{\left( {i-2} \right)}}} \hfill \\ \quad \left. {+\frac{{\bar{\boldsymbol{ \mathbf{\upxi}}}_{{(i)\left( {i-1} \right)}}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{{(i)\left( {i-1} \right)}}^n}}{{\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}_{{(i)\left( {i-1} \right)}}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{{(i)\left( {i-1} \right)}}^n} \right|}}\left( {{{\bar{s}}^n}_{{(i)\left( {i-1} \right)}}-{{\bar{T}}^n}_{{(i)\left( {i-1} \right)}}} \right){{\bar{V}}_{{\left( {i-1} \right)}}}} \right], \end{array} $$

(13.60)

where the nondimensional stretch is denoted by $\bar{s}_{(i)(j)}^n$, and it is defined as

$$ \bar{s}_{(i)(j)}^n=\frac{{\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}_{(i)(j)}^n+\bar{\boldsymbol{ \mathbf{\upeta}}}_{(i)(j)}^n} \right|-\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}_{(i)(j)}^n} \right|}}{{\left| {\bar{\boldsymbol{ \mathbf{\upxi}}}_{(i)(j)}^n} \right|}}. $$

(13.61)

The position of the ith and jth collocation points are given by ${{\bar{\mathbf{x}}}_{(i) }}$ and ${{\bar{\mathbf{x}}}_{(j) }},$ respectively, and, as such, the nondimensional relative initial position is defined as

$$ \bar{\boldsymbol{ \mathbf{\upxi}}}_{(i)(j)}^n={{\bar{\mathbf{x}}}_{(j) }}-{{\bar{\mathbf{x}}}_{(i) }}. $$

(13.62)

The nondimensional displacements of the ith and jth collocation points are given by $\bar{\mathbf{u}}_{(i)}^n$ and $\bar{\mathbf{u}}_{(j)}^n$, respectively. Therefore, the nondimensional relative displacement becomes

$$ \bar{\boldsymbol{ \mathbf{\upeta}}}_{(i)(j)}^n=\bar{\mathbf{u}}_{(j)}^n-\bar{\mathbf{u}}_{(i)}^n, $$

(13.63a)

and the term ${{\bar{T}}^n}_{(i)(j) }$ is defined as

$$ {{\bar{T}}^n}_{(i)(j) }=\frac{{{{\bar{T}}^n}_{(j) }+{{\bar{T}}^n}_{(i) }}}{2}. $$

(13.63b)

Based on Fig. 13.1, Eq. 13.55b can be discretized as

$$ \begin{array}{lll} \bar{\dot{T}}_{(i)}^n =\frac{2}{{{{\bar{\delta}}^2}\bar{A}}}\left[ {\left( {\frac{{{{\bar{\tau}}^n_{{(i)\left( {i+3} \right)}}}}}{{\left| {{{\bar{\mathbf{\upxi}}}^n_{{(i)\left( {i+3} \right)}}}} \right|}}-\epsilon\frac{{{{{\bar{\dot{e}}}}^n_{{(i)\left( {i+3} \right)}}}}}{2}} \right){{\bar{V}}_{{\left( {i+3} \right)}}}+\left( {\frac{{{{\bar{\tau}}^n_{{(i)\left( {i+2} \right)}}}}}{{\left| {{{\bar{\mathbf{\upxi}}}^n_{{(i)\left( {i+2} \right)}}}} \right|}}-\epsilon\frac{{{{{\bar{\dot{e}}}}^n_{{(i)\left( {i+2} \right)}}}}}{2}} \right){{\bar{V}}_{{\left( {i+2} \right)}}}} \right. \hfill \cr \quad+\left( {\frac{{{{\bar{\tau}}^n_{{(i)\left( {i+1} \right)}}}}}{{\left| {{{\bar{\mathbf{\upxi}}}^n_{{(i)\left( {i+1} \right)}}}} \right|}}-\epsilon\frac{{{{{\bar{\dot{e}}}}^n_{{(i)\left( {i+1} \right)}}}}}{2}} \right){{\bar{V}}_{{\left( {i+1} \right)}}}+\left( {\frac{{{{\bar{\tau}}^n_{{(i)\left( {i-3} \right)}}}}}{{\left| {{{\bar{\mathbf{\upxi}}}^n_{{(i)\left( {i-3} \right)}}}} \right|}}-\epsilon\frac{{{{{\bar{\dot{e}}}}^n_{{(i)\left( {i-3} \right)}}}}}{2}} \right){{\bar{V}}_{{\left( {i-3} \right)}}} \hfill \\ \quad\left. {+\left( {\frac{{{{\bar{\tau}}^n_{{(i)\left( {i-2} \right)}}}}}{{\left| {{{\bar{\mathbf{\upxi}}}^n_{{(i)\left( {i-2} \right)}}}} \right|}}-\epsilon \frac{{{{{\bar{\dot{e}}}}^n_{{(i)\left( {i-2} \right)}}}}}{2}} \right){{\bar{V}}_{{\left( {i-2} \right)}}}+\left( {\frac{{{{\bar{\tau}}^n_{{(i)\left( {i-1} \right)}}}}}{{\left| {{{\bar{\mathbf{\upxi}}}^n_{{(i)\left( {i-1} \right)}}}} \right|}}-\epsilon\frac{{{{{\bar{\dot{e}}}}^n_{{(i)\left( {i-1} \right)}}}}}{2}} \right){{\bar{V}}_{{\left( {i-1} \right)}}}} \right], \end{array} $$

