Thermodynamically consistent thermoelastic plate and shell formulations for small deformation and small strain using non-classical continuum mechanics incorporating internal rotations

In this paper, we derive non-classical continuum theory for physics of compressible and incompressible thermoviscous non-classical fluent continua using the conservation and balance laws (CBL) by incorporating additional physics of internal rotation rates arising from the velocity gradient tensor as well as their time varying rates and the rotational inertial effects. In this non-classical continuum theory time dependent deformation of fluent continua results in time varying rotation rates i.e., angular velocities and angular accelerations at material points. Resistance offered to these by deforming fluent continua results in additional moments, angular momenta and inertial effects due to rotation rates i.e., angular velocities and angular accelerations at the material points. Currently, this physics due to internal rotation rates and inertial effects is neither considered in classical continuum mechanics (CCM) nor in non-classical continuum mechanics (NCCM). In this paper, we present a derivation of conservation and balance laws in Eulerian description: conservation of mass (CM), balance of linear momenta (BLM), balance of angular momenta (BAM), balance of moment of moments (BMM), first and second laws of thermodynamics (FLT, SLT) that include: (i) Physics of internal rotation rates resulting from the velocity gradient tensor; (ii) New physics resulting due to angular velocities and angular accelerations due to spatially varying and time dependent rotation rates. The balance laws derived here are compared with those that only consider the rotational rates but neglect rotational inertial effects and angular accelerations to demonstrate the influence of the new physics. Constitutive variables and their argument tensors are established using conjugate pairs in the entropy inequality, additional desired physics and principle of equipresence when appropriate. Constitutive theories are derived using Helmholtz free energy density as well as representation theorem and integrity (complete basis). It is shown that the mathematical model consisting of the conservation and balance laws and constitutive theories presented in this paper has closure. Influence of new physics in the conservation and balance laws on compressible and incompressible thermoviscous fluent continua is demonstrated due to presence of angular velocities and angular accelerations arising from time varying rotation rates when the deforming fluent continua offer rotational inertial resistance. The fluent continua are considered homogeneous and isotropic. Model problem studies are considered in a follow-up paper.

In this work, we demonstrate the existence of rotational waves in deforming thermoelastic non-classical solid continua in which the conservation and balance laws consider internal rotations due to the deformation gradient tensor (Jacobian of deformation) as well as their time varying rates. In this non-classical continuum theory, time dependent deformation of solid continua results in time dependent varying rotations, angular velocities and angular accelerations at material points. Resistance to these by deforming continua results in additional moments due to rotations, angular momenta due to rotation rates and rotational inertial effects due to angular accelerations at the material points. Currently this physics is neither considered in classical continuum mechanics (CCM) nor in non-classical continuum mechanics (NCCM) based on internal and/or Cosserat rotations. In this paper, we present derivation of conservation and balance laws in Lagrangian description: conservation of mass, balance of linear momentum, balance of angular momentum, balance of moment of moments, first and second laws of thermodynamics that include: (i) physics due to internal rotations resulting from the displacement gradient tensor (ii) new physics associated with rotation rates (angular velocities) and angular accelerations resulting from the varying, time dependent internal rotations at the material points. The balance laws derived here are compared with those that only consider internal rotations and their rates in the absence of rotational inertial effects (Surana et al. in J Therm Eng 1(6):446–459, 2015; Int J Eng Res Ind Appl 8(2):77–106, 2015) to demonstrate the influence of new physics. Constitutive variables and their argument tensors are established using the conjugate pairs in the entropy inequality, additional desired physics and the principle of equipresence. The constitutive theories are derived using Helmholtz free energy density as well as representation theorem and integrity. It is shown that the mathematical model consisting of the conservation and balance laws and the constitutive theories has closure. Existence of rotational waves is demonstrated due to the presence of angular velocities and angular accelerations arising from the time varying rotations and their rates when deforming solid continua offer rotational inertial resistance. In this paper, we only consider isotropic and homogeneous solid continua with small strain, small deformation physics and reversible mechanical deformation. Model problem studies are presented in a follow up paper.

