Principal Invariants Research Articles

Physics-informed data-driven discovery of constitutive models with application to strain-rate-sensitive soft materials

AbstractA novel data-driven constitutive modeling approach is proposed, which combines the physics-informed nature of modeling based on continuum thermodynamics with the benefits of machine learning. This approach is demonstrated on strain-rate-sensitive soft materials. This model is based on the viscous dissipation-based visco-hyperelasticity framework where the total stress is decomposed into volumetric, isochoric hyperelastic, and isochoric viscous overstress contributions. It is shown that each of these stress components can be written as linear combinations of the components of an irreducible integrity basis. Three Gaussian process regression-based surrogate models are trained (one per stress component) between principal invariants of strain and strain rate tensors and the corresponding coefficients of the integrity basis components. It is demonstrated that this type of model construction enforces key physics-based constraints on the predicted responses: the second law of thermodynamics, the principles of local action and determinism, objectivity, the balance of angular momentum, an assumed reference state, isotropy, and limited memory. The three surrogate models that constitute our constitutive model are evaluated by training them on small-size numerically generated data sets corresponding to a single deformation mode and then analyzing their predictions over a much wider testing regime comprising multiple deformation modes. Our physics-informed data-driven constitutive model predictions are compared with the corresponding predictions of classical continuum thermodynamics-based and purely data-driven models. It is shown that our surrogate models can reasonably capture the stress–strain-strain rate responses in both training and testing regimes and improve prediction accuracy, generalizability to multiple deformation modes, and compatibility with limited data.

Learning nonlinear constitutive models in finite strain electromechanics with Gaussian process predictors

This paper introduces a metamodelling technique that employs gradient-enhanced Gaussian process regression (GPR) to emulate diverse internal energy densities based on the deformation gradient tensor F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{F}$$\\end{document} and electric displacement field D0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{D}_0$$\\end{document}. The approach integrates principal invariants as inputs for the surrogate internal energy density, enforcing physical constraints like material frame indifference and symmetry. This technique enables accurate interpolation of energy and its derivatives, including the first Piola-Kirchhoff stress tensor and material electric field. The method ensures stress and electric field-free conditions at the origin, which is challenging with regression-based methods like neural networks. The paper highlights that using invariants of the dual potential of internal energy density, i.e., the free energy density dependent on the material electric field E0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{E}_0$$\\end{document}, is inappropriate. The saddle point nature of the latter contrasts with the convexity of the internal energy density, creating challenges for GPR or Gradient Enhanced GPR models using invariants of F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{F}$$\\end{document} and E0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{E}_0$$\\end{document} (free energy-based GPR), compared to those involving F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{F}$$\\end{document} and D0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{D}_0$$\\end{document} (internal energy-based GPR). Numerical examples within a 3D Finite Element framework assess surrogate model accuracy across challenging scenarios, comparing displacement and stress fields with ground-truth analytical models. Cases include extreme twisting and electrically induced wrinkles, demonstrating practical applicability and robustness of the proposed approach.

Gradient enhanced gaussian process regression for constitutive modelling in finite strain hyperelasticity

This paper introduces a metamodelling technique that leverages gradient-enhanced Gaussian process regression (also known as gradient-enhanced Kriging), effectively emulating the response of diverse hyperelastic strain energy densities. The approach adopted incorporates principal invariants as inputs for the surrogate of the strain energy density. This integration enables the surrogate to inherently enforce fundamental physical constraints, such as material frame indifference and material symmetry, right from the outset. The proposed approach provides accurate interpolation for energy and the first Piola–Kirchhoff stress tensor (e.g. first order derivatives with respect to inputs). The paper presents three notable innovations. Firstly, it introduces the utilisation of Gradient-Enhanced Kriging to approximate a diverse range of phenomenological models, encompassing numerous isotropic hyperelastic strain energies and a transversely isotropic potential. Secondly, this study marks the inaugural application of this technique for approximating the effective response of composite materials. This includes rank-one laminates, for which analytical solutions are feasible. However, it also encompasses more complex composite materials characterised by a Representative Volume Element (RVE) comprising an elastomeric matrix with a centred spherical inclusion. This extension opens the door for future application of this technique to various RVE types, facilitating efficient three-dimensional computational analyses at the macro-scale of such composite materials, significantly reducing computational time compared to FEM2. The third innovation, facilitated by the integration of these surrogate models into a 3D Finite Element computational framework, lies in the assessment of these models scenarios encompassing intricate cases of extreme twisting and more importantly, buckling instabilities in thin-walled structures, thereby highlighting both the practical applicability and robustness of the proposed approach.