(13.64)

where

$$ {{\bar{\tau}}^n}_{(i)(j) }={{\bar{T}}^n}_{(j) }-{{\bar{T}}^n}_{(i) }, $$

(13.65a)

and the nondimensional rate of extension between the material points is given by

$$ \bar{\dot{e}}_{(i)(j)}^n=\frac{{\bar{\boldsymbol{ \mathbf{\upeta}}}_{(i)(j)}^n+\bar{\boldsymbol{ \mathbf{\upxi}}}_{(i)(j)}^n}}{{\left| {\bar{\boldsymbol{ \mathbf{\upeta}}}_{(i)(j)}^n+\bar{\boldsymbol{ \mathbf{\upxi}}}_{(i)(j)}^n} \right|}}\cdot \left( {\bar{\dot{\mathbf{u}}}_{(j)}^n-\bar{\dot{\mathbf{u}}}_{(i)}^n} \right). $$

(13.65b)

As explained in Sects. 7.3 and 12.9, the time integration of Eq. 13.60 can be performed by using explicit forward and backward difference techniques and Eq. 13.64 by forward difference time integration scheme.

13.6 Validation

The validity of the fully coupled PD thermomechanical equations is established by constructing PD solutions to previously considered problems. The first problem is a semi-infinite bar subjected to a transient thermal boundary condition. The second problem concerns the dynamic response of a thermoelastic bar with an initial sinusoidal velocity. The solutions to these problems are obtained by constructing one-dimensional PD models.

The third problem is a finite plate subjected to either a pressure shock or a thermal shock, and their combination. The solutions to these problems are obtained by constructing two-dimensional PD models. The fourth is a block of material subjected to a transient thermal boundary condition. The solution to this problem is obtained by constructing a three-dimensional PD model.

13.6.1 A Semi-infinite Bar Under Thermal Loading

A semi-infinite bar is subjected to the temperature boundary condition on the bounding end. The bounding end is stress free and is gradually heated. The stress-free condition on the bounding end is represented by not specifying any displacement or velocity conditions. The peridynamic discretization of the bar for thermal and deformational fields is shown in Fig. 13.2.

Fig. 13.2

Peridynamic model of the fields in the one-dimensional bar: (a) thermal, (b) deformation

The peridynamic predictions for the nondimensional temperature and displacement for the three different coupling scenarios are compared against the classical solution reported by Nickell and Sackman (1968). The coupling coefficient values of $\mathtt{\epsilon}=0,0.36,1$ are used to depict the decoupled, moderate, and strong coupling situations, respectively. The temperature boundary condition is imposed through the fictitious region ${{\mathcal{R}}_t}$, as explained in Chap. 12. The solution is obtained by specifying the geometric parameters, material properties, initial and boundary conditions, as well as the peridynamic discretization and time integration parameters as:

Geometric Parameters

• Length of bar: $\bar{L}=5$

• Area of cross section: $\bar{A}=6.25\times {10^{-4 }}$

Boundary Conditions

• $\bar{T}(0,\bar{t})=(\bar{t}/{{\bar{t}}_0})H({{\bar{t}}_0}-\bar{t})+H(\bar{t}-{{\bar{t}}_0})\kern0.75em,\ \mathrm{ with}\ {{\bar{t}}_o}=0.25$