In order to enhance currently used beam theories in $$\mathbb {R}^2$$ and $$\mathbb {R}^3$$ to include mechanisms of dissipation and memory, it is necessary to establish if the mathematical models for these theories can be derived using the conservation and the balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the beam mathematical models. Thermodynamic consistency of the currently used beam models will permit use of entropy inequality to establish constitutive theories in the presence of dissipation and memory mechanism in the currently used beam theories. This is the main motivation for the work presented in this paper. The currently used beam theories are derived based on kinematic assumptions related to the axial and transverse displacement fields. These are then used to derive strain measures followed by constitutive relations. For linear beam theories, strain measures are linear functions of displacement gradients and stresses are linear functions of strain measures. Using these stress and strain measures, energy functional is constructed over the volume of the beam consisting of kinetic energy, strain energy and potential energy of loads. The Euler’s equation(s) extracted from the first variation of this energy functional set to zero yields the differential equations describing the evolution of the deforming beam. Alternatively, principle of virtual work can also be used to derive mathematical models for beams. For linear elastic behavior with small deformation and small strain, the two approaches yield same mathematical models. In this paper we examine whether the currently used beam mathematical models with the corresponding kinematic assumption (i) can be derived using the conservation and balance laws of classical continuum mechanics or (ii) are the conservation and balance laws of non-classical continuum mechanics necessary in their derivation. In order to ensure that the mathematical models for various beam theories result in deformation that is in thermodynamic equilibrium we must establish the consistency of the beam theories with regard to the conservation and the balance laws of continuum mechanics, classical or non-classical in conjunction with their corresponding kinematic assumptions. Currently used Euler–Bernoulli and Timoshenko beam mathematical models that are representative of most beam mathematical models are investigated. This is followed by details of general and higher-order thermodynamically consistent beam theory that is free of kinematic assumptions and other approximations and remains valid for slender as well as deep beams. Model problem studies are presented for slender as well as deep beams. The new formulation presented in this paper ensures thermodynamic equilibrium as it is derived using the conservation and the balance laws of continuum mechanics and remains valid for slender as well as non-slender beams.

This paper considers dynamic behavior of non-classical thermoelastic solid continua. The mathematical model consists of the conservation and balance laws of non-classical continuum mechanics that incorporates additional physics of internal rotations arising due to deformation gradient tensor. We consider plane stress behavior with small deformation, small strain physics only. Galerkin Method with Weak Form (GM/WF) in space is considered to construct a space–time decoupled finite element formulation giving rise to ordinary differential equations (ODEs) in time containing mass matrix, stiffness matrix due to classical as well as non-classical physics and acceleration and displacement associated with nodal degrees of freedom. This formulation is utilized to: (1) study natural undamped modes of vibration (2) study transient dynamic response by time integrating the ODEs in time (3) study the transient dynamic response by transforming the ODEs in time to modal basis using eigenvectors of the undamped natural modes. The ODEs in modal basis are used to construct transient dynamic response by time integrating them as well as by considering their analytical solutions. The solutions of the model problem obtained using the mathematical model based on non-classical continuum mechanics with internal rotations are presented and are compared with those obtained using the mathematical model based on classical continuum mechanics to demonstrate the influence of new physics due to internal rotations on the dynamic response of solid continua.