Rheology of brain tissue and hydrogels: A novel hyperelastic and viscoelastic model for forensic applications

A new strain-energy function W, which possesses the strain energy expressible as a rational function of the principal invariants of the Cauchy tensor C, is proposed. It generates a hyperelastic constitutive equation with characteristics of brain tissues: a much stronger resistance to compression than to stretching and strongly nonlinear response in simple shear, including non-zero first and second normal differences. This model exponent α resembles the Ogden model in uniaxial stretching/compression and reveals plausible predictions for brain tissue with even values of α < 0 with sufficiently high magnitude (say, at α = −20). However, the dependence of the strain-energy function W on the principal invariants of C links it to hyperelastic hydrogel models (the Special and General Blatz–Ko models, neo-Hookean materials, incompressible Mooney–Rivlin and the Yeoh models). For α = −8, the present model reveals a compression/stretching behavior close to the tensorial Special Blatz–Ko model used for description of hydrogels. Furthermore, the present hyperelastic model is used as a kernel of the corresponding tensorial viscoelastic model with exponential fading memory. It belongs to the class of the integral Bernstein–Kearsley–Zapas (BKZ) models. In a number of important cases (the uniaxial stretching/compression, simple shear), it can be transformed into a differential viscoelastic model and predict viscoelastic liquid-like behavior under sustained deformations. The stress relaxation following an imposed strain reduces to the hyperelastic model with the elastic parameters exponentially fading in time. These tensorial hyperelastic and viscoelastic constitutive equations aim applications in modeling of blast-induced traumatic brain injuries and bullet penetration and spatter of brain tissue in forensic context.

Penetrating gunshots to the head after close-range shooting: Dynamics of waves and the effect of brain tissue rheology

Formation of the brain tissue backspatter after penetrating gunshots to the head is preceded and driven by formation and evolution of the bullet channel, which is filling with air and/or muzzle gases or issuing them with tissue fragments or without them. This process is explored here in a model situation in the framework of the dynamics of waves in brain tissue affected by its realistic rheological behavior, fragmentation, and gas dynamics in the evolving bullet channel. As a rheological model of the brain tissue, a new strain-energy function W, introduced in the accompanying work, is employed, which expresses the strain energy as a rational function of the principal invariants of the Cauchy tensor C. This strain-energy function W generates a hyperelastic constitutive equation, which resembles the behavior of brain tissues, i.e., reveals a much stronger resistance to compression than to stretching and strongly nonlinear response in simple shear. This new rheological model belongs to the class of hyperelastic models used for description of hydrogels. The equations of motion supplemented by this rheological model reveal the dynamics of the compression and rarefaction waves propagating through the brain tissue following the formation of the bullet channel. These waves are reflected from the skull and the bullet channel. In parallel, gas dynamics of air and/or muzzle gases flowing into or issued outward of the bullet channel, and stretching-driven fragmentation of the brain tissue are evolving in concert with the wave dynamics in the brain tissue. This allows for prediction of backspatter of the brain tissue resulting from a short-range shooting.

Crystal structure search with principal invariants

In this study, we propose a general way of finding a new crystal structure starting from a database. We also introduce a concise representation of the data based on principal invariants derived from the spatial arrangement of atoms in the crystal structure. Since the selection of the present representation does not depend on the size and selection method of the unit-cell defining the crystal structure, it is ideal for measuring similarity between two arbitrarily different crystal structures. For a more general search for crystal structures, the known crystal structure space is better defined using a variational autoencoder. Furthermore, we used either Pearson's correlation or the square root of the Jensen-Shannon divergence to directly search similar crystal structures for a wide variety of space groups. Our novel method uses extended database information, the similarity measure, and first-principles electronic structure calculation to predict several new metastable crystal structures of SnSe, P, and SnS.