Initial Conditions

• $\bar{u}(\bar{x},0)=\partial \bar{u}(\bar{x},0)/\partial \bar{t}=\bar{T}(\bar{x},0)=0$

PD Discretization Parameters

• Total number of material points in the $\bar{x}\text{--}$ direction: $200$

• Spacing between material points: $\bar{\Delta} =0.025$

• Incremental volume of material points: $\Delta \bar{V}=1.5625\times {10^{-5 }}$

• Volume of fictitious boundary layer: ${{\bar{V}}_{\bar{\delta}}}=(3)\times \Delta \bar{V}=4.6875\times {10^{-5 }}$

• Horizon: $\bar{\delta}=3.015\bar{\Delta}$

• Time step size: $\Delta {}{\bar t}=0.5\times {10^{-3 }}$

Numerical Results: Figure 13.3 provides a comparison of the temperature and displacement distribution predicted by the peridynamic simulation against the finite element predictions using ANSYS at $\bar{x} = 1$ for $\mathtt{\epsilon}=0,\ 0.36,\ 1.$ These results also agree extremely well with those reported by Nickell and Sackman (1968). It is evident that for all three degrees of coupling the temperature at $\bar{x} = 1$ increases with time in a very similar fashion while the displacement remains zero up until $\bar{t}=0.5$. At about time $\bar{t}=0.5$, the point $\bar{x} = 1$ starts to be displaced in the positive direction. The effects of coupling become apparent beyond $\bar{t}=0.5$. The temperature and displacement variation for the three degrees of coupling are no longer similar. The amplitudes of the temperature and displacement decrease as the strength of the coupling is increased. The coupling accelerates the diffusion of heat as there appears to be an increase in the amount of thermal and mechanical energy dissipated.

Fig. 13.3

For different coupling coefficients: (a) Displacement and (b) temperative predictions at $\bar{x}=1$

13.6.2 Thermoelastic Vibration of a Finite Bar

A bar of finite length is initially subjected to a sinusoidal velocity with zero displacement and temperature. The initial velocity is applied with a specified wavenumber. The ends of the bar are fixed with zero temperature and displacement. This particular thermoelastic vibration problem was considered by Armero and Simo (1992) using the finite element method. Construction of the PD solution is achieved by using the nondimensional form of the equations. The geometric parameters and the peridynamic discretization for the thermal and deformational fields are shown in Fig. 13.4. The temperature and displacement boundary conditions are imposed through the fictitious regions ${{\mathcal{R}}_t}$ and ${{\mathcal{R}}_u}$, respectively.

Fig. 13.4

Peridynamic model of the fields in the one-dimensional bar: (a) thermal, (b) deformation

The solution is obtained by specifying the geometric parameters, material properties, initial and boundary conditions, as well as the peridynamic discretization and time integration parameters as:

Geometric parameters

• Length of bar: $\bar{L}=100$

Boundary Conditions

• $\bar{T}(0,\bar{t})=\bar{T}(\bar{L},\bar{t})=0$

  $\bar{u}(0,\bar{t})=\bar{u}(\bar{L},\bar{t})=0$

Initial Conditions

• $\bar{u}(\bar{x},0)=\bar{T}(\bar{x},0)=0$

  $\partial \bar{u}(\bar{x},0)/\partial t=\sin (\pi \bar{x}/\bar{L})$

PD Discretization Parameters

• Total number of material points in the $\bar{x}\text{--}$ direction: $5000$

• Spacing between material points: $\bar{\Delta} =0.02$

• Incremental volume of material points: $\Delta \bar{V}=8\times {10^{-6 }}$

• Volume of fictitious boundary layer: ${{\bar{V}}_{\bar{\delta}}}=(3)\times \Delta \bar{V}=24\times {10^{-6 }}$

• Horizon: $\bar{\delta}=3.015\times \bar{\Delta}$

• Time step size: $\Delta {}\bar{t}=1\times {10^{-4 }}$

Numerical Results: The resulting elastic waves are progressive traveling waves. In the case of a fully coupled thermoelastic problem, there exist two types of waves: elastic and thermal. Both types of waves have been modified from their uncoupled forms. The modified elastic waves are attenuated, compared to the uncoupled elastic waves, and are subjected to dispersion and damping in time. The modified thermal waves also exhibit dispersion and damping in time. The peridynamic predictions for the temporal distribution of displacement and temperature at $\bar{x}=50$ and $\bar{x}=25$, respectively, are shown in Fig. 13.5 for coupling coefficients of $\mathtt{\epsilon}=0$ and $\mathtt{\epsilon}=1$. The peridynamic predictions are also compared with the classical finite element approximations given by Armero and Simo (1992).