This paper presents a thermodynamically consistent and kinematic assumption free formulation for dynamics of thermoviscoelastic plates/shells based on the conservation and balance laws of classical continuum mechanics (CCM) in which dissipation mechanism has been incorporated through ordered rate constitutive theory for deviatoric stress tensor. In this paper, we consider small deformation, small strain. The conservation and balance laws of CCM in [Formula: see text] in Lagrangian description using Cauchy stress tensor ([Formula: see text]) and linearized Green strain tensor ([Formula: see text]) constitute the mathematical model. The constitutive theory for the deviatoric Cauchy stress ([Formula: see text]) is derived using conjugate pairs in entropy inequality in conjunction with representation theorem. The argument tensors of [Formula: see text] are [Formula: see text] and rates of [Formula: see text] up to order [Formula: see text]. This yields a constitutive theory with dissipation mechanism based on rates of strain up to order [Formula: see text]. Constitutive theory for heat vector is also derived using the conjugate pairs in the entropy inequality and representation theorem. Finite element method is used to obtain solutions of the initial value problems descried by the balance of linear momenta (BLM), energy equation and the constitutive theories. The shell element geometry is described by the middle surface and the nodal vectors at the middle surface defining bottom and top surfaces of the element. The local approximation for the displacement field is [Formula: see text] - version hierarchical in the plane of the element as well as in the transverse direction. A space-time decoupled finite element formulation using Galerkin Method with Weak Form (GM/WF) in space is constructed for BLM as well as energy equation, both resulting in ordinary differential equations (ODEs) in time. The ordinary differential equations (ODEs) in time resulting from the finite element formulation of BLM are used to study: (i) natural undamped modes of vibration (ii) the transient dynamic response using the ODEs in time recast in modal basis: (a) using Rayleigh damping (b) using the ordered rate damping proposed in this paper. Time response is calculated using modal damping based on Rayleigh damping as well as using the proposed ordered rate damping mechanism. Model problem studies are presented to demonstrate: (1) accuracy of the natural frequencies obtained from the present formulation for thin and thick plates/shells (in which shear deformation is significant) and the results are compared with the currently used plate formulations (2) accuracy of damped transient response using proposed damping mechanism is compared with time response using Rayleigh damping (3) it is shown that Rayleigh damping has no physical basis and leads to spurious stationary states. The proposed damping yields accurate stationary states that are in exact agreement with the solution of corresponding BVP. A single formulation presented in this paper remains valid and accurate for very thin as well as very thick plates/shells and correctly simulates 3D state of deformation regardless of plate/shell thickness and is free of shear locking problems as well as need for shear corrections. When obtaining the time response, solution for an increment of time alternates between the solution of BLM followed by the solution of the energy equation. Details are presented in the paper.

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler's equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic as-How to cite this paper:

Marine structures are advanced material and structural assemblies that span over different length scales. The classical structural design approach is to separate these length scales. The used structural models are based on classical continuum mechanics. There are multiple situations where the classical theory breaks down. Non-classical effects tend arise when the size of the smallest repeating unit of a periodic structure is of the same order as the full structure itself. The aim of the present paper is to discuss representative problems from different length scales of ship structural design.

The paper presents constitutive theories for non-classical thermoviscoelastic fluids with dissipation and memory using a thermodynamic framework based on entirety of velocity gradient tensor. Thus, the conservation and the balance laws used in this work incorporate symmetric as well as antisymmetric part of the velocity gradient tensor. The constitutive theories derived here hold in coand contra-variant bases as well as in Jaumann rates and are derived using convected time derivatives of Green’s and Almansi strain tensors as well as the Cauchy stress tensor and its convected time derivatives in appropriate bases. The constitutive theories are presented in the absence as well as in the presence of the balance of moment of moments as balance law. It is shown that the dissipation mechanism and the fading memory in such fluids are due to stress rates as well as moment rates and their conjugates. The material coefficients are derived for the general forms of the constitutive theories based on integrity. Simplified linear (or quasi-linear) forms of the constitutive theories are also presented. Maxwell, Oldroyd-B and Giesekus constitutive models for non-classical thermoviscoelastic fluids are derived and are compared with those derived based on classical continuum mechanics. Both, compressible and incompressible thermoviscoelastic fluids are considered.

In the non-classical continuum theories for solid continua the presence of internal rotations and their gradients arising due to Jacobian of deformation and/or consideration of Cosserat rotations as additional unknown degrees of freedom at a material point necessitate existence of moment tensor. For small deformation, small strains theories, in Lagrangian description the Cauchy moment tensor and the rates of rotation gradients are rate of work conjugate pair in addition to the rate of work conjugate Cauchy stress tensor and the strain rate tensor. It is well established that in such non-classical theories the Cauchy stress tensor is non-symmetric and the antisymmetric components of the Cauchy stress tensor are balanced by gradients of the Cauchy moment tensor, the balance of angular momenta balance law. In the non-classical continuum theories incorporating internal rotations and conjugate moment tensor that are absent in the classical continuum theories, the fundamental question is “are the conservation and balance laws used in classical continuum mechanics sufficient to ensure dynamic equilibrium of the deforming volume of matter”. At this stage the Cauchy moment tensor remains non-symmetric if we only consider standard balance laws that are used in classical continuum theories. Thus, requiring constitutive theories for the symmetric as well as anti-symmetric Cauchy moment tensors. The work presented in this paper shows that when the thermodynamically consistent constitutive theories are used for symmetric as well as antisymmetric Cauchy moment tensor non physical and spurious solutions result even in simple model problems. This suggests that perhaps the additional conjugate tensors resulting due to presence of internal rotations, namely the Cauchy moment tensor and the antisymmetric part of the Cauchy stress stress tensor must obey some additional law or restriction so that the spurious behavior is precluded. This paper demonstrates that in the non-classical theory with internal rotations considered here the law of balance of moment of moments and the consideration of the equilibrium of moment of moments are in fact identical. When this balance law is considered the Cauchy moment tensor becomes symmetric, hence eliminating the constitutive theory for the antisymmetric Cauchy moment tensor and thereby eliminating spurious and non physical solutions. The necessity of this balance law is established theoretically and is also demonstrated through model problems using thermoelastic solids with small strain small deformation as an example. The findings reported in this paper hold for thermoviscoelastic solids with and without memory as well as when deformation and strains are small. Extensions of the concepts presented here for finite deformation and finite strain will be presented in a follow up paper.