Toward a robust detection of viscous and turbulent flow regions using unsupervised machine learning

We propose an invariant feature space for the detection of viscous-dominated and turbulent regions (i.e., boundary layers and wakes). The developed methodology uses the principal invariants of the strain and rotational rate tensors as input to an unsupervised Machine Learning Gaussian mixture model. The selected feature space is independent of the coordinate frame used to generate the processed data, as it relies on the principal invariants of the strain and rotational rate, which are Galilean invariants. This methodology allows us to identify two distinct flow regions: a viscous-dominated, rotational region (a boundary layer and a wake region) and an inviscid, irrotational region (an outer flow region). We have tested the methodology on a laminar and a turbulent (using Large Eddy Simulation) case for flows past a circular cylinder at Re = 40 and Re = 3900 and a laminar flow around an airfoil at Re=1×105. The simulations have been conducted using a high-order nodal Discontinuous Galerkin Spectral Element Method. The results obtained are analyzed to show that Gaussian mixture clustering provides an effective identification method of viscous-dominated and rotational regions in the flow. We also include comparisons with traditional sensors to show that the proposed clustering does not depend on the selection of an arbitrary threshold, as required when using traditional sensors.

Internally balanced hyperelastic constitutive model in terms of principal stretches

Internal variables are included in material constitutive models to capture realistic mechanical effects such as viscosity. Recent work for compressible hyperelastic material is developed based on applying the argument of calculus variation to two-factor multiplicative decomposition of the deformation gradient. Strain energy function is expressed in terms of principal invariants of the deformation gradient decomposed counterparts. The present work permits the use of strain energy function in terms principal stretches of the deformation gradient multiplicatively decomposed counterparts directly. This simplifies significantly the expression of second Piola–Kirchhoff stress and internal balance tensor.

Large Isotropic Elastic Deformations: On a Comprehensive Model to Correlate the Theory and Experiments for Incompressible Rubber-Like Materials

A comprehensive model, i.e., a model that: (i) suitably captures the mechanical behaviour of various types of rubber-like materials; (ii) describes the constitutive behaviour of a subject rubber-like specimen under different deformation modes via a single set of model parameter values, and (iii) is parent to many of the existing models of distinct types, is presented in this paper for application to the finite deformation of incompressible isotropic rubber-like materials. The model breaks away from Rivlin’s principal invariants I_{i} and Valanis-Landel’s separable function of principal stretches lambda _{i} representations, and instead adopts a general non-separable functional form W( lambda _{1}, lambda _{2}, lambda _{3} ), subject to the incompressibility constraint. By way of example, the application of the model to extant datasets from four types of rubber-like materials that exhibit discernible mechanical behaviours, namely natural unfilled and filled rubbers, hydrogels and (extremely) soft tissue specimens, will be considered. The favourable correlation between the model predictions and the considered experimental datasets, obtained via simultaneous fitting of the model to the data of various deformations, will be demonstrated. It will be shown that most of the landmark models in the literature including the W( I_{1}, I_{2} ) form Mooney-Rivlin, the Gent-like limiting chain extensibility, the nonaffine tube and the separable principal stretches-based type Ogden models are all a special sub-set of the presented parent model, and are all recovered from this model. An important implication of the non-separable functional form of the model will be conferred, through a specific dataset, where the predictions of the separable functions prove inherently inadequate. The consequences of these improvements for a more accurate modelling of the finite deformation of incompressible isotropic rubber-like materials will also be discussed.