Fig. 13.5

Variation of (a) displacement at $\bar{x}=50$ and (b) temperature at $\bar{x}=25$

13.6.3 Plate Subjected to a Shock of Pressure and Temperature, and Their Combination

The fully coupled nondimensional PD thermomechanical equations are further verified by solving a problem previously considered by Hosseini-Tehrani and Eslami (2000) using the Boundary Element Method. It concerns a square plate of isotropic material under either a pressure shock or a thermal shock, and their combination on the free edge in the positive $\bar{x}\text{--}\mathrm{ direction}.$ As shown in Fig. 13.6, it is clamped at the other edge and the insulated horizontal edges are free of any loading. The thermomechanical equations are solved for both uncoupled and coupled cases.

Fig. 13.6

Geometry and boundary conditions of the plate under pressure or thermal shock

Geometric Parameters

• Length: $\bar{L}=10$

• Width: $\bar{W}=10$

• Thickness: $\bar{H}=1$

Initial Conditions

• $\bar{T}\left( {\bar{x},\bar{y},0} \right)=0$

  ${{\bar{u}}_{\bar{x}}}\left( {\bar{x},\bar{y},0} \right)={{\bar{u}}_{\bar{y}}}\left( {\bar{x},\bar{y},0} \right)=0$

Boundary Conditions

• ${{\bar{T}}_{{,\bar{x}}}}\left( {\bar{x}=10,\bar{y},\bar{t}} \right)=0$

  ${{\bar{T}}_{{,\bar{y}}}}\left( {\bar{x},\bar{y}=\pm 5,\bar{t}} \right)=0$

  ${{\bar{u}}_{\bar{x}}}\left( {\bar{x}=10,\bar{y},\bar{t}} \right)={{\bar{u}}_{\bar{y}}}\left( {\bar{x}=10,\bar{y},\bar{t}} \right)=0$

  ${\sigma_{{\bar{y}\bar{y}}}}\left( {\bar{x},\bar{y}=\pm 5,\bar{t}} \right)={\sigma_{{\bar{x}\bar{y}}}}\left( {\bar{x},\bar{y}=\pm 5,\bar{t}} \right)=0$

• where $\bar{t}$ is the nondimensional time.

Pressure Shock

• $\bar{T}(\bar{x}=0,\bar{y},\bar{t})=0$

  ${\sigma_{{\bar{x}\bar{x}}}}(\bar{x}=0,\bar{y},\bar{t})=-P(\bar{t})=-5\bar{t}{e^{{-2\bar{t}}}}$

Thermal Shock

• $\bar{T}(\bar{x}=0,\bar{y},\bar{t})=5\bar{t}{e^{{-2\bar{t}}}}$

  ${\sigma_{{\bar{x}\bar{x}}}}(\bar{x}=0,\bar{y},\bar{t})=0$

Combined Pressure and Thermal Shock

• $\bar{T}(\bar{x}=0,\bar{y},\bar{t})=5\bar{t}{e^{-2\bar{t}}}$

  ${\sigma_{{\bar{x}\bar{x}}}}(\bar{x}=0,\bar{y},\bar{t})=-P(\bar{t})=-5\bar{t}{e^{-2\bar{t}}}$

PD Discretization Parameters

• Total number of material points in the $\bar{x}\text{--}$ direction: $200$

• Total number of material points in the $\bar{y}\text{--}$ direction: $200$

• Spacing between material points: $\bar{\Delta} =0.05$

• Incremental volume of material points: $\Delta \bar{V}=1.25\times {10^{-4 }}$

• Volume of fictitious boundary layer: ${{\bar{V}}_{\bar{\delta}}}=(3\times 200)\times \Delta \bar{V}=0.075$

• Volume of boundary layer: ${{\bar{V}}_{\bar{\Delta}}}=(1\times 200)\times \Delta \bar{V}=0.025$

• Horizon: $\bar{\delta}=3.015\times \bar{\Delta}$

• Time step size: $\Delta {}\bar{t}=0.5\times {10^{-3 }}$

The peridynamic discretization for the thermal field is shown in Fig. 13.7. The temperature boundary condition is imposed in fictitious region ${{\mathcal{R}}_t}$. The peridynamic discretization for the deformational field is shown in Fig. 13.8. The displacement boundary condition is imposed in fictitious region ${{\mathcal{R}}_u}$. The pressure is applied through boundary layer region ${{\mathcal{R}}_p}$.