3R1. Continuum Mechanics and Theory of Materials. - P Haupt (Inst of Mech, Univ of Kassel, Monchebergstr 7, Kassel, 34109, Germany). Springer-Verlag, Berlin. 2000. 583 pp. ISBN 3-540-66114-X. $82.00.Reviewed by JL Wegner (Dept of Mech Eng, Univ of Victoria, Eng Office Wing, Room 537, Victoria BC, V8W 3P6, Canada).The author attempts to portray the ideas and general principles of the theory of materials within the framework of phenomenological continuum mechanics. It is a well-written rigorous mathematical treatment of classical continuum mechanics and deals with such concepts such as elasticity, plasticity, viscoelasticity, and viscoplasticity in nonlinear materials. The volume consists of 13 chapters. Chapter 1 covers kinematics, that is the geometry of motion and the deformation of material bodies. The outline follows the script of most texts on classical continuum mechanics—the concepts of material bodies and the material derivative are introduced in Euclidean space. The deformation gradient tensor is introduced, and its physical meaning is described by the transformation of material line, surface, and volume elements. Similar to most treatments on classical continuum mechanics, the appropriate strain and stretch tensors are described. However, the introduction of convective coordinates in this volume is a departure from most treatises on classical continuum mechanics. In this volume, a treatment of strain rates in convective coordinates is provided with the argument that the choice of convective coordinates not only affords a deeper understanding of the strain tensors but also of their strain rates. Chapter 1 finishes with a section on incompatible configurations, that is when a material body can identify a configuration with a non-Euclidean space. Chapter 2 develops the classical balance relations of mechanics, for example: balance of mass in spatial and referential form, conservation of linear and rotational momentum in spatial and referential form. Here, the Cauchy, first Piola-Kirchhoff, weighted Cauchy, and the second Piola-Kirchhoff stress tensors are introduced, as well as their physical significance. This treatise finishes with the balance of mechanical energy and the balance of virtual work.A comprehensive theory of phenomenological material behavior, based on the general principles of thermomechanics, is developed and presented in this volume. This general theory is expounded in the remaining chapters, beginning in Chapter 3 where the classical balance relations of thermodynamics are presented. In Chapter 4, the terms frame of reference, change of frame, and objectivity are clarified, in preparation for the discussion of the constitutive equations in chapter five. Classical constitutive relations are presented in Chapter 5, that is, equations defining the perfect fluid, the linear-viscous fluid and the linear-elastic isotropic solid. Two extensions to the model of linear elasticity are also included in this treatise, namely the theories of linear viscoelasticity and plasticity. Chapter 6 contains the results of experimental testing of different materials such as steel and elastomers. The experimental results provide invaluable insight to the reader when compared to the classical theories of continuum mechanics. The classical constitutive models do reflect significant aspects of the material behavior observed. However, there are considerable discrepancies which cannot be resolved within the context of classical theories presented thus far in the volume. Hence, the motivation for a comprehensive theory of phenomenological material behavior based on the general principles of thermomechanics. The aim of material theory is to provide general principles and systematic methods for constructing mathematical models suitably representing the individual properties of material bodies. The general theory of material behavior, as it is developed in Chapter 7, is mainly due to W Noll. In Chapter 9, the constitutive relations for isotropic elastic and isotropic hyperelastic (compressible, and incompressible) solids are derived. Of interest, the one-dimensional stress-strain curves for the Mooney-Rivlin and Neo-Hookean models are plotted, along with a discussion of their limitations for applications to large deformations. What separates this volume from most on continuum mechanics is the treatise in Chapter 9 on constitutive relations for anisotropic hyperelastic solids. Chapter 10 considers nonlinear viscoelasticity, and Chapter 11 covers plasticity theory. Chapter 12 covers viscoplastisticity, which depicts rate-dependent material behavior with equilibrium hysteresis phenomena. Constitutive models, for all of these types of materials, in thermomechanics is discussed in Chapter 13. The author achieves his goals of presenting, in a rigorous manner, the ideas and general principles of the theory of materials within the framework of phenomenological continuum mechanics, providing the reader general theories of material behavior from which a reader can select the constitutive model that applies best. Continum Mechanics and Theory of Materials will be invaluable to advanced graduate students of materials science in engineering and in physics.