Constitutive modelling of fibre networks with stretch distributions. Part I: Theory and illustration

In this contribution, we propose a novel continuum mechanical concept to determine the homogenised mechanical response of random fibre networks. Their free energy is calculated as an average of the fibre free energy over the distribution of stretch, which forms the core of the new theory. The fibre-scale kinematic information contained in this distribution is intrinsic to the model and available for any state of macroscopic deformation in addition to the homogenised macroscopic response. In contrast to the various approaches that use directional averaging over the unit sphere, the presented model does not rely on a relation between fibre deformation and orientation, and therefore applies to highly non-affine networks as well as affine ones. In the present Part I of the work, the new concept is formally introduced, put into perspective with current approaches, and elaborated for the case of networks of elastic fibres with uniform orientation distribution, which generate a macroscopically isotropic hyperelastic response. The essential constitutive assumption of the theory establishes a relation between the macroscopic deformation of the network and the microscopic stretch distribution within. A phenomenological approach is used to illustrate the new concept, expressing statistical moments of the stretch distribution in terms of the macroscopic principal strain invariants. Application of the concept to 2D and 3D Voronoi networks with fibres of different properties finally exemplifies the accurate agreement of the model with discrete network simulations in terms of both the macroscopic and microscopic response. While the here presented phenomenological variant of the approach therefore represents an advance in the analytical multiscale modelling of network materials itself, the work also provides the basis for further developments, where the relation between the stretch distribution and macroscopic strain is derived from alternative principles.

Only two Betchov homogeneity constraints exist for isotropic turbulence

Statistically homogeneous flows obey exact kinematic relations. The Betchov homogeneity constraints (Betchov,J. Fluid Mech., vol. 1, 1956, pp. 497–504) for the average principal invariants of the velocity gradient are among the most well-known and extensively employed homogeneity relations. These homogeneity relations have far-reaching implications for the coupled dynamics of strain and vorticity, as well as for the turbulent energy cascade. Whether the Betchov homogeneity constraints are the only possible ones or whether additional homogeneity relations exist has not been proven yet. Here we show that the Betchov homogeneity constraints are the only homogeneity constraints for incompressible and statistically isotropic velocity gradient fields. Our analysis also applies to compressible/perceived velocity gradients, and it allows the derivation of homogeneity relations involving the velocity gradient and other dynamically relevant quantities, such as the pressure Hessian and viscous stresses.

Signal detection method using a single vector hydrophone in ocean acoustics.

This paper focuses on the problem of transient signal detection using a single vector hydrophone in a horizontal isotropic acoustic field. The covariance matrix of the three sensor output quantities is analyzed in terms of its trace and determinant principal invariants, and the mathematical relationship between the signal-to-noise ratio and these invariants is established. Both the signal power and the ambient-noise power can be accurately estimated directly from the covariance matrix of the received data, and the signal-to-noise ratio can further be obtained. A signal power detector is proposed based on the theoretical analysis, and simulation results show that (i) it can detect signals with low signal-to-noise ratio and (ii) its performance is greatly improved compared with that of an energy detector under the condition of a nonstationary ambient-noise field. An experiment was performed to verify the theoretical analysis and the conclusions of the numerical simulation, and the data-processing results show that this method has good applicability in signal detection.

First-Order Derivatives of Principal and Main Invariants of Magnetic Gradient Tensor of a Uniformly Magnetized Tesseroid and Spherical Shell

First-Order Derivatives of Principal and Main Invariants of Magnetic Gradient Tensor of a Uniformly Magnetized Tesseroid and Spherical Shell

A quadratic viscous fluid law for ice deduced from Steinemann's uni-axial compression and torsion experiments

AbstractThe response of ice to applied stress on ice-sheet flow timescales is commonly described by a non-linear incompressible viscous fluid, for which the deviatoric stress has a quadratic relation in the strain rate with two response coefficient functions depending on two principal strain-rate invariants I2 and I3. Commonly, a coaxial (linear) relation between the deviatoric stress and strain rate, with dependence on one strain-rate invariant I2 in a stress formulation, equivalently dependence on one deviatoric stress invariant in a strain-rate formulation, is adopted. Glen's uni-axial stress experiments determined such a coaxial law for a strain-rate formulation. The criterion for both uni-axial and shear data to determine the same relation is determined. Here, we apply Steinemann's uni-axial stress and torsion data to determine the two stress response coefficients in a quadratic relation with dependence on a single invariant I2. There is a non-negligible quadratic term for some ranges of I2; that is, a coaxial relation with dependence on one invariant is not valid. The data does not, however, rule out a coaxial relation with dependence on two invariants.