Fig. 13.7

Peridynamic model of the thermal field in a plate

Fig. 13.8

Peridynamic model of deformational field in a plate: (a) pressure shock, (b) thermal shock

Numerical results: Figure 13.9 shows the temperature and displacement variations at $\bar{y}=0$ due to the pressure shock at times $\bar{t}=3$ and $\bar{t}=6$. When the coupling coefficient is zero, no temperature change is expected. However, when the coupled effect is included, even though mechanical loading is applied, temperature change is expected. The compressive stress along the boundary causes a temperature rise. As observed in this figure, the peak of the temperature distribution moves to the right as time progresses. Figure 13.9 also shows the axial displacement along the $\bar{x}$ -axis. The PD results are also in close agreement with the BEM results (Hosseini-Tehrani and Eslami 2000). Figure 13.10 shows the temperature and displacement variations at $\bar{y}=0$ due to thermal shock at times $\bar{t}=3$ and $\bar{t}=6$. As observed, the coupling term in the thermal field causes a temperature drop, and the peridynamic predictions are in close agreement with the BEM solution published by Hosseini-Tehrani and Eslami (2000). Figure 13.11 shows the temperature and displacement variations at $\bar{y}=0$ due to combined pressure and thermal shock at times $\bar{t}=3$ and $\bar{t}=6$. The PD predictions are in close agreement with the BEM results by Hosseini-Tehrani and Eslami (2000).

Fig. 13.9

Variations along the centerline in the plate for uncoupled ($\mathtt{\epsilon}=0$) and coupled ($\mathtt{\epsilon}\ne 0$) cases under pressure shock loading: (a) temperature, and (b) displacement

Fig. 13.10

Variations along the centerline in the plate for uncoupled ($\mathtt{\epsilon}=0$) and coupled ($\mathtt{\epsilon}\ne 0$) cases under thermal shock loading: (a) temperature, and (b) displacement

Fig. 13.11

Variations along the centerline in the plate for uncoupled ($\mathtt{\epsilon}=0$) and coupled ($\mathtt{\epsilon}\ne 0$) cases under combined thermal and pressure shock loading: (a) temperature, and (b) displacement

13.6.4 A Block of Material Under Thermal Loading

A three-dimensional finite block of material is gradually heated at one end, and the remaining surfaces are insulated. As shown in Fig. 13.12, it is clamped at the other end without any other type of loading. The PD discretization of the thermal and deformational fields is shown in Fig. 13.13

Fig. 13.12

Geometry and boundary conditions of the block under thermal loading

Fig. 13.13

Three-dimensional peridynamic model of the fields: (a) thermal, (b) deformation

The solution is obtained by specifying the geometric parameters, initial and boundary conditions, as well as the peridynamic discretization and time integration parameters as:

Geometric Parameters

• Length: $\bar{L}=5$

• Width: $\bar{W}=0.15$

• Thickness: $\bar{H}=0.15$

Initial Conditions

• $\bar{u}(\bar{x},\bar{y},\bar{z},0)=\partial \bar{u}(\bar{x},\bar{y},\bar{z},0)/\partial \bar{t}=\bar{T}(\bar{x},\bar{y},\bar{z},0)=0$

Boundary Conditions

• $\begin{array}{ll} \bar{T}(0,\bar{y},\bar{z},\bar{t})=(\bar{t}/{{\bar{t}}_0})H({{\bar{t}}_0}-\bar{t})+H(\bar{t}-{{\bar{t}}_0})\kern0.5em \end{array}$

  ${{\bar{T}}_{{,\bar{x}}}}\left( {\bar{x}=\bar{L},\bar{y},\bar{z},\bar{t}} \right)=0$

  ${{\bar{T}}_{{,\bar{y}}}}\left( {\bar{x},\bar{y}=0,\bar{z},\bar{t}} \right)=0$

  ${{\bar{T}}_{{,\bar{y}}}}\left( {\bar{x},\bar{y}=\bar{W},\bar{z},\bar{t}} \right)=0$