This article develops a unified variational framework for configurational (or material) forces in both Classical (3D, non-relativistic) and Relativistic (4D) Continuum Mechanics. Configurational forces describe the evolution of material defects—such as cracks, dislocations, and interfaces—which move relative to the material rather than through physical space. In the classical setting of hyperelasticity, the authors revisit the balance of configurational forces using an intrinsic Lagrangian formulation, where the material body is modeled as an abstract three-dimensional manifold. By treating the reference configuration as a variable and performing a Lagrangian variation with respect to it, they show that the configurational forces balance naturally emerges. Importantly, this balance equation is not independent: it is equivalent to the standard balance of linear momentum combined with constitutive relations, and it is expressed through the Eshelby stress tensor on the reference configuration. The framework is then extended to Relativistic Hyperelasticity within General Relativity. Matter is described by a matter field, a vector valued function, defined on the four-dimensional Universe, and the Lagrangian ( i.e , Action) includes both matter and gravitational contributions. Two stress–energy tensors arise: the Noether stress–energy tensor (from variations with respect to the matter field) and the Hilbert stress–energy tensor (from variations with respect to the Universe metric). Assuming General Covariance, the authors prove that these tensors and their associated balance laws are equivalent. By introducing the notion of an observer and specializing to static spacetimes, the authors define a relativistic generalization of the deformation and derive a four-dimensional Eshelby tensor. They show that in Special Relativity, as in Classical Continuum Mechanics, the relativistic configurational forces balance is not a new equation but follows from the conservation laws of the Noether stress–energy tensor. Finally, they recover the classical configurational forces balance as the non-relativistic limit of the relativistic theory. Overall, the paper provides a rigorous geometric and variational interpretation of configurational forces, unifying classical and relativistic formulations and clarifying their deep connection with standard equilibrium equations.

The work presented in this paper extends the kinematic assumption free and thermodynamically consistent formulation for bending of thermoelastic beams presented by Surana et al. for bending of thermoviscoelastic beams with dissipation mechanism without memory. We consider small strain, small deformation physics in Lagrangian description. Conservation and balance laws of classical continuum mechanics (CCM) constitute the mathematical model for the physics considered in this paper. Constitutive variables and their argument tensors are established using conjugate pairs in the entropy inequality, additional desired physics and the principle of equipresence. Cauchy stress tensor is decomposed into equilibrium stress tensor () and deviatoric stress tensor (). Constitutive theory for is derived using Helmholtz free energy density in conjunction with incompressibility condition. The constitutive theory for is derived in by first establishing its argument tensors using conjugate pairs in the entropy inequality and other desired physics and then using the representation theorem with complete basis (integrity). The constitutive theory for is a nonlinear constitutive theory in terms of strain tensor containing up to fifth degree terms in the components of the strain tensor and an ordered rate theory in the strain rate tensors up to order n. Simplified linear ordered rate theory for is also presented. The formulation presented here for thermoviscoelastic beams is based on the conservation and balance laws of classical continuum mechanics and construction of beam finite element formulation using hpk framework with variationally consistent integral form. In this approach the mathematical model consist of true conservation and balance laws and the choice of local approximations for the beam finite elements facilitates incorporation of required kinematic description based on the application. This approach is free of the a priori assumptions of kinematic relations, computations are unconditionally stable and the local approximations can be of higher degree (p) and of higher order (k). This approach addresses slender as well as deep beam physics and can be used to measure error in the computed solution through residual functional. The rate constitutive theory is based on the second law thermodynamics consistent with physics of deformation. Mathematical details of new formulation and the model problem studies and comparisons with currently used beam models are presented for slender as well as deep beams. This approach permits consideration of reversible (thermoelastic) as well as irreversible (thermoviscoelastic) processes. It is shown that Rayleigh damping used currently to derive modal damping has no physical basis and can lead to spurious solution. The dissipation mechanism presented here has physical basis and yields non-spurious and valid solutions Model problem studies are presented for undamped natural modes of vibration, damped and undamped transient dynamic response. The results obtained from the formulation presented here are compared with published works.