First-order derivatives of principal and main invariants of gravity gradient tensor of the tesseroid and spherical shell

First-order derivatives of principal and main invariants of gravity gradient tensor of the tesseroid and spherical shell

Elastic dipole tensors and relaxation volumes of point defects in concentrated random magnetic Fe-Cr alloys

Point defects in body-centred cubic Fe, Cr and concentrated random magnetic Fe-Cr are investigated using density functional theory and theory of elasticity. The volume of a substitutional Cr atom in ferromagnetic bcc Fe is approximately 18% larger than the volume of a host Fe atom, whereas the volume of a substitutional Fe atom in antiferromagnetic bcc Cr is 5% smaller than the volume of a host Cr atom. In an alloy, elastic dipole P and relaxation volume Ω tensors of vacancies and self-interstitial atom (SIA) defects exhibit large fluctuations, with vacancies having negative and SIA large positive relaxation volumes. Dipole tensors of vacancies are nearly isotropic across the entire alloy composition range, with diagonal elements Pii decreasing as a function of Cr content. Fe-Fe and Fe-Cr SIA dumbbells are more anisotropic than Cr-Cr dumbbells. We find that fluctuations of elastic dipole tensors of SIA defects are primarily associated with the variable crystallographic orientations of the dumbbells. Statistical properties of tensors P and Ω are analysed using their principal invariants, suggesting that point defects differ significantly in alloys containing below and above 10% at. Cr. The von Mises stresses caused by dumbbells are notably larger than those caused by vacancies. The relaxation volume of a vacancy depends sensitively on whether it occupies a Fe or a Cr lattice site. A correlation between elastic relaxation volumes and magnetic moments of defects found in this study suggests that magnetism is a significant factor influencing elastic fields of defects in Fe-Cr alloys.

Stress formalism for elastic waves in anisotropic solids

Traditionally, the propagation of bulk elastic waves in anisotropic crystals or solids is governed by the displacement form of the elastodynamic wave equation. Phase velocities follow after transforming the wave equation into an eigenvalue problem, which gives the Christoffel equations, and solving for the roots of the characteristic polynomial. Alternatively, only a few researchers have considered a pure stress wave equation of motion to describe the propagation of waves in elastic solids. However, to our knowledge, the dual characteristic polynomial and its solutions based on a pure stress formalism has not been established previously. This presentation will outline the needed steps to reach and solve the dual characteristic polynomial. Phase velocity solutions follow from the formation of new principal invariants, which are functions of the elastic compliance constants and propagation directions. The dual formalism is proven consistent to the traditional displacement formalism through the use of sixth-rank Levi-Civitia identities. Lastly, possible extensions of this work will be highlighted.

On the relationship between volcanic pointlike features and the crust-mantle boundary at the Pinacate Volcanic Field, Sonora, México

A description of numerical methods is presented, which applied to data from a digital elevation model, helps to enhance the presence of pointlike features in a volcanic field. It was assumed in the development of the method, that the scalar field representing the topography in a digital elevation model is the vertical gradient of a harmonic function u. The second derivatives of u, in orthogonal directions, x, y, z, are computed in the wavenumber domain, via the two-dimensional Fourier transform, to obtain the gradient tensor of u. At every point in the digital elevation model, the gradient tensor components form a 3 by 3 symmetric matrix. I use the singular value decomposition of these matrices to calculate its eigenvalues to obtain their principal invariants. The square of the second invariant is found as an excellent choice to select pointlike features, likely related to volcanic vents. The method is fully described and illustrated using a digital elevation model with a horizontal resolution of 25 m from the Pinacate volcanic field in northwestern Sonora, where we have selected 353 pointlike features. Fundamental statistical analysis of the azimuth and location of the central point between every two points was carried in the range (0–5] km to obtain a rose diagram, where the azimuth of maximum and minimum frequencies are separated apart 90°, suggesting that most of the magma, that built the Pinacate Volcanic Field, was emplaced along the N25 ± 5°W direction. This direction is nearly parallel to the trend of the Crust-Mantle interface, inferred by nonlinear inversion of isostatic residual gravity data, which lies at a depth of about 18.5 km beneath the Santa Clara Volcano. The magnetization between 18 and 21 km-depth, inferred by linear inversion of aeromagnetic data, may correspond with mafic rocks of gabbroic composition, which led to monogenetic volcanism of basaltic composition. It is suggested that monogenetic volcanic activity in the PVF was accommodated mainly in a direction normal to a major lateral change in density.