  ${{\bar{T}}_{{,\bar{z}}}}\left( {\bar{x},\bar{y},\bar{z}=0,\bar{t}} \right)=0$

  ${{\bar{T}}_{{,\bar{z}}}}\left( {\bar{x},\bar{y},\bar{z}=\bar{H},\bar{t}} \right)=0$

  ${{\bar{u}}_{\bar{x}}}\left( {\bar{x}=\bar{L},\bar{y},\bar{z},\bar{t}} \right)={{\bar{u}}_{\bar{y}}}\left( {\bar{x}=\bar{L},\bar{y},\bar{z},\bar{t}} \right)={{\bar{u}}_{\bar{z}}}\left( {\bar{x}=\bar{L},\bar{y},\bar{z},\bar{t}} \right)=0$

  ${\sigma_{{\bar{x}\bar{x}}}}\left( {\bar{x}=0,\bar{y},\bar{z},\bar{t}} \right)={\sigma_{{\bar{x}\bar{y}}}}\left( {\bar{x}=0,\bar{y},\bar{z},\bar{t}} \right)={\sigma_{{\bar{x}\bar{z}}}}\left( {\bar{x}=0,\bar{y},\bar{z},\bar{t}} \right)=0$

  ${\sigma_{{\bar{y}\bar{y}}}}\left( {\bar{x},\bar{y}=0,\bar{z},\bar{t}} \right)={\sigma_{{\bar{x}\bar{y}}}}\left( {\bar{x},\bar{y}=0,\bar{z},\bar{t}} \right)={\sigma_{{\bar{y}\bar{z}}}}\left( {\bar{x},\bar{y}=0,\bar{z},\bar{t}} \right)=0$

  ${\sigma_{{\bar{y}\bar{y}}}}\left( {\bar{x},\bar{y}=\bar{W},\bar{z},\bar{t}} \right)={\sigma_{{\bar{x}\bar{y}}}}\left( {\bar{x},\bar{y}=\bar{W},\bar{z},\bar{t}} \right)={\sigma_{{\bar{y}\bar{z}}}}\left( {\bar{x},\bar{y}=\bar{W},\bar{z},\bar{t}} \right)=0$

  ${\sigma_{{\bar{z}\bar{z}}}}\left( {\bar{x},\bar{y},\bar{z}=0,\bar{t}} \right)={\sigma_{{\bar{x}\bar{z}}}}\left( {\bar{x},\bar{y},\bar{z}=0,\bar{t}} \right)={\sigma_{{\bar{y}\bar{z}}}}\left( {\bar{x},\bar{y},\bar{z}=0,\bar{t}} \right)=0$

  ${\sigma_{{\bar{z}\bar{z}}}}\left( {\bar{x},\bar{y},\bar{z}=\bar{H},\bar{t}} \right)={\sigma_{{\bar{x}\bar{z}}}}\left( {\bar{x},\bar{y},\bar{z}=\bar{H},\bar{t}} \right)={\sigma_{{\bar{y}\bar{z}}}}\left( {\bar{x},\bar{y},\bar{z}=\bar{H},\bar{t}} \right)=0$

PD Discretization Parameters

• Total number of material points in the $\bar{x}\text{--}$ direction: $200$

• Total number of material points in the $\bar{\mathrm{ y}} \text{--}$ direction: $6$

• Total number of material points in the $\bar{\mathrm{ z}} \text{--}$ direction: $6$

• Spacing between material points: $\bar{\Delta} =0.025$

• Incremental volume of material points: $\Delta \bar{V}=1.5625\times {10^{-5 }}$

• Volume of fictitious boundary layer: ${{\bar{V}}_{\bar{\delta}}}=(3\times 6\times 6)\times \Delta \bar{V}=1.6875\times {10^{-3 }}$

• Horizon: $\bar{\delta}=3.015\times \bar{\Delta}$

• Time step size: $\Delta \bar{t}=1\times {10^{-4 }}$

Numerical Results: As shown in Fig. 13.14, the PD predictions for temperature and displacement variations along the length of the block are compared with the FEA results from ANSYS at $\bar{t}=1$ and 2 for $\mathtt{\epsilon}=0$ and $\mathtt{\epsilon}=1$. The comparison indicates excellent agreement.

Fig. 13.14

Predictions at $(\bar{y}=\bar{W}/2,\bar{z}=\bar{H}/2)$ for coupled and uncoupled cases: (a) displacement, and (b) temperature