A clear-cut definition of non-classical continuum mechanics can be given only by a negation, so that we need recall what is understood (by us) by “classical continuum mechanics”.

ABSTRACTThis paper presents two specific thermodynamically consistent non-classical continuum theories for solid and fluent continua. The first non-classical continuum theory for solid continua incorporates Jacobian of deformation in its entirety in the conservation and the balance laws and the derivation of the constitutive theories. The second non-classical continuum theory for solid continua considers Jacobian of deformation in its entirety as well as the Cosserat rotations in the conservation and balance laws as well as the constitutive theories. The first non-classical continuum theory for fluent continua presented here considers velocity gradient tensor in its entirety. The second non-classical continuum theory for fluent continua considers velocity gradient tensor in its entirety as well as Cosserat rotation rates in the derivation of the conservation and balance laws and the constitutive theories. Since the non-classical continuum theories for solid and fluent continua considered here incorporate additional physics of deformation due to rotations and rotation rates compared to classical continuum mechanics, the conservation and balance laws of classical continuum mechanics are shown to require modification as well as a new balance law balance of moment of moments is required to accommodate the new physics due to rotations and rotation rates. Eringen’s micropolar, micromorphic and microstretch theories, couple stress theories and nonlocal theories are also discussed within the context of the non-classical theories presented here for solid and fluent continua. Some applications of these theories are also discussed.

This report summarizes activities undertaken during FY08-FY10 for the LDRD Peridynamics as a Rigorous Coarse-Graining of Atomistics for Multiscale Materials Design. The goal of our project was to develop a coarse-graining of finite temperature molecular dynamics (MD) that successfully transitions from statistical mechanics to continuum mechanics. The goal of our project is to develop a coarse-graining of finite temperature molecular dynamics (MD) that successfully transitions from statistical mechanics to continuum mechanics. Our coarse-graining overcomes the intrinsic limitation of coupling atomistics with classical continuum mechanics via the FEM (finite element method), SPH (smoothed particle hydrodynamics), or MPM (material point method); namely, that classical continuum mechanics assumes a local force interaction that is incompatible with the nonlocal force model of atomistic methods. Therefore FEM, SPH, and MPM inherit this limitation. This seemingly innocuous dichotomy has far reaching consequences; for example, classical continuum mechanics cannot resolve the short wavelength behavior associated with atomistics. Other consequences include spurious forces, invalid phonon dispersion relationships, and irreconcilable descriptions/treatments of temperature. We propose a statistically based coarse-graining of atomistics via peridynamics and so develop a first of a kind mesoscopic capability to enable consistent, thermodynamically sound, atomistic-to-continuum (AtC) multiscale material simulation. Peridynamics (PD) is a microcontinuum theory that assumes nonlocal forces for describing long-range material interaction. The force interactions occurring at finite distances are naturally accounted for in PD. Moreover, PDs nonlocal force model is entirely consistent with those used by atomistics methods, in stark contrast to classical continuum mechanics. Hence, PD can be employed for mesoscopic phenomena that are beyond the realms of classical continuum mechanics and atomistic simulations, e.g., molecular dynamics and density functional theory (DFT). The latter two atomistic techniques are handicapped by the onerous length and time scales associated with simulating mesoscopic materials. Simulating such mesoscopic materials is likely to require, and greatly benefit from multiscale simulations coupling DFT, MD, PD, and explicit transient dynamic finite element methods FEM (e.g., Presto). The proposed work fills the gap needed to enable multiscale materials simulations.