## 1 Introduction  
The size-dependent mechanical behavior of crystalline materials has been clearly shown by various experimental findings in the literature [1, 2, 3, 4, 5]. Experiments performed on nano/micro sized metallic specimens (500 nm-50 \(\upmu\)m) clearly demonstrate the size-dependent nature of their strength which results in inhomogeneous plastic deformation. Lacking an intrinsic length scale, conventional plasticity theories are unable to demonstrate such a size-dependent behavior; therefore, models introducing strain gradient terms in the constitutive relations are widely proposed [6, 7, 8, 9]. Earliest numerical analysis has been performed on theories based on Aifantis yield condition [10, 11, 12, 13]. The Aifantis approach is further developed and established as explicit gradient theories by [PERSON] and [PERSON] and the numerical issues have been addressed by the proper regularization of localized deformation with an alternative approach so-called implicit gradient theory [14, 15, 16].  
A significant contribution to introduce a thermodynamically consistent theory for strain gradient plasticity has been done by [PERSON] and [PERSON] [17, 18, 19]. It has been shown that such a thermodynamically consistent theory can also be established by using [PERSON]'s micromorphic theories of continua [20]. By [PERSON]'s theory the gradient terms are naturally introduced through higher order degrees of freedom and thermodynamical consistency can be achieved in a systematic fashion by using the principle of virtual power. That track of micromorphic approach is further adopted to several strain gradient problems by [PERSON] and [PERSON] [21, 22, 23, 24, 25] and unified to [PERSON]'s thermodynamics [26] by [PERSON] et al. [8].  
The micromorphic variable defined as an additional degree of freedom permits the application of unique boundary conditions which brings certain ease to mimic a hard elastic phase, especially for crystalline materials where hardening behavior might strictly depend on geometrically necessary dislocations. Moreover, numerical implementation of such gradient approaches and application of higher order boundary conditions in numerical simulations are very demanding; therefore, reference analytical or semi-analytical solutions are necessary to verify the robustness of the implementation. Even tough, the capability of the micromorphic approach predicting the size effects and inhomogeneous plastic fields promoted by the gradient terms are shown numerically, analytical solutions are provided for only a few model problems in the literature [8; 21; 22].  
The main aim of this paper is to provide analytical solutions for several 2D model problems to demonstrate the unique plastic fields generated by the implementation of micromorphic approach for gradient plasticity. For that purpose, first the approach is presented providing the basic kinematics within the finite strain framework and as an additional degree of freedom to be coupled with the equivalent plastic strain, a microdamage variable \(e^{p}\) is introduced in Section 2. The balance equations are obtained from the principle of virtual power and balance laws are derived. Section 3 presents the micromorphic constitutive equations and provides the general form of analytical solutions for softening and inhomogeneous size-dependent strain hardening. In Section 4, several model problems for a 2d plate under simple shear is scrutinized and the analytical solutions are provided. The analytical results are also shown to match with the numerical results obtained from the implemented user element subroutine (UEL) to the commercial finite element software Abaqus/Standard. The implemented element subroutine is based on a standard four-node isoparametric element on the interpolated fields with \(C_{0}\) continuity. Finally, we close in Section 5 with the final remarks.