${\mathbf{U}} = \mathbf{C}^{1/2}$ and Its Invariants in Terms of $\mathbf{C}$ and Its Invariants

We consider $N\times N$ tensors for $N= 3,4,5,6$. In the case $N=3$, it is desired to find the three principal invariants $i_1, i_2, i_3$ of $\bf U$ in terms of the three principal invariants $I_1, I_2, I_3$ of ${\bf C}={\bf U}^2$. Equations connecting the $i_\alpha$ and $I_\alpha$ are obtained by taking determinants of the factorisation \[\lambda^2{\bf I}- {\bf C} = (\lambda{\bf I}- {\bf U}) (\lambda{\bf I}+ {\bf U})\] and comparing coefficients. On eliminating $i_2$ we obtain a quartic equation with coefficients depending solely on the $I_\alpha$ whose largest root is $i_1$. Similarly, we may obtain a quartic equation whose largest root is $i_2$. For $N=4$ we find that $i_2$ is once again the largest root of a quartic equation and so all the $i_\alpha$ are expressed in terms of the $I_\alpha$. Then $\bf U$ and ${\bf U}^{-1}$ are expressed solely in terms of $\bf C$, as for $N=3$. For $N= 5$ we find, but do not exhibit, a twentieth degree polynomial of which $i_1$ is the largest root and which has four spurious zeros. We are unable to express the $i_\alpha$ in terms of the $I_\alpha$ for $N=5$. Nevertheless, $\bf U$ and ${\bf U}^{-1}$ are expressed in terms of powers of $\bf C$ with coefficients now depending on the $i_\alpha$. For $N=6$ we find, but do not exhibit, a 32 degree polynomial which has largest root $i_1^2$. Sixteen of these roots are relevant but the other 16, which we exhibit, are spurious. $\bf U$ and ${\bf U}^{-1}$ are expressed in terms of powers of $\bf C$. The cases $N>6$ are discussed. Keywords: Continuum mechanics, polar decomposition, tensor square roots, principal invariants, cubic equations, quartic equations, equations of degree 16

Mechanical Properties of Layered Composite Coal–Rock Subjected to True Triaxial Stress

Underground mining or tunnelling activity is always associated with the composite geological formations. The mechanical properties of layered composite coal–rock subjected to true triaxial stress conditions are significantly different from those under conventional triaxial or uniaxial stress conditions. In this work, we conducted a series of true triaxial tests using the self-developed true-triaxial apparatus to investigate the mechanical response (e.g., deformation, strength, and failure characteristics) of the layered composite coal–rock (CCR). The results show that the uniaxial strength of CCR lies between the strength of pure sandstone and coal, and the direction of the bedding affects the overall strength of the samples. The true triaxial strength of both the pure rock and CCR increases first and then decreases with the increase of the intermediate principal stress. Moreover, for a given loading direction, as the thickness of the sandstone layer increased, the strength of the CCR increases. The deformation of the CCR shows more obvious plasticity than that of the pure sandstone due to the coordinated deformation of the coal and sandstone layers. In addition, a new true triaxial strength criterion expressed by the first and third equivalent principal stress invariants was proposed, which can well describe the strength characteristics of different coal rocks. The stress states, weak structural planes, and localized stress have a great influence on the failure modes of CCR. The local stress concentration near the contact surface promotes the development of the secondary failure fractures. These findings are of great significance in stability designing in deep underground engineering.