Modern material systems with properly designed microstructures offer new avenues for engineering materials with advantageous mechanical properties and functionalities in various applications. With the increasing desire to design structures that have complex shapes or that are simply cheaper and more efficient, the demand for complex numerical simulations intensifies. Capturing the behaviour of composites relies on the interaction between the macroscopic and microscopic components, which can make direct numerical simulation too computationally expensive. Concurrent finite element analysis (FE²) links the different length scales in a multiscale approach. In each integration point of the macroscale a representative volume element (RVE) is embedded to address the microscopic heterogeneity. However, the need to evaluate the micromodel many times will come with the drawback of high computational costs. Various surrogate models have been proposed to accelerate computational models. One possible approach is employing machine learning techniques to substitute the evaluation of the RVE. Currently, artificial neural networks are being used in a wide variety of fields, due to their exceptional ability to recognize patterns. Although these techniques are able to construct models for complex input–output relations, their applications to mechanics of materials are still limited. These techniques are usually problem-dependent and may suffer from the danger of extrapolation beyond the original sampling space, e.g. different loading paths. This is not naturally resolved, mainly due to the loss of physics in the current machine learning models. The newly proposed Deep Material Network (DMN) by Liu et al. is a data-driven multiscale material modelling method that aims to resolve this issue by composing a network constructed from an assembly of mechanistic building blocks, where each building block resembles an analytical homogenization. Because the fitting parameters of this network are interpretable with physical meaning and because sampling the DMN still involves evaluation of classical constitutive relations, the loss of essential physics in minimized. The material network can effectively be trained using stochastic gradient descent with backpropagation, based on linear elastic data of an RVE. The extrapolation capabilities to unknown loading paths are investigated with a matrix-inclusion composite, with the focus on the accuracy of the model for loading-unloading scenarios, and an extension to the training process is proposed where the fidelity of a trained DMN is assessed by generating and grading a yield stress envelope. The results show that a well trained DMN is capable of predicting the response for unseen loading paths involving unloading exceptionally well. The complete learning and extrapolation procedures of the DMN establish a reliable data-driven framework for concurrent multiscale material modelling and design.