We develop an error-in-constitutive-relation (ECR) approach toward the full-field characterization of linear viscoelastic solids described within the framework of standard generalized materials. To this end, we formulate the viscoelastic behavior in terms of the (Helmholtz) free energy potential and a dissipation potential. Assuming the availability of full-field interior kinematic data, the constitutive mismatch between the kinematic quantities (strains and internal thermodynamic variables) and their “stress” counterparts (Cauchy stress tensor and that of thermodynamic tensions), commonly referred to as the ECR functional, is established with the aid of Legendre–Fenchel gap functionals linking the thermodynamic potentials to their energetic conjugates. We then proceed by introducing the modified ECR (MECR) functional as a linear combination between its ECR parent and the kinematic data misfit, computed for a trial set of constitutive parameters. The affiliated stationarity conditions then yield two coupled evolution problems, namely (i) the forward evolution problem for the (trial) displacement field driven by the constitutive mismatch, and (ii) the backward evolution problem for the adjoint field driven by the data mismatch. This allows us to establish compact expressions for the MECR functional and its gradient with respect to the viscoelastic constitutive parameters. For generality, the formulation is established assuming both time-domain (i.e. transient) and frequency-domain data. We illustrate the developments in a two-dimensional setting by pursuing the multi-frequency MECR reconstruction of (i) piecewise-homogeneous standard linear solid, and (b) smoothly-varying Jeffreys viscoelastic material.

中文翻译：

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用于表征线性粘弹性固体的修正本构关系误差 (MECR) 框架


我们开发了一种本构关系误差（ECR）方法，用于对标准广义材料框架内描述的线性粘弹性固体进行全场表征。为此，我们根据（亥姆霍兹）自由能势和耗散势来制定粘弹性行为。假设全场内部运动学数据可用，运动学量（应变和内部热力学变量）与其“应力”对应物（柯西应力张量和热力学张力）之间的本构失配，通常称为 ECR 泛函，是在勒让德-芬切尔间隙泛函的帮助下建立的，该泛函将热力学势与其能量共轭联系起来。然后，我们继续引入修改后的 ECR (MECR) 函数作为其 ECR 父代与运动学数据失配之间的线性组合，针对一组试验本构参数进行计算。然后，附属平稳性条件产生两个耦合演化问题，即（i）由本构失配驱动的（试验）位移场的前向演化问题，以及（ii）由数据失配驱动的伴随场的后向演化问题。这使我们能够建立 MECR 泛函及其相对于粘弹性本构参数的梯度的紧凑表达式。出于一般性考虑，该公式是假设时域（即瞬态）和频域数据而建立的。我们通过对 (i) 分段均匀标准线性固体和 (b) 平滑变化的 Jeffreys 粘弹性材料进行多频 MECR 重建来说明二维环境中的发展。